Description
Carbon pricing is the generic term for putting a price on carbon through either subsidies, a carbon tax, or an emissions trading ("cap-and-trade") system.
ABSTRACT
Title of Document: PRICING CARBON: ALLOWANCE PRICE
DETERMINATION IN THE EU ETS
Beat Hintermann, Ph.D., 2008
Directed By: Prof. A. Lange, Department of Agricultural and
Resource Economics (AREC)
The allowance price in Phase I of the European Union Emissions Trading
Scheme (EU ETS) followed a peculiar path, increasing from €7 in 2005 to over €30
in 2006, before crashing, recovering and ultimately finishing at zero by the end of
2007. I examine if the price can be explained by marginal abatement costs as
predicted by economic theory, or if there were other price determinants. This has
important policy implications, since the least-cost solution depends on the equality of
permit price and marginal abatement costs and is the main argument in favor of
permit markets.
I start with a model that incorporates the most commonly cited market
fundamentals and find that the latter only explain a small part of the allowance price
variation, raising the question of a bubble. I carry out two different bubbles tests, the
results of both of which are consistent with the presence of an allowance price
bubble.
I then address whether market manipulation by dominant power generators
could have lead to the initial allowance price increase. I extend economic theory to
include the interaction between output and permit markets. I derive a threshold of
free allocation beyond which firms find it profitable to manipulate the permit price
upwards, even if they are net allowance buyers. Market data indicates that this
threshold was exceeded for EU power generators.
Finally, I investigate the possibility that due to the speed at which the market
was set up, firms may have been unable to engage in effective abatement before the
end of Phase I. I develop a model under the assumption of no abatement, where firms
aim to reach compliance exclusively by purchasing allowances on the market. Thus,
the allowance payoff becomes that of a binary option, for which I derive a pricing
formula. The model fits daily data from the years 2006-7 well.
I conclude that the allowance price in Phase I was not driven by marginal
abatement costs, but by a combination of price manipulation, self-fulfilling
expectations and/or the penalty for noncompliance weighted by the probability of a
binding cap.
THE PRICE OF CARBON: ALLOWANCE PRICE DEVELOPMENT IN THE EU
ETS
By
Beat Hintermann
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park, in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2008
Advisory Committee:
Professor Andreas Lange, Chair
Professor Albert S. Kyle
Professor Marc L. Nerlove
Professor John K. Horowitz
Professor S. Boragan Aruoba
© Copyright by
Beat Hintermann
2008
ii
Preface
This dissertation is a hybrid between a three-paper dissertation and a unified
dissertation. On the one hand, it is based on three separate papers, two of which I
already submitted to an economic journal; they are currently under review. I hope to
submit the third paper shortly.
On the other hand, the three papers all treat the same subject matter: What
drove the allowance price during the first phase of the EU ETS?
Chapters 1 and 6 contain a general introduction and conclusion, respectively,
that are applicable to all three articles, and I removed the description of the market
from the individual papers and concentrated it in a separate section (Chapter 2). I
also made minor changes such as renaming equations, tables and figures. In spite of
this, Chapters 3-5 remain largely self-contained and may give rise to some
redundancy. They also are independent in terms of notation, such that a symbol or
letter appearing in two different chapters may represent two different variables or
parameters.
iii
Dedication
To Mary.
iv
Acknowledgements
I would like to thank all members of my dissertation committee for their
support and guidance in this project. Special thanks go to the committee chair,
Andreas Lange.
Many thanks also to faculty and fellow students at UMD-AREC for valuable
insights and suggestions.
I gratefully acknowledge the Swiss National Fund, which provided funds
during the last six months of this dissertation.
v
Table of Contents
Preface........................................................................................................................... ii
Dedication .................................................................................................................... iii
Acknowledgements ...................................................................................................... iv
Table of Contents .......................................................................................................... v
List of Tables .............................................................................................................. vii
List of Figures ............................................................................................................ viii
Chapter 1: Introduction ................................................................................................. 1
Chapter 2: The EU ETS ................................................................................................ 7
Chapter 3: Price Drivers and CO2 Bubbles in the EU ETS ........................................ 12
Abstract ................................................................................................................... 12
3.1. Introduction ...................................................................................................... 13
3.2. Allowance market model ................................................................................. 16
3.2.a.) Literature ................................................................................................. 16
3.2.b.) Base model ............................................................................................. 19
3.2.c.) Introducing dynamic expectations of fundamental prices ...................... 27
3.2.d.) Introducing lagged EUA price changes .................................................. 29
3.3. Results .............................................................................................................. 31
3.4. A CO
2
Bubble? ............................................................................................... 36
3.4.a.) Some bubble background ........................................................................ 36
3.4.b.) Bubble tests............................................................................................. 38
3.4.c.) Cointegration test .................................................................................... 40
3.4.d.) Regime-switching test ............................................................................ 42
3.5. Conclusions ...................................................................................................... 45
Chapter 4: Market Power and Windfall Profits in Emission Permit Markets............. 48
Abstract ................................................................................................................... 48
4.1. Introduction ...................................................................................................... 49
4.2. Market power in output and permit market ..................................................... 54
4.3. Application to the EU ETS .............................................................................. 64
4.3.a.) General applicability ................................................................................ 64
4.3.b.) The UK power market ............................................................................. 66
4.3.c.) The German power market ...................................................................... 70
4.4. Conclusions ..................................................................................................... 72
Chapter 5: An Options Pricing Approach to CO
2
Allowances in the EU ETS.......... 75
Abstract ................................................................................................................... 75
5.1 Introduction ...................................................................................................... 76
5.2. Emission allowance pricing in the absence of abatement ............................... 80
5.2.a.) Literature on permit pricing ..................................................................... 80
5.2.b.) Derivation of option pricing formula....................................................... 81
vi
5.3. Deriving the mean and standard deviation of future emissions ...................... 85
5.3.a.) CO
2
emissions as a function of exogenous stochastic processes ............. 85
5.3.b.) Properties of the stochastic processes c
t
and h
t
........................................ 90
5.4. Estimation ....................................................................................................... 95
5.4.a.) Data .......................................................................................................... 96
5.4.b.) Parameter estimation for electricity consumption and precipitation ....... 99
5.4.c.) Evaluation of the options pricing formula ............................................. 102
5.5 Conclusions .................................................................................................... 107
Chapter 6: Conclusions ............................................................................................ 110
Tables ........................................................................................................................ 114
Figures....................................................................................................................... 122
Appendices ................................................................................................................ 130
Appendix A: Proof of Equation (3.8) ................................................................... 130
Appendix B: Proof of Lemma 1 and Eq. (4.9) ...................................................... 132
Appendix C: Derivation of variance and covariance of future emissions ............ 137
References ................................................................................................................. 141
vii
List of Tables
Table 2.1: Summary results for Phase I of the EU ETS ........................................... 114
Table 3.1: Results from estimating Equation (3.10) ................................................. 115
Table 3.2: Results from estimating equation (3.11) .................................................. 116
Table 3.3: Cointegration test results ........................................................................ 117
Table 3.4: Results from regime-switching tests ....................................................... 117
Table 5.1: Data availability and installed hydroelectric capacity by country ........... 118
Table 5.2: Parameter estimates for diffusion processes ............................................ 119
Table 5.3: Correlation coefficients
a
among different series...................................... 120
Table 5.4: Parameter estimates from options pricing formula .................................. 121
viii
List of Figures
Figure 1.1: EUA price and trading volumes, Phase I ............................................... 122
Figure 2.2: Allowance allocation and emissions by sector (total values) ................. 123
Figure 2.3: Allowance allocation and emissions by sector (percent of total) ........... 124
Figure 3.1: EUA, coal, gas, DAX and reservoir levels ............................................. 125
Figure 4.2: Dark spreads and green dark spreads in Germany ................................. 126
Figure 5.1: Available electricity consumption data, pre-2006 .................................. 127
Figure 5.2: Weighted average precipitation in the EU ............................................. 128
Figure 5.3: EUA price, prediction and forward price for Phase II ........................... 129
Figure 5.4: Sensitivity of prediction to variance of future electricity demand ......... 129
Chapter 1: Introduction
1
Chapter 1: Introduction
On January 1, 2005, the world’s first non-voluntary CO
2
emissions market
opened for business. The European Union Emissions Trading Scheme (EU ETS) is
the European Union’s prime instrument to achieve its Kyoto targets. The system
covers emissions from energy-intensive industries that are responsible for about 45%
of the EU’s total CO
2
emissions. The EU ETS is by far the largest regulated
emissions permit market to date, dwarfing other markets in terms of total emissions,
included installations and market value.
The market is organized into distinct phases that each have different caps and
rules. The first phase covered the years 2005-2007, followed by the second phase,
which coincides with the 2008-2012 Kyoto compliance period. No banking was
allowed between the first and second phases, such that the first phase was a self-
contained market with a finite time horizon. First-phase emission permits (called EU
allowances, or EUAs) lost their value if unused for compliance.
The EUA price during the first phase followed a rather peculiar path, shown in
Figure 1.1. It started around €7 but rapidly increased to levels above €20, even
surpassing €30 at some point, before crashing to half of its value in April 2006,
stabilizing again in the €13-18 region and finally decreasing to zero by mid 2007,
where it remained for the rest of the market.
Chapter 1: Introduction
2
The initially very high allowance price is surprising considering that the first
phase was understood to be a pilot run for the second phase, with the cap not
expected to be very stringent. Even more surprising is the stabilization after the price
crash, followed by a slow march to zero. Given that the market turned out to be
oversupplied with permits, the price should have dropped to zero immediately, rather
than over the course of a year, if it equals marginal abatement costs (unless, of course,
the fundamentals driving marginal abatement costs also slowly declined over time).
The three main chapters in this dissertation (Chapters 3-5) are self-contained
and different in their methods, tools, and even notation to some extent, but they all
aim to answer the same question: What drove the allowance price during the first
phase of the EU ETS?
This question is interesting from an economic theory perspective, but it is
equally important in terms of policy implications. The main reason to institute a cap-
and-trade market, as opposed to using command-and-control methods, is that it yields
the lowest overall cost to reduce emissions to a specific target.
1
But if allowance
prices are too high (i.e. above marginal abatement costs), then the least-cost argument
vanishes and overall welfare might be better served using a different policy. Note
that firms will pass on much of the marginal cost of carbon as they do with any other
1
It shares this property with an emissions tax. In a world without uncertainty and full
auctioning of permits (or at least no updating of free permit allocations based on events
during the market), the permit price should equal the emissions tax for the same amount of
emissions reduction.
Chapter 1: Introduction
3
input to production,
2
so in the end it will be consumers that pay too much for a given
emissions reduction if allowance prices are above their efficient level.
In Chapter 3, I focus on widely identified fundamental price drivers and test
whether and to what extent they are able to explain the observed allowance price
path. A deviation between price and fundamentals is commonly referred to as a price
bubble. To the extent that the existence of a bubble can never be fully proven due to
an identification problem, I carry out two tests that examine whether or not the data is
consistent with the presence of a bubble. The first test relies on a market model that
assumes that the allowance price equals marginal abatement costs. I estimate this
model using a regime-switching approach that allows the allowance price to depend
differently on market fundamentals during different time periods. A likelihood ratio
test reveals that the two regimes are not mutually independent, but that the state in the
current period has an impact on the state probability in the next period (Markov
switching). This is consistent with a bubble, or a series of stochastically crashing
bubbles with interchanging boom and bust phases.
The second test relies on cointegration. If the allowance price and the
supposed market fundamentals exhibit cointegration, this would be evidence that the
price was indeed driven by fundamentals, and that therefore there was no bubble. I
2
The degree of cost pass-through depends on the price elasticity of consumer demand. With
completely inelastic demand, costs are fully passed through.
Chapter 1: Introduction
4
find no such cointegration in the data. Thus, while not conclusive, the results from
both tests are consistent with the bubbles hypothesis.
Emission permits were initially given away for free in accordance with the
European Commission’s mandate that countries could sell at most 5 % of their total
allowances, a measure taken to obtain industry support for the Directive. Chapter 4
examines one potential effect of free allocation on the allowance price. If firms were
able to pass through the marginal cost of CO
2
to consumers, they were in a position to
reap large windfall profits. In a competitive and efficient market, windfall profits
should not lead to permit price distortions. However, this changes in the presence of
market power. I examine how the initial allowance allocation affects the permit price
under the assumption of market power in both the output and the permit market. This
is an extension of Hahn’s (1986) results, which prescribe that a dominant firm will
manipulate the permit price upwards (downwards) if it is a net seller (buyer) of
permits. I show that when taking the interactions between the output and permit
market into account, this prescription changes in a significant way, meaning that the
largest permit holders in the EU ETS (i.e. power producers) would have found it
profitable to drive up the permit price despite the fact that they were net allowance
buyers, assuming that they had some market power.
Although market power per se is unrelated to the issue of bubbles, every price
bubble has to get started somehow. In addition to the basic subject matter that is the
thread throughout this dissertation, this provides another link between Chapters 3 and
Chapter 1: Introduction
5
4: Market-power related price manipulation could have driven the allowance price
upwards, leading to the beginning of an eventually self-sustaining price bubble.
Chapter 5 is motivated by the possibility that firms may have been unable to
engage in significant abatement in time for the first phase. The market was set up at
breakneck speed relative to “normal” time frames for instituting such markets,
3
providing little time for countries to determine historic emissions, define their caps
and distribute them among different installations. Firms had even less time to prepare
for the market, given that their individual permit allocations, the country-level caps
and thus the total cap was not known until the market had already come into effect.
Because CO
2
cannot be captured in a cost-effective manner today, the main
sources of abatement are changes in production technology towards less emission-
intensive processes, and the substitution of fuels with a lower emission factor per unit
of output. Since a change in production technology requires significant planning and
construction time combined with a minimum level of price certainty, fuel switching
4
was commonly assumed to be the abatement method of choice in the first phase of the
market. However, energy-intensive industries are generally locked into long-term
fuel contracts and may have been unable to switch, or unwilling to do so until the
price signal was more stable.
3
The directive that mandated the EU ETS (Directive 2003/87/EC) was issued in October
2003, just over a year before the market started.
4
This is not necessarily a substitution within the same plant, but a substitution across plants.
For example, more of the total electricity demand could be produced using gas-fired
generation while reducing the output of coal-fired generators.
Chapter 1: Introduction
6
If abatement is not feasible, firms will aim to reach compliance exclusively by
purchasing allowances on the market.
5
With stochastic emissions the exact number of
allowances needed for compliances is unknown ex ante. Firms face a situation where
an additional allowance will be worth the same as the penalty for noncompliance if
overall emissions exceed the cap at the end of the market, but be worthless if the cap
turns out not to be binding. Thus, the payoff of an allowance becomes that of a
binary cash-or-nothing option. In Chapter 5, I set up an options pricing model under
the assumption of no abatement that expresses the allowance price purely as a
function of the penalty and the probability of a binding cap, and fit the resulting
options pricing formula to market data.
Chapter 6 draws conclusions on the combined findings in this dissertation.
5
Note that the electricity sector cannot reduce output below demand, otherwise the grid
would crash. The other sectors could in theory reduce output in order to curb emissions, but
this is generally assumed to be a costlier measure than buying permits and/or paying the
penalty for noncompliance.
Chapter 2: The EU ETS
7
Chapter 2: The EU ETS
In the following I describe the main features of the European Union Emissions
Trading Scheme (EU ETS). For a more detailed introduction to the market, see
Kruger & Pizer (2004) and the White Paper by the PEW Institute (2005).
The EU ETS covers CO
2
emissions from 6 broadly defined industry groups in
all countries of the EU. These sectors are power & heat, metals and coke ovens, oil
refineries, glass & ceramics, cement & lime, and paper & pulp. The emissions
included in the market account for almost half of the EU’s total CO
2
output. In the
first phase, about 11,000 individual installations received a total of 2.1 billion EU
allowances (EUAs) annually, mostly at no cost. One EUA gives the bearer a one-
time right to emit one ton of CO
2
.
The market is organized into distinct trading phases. The first phase spanned
the years 2005-2007
6
and was considered a pilot run for the second phase, which
coincides with the Kyoto compliance period of 2008-2012. Pilot phase allowances
could not be banked into the second phase and lost their value if unused for
compliance. Future phases are planned to last five years each, with no banking
restrictions from one phase to the next. On the other hand, borrowing is not allowed
between any two phases. But because firms receive annual allowances in March of
6
The first trade of the EU ETS was made on February 27, 2003, between Shell and NUON (a
Dutch utility) under a forward market.
Chapter 2: The EU ETS
8
every year but don’t have to surrender allowances until the end of April, they can
effectively bank and borrow across time within a trading phase.
Firms can trade allowances freely within the EU. Trades may occur
bilaterally, through brokers (over-the-counter or OTC trades) or on one of six
exchanges.
7
By April 31 of each year, firms have to surrender permits corresponding
to their emissions in the previous calendar year. For every ton of CO
2
emissions for
which firms cannot surrender an allowance, they are fined a penalty of €40 in the first
phase, and of €100 in the second phase. In addition, they have to surrender the
missing allowances in the following year.
Jurisdiction in the EU ETS is divided between the EC and the member states.
The latter are required to submit detailed national allocation plans (NAPs) to the EC
for every phase anew (in other words, the cap changes in every phase). This is a two-
step procedure: First, member states have to decide how much of their overall
emissions reduction burden (as defined by their individual Kyoto commitments) they
want to assign to the EU ETS sectors within their countries, with the remainder of the
burden falling on other sectors such as transportation and households. In a second
step, the allowances have to be distributed among the individual installations. All
NAPs have to be approved by the EC in order to minimize competitive distortions
among similar companies in different member states.
8
7
These are ECX, EEX, EXAA, Climex, Nordpool and Powernext.
8
Although the Trading Directive defines both least-cost achievement of the Kyoto targets and
harmonization between member states as explicit goals, Boehringer and Lange (2005a) show
that both cannot be achieved simultaneously, given the constraint of free permit allocation.
Chapter 2: The EU ETS
9
The scheme is based on Directive 2003/87/EC, which became law on October
25, 2003. This left little time for firms and EU member countries to prepare for the
market. In setting up the first-phase NAPs, countries were faced with the problem that
they had very little information about firms’ historic emissions. Unlike US power
plants that were subject to emissions regulations since at least the mid 1990’s, most
firms in the EU had never had to disclose emissions of other than local pollutants.
The member countries addressed this lack of data by using industry projections
generated by the firms themselves. In addition, most market participants expected
that second-phase NAPs were going to be based on verified 2005 emissions (which
was vehemently opposed by the EC, but which took place nonetheless given that this
was the best data available to the member countries). Using industry projections and
defining second-phase allowance allocations based on first-phase emissions clearly
introduces incentive problems in the sense that firms were encouraged to over-state
their expected emissions (in order to receive more first-phase allowances), and to
under-abate (in order to receive more second-phase allowances).
9
Permit allocations, trades and actual emissions are recorded in national
registries run by each Member State, where all installations that are subject to the EU
ETS have their individual accounts. The Central Administrator of the EU runs a
central registry, called the Community Independent Transaction Log (CITL), which
connects the 27 national registries and checks the recorded transactions for
Thus, there is a tradeoff between efficiency and fairness in terms of a “level playing field”
between similar firms located in different member states.
9
For the effects of updating on firms’ decisions, see Boehringer and Lange (2005b).
Chapter 2: The EU ETS
10
irregularities. It is the duty of member states to establish and/or verify firms’ actual
emissions by multiplying energy inputs with appropriate conversion factors.
The EU ETS is linked to other carbon markets in the sense that certificates
from Kyoto’s Clean Development Mechanism (CDM), called CERs, and from Joint
Implementation (JI), called ERUs, can be surrendered instead of EUAs. Some
countries imposed a limit as to what the percentage of a firm’s emissions can be
covered with such non-EU based emission currencies, but these are still being worked
out. In any case, neither CERs nor ERUs were actually available throughout the first
phase, so for all practical purposes, the first phase of the EU ETS was self-contained.
Figure 2.1 shows allowances and emissions by EU member country for the
year 2006. The largest six countries account for over 70%, both in terms of allocation
and emissions. Allocation and emissions by sector are shown in Figures 2.2 and 2.3.
The power & heat sector received nearly 70% of the total allocation. At the same
time, this was the only sector with a net shortage of allowances, with all other sectors
acting as net allowance suppliers.
10
In terms of installation size, about 90 % of the
covered firms are relatively small (<1 Mt CO
2
/y) and received about 19% of the total
allocation. On the other extreme of the spectrum are the very large emitters (>10
Mt/y), which make up less than one percent of all installations in number but received
more than a third of all allowances. Most of these large emitters are power plants.
10
Note that these are aggregate numbers; individually, there were power stations with an
allowance surplus in 2005 and 2006, and many industrial firms with a shortfall.
Chapter 2: The EU ETS
11
Pre-market expectations of the allowance price were generally very low,
11
and
the steep price increase took many observers by surprise. For over a year, the
allowance price was above €20, and at its peak it reached over €31 in April 2006.
The April price crash was triggered by the first round of emissions verifications,
which revealed that 2005 emissions were 94 MT below the cap.
12
The second round
of emissions verifications in May 2007 again found an allowance surplus, but this no
longer had a significant impact since prices had decreased to a few cents. Liquidity
was overall high, and a significant amount of the total allocation was traded even in
the first year. Table 2.1 shows a market summary of the first phase.
11
In a simulation-based analysis of the EU ETS, Reilly and Paltsev (2005) calculated market-
clearing marginal abatement costs to be € 0.6-0.9 for their base scenario, with prices in even
the most extreme scenarios below €7. Medium price estimates by brokers were somewhat
higher, around of €5.00 for the first phase (PEW, 2005).
12
Emissions verification numbers were planned to be announced in May, but in late April
reports were leaked that Belgium, France, the Czech Republic, the Netherlands and Estonia
all had allowance surpluses, and the allowance shortage in Spain was much smaller than
anticipated. By early May, the market was found to be 63.6 Mt long, with 21 countries
reporting. It is interesting to note that the announcement of the Polish surplus of another 26
Mt in September 2006 did not affect prices very much.
Chapter 3: Bubbles
12
Chapter 3: Price Drivers and CO2 Bubbles in the EU ETS
(Paper submitted to JEEM in July 2008)
Abstract
In the first phase of the EU Emissions Trading Scheme (EU ETS), the price
per ton of CO
2
rose to over €30 before decreasing to zero by mid 2007. I examine to
what extent this variation can be explained by market fundamentals, and whether
there was a price bubble. The presence of the latter would question the main
argument in favor of permit markets, which is to achieve a given emissions cap at
least cost. I derive a structural model of the allowance price under the assumption
efficient markets, which I gradually relax by allowing for delayed adjustment of price
to fundamentals, as well as by introducing lagged LHS variables. The pattern of the
results suggests that a price bubble is at least possible. I then pursue this hypothesis
further by carrying out two different bubbles tests, both of which are consistent with
the presence of a bubble.
Keywords: Emissions permit markets, air pollution, climate change, bubble,
speculation, CO
2
, asset pricing, EU ETS.
JEL classification: D84, G12, G14, Q52-54
Chapter 3: Bubbles
13
3.1. Introduction
The allowance price per ton of carbon dioxide (CO
2
) in first phase of the
European Union Emissions Trading Scheme (EU ETS) exhibited tremendous
variation. It started around €5 but quickly increased to a range of €20-30 where it
remained for over a year. The price crashed after the first round of emissions
verifications that showed the market to be long, but recovered somewhat and
remained around €15 for another few months, before starting a gradual decline. By
mid 2007, an allowance was virtually worthless.
Market analysts and economists alike have been looking for reasons behind
the peculiar allowance price movement. Some have pointed to market fundamentals
such as fuel prices and the weather (Alberola et al., 2008, Bunn and Fezzi, 2007,
Mansanet-Bataller et al., 2007, Rickels et al., 2007) but others found no such
correlation and confined themselves to forecasting based on pure time-series
approaches (Chesney and Taschini, 2008, Paolella and Taschini, 2006). Whereas the
April 2006 crash can be explained by the lower-than-expected overall emission
reports, it is not clear why the allowance price was driven that high in the first place.
13
Also, the fact that it did not collapse completely but remained at a (in hindsight) very
high level through 2006 lacks a satisfactory explanation.
13
In theory, market participants need not know aggregate emissions for the price to be
efficient. If every firm with a permit surplus (deficit) sells (buys) permits on the market, the
price should marginal abatement costs regardless of emissions verifications.
Chapter 3: Bubbles
14
In this paper I examine if and to what extent the allowance price in the first
phase of the EU ETS was based on market fundamentals. I first set up an economic
model that specifies allowance price changes as a function of a set of widely accepted
price drivers (fuel prices, temperature, reservoir levels, economic indicators and
announcements of verified emissions), under the assumption of efficient markets and
using the best data available. I then relax this model by introducing lagged
fundamentals to account for non-immediate adjustment of allowance prices to
changes in fundamentals and to proxy for unobserved expectations about
fundamentals. In a further step I add past price changes as predictors of current price
changes and gage their importance relative to the fundamentals.
I find that only a small portion of the allowance price variation can be
explained by market fundamentals, even when taking into account dynamic
expectations about fundamentals. A situation where an asset price is driven by
expectations about future increases in a manner that is detached from fundamentals is
commonly referred to as a price bubble. All tests to identify bubbles are inherently
plagued by an identification problem, since the researcher can never know whether a
difference between the price and the “true” value of an asset is due to a bubble, or to a
misspecification of the market structure when calculating the intrinsic value (Flood
and Garber, 1980, Garber, 1989, Gurkaynak, 2005). However, a permit market seems
an especially appropriate place to investigate the presence of a bubble because the
asset in question has a clearly defined value: One allowance is worth the cost of
reducing aggregate CO
2
emissions to one ton below the aggregate cap. This is in
Chapter 3: Bubbles
15
contrast to stock prices, where it is often unclear what a stock is really worth and
should lead to a fairly trustworthy estimate of the intrinsic value, thus reducing the
danger of identification error considerably.
The contribution of this section of the dissertation is threefold: First, the
stepwise procedure of starting with economic theory and then relaxing the most
stringent constraints allows insights into the determinants of the EU ETS allowance
price that go beyond existing analyses, which typically do not start from a rigorous
economic model and determine model specification on a mostly ad-hoc basis.
Second, I use a dataset of daily weather measurements in dozens of monitoring
stations across Europe reaching back over three decades, which I combine with
detailed information about regional population density to account for population-
weighed temperature deviations from their long-term expectations. No dataset of
comparable quality has been used in the literature address the influence of weather
shocks on the allowance price. Last, to my knowledge no permit market has ever
been tested for the presence of a price bubble, although such markets appears to offer
more favorable conditions than stock markets to mitigate the identification problem
encountered in any bubbles test.
In the next section I review the literature and derive the market model. In
Section 3.3, I introduce the data and present the estimation results for the proposed
models. Section 3.4 contains two bubbles tests, one based on regime switching and
the other on cointegration between allowance prices and market fundamentals, and
Section 3.5 concludes.
Chapter 3: Bubbles
16
3.2. Allowance market model
3.2.a.) Literature
There is a large volume of empirical work about the SO
2
permit trading
system in the USA (Carlson, 2000, Joskow et al., 1998, Montero, 1999, Schmalensee,
1998, Stavins, 1998) and more recently (Burtraw et al., 2005, Kosobud et al., 2005),
to name only a few. For the EU ETS, the empirical literature is scarcer because it is a
much newer market, and because fewer data are available in general since the
involved firms previously faced very little regulation.
14
There exists a number of
studies that model the EUA price and its volatility mainly for risk management and
forecasting purposes (Benz and Trueck, 2006, Chesney and Taschini, 2008, Fehr and
Hinz, 2006, Seifert et al., 2008). While useful for companies that need to hedge
against the risk embedded in carbon prices, they do not shed much light on
fundamental price drivers.
I am aware of four papers that explicitly aim to determine the impact of
market fundamentals: Bunn and Fezzi (2007) use a cointegrated VAR model with
allowances, electricity and gas in the UK and daily temperature in London as an
exogenous variable, and impose the necessary identifying restrictions using auxiliary
14
In the USA, historic emissions and information about production, fuel use and abatement
are readily available, which is not the case in the EU.
Chapter 3: Bubbles
17
regressions. They find that the gas price influences the EUA price, and that both gas
and EUA prices help determine the electricity price.
Mansanet-Batallet et al. (2007) focused on EU-wide fuel prices and a weather
index comprised of several cities. They focus on the first year of the market only and
include dummies for the six largest price changes, which end up accounting for most
of the explanatory power of their model.
15
This sidesteps the question of what
actually drives the allowance price. In addition, they include regressors that are not
obviously related to allowance prices, such as the Brent oil price.
16
They find that oil
prices, natural gas prices and temperature in Germany are the only significant
allowance price drivers, whereas other determinants (such as the EU weather index
and coal prices) turned out to be uncorrelated.
Alberola et al. (2008) use temperatures in capital cities of six EU countries,
along with a number of EU-wide energy variables, and extent the analysis to the first
two years of the market. Unfortunately, they treat highly endogenous variables such
as electricity prices, clean dark and clean spark spreads
17
as exogenous determinants
15
It is not clear what the explanatory power of the fundamentals themselves is since no
estimation results are presented for a model without dummies.
16
The explanation given for including oil prices is not very clear. They cite a study by
Christiansen et al. (2005) which looked at very general determinants for greenhouse gas
markets, but is not specific to fuel switching in the EU’s power sector. Very little power is
generated using oil in Europe, so a switch from oil to gas is not likely to be the marginal
abatement activity.
17
The dark spreads is the theoretical gross profit of a power plant to generate a unit of
electricity using coal, having bought the fuel necessary to produce it: Dark spread = power
price-fuel price*heat rate. The heat rate is the efficiency at which a power plant converges
energy in fuel into electric output. The clean dark spread is the dark spread minus CO
2
costs
embedded in producing a unit of electricity: Clean dark spread=dark spread-CO
2
Chapter 3: Bubbles
18
of the allowance price.
18
This is no problem for forecasting, but endogeneity will lead
to biased coefficient estimates of the price drivers.
Rickels et al. (2007) build on Mansanet-Bataller et al. (2007) but include data
through 2006. They separate allowance price determinants into supply and demand
side, and their choice of market fundamentals seems more appropriate than that of
Alberola et al. (2008) and more complete than that of Bunn & Fezzi (2007).
However, their econometric specification is questionable: Although they check for
cointegration between allowance and fuel prices (they find none) and thus implicitly
acknowledge the presence of unit roots in the price data, they specify their model in
levels as opposed to differences (or returns) to render the data stationary. A
nonstationary error may lead to untrustworthy coefficient estimates.
All four studies are valuable contributions to finding allowance price
determinants, but neither is based on a rigorous economic market model. The
inclusion/exclusion of market fundamentals as well as the econometric specification
is mostly ad-hoc, which leads to the aforementioned problems. They also do not take
into account the no-banking provision from the first into the second phase, and they
include lagged market fundamentals as well as lagged EUA prices as allowance price
fundamentals from the outset without discussing the economic meaning of this. If
yesterday’s price change determines today’s, then what determines yesterday’s price
price*emission intensity. The spark spread and clean spark spread are analogous measures
for gas-generated electricity.
18
Although the electricity price, and thus spreads, is correlated with the EUA, the causation is
very likely the other way around, because electricity producers pass carbon costs (at least
partly) through to consumers.
Chapter 3: Bubbles
19
change, and what are the true determinants of allowance prices? In the following I set
up a simple economic model of the allowance price that explicitly addresses these
issues.
3.2.b.) Base model
In order to incorporate the uncertainty inherent in the demand and supply of
allowances as well as the fixed time horizon of the first market phase I follow an
approach by Maeda (2004), who analyzed the effect of uncertainty in business-as-
usual (BAU) emissions (referring to emissions in the absence of a carbon cost). I
extend his model to T periods, where each period represents one day and T
corresponds to the number of business days during the first phase of the EU ETS. Let
BAU
it
represent firm i’s random BAU emissions in period t, which depend on a
vector of normally distributed risk factors ? shared by all N firms in the market:
(3.1)
( )
j i E E
Var
BAU Cov
E BAU E BAU
jt it it t
t
t it
it
it t t t it t it t t it
? = = ?
?
?
=
+ ? ? ? + ? = ?
? ?
, 0 ] [ ] [ ;
) (
) , (
] [ )] ( [ ) (
1 1
? ? ? ?
? ?
Firm i’s BAU emissions in the current period are the sum of expected
emissions and an adjustment term that is proportional to a shock in ?
t
which contains
exogenous variables that influence either demand or supply of emissions. Abatement
is defined as the difference between firm i’s BAU and actual emissions e:
Chapter 3: Bubbles
20
(3.2)
it t it it
e BAU a ? ? = ) (
Abatement has a cost defined by a firm’s abatement cost function or its
derivative, the marginal abatement cost (MAC) function. As is well known from
permit market theory, each firm chooses abatement such that its MAC is equal to the
permit price in every period, which implicitly defines the optimal amount of
abatement:
)) ( , , ( )) ( , , (
1 * *
t it t t it it t it t it it t
BAU X MAC a BAU X a MAC ? = ? ? =
?
? ? ,
where X
t
refers to a vector of variables that determines the MAC function. To clear
the market, aggregate abatement has to equal the difference between overall BAU
emissions and the emissions cap S:
(3.3)
?? ??
= = = =
? =
T
k
N
i
ik
T
k
N
i
ik
S BAU a
1 1 1 1
*
Because firms involved in the production of power & heat are dominant
within the EU ETS, it makes sense to focus on emissions and abatement in this sector.
I will further assume that the predominant method of abatement is a (marginal) shift
in the generation dispatch order away from coal towards gas,
as the former is more
than twice as emissions-intensive per unit of output than the latter.
19
Fuel switching is
19
This shift will take place in the medium load spectrum, as peak load is already generated
using gas (and hydro) and base load is generated using nuclear, lignite and coal. Most likely
Chapter 3: Bubbles
21
generally considered to be important in the EU ETS (Alberola, et al., 2008,
Christiansen, et al., 2005, Delarue and D'haeseleer, 2007, Fehr and Hinz, 2006).
This
means that in addition to BAU emissions, abatement costs in the EU ETS depend on
gas and coal prices, which I will denominate as
t
G and
t
C . If aggregate marginal
abatement costs are approximately linear over the range where fuel switching is
feasible,
20
the market’s abatement cost function can be written as
21
(3.4)
? ? ? ?
= = = =
+ + + + =
N
i
it t t
N
i
it
N
i
it t t
N
i
it t
BAU g C d G d a b c BAU C G a MAC
1
2 1
1 1 1
) , , , (
The coefficients c>0 and b>0 indicate nonzero and increasing marginal
abatement costs, respectively. Aggregate MAC increases with gas prices and
decreases with coal prices, such that d
1
> 0 and d
2
< 0. Increased BAU emissions
translate in more necessary abatement to achieve the fixed cap S, which means that
0 > g .
In equilibrium, allowance demand must equal supply and the aggregate MAC
has to equal the permit price. This allows me to solve for the optimal aggregate
abatement:
it would entail the replacement of some very inefficient coal generators by combined cycle
gas generators (CCGTs).
20
In reality, the MAC functions of individual firms are step functions. However, aggregate
MACs on a sectoral level will be almost continuous over a certain range due to the range in
different generator efficiencies.
21
I also tried a specification where prices enter as logs, as is usually done in the finance
literature. It is not obvious in this case which specification is more appropriate. In any case,
the final results turned out to be very similar.
Chapter 3: Bubbles
22
(3.5)
b
BAU g dF c
b
a
N
i
it t
t
N
i
it
?
?
=
=
+ +
? =
1
1
*
?
where I set
t t t
C d G d dF
2 1
+ ? for notational convenience. Substituting (3.5) into
(3.3) yields
(3.6)
? ? ? ? ? ?
= = = = = =
? = ? ? ?
T
k
ik
N
i
T
k
ik
N
i
T
k
k
T
k
k
S BAU BAU
b
g
F
b
d
b
cT
b
1 1 1 1 1 1
1
?
I now take expectations at time t, subtract them from (3.6) and simplify:
( ) ( ) ( ) ( )
? ? ? ?
= + =
?
+ =
?
+ =
? + + ? = ?
N
i
T
t k
ik k ik
T
t k
k k k
T
t k
k k
BAU E BAU b g F E F d E
1 1
1
1
1
1
] [ ] [ ] [? ?
Entries for periods before t cancel out because their ex-post expectation is the
same as their realization. Likewise, the terms b cT / and S do not vary over time and
cancel. Substituting (3.1) and dividing by N yields
( ) ( ) ( )
? ? ? ? ? ?
= + = = + =
?
+ =
?
+ =
?
+
+ ? ? ?
+
+ ? = ?
N
i
T
t k
it
N
i
T
t k
k k k it
T
t k
k k k
T
t k
k k k
N
b g
E
N
b g
F E F
N
d
E
N
1 1 1 1
1
1
1
1
1
] [ ] [ ] [
1
? ? ? ?
Provided that the error is stationary, the last term’s mean and variance go to
zero as N goes to infinity. The intuition behind this is that uncorrelated, firm-specific
shocks cancel each other out in a large market, i.e. only shocks that affect all firms
simultaneously have an impact on BAU emissions (and thus on marginal abatement
costs). Simplifying the notation and solving for allowance prices results in
Chapter 3: Bubbles
23
(3.7) ( ) ( ) ( )
? ? ? ?
+ =
?
+ =
?
+ =
?
+ =
? ? ? + + ? + =
T
t k
k k k
T
t k
k k k
T
t k
k k
T
t k
k
E b g N F E F d E
1
1
1
1
1
1
1
] [ * ] [ ] [ ? ? ?
where ? is the average covariance between aggregate BAU emissions and ?
t
under
the assumption that this relationship is time-invariant, i.e. t
N
N
i
it t
? = =
?
=1
1
? ? ? .
If markets are efficient, prices incorporate changes in underlying
fundamentals immediately (Malkiel, 2007), implying that
t t t t
P P r P E ? ? + =
+
) 1 ( ] [
1
,
where r is the interest rate.
Equation (3.7) can be solved recursively (see Appendix
A) to
(3.8) ( )
( )
? ?
?
?
?
?
?
? ? ?
+ +
?
+ =
T
t
k T
t k t
T
t
k T
t t
t t
E
b g N
F F
d
?
?
?
?
?? ?
] [
1 1
1
The allowance price is determined by the previous-day price, changes in fuel
prices and shocks to
t
? . The summation term in the denominators of the RHS
decreases through time and indicates that shocks to exogenous variables increasingly
affect the permit price. This makes intuitive sense: In the beginning of the market, a
shock to emissions should not influence the permit price much, as it will likely be
neutralized by shock in the opposite direction later on. As time progresses this
probability diminishes. This means that in theory, fluctuations in the allowance price
should increase towards the end of the market. In practice, there is also an opposite
effect: New markets typically show more volatility than mature ones because market
Chapter 3: Bubbles
24
participants learn. Combined with the (in hindsight) apparent allowance surplus
which drove the price down to transaction costs by mid 2007, it seems likely that this
effect overshadows the inherent increasing uncertainty in a time-limited permit
market without banking.
In theory, the discount rate in equation (3.8) could be estimated directly using
nonlinear tools. However, in practice the day-to-day discount rate is very close to
zero. I therefore simplify the equation (3.8) to
(3.9) ( )
( )
? ?
?
?
?
? ? ?
+ +
?
= ?
T
t
k T
t t t
T
t
k T
t
t
E
b g N
F
d
?
?
?
?
] [
1
where ? refers to the first-difference operator. To keep the estimation linear I use an
annual discount rate of 10% to calculate the denominators on the RHS.
22
I assume that consumer demand is inelastic in the short term. Because
demand must meet supply at all times in the electricity grid,
t
? includes factors that
determine either demand or supply of BAU emissions. Specifically, I will include
temperatures across Europe, reservoir levels in the Nordic countries, the DAX and a
dummy indicating the first round of emissions verifications. The assumptions behind
this choice are the following: Temperatures affect consumer demand through
increased changes in heating (winter) or cooling (summer); reservoir levels influences
22
The choice of discount rate is more important for the RHS because small changes can cause
significant differences in the numerical value for the summation term. However, using
discount rates of 5% and 20% did not significantly change the results.
Chapter 3: Bubbles
25
emissions on the supply side through the availability of renewable energy;
23
and the
DAX is a proxy for overall economic performance in the EU’s largest economy. This
leads to the following econometric specification:
t t t t t
t t
t
t
t
t t
t
t t
t t
t
t
t
F
t
t
u v v N
D
DX R E R T E T
S W
C G
+ + =
+ +
?
?
+
?
?
+
?
?
+ +
?
?
+
?
?
= ?
?
2
1 1 0
2 2
6 5 4 3 2 1
); , 0 ( ~
] [ ] [
) (
? ? ? ?
? ? ? ? ? ? ? ? ?
(3.10)
noise white
otherwise 0 , 06 / 28 / 4 06 / 25 / 4 on 1 : dummy ver. emissions
DAX
levels reservoir nordic
otherwise 0 , Sep to Jun 1 : dummy summer
otherwise 0 , Mar to Nov 1 : dummy winter
e temperatur daily average
Europe rn Northweste for marker coal
gas natural UK for price forward month one
t
t t t
t
t
t t t
t t t
t
t
F
t
T
t
k T
t
u
D D D
DX
R
S S S
W W W
T
C
G
= ? =
= =
= =
?
?
?
?
?
Although the error ?
t
is uncorrelated over time, I allow its conditional
variance to change using an ARCH(1) specification, which is standard procedure in
the analysis of price series as price changes tend not to follow a normal distribution
with a fixed variance.
24
23
The more hydro and wind power available, the less power has to be produced using fossil
fuels, and thus the lower are BAU emissions. See also Christiansen (2005).
24
The alternative would be to drop the assumptions of Gaussian errors altogether, which has
been forcefully advocated by Mandelbrot (1997, 2004). In my regime-switching bubbles test
(see below) I use a t-distribution.
Chapter 3: Bubbles
26
In order to define shocks to reservoir levels and the weather, I need a measure
of what “normal” levels are. For reservoir levels, I use weekly median levels based
on the years 1991-2006. Because reservoir levels are cumulative by nature, a level of
one TWh below the median level today will lead to an expectation of tomorrow’s
level also to be one TWh too low, assuming that precipitation is “normal”. To
represent an unanticipated shock in reservoir levels I form first differences, such that
) ( ] [
1
med
t t t t t t
R R R R E R ? ? = ? ? ?
?
)
.
For temperature, I construct daily expectations using 30-y means, i.e.
?
=
?
=
2004
1975
1
* 30 / 1 ] [
y
dy t t
T T E , where d refers to the calendar day corresponding to day t
and y to years. Because traders are likely to take weather forecasts into account and
the weather over the weekends should influence Monday trades I calculate 5-day
moving averages of temperature minus its expectation centered on the current day:
( ) 5 / ] [ ] [
2
2
5 5
1 ?
+
? =
?
? = ? ?
t
t k
k k
d
t
d
t t t
T E T T T E T
)
. An alternative would be to use first
differences for reasons analogous to those discussed for reservoir levels, but the
problem with this approach is that smoothing combined with differencing leads to
very low variation, possibly diminishing any real signal below noise.
Comparing (3.9) to (3.10) shows that the latter is a reduced form of the former
because the parameters b , g and ? are not individually identified, only their
combined impact. While specification (3.10) is well grounded in economic theory, it
is based on two rather strong assumptions. First, expectations of tomorrow’s
Chapter 3: Bubbles
27
fundamental prices are today’s fundamental prices. Second, allowance price changes
are not autocorrelated over time. In the following two subsections I relax these
assumptions.
3.2.c.) Introducing dynamic expectations of fundamental prices
So far I have assumed that EUA prices have the Markov property in the sense
that tomorrow’s price is a function only of today’s price, but not of the preceding
price path. The Markov property is the centerpiece of asset pricing for stocks and
derivatives, and essentially implies that there are no arbitrage opportunities.
25
If all
traders have rational expectations and access to the same information, a belief (for
whatever reason) that a price will reach a certain level in the future will push the price
to that level today. The Markov property also implies that without storage costs, spot
and futures prices are equal, and that spot and forward prices differ exactly by the rate
of interest (Hull, 2002).
In reality, however, the relationship between spot and futures prices can be
quite different, even when taking storage costs into account. Whatever the reason
(asymmetric information, risk aversion, fixed contracts or bounded rationality), it is
possible that traders form their expectations about prices for EUA fundamentals not
only based on today’s prices, but also on past prices and a combination of spot and
futures prices.
25
If prices were a function of the past price path, chartist traders could use this information to
their advantage. However, there is little evidence that they are in fact able to do so.
Chapter 3: Bubbles
28
The problem is that it is impossible to know a priori which variables that
model traders’ unobserved expectations best. In order to search for the combination
of variables that best explains EUA price changes I set up the following specification:
(3.11)
t
t
q
t
p
t
o
t t
t
n
t t
t
m
t
l
t
k
t
j
t
i
t t
R R T
S W
T
S W
C C G G G
D
? ? ? ? ? ? ?
? ? ? ? ? ? ?
+
?
+
?
+
?
+ +
?
+ +
?
+
?
+
?
+
?
+
?
+ = ?
) ) ) )
11 10 9 8 7 6
5 4 3 2 1
) ( ) (
The indices (i, …, q) refer to a draw from a set of candidate variables.
Specifically,
( )
( )
( )
( )
20 5
,
5 ,
20 5 1
,
5 2 1 5 2 1
, ,
, ,
,
, , ,
, , , , , , ,
? ?
? ? ?
? ? ? ? ? ?
? ? ? ?
?
? ? ? ? ?
? ? ? ? ? ? ? ? ?
t t t t t
q p
M
t
d
t
o n
t t t t t t
m l
S
t
S
t
S
t
S
t
S
t
F
t
F
t
F
t
F
t
F
t
k j i
R R R R R R
T T T
C C C C C C C
G G G G G G G G G G G
) ) ) ) ) )
) ) )
where
S
t
G refers to the spot (day-ahead, to be exact) price for UK natural gas,
M
t
T is
the monthly deviation of temperature from the expected value and all other variables
are defined as above. I estimate (3.11) for each possible combination of the indices
(i, …, q) using an ARCH(1) model and choose the specification that yields the lowest
BIC.
26
The best-fitting specifications for the full, pre-crash and post crash periods are
the following:
26
Bayesian Information Criterion, also known as Schwartz’ Information Criterion. This
criterion trades off model fit and model parsimony and puts more weight on the latter than
Akaike’s Information Criterion. This procedure results in estimating 8*8*8*4*4*2*2*3*3=
294,912 ARCH regressions, and it took the 24 computers in AREC-UMD’s experimental lab
Chapter 3: Bubbles
29
20 5 20
20 1 2 1
20 5 20 2
, ; ; ; , :
; ; ; , , :
, ; ; ; , :
? ? ?
? ? ? ?
? ? ? ?
? = ? = = = ? = = ? = = ? = ?
? = = = = ? = = ? = ? = ? = ?
? = ? = = = ? = = ? = = ? =
t t t t
M
t t t
S
t
F
t
t t
M
t t
F
t
F
t
F
t
t t t t
M
t t t
F
t
F
t
R R q R R p T o n C C m l G k j G i crash Post
R R q p T o n C m l G k G j G i crash Pre
R R q R R p T o n C C m l G k j G i period Full
3.2.d.) Introducing lagged EUA price changes
Although the variance in specifications (3.10) and (3.11) is allowed to vary
over time due to the ARCH(1) term, the error itself is still supposed to be
uncorrelated over time. If the residuals are autocorrelated, it is common to either
specify an ARMA (p, q) error or to introduce lagged LHS variables on the RHS of the
equation.
27
Inclusion of lagged LHS variables can reduce or eliminate autocorrelation of
the residuals and increase the overall fit of the model. This is the reason why most
analyses of time series include either lagged prices or AR terms in the error.
However, this comes at a price: Because of different possible causes for
autocorrelation, the interpretation of the regression coefficients can become difficult.
over two days to complete this task. Including either more variables in (3.11) or widening the
candidate sets would be very challenging in terms of raw computing power (in May 2008,
that is). Note that if either of (i, j, k), (l, m), (n, o) and (p, q) draw the same variable from
their respective candidate set, one of them is dropped due to multicollinearity, which means
that the number of regressors included in (3.11) ranges from a minimum of 6 (5 plus a
dummy) to a maximum of 12.
27
Without any exogenous variables, AR (p) in the error term and p lagged LHS variables are
equivalent specifications, but this changes in the presence of exogenous variables (Bauwens
et al., 1999, p. 144). Also note that including lagged LHS variables and MA(q) error terms at
the same time will lead to biased estimation, because the regressors are no longer independent
of the error term.
Chapter 3: Bubbles
30
This is not a problem if the main goal is price forecasting, but what is the meaning of
a lagged LHS variable in a structural equation that seeks to define price determinants?
This question is routinely ignored, but in this context this would be inappropriate.
Autocorrelation in the residuals from estimating (3.11) could in principle be
caused by three different reasons: First, there could be an omitted fundamental
variable that is related to the allowance price, and which exhibits autocorrelation. In
this case, lagged LHS variables would serve as instruments for the omitted exogenous
variable. Second, expectations about future fundamental prices could not be captured
adequately by the additional terms in (3.11), and the true expectations exhibit
autocorrelation. Third, allowance prices could increase simply because they are
expected to do so based on past increases, regardless of fundamentals. This would be
the case of a price bubble. The coexistence of these three possibilities has made
conclusive testing for the existence of a price bubbles an almost impossible task,
especially in the absence of clearly defined market fundamentals. I will examine this
issue further in section 3.4.
Because the residuals from (3.11) indeed exhibit autocorrelation (see below), I
include five lags of allowance price changes in (3.11) but leave the equation
otherwise unchanged. I discuss the economic meaning of including lagged LHS
variables in a regression below.
Chapter 3: Bubbles
31
3.3. Results
The type and provenance of my data is as follows:
EUA prices: Daily series of over-the-counter (OTC) prices, Point Carbon.
28
Gas prices: Daily series of ICE month-ahead futures and Zeebrugge day-ahead prices
for UK natural gas.
Coal prices: McCloskey coal marker for North-Western Europe, which incorporates
information on all trades concerning coal that enters Europe from abroad and which
reach maturity within 3 months. It is an average of actual transactions or, in their
absence, an assessment of fair value by traders. This marker is published weekly.
Temperature: From the European Climate Assessment & Dataset
29
, which has daily
entries from a large number of monitoring locations across Europe. I weighted
temperature deviations by population around each monitoring location, using a World
Bank city area population dataset.
Nordic reservoir levels: Weekly reservoir levels (in TWh) and median levels based on
1991-2006 taken from Nordpool exchange. The Nordpool market (Norway, Sweden
and Finland) is the main hydropower-producing region in Europe.
28
Available at www.pointcarbon.com, last accessed in February 2008.
29
Klein Tank et al., “Daily Dataset of 20
th
-century surface air temperature and precipitation
series for the European Climate Assessment”, 2007, available at http://eca.knmi.nl.
Chapter 3: Bubbles
32
I estimate equation (3.10) separately for the January 2005 to June 2007,
30
as
well as for the period before the price crash induced by the first round of emissions
verifications in April 2006 (“pre-crash”) and after (“post-crash”). Visual inspection
of the price graph as well as previous analyses (Alberola, et al., 2008, Bunn and
Fezzi, 2007) indicate that the relationship between market fundamentals and the
allowance price likely changed after the price adjustment. Because the DAX lacked
correlation in this as well as all following models, and its inclusion increased both
Akaike’s and Schwarz’ information criteria, I removed it from all regressions and re-
estimated them without it.
The results are presented in Table 3.1. The coefficient estimates are
significantly different before and after the price crash based on an LR test, confirming
the suspected structural break. Gas prices are positive and significant for all periods,
and summer temperatures are significant in the two subperiods. The crash dummy
and the ARCH terms (not shown) are highly significant, as expected, but none of the
other variables appears to be associated with EUA price changes. The residuals from
all three regressions exhibit autocorrelation.
The goodness of fit of the model (calculated as the model sum of squares
divided by the total sum of squares) shows that the overwhelming part of the model’s
predictive power is due to the emissions verification dummy. Less than 4 % of the
30
As can be seen in Fig. 2. 2, prices reached transaction costs of a few cents by mid 2007,
which makes the inclusion of the second half of 2007 pointless.
Chapter 3: Bubbles
33
variation is explained in the pre-crash model, whereas the fit in the post-crash period
is even worse with 1%.
Results from estimating (3.11) are shown in the first two columns of Table
3.2. Once again, the coefficients before and after the price crash are significantly
different, confirming the structural break. Autocorrelation persists in the residuals
from (3.11), although to a lesser degree than for (3.10). Due to the presence of
various “flavors” of the same type of variable, the coefficients and associated p-
values of the individual lags lose their straightforward interpretation. For example,
the two coefficients on the reservoir level variables are of opposite sign but similar
magnitude for the full and the post-crash period. In contrast, gas prices are
consistently positive except for the coefficient on spot prices in the post-crash period.
Winter temperatures now have the expected sign in all periods (and are significant in
the full and post-crash period), whereas summer temperatures are positive for the full
and pre-crash period but negative and for the post-crash period. In any case, the main
focus here is on model fit, which has improved in terms of BIC by construction, but
not by very much, in spite of the serious data-mining exercise to find the “best”
specification:
31
The model predicts about 11% of the variation in the pre-crash
period, and less 3 % for the period after the crash.
31
I want to emphasize that the point of data-mining equation (3.11) for the best fit is to show
that even so, the model does not explain much of the variation in EUA prices, and is highly
sensitive to the inclusion of lagged LHS variables. A good fit from such a procedure would
not prove anything.
Chapter 3: Bubbles
34
Introducing 5 lags of allowance price changes into specification (3.11) yields
the results presented in columns 3 and 4 of Table 3.2. Lagged EUA price changes are
very effective in explaining current price changes in all periods. However, the
inclusion of the LHS lags takes its toll on the explanatory power of the exogenous
variables. Other than gas and the crash dummy, none of the coefficients is
statistically significant. Note that in spite of the lagged price changes, autocorrelation
persists, as well as evidence for a structural break.
Taken together, the results shown in Tables 3.1-3.2 imply the following:
1.) Allowance prices appear to violate the Markov property. Market
fundamentals are not immediately internalized, as past changes in fundamentals help
explain a portion of the price movements. Furthermore, price changes exhibit
autocorrelation.
2.) There was a structural break in the allowance price series after the price
adjustment due to the emissions verifications in April 2006. The coefficients on
market fundamentals from estimating the subperiod before and after the price crash
are significantly different.
3.) UK Gas prices are consistently associated with the allowance price before
the price adjustment in April 2006. This is consistent with results obtained by Bunn
& Fezzi (2007) and Alberola et al. (2008).
4.) Coal prices do not appear to be significantly correlated with the EUA
price, even though they should be important if fuel switching is an important form of
Chapter 3: Bubbles
35
abatement. This is probably due to the fact that they exhibit much less variation than
gas prices.
5.) Temperatures and reservoir levels help determine the allowance price, but
they are sensitive to the model specification and time period.
6.) When lagged EUA price changes are introduced into the model, they
absorb most of the price variation while all exogenous variables lose their
significance, with the exception of gas prices. The overall model fit greatly increases,
but some autocorrelation persists.
As discussed in more detail above, persistent autocorrelation and the
dominance of lagged LHS variables in explaining the EUA price variation could be
due to omitted exogenous (autocorrelated) variables, expectations about fundamentals
or a bubble-like phenomenon. However, the pattern of results gives some indications
as to which possibility is more likely.
My model contains all of the variables that are widely considered to drive the
allowance price. In order for an unobserved exogenous variable to drive the results,
this variable would have to be very important, exhibit strong autocorrelation and a
pattern similar to the very distinct price movement of the EUA. The existence of
such an exogenous variable seems unlikely.
I proxied expectations about fundamentals by introducing various time lags of
these variables into the model. Although this method is certainly an imperfect
measure of expectations, even the best-fitting combination of fundamentals out of
Chapter 3: Bubbles
36
almost 300,000 possibilities did not provide a very good model of EUA price
changes. Naturally there are more variables and lags that could be included in the
model, but based on my results so far it seems unlikely that expectations about market
fundamentals are behind the very distinct EUA price path.
The overall poor performance of the models without lagged EUA price
changes, the instability of the exogenous variables across models and time periods,
the fact that all fundamentals except for UK gas prices lose their significance as soon
as lagged prices are introduced and the persistence of autocorrelation across models
are all consistent with self-fulfilling expectations about the allowance price as a main
price driver, as would be the case in a price bubble. In the next section I examine this
issue further and carry out two bubbles tests.
3.4. A CO
2
Bubble?
3.4.a.) Some bubble background
Regardless of whether or not prices have the Markov property, they are
supposed to be driven by market fundamentals (hence the name) according to the
Capital Asset Pricing Model and Modern Portfolio Theory, two workhorses of
modern financial economics. Generally, an asset’s price P
t
can be represented as
t t t t
B F P ? + + = (Diba and Grossman, 1987, 1988, Flood and Garber, 1980), where
t
F represents the market fundamental (or intrinsic asset value),
t
B is a bubble term
Chapter 3: Bubbles
37
and
t
? white noise. Prices that are determined exclusively by
t
F are a special case
where the bubble term is set to zero.
A positive bubble term is due to traders’ expectations. If traders believe that
prices will increase, or believe that most other traders believe this (and so forth ad
infinitum), any random price expectation can be self-fulfilling.
32
After a time of
expectation-driven price increases, market participants will eventually realize that the
price is too high.
33
This will set in motion a positive feedback loop of asset sales,
downward adjustment of future price expectations and yet more sales. Note that in a
bubble, traders know that an asset’s price is above its intrinsic value, but they buy (or
hold) the asset anyway because they expect further price increases.
There exists an ample literature about price bubbles, an illuminating survey of
which is given by Camerer (1989) and, more recently by Abreu & Brunnermeier
(2003). The cornerstone of bubble theory is something called a growing rational
bubble, which refers to a constant term that appears in solutions to difference
equations that describe price formation in a market. Growing rational bubbles
increase exponentially at the rate of interest.
34
Under the standard assumption of
rational expectations, growing rational bubbles cannot exist with a finite number of
32
The assumption that prices are not driven by fundamentals but by traders’ beliefs is the
basis for technical or “chartist” analysis. Chartists aim to predict future price movements
based on past price alone, and have come up with a range of tools and indicators such as price
floors and ceilings, “head-and-shoulders”, turning points and more.
33
This can be triggered by an event such as one large seller coming to the market, a bearish
news report, or a round of emissions verifications as was the case in April 2006.
34
Blanchard and Watson (1982) developed the theory for stochastically crashing bubbles,
where traders know that the bubble will burst but not when. Stochastically crashing bubbles
have to grow at a rate greater than the rate of interest.
Chapter 3: Bubbles
38
agents trading a limited number of assets in a discrete time setting. The reasons for
this were formally developed by Tirole (1982), but to put it simply, it will be
irrational to hold the asset just before the bubble bursts, and therefore by backward
induction it will be irrational to hold the asset in any earlier period.
A widely used approach to solving the bubble existence problem has been to
allow for either some sort of irrationality or for incomplete information (Black, 1986,
Day and Huang, 1990, Frankel and Froot, 1990, Friedman and Aoki, 1992), and
bubbles have been shown to persist in lab experiments even with experienced subjects
(Hussam et al., 2008). In the following I will sidestep the question of theoretical
existence and focus on investigating whether a bubble did exist.
3.4.b.) Bubble tests
Bubbles tests have been largely confined to stocks that pay dividends. In such
a setup, the market fundamental can be shown to be the expected current value of the
dividend stream. Gurkaynak (2005) reviewed a series of bubbles tests, all of which
base their analysis on S&P 500 prices and dividends going back to 1871. Some
appear not to be appropriate to test for the presence of a bubble either on conceptual
and/or econometric grounds,
35
whereas others are more convincing (West, 1987) but
35
For example, variance bounds tests as introduced by Shiller (1981) and LeRoy and Porter
(1981) rely on dividends observed into the infinite future, which clearly cannot be
implemented. However, ways of getting around this problem (such as using the last observed
dividend as the terminal price) void the test of its meaning, as rejection of the null is no
longer linked to the presence of a bubble (Flood et al., 1994).
Chapter 3: Bubbles
39
cannot be implemented for an asset that has no stream of returns such as one-time
allowances. The classes of bubbles tests that can most readily be implemented in the
EUA context involve cointegration and regime switching tests.
Cointegration tests are based on the assumption that any bubble term would
have to increase faster than the underlying fundamentals. If the price and
fundamental series are cointegrated, this is clearly not the case. Therefore, a test for
cointegration between asset price and fundamental amounts to a bubbles test. Such
tests have been proposed by Diba & Grossman (1987, 1988) and by Hamilton and
Whiteman (1985).
Bubbles tests based on cointegration have been attacked on two grounds
(Evans, 1986): Whereas bubble theory predicts that a bubble can never be rendered
stationary no matter how many times it is differenced, this may not be true with real
data and small samples. In reality, prices do not follow a purely exponential increase
because they are influenced by too many other observed and unobserved factors.
Hence, any real price will eventually become (or at least appear) stationary when
differenced a sufficient number of times.
Second, a price that contains a series of stochastically crashing bubbles may
well pass a stationarity test, especially if the magnitude of price increases and
decreases is roughly equal (Evans, 1986, Hall et al., 1999). To solve this problem,
regime switching methods have been developed (Engel and Hamilton, 1990, Hall, et
al., 1999, Kim et al., 2008, Schaller and van Norden, 1997). Such models allow the
Chapter 3: Bubbles
40
data to be in more than one state. The presence of different regimes is then
interpreted as evidence for a series of bubbles.
3.4.c.) Cointegration test
Cointegration between EUA prices and market fundamentals would imply that
price movements can be accounted for by changes in fundamentals, and that therefore
there cannot be a bubble. Note that this is not the same as the estimations described
in section 4, as the presence of cointegration would imply that eqs. (3.10-11) are
misspecified, because they are written entirely in terms of differences but not levels
of the integrated variables.
Unit root tests indicate that the EUA price series, fuel prices, reservoir level
deviations and the DAX each are integrated of order 1, whereas temperature
deviations are stationary. The five integrated variables are plotted in Figure 3.1.
To test for cointegration, I start by re-specifying (3.10) as an Autoregressive
Distributed Lag (ADL) model while dropping the denominator terms on the RHS:
(3.12)
t t t t
t t t t t
D TS L C TW L C
DX L B R L B C L B G L B m L A
? ?
?
+ + + +
+ + + + =
) ( ) (
) ( ) ( ) ( ) ( ) (
2 1
4 3 2 1
) (L A is a lag polynomial defined by
p t p t t t
L A
? ? ?
? ? ? ? = ? ? ? ? ? ? ? ... 1 ) (
2 2 1 1
; ) (L B
i
and ) (L C
j
, ) 4 , 3 , 2 , 1 ( ? i and
Chapter 3: Bubbles
41
) 2 , 1 ( ? j , are lag polynomials of potentially different order each. I choose 5 lags for
all variables in order to incorporate weekly cycles. The cointegration vector
(3.13)
t t t t t t
DX
A
B
R
A
B
C
A
B
G
A
B
A
m
z
) 1 (
) 1 (
) 1 (
) 1 (
) 1 (
) 1 (
) 1 (
) 1 (
) 1 (
4 3 2 1
? ? ? ? ? = ?
is stationary if the EUA price is cointegrated with the fundamentals,
36
with ) 1 ( A and
) 1 (
i
B referring to the sum of all coefficients of the corresponding polynomials. The
variable
t
z measures how much out of equilibrium the cointegrated variables are in
any given period, with 0 =
t
z indicating the long-term equilibrium. If 0 >
t
z ,
t
? is
too high in relation to the other variables and will tend to decrease towards the
equilibrium (provided that 0 ) 1 ( > A ), and vice versa for 0 <
t
z . The more out of
equilibrium the cointegrated variables are, the stronger the forces that push them back
towards it.
Testing for cointegration is equivalent to testing whether
t
z has a unit root.
There are two different ways of obtaining an estimate for
t
z . One is to estimate
(3.12) and compute an estimate
t
z with the parameter estimates. Another is to
simply fit a linear regression of the allowance price on a constant, fuel prices and
reservoir level deviations.
37
I will label he residuals of this regression
t
z
(
. This
36
For a derivation, see Johnston and Di Nardo (1997)
37
This approach relies on the concept of superconsistency introduced by Engle and Granger
(1991). A problem with this could be that superconsistency is strongly based on asymptotic
Chapter 3: Bubbles
42
approach neglects all stationary exogenous variables, but (in theory) adding a
stationary variable should have no impact on a unit root test.
Standard critical values for unit root tests cannot be used in either case,
because I don’t know
t
z but only its prediction. MacKinnon (1991) derived relevant
critical values using Monte Carlo simulations. For a system of 5 potentially
integrated variables and a constant term, the critical values are -4.13 (10%), -4.42
(5%) and -4.96 (1%).
The cointegration test results are presented in Table 3.3. The two approaches
to compute the estimate for z
t
yield different results in terms of the actual values for
t
z and
t
z
(
, but for both series the null of a unit root cannot be rejected by a wide
margin. There appears to exist no cointegrating vector between the EUA and
integrated fundamentals. While no definitive proof, lack of cointegration between
price and fundamentals is certainly consistent with a price bubble.
3.4.d.) Regime-switching test
I carry out a regime switching test outlined by Hamilton (1989) and Engel and
Hamilton (1990), and, among others, applied by van Norden and Schaller (1997,
1996). Kim and Nelson (1999) extended this class of tests to a state-space framework
and a Bayesian analysis, but for the purpose of this paper, the traditional approach
properties, which may not be a practical assumption in finite samples (Johnston and DiNardo,
1997).
Chapter 3: Bubbles
43
suffices. The basic idea of a regime switching test is that not one, but two (or more)
different distributions govern price changes. In the bubbles context, a different
distribution would be expected in the growth and in the bust phase. Regime switching
is especially useful to detect a sequence of stochastically crashing bubbles where the
transition between growth and bust is not known, but it can also be used to detect a
single bubble as long as both the growth and bust phase are included in the data. Let
the variable ) 2 , 1 ( ?
t
S refer to one of two states. Allowance price changes can then
be written as
(3.14) ) 1 , 0 ( ~
2
2
1
1
1
N X X EUA
t t
S
t
S
t t
t
? ? ? ? ? + + = ?
The variables whose influence on the allowance price is different in the two
regimes are collected in the vector
1
t
X , whereas those with a stable impact across
states are represented by
2
t
X . This means that
2
2
2
1
2
? ? ? = = , but that the vector
1
1
S
?
is different for different values of
t
S . Likewise, I allow the variance to vary across
states. The transition between states is governed by a first-order Markov process:
q s s
q s s
p s s
p s s
t t
t t
t t
t t
? = = =
= = =
? = = =
= = =
?
?
?
?
1 ] 0 | 1 Pr[
] 0 | 0 Pr[
1 ] 1 | 0 Pr[
] 1 | 1 Pr[
1
1
1
1
The system can be solved for the parameter vector ) , , , , , , (
2 1
2
2
1
1
1
q p ? ? ? ? ? ? =
by maximum likelihood using numerical methods as shown by (Engel and Hamilton,
Chapter 3: Bubbles
44
1990, Hamilton, 1989, Kim and Nelson, 1999). Under the null hypothesis of no
bubble, the state in period t-1 has no impact on the state in period t, which means that
p q ? = 1 . The likelihood ratio statistic to test the null is
( ) ( ) | | ) ( ~ log log 2
2
r L L LR
R U
? ? =
where q is forced to equal p ? 1 in the restricted (R) model but is left as a free
parameter in the unrestricted (U) model, and r is the number of parameters in the
restricted model, in this case 3 ) ( length ) ( length * 2
2 1
+ + = ? ? r .
I estimate (3.14) based on model (3.10) plus an intercept for the entire period
as well as the pre-crash and post-crash subperiods. I allow all fundamentals to
influence the allowance price differently in the two states with the exception of the
emissions verification dummy. Results for the transition probabilities and the LR
statistic are given in Table 3.4.
38
The null of no state dependence is clearly rejected
for all periods.
These results imply the existence of (at least two) distinct regimes, and that
the sequence of regimes is nonrandom. Like the cointegration test, this is no
conclusive proof for a bubble, but it is consistent with the presence of a one or a
series of stochastically crashing bubbles.
38
Full results available from the author upon request.
Chapter 3: Bubbles
45
3.5. Conclusions
In the first phase of the EU ETS, the allowance price exhibited high volatility
and followed a peculiar path. While the crash in April 2006 was clearly caused by an
adjustment of expectations about aggregate emissions, it is not obvious what drove
the price that high in the first place, and why it took so long to finally decrease to
zero. In this paper I examine if and to what extent the allowance price was
consistently driven by market fundamentals.
I set up a market model that relates the change in the allowance price to
changes in fuel prices, temperature and reservoir level, under the assumption of
efficient markets. I estimate this model for the entire period, as well as for the
subperiods before and after the allowance price crash. I then relax the model by first
allowing for delayed adjustment to fundamentals, and then by including lagged price
changes as predictors for current price changes.
The specification that relies exclusively on contemporaneous and exogenous
price drivers performs quite poorly, in spite of the fact that I’m using the best data
available. The introduction of lagged exogenous variables improves price predictions
for the period before the price crash, but as soon as lagged EUA price changes are
allowed in the model, all explanatory variables lose their significance with the
exception of UK gas prices.
Although lagged LHS variables are routinely used in time series analysis, it is
important to ask what exactly a dependence of price on its own past means if the goal
Chapter 3: Bubbles
46
is an analysis of price drivers. Autocorrelation could be caused by an omitted
autocorrelated exogenous variable, expectations about fundamentals or self-fulfilling
expectations about future price changes that are not related to fundamentals. The last
situation is equivalent to a price bubble.
In a price bubble, firms that held surplus allowances would be reluctant to sell
because they expected future prices to increase. For the same reason, buyers wanted
to buy sooner rather than later, driving prices to whatever level expectations
happened to be. The presence of a price bubble would in effect destroy the prime
advantage of a permit market, which is the achievement of a given emissions cap at
least cost, because the inflated permit price is at least partially passed on to
consumers.
I examine this hypothesis further by carrying out two bubbles test, one based
on cointegration between the EUA price and market fundamentals and the other on
regime switching. Both tests indicate that a price bubble, or a series of bubbles, is
consistent with the data. The positive test results add another layer of evidence to the
bubbles hypothesis, especially since market fundamentals in a permit market are
better known than in the typical context of bubbles tests, which so far have been
almost exclusively been applied to stock markets.
To formulate it the other way around, in order for these results not to indicate
the presence of a price bubble in EU ETS allowances, there must either exist a crucial
but as of yet unrecognized fundamental price driver whose realizations tally with the
peculiar price movement of the EUA, or expectations about fundamentals had to be
Chapter 3: Bubbles
47
extreme enough (and, in hindsight, far away from actual realizations) to account for
this price variation. In the absence of either of these two –in my view unlikely-
scenarios, one would have to conclude that the first phase of the EU ETS indeed was
characterized by one or a series of speculative price bubbles.
Chapter 4: Market Power
48
Chapter 4: Market Power and Windfall Profits in Emission
Permit Markets
Abstract
Although market power in permit markets has been examined in detail
following the seminal work of Hahn (1984), the effect of free allocation on price
manipulation with market power in both output and permit market has not specifically
been addressed. I show that in this case, the threshold for free allocation above which
dominant firms find it profitable to increase the permit price is below their emissions.
In addition to being of general economic interest, this issue is relevant in the context
of the EU ETS, where it appears that power producers profited from a high permit
price. Because power producers were net permit buyers, Hahn’s results imply that
market power in this sector could not have been involved. My results change this
conclusion. Using data from the UK and German power markets, I find that power
generators received free allowances well in excess of the derived threshold.
Keywords: Market power, emissions permit markets, air pollution, EU ETS, CO
2
,
electricity generation, permit allocation, windfall profits, cost pass-through.
JEL classification: H23, L11-13, L94, L98, Q48, Q52-54, Q58
Chapter 4: Market Power
49
4.1. Introduction
During the first eighteen months of the European Union Emissions Trading
Scheme (EU ETS), the allowance price per ton of CO2 was far above ex-ante
expectations. It fell to one-half of its value in April 2006 after the first round of
emissions verifications showed the market to be long and eventually reached zero by
mid 2007, but it is not clear what drove the price so high in the first place. A series of
studies (Alberola, et al., 2008, Bunn and Fezzi, 2007, Mansanet-Bataller, et al., 2007,
Rickels, et al., 2007) has tried to empirically explain the price path by market
fundamentals such as fuel prices and weather variables, but only with limited success
as fundamentals appear to only account for a small fraction of the allowance price
variation. Especially the very high price levels before the April price crash lack a
satisfactory explanation.
An inflated permit price in the sense that it is above marginal abatement costs
of the market as a whole destroys the most powerful argument in favor of instituting
pollution permit markets, which is to achieve a given emissions target at least cost.
The increased costs are due to over-abatement on behalf of any firm that does set its
marginal abatement cost equal to the permit price, and to consumers paying too high
prices for pollution control if permit prices are passed through in the output market.
In this paper, I examine whether price manipulation within the EU?s power &
heat sector could have been a cause of the apparent allowance price inflation. I extend
economic theory by setting up a model that allows for market power in both the
Chapter 4: Market Power
50
output and the permit market and explicitly accounts for a link between these
markets. I derive the conditions under which a dominant firm will exercise its market
power to increase the permit price in order to maximize overall profits in both
markets. Finally, I apply my theory results to data from the EU ETS and show that
these conditions were fulfilled, i.e. provided that such “double” market power existed,
it would have led to an inflation of both permit and output price.
The interplay between permit and output market is at the root of what has
become known as “windfall profits” in the empirical literature. If firms are able to
pass through pollution costs to consumers but receive most (or all) permits allocated
for free, they get reimbursed for costs they never had to incur. Windfall profits have
been identified as an issue in permit markets in general (Bovenberg and Goulder,
2000, Vollebergh et al., 1997), and in particular in the EU ETS (Grubb and Neuhoff,
2006, Hepburn et al., 2006, Neuhoff et al., 2006, Sijm et al., 2006, Smale et al.,
2006). Such profits constitute a wealth transfer from consumers to firms but they do
not impact efficiency directly
39
nor affect the permit price in a competitive market.
This no longer holds under the presence of market power in both the output and
permit market, because a price-setting firm will take windfall profits into account
when making its production and permit purchase decisions.
One of the best-known results about market power in permit markets is Hahn?s
(1984) finding that the permit price is an increasing function of the dominant firm?s
39
Handing out permits for free impacts efficiency through existing distortions such as income
taxes. In theory, the revenue from a tax or selling permits has to be recycled through lower
distortionary taxes to achieve (Bovenberg and Goulder, 1996, Parry, 1995).
Chapter 4: Market Power
51
permit allocation. If this firm is a net buyer of permits, it will exert its power to
decrease the permit price in order to minimize compliance costs, and vice versa.
Other studies have confirmed and extended these findings (Isaac and Holt, 1999,
Liski and Montero, 2005, Maeda, 2003, Westskog, 1996). Hahn?s results imply that a
dominant firm in the power & heat sector could not possibly have used its market
power to increase the allowance price, because this sector was underprovided with
permits and thus, if anything, would have used its power to decrease the price.
Hahn derived his results by focusing exclusively on the permit market while
ignoring any distortions in the output market. However, if a firm has market power in
the permit market, it is likely to also perceive market power in the output market.
40
Misiolek and Elder (1989) introduced exclusionary manipulation whereby the
dominant firm intends to drive competitors out of the output market by manipulating
the permit price, an approach also followed by von der Fehr (1993) and Godby
(2000). A series of lab experiments empirically tested the relevance of combining
market power in the output and permit markets (Cason et al., 2003, Godby, 2002,
Muller and Mestelman, 1998). These studies found that a combination of market
power in both markets increased the dominant firm?s power to manipulate prices and
that the overall effect on industry profits and consumer welfare depended on firms?
relative efficiencies and permit allocation and thus was ambiguous. However, they
did not address whether and how Hahn?s threshold of “neutral” allocation is altered
40
If a permit market covers several industrial sectors, the firm’s market share in the permit
market will be lower than its share in the output market, which implies market power in the
latter market given market power in the former.
Chapter 4: Market Power
52
by the presence of market power in both permit and output market which are
explicitly linked.
The issue of double market power is closely related to the literature pertaining
to “raising rivals? costs” (Hart and Tirole, 1990, Krattenmaker and Salop, 1986a, b,
Ordover et al., 1990, Salop and Scheffman, 1987, 1983). The focus of this literature is
the theory that predatory firms may increase their market share and overall profits by
artificially increasing industry costs, given certain assumptions. This can take many
forms, including the institution of mandatory standards, labeling, advertising etc, all
of which are expected to be less costly on a per-output basis for the dominant firm
than for the price-taking fringe. One particular version of raising rivals? costs is to
over-purchase necessary inputs of production (Salop and Scheffman, 1987), which is
a profitable strategy if the output price increase from this manipulation exceeds the
firm?s average cost increase. Sartzetakis (1997) applied this framework specifically
to emissions permits as a necessary input to production, but he refrained from
examining how free allocation determines the existence and direction of price
manipulation.
41
However, certain aspects of the interplay between market power in the output
and an associated pollution permit market are not well captured by this literature.
First, raising rivals? costs focuses on increasing profits of a dominant firm at the
expense of rivals while decreasing overall industry profits. Profits from jointly
41
Indeed, he mentions that a policy based on such a threshold would require full information
and the “willingness to base permits allocation on efficiency rather than distributional
considerations”.
Chapter 4: Market Power
53
manipulating output and permit prices on the other hand accrue to all firms in the
industry and they come at the expense of consumers and taxpayers. In fact, fringe
firms can free ride on the manipulative actions of the dominant firm as they enjoy
increased profits without incurring the costs of price manipulation. Second, there are
no strong assumptions needed about a dominant firm?s efficiency relative to that of
the fringe in order for price manipulation to be profitable: As I show below, even a
dominant firm that is very inefficient at abating or producing can find it optimal to
increase the permit price, given that it receives a sufficiently generous free allocation.
In the next section I derive a threshold of free allocation beyond which a
dominant firm will find it profitable to increase the permit price. The threshold is a
function of the firm?s market power in both markets as well as its emission intensity
but is always below the full-allocation threshold defined by Hahn. This is my core
result and means that a dominant firm may find it optimal to increase the permit price
even if it is a net buyer of permits. I then apply this finding to the EU ETS and
examine whether firms in the power & heat sector likely received a free allocation in
excess of this threshold, and therefore whether market power in this sector could be a
cause for the high allowance price. Section 4.4 concludes.
Chapter 4: Market Power
54
4.2. Market power in output and permit market
In the following I set up a simple model for an industry sector containing N
firms that is subject to an emissions permit market.
42
I define the cost function for
firm i=1, ..., N as ) , (
i i
i
e q C , a continuous function which depends on output
i
q and
emissions e
i
and is twice differentiable in both arguments. Costs are increasing in
output, decreasing in emissions and convex in both arguments, such that 0 >
i
q
C ,
0 >
i
qq
C , 0 <
i
e
C , 0 >
i
ee
C , 0 <
i
qe
C and ( ) 0
2
> ?
i
qe
i
ee
i
qq
C C C . I assume that firm 1 has
market power in both the output and the permit market.
43
To study the equilibrium, I start by analyzing the behavior of firms i=2, …, N
that comprise the price-taking fringe, before I move on to the dominant firm. The
fringe’s profit maximization problem is
(4.1)
i i
i i i i
i
i i
x e q
x e t s
x x e q C pq
?
? ? ? = ?
. .
) ( ) , ( max
, ,
?
where p is the output price, ? the permit price,
i
x refers to permit purchases and
i
x
is firm i’s free permit allocation. With a binding cap, I can substitute the constraint
into the objective function and arrive at the familiar first-order conditions that
42
This permit market may also include other sectors, but for simplicity I will confine the
analysis to one sector.
43
With a permit market that covers just one sector, assuming market power in one market but
not the other seems arbitrary. If the permit market covers many other sectors as well, then it
is conceivable that a firm has market power in the output market but not the permit market.
The converse, however, would not make economic sense.
Chapter 4: Market Power
55
marginal production costs equal the output price, and marginal abatement costs equal
the permit price. This implicitly defines the fringe’s optimal output, emissions and
permit purchase decisions:
(4.2)
) , (
) , (
) (
) (
* * *
* *
?
?
? p x x e
p q q
C
C p
i i i
i i
i
e
i
q
= =
=
?
? ? =
? =
The dominant firm takes (4.2) into account when maximizing its own profits.
It faces an inverse demand function and a permit market-clearing condition of
(4.3)
( )
) , (
) , ( ) (
2
*
1
2
*
1
?
?
p x x S
p q q P Q P p
N
i
i
N
i
i
?
?
=
=
+ =
+ = =
where S is the overall emissions cap and q
1
and x
1
are the dominant firm’s output and
permit purchase decisions, respectively. This system of equations describes a fixed
point with a mapping of | | ( ) ) , ( ), , ( ) , ( ), , (
1 1 1 1 1 1 1 1
x q x q p x q x q p F ? ? ? . A unique
solution exists if the vector ) , ( ? p belongs to a convex set (which is trivially true for
prices), and ] [? F is upper-semicontinuous and monotone, which is assured by the
continuity and monotonicity of the demand function ) (Q P and the cost functions
) , (
i i
i
e q C .
From equations (4.1)-(4.3) it follows that the output price and the permit price
are both a function of the dominant firm’s output and permit purchase decisions:
Chapter 4: Market Power
56
) , (
) , (
1 1
1 1
x q
x q p p
? ? =
=
The impact of the dominant firm’s output and permit purchase decisions on
the output and permit price can be assessed using comparative static calculations and
is summarized in the following Lemma:
Lemma 1:
The dominant firm’s output and permit purchase decisions will influence
output and permit price jointly such that
0
x
0;
q
0
x
p
0;
q
p
1 1
1 1
> <
> <
?
??
?
??
?
?
?
?
(Proof: Appendix B)
The dominant firm’s profit maximization problem and the resulting first-order
conditions are
(4.4) ) ( ) , ( ) ( ) , ( ) , ( max
1 1 1 1 1 1 1 1
1
1 1 1 1
, ,
1 1 1
e x x q x x e q C q x q p
e x q
? + ? ? ? = ? ? ?
(4.4a) 0 ) ( ) ( ) (
1
1 1
1
1
1
= ? ? ? ? + ?
q
x x C q
q
p
p
q
?
??
?
?
(q
1
> 0)
Chapter 4: Market Power
57
(4.4b) 0 ) ( ) (
1
1 1 1
1
= + ? ? ? ? ?
?
??
?
?
?
x
x x q
x
p
(x
1
> 0)
(4.4c) ? = ? ? ) (
1
e
C
(4.4d) 0 ) ( ; 0 ;
1 1 1 1
= ? ? ? e x e x ? ?
The last first-order condition implies that the constraint may or may not be
binding. To analyze the incentive of the firm to manipulate the permit price in either
direction I combine (4.4b) and (4.4c) to get
(4.5)
*
1
1 1
1
*
1
1
) ( ) ( ) ( q
x
p
x
x x C
e
?
?
?
??
? ? ? + ? = ? ?
where the asterisks indicate that the permit purchases and output are chosen optimally
by the firm according to (4.4). If with a permit price increase the additional revenue
from cost pass-through (the last term on the RHS) outweighs the higher permit
purchase costs (the second term), then the firm?s marginal abatement costs are below
the permit price. This means that it will under-abate -or, equivalently, over-purchase
permits-relative to the situation where it perceives no price-setting power through its
permit purchase decision in either market
44
( 0 / /
1 1
= = x p x ? ? ? ?? ) and thus push up
the permit price. Moreover, if the revenue effect outweighs the compliance cost
effect to the point where 0
1
= ?
e
C , then it will not abate at all and
*
1 1
*
1
x e e
BAU
? = ,
44
Note that it still may perceive market power through its output decision. Equation (4.5)
strictly applies to output and permit price manipulation through the permit purchase pathway.
Chapter 4: Market Power
58
where
BAU
e
1
refers to business-as-usual (BAU) emissions in the absence of a permit
market. Conversely, if compliance costs outweigh increased revenue the firm will
find it optimal to under-purchase permits in order to depress the permit price and
over-abate accordingly and over-abate accordingly. This can be summarized as
(4.6)
?
?
?
?
??
?
?
> ?
= ?
< ?
? ?
<
=
>
1
1
1
1
1
*
1
*
1
1
) (
e
e
e
C
C
C
x
x x q
x
p
Condition (4.6) implies that there is a specific amount of free allocation that
will cause the dominant firm to set its marginal abatement costs equal to the permit
price. Solving (4.6) for this threshold allocation
0
1
x yields
(4.7)
*
1
1
1 *
1
0
1
/
/
q
x
x p
x x
? ??
? ?
? =
This quantity is unambiguously smaller than the firm’s optimal permit
purchases, provided that 0
1
<
qe
C .
45
Note that the firm’s optimal permit purchases and
output are a function of its allocation, such that the threshold in (4.7) is difficult to
compute ex-ante, except for very simple functional forms of the cost function and
permit and electricity demand. However, the threshold can be evaluated relatively
45
If 0
1
=
qe
C , then 0 / /
1 1
= = q x p ? ?? ? ? (see Appendix A), and
*
1
0
1
x x =
Chapter 4: Market Power
59
easily ex-post when making some simplifying assumptions about consumer demand
response (see below).
46
Equations (4.6)-(4.7) lead to the following result:
Result 1:
After the market has been instituted and firms’ allocation, emissions and output
decisions have been observed, we can infer that:
a. If the dominant firm received a free permit allocation equal to
0
1
x , it
acted as a price taker in the permit market in the sense that it set its
marginal abatement costs equal to the permit price.
b. If the dominant firm’s allocation was greater (smaller) than
0
1
x its
marginal abatement costs were below (above) the permit price and it
manipulated the permit price upwards (downwards) by over- (under-)
purchasing permits.
c. The threshold allocation
0
1
x is smaller than the firm’s emissions and
necessarily makes the firm a net buyer of permits.
46
This caveat applies to some extent also to Hahn’s results. Only if the firm’s cost function
is known can the regulator compute its efficient emissions and thus determine
H
x
1
. The
difference is that in my setup, the regulator also needs to know the firm’s degree of market
power and find a closed-form or numerical solution for ) (
1
*
1
x x . I will leave the proof for
the existence and uniqueness of such a solution for future work.
Chapter 4: Market Power
60
Result 1 is the core finding of this paper and states that even if the dominant
firm is a net buyer of permits it can find it in its interest to manipulate the permit price
upwards, provided that its allocation is sufficiently high.
Note that this is a generalization of Hahn?s result, which I will denominate as
1 1
x x
H
= : A dominant firm will only abstain from manipulating the price if it
receives exactly the number of allowances necessary to cover its emissions and
therefore does not trade. To see this, simply set 0 /
1
= x p ? ? in (4.6) or (4.7), thus
eliminating the link between output and permit markets. Also note that if the second
term on the RHS on (4.7) is sufficiently large (i.e. if the impact of the firm’s permit
purchases on output and permit price is sufficiently strong) then 0
0
1
< x . In this case,
even full auctioning would lead the firm to choose a permit price that is greater than
its abatement costs.
On the other hand, if a firm has been observed to emit more (less) than its
initial permit allocation, Hahn's model would imply that the firms marginal abatement
costs are below (above) the market price. My model therefore shows that this
conclusion is premature if output markets are taken into account.
So far I have focused on the effect of permit allocation on the permit price.
However, as is clear from (4.3) and (4.4), the dominant firm?s allocation also has an
impact on the output price. I start by re-writing (4a) as
(4.8)
1
1
*
1
*
1
1
1
) ( ) ( ) (
q
x x q
q
p
C p
q
?
??
?
?
? + ? ? = ?
Chapter 4: Market Power
61
With neither market power nor a permit market there would be the standard
outcome that price equals marginal production cost, i.e.
1
q
C p = . Market power in the
output market increases the output price by the second term on the RHS, which is also
a familiar result. The last term describes the effect of linking a permit market to the
output market. Because 0 /
1
< q ? ?? , this term decreases (increases) the output price
if the firm is a net buyer (seller) of permits. Substituting Hahn’s result of
*
1 1
x x
H
=
would cancel this third term, but it would not remove the output price distortion
introduced by the second term. To see how my generalized threshold
0
1
x performs in
this case, I solve (4.5) for
1
*
1
x x ? and substitute into (4.8) to get
(4.9) ) (
/
/
/
/
) ( ) (
1
1
1 *
1
1 1
1 *
1
1
1
?
? ??
? ??
?
??
? ??
? ?
?
?
? ? + + ? ? = ?
e q
C
x
q
q
q x
x p
q
q
p
C p
By construction, allocating
0
1
x to the dominant firm eliminates the last term,
as in this case the marginal abatement costs are equal to the permit price. The third
term on the RHS is negative and thus decreases output price distortion. However, the
price distortion is not fully removed because it can be shown that
(4.10)
1 1 1
1
/
/
q
p
q x
x p
?
?
?
??
? ??
? ?
> (Proof: Appendix B)
It follows immediately that the output price can be brought to its efficient
level only by allocating less than
0
1
x to the dominant firm, because in this case the
Chapter 4: Market Power
62
last term will be negative. The threshold allocation to the dominant firm that yields
1
q
C p = can be computed using (4.8) and is
(4.11)
*
1
1
1 *
1
00
1
/
/
q
q
q p
x x
? ??
? ?
? =
The fact that
0
1
00
1
x x < can easily be verified by using the inequality in (4.10).
As before, due to the dependence of
*
1
x on
1
x , this threshold can be evaluated ex-
ante only under very simple functional forms. This leads to the following result:
Result 2:
After the market has been instituted and firms’ allocation, emissions and output
decisions have been observed, we can infer that:
a. If the dominant firm received an allocation of
00
1
x , its marginal
production costs were equal to the output price.
b. If the firm received more (less) than
00
1
x , marginal production costs
were greater (smaller) than the output price.
c. The threshold allocation x
1
00
is smaller than
0
1
x .
Chapter 4: Market Power
63
The fact that marginal costs are equal to price in the two markets at two
different levels means that efficiency cannot be restored completely. Either the
permit price is distorted, or the output price, or both:
Result 3:
a. The first-best solution in the sense that both the output and the permit
price are at their competitive levels cannot be achieved by means of
permit allocation alone, because
0
1
00
1
x x < .
b. If the firm received more than
0
1
x (less than
00
1
x ), both output and
permit price were distorted upwards (downwards) relative to
marginal costs. If the firm’s allocation was
0
1 1
00
1
x x x < < , the output
price was above and the permit price below marginal costs.
Results 1-3 imply that under the assumption of market power in both markets,
the amount of free allocation is crucial for price distortion, and that Hahn?s
“neutralizing” allocation prescription will result in an inflation of both output and
permit price. In the following section I will empirically address the relevance of
these findings in the context of the EU ETS.
Chapter 4: Market Power
64
4.3. Application to the EU ETS
4.3.a.) General applicability
Although the EU ETS covers six broad industry sectors, the main players both
in terms of allocation and emissions are firms within the power & heat sector. There
is evidence that this sector was subject to significant windfall profits (Grubb and
Neuhoff, 2006, Hepburn, et al., 2006, Neuhoff, et al., 2006, Sijm, et al., 2006), which
sets the stage for price manipulation as analyzed in the previous section. According to
market observers (e.g. Point Carbon), it was the sustained allowance purchases from
power & heat, combined with a relatively short allowance supply from the other
sectors, that drove the price to the –in hindsight-very high level. There are a number
of very large power producers for which the assumption of some market power seems
at least possible.
On aggregate, firms in the power & heat sector were net demanders of
allowances, whereas the other sectors covered by the EU ETS were over-allocated as
a whole (Figs. 2.2-3). Hahn?s results imply that in this case, price manipulation by
dominant firms within this sector could not have been behind the allowance price
increase, but as I show in section 4.2, this does not hold if firms are able to influence
both the output and the allowance price. According to Result 1, a dominant firm
would have found it optimal to use its market power to increase the allowance price
even if it was a net buyer of allowances, as long as its free allocation exceeded x
1
0
.
Chapter 4: Market Power
65
I will now address the question whether there is any evidence relating firms’
actual allocation to x
1
0
. To do this, it will be convenient to introduce a substitution in
notation. At x
1
= x
1
0
the emissions constraint will be binding such that x
1
= e
1
.
Defining y
1
= x
1
/e
1
to be the proportion of actual emissions that the dominant firm
receives allocated for free, I can re-state (4.7) in terms of the corresponding y
1
0
:
(4.7’)
1
1
1
1
1
1
0
1
1
/
/
1
q
e x
x p
y ? < ? = ?
?
? ??
? ?
The threshold in (4.7’) is exactly equivalent to
0
1
x as defined by (4.7), but
instead of depending on the dominant firm’s output
1
q it now contains its emission
intensity (average emissions per unit of output), denoted
1
? .
The main difficulty to empirical assessment of (4.7) is to determine the effect
of a dominant firm’s permit purchase decisions on the output and the permit price.
To get around this problem, I will use the fact that the numerator on the RHS of (4.7’)
is equivalent to the impact of the permit price on the output price if there is no
demand response:
47
47
Totally differentiating output and permit price and dividing yields
1 1 1 1
1 1 1 1
* / * /
* / * /
dx x dq q
dx x p dq q p
d
dp
? ?? ? ??
? ? ? ?
? +
+
=
Chapter 4: Market Power
66
(4.12)
0 1
1
1
/
/
=
=
dq
d
dp
x
x p
? ? ??
? ?
I will argue that the short-term demand response for electricity by households
is very small, as the most efficient means to reduce demand is to make changes in the
portfolio of household appliances towards more energy-efficient items, which takes
time. Assuming no consumer demand response during the first 18 months of the EU
ETS, I can substitute (4.12) into (4.7’). The effect of the allowance price on the
electricity price ( ? d dp/ ) can be estimated by looking at the electricity spreads
before and after the institution of the permit market relative to the allowance price. I
will do this separately using market data from the UK and Germany, two of the
largest players in the EU ETS.
4.3.b.) The UK power market
Gas-fired power plants are at the margin during medium and peak hours in the
UK and are therefore price setting during these loads (Grubb and Newbery, 2007).
48
Figure 4.1 shows year-ahead spark spreads and clean spark spreads in the UK. The
spark spread is the theoretical gross profit of a gas-fired power plant from selling a
unit of electricity, having bought the fuel necessary to generate it:
48
During base loads, the marginal generator is coal and nuclear for most hours, as they are
generally ranked lower in the merit order than gas plants.
Chapter 4: Market Power
67
(4.13)
) / ( ) / ( ) / ( ) / (
*
e MWh g MWh g MWh Euro
gas
e MWh Euro e MWh Euro
p p spread Spark ? ? =
Here, p refers to the power price as before,
gas
p to the price for natural gas
and ? to the heat rate (or efficiency). The clean spark spread, also called green spark
spread, further adjusts the revenue stream by the CO
2
costs embedded in power
generation:
(4.14)
gas
spread spark spread spark Clean ? ? * ? =
where
gas
? denotes the emission intensity of a typical gas plant. Solving (4.14) for
the electricity price and its derivative with respect to the permit price yields
(4.15)
gas gas gas
d
dp
p spread spark clean p ?
?
? ? ? = ? ? + = *
At first sight, this seems to imply that the extent to which the electricity price
is increased due to CO
2
costs is simply the emissions intensity used to compute the
clean dark spread. However, these spreads are created as benchmarks and don’t
imply that the emission intensity used in their calculation is necessarily the average
emission intensity of the marginal generator in the market. In theory, one could
calculate the clean spark spread for any type of generator.
Chapter 4: Market Power
68
To see how
gas
? relates to ? d dp / it is useful to compare pre-market spark
spreads with post-market clean spark spreads. With full cost pass-through,
49
constant
demand and stable technology, the clean spark spread will be equal to the pre-market
spread if the “correct” emission factor is used in (4.14), i.e. if ? ? d dp
gas
/ = . In
other words, theoretical gross profits
50
of a power producer will not change if the
power price is increased by an amount exactly equivalent to the embedded carbon
cost.
Figure 4.1 shows almost precisely such a market. Before the carbon market,
UK spark spreads fluctuated around a level of £6-8. In January 2004, the year-ahead
spreads immediately incorporated the carbon cost of producing electricity based on
the year-ahead forward price for CO
2
, with the spark spread increasing and the clean
spark spread taking the place of the spark spread. The equalization of pre-market
spark spreads and post-market clean spark spreads implies that the CO
2
costs
embedded in the production of peak electricity are those that correspond to the
allowance price multiplied by ?
g
used to calculate the clean dark spread, which is
0.41 tCO
2
/MWh, the emission intensity of a typical Combined Cycle Gas Turbine
(CCGT) (PointCarbon, 2007). This means that in the UK,
e MWh tCO d dp
g
/ 41 . 0 /
2
= ? ? ? . Naturally, this is an approximation as the clean
spark spread has its own variation over time, and factors other than the allowance
49
Assuming a completely inelastic demand, profit-maximizing firms will pass their costs
fully through to consumers.
50
As discussed above, there may well be windfall profits due to the institution of the market
because of free allocation.
Chapter 4: Market Power
69
price could potentially have affected the spreads. However, the figure and other
evidence (Grubb and Newbery, 2007, Vorspools, 2006) implies that electricity prices
increased roughly by the amount that corresponds to carbon costs incurred by an
average CCGT.
I will now do the following back-of-the envelope calculation: Firms in the UK
power sector on average received 77.1% of their emissions allocated for free.
51
Substituting a value of 0.77 for the UK, setting 41 . 0 = ? d dp and solving (4.7’) for
the emission intensity, this means that it would have been profitable for dominant UK
power firms to use their market power to increase the allowance price (and with it the
electricity price) by over-purchasing allowances if
(4.16) e MWh tCO / 78 . 1 ) 77 . 0 1 ( 41 . 0
2
= ? < ?
In comparison, the emission intensity of an anthracite coal power plant (the
most emission-intensive method of power generation used today in Europe) with a
heat rate of 33.3%,
52
is 1.06 tCO
2
per MWh of output. In other words, even the least
efficient of all power production companies received an allocation in excess of
0
1
y (or
0
1
x ) and would have found it profitable to manipulate the permit price upwards,
provided that it had some market power.
51
Power & heat generators with an allocation of at least 100,000 allowances, based on 2005
numbers. This list is a subset of installations with activity code 1 (combustion) in the
Community Independent Transaction Log (CITL). Many code-1 installations produce process
power & heat only and are much smaller than power plants. For all code 1 installations
combined, the fraction of free allocation is even larger.
52
Power plants with such a low heat rate are most likely not allowed to operate in the UK.
Chapter 4: Market Power
70
Note that this result is quite robust to what cost pass-through rate I substitute
into the numerator of (4.7?). The fact that companies received a free allocation
covering around 77 % of their emissions means that the inequality in (4.16) holds as
long as a firm?s emission intensity is smaller than (1/0.23=) 4.3 times the cost pass-
through rate. Also note that power companies in the UK received the least generous
allowance allocation in the EU ETS.
4.3.c.) The German power market
Large power producers in Germany received an average of 99.5% of their
actual emissions allocated for free, much more than their UK counterparts.
Differences in allocation for firms in the same sector but different countries have
raised discussions about the ability to achieve economic efficiency in a system where
each member country is able to independently define its own National Allocation
Plan (Boehringer and Lange, 2005a).
In Germany, the marginal generator is a coal-fired power plant during most
hours of the year, including the entire base load (Grubb and Newbery, 2007, Sijm, et
al., 2006). German dark spreads and clean dark spreads are presented in Figure 4.2,
along with their average (middle line). The dark spread is equivalent to the spark
spread described above, but instead of gas it is applied to coal generation. Likewise,
the clean dark spread is the dark spread adjusted for the CO
2
emission costs inherent
in coal generation. The heat rate and emission intensity used to calculate dark spreads
Chapter 4: Market Power
71
and clean dark spreads are 35% and 0.96 tCO2/MWh of output, respectively
(PointCarbon, 2007).
Cost pass-through was not immediate as in the UK, but 50% of carbon costs
were passed through after one year, as indicated by the center line reaching the pre-
market level of €10/MWh. By January 2006, the clean dark spread reached the pre-
market dark spread level, implying that by then the carbon costs embedded in coal
generation had been fully passed through to consumers. Again, the movement of
clean dark spreads towards pre-market dark spread levels could include reasons other
than CO2 cost, but the figure implies that some cost pass-through was very likely.
Suppose now that cost pass-through on average was 0.5*
c
? , where
c
? is the
carbon intensity of coal generation used for the calculation of the green dark spread.
Substituting this value along with an allocation of 99.5 % into (4.7’) reveals that
German power firms with market power would have found it profitable to inflate the
allowance price if their emission intensity was below
e MWh tCO / 96 ) 995 . 0 1 ( 96 . 0 * 5 . 0
2
= ? < ?
which is trivially the case for any power company. The choice of pass-through rate
(and the assumption of zero demand response) is even less important here than for the
UK, because almost any positive cost pass-through rate multiplied by (1/0.005=) 200
will lead to a number that exceeds real-world emission intensities.
Chapter 4: Market Power
72
Naturally, these calculations do not show in any way that electricity producers
in the UK or in Germany actually had market power in both the output and permit
market, and used this power to inflate the allowance price. But they do show that
assuming that some firms had market power, 1.) they would have found it profitable
to over-purchase allowances and under-abate emissions in order to inflate the permit
price, because the ensuing increase in the electricity price would have more than
compensated them for increased allowance purchase costs, and 2.) this would have
lead to an electricity price increase relative to a situation with less free allocation.
4.4. Conclusions
There is a large literature about market power in permit markets, but, to my
knowledge, no paper has directly addressed the effect of free allocation on price
manipulation in the presence of market power in both permit market as well as the
linked output market. Besides being of general economic interest, this particular
question is motivated by a very high (in hindsight too high) allowance price during
the first phase of the EU ETS, which reportedly led to large windfall profits
especially for firms in the power & heat sector. These firms received most of their
allowances for free but were able to pass through a large part of the opportunity costs
to consumers. The reason for the apparent price inflation is not clear to date, but the
presence of windfall profits (which are increasing in the permit price) and the history
of imperfect competition in the power & heat sector raises the question whether
Chapter 4: Market Power
73
dominant power producers could have used their market weight in order to increase
the permit price.
According to Hahn?s (1984) well-known results, the answer to this question is
clearly negative, because power & heat is the only sector that was under-allocated
with permits and thus was a net allowance buyer. In Hahn?s framework, any dominant
permit buyer would depress rather than inflate the permit price, and would act
competitively only when given the exact amount of free allocation that covers its
emissions.
In this paper, I show that Hahn?s results no longer hold with market power in
both markets. I derive the threshold of free allocation above which the dominant firm
finds it profitable to under-abate and over-purchase allowances in order to push up
the permit price. This threshold is a function of cost pass-through and firms? average
emission intensity and is always less than a firm?s emissions were it to set its marginal
abatement costs equal to the permit price.
These findings are not subject to stringent assumptions about relative
efficiency in production and/or abatement among firms, as is typically the case in the
raising rivals? costs literature that discusses market manipulation in input and output
markets. Firms do not profit at the expense of their industry rivals but that of
consumers via the increased output price. In fact, the industry fringe profits from
market manipulation on behalf of the dominant firm, as its revenue increases as well.
I apply my theoretical results to the UK and German power market. Using
market evidence, I show that power generators in these countries received an
Chapter 4: Market Power
74
allocation in excess of x
1
0
and would therefore have been interested in increasing the
allowance price, provided they had the ability to do so. In the UK, this result is due to
an almost complete and immediate cost pass-through, whereas in Germany costs were
passed through more slowly and/or less completely. However, because of the very
generous allocation to German power generators, almost any positive pass-through
rate would have made it profitable for them to manipulate the permit price upwards.
Chapter 5: Options Pricing
75
Chapter 5: An Options Pricing Approach to CO
2
Allowances in
the EU ETS
Abstract
The EU ETS was set up very quickly, which could have made it impossible
for firms to adjust their production technology before the end of Phase I. I derive an
allowance pricing formula under the assumption that abatement was infeasible, which
renders the allowance price a function of the penalty for noncompliance and the
probability that the cap turns out to be binding. This is the pricing formula of a
binary option, with the underlying process being CO
2
emissions. The options pricing
formula depends on the mean and variance of future emissions.
I define the processes driving (stochastic) emissions and estimate their
parameters using market data. I then incorporate these parameter estimates into the
options pricing formula and estimate the remaining free parameters. The results
make economic sense, and the model fits the data reasonably well. This implies that
allowance prices may indeed not have been determined by marginal abatement costs
during the first market phase.
Keywords: Emissions permit markets, air pollution, CO
2
, climate change, options
pricing, asset pricing, EU ETS.
JEL classification: G12, G14, G18, Q52, Q53, Q54
Chapter 5: Options Pricing
76
5.1 Introduction
The centerpiece of emissions permit market theory is that firms equate their
marginal abatement costs to the permit price. Intuitively, if a firm finds it cheaper to
abate an additional unit of emissions than what a permit is worth, it will make a profit
from abating and either buy one fewer or sell one more permit on the market.
Likewise, if the firm finds that purchasing a permit on the market is cheaper than to
abate another unit of emissions, it will not abate but use the market to reach
compliance. The efficient solution of this arbitrage game is that all firms abate exactly
to the point where their marginal abatement costs are equal to the permit price.
However, this leaves out two important possibilities: For one, firms don’t
know exactly what their emissions are going to be, even if they are engaging in
abatement, if abatement consists of reducing emissions per unit of (stochastic) output.
And second, abatement may not be feasible, or at least not practical, for the involved
firms in the short run. Most permit markets to date, including the EU ETS, impose a
penalty for noncompliance: For every unit (usually a ton) of emissions for which the
firm cannot surrender a permit, it is fined a penalty; in addition, the missing permits
have to be surrendered in the following year.
In this paper I will address precisely this question: Was the allowance price in
the EU ETS determined predominantly by firms looking with one eye to the penalty
Chapter 5: Options Pricing
77
and with the other on the realized emissions to date and the expected emissions to
come?
I develop an allowance pricing formula using options pricing techniques that
does not incorporate abatement, but instead relies on the penalty and expected overall
emissions levels as price drivers. While eliminating abatement from the problem is
mainly intended to simplify the calculations, one can also argue that this assumption
is quite realistic for the following reasons:
First, the timely construction of cleaner production technology (e.g. more
efficient power plants) was largely infeasible before the end of the first phase. The
market was set up at a very rapid speed, with only a little over year between its legal
conception and the start of the first phase. Some EU member countries did not
finalize their national allocation plans (NAPs) until mid 2005, which created
considerable uncertainty about the total cap and the resulting allowance price.
Uncertainty over the return of irreversible investment delays such investment.
53
Besides, even under complete certainty, many new plants simply take longer than 4
years (the time between the inception of the market and the end of the first phase) to
plan and construct.
In the absence of cost-effective technology that filters CO
2
from exhaust
gases, this leaves essentially only fuel switching as a method of abatement (Alberola,
et al., 2008, Bunn and Fezzi, 2007, Mansanet-Bataller, et al., 2007, Rickels, et al.,
53
For a thorough treatment of investment under uncertainty and the ensuing option value of
waiting, see Dixit and Pindyck (1994).
Chapter 5: Options Pricing
78
2007, Sijm, et al., 2006). However, energy-intensive industries are typically locked
into long-term contracts. It is questionable whether firms were able to adjust these
contracts in time (Chesney and Taschini, 2008), and/or whether they were willing to
do so, considering the large uncertainty about future caps.
Second, firms probably anticipated that their first-phase emissions were going
to be used to guide the distribution of second-phase allowances. The EC vehemently
argued against this and repeatedly promised that this was not going to be the case, but
it happened nevertheless: Most EU member countries based their second-phase NAPs
on verified 2005 emissions. One possible reason for this –from an economic point of
view highly inefficient- choice is the scarcity of information about historic emissions
in the EU, which made it almost impossible for the EU member countries to fight the
temptation to use the information gained from the first round of emissions
verification. Basing future allocation on current emissions creates a disincentive to
abate, because every unit of abatement comes causes not only costs in the current
period (e.g. due to fuel switching) but also a reduction in future free allocation
(Boehringer and Lange, 2005b).
Third, aggregate emissions were below the total cap for each individual year
of the market.
54
An allowance surplus by itself does not automatically mean that there
was no abatement, especially during the first 16 months of the market when the
allowance price was very high and actual emissions had yet to be verified (Ellerman
54
During the first phase as a whole, the cap and total emissions were 6.250 and 6.081 billion
tons, respectively. The allowance surplus of 168 million tons corresponds to about 2.7 % of
the cap.
Chapter 5: Options Pricing
79
and Buchner, 2006). On the other hand, it is precisely the beginning of the market
that was especially constrained in terms of building new infrastructure and adjusting
long-term fuel contracts. And last but not least, I found no consistent correlation
between allowance prices and commonly identified abatement price drivers (see
Chapter 3 of this Dissertation). For all these reasons, the assumption of no abatement
during the first phase may not be far from the truth.
The purchase of an allowance gives the bearer the option to use it for
compliance at the end of the period, or, equivalently, to sell it. However, if the cap
turns out to be not binding, the bearer can retire the allowance. In other words, the
holder has the right, but not the obligation to use the allowance. This makes an
allowance a financial option. Specifically, the payoff function is that of a type of
binary option called a cash-or-nothing call option. There exist well-established
pricing formulae for binary options if the underlying asset follows a lognormal
distribution. I develop a pricing formula based on two normally distributed
underlying processes, namely electricity consumption and precipitation.
The fact that allowances can be viewed as financial options is neither
remarkable in itself nor new. A handful of studies have used financial methods to
either predict allowance prices themselves, or to price options on allowances.
However, to my knowledge, no one has used observed allowance prices and then
used options pricing formulas to back out the underlying parameters.
In Section 5.2 I give some more background and derive an options pricing
formula for EU ETS allowances as a function of past and future emissions in the
Chapter 5: Options Pricing
80
power & heat sector, the cap, the penalty for noncompliance and a set of free
parameters. This pricing formula contains the mean and variance of expected future
emissions between the current day and the end of the market, which I derive in
Section 5.3 as a function of exogenous stochastic processes. Section 5.4 contains
empirical estimates for these underlying processes as well as estimates for the free
parameters in the options pricing formula. Section 5.5 concludes.
5.2. Emission allowance pricing in the absence of abatement
5.2.a.) Literature on permit pricing
Historically, permit pricing formulas were derived by solving an optimal
control problem, originating with Montgomery (1972). This was later extended to the
dynamic case (Leiby and Rubin, 2001), to incorporate banking and borrowing
(Cronshaw and Brown-Kruse, 1996, Rubin, 1996) and to address uncertainty
(Schennach, 2000, Zhao, 2003) and to some extent volatility (Newell et al., 2005).
But it was not until recently that financial methods have been employed to derive
emissions permit prices.
Kosobud et al. (2005) introduced financial tools to the analysis of SO
2
permits
in the US Acid Rain program. Other contributions that approach permit markets and
the efficient price of an allowance from a financial perspective include Benz and
Trueck (2006) and Fehr and Hinz (2006). Seifert et al (2008) explicitly mention the
Chapter 5: Options Pricing
81
option value of a permit when compared with the alternative of irreversible
investment in emissions abatement. Chesney and Taschini (2008) go one step further
and define EU ETS allowances to be financial (as opposed to “real” options) and
derive a pricing formula that comes close to options pricing. However, all of these
approaches start with the definition of underlying pollution processes, and then derive
a market-clearing permit price by method of simulation. This allows for valuable
insights in terms of price volatility and hedging strategies, but it does not address the
question of what actually drives permit prices. My paper intends to fill this gap by
taking the allowance price series in the EU ETS as given and test whether it is
consistent with a model that is based on options pricing techniques, and, in particular,
whether it was driven by the penalty of noncompliance and the probability that the
cap would turn out to be binding.
5.2.b.) Derivation of option pricing formula
Let
t
P be the closing price for an allowance on day t, with the day index
) ,..., 2 , 1 ( T t = starting on January 1, 2005, and ending at t =T on December 31, 2007.
Let
t
g represent CO
2
emissions on day t,
?
=
?
t
k
k
t
g G
1
1
cumulative emissions
since the beginning of the market, and
?
+ =
?
T
t k
k
T
t
g G
1
cumulative future emissions
until the end of the market. It follows from these definitions that at time t,
t
G
1
is
observed and
T
t
G is stochastic. Furthermore, let P be the penalty for noncompliance
Chapter 5: Options Pricing
82
and
0
S be the total emissions cap over the entire market period imposed by the
regulator. Finally, it will also be useful to define
t
t
G S S
1 0
? = to be the “remaining
cap” until the end of the market.
At time T, the price of an allowance is zero if emissions are below the cap, or
equal to the penalty if the cap turned out to be binding. Naturally, if the cap is
already exceeded at time t<T, the probability that the allowance price is equal to the
penalty is one:
(5.1)
0
0 0
?
>
¹
´
¦
=
T
T
T
T
S
S
if
if
P
P
The penalty is the sum of the per-unit penalty and the cost of buying an
additional permit for the second phase, for which I use the forward price of second-
phase allowances
II Phase
t
P :
=
T
P €
II Phase
T
P + 40
At t<T, it is not known with certainty whether the cap will be exceeded,
provided that it has not been exceeded already. The expected price is
(5.2)
] [ ] 0 | [
) ( * ] [ ] 0 | [
T t t T t
S
T
t
T
t t T t t T t
P E S P E
dG G P E S P E
t
= ?
= >
?
?
?
Chapter 5: Options Pricing
83
where ) (
T
t t
G ? denotes the probability density function over cumulative future CO
2
emissions and ] [?
t
E stands for the expectation taken using all information available at
time t. Note that equation (5.2) is very similar to an equation derived by Chesney and
Taschini (2008).
I will assume that daily emissions g
t
are normally distributed.
55
As I will
argue below, emissions are a linear function of underlying normally distributed AR(1)
processes, meaning that the assumption of normally distributed emissions is really an
assumption about normally distributed underlying processes. Because
T
t
G represents
the summation of random events
k
g for t k > , it follows that
T
t
G is normally
distributed as well. I will denote the mean and standard deviation as
t
µ and
t
s ,
respectively. It follows that
) 1 , 0 ( ~ N
s
G
Q
t
t
T
t
t
µ ?
?
has a standard normal distribution. Let ) (? ? and ) (? ? be the probability density
function (pdf) and cumulative probability density function (cdf) of the standard
55
In theory, the choice of a normal distribution makes a truncation at zero necessary since
negative emissions are not defined. But because CO
2
emissions in the EU are many standard
deviations away from zero, the correction implied by the truncation is very small, such that
for the remainder of this paper I will neglect the truncation issue and assume that emissions
are normally distributed. Note that this assumption is similar to that of demographers who
assume that people’s height is normally distributed.
Chapter 5: Options Pricing
84
normal distribution, respectively. I now convert the integral in (5.3) into an integral
over
t
Q :
(5.4)
?
?
?
= >
t t t
s S
t t T t t T t
dQ Q P E S P E
/ ) (
) ( * ] [ ] 0 | [
µ
?
which is equivalent to
(5.5)
|
|
¹
|
\
| ?
? = >
t
t t
T t t T t
s
S
P E S P E
µ
* ] [ ] 0 | [
Arbitrage considerations dictate that the price at time t be equal to the
expected price at T, discounted by the risk-free rate of interest r.
56
The same
reasoning applies to the forward price for Phase II allowances, such that
) (
* ] [
t T r II Phase
t
II Phase
T t
e P P E
?
= . This means that the allowance price is a martingale
defined by
(5.6)
] 40 [ 0 |
* ] 40 [ 0 |
) (
) (
II Phase
t
t T r
t t
t
t t II Phase
t
t T r
t t
P e S P
s
S
P e S P
+ = ?
|
|
¹
|
\
| ?
? + = >
? ?
? ?
µ
56
Real-world markets are typically not risk-neutral, but option prices based on risk neutrality
nevertheless yield the correct (meaning no-arbitrage) solution for traded assets (Hull, 2002).
Risk aversion may be more important for the pricing of non-traded assets such as the weather
or electricity demand, but the price of market risk can never be measured with a sufficient
degree of confidence in order to make its inclusion in a pricing formula worthwhile, due to
measurement and identification issues (e.g. a greater market fundamental and a higher price
of risk have the same effect on the price). In absence of a convincing prior for market risk, I
will omit the latter in my analysis.
Chapter 5: Options Pricing
85
This is the option pricing formula for a cash-or-nothing call option based on a
normally distributed underlying asset or process.
Given knowledge of past emissions
t
G
0
, the overall cap
0
S , the forward price
penalty
II Phase
t
P and the interest rate r, and estimates for the mean
t
µ and standard
deviation
t
s of future emissions at every point in time, an estimate for the allowance
price
t
P
ˆ
can be computed for reach day, and the resulting time series compared with
the observed series
t
P for T t ..., , 2 , 1 = . A situation where the two series correspond to
one another would be interpreted as evidence that the allowance price was driven by
emissions and the penalty, but not abatement.
What remains to be determined in order to evaluate (5.6) are past emissions
and the mean and standard deviation of cumulative future emissions. These are not
directly observed, but have to be derived from underlying processes and ultimately
estimated using market data. This is the subject of the following section.
5.3. Deriving the mean and standard deviation of future emissions
5.3.a.) CO
2
emissions as a function of exogenous stochastic processes
Emissions are verified only once a year, and there exists no direct data about
daily emissions. However, for the power and heat sector (which is by far the most
Chapter 5: Options Pricing
86
important in the EU ETS) here is something that comes close: Daily electricity
consumption.
Electricity is special in the sense that demand has to be met with a matching
supply at all times in order for the grid not to crash. I will make the simplifying
assumption of zero short-term demand elasticity of electricity. This makes electricity
demand an exogenous process driven by stochastic elements such as the weather.
57
More precisely, it is not generation of electricity in general that drives CO
2
emissions, but generation of electricity using conventional thermal generators by
burning fossil fuels like coal and natural gas. Thus, daily emissions are a function of
consumer demand, as well as the availability of “clean” (i.e. non-CO
2
-emitting)
sources of energy, mainly hydroelectric and nuclear power.
58
Hydroelectric
generation depends on rainfall and varies within and between years, but nuclear
generation is largely constant due to prohibitively high start-up costs.
Let c
t
represent overall electricity consumption; c
t
c
consumption of
conventional fossil-fueled generation; n nuclear power generation (all in Giga-Watt-
hours (GWh) per day); and h
t
rainfall in the EU in millimeters (mm) per day,
weighted by installed hydroelectric capacity. Assuming that all available
57
In the long run, consumers will react to higher electricity prices by changing their
consumption habits and appliance portfolio, such that electricity demand is also a function of
the electricity price. But regardless of the time horizon, exogenous weather shocks will
always drive short-term consumption.
58
Although wind generation has increased rapidly during the past few years, it still accounts
for a relatively small fraction of total power production.
Chapter 5: Options Pricing
87
hydroelectric and nuclear power is used (i.e. that they are lowest in the merit order
59
),
demand for conventional generation can be expressed as
(5.7) n h c c
t t
c
t
? ? = ?
where ? is a fixed coefficient translating precipitation into hydroelectric power.
Since precipitation can be stored to some extent, either in reservoirs or as
snow in the mountains, there is no immediate relationship between precipitation and
hydro generation on any given day, which makes a regression of hydro output on
daily precipitation impractical. On the long run, however, all net hydro generation is
ultimately due to precipitation, and even though rainfall today may not translate into
more generation today, it nevertheless reduces expected conventional generation
needed to satisfy consumer electricity demand until the end of the market. I compute
the precipitation-to-rainfall conversion factor ? by dividing the EU’s total hydro
generation
60
in 1990-2005 of 4,852,339 GWh by cumulative weighted precipitation
over the same period of 9,775.28 mm, using installed hydroelectric capacity per
country as weights. This results in a conversion factor of ?=496.389 GWh/mm.
In the EU, 12 member countries have nuclear power plants (BE, CZ, DE, ES,
FI, FR, HU, NL, SK, SL, SW, UK). Their average total output in the years 2003-
2005 was 2,679 GWh per day, which I will use as a measure for n.
59
The merit order is the sequence by which individual generators are brought online and is
usually based on marginal cost.
60
From World Development Indicators database, World Bank.
Chapter 5: Options Pricing
88
The emission intensity (in CO
2
/GWh) of the marginal generator varies with
demand. The correct way to express emissions in Europe’s power & heat sector is
(5.8)
?
? =
c
t
c
t
dy y g
0
) (
where ) (
c
t
c ? is an emission intensity function transforming conventional thermal
power generation into CO
2
emissions. To compute the integral in (5.8) I would need
to know the exact dispatch order and the marginal emission intensity of all generators
involved, which is information that is not readily available. Instead, I assume that the
emission intensity of the marginal generators (i.e. all generators that are not running
all the time) follow a quadratic function. This allows me to express (5.8) as
(5.9)
) min( *
*
min c
t
c
t t
c g
c g
?
?
? ? ?
+ ? ?
The parameter ? translates fossil-fueled electricity generation into CO
2
emissions. ? is a constant defined as the difference between CO
2
emissions
associated with minimum thermal generation
min
g and the (theoretical) emissions if
the emission intensity ? were applicable to the inframarginal generation.
61
61
In theory, the average emission intensity of inframarginal generation could be greater or
smaller than the emission intensity of marginal generation. For example, if inframarginal
generation consists to a large part of lignite or anthracite coal power plants, then K>0, but if it
exists largely of efficient generators such as combined cycle gas turbines (CCGTs) that are
low in the dispatch order due to their small marginal costs, then K<0. In the EU, the former
Chapter 5: Options Pricing
89
Combining (5.7) and (5.9), daily CO
2
emissions can be expressed as
(5.10) ) ( * n h c g
t t t
? ? + ? = ? ?
In this specification, emissions are a function of a set of parameters and the
two stochastic and exogenous processes
t
c and
t
h . The properties of
t
g , and thus of
t
µ and
t
s , are therefore a function of the properties of
t
c and
t
h . At time s, the
mean of future CO
2
emissions at s t ? is defined by
(5.11)
( )
?
?
?
+ =
+ =
+ =
? ? + ? =
(
¸
(
¸
? ? + =
(
¸
(
¸
= =
T
t k
t s t s
T
t k
t t s
T
t k
k s
T
t s t
n h E c E K t T
n h c K E
g E G E
1
1
1
] [ ] [ * ) (
) ( *
] [
? ?
? ?
µ
The calculation of the variance of future emissions is a little more
complicated. In Appendix C (Result 1), I show that at time u t s ? ? , the variance is
(5.12)
( )
( )
? ?
?
? ? ?
+ = + =
+ =
+ = + = + =
? ? + +
+ ? =
+ = =
T
t k
T
k u
u k s u k s u k s u k s
T
t k
k s k k s k s
T
t k
T
k u
u k s
T
t k
k s
T
t s t
c h Cov h c Cov h h Cov c c Cov
h Var h c Cov c Var
g g Cov g Var G Var s
1 1
2 2
1
2 2
1 1 1
2
] , [ ] , [ ] , [ ] , [ 2
] [ ] , [ 2 ] [
] , [ 2 ] [ ] [
? ? ? ?
? ? ?
is much more likely given the large number of lignite plants in Germany and the new EU
member countries from Eastern Europe.
Chapter 5: Options Pricing
90
Both expressions are functions of the constants ? and n, the parameters K
and ? , the mean and variance of electricity consumption and precipitation, and the
covariance of electricity consumption and precipitation between different days.
I defined the constants ? and n above and will treat them as known; K and
? will enter the estimation as free parameters. The derivation of the mean, variance
and covariance of electricity consumption and rainfall is the subject of the next
subsection.
5.3.b.) Properties of the stochastic processes c
t
and h
t
For the definition of the stochastic processes of electricity demand and
precipitation, I will draw extensively from a paper by Peter Alaton, Boualem
Djehiche and David Stillberger (2002). Although their analysis focuses on pricing a
weather option over heating-degree days with the underlying process being
temperature, it is very similar in principle to both electricity demand and
precipitation, as both are exogenously driven stochastic processes that contain
deterministic annual fluctuation and long-term trends. The contribution of my paper
is not the derivation of the property of such processes, but the application of these
methods to model CO
2
emissions and, ultimately, to CO
2
allowance pricing.
Chapter 5: Options Pricing
91
I will model both electricity consumption and precipitation diffusion
processes
62
consisting of a deterministic mean and a stochastic part, and which exhibit
mean-reversion.
63
For mathematical tractability, I include the stochastic element in
the form of a generalized Wiener process. Combining the processes in the index x,
they can be described as
(5.13) h c x dW t i dt x x a
dt
dx
dx
x
t x t
m
t x
m
t
t
, ; )] ( [ ) ( * = +
(
¸
(
¸
? + = ?
This is known as an Ornstein-Uhlenbeck process with a non-zero mean and
time-varying volatility.
64
The term in brackets represents the drift of the processes,
followed by the diffusion term defined by the standard Wiener process
x
t
dW times
the corresponding volatility. The first element of the drift term in (5.13) is due to the
fact that mean consumption and precipitation change throughout the year. The mean
reversion parameters a
x
measure the speed at which the processes revert back to their
long-term mean.
I constrain the volatility to be constant within each calendar month, but allow
it to differ across months. The index i labels the month to which the time index t
62
A diffusion process is the solution to a stochastic differential equation. In particular, it is a
continuous-time Markov-process with a continuous sample path. This is a realistic
description for electricity consumption and precipitation and makes the derivation and
exposition easier, but the market and weather data discrete are naturally only available for
discrete points in time.
63
Mean reversion is a commonly observed characteristic in many naturally occurring
processes, as they generally do not grow without bounds and eventually return to their long-
term mean.
64
See, for example, Bibby and Sorensen (1995).
Chapter 5: Options Pricing
92
refers. For reasons of data availability (see below) I will start this index at 1 in
January 1976 and finish at 384 in December 2007. Thus,
2007 Dec 384
1976 Feb 2
1976 Jan 1 ) (
? =
? =
? =
t if
t if
t if t i
?
Because I assume that the volatility is the same for each calendar month, it
must be that ] 12 * [ ] [ k i i + =? ? for any integer k .
I define the long-term mean of electricity consumption and precipitation as
(5.14)
] 365 / 2 sin[ * *
* ] 365 / 2 sin[ * *
2 1 0
2 1 0
h h h h m
t
t
c c c c c m
t
t t h
WD D t t c
? ? ? ? ?
? ? ? ? ?
+ + + =
+ + + + =
The parameters
x
0
? and
x
1
? ( h c x , = ) describe the level and trend of the two
process, respectively, whereas
x
2
? describe the amplitudes of the respective sine
waves. The phase angles
x
? shift the oscillation of the two processes to their correct
position. Lastly, the vector of coefficients
c
D (not applicable to rainfall) accounts
differences in electricity consumption across different weekdays, and
t
WD is a vector
of weekday dummies.
Equation (5.13) describes two stochastic differential equations. At time t s ? ,
their solution is
65
65
See, for example, Øksendahl (2007) Chapter 2.
Chapter 5: Options Pricing
93
(5.15) h c x dW i e x e x x x
x
x
t
s
t a m
t
s t a m
s s t
x x
, ; )] ( [ ) (
) ( ) (
= + + ? =
?
? ? ? ?
?
?
? ?
The first term on the RHS is the deviation of actual consumption/precipitation
at the present time s from its mean. As time goes on, the impact of this deviation will
diminish due to the mean-reversion property of both processes, measured by the
exponent. If one of the processes is at its average at time s, or if s t >> , then the first
term will drop out, and the expectation at time t simply becomes the mean
expectation
m
t
x defined by (5.14).
The mean and variance of electricity demand and precipitation can be
computed as
(5.16) h c x x e x x x E
m
t
s t a m
s s t s
x
, ; ] [ ] [
) (
= + ? =
? ?
(5.17)
( ) | |
( )
h c x dy e y i
dW e i E
x E x E x Var
t
s
y t a
x
t
s
t a
x s
t s t s t s
x
x
, ; )] ( [
)] ( [
] [ ] [
) *( 2 2
2
2 ) *( 2 2
2
= =
(
¸
(
¸
=
? =
?
?
? ?
? ?
?
? ?
?
?
The second equality follows from the fact that | | dt dW E
x
t
=
2
) ( . If the
volatility does not change between s and time t, (5.17) can be solved to
(5.18) ( ) ) ( ) ( ; , ; 1
2
)] ( [
] [
) ( 2
2
t i s i h c x e
a
t i
x Var
s t a
x
x
t s
x
= = ? =
?
?
Chapter 5: Options Pricing
94
If s and t are not within the same month, the expression becomes more
complicated. I will denote the first day of each month as { } i t i t t i t = = ) ( : min )] ( [
min
.
In Appendix C (Result 2) I show that for h c x , = and ) ( ) ( t i s i ? , the general
expression for the variance is
(5.19) ( )
)
`
¹
¹
´
¦
? + + ? =
? ? + ? ?
?
=
?
)] ( [ )] ( [ ] 1 [ ] [
2
1
] [
2 ) ( 2 2 ]) 1 [ ( 2
1 ) (
) (
2 2
min
s i e t i e k k
a
x Var
x
s t a
x
k t t a
t i
s i k
x x
x
t s
x x
? ? ? ?
It is easy to verify that if the volatility is the same for each month, (5.19)
collapses to (5.18).
To calculate the covariance between electricity consumption and rainfall on
the same day, note that dt dW dW E
ch h
t
c
t
? = ] [ , where
] [ * ] [ / ] , [
t t t t
ch
h Var c Var h c Cov ? ? is the correlation coefficient between the two
processes. Thus,
( )( ) | |
?
?
? + ?
? + ?
=
(
¸
(
¸
=
? ? =
t
s
y t a a
h c
ch
t
s
h c t a a
h c s
t s t t s t s t t s
dy e y i y i
dW dW e i i E
h E h c E c E h c Cov
h c
h c
) )*( (
) )*( (
)] ( [ )] ( [
)] ( [ )] ( [
] [ ] [ ] , [
? ? ?
? ? ? ?
? ?
?
Analogous to the procedure used for the variance, this can be solved to
Chapter 5: Options Pricing
95
(5.20)
( )
¦
)
¦
`
¹
¦
¹
¦
´
¦
? +
+ + ?
+
=
? + ?
+ ? + ?
?
=
?
)] ( [ )] ( [ )] ( [ )] ( [
] 1 [ ] 1 [ ] [ ] [
] , [
) )( (
]) 1 [ )( (
1 ) (
) (
min
s i s i e t i t i
e k k k k
a a
h c Cov
h c
s t a a
h c
k t t a a
t i
s i k
h c h c
h c
ch
t t s
h c
h c
? ? ? ?
? ? ? ?
?
Lastly, the covariance between electricity consumption/precipitation on day t
and u for s ? t ? u is defined by (see Appendix C, Result 3):
(5.21)
] , [ * ] , [
] , [ * ] , [
, ]; [ * ] , [
) *(
) *(
) *(
t t s
t u a
u t s
t t s
t u a
u t s
t s
t u a
u t s
h c Cov e c h Cov
h c Cov e h c Cov
h c x x Var e x x Cov
c
h
x
? ?
? ?
? ?
=
=
= =
Expressions (5.16) and (5.19)-(5.21) can now be substituted into (5.11) and
(5.12). In the following section I obtain empirical parameter estimates for
ch
x
c x x x x
a D ? ? ? ? ? , , , , , ,
2 1 0
and ) (i
x
? .
5.4. Estimation
There are two different steps in the estimation. The final goal is to express
equation (5.6) as a function of data, known constants, and a set of free parameters,
and calculate these free parameters using market data. Since most countries lack
daily electricity consumption data prior to 2006, I will evaluate (5.6) for the period
between January 1, 2006 and December 31, 2007.
Because in January 2006, realizations of electricity consumption and
precipitation for later days were not yet known, I estimate
t
µ and
t
s using data
Chapter 5: Options Pricing
96
through 2005 only, with some exceptions where necessary. In theory I could update
the estimates for every day, but for simplicity I will use pre-2006 data.
5.4.a.) Data
Daily data about electricity consumption is available from the Union for the
Coordination of Transmission of Electricity (UCTE)
66
for continental European
countries, including all EU member states except for the Nordic countries,
67
the UK,
Ireland, the Baltic States, Malta and Cyprus. Electricity consumption has been
measured on every third Wednesday of each month
68
since 1994 for 9 EU countries,
since 1996 for Germany and since 1999 for another 5 EU countries. Weekend
consumption is available for every Weekend following the third Wednesday of each
month in the year 2000. Starting in January 2006, electricity consumption is
available on a daily basis for all UCTE countries. To supplement the UCTE data I
obtained all available historic electricity consumption data directly from the
transmission system operators (TSOs) in the UK, Ireland and the Nordic countries.
69
66
Available at www.ucte.org, last accessed in September 2008.
67
Sweden, Denmark and Finland. Note that Norway is not part of the EU, and although it is
now linked to the EU ETS, this was not the case during the first phase of the market.
68
Wednesdays are supposed to be the most typical weekdays (as opposed to Mondays and
Fridays, which may be slightly different), and the third week is supposed to be the typical
week of a month.
69
UK: Daily data since 2001 from the National grid, available at
http://www.nationalgrid.com/uk/Electricity/Data/; Ireland: Daily data since 2002 from
Eirgrid, available at http://www.eirgrid.com; Denmark: Daily data since 2000 from
Energinet, available at http://www.energinet.dk; Finland: Daily data since 2004 from Fingrid,
Chapter 5: Options Pricing
97
I exclude Malta, Cyprus and the Baltic States from the analysis, because the former
are not integrated into Europe’s electricity grid and no daily electricity consumption
data for the latter is available. In terms of annual electricity production, the 20
countries included account for 99% of total production in the EU-25.
70
The EU produces nearly all of the electricity it consumes, with net
imports/exports accounting for less than 0.1 percent overall consumption. I therefore
exclude imports/exports in my calculations and set consumption equal to production.
In order to accommodate the variation in type and provenance of the data I
will carry out the analyses separately for each group of countries for which the
available data is of the same type (e.g. daily vs. monthly) and covers the same time
period. The six groups are listed in Table 5.1. Figures 5.1a-f show the available pre-
2006 electricity consumption data by group. All countries have daily data for the
years 2006 and 2007.
For precipitation, I use the European Climate Assessment and Dataset.
71
This
dataset contains daily data for 1,048 monitoring stations located in 42 countries. The
length of the series varies from a few years to >150 years, with most series spanning
available at http://www.fingrid.fi; Sweden: Daily data since 2000 from Svenska Kraftnät,
available at http://www.svk.se/web/Page.aspx?id=5794.
70
In 2007, Romania and Bulgaria joined the Union to make it the EU-27. However, because
they were not part of the market during the first two years, and their registries were not ready
until the end of 2007, they can be excluded from Phase I.
71
Klein Tank et al. (2007): “Daily Dataset of 20
th
-Century Surface Air Temperature and
Precipitation Series for the European Climate Assessment”, available at eca.knmi.nl, last
accessed in September 2008.
Chapter 5: Options Pricing
98
several decades. To model the stochastic process underlying precipitation, I use data
covering the years 1976-2005.
The conversion of precipitation into hydroelectric power is location-specific.
For example, rainfall in the Netherlands or in Denmark is largely irrelevant for power
generation because these countries have very little installed hydroelectric generation
capacity, whereas hydro generation constitutes a large share of total power production
in Alpine and Scandinavian countries. I average station entries by country,
72
and then
create a weighted European average using installed hydroelectric capacity in 2006 as
weights.
73
Installed hydro generation is given in the last column of Table 5.1.
Weighted precipitation in millimeters (mm) is shown in Figure 5.2 for a
subset of the sample period. Whereas it is difficult to visually discern a pattern in the
raw data (Fig. 5.2a), using moving 7-day-average (Fig. 5.2b) reveals a clear
seasonality.
72
For low-lying countries such as Belgium and Luxembourg, I simply take an average of all
monitoring stations. However, since hydro generation in the Alps and in Scandinavia is
highly location-specific, I take an average of the subset of monitoring stations that are located
in or near mountains. A full list of the selected stations is available from the author upon
request.
73
This data comes from UCTE (www.ucte.org) for continental Europe; from Nordpool
(www.nordel.org) for Scandinavia; from the Austrian Energy Agency
(www.energyagency.at/enercee/) for the Baltic States; from Harrison (2005) for the UK; and
from the Electricity Supply Board (ESB, available at
http://www.esb.ie/main/about_esb/power_stations_intro.jsp) for Ireland; all accessed in
September 2008.
Chapter 5: Options Pricing
99
5.4.b.) Parameter estimation for electricity consumption and precipitation
I estimate the parameters
x x x x x
D , , , ,
2 1 0
? ? ? ? and ] [i
x
? with a model that
features an autoregressive error to account for mean-reversion and multiplicative
heteroskedasticity to allow the variance to differ across months:
(5.22)
h c c c x Nov Jan t i
t i N u
u
WD D t t t x
t
x
t
x x
x
x
x
t
x
t
x
t x
x
t
x
t t
x x x x x
t
, ..., , , ; } * ... * exp{ )] ( [
)] ( [ , 0 ( ~
*
* ) 365 / 2 cos( * ) 365 / 2 sin( * *
6 2 1
11 1 0
2
2
1
2 1 1 0
= + + + =
+ =
+ + + + + =
?
? ? ? ?
?
? ? ?
? ? ? ? ? ? ?
Note that the index x now covers six different electricity consumption series,
plus the (weighted) precipitation series, all of which are estimated separately by
maximum likelihood.
The parameters
x x
1 0
, ? ? and
x
D are the same as in (5.14) and are estimated
directly. The transformation of the sine wave plus the phase angle into a sine and
cosine wave is based on a Fourier transform and serves to linearize the equation. The
parameters
x
2
? and
x
? can be computed using the estimates of
x
1
? and
x
2
? :
74
(5.23)
] / arctan[
) ( ) (
1 2
2
2
2
1 2
x x x
x x x
? ? ?
? ? ?
=
+ =
; h c c c x , ..., , ,
6 2 1
=
The t-statistics and confidence intervals have to be calculated using the delta
method.
74
See, for example, Beckwith et al. (1995), p. 131.
Chapter 5: Options Pricing
100
I estimate the daily variance ] [
2
i
x
? from the autocorrelation parameters
x
?
and the variance of the white noise ] [
2
i
x
? .
75
For a stationary AR(1) process, the
variance is given as
(5.24) ( ) | |
x
x
t t t x
i
E x E x E i
?
?
? ?
?
= = ? =
1
] [
] [ ] [ ] [
2
2 2 2
The mean-reversion parameters
x
a measure the speed at which a shock to
t
x
is felt at later times. From (5.16), the expectation of future electricity consumption or
precipitation is
h c x x e
x e x x x E
m
t
s t a x
s
m
t
s t a m
s s t s
x
x
, *
] [ ] [
) (
) (
= + =
+ ? =
? ?
? ?
?
This makes it clear that the term
) ( s t a
x
e
? ?
is equivalent to the impulse-response
function of the AR(1) process defined by
76
s t
s t
?
= ? ? ) , (
75
Because I cannot estimate an AR(1) parameter with data that only contains entries for
every 3
rd
Wednesday per month, I use the 2006-7 data to estimate this parameter for Series 1-
3. Likewise, the estimate of the variance is sensitive to the frequency of measurement
(Hayashi and Yoshida, 2005) and generally improves with greater frequency. I therefore also
use the 2006-7 data to estimate the variance and the correlation coefficients (see below).
Note that for all other parameters, I use pre-2006 data only. Note that the daily variance and
mean reversion parameter for Series 4-6 are not significantly different between pre- and post-
2006 data.
76
See Hamilton (1994) p. 53-54.
Chapter 5: Options Pricing
101
which measures the impact of an exogenous shock occurring in period s on the
variable in period t. Equating the two and solving yields
(5.25) ) ln(
x x
a ? ? =
All parameter estimates are given in Table 5.2.
I compute the correlation coefficients among the different series
kl
?
) ..., , , , (
1 1
h c c l k = by using the data from 2006-2007, for which all series have daily
entries.
77
The results in Table 5.3 show that electricity consumption across the six
different regions is highly correlated, but that precipitation weighted by available
hydroelectric power and electricity consumption is not. Because the correlation
coefficient between precipitation and all six electricity consumption series is zero, I
will set j h c Cov h c Cov
t t s t
j
t s
? = = 0 ] , [ ] , [ .
I derived the expressions for the variance and covariance in (5.16), (5.19) and
(5.21) for total electricity consumption. Due to the six different data groups the parts
77
Hayashi and Yoshida (2005) developed an unbiased estimator to compute the correlation
coefficient between time series of different measuring intervals, but that estimator is not
bounded by unity in magnitude, relying on truncation instead. Also, this would only address
the problem of differing frequencies within the same time period, but not that of different
time periods.
Using the much higher-frequency data for 2006-2007 is equivalent to assuming that the
covariance between electricity consumption in the six different regions and EU-wide
weighted precipitation is the same before and after January 1, 2006. For the two groups for
which ample data is available (groups 4-6), this assumption appears to hold. Note that the
market participants very likely have much better information about these covariances than
what would be gleaned based on a few monthly data points from pre-2006 data available to
the researcher.
Chapter 5: Options Pricing
102
involving electricity consumption that are based on overall consumption have to be
adjusted to
(5.16’)
?
=
=
6
1
] [ ] [
j
j
t s t s
c E c E
(5.19’)
( )
¦
)
¦
`
¹
¦
¹
¦
´
¦
? +
+ + ?
+
=
+ =
? + ?
+ ? + ?
?
=
= + = =
?
? ? ?
)] ( [ )] ( [ )] ( [ )] ( [
] 1 [ ] 1 [ ] [ ] [
] , [
] , [ 2 ] [ ] [
) )( (
]) 1 [ )( (
1 ) (
) (
6
1
6
1
6
1
min
s i s i e t i t i
e k k k k
a a
c c Cov
c c Cov c Var c Var
l j
l
c
j
c
l j
l
c
j
c
l j l j
l j
c c
s t a a
c c
k t t a a
t i
s i k
c c c c
c c
l j
l
t
j
t s
j j l
l
t
j
t s
j
j
t s t s
? ? ? ?
? ? ? ?
?
(5.21’) ( )
? ? ?
= + =
? ? ? ?
=
? ?
+ + =
6
1
6
1
) *( ) *(
6
1
) *(
] , [ ] [ * ] , [
j j l
l
t
j
t s
t u a t u a
j
j
t s
t u a
u t s
c c Cov e e c Var e c c Cov
l
c
j
c
j
c
5.4.c.) Evaluation of the options pricing formula
With these parameter estimates, I can now proceed to evaluating the options
pricing formula. Because emissions were below the total cap at the end of the
market, as well as for each year individually, I will disregard the second line of
equation (5.6). I use first- and second-phase over-the-counter (OTC) allowance
prices from Point Carbon.
The mean and standard deviation of future emissions are a function of free
parameters and estimates of the mean, variance and covariance of the processes for
Chapter 5: Options Pricing
103
electricity consumption and precipitation. Substituting (5.11) and (5.12) into (5.6)
and simplifying gives
(5.26)
( ) ( )
2 / 1
1
2
1
2
1
1 1
0 ) (
] , [ ] , [ 2 ] [ ] [
] [ ] [
(
* ] 40 [
|
¹
|
\
|
+ + + ?
? ? + ? ? ?
|
|
¹
|
\
| ? +
? + =
? ? ?
? ?
+ = + = + =
= + =
? ?
T
t k
u k s
T
k u
u k s k s k
T
t k
s t
t
k
k k
T
t k
k s k s t
t
t II Phase
t
t T r
t
h h Cov c c Cov h Var c Var B
n h c n h E c E A
with
B
S A TK
P e P
? ?
? ?
?
?
where A
t
and B
t
are known functions of the parameters of the diffusion processes
for electricity consumption and precipitation.
To account for the price crash after the first round of emissions verifications, I
add dummy variable to allow for the updating of firms’ expectation about emissions
from other sectors (and emissions of other firms in the power & heat sector for that
matter, but these parameters cannot be individually identified). Simplifying leads to
(5.27)
V S TK K
A
P e
P
B Y with
S D K
Y
t
II Phase
t
t T r
t
t t
EV EV
t
t
+ ? ?
?
|
|
¹
|
\
|
+
? ?
?
=
? ?
?
0
) (
1
40
*
?
Chapter 5: Options Pricing
104
where
1 ?
? refers to the c.d.f of the standard normal distribution and
EV
t
D is a
dummy that takes on the value of zero before the first round of emissions
verifications, and of one thereafter. Given the price crash in April 2006,
EV
S has to
be positive, implying that firms updated their expectation of the total number of
remaining available permits upwards. I use an annual interest rate of 10% for the
calculation of the discounted penalty.
78
Total emissions from other sectors or emissions during 2005 are contained in
V S TK K + ? ?
0
, where V stands for any time-invariant parameter that shifts the
amount of permits available to the power & heat sector for the years 2006-2007.
Note that K has to be negative, since
0
S is the total cap and the number of permits
available to power generators (net of the correction associated with minimum
generation TK ) has to be positive.
Estimates for the free parameters K , ? and
EV
S can be computed by taking
averages of
t
Y for the period before and after the allowance price crash. Note that the
parameters are not individually identified, such that one of them has to be held fixed
in order to calculate the others. For example, when holding
EV
S fixed, K
ˆ
and ?ˆ are
defined as
78
Use of 0% and 20% did not alter the results significantly.
Chapter 5: Options Pricing
105
(5.28)
?
?
?
?
EV
EV
t t
EV
t t
S K
D Y
K
D Y
?
= = ?
= = ?
) 1 ( |
) 0 ( |
2
1
2 1 1 2
ˆ ;
/ 1
ˆ
? ?
?
? ? ?
=
?
= ?
EV EV
S S
K
where ) ( | ?
t
Y refers to the conditional sample average. Results for fixed values of
EV
S are shown in the left panel of Table 5.4. The shaded areas are the parameter
combinations that make economic sense. The emission intensity ? has to be
somewhere between 600 and 900 tCO
2
/GWh,
79
which is the case for
MT S MT
EV
150 100 < < , and for MT K MT 770 , 1 660 , 2 ? < < ? . Both ranges are
plausible: The range for
EV
S implies that firms expected 2005 emissions to exceed the
annual cap by 6-56 MT, but instead they turned out to be 94 MT below.
80
The range
for K means that of the about 4,200 MT of permits issued for the years 2006 and
2007, between 42-63 % were used to cover emissions of power generators.
81
The
goodness of fit (defined by the model sum of squares divided by total sum of squares)
is 0.81, much larger than the corresponding values for the model presented in Chapter
3 (see Tables 3.1-2).
79
The average emissions intensity of the marginal generators in the EU will not exceed that
of a coal-fired power plant, which emits about 920 tCO2/GWh. The emission intensity of
Combined Cycle Gas Turbines (CCGTs) is about half of this, but coal is at the margin for the
majority of the load in the EU.
80
The first round of emissions verifications found emissions to be 94 MT below the total
2005 allocation.
81
Note that the sector defined as “Power & Heat” accounts for about 70% of total emissions,
but this includes production of heat as well as industrial process combustion, not just
electricity producers.
Chapter 5: Options Pricing
106
Using the estimates
EV
S K | ˆ ,
ˆ
? I compute the estimated price series
t
P
ˆ
:
(5.29)
|
|
¹
|
\
|
+
?
? + =
? ?
t
t
t
EV EV
t II Phase
t
t T r
t
B
A
B
S D K
P e P
?ˆ
ˆ
(
* ] 40 [
ˆ
) (
Figure 5.3 shows the predicted price series, along with the actual allowance
price and the forward price for second-phase allowances. The estimated series
follows the data quite well until the April 2006 price crash, after which it falls below
the actual price, before crossing it and finishing the period slightly too high.
Importantly, the prediction shows the post-crash stabilization followed by a gradual
decline to zero observed in the real price series, which has puzzled market observers.
This is the result of the probability of a binding cap slowly approaching zero, as time
progresses and actual emissions are observed.
A striking difference between prediction and actual price is the volatility. The
allowance price series appears to fluctuate much more than the prediction, which of
course could be due to shocks in unobserved variables that drive emissions that end
up in the model residual. However, there is an alternative explanation: If power
generators had a better idea about the variance of future demand than the researcher
(because they have access to better data, especially for the area which they have
exclusively serviced for decades before market liberalization), then I would have
overestimated the standard deviation of future generation denominated by B
t
in
Chapter 5: Options Pricing
107
(5.27). A too large standard deviation would attenuate demand shocks, leading to a
prediction that is too smooth and too inert to electricity demand shocks.
To evaluate this possibility, I test the results for their sensitivity to
t
B . I
divide
t
B by 10 and 10, corresponding to a factor of 10 resp. 100 by which my
estimate for the variance exceeds the estimate used by the market participants.
82
The
predictions are shown in Figure 5.4. It is clear that the results are greatly influenced
by the uncertainty embedded in future emissions. Whereas the larger correction
appears to overshoot and lead to excessive price volatility, the more modest
correction by one order of magnitude fits the data much better, with a goodness of fit
of 0.92. The corresponding estimates for K , ? and
EV
S are shown in the right panel
of Table 5.4. Again, for sensible emission intensities (shaded region), the associated
values for
EV
S and K are plausible.
5.5 Conclusions
In this paper I derive an allowance pricing formula based on the assumption
that firms were not able to engage in significant abatement during the first three years
of the EU ETS, and therefore had no control over their emissions. In this case, the
value of an allowance can be characterized by an options pricing formula for a cash-
82
Note that when computing the standard deviation of future consumption using the 2006-7
data and adding all the series up, I get a result that is about 1.4 times lower than the standard
deviation calculated using separate series.
Chapter 5: Options Pricing
108
or-nothing call option. This formula contains the mean and standard deviation of CO
2
emissions for the remainder of the market.
I calculate daily emissions based on daily demand for electricity generated by
conventional thermal combustion. This is a function of total electricity consumption
and the availability of non-emitting sources for electricity such as hydro and nuclear.
I assume that these processes are characterized by diffusions and estimate the
diffusion parameters using market data. This allows me to express the allowance
price as a function of data, estimates for future emissions and three free parameters.
The parameter estimates are highly significant and make economic sense. The
predicted allowance price series fits the actually observed prices quite well, especially
when adjusting the estimate for future electricity demand volatility downwards, based
on the hypothesis that power generators had a better idea about it. Importantly, the
model is able to explain the price stabilization after the price crash, followed by a
long and steady decline towards zero, which can be explained by a declining
probability that the cap was going to be binding. A model based on abatement
parameters would only be able to explain such a movement if the price of
fundamentals related to abatement also expressed such a steady decline, which was
not the case (see Chapter 3).
I conclude that the allowance price during the first phase of the EU ETS was
to a large extent driven by the penalty for noncompliance and the probability of a
binding cap, at least for the years 2006 and 2007. This could be due to the speed of
market-setup, which did not allow the involved firms to adjust their emissions in
Chapter 5: Options Pricing
109
time, and/or the realization after the April 2006 price crash that the market was likely
to be oversupplied with permits and that therefore abatement measures would not be
profitable.
Chapter 6: Conclusions
110
Chapter 6: Conclusions
In my dissertation I examine the relationship between first-phase allowances
in the EU ETS and various price drivers. In Chapter 3 I start with the most
commonly cited market fundamentals that are assumed to drive marginal abatement
costs, but find them to have little explanatory power for the allowance price path,
despite using the best available data. I then go on to test the allowance market for
the presence of a price bubble. The results are consistent with the presence of a
bubble (or a series of bubbles), although a conclusive proof of the existence of a
bubble is impossible on theoretical grounds.
In my analysis I skip the issue of how the bubble(s), if any, got started, and
focus exclusively on their presence. But something must drive the price up initially
before self-fulfilling expectations can form and take over. In Chapter 4 I focus on the
group that profited most from the high allowance price: Power producers, due to free
allocation and cost pass-through to electricity prices. Although no reason for price
inflation in competitive markets, the presence of market power can change this. The
problem with this hypothesis is that power producers were net buyers of allowances,
and existing economic theory predicts that they would decrease, rather than increase,
the allowance price, provided that they had market power. I show that when taking
the interaction between output and permit market into account, this prescription no
Chapter 6: Conclusions
111
longer holds true. Actual allocation amounts and cost pass-through rates in the EU
ETS indicate that dominant power producers would indeed have found it profitable to
inflate the allowance price.
Chapter 5 is to some extent complementary to Chapter 3: If allowance prices
were not driven by marginal abatement costs, is there an alternative explanation to a
bubble? There is some evidence supporting the hypothesis that firms simply did not
have enough time to adjust their process emissions in time for the first phase. In this
case, the allowance price would be driven by the penalty for noncompliance and the
probability that the overall cap turns out to be binding. I set up an options pricing
model that has these characteristics and find that it fits the data well, better indeed
than the market model in Chapter 3. One caveat for the findings of the options
pricing paper is that they only apply to the years 2006-7.
Taken together, my findings imply the following:
First, the allowance price during the first phase of the EU ETS was not equal
to marginal abatement costs. This could be due to the formation of a price bubble,
market power in the power sector or firms’ inability to engage in timely and
significant abatement. It is quite possible that all three of these reasons were involved
simultaneously or sequentially: A bubble needs to get started somehow, which would
reconcile the first two reasons. Also, the bubbles results are much stronger for the
period before the crash, whereas the options pricing story may apply to the latter part
of the market only. In this case, it would not have been so much inability, but rather
Chapter 6: Conclusions
112
unwillingness to abate on the part of power producers, once they were fairly certain
that the market was going to be long after the price crash. Thus, my findings are
consistent with market power starting a bubble, which then sustained itself for a while
until it was popped by the first round of emissions verifications. From then on, firms
were no longer concentrating on abatement but rather on optimizing their allowance
portfolio, taking the stochastic nature of emissions into account.
Second, regardless of the exact combination of reasons, the fact that first-
phase allowance prices were not equal to marginal abatement costs means that the
first phase of the EU ETS failed from the perspective of reaching a given emissions
goal at least cost. It is possible that the prime goal of the EU was not the design of an
efficient policy instrument for 2005-2007, but instead to prepare the EU for the Kyoto
compliance period of 2008-2012. But this could arguably have been achieved at a
lower cost to consumers and the economy as a whole, considering the large increase
in output prices without the benefit of an emissions reduction.
Future cap-and-trade markets should be set up such that they avoid the
problems encountered during the first phase of the EU ETS. Specifically, regulators
of future markets should consider auctioning most if not all allowances, which would
avoid the problem of market manipulation while giving the regulator the opportunity
to reimburse consumers for higher output prices from carbon cost pass-through.
Further, more frequent rounds of emissions verifications would prick any price
bubble sooner by breaking the cycle of self-fulfilling expectations and bring the price
back to its fundamental value. Lastly, companies should be given sufficient time and
Chapter 6: Conclusions
113
regulatory certainty to engage in large-scale abatement decisions. If the time is too
short, or the price signal to uncertain, the allowance market may deteriorate into a
betting game where firms aim to reach compliance exclusively by buying allowances
on the market, rather than abating emissions.
Future research is needed in this area. The second phase of the EU ETS, as
well as upcoming carbon markets in the eastern USA, Japan, Australia and Canada
should be examined as to whether the permit price is truly driven by marginal
abatement costs. Another promising area of research would be the design of
sophisticated auctioning schemes that would allow the regulator to gather more
information about firms’ marginal abatement costs. This knowledge would aid in
setting the cap as well as in determining whether there is a discrepancy between
permit price and marginal costs, once the market is under way.
114
Tables
Table 2.1: Summary results for Phase I of the EU ETS
2005 2006 2007
Total Phase I
Price (time average) € 18.40 € 18.05 € 0.72
€ 12.39
Trading volume
a
262 Mt 817 Mt 1,364 Mt
2,443 Mt
Trading value
a
€ 5.4 billion € 14.6 billion € 28.0 billion
€ 48.0 billion
Allocation 2,099 Mt 2,072 Mt 2,079 Mt
6,250 Mt
Emissions 2,010 Mt 2,031 Mt 2,041 Mt
6,081 Mt
Surplus (volume) 89 Mt 41 Mt 39 Mt
168 Mt
Surplus (%) 4.22 % 1.98 % 1.85 %
2.69 %
a: OTC and exchange trading for phase I and II, but excluding bilateral trades
115
Table 3.1: Results from estimating Equation (3.10)
Full period Pre-crash Post-crash
D.Gas
F
26.0812*** 37.5707*** 20.0823***
p <0.0001 <0.0001 <0.0001
D.Coal -26.1478 333.9588 -22.2437
p 0.8549 0.2534 0.7444
Temp
5d
W -3.7832 -7.2270 -1.7798
p 0.5922 0.6950 0.6378
Temp
5d
S 11.2833 65.2161* 13.3100**
p 0.3355 0.0606 <0.0001
D.Res 0.4370 -29.7230 0.2881
p 0.9744 0.4210 0.9652
Crash -4.1946***
p <0.0001
N 609 333 272
Chi
2
2143.59 29.95 54.23
p <0.0001 <0.0001 <0.0001
LL -415.91 -244.37 -143.59
AIC 847.81 502.74 301.17
BIC 883.11 529.39 326.41
Goodness of fit
#
0.2481 0.0376 0.0105
*: p<0.1; **: p<0.05; ***
<0.01; all variables defined in text
#: Model sum of squares/total sum of squares
116
Table 3.2: Results from estimating equation (3.11)
(1) (2) (3) (4)
Pre-crash Post-crash Pre-crash Post-crash
D.GasF 53.1555*** 28.7834*** 42.2153*** 32.4172***
p <0.0001 <0.0001 0.0001 <0.0001
L.D.GasF 30.1149*** 20.3575***
p <0.0001 0.0100
L2.D.GasF 9.0277 -10.7088
p 0.4483 0.4630
D.GasS -17.1810*** -7.4000**
p <0.0001 0.0182
L.D.Coal 268.0765 93.2723
p 0.2671 0.7679
D20.Coal -16.3700 4.3739
p 0.1694 0.8020
Temp1M W -0.0008 -0.0010** -0.0007 0.0002
p 0.2149 0.0171 0.3869 0.7611
Temp1M S 0.0074*** -0.0015*** -0.0025* -0.0003
p <0.0001 0.0010 0.0696 0.5886
D5.Res -18.2342*** 2.2228
p <0.0001 0.6021
D20.Res -0.1130 18.6926*** -0.0460 -2.2647
p 0.2287 <0.0001 0.7251 0.6056
L.D.EUA 0.1724*** 0.4123***
p 0.0085 <0.0001
L2.D.EUA -0.0827* -0.1305***
p 0.0756 0.0043
L3.D.EUA 0.1140*** 0.0973***
p 0.0008 0.0001
L4.D.EUA 0.1699*** 0.0379**
p <0.0001 0.0358
L5.D.EUA -0.0224 0.1149***
p 0.6284 <0.0001
N 329 272 318 253
Chi2 152.08 293.00 260.52 188.26
p <0.0001 <0.0001 <0.0002 <0.0003
AIC 469.88 234.24 439.52 162.35
BIC 504.04 266.69 492.18 211.82
Goodness of fit 0.1092 0.0278 0.1325 0.2374
*: p<0.1; **: p<0.05; ***<0.01; all variables defined in text
117
Table 3.3: Cointegration test results
Full period Pre-crash Post-crash
A(1) -0.09517 -0.49985 0.38262
p 0.247 <0.001 0.006
B1(1) (Gas) -0.00016 0.00146 0.00225
p 0.971 0.822 0.868
B2(1) (Coal) -0.03021 -0.31725 0.10131
p 0.764 0.018 0.749
B3(1) (Reservoirs) 0.00210 -0.03683 0.00839
p 0.497 0.005 0.350
B4(1) (DAX) -0.00002 0.00032 -0.00043
p 0.671 0.024 0.107
_const 0.3711 1.1521 2.3260
p 0.589 0.152 0.412
Unit Root test on z
t
-0.391 -0.974 -0.543
Unit Root test on
(
z
t
-2.241 -1.915 -2.751
Table 3.4: Results from regime-switching tests
Full period pre-crash post-crash
p 0.979 0.984 0.995
q 0.994 0.931 1.000
LR statistic 118.320 38.940 60.360
p <0.0001 0.0007 <0.0001
118
Table 5.1: Data availability and installed hydroelectric capacity by country
Country per Start of data series
a
Hydro capacity
data series Type Year Source
b
in 2006 (MW)
Series 1
Austria 3rd Wed. 1994 UCTE 11,811
Belgium 3rd Wed. 1994 UCTE 1,411
France 3rd Wed. 1994 UCTE 25,457
Greece 3rd Wed. 1994 UCTE 3,133
Italy 3rd Wed. 1994 UCTE 21,070
Luxembourg 3rd Wed. 1994 UCTE 1,128
Netherlands 3rd Wed. 1994 UCTE 37
Portugal 3rd Wed. 1994 UCTE 4,948
Spain 3rd Wed. 1994 UCTE 20,714
Series 2
Germany 3rd Wed. 1996 UCTE 9,100
Series 3
Czech Republic 3rd Wed. 1999 UCTE 2,175
Hungary 3rd Wed. 1999 UCTE 46
Poland 3rd Wed. 1999 UCTE 2,324
Slovak Republic 3rd Wed. 1999 UCTE 2,429
Slovenia 3rd Wed. 1999 UCTE 873
Series 4
UK daily 2002 Country TSO 4,256
Ireland Daily 2002 Country TSO 512
Series 5
Sweden daily 2001 Country TSO 16,180
Denmark daily 2000 Country TSO 10
Series 6
Finland daily 2004 Country TSO 3,044
a: All countries have daily data starting in 2006
b: UCTE: Union for the Coordination of transmission of electricity;
TSO: Transmission system operator
119
Table 5.2: Parameter estimates for diffusion processes
c1 c2 c3 c4 c5 c6 h
N 168 144 108 1,460 2,190 730 10,950
Const. 1486.06 1248.56 654.25 763.47 569.68 207.54 23.45
z 22.73 36.44 17.47 16.92 25.12 1.95 44.10
Trend 86.98 5.33 4.84 9.06 -2.07 1.20 -0.01
z 32.44 3.92 3.42 5.57 -2.44 0.33 -0.28
Mo n/a n/a n/a -20.84 -3.51 0.66 n/a
z n/a n/a n/a -22.31 -5.98 1.60 n/a
Fr n/a n/a n/a -20.31 -13.98 1.01 n/a
z n/a n/a n/a -20.31 -22.31 2.31 n/a
Sa -416.47 -207.72 -72.13 -128.22 -67.15 -15.71 n/a
z 32.44 3.92 3.42 -101.66 -97.18 -28.56 n/a
Su -750.21 -328.49 -128.43 -157.64 -72.43 -21.45 n/a
z -13.80 -26.70 -12.23 -133.32 -103.87 -43.08 n/a
XNY n/a n/a n/a -86.72 -37.25 -11.54 n/a
z n/a n/a n/a -20.08 -12.89 -4.85 n/a
?
2
x
(sine)
375.85 145.36 116.96 134.10 104.99 36.98 3.00
z 18.99 25.19 32.22 35.09 41.12 10.91 7.06
?
x
(phase) 1.33 1.39 1.41 1.23 1.34 1.35 -0.40
z 42.06 49.94 46.71 38.54 47.09 14.26 -2.96
AR(1)* 0.58 0.39 0.59 0.84 0.86 0.91 0.52
z 18.95 11.32 21.52 95.98 87.02 74.92 103.92
a* 0.54 0.94 0.53 0.18 0.15 0.09 0.65
z 10.24 10.68 11.38 17.15 13.20 6.94 68.02
?*
Jan 499.71 133.17 88.72 65.21 46.42 23.69 17.21
Feb 316.67 94.92 53.10 45.35 42.27 23.84 16.15
Mar 366.30 119.39 64.96 67.47 41.32 21.19 19.15
Apr 453.41 142.96 79.08 79.32 51.52 31.36 14.49
May 400.48 135.97 55.56 92.75 48.67 33.66 16.28
Jun 387.02 132.80 59.94 45.64 46.51 34.84 16.54
Jul 427.82 116.79 55.50 20.17 30.71 12.12 18.07
Aug 305.51 97.78 50.71 69.65 12.10 7.77 20.91
Sep 389.43 122.02 61.45 23.50 18.56 7.65 20.21
Oct 387.15 120.30 64.95 31.66 27.65 11.17 22.63
Nov 432.38 108.56 73.35 39.00 35.50 19.57 21.50
Dec 414.69 163.85 85.40 96.82 55.91 42.84 17.68
*For series 1-3, based on 2006-7 data; all other estimates based on pre-2006 data
120
Table 5.3: Correlation coefficients
a
among different series
c1 c2 c3 c4 c5 c6 h
c1 1.000
c2 0.8814* 1.000
c3 0.9016* 0.8730* 1.000
c4 0.4554* 0.2976* 0.4927* 1.000
c5 0.5170* 0.3897* 0.6032* 0.9231* 1.000
c6 0.4588* 0.3672* 0.5573* 0.8496* 0.9418* 1.000
h -0.067 0.014 -0.036 -0.038 -0.033 -0.020 1.000
*p<0.05; all coefficients based on 2006-7 data
a: The correlation coefficient between series x
t
i
and x
t
j
and the corresponding p-value
are computed as
ˆ ? =
(x
t
i
? x
i
)(x
t
j
? x
j
)
t=1
T
i , j
?
(x
t
i
? x
i
)
2
t=1
T
i, j
?
(x
t
j
? x
j
)
2
t=1
T
i, j
?
; p = 2*ttail T
i, j
?2, ˆ ? T
i, j
?2 / 1? ˆ ?
2
( )
where T
i, j
refers to the number of days for which both series have valid entries.
121
Table 5.4: Parameter estimates from options pricing formula
Original Model using B
t
Model using B
t
/ 10
1
?
-2,967,479 1
?
-2,955,860
2
?
-3,134,951 2
?
-2,991,328
Good. of fit:
a
0.8055 Good. of fit:
a
0.9232
EV
S (MT) K
ˆ
(MT)
?ˆ
EV
S (MT) K
ˆ
(MT)
?ˆ
10 -177 60 10 -833 282
20 -354 119 20 -1,667 564
30 -532 179 30 -2,500 846
40 -709 239 40 -3,334 1,128
50 -886 299 50 -4,167 1,410
60 -1,063 358 60 -5,000 1,692
70 -1,240 418 70 -5,834 1,974
80 -1,418 478 80 -6,667 2,256
90 -1,595 537 90 -7,500 2,537
100 -1,772 597 100 -8,334 2,819
110 -1,949 657 110 -9,167 3,101
120 -2,126 717 120 -10,001 3,383
130 -2,304 776 130 -10,834 3,665
140 -2,481 836 140 -11,667 3,947
150 -2,658 896 150 -12,501 4,229
160 -2,835 955 160 -13,334 4,511
170 -3,012 1,015 170 -14,168 4,793
180 -3,189 1,075 180 -15,001 5,075
190 -3,367 1,135 190 -15,834 5,357
200 -3,544 1,194 200 -16,668 5,639
Shaded areas: Economically plausible parameter ranges for ?
a: Goodness of fit=
?
=
? ?
T
t
t t t t
P E P P E P
1
2 2
]) [ /( ])
ˆ
[
ˆ
(
Figures
Figure 1.1: EUA price and trading volumes, Phase I
122
Figure 1.1: EUA price and trading volumes, Phase I
Figure 2.1: Allowance allocation and emissions by EU member
Figure 2.2: Allowance allocation and emissions by sector (total values)
0
50
100
150
200
250
300
350
400
450
500
D
E
P
L
G
B
I
T
E
S
M
i
l
l
i
o
n
t
o
n
s
C
O
2
0
200
400
600
800
1000
1200
1400
1600
Power & Heat
M
i
l
l
i
o
n
t
o
n
s
C
O
2
123
Figure 2.1: Allowance allocation and emissions by EU member country
Figure 2.2: Allowance allocation and emissions by sector (total values)
E
S
F
R
C
Z
N
L
G
R
B
E
F
I
P
T
A
T
S
K
H
U
D
K
S
E
I
E
E
E
L
T
S
I
Country
Power & Heat Metals Cement &
lime
Oil & Gas Glass &
ceramics
C
Y
L
V
L
U
M
T
Allocation
Emissions
Pulp & Paper
Allocation
Emissions
Figure 2.3: Allowance allocation and emissions by sector (percent of total)
9.6
9.0
7.6
a.) Allocation by sector (%)
8.3
8.9
7.3
124
Figure 2.3: Allowance allocation and emissions by sector (percent of total)
69.9
2.0 1.8
a.) Allocation by sector (%)
Power & Heat
Metals
Cement & lime
Oil & Gas
Glass & ceramics
Pulp & Paper
72.3
1.7 1.5
b.) Emissions by sector (%)
Power & Heat
Metals
Cement & lime
Oil & Gas
Glass & ceramics
Pulp & Paper
Figure 2.3: Allowance allocation and emissions by sector (percent of total)
Power & Heat
Cement & lime
Glass & ceramics
Pulp & Paper
Power & Heat
Cement & lime
Glass & ceramics
Pulp & Paper
125
Figure 3.1: EUA, coal, gas, DAX and reservoir levels
Figure 4.1: Spark spread
Figure 4.2: Dark spreads and green dark spreads in Germany
126
Figure 4.1: Spark spread and green (clean) spark spread in the UK
Figure 4.2: Dark spreads and green dark spreads in Germany
127
Figure 5.1: Available electricity consumption data, pre-2006
128
Figure 5.2: Weighted average precipitation in the EU
129
Figure 5.3: EUA price, prediction and forward price for Phase II
Figure 5.4: Sensitivity of prediction to variance of future electricity demand
Appendix A
130
Appendices
Appendix A: Proof of Equation (3.8)
Derivation of Equation (3.8)
I start by restating equation (3.7):
(3.7) ( ) ( ) ( )
? ? ? ?
+ =
?
+ =
?
+ =
?
+ =
? ? ? + + ? + =
T
t k
k k k
T
t k
k k k
T
t k
k k
T
t k
k
E b g N F E F d E
1
1
1
1
1
1
1
] [ * ] [ ] [ ? ? ?
If allowance and fuel prices have the Markov property such that
t t t
P P E ? =
+
] [
1
where
t
P represents any price, r + = 1 ? the discount factor and r the interest rate, at k=T-1
it must be that
(A1) ( ) ( ) ] [
1 1 1 T T T T T T T
E h F F d ? ? ? + ? + =
? ? ?
? ?? ?
where ? ) ( b g N h + ? . Now I move one period back to k=T-2:
( ) ( ) ] [ ] [ ) (
1 1 2 1 1 2 1 2
2
1 T T T T T T T T T T t T T
E E h F F F F d ? ? ? + ? ? ? + ? + ? + + = +
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
Substituting (A1) for ?
T
and rearranging yields
(A2) ( ) ( ) ] [ ) ( ) 1 (
1 2 1 2 1 2
2
1 ? ? ? ? ? ? ?
? ? ? + ? + + = +
T T T T T T T
E h F F d ? ? ? ? ? ?
Appendix A
131
Moving another period back to k=T-3:
( )
( ) ] [ ] [ ] [
) (
1 1 2 1 2 3 2
1 2 1 3 2 3
2 3
1 2
T T T T T T T T T
T T T T T T T T T T
E E E h
F F F F F F d
? ? ? + ? ? ? + ? ? ? +
? + ? + ? + + + = + +
? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
As before, I substitute (A1) for
T
? and simplify:
( )
( ) ] [ ] [
) ( ) 1 (
1 2 1 2 3 2
2 1 3 2 3
2 3
1 2
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? + ? ? ? +
? + ? + + + = + +
T T T T T T
T T T T T T T
E E h
F F F F d ? ? ? ? ? ? ? ? ?
Now I further substitute (A2) for
1
) 1 (
?
+
T
? ? to get
(A3) ( ) ( ) ] [ ) ( ) 1 (
2 3 2 3 2 3
2 3
2
2
? ? ? ? ? ? ?
? ? ? + ? + + + = + +
T T T T T T T
E h F F d ? ? ? ? ? ? ? ?
The next step would be to move to period k=T-4 and successively substituting (A1),
(A2) and (A3). However, the general solution is apparent:
(A4) ( ) ] [ ) (
1 1
1
1 t t t
GC
t
GC
t
T
t k
k T
t
T
t k
k T
t
E h F F d ? ? ? + ? + =
? ?
=
+ ?
?
=
?
? ?
? ? ? ?
The first term on the RHS can be re-written as
?
=
?
?
T
t k
k T
t
? ? ? * *
1
. Dividing (A4)
by the summation term on the LHS yields the result:
(3.8) ( )
( )
? ?
?
?
?
?
?
? ? ?
+ +
?
+ =
T
t
k T
t t t
T
t
k T
t t
t t
E
b g N
F F
d
?
?
?
?
?? ?
] [
1 1
1
Appendix B
132
Appendix B: Proof of Lemma 1 and Eq. (4.9)
Proof of Lemma 1
Differentiating (4.2) w.r.t. p and rearranging gives
(
¸
(
¸
=
(
¸
(
¸
(
(
¸
(
¸
0
1
/
/
p e
p q
C C
C C
i
i
i
ee
i
qe
i
qe
i
qq
? ?
? ?
Solving for the effect of a price change on output and emissions yields
(B1)
( ) 0 0
0
2
> ? ? ? >
?
?
=
>
?
=
i
qe
i
ee
i
qq
i
i
i
qe
i
i
i
ee i
C C C
C
p
e
C
p
q
?
?
?
?
Similarly, differentiating (4.2) w.r.t. the permit price and solving yields
(B2)
0
0
<
?
?
=
<
?
=
i
i
qq
i
i
i
qe
i
C
e
C
q
??
?
??
?
To analyze the effect of the dominant firm’s output on output price p and
permit price ?, differentiate (4.3) w.r.t. q
1
and rearrange:
Appendix B
133
(B3)
(
¸
(
¸
? ?
=
(
(
(
(
¸
(
¸
(
(
(
(
¸
(
¸
|
|
¹
|
\
|
? ? ?
? ?
? ?
= =
= =
0
1
1
1
2 2
2 2
P
q
p
q
p
x x
p
q
P
q
P
N
i
i
N
i
i
N
i
i
N
i
i
?
?
?
??
?
?
??
?
?
?
??
?
Solving for the effect of q
1
on the permit price:
(B4)
(
¸
(
¸
? ? +
? ?
=
? ? ? ? ?
?
= = = = =
=
N
i
N
i
N
i
i i
N
i
i i
N
i
i
N
i
i
x
p
q
p
x q
P
x
p
x
P
q
2 2 2 2 2
2
1
??
?
?
?
?
?
??
?
??
?
?
?
?
??
Because P’<0 and e
i
=x
i
for i=(2,…,N), it follows immediately from (B1) that
the numerator is positive. As for the denominator, the fist term is negative from (B2).
In order to show that (B4) is negative I have to show that the term in the brackets is
positive, i.e. that
(B5) 0
?
2 2 2 2
> ? ? ?
? ? ? ?
= = = =
N
i
N
i
i i
N
i
N
i
i i
x
p
q
p
x q
??
?
?
?
?
?
??
?
Substituting (B1) and (B2), this is equivalent to showing that
(B6) 0
?
2 2 2 2
>
?
?
?
?
?
?
?
? ? ? ?
= = = =
N
i
i
i
qq
N
i
i
i
ee
N
i
i
i
qe
N
i
i
i
qe
C
C
C C
to prove the inequality in (B5). Separating out the a single firm, it is clear that
Appendix B
134
( )
( )
( )
0
1
2
2
2
>
?
=
?
?
?
i
i
i
qe
i
i
ee
i
qq
C C C
which enables me to express (B6) as
0
1
?
2
>
?
?
+
?
?
?
=
N
j i
i
i
j
qe
i
qe
j
ee
i
qq
i
C C C C
Noting the symmetry between i/j and j/i multiplications and dropping the first
(positive) term, I can express this as
(B7) 0
2
?
2
>
?
? +
?
< ?
N
j i
i
j
qe
i
qe
i
ee
j
qq
j
ee
i
qq
C C C C C C
Squaring both sides of the numerator in (B7) yields
( ) ( ) ( )
2
?
2 2
4 4 2
j
qe
i
qe
j
ee
j
qq
i
ee
i
qq
i
ee
j
qq
i
ee
j
qq
j
ee
i
qq
j
ee
i
qq
C C C C C C C C C C C C C C > > + +
where the second inequality comes from the fact that
( ) ( )
2 2
j
qe
i
qe
j
ee
j
qq
i
ee
i
qq
i
qe
i
ee
i
qq
C C C C C C i C C C > ? ? >
Subtracting the RHS of the first inequality completes the proof:
(B8) ( ) ( ) ( ) 0 0 2
! !
2 2 2
> ? ? > ? = + ?
i
ee
j
qq
j
ee
i
qq
i
ee
j
qq
i
ee
j
qq
j
ee
i
qq
j
ee
i
qq
C C C C C C C C C C C C
Appendix B
135
Now I derive the sign of the other three expressions in Lemma 1 by solving
(B3) for the effect of firm 1’s output on the output price and using (B2) and (B5):
(B9) 0
/
2
2
1
<
? ? +
?
=
?
?
=
=
P
x
x P
q
p
N
i
i
N
i
i
??
?
?? ?
?
?
because the numerator is positive. Finally, differentiating (4.3) w.r.t. x
1
gives
(
¸
(
¸
?
=
(
(
(
(
¸
(
¸
(
(
(
(
¸
(
¸
? ?
? ?
? ?
= =
= =
1
0
1
1
2 2
2 2
x
p
x
p
x x
p
q
P
q
P
M
i
i
M
i
i
N
i
i
N
i
i
?
?
?
??
?
?
??
?
?
?
??
?
Solving this yields
(B10) 0
/
2
1
>
?
=
?
=
N
i
i
p q
x
? ?
?
??
(B11) 0
/
2
1
>
?
?
=
?
=
N
i
i
q
x
p
?? ?
?
?
Proof of equation (4.9)
Keeping in mind that 0 /
1
< q ? ?? , I re-write (4.9) as
1
1
?
1
1
/
/
/
/
q
q p
x
x p
? ??
? ?
? ??
? ?
<
Appendix B
136
Substituting (B4) and (B9)-(B10) into this expression and simplifying yields
(B12)
?
?
?
?
=
=
=
=
?
<
?
N
i
i
N
i
i
N
i
i
N
i
i
p x
x
p q
q
2
2
?
2
2
/
/
/
/
? ?
?? ?
? ?
?? ?
Multiplying both sides by the two denominators (again reversing the
inequality) and bringing both terms to the left hand side gives
(B13) 0
!
2 2 2 2
> ? = ?
? ? ? ?
= = = =
N
i
N
i
i i
N
i
N
i
i i
x
p
q
p
x q
??
?
?
?
?
?
??
?
which I prove above.
Appendix C
137
Appendix C: Derivation of variance and covariance of future emissions
Result 1: Variance of future CO
2
emissions
The variance of
T
t
G is defined by
(C1)
? ? ?
+ = + = + =
+ = =
T
t k
T
k u
u k s
T
t k
k s
T
t s t
g g Cov g Var G Var s
1 1 1
2
] , [ 2 ] [ ] [
The variance of g
t
and the covariance between g
t
and g
u
are
(C2)
| |
| |
( ) ] [ ] , [ 2 ] [
] [
2 2
1
2
1
1
t s t t s t s
c
t s
c
t s t s
h Var h c Cov c Var
c Var
c K Var g Var
? ? ?
?
?
+ ? =
=
+ =
(C3)
( )( ) | |
( )( ) | |
{ }{ } | |
( )( ) | | ( )( ) | |
( )( ) | | ( )( ) | |
( ) ] , [ ] , [ ] , [ ] , [
] [ ] [ ] [ ] [
] [ ] [ ] [ ] [
]) [ ( ] [ ]) [ ( ] [
] [ ] [
] [ ] [ ] , [
2 2
2 2
2 2 2
2
2
u t s u t s u t s u t s
u s u t s t s u s u t s t s
u s u t s t s u s u t s t s
u s u u s u t s t t s t s
c
u s
c
u
c
t s
c
t s
u s u t s t s u t s
c h Cov h c Cov h h Cov c c Cov
c E c h E h E h E h c E c E
h E h h E h E c E c c E c E
h E h c E c h E h c E c E
c E c c E c E
g E g g E g E g g Cov
? ? ? ?
?? ??
? ? ?
? ? ?
?
? ? + =
? ? ? ? ? ?
? ? + ? ? =
? ? ? ? ? ? =
? ? =
? ? =
Combining (C2) and (C3) establishes the result shown in equation (5.12)
Appendix C
138
(5.12)
( )
( )
? ?
?
+ = + =
+ =
? ? + +
+ =
T
t k
T
k u
u t s u t s u t s u t s
T
t k
t s t s t
c h Cov h c Cov h h Cov c c Cov
h Var c Var s
1 1
2 2
1
2 2 2
] , [ ] , [ ] , [ ] , [ 2
] [ ] [
? ? ? ?
? ?
Result 2: Generalization of the variance for different volatilities
I start by restating the equation (5.17): The variance of
t
c and
t
h for t s ? ? 0
is
(5.17) h c x dy y i e x Var
x
t
s
t a
t s
x
, )] ( [ ] [
2 ) ( 2
= =
?
? ?
?
?
Suppose that at time s, we’re in month 5 and want to calculate the variance of
consumption/precipitation in month 8. Using the notation defined in the text that
{ } i t i t t i t = = ) ( : min )] ( [
min
, we have that ] 9 [ ] 8 [ ] 7 [ ] 6 [
min min min min
t t t t t s < < < < < . I
now split up the integral in (17) into four integrals with constant volatility:
dy e dy e
dy e dy e x Var
x
t
t
y t a
x
t
t
y t a
x
t
t
y t a
x
t
s
y t a
t s
x x
x x
] 8 [ ] 7 [
] 6 [ ] 5 [ ] [
2
] 8 [
) ( 2 2
] 8 [
] 7 [
) ( 2
2
] 7 [
] 6 [
) ( 2 2
] 6 [
) ( 2
min
min
min
min
min
min
? ?
? ?
? ?
? ?
? ? ? ?
? ? ? ?
+ +
+ =
Next, I split the exponents such that they match with the new upper limits of
the integrals and move the remainder (a constant) in front:
Appendix C
139
dy e dy e e
dy e e dy e e x Var
x
t
t
y t a
x
t
t
y t a t t a
x
t
t
y t a t t a
x
t
s
y t a t t a
t s
x x x
x x x x
] 8 [ ] 7 [
] 6 [ ] 5 [ ] [
2
] 8 [
) ( 2 2
] 8 [
] 7 [
) ] 8 [ ( 2 ]) 8 [ ( 2
2
] 7 [
] 6 [
) ] 7 [ ( 2 ]) 7 [ ( 2 2
] 6 [
) ] 6 [ ( 2 ]) 6 [ ( 2
min
min
min
min min
min
min
min min
min
min min
? ?
? ?
? ?
? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ?
+ +
+ =
Because the volatilities are constant within each integral, each of them can be
easily solved:
( ) ( )
( ) ( )
]) 7 [ ( 2
2
]) 7 [ ] 8 [ ( 2
2
]) 8 [ ( 2
]) 6 [ ] 7 [ ( 2
2
]) 7 [ ( 2 ) ] 6 [ ( 2
2
]) 6 [ ( 2
min min min min
min min min min min
1 *
2
] 8 [
1 *
2
] 7 [
*
1 *
2
] 6 [
* 1 *
2
] 5 [
* ] [
t t a
x
x t t a
x
x t t a
t t a
x
x t t a s t a
x
x t t a
t s
x x x
x x x x
e e e
e e e e x Var
? ? ? ? ? ?
? ? ? ? ? ? ? ?
? + ? +
? + ? =
?
?
?
?
?
?
?
?
Multiplying out and some rearranging gives
( ) ( )
( ) ¦
)
¦
`
¹
¦
¹
¦
´
¦
? + ? +
? + ?
=
? ? ? ?
? ? ? ?
) ( 2 2 2 ]) 8 [ ( 2 2 2
]) 7 [ ( 2 2 2 ]) 6 [ ( 2 2 2
] 5 [ ] 8 [ ] 8 [ ] 7 [
] 7 [ ] 6 [ ] 6 [ ] 5 [
2
1
] [
min
min min
s t a
x x
t t a
x x
t t a
x x
t t a
x x
x
t s
x x
x x
e e
e e
a
x Var
? ? ? ?
? ? ? ?
which can be generalized to
(5.19) ( )
)
`
¹
¹
´
¦
? + + ? =
? ? + ? ?
?
=
?
)] ( [ )] ( [ ] 1 [ ] [
2
1
] [
2 ) ( 2 2 ]) 1 [ ( 2
1 ) (
) (
2 2
min
s i e t i e k k
a
x Var
x
s t a
x
k t t a
t i
s i k
x x
x
t s
x x
? ? ? ?
Result 3: Covariance of x on two different days
The covariance between
t
x and
u
x , for h c x , = and u t s ? ? is given by
Appendix C
140
( )( ) | |
(
¸
(
¸
=
? ? =
? ?
? ? ? ?
?
? ?
?
? ?
? ? ? ? dW i e dW i e E
x E x x E x E x x Cov
x
u
s
u
x
t
s
t
s
u s u t s t s u t s
x x
)] ( [ * )] ( [
] [ ] [ ] , [
) ( ) (
I split up the second integral into two parts and pull out the constant term:
(
(
¸
(
¸
|
|
¹
|
\
|
+ =
? ? ?
? ? ? ? ? ? ? ?
?
? ?
?
? ? ?
?
? ?
? ? ? ? ? ? dW i e dW i e e dW i e E x x Cov
x
u
t
u
x
t
s
t t u
x
t
s
t
s u t s
x x x x
)] ( [ )] ( [ * )] ( [ ] , [
) ( ) ( ) ( ) (
Multiplying out gives
(
¸
(
¸
+
(
¸
(
¸
=
? ?
? ?
? ? ? ?
? ? ? ? ? ?
?
? ?
?
? ?
?
? ?
?
? ? ?
? ? ? ?
? ? ? ?
dW i e dW i e E
dW i e dW i e E e x x Cov
x
u
t
u
x
t
s
t
s
x
t
s
t
x
t
s
t
s
t u
u t s
x x
x x x
)] ( [ * )] ( [
)] ( [ * )] ( [ ] , [
) ( ) (
) ( ) ( ) (
The second term is the expectation of the product of two stochastic processes
occurring during non-overlapping time periods. Because a Wiener process is iid, this
term drops out. Using the fact that dt dW =
2
) ( establishes the result:
(5.20)
] [ *
) )]( ( [
)] ( [ * )] ( [ ] , [
) (
2 2 ) ( 2 ) (
) ( ) ( ) (
t s
t u
x
t
s
t
s
t u
x
t
s
t
x
t
s
t
s
t u
u t s
x Var e
dW i e E e
dW i e dW i e E e x x Cov
x
x x
x x x
? ?
? ? ? ?
? ? ? ? ? ?
=
(
¸
(
¸
=
(
¸
(
¸
=
?
? ?
?
?
? ? ?
?
? ?
?
? ? ?
? ?
? ? ? ?
141
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doc_228620669.pdf
Carbon pricing is the generic term for putting a price on carbon through either subsidies, a carbon tax, or an emissions trading ("cap-and-trade") system.
ABSTRACT
Title of Document: PRICING CARBON: ALLOWANCE PRICE
DETERMINATION IN THE EU ETS
Beat Hintermann, Ph.D., 2008
Directed By: Prof. A. Lange, Department of Agricultural and
Resource Economics (AREC)
The allowance price in Phase I of the European Union Emissions Trading
Scheme (EU ETS) followed a peculiar path, increasing from €7 in 2005 to over €30
in 2006, before crashing, recovering and ultimately finishing at zero by the end of
2007. I examine if the price can be explained by marginal abatement costs as
predicted by economic theory, or if there were other price determinants. This has
important policy implications, since the least-cost solution depends on the equality of
permit price and marginal abatement costs and is the main argument in favor of
permit markets.
I start with a model that incorporates the most commonly cited market
fundamentals and find that the latter only explain a small part of the allowance price
variation, raising the question of a bubble. I carry out two different bubbles tests, the
results of both of which are consistent with the presence of an allowance price
bubble.
I then address whether market manipulation by dominant power generators
could have lead to the initial allowance price increase. I extend economic theory to
include the interaction between output and permit markets. I derive a threshold of
free allocation beyond which firms find it profitable to manipulate the permit price
upwards, even if they are net allowance buyers. Market data indicates that this
threshold was exceeded for EU power generators.
Finally, I investigate the possibility that due to the speed at which the market
was set up, firms may have been unable to engage in effective abatement before the
end of Phase I. I develop a model under the assumption of no abatement, where firms
aim to reach compliance exclusively by purchasing allowances on the market. Thus,
the allowance payoff becomes that of a binary option, for which I derive a pricing
formula. The model fits daily data from the years 2006-7 well.
I conclude that the allowance price in Phase I was not driven by marginal
abatement costs, but by a combination of price manipulation, self-fulfilling
expectations and/or the penalty for noncompliance weighted by the probability of a
binding cap.
THE PRICE OF CARBON: ALLOWANCE PRICE DEVELOPMENT IN THE EU
ETS
By
Beat Hintermann
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park, in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2008
Advisory Committee:
Professor Andreas Lange, Chair
Professor Albert S. Kyle
Professor Marc L. Nerlove
Professor John K. Horowitz
Professor S. Boragan Aruoba
© Copyright by
Beat Hintermann
2008
ii
Preface
This dissertation is a hybrid between a three-paper dissertation and a unified
dissertation. On the one hand, it is based on three separate papers, two of which I
already submitted to an economic journal; they are currently under review. I hope to
submit the third paper shortly.
On the other hand, the three papers all treat the same subject matter: What
drove the allowance price during the first phase of the EU ETS?
Chapters 1 and 6 contain a general introduction and conclusion, respectively,
that are applicable to all three articles, and I removed the description of the market
from the individual papers and concentrated it in a separate section (Chapter 2). I
also made minor changes such as renaming equations, tables and figures. In spite of
this, Chapters 3-5 remain largely self-contained and may give rise to some
redundancy. They also are independent in terms of notation, such that a symbol or
letter appearing in two different chapters may represent two different variables or
parameters.
iii
Dedication
To Mary.
iv
Acknowledgements
I would like to thank all members of my dissertation committee for their
support and guidance in this project. Special thanks go to the committee chair,
Andreas Lange.
Many thanks also to faculty and fellow students at UMD-AREC for valuable
insights and suggestions.
I gratefully acknowledge the Swiss National Fund, which provided funds
during the last six months of this dissertation.
v
Table of Contents
Preface........................................................................................................................... ii
Dedication .................................................................................................................... iii
Acknowledgements ...................................................................................................... iv
Table of Contents .......................................................................................................... v
List of Tables .............................................................................................................. vii
List of Figures ............................................................................................................ viii
Chapter 1: Introduction ................................................................................................. 1
Chapter 2: The EU ETS ................................................................................................ 7
Chapter 3: Price Drivers and CO2 Bubbles in the EU ETS ........................................ 12
Abstract ................................................................................................................... 12
3.1. Introduction ...................................................................................................... 13
3.2. Allowance market model ................................................................................. 16
3.2.a.) Literature ................................................................................................. 16
3.2.b.) Base model ............................................................................................. 19
3.2.c.) Introducing dynamic expectations of fundamental prices ...................... 27
3.2.d.) Introducing lagged EUA price changes .................................................. 29
3.3. Results .............................................................................................................. 31
3.4. A CO
2
Bubble? ............................................................................................... 36
3.4.a.) Some bubble background ........................................................................ 36
3.4.b.) Bubble tests............................................................................................. 38
3.4.c.) Cointegration test .................................................................................... 40
3.4.d.) Regime-switching test ............................................................................ 42
3.5. Conclusions ...................................................................................................... 45
Chapter 4: Market Power and Windfall Profits in Emission Permit Markets............. 48
Abstract ................................................................................................................... 48
4.1. Introduction ...................................................................................................... 49
4.2. Market power in output and permit market ..................................................... 54
4.3. Application to the EU ETS .............................................................................. 64
4.3.a.) General applicability ................................................................................ 64
4.3.b.) The UK power market ............................................................................. 66
4.3.c.) The German power market ...................................................................... 70
4.4. Conclusions ..................................................................................................... 72
Chapter 5: An Options Pricing Approach to CO
2
Allowances in the EU ETS.......... 75
Abstract ................................................................................................................... 75
5.1 Introduction ...................................................................................................... 76
5.2. Emission allowance pricing in the absence of abatement ............................... 80
5.2.a.) Literature on permit pricing ..................................................................... 80
5.2.b.) Derivation of option pricing formula....................................................... 81
vi
5.3. Deriving the mean and standard deviation of future emissions ...................... 85
5.3.a.) CO
2
emissions as a function of exogenous stochastic processes ............. 85
5.3.b.) Properties of the stochastic processes c
t
and h
t
........................................ 90
5.4. Estimation ....................................................................................................... 95
5.4.a.) Data .......................................................................................................... 96
5.4.b.) Parameter estimation for electricity consumption and precipitation ....... 99
5.4.c.) Evaluation of the options pricing formula ............................................. 102
5.5 Conclusions .................................................................................................... 107
Chapter 6: Conclusions ............................................................................................ 110
Tables ........................................................................................................................ 114
Figures....................................................................................................................... 122
Appendices ................................................................................................................ 130
Appendix A: Proof of Equation (3.8) ................................................................... 130
Appendix B: Proof of Lemma 1 and Eq. (4.9) ...................................................... 132
Appendix C: Derivation of variance and covariance of future emissions ............ 137
References ................................................................................................................. 141
vii
List of Tables
Table 2.1: Summary results for Phase I of the EU ETS ........................................... 114
Table 3.1: Results from estimating Equation (3.10) ................................................. 115
Table 3.2: Results from estimating equation (3.11) .................................................. 116
Table 3.3: Cointegration test results ........................................................................ 117
Table 3.4: Results from regime-switching tests ....................................................... 117
Table 5.1: Data availability and installed hydroelectric capacity by country ........... 118
Table 5.2: Parameter estimates for diffusion processes ............................................ 119
Table 5.3: Correlation coefficients
a
among different series...................................... 120
Table 5.4: Parameter estimates from options pricing formula .................................. 121
viii
List of Figures
Figure 1.1: EUA price and trading volumes, Phase I ............................................... 122
Figure 2.2: Allowance allocation and emissions by sector (total values) ................. 123
Figure 2.3: Allowance allocation and emissions by sector (percent of total) ........... 124
Figure 3.1: EUA, coal, gas, DAX and reservoir levels ............................................. 125
Figure 4.2: Dark spreads and green dark spreads in Germany ................................. 126
Figure 5.1: Available electricity consumption data, pre-2006 .................................. 127
Figure 5.2: Weighted average precipitation in the EU ............................................. 128
Figure 5.3: EUA price, prediction and forward price for Phase II ........................... 129
Figure 5.4: Sensitivity of prediction to variance of future electricity demand ......... 129
Chapter 1: Introduction
1
Chapter 1: Introduction
On January 1, 2005, the world’s first non-voluntary CO
2
emissions market
opened for business. The European Union Emissions Trading Scheme (EU ETS) is
the European Union’s prime instrument to achieve its Kyoto targets. The system
covers emissions from energy-intensive industries that are responsible for about 45%
of the EU’s total CO
2
emissions. The EU ETS is by far the largest regulated
emissions permit market to date, dwarfing other markets in terms of total emissions,
included installations and market value.
The market is organized into distinct phases that each have different caps and
rules. The first phase covered the years 2005-2007, followed by the second phase,
which coincides with the 2008-2012 Kyoto compliance period. No banking was
allowed between the first and second phases, such that the first phase was a self-
contained market with a finite time horizon. First-phase emission permits (called EU
allowances, or EUAs) lost their value if unused for compliance.
The EUA price during the first phase followed a rather peculiar path, shown in
Figure 1.1. It started around €7 but rapidly increased to levels above €20, even
surpassing €30 at some point, before crashing to half of its value in April 2006,
stabilizing again in the €13-18 region and finally decreasing to zero by mid 2007,
where it remained for the rest of the market.
Chapter 1: Introduction
2
The initially very high allowance price is surprising considering that the first
phase was understood to be a pilot run for the second phase, with the cap not
expected to be very stringent. Even more surprising is the stabilization after the price
crash, followed by a slow march to zero. Given that the market turned out to be
oversupplied with permits, the price should have dropped to zero immediately, rather
than over the course of a year, if it equals marginal abatement costs (unless, of course,
the fundamentals driving marginal abatement costs also slowly declined over time).
The three main chapters in this dissertation (Chapters 3-5) are self-contained
and different in their methods, tools, and even notation to some extent, but they all
aim to answer the same question: What drove the allowance price during the first
phase of the EU ETS?
This question is interesting from an economic theory perspective, but it is
equally important in terms of policy implications. The main reason to institute a cap-
and-trade market, as opposed to using command-and-control methods, is that it yields
the lowest overall cost to reduce emissions to a specific target.
1
But if allowance
prices are too high (i.e. above marginal abatement costs), then the least-cost argument
vanishes and overall welfare might be better served using a different policy. Note
that firms will pass on much of the marginal cost of carbon as they do with any other
1
It shares this property with an emissions tax. In a world without uncertainty and full
auctioning of permits (or at least no updating of free permit allocations based on events
during the market), the permit price should equal the emissions tax for the same amount of
emissions reduction.
Chapter 1: Introduction
3
input to production,
2
so in the end it will be consumers that pay too much for a given
emissions reduction if allowance prices are above their efficient level.
In Chapter 3, I focus on widely identified fundamental price drivers and test
whether and to what extent they are able to explain the observed allowance price
path. A deviation between price and fundamentals is commonly referred to as a price
bubble. To the extent that the existence of a bubble can never be fully proven due to
an identification problem, I carry out two tests that examine whether or not the data is
consistent with the presence of a bubble. The first test relies on a market model that
assumes that the allowance price equals marginal abatement costs. I estimate this
model using a regime-switching approach that allows the allowance price to depend
differently on market fundamentals during different time periods. A likelihood ratio
test reveals that the two regimes are not mutually independent, but that the state in the
current period has an impact on the state probability in the next period (Markov
switching). This is consistent with a bubble, or a series of stochastically crashing
bubbles with interchanging boom and bust phases.
The second test relies on cointegration. If the allowance price and the
supposed market fundamentals exhibit cointegration, this would be evidence that the
price was indeed driven by fundamentals, and that therefore there was no bubble. I
2
The degree of cost pass-through depends on the price elasticity of consumer demand. With
completely inelastic demand, costs are fully passed through.
Chapter 1: Introduction
4
find no such cointegration in the data. Thus, while not conclusive, the results from
both tests are consistent with the bubbles hypothesis.
Emission permits were initially given away for free in accordance with the
European Commission’s mandate that countries could sell at most 5 % of their total
allowances, a measure taken to obtain industry support for the Directive. Chapter 4
examines one potential effect of free allocation on the allowance price. If firms were
able to pass through the marginal cost of CO
2
to consumers, they were in a position to
reap large windfall profits. In a competitive and efficient market, windfall profits
should not lead to permit price distortions. However, this changes in the presence of
market power. I examine how the initial allowance allocation affects the permit price
under the assumption of market power in both the output and the permit market. This
is an extension of Hahn’s (1986) results, which prescribe that a dominant firm will
manipulate the permit price upwards (downwards) if it is a net seller (buyer) of
permits. I show that when taking the interactions between the output and permit
market into account, this prescription changes in a significant way, meaning that the
largest permit holders in the EU ETS (i.e. power producers) would have found it
profitable to drive up the permit price despite the fact that they were net allowance
buyers, assuming that they had some market power.
Although market power per se is unrelated to the issue of bubbles, every price
bubble has to get started somehow. In addition to the basic subject matter that is the
thread throughout this dissertation, this provides another link between Chapters 3 and
Chapter 1: Introduction
5
4: Market-power related price manipulation could have driven the allowance price
upwards, leading to the beginning of an eventually self-sustaining price bubble.
Chapter 5 is motivated by the possibility that firms may have been unable to
engage in significant abatement in time for the first phase. The market was set up at
breakneck speed relative to “normal” time frames for instituting such markets,
3
providing little time for countries to determine historic emissions, define their caps
and distribute them among different installations. Firms had even less time to prepare
for the market, given that their individual permit allocations, the country-level caps
and thus the total cap was not known until the market had already come into effect.
Because CO
2
cannot be captured in a cost-effective manner today, the main
sources of abatement are changes in production technology towards less emission-
intensive processes, and the substitution of fuels with a lower emission factor per unit
of output. Since a change in production technology requires significant planning and
construction time combined with a minimum level of price certainty, fuel switching
4
was commonly assumed to be the abatement method of choice in the first phase of the
market. However, energy-intensive industries are generally locked into long-term
fuel contracts and may have been unable to switch, or unwilling to do so until the
price signal was more stable.
3
The directive that mandated the EU ETS (Directive 2003/87/EC) was issued in October
2003, just over a year before the market started.
4
This is not necessarily a substitution within the same plant, but a substitution across plants.
For example, more of the total electricity demand could be produced using gas-fired
generation while reducing the output of coal-fired generators.
Chapter 1: Introduction
6
If abatement is not feasible, firms will aim to reach compliance exclusively by
purchasing allowances on the market.
5
With stochastic emissions the exact number of
allowances needed for compliances is unknown ex ante. Firms face a situation where
an additional allowance will be worth the same as the penalty for noncompliance if
overall emissions exceed the cap at the end of the market, but be worthless if the cap
turns out not to be binding. Thus, the payoff of an allowance becomes that of a
binary cash-or-nothing option. In Chapter 5, I set up an options pricing model under
the assumption of no abatement that expresses the allowance price purely as a
function of the penalty and the probability of a binding cap, and fit the resulting
options pricing formula to market data.
Chapter 6 draws conclusions on the combined findings in this dissertation.
5
Note that the electricity sector cannot reduce output below demand, otherwise the grid
would crash. The other sectors could in theory reduce output in order to curb emissions, but
this is generally assumed to be a costlier measure than buying permits and/or paying the
penalty for noncompliance.
Chapter 2: The EU ETS
7
Chapter 2: The EU ETS
In the following I describe the main features of the European Union Emissions
Trading Scheme (EU ETS). For a more detailed introduction to the market, see
Kruger & Pizer (2004) and the White Paper by the PEW Institute (2005).
The EU ETS covers CO
2
emissions from 6 broadly defined industry groups in
all countries of the EU. These sectors are power & heat, metals and coke ovens, oil
refineries, glass & ceramics, cement & lime, and paper & pulp. The emissions
included in the market account for almost half of the EU’s total CO
2
output. In the
first phase, about 11,000 individual installations received a total of 2.1 billion EU
allowances (EUAs) annually, mostly at no cost. One EUA gives the bearer a one-
time right to emit one ton of CO
2
.
The market is organized into distinct trading phases. The first phase spanned
the years 2005-2007
6
and was considered a pilot run for the second phase, which
coincides with the Kyoto compliance period of 2008-2012. Pilot phase allowances
could not be banked into the second phase and lost their value if unused for
compliance. Future phases are planned to last five years each, with no banking
restrictions from one phase to the next. On the other hand, borrowing is not allowed
between any two phases. But because firms receive annual allowances in March of
6
The first trade of the EU ETS was made on February 27, 2003, between Shell and NUON (a
Dutch utility) under a forward market.
Chapter 2: The EU ETS
8
every year but don’t have to surrender allowances until the end of April, they can
effectively bank and borrow across time within a trading phase.
Firms can trade allowances freely within the EU. Trades may occur
bilaterally, through brokers (over-the-counter or OTC trades) or on one of six
exchanges.
7
By April 31 of each year, firms have to surrender permits corresponding
to their emissions in the previous calendar year. For every ton of CO
2
emissions for
which firms cannot surrender an allowance, they are fined a penalty of €40 in the first
phase, and of €100 in the second phase. In addition, they have to surrender the
missing allowances in the following year.
Jurisdiction in the EU ETS is divided between the EC and the member states.
The latter are required to submit detailed national allocation plans (NAPs) to the EC
for every phase anew (in other words, the cap changes in every phase). This is a two-
step procedure: First, member states have to decide how much of their overall
emissions reduction burden (as defined by their individual Kyoto commitments) they
want to assign to the EU ETS sectors within their countries, with the remainder of the
burden falling on other sectors such as transportation and households. In a second
step, the allowances have to be distributed among the individual installations. All
NAPs have to be approved by the EC in order to minimize competitive distortions
among similar companies in different member states.
8
7
These are ECX, EEX, EXAA, Climex, Nordpool and Powernext.
8
Although the Trading Directive defines both least-cost achievement of the Kyoto targets and
harmonization between member states as explicit goals, Boehringer and Lange (2005a) show
that both cannot be achieved simultaneously, given the constraint of free permit allocation.
Chapter 2: The EU ETS
9
The scheme is based on Directive 2003/87/EC, which became law on October
25, 2003. This left little time for firms and EU member countries to prepare for the
market. In setting up the first-phase NAPs, countries were faced with the problem that
they had very little information about firms’ historic emissions. Unlike US power
plants that were subject to emissions regulations since at least the mid 1990’s, most
firms in the EU had never had to disclose emissions of other than local pollutants.
The member countries addressed this lack of data by using industry projections
generated by the firms themselves. In addition, most market participants expected
that second-phase NAPs were going to be based on verified 2005 emissions (which
was vehemently opposed by the EC, but which took place nonetheless given that this
was the best data available to the member countries). Using industry projections and
defining second-phase allowance allocations based on first-phase emissions clearly
introduces incentive problems in the sense that firms were encouraged to over-state
their expected emissions (in order to receive more first-phase allowances), and to
under-abate (in order to receive more second-phase allowances).
9
Permit allocations, trades and actual emissions are recorded in national
registries run by each Member State, where all installations that are subject to the EU
ETS have their individual accounts. The Central Administrator of the EU runs a
central registry, called the Community Independent Transaction Log (CITL), which
connects the 27 national registries and checks the recorded transactions for
Thus, there is a tradeoff between efficiency and fairness in terms of a “level playing field”
between similar firms located in different member states.
9
For the effects of updating on firms’ decisions, see Boehringer and Lange (2005b).
Chapter 2: The EU ETS
10
irregularities. It is the duty of member states to establish and/or verify firms’ actual
emissions by multiplying energy inputs with appropriate conversion factors.
The EU ETS is linked to other carbon markets in the sense that certificates
from Kyoto’s Clean Development Mechanism (CDM), called CERs, and from Joint
Implementation (JI), called ERUs, can be surrendered instead of EUAs. Some
countries imposed a limit as to what the percentage of a firm’s emissions can be
covered with such non-EU based emission currencies, but these are still being worked
out. In any case, neither CERs nor ERUs were actually available throughout the first
phase, so for all practical purposes, the first phase of the EU ETS was self-contained.
Figure 2.1 shows allowances and emissions by EU member country for the
year 2006. The largest six countries account for over 70%, both in terms of allocation
and emissions. Allocation and emissions by sector are shown in Figures 2.2 and 2.3.
The power & heat sector received nearly 70% of the total allocation. At the same
time, this was the only sector with a net shortage of allowances, with all other sectors
acting as net allowance suppliers.
10
In terms of installation size, about 90 % of the
covered firms are relatively small (<1 Mt CO
2
/y) and received about 19% of the total
allocation. On the other extreme of the spectrum are the very large emitters (>10
Mt/y), which make up less than one percent of all installations in number but received
more than a third of all allowances. Most of these large emitters are power plants.
10
Note that these are aggregate numbers; individually, there were power stations with an
allowance surplus in 2005 and 2006, and many industrial firms with a shortfall.
Chapter 2: The EU ETS
11
Pre-market expectations of the allowance price were generally very low,
11
and
the steep price increase took many observers by surprise. For over a year, the
allowance price was above €20, and at its peak it reached over €31 in April 2006.
The April price crash was triggered by the first round of emissions verifications,
which revealed that 2005 emissions were 94 MT below the cap.
12
The second round
of emissions verifications in May 2007 again found an allowance surplus, but this no
longer had a significant impact since prices had decreased to a few cents. Liquidity
was overall high, and a significant amount of the total allocation was traded even in
the first year. Table 2.1 shows a market summary of the first phase.
11
In a simulation-based analysis of the EU ETS, Reilly and Paltsev (2005) calculated market-
clearing marginal abatement costs to be € 0.6-0.9 for their base scenario, with prices in even
the most extreme scenarios below €7. Medium price estimates by brokers were somewhat
higher, around of €5.00 for the first phase (PEW, 2005).
12
Emissions verification numbers were planned to be announced in May, but in late April
reports were leaked that Belgium, France, the Czech Republic, the Netherlands and Estonia
all had allowance surpluses, and the allowance shortage in Spain was much smaller than
anticipated. By early May, the market was found to be 63.6 Mt long, with 21 countries
reporting. It is interesting to note that the announcement of the Polish surplus of another 26
Mt in September 2006 did not affect prices very much.
Chapter 3: Bubbles
12
Chapter 3: Price Drivers and CO2 Bubbles in the EU ETS
(Paper submitted to JEEM in July 2008)
Abstract
In the first phase of the EU Emissions Trading Scheme (EU ETS), the price
per ton of CO
2
rose to over €30 before decreasing to zero by mid 2007. I examine to
what extent this variation can be explained by market fundamentals, and whether
there was a price bubble. The presence of the latter would question the main
argument in favor of permit markets, which is to achieve a given emissions cap at
least cost. I derive a structural model of the allowance price under the assumption
efficient markets, which I gradually relax by allowing for delayed adjustment of price
to fundamentals, as well as by introducing lagged LHS variables. The pattern of the
results suggests that a price bubble is at least possible. I then pursue this hypothesis
further by carrying out two different bubbles tests, both of which are consistent with
the presence of a bubble.
Keywords: Emissions permit markets, air pollution, climate change, bubble,
speculation, CO
2
, asset pricing, EU ETS.
JEL classification: D84, G12, G14, Q52-54
Chapter 3: Bubbles
13
3.1. Introduction
The allowance price per ton of carbon dioxide (CO
2
) in first phase of the
European Union Emissions Trading Scheme (EU ETS) exhibited tremendous
variation. It started around €5 but quickly increased to a range of €20-30 where it
remained for over a year. The price crashed after the first round of emissions
verifications that showed the market to be long, but recovered somewhat and
remained around €15 for another few months, before starting a gradual decline. By
mid 2007, an allowance was virtually worthless.
Market analysts and economists alike have been looking for reasons behind
the peculiar allowance price movement. Some have pointed to market fundamentals
such as fuel prices and the weather (Alberola et al., 2008, Bunn and Fezzi, 2007,
Mansanet-Bataller et al., 2007, Rickels et al., 2007) but others found no such
correlation and confined themselves to forecasting based on pure time-series
approaches (Chesney and Taschini, 2008, Paolella and Taschini, 2006). Whereas the
April 2006 crash can be explained by the lower-than-expected overall emission
reports, it is not clear why the allowance price was driven that high in the first place.
13
Also, the fact that it did not collapse completely but remained at a (in hindsight) very
high level through 2006 lacks a satisfactory explanation.
13
In theory, market participants need not know aggregate emissions for the price to be
efficient. If every firm with a permit surplus (deficit) sells (buys) permits on the market, the
price should marginal abatement costs regardless of emissions verifications.
Chapter 3: Bubbles
14
In this paper I examine if and to what extent the allowance price in the first
phase of the EU ETS was based on market fundamentals. I first set up an economic
model that specifies allowance price changes as a function of a set of widely accepted
price drivers (fuel prices, temperature, reservoir levels, economic indicators and
announcements of verified emissions), under the assumption of efficient markets and
using the best data available. I then relax this model by introducing lagged
fundamentals to account for non-immediate adjustment of allowance prices to
changes in fundamentals and to proxy for unobserved expectations about
fundamentals. In a further step I add past price changes as predictors of current price
changes and gage their importance relative to the fundamentals.
I find that only a small portion of the allowance price variation can be
explained by market fundamentals, even when taking into account dynamic
expectations about fundamentals. A situation where an asset price is driven by
expectations about future increases in a manner that is detached from fundamentals is
commonly referred to as a price bubble. All tests to identify bubbles are inherently
plagued by an identification problem, since the researcher can never know whether a
difference between the price and the “true” value of an asset is due to a bubble, or to a
misspecification of the market structure when calculating the intrinsic value (Flood
and Garber, 1980, Garber, 1989, Gurkaynak, 2005). However, a permit market seems
an especially appropriate place to investigate the presence of a bubble because the
asset in question has a clearly defined value: One allowance is worth the cost of
reducing aggregate CO
2
emissions to one ton below the aggregate cap. This is in
Chapter 3: Bubbles
15
contrast to stock prices, where it is often unclear what a stock is really worth and
should lead to a fairly trustworthy estimate of the intrinsic value, thus reducing the
danger of identification error considerably.
The contribution of this section of the dissertation is threefold: First, the
stepwise procedure of starting with economic theory and then relaxing the most
stringent constraints allows insights into the determinants of the EU ETS allowance
price that go beyond existing analyses, which typically do not start from a rigorous
economic model and determine model specification on a mostly ad-hoc basis.
Second, I use a dataset of daily weather measurements in dozens of monitoring
stations across Europe reaching back over three decades, which I combine with
detailed information about regional population density to account for population-
weighed temperature deviations from their long-term expectations. No dataset of
comparable quality has been used in the literature address the influence of weather
shocks on the allowance price. Last, to my knowledge no permit market has ever
been tested for the presence of a price bubble, although such markets appears to offer
more favorable conditions than stock markets to mitigate the identification problem
encountered in any bubbles test.
In the next section I review the literature and derive the market model. In
Section 3.3, I introduce the data and present the estimation results for the proposed
models. Section 3.4 contains two bubbles tests, one based on regime switching and
the other on cointegration between allowance prices and market fundamentals, and
Section 3.5 concludes.
Chapter 3: Bubbles
16
3.2. Allowance market model
3.2.a.) Literature
There is a large volume of empirical work about the SO
2
permit trading
system in the USA (Carlson, 2000, Joskow et al., 1998, Montero, 1999, Schmalensee,
1998, Stavins, 1998) and more recently (Burtraw et al., 2005, Kosobud et al., 2005),
to name only a few. For the EU ETS, the empirical literature is scarcer because it is a
much newer market, and because fewer data are available in general since the
involved firms previously faced very little regulation.
14
There exists a number of
studies that model the EUA price and its volatility mainly for risk management and
forecasting purposes (Benz and Trueck, 2006, Chesney and Taschini, 2008, Fehr and
Hinz, 2006, Seifert et al., 2008). While useful for companies that need to hedge
against the risk embedded in carbon prices, they do not shed much light on
fundamental price drivers.
I am aware of four papers that explicitly aim to determine the impact of
market fundamentals: Bunn and Fezzi (2007) use a cointegrated VAR model with
allowances, electricity and gas in the UK and daily temperature in London as an
exogenous variable, and impose the necessary identifying restrictions using auxiliary
14
In the USA, historic emissions and information about production, fuel use and abatement
are readily available, which is not the case in the EU.
Chapter 3: Bubbles
17
regressions. They find that the gas price influences the EUA price, and that both gas
and EUA prices help determine the electricity price.
Mansanet-Batallet et al. (2007) focused on EU-wide fuel prices and a weather
index comprised of several cities. They focus on the first year of the market only and
include dummies for the six largest price changes, which end up accounting for most
of the explanatory power of their model.
15
This sidesteps the question of what
actually drives the allowance price. In addition, they include regressors that are not
obviously related to allowance prices, such as the Brent oil price.
16
They find that oil
prices, natural gas prices and temperature in Germany are the only significant
allowance price drivers, whereas other determinants (such as the EU weather index
and coal prices) turned out to be uncorrelated.
Alberola et al. (2008) use temperatures in capital cities of six EU countries,
along with a number of EU-wide energy variables, and extent the analysis to the first
two years of the market. Unfortunately, they treat highly endogenous variables such
as electricity prices, clean dark and clean spark spreads
17
as exogenous determinants
15
It is not clear what the explanatory power of the fundamentals themselves is since no
estimation results are presented for a model without dummies.
16
The explanation given for including oil prices is not very clear. They cite a study by
Christiansen et al. (2005) which looked at very general determinants for greenhouse gas
markets, but is not specific to fuel switching in the EU’s power sector. Very little power is
generated using oil in Europe, so a switch from oil to gas is not likely to be the marginal
abatement activity.
17
The dark spreads is the theoretical gross profit of a power plant to generate a unit of
electricity using coal, having bought the fuel necessary to produce it: Dark spread = power
price-fuel price*heat rate. The heat rate is the efficiency at which a power plant converges
energy in fuel into electric output. The clean dark spread is the dark spread minus CO
2
costs
embedded in producing a unit of electricity: Clean dark spread=dark spread-CO
2
Chapter 3: Bubbles
18
of the allowance price.
18
This is no problem for forecasting, but endogeneity will lead
to biased coefficient estimates of the price drivers.
Rickels et al. (2007) build on Mansanet-Bataller et al. (2007) but include data
through 2006. They separate allowance price determinants into supply and demand
side, and their choice of market fundamentals seems more appropriate than that of
Alberola et al. (2008) and more complete than that of Bunn & Fezzi (2007).
However, their econometric specification is questionable: Although they check for
cointegration between allowance and fuel prices (they find none) and thus implicitly
acknowledge the presence of unit roots in the price data, they specify their model in
levels as opposed to differences (or returns) to render the data stationary. A
nonstationary error may lead to untrustworthy coefficient estimates.
All four studies are valuable contributions to finding allowance price
determinants, but neither is based on a rigorous economic market model. The
inclusion/exclusion of market fundamentals as well as the econometric specification
is mostly ad-hoc, which leads to the aforementioned problems. They also do not take
into account the no-banking provision from the first into the second phase, and they
include lagged market fundamentals as well as lagged EUA prices as allowance price
fundamentals from the outset without discussing the economic meaning of this. If
yesterday’s price change determines today’s, then what determines yesterday’s price
price*emission intensity. The spark spread and clean spark spread are analogous measures
for gas-generated electricity.
18
Although the electricity price, and thus spreads, is correlated with the EUA, the causation is
very likely the other way around, because electricity producers pass carbon costs (at least
partly) through to consumers.
Chapter 3: Bubbles
19
change, and what are the true determinants of allowance prices? In the following I set
up a simple economic model of the allowance price that explicitly addresses these
issues.
3.2.b.) Base model
In order to incorporate the uncertainty inherent in the demand and supply of
allowances as well as the fixed time horizon of the first market phase I follow an
approach by Maeda (2004), who analyzed the effect of uncertainty in business-as-
usual (BAU) emissions (referring to emissions in the absence of a carbon cost). I
extend his model to T periods, where each period represents one day and T
corresponds to the number of business days during the first phase of the EU ETS. Let
BAU
it
represent firm i’s random BAU emissions in period t, which depend on a
vector of normally distributed risk factors ? shared by all N firms in the market:
(3.1)
( )
j i E E
Var
BAU Cov
E BAU E BAU
jt it it t
t
t it
it
it t t t it t it t t it
? = = ?
?
?
=
+ ? ? ? + ? = ?
? ?
, 0 ] [ ] [ ;
) (
) , (
] [ )] ( [ ) (
1 1
? ? ? ?
? ?
Firm i’s BAU emissions in the current period are the sum of expected
emissions and an adjustment term that is proportional to a shock in ?
t
which contains
exogenous variables that influence either demand or supply of emissions. Abatement
is defined as the difference between firm i’s BAU and actual emissions e:
Chapter 3: Bubbles
20
(3.2)
it t it it
e BAU a ? ? = ) (
Abatement has a cost defined by a firm’s abatement cost function or its
derivative, the marginal abatement cost (MAC) function. As is well known from
permit market theory, each firm chooses abatement such that its MAC is equal to the
permit price in every period, which implicitly defines the optimal amount of
abatement:
)) ( , , ( )) ( , , (
1 * *
t it t t it it t it t it it t
BAU X MAC a BAU X a MAC ? = ? ? =
?
? ? ,
where X
t
refers to a vector of variables that determines the MAC function. To clear
the market, aggregate abatement has to equal the difference between overall BAU
emissions and the emissions cap S:
(3.3)
?? ??
= = = =
? =
T
k
N
i
ik
T
k
N
i
ik
S BAU a
1 1 1 1
*
Because firms involved in the production of power & heat are dominant
within the EU ETS, it makes sense to focus on emissions and abatement in this sector.
I will further assume that the predominant method of abatement is a (marginal) shift
in the generation dispatch order away from coal towards gas,
as the former is more
than twice as emissions-intensive per unit of output than the latter.
19
Fuel switching is
19
This shift will take place in the medium load spectrum, as peak load is already generated
using gas (and hydro) and base load is generated using nuclear, lignite and coal. Most likely
Chapter 3: Bubbles
21
generally considered to be important in the EU ETS (Alberola, et al., 2008,
Christiansen, et al., 2005, Delarue and D'haeseleer, 2007, Fehr and Hinz, 2006).
This
means that in addition to BAU emissions, abatement costs in the EU ETS depend on
gas and coal prices, which I will denominate as
t
G and
t
C . If aggregate marginal
abatement costs are approximately linear over the range where fuel switching is
feasible,
20
the market’s abatement cost function can be written as
21
(3.4)
? ? ? ?
= = = =
+ + + + =
N
i
it t t
N
i
it
N
i
it t t
N
i
it t
BAU g C d G d a b c BAU C G a MAC
1
2 1
1 1 1
) , , , (
The coefficients c>0 and b>0 indicate nonzero and increasing marginal
abatement costs, respectively. Aggregate MAC increases with gas prices and
decreases with coal prices, such that d
1
> 0 and d
2
< 0. Increased BAU emissions
translate in more necessary abatement to achieve the fixed cap S, which means that
0 > g .
In equilibrium, allowance demand must equal supply and the aggregate MAC
has to equal the permit price. This allows me to solve for the optimal aggregate
abatement:
it would entail the replacement of some very inefficient coal generators by combined cycle
gas generators (CCGTs).
20
In reality, the MAC functions of individual firms are step functions. However, aggregate
MACs on a sectoral level will be almost continuous over a certain range due to the range in
different generator efficiencies.
21
I also tried a specification where prices enter as logs, as is usually done in the finance
literature. It is not obvious in this case which specification is more appropriate. In any case,
the final results turned out to be very similar.
Chapter 3: Bubbles
22
(3.5)
b
BAU g dF c
b
a
N
i
it t
t
N
i
it
?
?
=
=
+ +
? =
1
1
*
?
where I set
t t t
C d G d dF
2 1
+ ? for notational convenience. Substituting (3.5) into
(3.3) yields
(3.6)
? ? ? ? ? ?
= = = = = =
? = ? ? ?
T
k
ik
N
i
T
k
ik
N
i
T
k
k
T
k
k
S BAU BAU
b
g
F
b
d
b
cT
b
1 1 1 1 1 1
1
?
I now take expectations at time t, subtract them from (3.6) and simplify:
( ) ( ) ( ) ( )
? ? ? ?
= + =
?
+ =
?
+ =
? + + ? = ?
N
i
T
t k
ik k ik
T
t k
k k k
T
t k
k k
BAU E BAU b g F E F d E
1 1
1
1
1
1
] [ ] [ ] [? ?
Entries for periods before t cancel out because their ex-post expectation is the
same as their realization. Likewise, the terms b cT / and S do not vary over time and
cancel. Substituting (3.1) and dividing by N yields
( ) ( ) ( )
? ? ? ? ? ?
= + = = + =
?
+ =
?
+ =
?
+
+ ? ? ?
+
+ ? = ?
N
i
T
t k
it
N
i
T
t k
k k k it
T
t k
k k k
T
t k
k k k
N
b g
E
N
b g
F E F
N
d
E
N
1 1 1 1
1
1
1
1
1
] [ ] [ ] [
1
? ? ? ?
Provided that the error is stationary, the last term’s mean and variance go to
zero as N goes to infinity. The intuition behind this is that uncorrelated, firm-specific
shocks cancel each other out in a large market, i.e. only shocks that affect all firms
simultaneously have an impact on BAU emissions (and thus on marginal abatement
costs). Simplifying the notation and solving for allowance prices results in
Chapter 3: Bubbles
23
(3.7) ( ) ( ) ( )
? ? ? ?
+ =
?
+ =
?
+ =
?
+ =
? ? ? + + ? + =
T
t k
k k k
T
t k
k k k
T
t k
k k
T
t k
k
E b g N F E F d E
1
1
1
1
1
1
1
] [ * ] [ ] [ ? ? ?
where ? is the average covariance between aggregate BAU emissions and ?
t
under
the assumption that this relationship is time-invariant, i.e. t
N
N
i
it t
? = =
?
=1
1
? ? ? .
If markets are efficient, prices incorporate changes in underlying
fundamentals immediately (Malkiel, 2007), implying that
t t t t
P P r P E ? ? + =
+
) 1 ( ] [
1
,
where r is the interest rate.
Equation (3.7) can be solved recursively (see Appendix
A) to
(3.8) ( )
( )
? ?
?
?
?
?
?
? ? ?
+ +
?
+ =
T
t
k T
t k t
T
t
k T
t t
t t
E
b g N
F F
d
?
?
?
?
?? ?
] [
1 1
1
The allowance price is determined by the previous-day price, changes in fuel
prices and shocks to
t
? . The summation term in the denominators of the RHS
decreases through time and indicates that shocks to exogenous variables increasingly
affect the permit price. This makes intuitive sense: In the beginning of the market, a
shock to emissions should not influence the permit price much, as it will likely be
neutralized by shock in the opposite direction later on. As time progresses this
probability diminishes. This means that in theory, fluctuations in the allowance price
should increase towards the end of the market. In practice, there is also an opposite
effect: New markets typically show more volatility than mature ones because market
Chapter 3: Bubbles
24
participants learn. Combined with the (in hindsight) apparent allowance surplus
which drove the price down to transaction costs by mid 2007, it seems likely that this
effect overshadows the inherent increasing uncertainty in a time-limited permit
market without banking.
In theory, the discount rate in equation (3.8) could be estimated directly using
nonlinear tools. However, in practice the day-to-day discount rate is very close to
zero. I therefore simplify the equation (3.8) to
(3.9) ( )
( )
? ?
?
?
?
? ? ?
+ +
?
= ?
T
t
k T
t t t
T
t
k T
t
t
E
b g N
F
d
?
?
?
?
] [
1
where ? refers to the first-difference operator. To keep the estimation linear I use an
annual discount rate of 10% to calculate the denominators on the RHS.
22
I assume that consumer demand is inelastic in the short term. Because
demand must meet supply at all times in the electricity grid,
t
? includes factors that
determine either demand or supply of BAU emissions. Specifically, I will include
temperatures across Europe, reservoir levels in the Nordic countries, the DAX and a
dummy indicating the first round of emissions verifications. The assumptions behind
this choice are the following: Temperatures affect consumer demand through
increased changes in heating (winter) or cooling (summer); reservoir levels influences
22
The choice of discount rate is more important for the RHS because small changes can cause
significant differences in the numerical value for the summation term. However, using
discount rates of 5% and 20% did not significantly change the results.
Chapter 3: Bubbles
25
emissions on the supply side through the availability of renewable energy;
23
and the
DAX is a proxy for overall economic performance in the EU’s largest economy. This
leads to the following econometric specification:
t t t t t
t t
t
t
t
t t
t
t t
t t
t
t
t
F
t
t
u v v N
D
DX R E R T E T
S W
C G
+ + =
+ +
?
?
+
?
?
+
?
?
+ +
?
?
+
?
?
= ?
?
2
1 1 0
2 2
6 5 4 3 2 1
); , 0 ( ~
] [ ] [
) (
? ? ? ?
? ? ? ? ? ? ? ? ?
(3.10)
noise white
otherwise 0 , 06 / 28 / 4 06 / 25 / 4 on 1 : dummy ver. emissions
DAX
levels reservoir nordic
otherwise 0 , Sep to Jun 1 : dummy summer
otherwise 0 , Mar to Nov 1 : dummy winter
e temperatur daily average
Europe rn Northweste for marker coal
gas natural UK for price forward month one
t
t t t
t
t
t t t
t t t
t
t
F
t
T
t
k T
t
u
D D D
DX
R
S S S
W W W
T
C
G
= ? =
= =
= =
?
?
?
?
?
Although the error ?
t
is uncorrelated over time, I allow its conditional
variance to change using an ARCH(1) specification, which is standard procedure in
the analysis of price series as price changes tend not to follow a normal distribution
with a fixed variance.
24
23
The more hydro and wind power available, the less power has to be produced using fossil
fuels, and thus the lower are BAU emissions. See also Christiansen (2005).
24
The alternative would be to drop the assumptions of Gaussian errors altogether, which has
been forcefully advocated by Mandelbrot (1997, 2004). In my regime-switching bubbles test
(see below) I use a t-distribution.
Chapter 3: Bubbles
26
In order to define shocks to reservoir levels and the weather, I need a measure
of what “normal” levels are. For reservoir levels, I use weekly median levels based
on the years 1991-2006. Because reservoir levels are cumulative by nature, a level of
one TWh below the median level today will lead to an expectation of tomorrow’s
level also to be one TWh too low, assuming that precipitation is “normal”. To
represent an unanticipated shock in reservoir levels I form first differences, such that
) ( ] [
1
med
t t t t t t
R R R R E R ? ? = ? ? ?
?
)
.
For temperature, I construct daily expectations using 30-y means, i.e.
?
=
?
=
2004
1975
1
* 30 / 1 ] [
y
dy t t
T T E , where d refers to the calendar day corresponding to day t
and y to years. Because traders are likely to take weather forecasts into account and
the weather over the weekends should influence Monday trades I calculate 5-day
moving averages of temperature minus its expectation centered on the current day:
( ) 5 / ] [ ] [
2
2
5 5
1 ?
+
? =
?
? = ? ?
t
t k
k k
d
t
d
t t t
T E T T T E T
)
. An alternative would be to use first
differences for reasons analogous to those discussed for reservoir levels, but the
problem with this approach is that smoothing combined with differencing leads to
very low variation, possibly diminishing any real signal below noise.
Comparing (3.9) to (3.10) shows that the latter is a reduced form of the former
because the parameters b , g and ? are not individually identified, only their
combined impact. While specification (3.10) is well grounded in economic theory, it
is based on two rather strong assumptions. First, expectations of tomorrow’s
Chapter 3: Bubbles
27
fundamental prices are today’s fundamental prices. Second, allowance price changes
are not autocorrelated over time. In the following two subsections I relax these
assumptions.
3.2.c.) Introducing dynamic expectations of fundamental prices
So far I have assumed that EUA prices have the Markov property in the sense
that tomorrow’s price is a function only of today’s price, but not of the preceding
price path. The Markov property is the centerpiece of asset pricing for stocks and
derivatives, and essentially implies that there are no arbitrage opportunities.
25
If all
traders have rational expectations and access to the same information, a belief (for
whatever reason) that a price will reach a certain level in the future will push the price
to that level today. The Markov property also implies that without storage costs, spot
and futures prices are equal, and that spot and forward prices differ exactly by the rate
of interest (Hull, 2002).
In reality, however, the relationship between spot and futures prices can be
quite different, even when taking storage costs into account. Whatever the reason
(asymmetric information, risk aversion, fixed contracts or bounded rationality), it is
possible that traders form their expectations about prices for EUA fundamentals not
only based on today’s prices, but also on past prices and a combination of spot and
futures prices.
25
If prices were a function of the past price path, chartist traders could use this information to
their advantage. However, there is little evidence that they are in fact able to do so.
Chapter 3: Bubbles
28
The problem is that it is impossible to know a priori which variables that
model traders’ unobserved expectations best. In order to search for the combination
of variables that best explains EUA price changes I set up the following specification:
(3.11)
t
t
q
t
p
t
o
t t
t
n
t t
t
m
t
l
t
k
t
j
t
i
t t
R R T
S W
T
S W
C C G G G
D
? ? ? ? ? ? ?
? ? ? ? ? ? ?
+
?
+
?
+
?
+ +
?
+ +
?
+
?
+
?
+
?
+
?
+ = ?
) ) ) )
11 10 9 8 7 6
5 4 3 2 1
) ( ) (
The indices (i, …, q) refer to a draw from a set of candidate variables.
Specifically,
( )
( )
( )
( )
20 5
,
5 ,
20 5 1
,
5 2 1 5 2 1
, ,
, ,
,
, , ,
, , , , , , ,
? ?
? ? ?
? ? ? ? ? ?
? ? ? ?
?
? ? ? ? ?
? ? ? ? ? ? ? ? ?
t t t t t
q p
M
t
d
t
o n
t t t t t t
m l
S
t
S
t
S
t
S
t
S
t
F
t
F
t
F
t
F
t
F
t
k j i
R R R R R R
T T T
C C C C C C C
G G G G G G G G G G G
) ) ) ) ) )
) ) )
where
S
t
G refers to the spot (day-ahead, to be exact) price for UK natural gas,
M
t
T is
the monthly deviation of temperature from the expected value and all other variables
are defined as above. I estimate (3.11) for each possible combination of the indices
(i, …, q) using an ARCH(1) model and choose the specification that yields the lowest
BIC.
26
The best-fitting specifications for the full, pre-crash and post crash periods are
the following:
26
Bayesian Information Criterion, also known as Schwartz’ Information Criterion. This
criterion trades off model fit and model parsimony and puts more weight on the latter than
Akaike’s Information Criterion. This procedure results in estimating 8*8*8*4*4*2*2*3*3=
294,912 ARCH regressions, and it took the 24 computers in AREC-UMD’s experimental lab
Chapter 3: Bubbles
29
20 5 20
20 1 2 1
20 5 20 2
, ; ; ; , :
; ; ; , , :
, ; ; ; , :
? ? ?
? ? ? ?
? ? ? ?
? = ? = = = ? = = ? = = ? = ?
? = = = = ? = = ? = ? = ? = ?
? = ? = = = ? = = ? = = ? =
t t t t
M
t t t
S
t
F
t
t t
M
t t
F
t
F
t
F
t
t t t t
M
t t t
F
t
F
t
R R q R R p T o n C C m l G k j G i crash Post
R R q p T o n C m l G k G j G i crash Pre
R R q R R p T o n C C m l G k j G i period Full
3.2.d.) Introducing lagged EUA price changes
Although the variance in specifications (3.10) and (3.11) is allowed to vary
over time due to the ARCH(1) term, the error itself is still supposed to be
uncorrelated over time. If the residuals are autocorrelated, it is common to either
specify an ARMA (p, q) error or to introduce lagged LHS variables on the RHS of the
equation.
27
Inclusion of lagged LHS variables can reduce or eliminate autocorrelation of
the residuals and increase the overall fit of the model. This is the reason why most
analyses of time series include either lagged prices or AR terms in the error.
However, this comes at a price: Because of different possible causes for
autocorrelation, the interpretation of the regression coefficients can become difficult.
over two days to complete this task. Including either more variables in (3.11) or widening the
candidate sets would be very challenging in terms of raw computing power (in May 2008,
that is). Note that if either of (i, j, k), (l, m), (n, o) and (p, q) draw the same variable from
their respective candidate set, one of them is dropped due to multicollinearity, which means
that the number of regressors included in (3.11) ranges from a minimum of 6 (5 plus a
dummy) to a maximum of 12.
27
Without any exogenous variables, AR (p) in the error term and p lagged LHS variables are
equivalent specifications, but this changes in the presence of exogenous variables (Bauwens
et al., 1999, p. 144). Also note that including lagged LHS variables and MA(q) error terms at
the same time will lead to biased estimation, because the regressors are no longer independent
of the error term.
Chapter 3: Bubbles
30
This is not a problem if the main goal is price forecasting, but what is the meaning of
a lagged LHS variable in a structural equation that seeks to define price determinants?
This question is routinely ignored, but in this context this would be inappropriate.
Autocorrelation in the residuals from estimating (3.11) could in principle be
caused by three different reasons: First, there could be an omitted fundamental
variable that is related to the allowance price, and which exhibits autocorrelation. In
this case, lagged LHS variables would serve as instruments for the omitted exogenous
variable. Second, expectations about future fundamental prices could not be captured
adequately by the additional terms in (3.11), and the true expectations exhibit
autocorrelation. Third, allowance prices could increase simply because they are
expected to do so based on past increases, regardless of fundamentals. This would be
the case of a price bubble. The coexistence of these three possibilities has made
conclusive testing for the existence of a price bubbles an almost impossible task,
especially in the absence of clearly defined market fundamentals. I will examine this
issue further in section 3.4.
Because the residuals from (3.11) indeed exhibit autocorrelation (see below), I
include five lags of allowance price changes in (3.11) but leave the equation
otherwise unchanged. I discuss the economic meaning of including lagged LHS
variables in a regression below.
Chapter 3: Bubbles
31
3.3. Results
The type and provenance of my data is as follows:
EUA prices: Daily series of over-the-counter (OTC) prices, Point Carbon.
28
Gas prices: Daily series of ICE month-ahead futures and Zeebrugge day-ahead prices
for UK natural gas.
Coal prices: McCloskey coal marker for North-Western Europe, which incorporates
information on all trades concerning coal that enters Europe from abroad and which
reach maturity within 3 months. It is an average of actual transactions or, in their
absence, an assessment of fair value by traders. This marker is published weekly.
Temperature: From the European Climate Assessment & Dataset
29
, which has daily
entries from a large number of monitoring locations across Europe. I weighted
temperature deviations by population around each monitoring location, using a World
Bank city area population dataset.
Nordic reservoir levels: Weekly reservoir levels (in TWh) and median levels based on
1991-2006 taken from Nordpool exchange. The Nordpool market (Norway, Sweden
and Finland) is the main hydropower-producing region in Europe.
28
Available at www.pointcarbon.com, last accessed in February 2008.
29
Klein Tank et al., “Daily Dataset of 20
th
-century surface air temperature and precipitation
series for the European Climate Assessment”, 2007, available at http://eca.knmi.nl.
Chapter 3: Bubbles
32
I estimate equation (3.10) separately for the January 2005 to June 2007,
30
as
well as for the period before the price crash induced by the first round of emissions
verifications in April 2006 (“pre-crash”) and after (“post-crash”). Visual inspection
of the price graph as well as previous analyses (Alberola, et al., 2008, Bunn and
Fezzi, 2007) indicate that the relationship between market fundamentals and the
allowance price likely changed after the price adjustment. Because the DAX lacked
correlation in this as well as all following models, and its inclusion increased both
Akaike’s and Schwarz’ information criteria, I removed it from all regressions and re-
estimated them without it.
The results are presented in Table 3.1. The coefficient estimates are
significantly different before and after the price crash based on an LR test, confirming
the suspected structural break. Gas prices are positive and significant for all periods,
and summer temperatures are significant in the two subperiods. The crash dummy
and the ARCH terms (not shown) are highly significant, as expected, but none of the
other variables appears to be associated with EUA price changes. The residuals from
all three regressions exhibit autocorrelation.
The goodness of fit of the model (calculated as the model sum of squares
divided by the total sum of squares) shows that the overwhelming part of the model’s
predictive power is due to the emissions verification dummy. Less than 4 % of the
30
As can be seen in Fig. 2. 2, prices reached transaction costs of a few cents by mid 2007,
which makes the inclusion of the second half of 2007 pointless.
Chapter 3: Bubbles
33
variation is explained in the pre-crash model, whereas the fit in the post-crash period
is even worse with 1%.
Results from estimating (3.11) are shown in the first two columns of Table
3.2. Once again, the coefficients before and after the price crash are significantly
different, confirming the structural break. Autocorrelation persists in the residuals
from (3.11), although to a lesser degree than for (3.10). Due to the presence of
various “flavors” of the same type of variable, the coefficients and associated p-
values of the individual lags lose their straightforward interpretation. For example,
the two coefficients on the reservoir level variables are of opposite sign but similar
magnitude for the full and the post-crash period. In contrast, gas prices are
consistently positive except for the coefficient on spot prices in the post-crash period.
Winter temperatures now have the expected sign in all periods (and are significant in
the full and post-crash period), whereas summer temperatures are positive for the full
and pre-crash period but negative and for the post-crash period. In any case, the main
focus here is on model fit, which has improved in terms of BIC by construction, but
not by very much, in spite of the serious data-mining exercise to find the “best”
specification:
31
The model predicts about 11% of the variation in the pre-crash
period, and less 3 % for the period after the crash.
31
I want to emphasize that the point of data-mining equation (3.11) for the best fit is to show
that even so, the model does not explain much of the variation in EUA prices, and is highly
sensitive to the inclusion of lagged LHS variables. A good fit from such a procedure would
not prove anything.
Chapter 3: Bubbles
34
Introducing 5 lags of allowance price changes into specification (3.11) yields
the results presented in columns 3 and 4 of Table 3.2. Lagged EUA price changes are
very effective in explaining current price changes in all periods. However, the
inclusion of the LHS lags takes its toll on the explanatory power of the exogenous
variables. Other than gas and the crash dummy, none of the coefficients is
statistically significant. Note that in spite of the lagged price changes, autocorrelation
persists, as well as evidence for a structural break.
Taken together, the results shown in Tables 3.1-3.2 imply the following:
1.) Allowance prices appear to violate the Markov property. Market
fundamentals are not immediately internalized, as past changes in fundamentals help
explain a portion of the price movements. Furthermore, price changes exhibit
autocorrelation.
2.) There was a structural break in the allowance price series after the price
adjustment due to the emissions verifications in April 2006. The coefficients on
market fundamentals from estimating the subperiod before and after the price crash
are significantly different.
3.) UK Gas prices are consistently associated with the allowance price before
the price adjustment in April 2006. This is consistent with results obtained by Bunn
& Fezzi (2007) and Alberola et al. (2008).
4.) Coal prices do not appear to be significantly correlated with the EUA
price, even though they should be important if fuel switching is an important form of
Chapter 3: Bubbles
35
abatement. This is probably due to the fact that they exhibit much less variation than
gas prices.
5.) Temperatures and reservoir levels help determine the allowance price, but
they are sensitive to the model specification and time period.
6.) When lagged EUA price changes are introduced into the model, they
absorb most of the price variation while all exogenous variables lose their
significance, with the exception of gas prices. The overall model fit greatly increases,
but some autocorrelation persists.
As discussed in more detail above, persistent autocorrelation and the
dominance of lagged LHS variables in explaining the EUA price variation could be
due to omitted exogenous (autocorrelated) variables, expectations about fundamentals
or a bubble-like phenomenon. However, the pattern of results gives some indications
as to which possibility is more likely.
My model contains all of the variables that are widely considered to drive the
allowance price. In order for an unobserved exogenous variable to drive the results,
this variable would have to be very important, exhibit strong autocorrelation and a
pattern similar to the very distinct price movement of the EUA. The existence of
such an exogenous variable seems unlikely.
I proxied expectations about fundamentals by introducing various time lags of
these variables into the model. Although this method is certainly an imperfect
measure of expectations, even the best-fitting combination of fundamentals out of
Chapter 3: Bubbles
36
almost 300,000 possibilities did not provide a very good model of EUA price
changes. Naturally there are more variables and lags that could be included in the
model, but based on my results so far it seems unlikely that expectations about market
fundamentals are behind the very distinct EUA price path.
The overall poor performance of the models without lagged EUA price
changes, the instability of the exogenous variables across models and time periods,
the fact that all fundamentals except for UK gas prices lose their significance as soon
as lagged prices are introduced and the persistence of autocorrelation across models
are all consistent with self-fulfilling expectations about the allowance price as a main
price driver, as would be the case in a price bubble. In the next section I examine this
issue further and carry out two bubbles tests.
3.4. A CO
2
Bubble?
3.4.a.) Some bubble background
Regardless of whether or not prices have the Markov property, they are
supposed to be driven by market fundamentals (hence the name) according to the
Capital Asset Pricing Model and Modern Portfolio Theory, two workhorses of
modern financial economics. Generally, an asset’s price P
t
can be represented as
t t t t
B F P ? + + = (Diba and Grossman, 1987, 1988, Flood and Garber, 1980), where
t
F represents the market fundamental (or intrinsic asset value),
t
B is a bubble term
Chapter 3: Bubbles
37
and
t
? white noise. Prices that are determined exclusively by
t
F are a special case
where the bubble term is set to zero.
A positive bubble term is due to traders’ expectations. If traders believe that
prices will increase, or believe that most other traders believe this (and so forth ad
infinitum), any random price expectation can be self-fulfilling.
32
After a time of
expectation-driven price increases, market participants will eventually realize that the
price is too high.
33
This will set in motion a positive feedback loop of asset sales,
downward adjustment of future price expectations and yet more sales. Note that in a
bubble, traders know that an asset’s price is above its intrinsic value, but they buy (or
hold) the asset anyway because they expect further price increases.
There exists an ample literature about price bubbles, an illuminating survey of
which is given by Camerer (1989) and, more recently by Abreu & Brunnermeier
(2003). The cornerstone of bubble theory is something called a growing rational
bubble, which refers to a constant term that appears in solutions to difference
equations that describe price formation in a market. Growing rational bubbles
increase exponentially at the rate of interest.
34
Under the standard assumption of
rational expectations, growing rational bubbles cannot exist with a finite number of
32
The assumption that prices are not driven by fundamentals but by traders’ beliefs is the
basis for technical or “chartist” analysis. Chartists aim to predict future price movements
based on past price alone, and have come up with a range of tools and indicators such as price
floors and ceilings, “head-and-shoulders”, turning points and more.
33
This can be triggered by an event such as one large seller coming to the market, a bearish
news report, or a round of emissions verifications as was the case in April 2006.
34
Blanchard and Watson (1982) developed the theory for stochastically crashing bubbles,
where traders know that the bubble will burst but not when. Stochastically crashing bubbles
have to grow at a rate greater than the rate of interest.
Chapter 3: Bubbles
38
agents trading a limited number of assets in a discrete time setting. The reasons for
this were formally developed by Tirole (1982), but to put it simply, it will be
irrational to hold the asset just before the bubble bursts, and therefore by backward
induction it will be irrational to hold the asset in any earlier period.
A widely used approach to solving the bubble existence problem has been to
allow for either some sort of irrationality or for incomplete information (Black, 1986,
Day and Huang, 1990, Frankel and Froot, 1990, Friedman and Aoki, 1992), and
bubbles have been shown to persist in lab experiments even with experienced subjects
(Hussam et al., 2008). In the following I will sidestep the question of theoretical
existence and focus on investigating whether a bubble did exist.
3.4.b.) Bubble tests
Bubbles tests have been largely confined to stocks that pay dividends. In such
a setup, the market fundamental can be shown to be the expected current value of the
dividend stream. Gurkaynak (2005) reviewed a series of bubbles tests, all of which
base their analysis on S&P 500 prices and dividends going back to 1871. Some
appear not to be appropriate to test for the presence of a bubble either on conceptual
and/or econometric grounds,
35
whereas others are more convincing (West, 1987) but
35
For example, variance bounds tests as introduced by Shiller (1981) and LeRoy and Porter
(1981) rely on dividends observed into the infinite future, which clearly cannot be
implemented. However, ways of getting around this problem (such as using the last observed
dividend as the terminal price) void the test of its meaning, as rejection of the null is no
longer linked to the presence of a bubble (Flood et al., 1994).
Chapter 3: Bubbles
39
cannot be implemented for an asset that has no stream of returns such as one-time
allowances. The classes of bubbles tests that can most readily be implemented in the
EUA context involve cointegration and regime switching tests.
Cointegration tests are based on the assumption that any bubble term would
have to increase faster than the underlying fundamentals. If the price and
fundamental series are cointegrated, this is clearly not the case. Therefore, a test for
cointegration between asset price and fundamental amounts to a bubbles test. Such
tests have been proposed by Diba & Grossman (1987, 1988) and by Hamilton and
Whiteman (1985).
Bubbles tests based on cointegration have been attacked on two grounds
(Evans, 1986): Whereas bubble theory predicts that a bubble can never be rendered
stationary no matter how many times it is differenced, this may not be true with real
data and small samples. In reality, prices do not follow a purely exponential increase
because they are influenced by too many other observed and unobserved factors.
Hence, any real price will eventually become (or at least appear) stationary when
differenced a sufficient number of times.
Second, a price that contains a series of stochastically crashing bubbles may
well pass a stationarity test, especially if the magnitude of price increases and
decreases is roughly equal (Evans, 1986, Hall et al., 1999). To solve this problem,
regime switching methods have been developed (Engel and Hamilton, 1990, Hall, et
al., 1999, Kim et al., 2008, Schaller and van Norden, 1997). Such models allow the
Chapter 3: Bubbles
40
data to be in more than one state. The presence of different regimes is then
interpreted as evidence for a series of bubbles.
3.4.c.) Cointegration test
Cointegration between EUA prices and market fundamentals would imply that
price movements can be accounted for by changes in fundamentals, and that therefore
there cannot be a bubble. Note that this is not the same as the estimations described
in section 4, as the presence of cointegration would imply that eqs. (3.10-11) are
misspecified, because they are written entirely in terms of differences but not levels
of the integrated variables.
Unit root tests indicate that the EUA price series, fuel prices, reservoir level
deviations and the DAX each are integrated of order 1, whereas temperature
deviations are stationary. The five integrated variables are plotted in Figure 3.1.
To test for cointegration, I start by re-specifying (3.10) as an Autoregressive
Distributed Lag (ADL) model while dropping the denominator terms on the RHS:
(3.12)
t t t t
t t t t t
D TS L C TW L C
DX L B R L B C L B G L B m L A
? ?
?
+ + + +
+ + + + =
) ( ) (
) ( ) ( ) ( ) ( ) (
2 1
4 3 2 1
) (L A is a lag polynomial defined by
p t p t t t
L A
? ? ?
? ? ? ? = ? ? ? ? ? ? ? ... 1 ) (
2 2 1 1
; ) (L B
i
and ) (L C
j
, ) 4 , 3 , 2 , 1 ( ? i and
Chapter 3: Bubbles
41
) 2 , 1 ( ? j , are lag polynomials of potentially different order each. I choose 5 lags for
all variables in order to incorporate weekly cycles. The cointegration vector
(3.13)
t t t t t t
DX
A
B
R
A
B
C
A
B
G
A
B
A
m
z
) 1 (
) 1 (
) 1 (
) 1 (
) 1 (
) 1 (
) 1 (
) 1 (
) 1 (
4 3 2 1
? ? ? ? ? = ?
is stationary if the EUA price is cointegrated with the fundamentals,
36
with ) 1 ( A and
) 1 (
i
B referring to the sum of all coefficients of the corresponding polynomials. The
variable
t
z measures how much out of equilibrium the cointegrated variables are in
any given period, with 0 =
t
z indicating the long-term equilibrium. If 0 >
t
z ,
t
? is
too high in relation to the other variables and will tend to decrease towards the
equilibrium (provided that 0 ) 1 ( > A ), and vice versa for 0 <
t
z . The more out of
equilibrium the cointegrated variables are, the stronger the forces that push them back
towards it.
Testing for cointegration is equivalent to testing whether
t
z has a unit root.
There are two different ways of obtaining an estimate for
t
z . One is to estimate
(3.12) and compute an estimate
t
z with the parameter estimates. Another is to
simply fit a linear regression of the allowance price on a constant, fuel prices and
reservoir level deviations.
37
I will label he residuals of this regression
t
z
(
. This
36
For a derivation, see Johnston and Di Nardo (1997)
37
This approach relies on the concept of superconsistency introduced by Engle and Granger
(1991). A problem with this could be that superconsistency is strongly based on asymptotic
Chapter 3: Bubbles
42
approach neglects all stationary exogenous variables, but (in theory) adding a
stationary variable should have no impact on a unit root test.
Standard critical values for unit root tests cannot be used in either case,
because I don’t know
t
z but only its prediction. MacKinnon (1991) derived relevant
critical values using Monte Carlo simulations. For a system of 5 potentially
integrated variables and a constant term, the critical values are -4.13 (10%), -4.42
(5%) and -4.96 (1%).
The cointegration test results are presented in Table 3.3. The two approaches
to compute the estimate for z
t
yield different results in terms of the actual values for
t
z and
t
z
(
, but for both series the null of a unit root cannot be rejected by a wide
margin. There appears to exist no cointegrating vector between the EUA and
integrated fundamentals. While no definitive proof, lack of cointegration between
price and fundamentals is certainly consistent with a price bubble.
3.4.d.) Regime-switching test
I carry out a regime switching test outlined by Hamilton (1989) and Engel and
Hamilton (1990), and, among others, applied by van Norden and Schaller (1997,
1996). Kim and Nelson (1999) extended this class of tests to a state-space framework
and a Bayesian analysis, but for the purpose of this paper, the traditional approach
properties, which may not be a practical assumption in finite samples (Johnston and DiNardo,
1997).
Chapter 3: Bubbles
43
suffices. The basic idea of a regime switching test is that not one, but two (or more)
different distributions govern price changes. In the bubbles context, a different
distribution would be expected in the growth and in the bust phase. Regime switching
is especially useful to detect a sequence of stochastically crashing bubbles where the
transition between growth and bust is not known, but it can also be used to detect a
single bubble as long as both the growth and bust phase are included in the data. Let
the variable ) 2 , 1 ( ?
t
S refer to one of two states. Allowance price changes can then
be written as
(3.14) ) 1 , 0 ( ~
2
2
1
1
1
N X X EUA
t t
S
t
S
t t
t
? ? ? ? ? + + = ?
The variables whose influence on the allowance price is different in the two
regimes are collected in the vector
1
t
X , whereas those with a stable impact across
states are represented by
2
t
X . This means that
2
2
2
1
2
? ? ? = = , but that the vector
1
1
S
?
is different for different values of
t
S . Likewise, I allow the variance to vary across
states. The transition between states is governed by a first-order Markov process:
q s s
q s s
p s s
p s s
t t
t t
t t
t t
? = = =
= = =
? = = =
= = =
?
?
?
?
1 ] 0 | 1 Pr[
] 0 | 0 Pr[
1 ] 1 | 0 Pr[
] 1 | 1 Pr[
1
1
1
1
The system can be solved for the parameter vector ) , , , , , , (
2 1
2
2
1
1
1
q p ? ? ? ? ? ? =
by maximum likelihood using numerical methods as shown by (Engel and Hamilton,
Chapter 3: Bubbles
44
1990, Hamilton, 1989, Kim and Nelson, 1999). Under the null hypothesis of no
bubble, the state in period t-1 has no impact on the state in period t, which means that
p q ? = 1 . The likelihood ratio statistic to test the null is
( ) ( ) | | ) ( ~ log log 2
2
r L L LR
R U
? ? =
where q is forced to equal p ? 1 in the restricted (R) model but is left as a free
parameter in the unrestricted (U) model, and r is the number of parameters in the
restricted model, in this case 3 ) ( length ) ( length * 2
2 1
+ + = ? ? r .
I estimate (3.14) based on model (3.10) plus an intercept for the entire period
as well as the pre-crash and post-crash subperiods. I allow all fundamentals to
influence the allowance price differently in the two states with the exception of the
emissions verification dummy. Results for the transition probabilities and the LR
statistic are given in Table 3.4.
38
The null of no state dependence is clearly rejected
for all periods.
These results imply the existence of (at least two) distinct regimes, and that
the sequence of regimes is nonrandom. Like the cointegration test, this is no
conclusive proof for a bubble, but it is consistent with the presence of a one or a
series of stochastically crashing bubbles.
38
Full results available from the author upon request.
Chapter 3: Bubbles
45
3.5. Conclusions
In the first phase of the EU ETS, the allowance price exhibited high volatility
and followed a peculiar path. While the crash in April 2006 was clearly caused by an
adjustment of expectations about aggregate emissions, it is not obvious what drove
the price that high in the first place, and why it took so long to finally decrease to
zero. In this paper I examine if and to what extent the allowance price was
consistently driven by market fundamentals.
I set up a market model that relates the change in the allowance price to
changes in fuel prices, temperature and reservoir level, under the assumption of
efficient markets. I estimate this model for the entire period, as well as for the
subperiods before and after the allowance price crash. I then relax the model by first
allowing for delayed adjustment to fundamentals, and then by including lagged price
changes as predictors for current price changes.
The specification that relies exclusively on contemporaneous and exogenous
price drivers performs quite poorly, in spite of the fact that I’m using the best data
available. The introduction of lagged exogenous variables improves price predictions
for the period before the price crash, but as soon as lagged EUA price changes are
allowed in the model, all explanatory variables lose their significance with the
exception of UK gas prices.
Although lagged LHS variables are routinely used in time series analysis, it is
important to ask what exactly a dependence of price on its own past means if the goal
Chapter 3: Bubbles
46
is an analysis of price drivers. Autocorrelation could be caused by an omitted
autocorrelated exogenous variable, expectations about fundamentals or self-fulfilling
expectations about future price changes that are not related to fundamentals. The last
situation is equivalent to a price bubble.
In a price bubble, firms that held surplus allowances would be reluctant to sell
because they expected future prices to increase. For the same reason, buyers wanted
to buy sooner rather than later, driving prices to whatever level expectations
happened to be. The presence of a price bubble would in effect destroy the prime
advantage of a permit market, which is the achievement of a given emissions cap at
least cost, because the inflated permit price is at least partially passed on to
consumers.
I examine this hypothesis further by carrying out two bubbles test, one based
on cointegration between the EUA price and market fundamentals and the other on
regime switching. Both tests indicate that a price bubble, or a series of bubbles, is
consistent with the data. The positive test results add another layer of evidence to the
bubbles hypothesis, especially since market fundamentals in a permit market are
better known than in the typical context of bubbles tests, which so far have been
almost exclusively been applied to stock markets.
To formulate it the other way around, in order for these results not to indicate
the presence of a price bubble in EU ETS allowances, there must either exist a crucial
but as of yet unrecognized fundamental price driver whose realizations tally with the
peculiar price movement of the EUA, or expectations about fundamentals had to be
Chapter 3: Bubbles
47
extreme enough (and, in hindsight, far away from actual realizations) to account for
this price variation. In the absence of either of these two –in my view unlikely-
scenarios, one would have to conclude that the first phase of the EU ETS indeed was
characterized by one or a series of speculative price bubbles.
Chapter 4: Market Power
48
Chapter 4: Market Power and Windfall Profits in Emission
Permit Markets
Abstract
Although market power in permit markets has been examined in detail
following the seminal work of Hahn (1984), the effect of free allocation on price
manipulation with market power in both output and permit market has not specifically
been addressed. I show that in this case, the threshold for free allocation above which
dominant firms find it profitable to increase the permit price is below their emissions.
In addition to being of general economic interest, this issue is relevant in the context
of the EU ETS, where it appears that power producers profited from a high permit
price. Because power producers were net permit buyers, Hahn’s results imply that
market power in this sector could not have been involved. My results change this
conclusion. Using data from the UK and German power markets, I find that power
generators received free allowances well in excess of the derived threshold.
Keywords: Market power, emissions permit markets, air pollution, EU ETS, CO
2
,
electricity generation, permit allocation, windfall profits, cost pass-through.
JEL classification: H23, L11-13, L94, L98, Q48, Q52-54, Q58
Chapter 4: Market Power
49
4.1. Introduction
During the first eighteen months of the European Union Emissions Trading
Scheme (EU ETS), the allowance price per ton of CO2 was far above ex-ante
expectations. It fell to one-half of its value in April 2006 after the first round of
emissions verifications showed the market to be long and eventually reached zero by
mid 2007, but it is not clear what drove the price so high in the first place. A series of
studies (Alberola, et al., 2008, Bunn and Fezzi, 2007, Mansanet-Bataller, et al., 2007,
Rickels, et al., 2007) has tried to empirically explain the price path by market
fundamentals such as fuel prices and weather variables, but only with limited success
as fundamentals appear to only account for a small fraction of the allowance price
variation. Especially the very high price levels before the April price crash lack a
satisfactory explanation.
An inflated permit price in the sense that it is above marginal abatement costs
of the market as a whole destroys the most powerful argument in favor of instituting
pollution permit markets, which is to achieve a given emissions target at least cost.
The increased costs are due to over-abatement on behalf of any firm that does set its
marginal abatement cost equal to the permit price, and to consumers paying too high
prices for pollution control if permit prices are passed through in the output market.
In this paper, I examine whether price manipulation within the EU?s power &
heat sector could have been a cause of the apparent allowance price inflation. I extend
economic theory by setting up a model that allows for market power in both the
Chapter 4: Market Power
50
output and the permit market and explicitly accounts for a link between these
markets. I derive the conditions under which a dominant firm will exercise its market
power to increase the permit price in order to maximize overall profits in both
markets. Finally, I apply my theory results to data from the EU ETS and show that
these conditions were fulfilled, i.e. provided that such “double” market power existed,
it would have led to an inflation of both permit and output price.
The interplay between permit and output market is at the root of what has
become known as “windfall profits” in the empirical literature. If firms are able to
pass through pollution costs to consumers but receive most (or all) permits allocated
for free, they get reimbursed for costs they never had to incur. Windfall profits have
been identified as an issue in permit markets in general (Bovenberg and Goulder,
2000, Vollebergh et al., 1997), and in particular in the EU ETS (Grubb and Neuhoff,
2006, Hepburn et al., 2006, Neuhoff et al., 2006, Sijm et al., 2006, Smale et al.,
2006). Such profits constitute a wealth transfer from consumers to firms but they do
not impact efficiency directly
39
nor affect the permit price in a competitive market.
This no longer holds under the presence of market power in both the output and
permit market, because a price-setting firm will take windfall profits into account
when making its production and permit purchase decisions.
One of the best-known results about market power in permit markets is Hahn?s
(1984) finding that the permit price is an increasing function of the dominant firm?s
39
Handing out permits for free impacts efficiency through existing distortions such as income
taxes. In theory, the revenue from a tax or selling permits has to be recycled through lower
distortionary taxes to achieve (Bovenberg and Goulder, 1996, Parry, 1995).
Chapter 4: Market Power
51
permit allocation. If this firm is a net buyer of permits, it will exert its power to
decrease the permit price in order to minimize compliance costs, and vice versa.
Other studies have confirmed and extended these findings (Isaac and Holt, 1999,
Liski and Montero, 2005, Maeda, 2003, Westskog, 1996). Hahn?s results imply that a
dominant firm in the power & heat sector could not possibly have used its market
power to increase the allowance price, because this sector was underprovided with
permits and thus, if anything, would have used its power to decrease the price.
Hahn derived his results by focusing exclusively on the permit market while
ignoring any distortions in the output market. However, if a firm has market power in
the permit market, it is likely to also perceive market power in the output market.
40
Misiolek and Elder (1989) introduced exclusionary manipulation whereby the
dominant firm intends to drive competitors out of the output market by manipulating
the permit price, an approach also followed by von der Fehr (1993) and Godby
(2000). A series of lab experiments empirically tested the relevance of combining
market power in the output and permit markets (Cason et al., 2003, Godby, 2002,
Muller and Mestelman, 1998). These studies found that a combination of market
power in both markets increased the dominant firm?s power to manipulate prices and
that the overall effect on industry profits and consumer welfare depended on firms?
relative efficiencies and permit allocation and thus was ambiguous. However, they
did not address whether and how Hahn?s threshold of “neutral” allocation is altered
40
If a permit market covers several industrial sectors, the firm’s market share in the permit
market will be lower than its share in the output market, which implies market power in the
latter market given market power in the former.
Chapter 4: Market Power
52
by the presence of market power in both permit and output market which are
explicitly linked.
The issue of double market power is closely related to the literature pertaining
to “raising rivals? costs” (Hart and Tirole, 1990, Krattenmaker and Salop, 1986a, b,
Ordover et al., 1990, Salop and Scheffman, 1987, 1983). The focus of this literature is
the theory that predatory firms may increase their market share and overall profits by
artificially increasing industry costs, given certain assumptions. This can take many
forms, including the institution of mandatory standards, labeling, advertising etc, all
of which are expected to be less costly on a per-output basis for the dominant firm
than for the price-taking fringe. One particular version of raising rivals? costs is to
over-purchase necessary inputs of production (Salop and Scheffman, 1987), which is
a profitable strategy if the output price increase from this manipulation exceeds the
firm?s average cost increase. Sartzetakis (1997) applied this framework specifically
to emissions permits as a necessary input to production, but he refrained from
examining how free allocation determines the existence and direction of price
manipulation.
41
However, certain aspects of the interplay between market power in the output
and an associated pollution permit market are not well captured by this literature.
First, raising rivals? costs focuses on increasing profits of a dominant firm at the
expense of rivals while decreasing overall industry profits. Profits from jointly
41
Indeed, he mentions that a policy based on such a threshold would require full information
and the “willingness to base permits allocation on efficiency rather than distributional
considerations”.
Chapter 4: Market Power
53
manipulating output and permit prices on the other hand accrue to all firms in the
industry and they come at the expense of consumers and taxpayers. In fact, fringe
firms can free ride on the manipulative actions of the dominant firm as they enjoy
increased profits without incurring the costs of price manipulation. Second, there are
no strong assumptions needed about a dominant firm?s efficiency relative to that of
the fringe in order for price manipulation to be profitable: As I show below, even a
dominant firm that is very inefficient at abating or producing can find it optimal to
increase the permit price, given that it receives a sufficiently generous free allocation.
In the next section I derive a threshold of free allocation beyond which a
dominant firm will find it profitable to increase the permit price. The threshold is a
function of the firm?s market power in both markets as well as its emission intensity
but is always below the full-allocation threshold defined by Hahn. This is my core
result and means that a dominant firm may find it optimal to increase the permit price
even if it is a net buyer of permits. I then apply this finding to the EU ETS and
examine whether firms in the power & heat sector likely received a free allocation in
excess of this threshold, and therefore whether market power in this sector could be a
cause for the high allowance price. Section 4.4 concludes.
Chapter 4: Market Power
54
4.2. Market power in output and permit market
In the following I set up a simple model for an industry sector containing N
firms that is subject to an emissions permit market.
42
I define the cost function for
firm i=1, ..., N as ) , (
i i
i
e q C , a continuous function which depends on output
i
q and
emissions e
i
and is twice differentiable in both arguments. Costs are increasing in
output, decreasing in emissions and convex in both arguments, such that 0 >
i
q
C ,
0 >
i
C , 0 <
i
e
C , 0 >
i
ee
C , 0 <
i
qe
C and ( ) 0
2
> ?
i
qe
i
ee
i
C C C . I assume that firm 1 has
market power in both the output and the permit market.
43
To study the equilibrium, I start by analyzing the behavior of firms i=2, …, N
that comprise the price-taking fringe, before I move on to the dominant firm. The
fringe’s profit maximization problem is
(4.1)
i i
i i i i
i
i i
x e q
x e t s
x x e q C pq
?
? ? ? = ?
. .
) ( ) , ( max
, ,
?
where p is the output price, ? the permit price,
i
x refers to permit purchases and
i
x
is firm i’s free permit allocation. With a binding cap, I can substitute the constraint
into the objective function and arrive at the familiar first-order conditions that
42
This permit market may also include other sectors, but for simplicity I will confine the
analysis to one sector.
43
With a permit market that covers just one sector, assuming market power in one market but
not the other seems arbitrary. If the permit market covers many other sectors as well, then it
is conceivable that a firm has market power in the output market but not the permit market.
The converse, however, would not make economic sense.
Chapter 4: Market Power
55
marginal production costs equal the output price, and marginal abatement costs equal
the permit price. This implicitly defines the fringe’s optimal output, emissions and
permit purchase decisions:
(4.2)
) , (
) , (
) (
) (
* * *
* *
?
?
? p x x e
p q q
C
C p
i i i
i i
i
e
i
q
= =
=
?
? ? =
? =
The dominant firm takes (4.2) into account when maximizing its own profits.
It faces an inverse demand function and a permit market-clearing condition of
(4.3)
( )
) , (
) , ( ) (
2
*
1
2
*
1
?
?
p x x S
p q q P Q P p
N
i
i
N
i
i
?
?
=
=
+ =
+ = =
where S is the overall emissions cap and q
1
and x
1
are the dominant firm’s output and
permit purchase decisions, respectively. This system of equations describes a fixed
point with a mapping of | | ( ) ) , ( ), , ( ) , ( ), , (
1 1 1 1 1 1 1 1
x q x q p x q x q p F ? ? ? . A unique
solution exists if the vector ) , ( ? p belongs to a convex set (which is trivially true for
prices), and ] [? F is upper-semicontinuous and monotone, which is assured by the
continuity and monotonicity of the demand function ) (Q P and the cost functions
) , (
i i
i
e q C .
From equations (4.1)-(4.3) it follows that the output price and the permit price
are both a function of the dominant firm’s output and permit purchase decisions:
Chapter 4: Market Power
56
) , (
) , (
1 1
1 1
x q
x q p p
? ? =
=
The impact of the dominant firm’s output and permit purchase decisions on
the output and permit price can be assessed using comparative static calculations and
is summarized in the following Lemma:
Lemma 1:
The dominant firm’s output and permit purchase decisions will influence
output and permit price jointly such that
0
x
0;
q
0
x
p
0;
q
p
1 1
1 1
> <
> <
?
??
?
??
?
?
?
?
(Proof: Appendix B)
The dominant firm’s profit maximization problem and the resulting first-order
conditions are
(4.4) ) ( ) , ( ) ( ) , ( ) , ( max
1 1 1 1 1 1 1 1
1
1 1 1 1
, ,
1 1 1
e x x q x x e q C q x q p
e x q
? + ? ? ? = ? ? ?
(4.4a) 0 ) ( ) ( ) (
1
1 1
1
1
1
= ? ? ? ? + ?
q
x x C q
q
p
p
q
?
??
?
?
(q
1
> 0)
Chapter 4: Market Power
57
(4.4b) 0 ) ( ) (
1
1 1 1
1
= + ? ? ? ? ?
?
??
?
?
?
x
x x q
x
p
(x
1
> 0)
(4.4c) ? = ? ? ) (
1
e
C
(4.4d) 0 ) ( ; 0 ;
1 1 1 1
= ? ? ? e x e x ? ?
The last first-order condition implies that the constraint may or may not be
binding. To analyze the incentive of the firm to manipulate the permit price in either
direction I combine (4.4b) and (4.4c) to get
(4.5)
*
1
1 1
1
*
1
1
) ( ) ( ) ( q
x
p
x
x x C
e
?
?
?
??
? ? ? + ? = ? ?
where the asterisks indicate that the permit purchases and output are chosen optimally
by the firm according to (4.4). If with a permit price increase the additional revenue
from cost pass-through (the last term on the RHS) outweighs the higher permit
purchase costs (the second term), then the firm?s marginal abatement costs are below
the permit price. This means that it will under-abate -or, equivalently, over-purchase
permits-relative to the situation where it perceives no price-setting power through its
permit purchase decision in either market
44
( 0 / /
1 1
= = x p x ? ? ? ?? ) and thus push up
the permit price. Moreover, if the revenue effect outweighs the compliance cost
effect to the point where 0
1
= ?
e
C , then it will not abate at all and
*
1 1
*
1
x e e
BAU
? = ,
44
Note that it still may perceive market power through its output decision. Equation (4.5)
strictly applies to output and permit price manipulation through the permit purchase pathway.
Chapter 4: Market Power
58
where
BAU
e
1
refers to business-as-usual (BAU) emissions in the absence of a permit
market. Conversely, if compliance costs outweigh increased revenue the firm will
find it optimal to under-purchase permits in order to depress the permit price and
over-abate accordingly and over-abate accordingly. This can be summarized as
(4.6)
?
?
?
?
??
?
?
> ?
= ?
< ?
? ?
<
=
>
1
1
1
1
1
*
1
*
1
1
) (
e
e
e
C
C
C
x
x x q
x
p
Condition (4.6) implies that there is a specific amount of free allocation that
will cause the dominant firm to set its marginal abatement costs equal to the permit
price. Solving (4.6) for this threshold allocation
0
1
x yields
(4.7)
*
1
1
1 *
1
0
1
/
/
q
x
x p
x x
? ??
? ?
? =
This quantity is unambiguously smaller than the firm’s optimal permit
purchases, provided that 0
1
<
qe
C .
45
Note that the firm’s optimal permit purchases and
output are a function of its allocation, such that the threshold in (4.7) is difficult to
compute ex-ante, except for very simple functional forms of the cost function and
permit and electricity demand. However, the threshold can be evaluated relatively
45
If 0
1
=
qe
C , then 0 / /
1 1
= = q x p ? ?? ? ? (see Appendix A), and
*
1
0
1
x x =
Chapter 4: Market Power
59
easily ex-post when making some simplifying assumptions about consumer demand
response (see below).
46
Equations (4.6)-(4.7) lead to the following result:
Result 1:
After the market has been instituted and firms’ allocation, emissions and output
decisions have been observed, we can infer that:
a. If the dominant firm received a free permit allocation equal to
0
1
x , it
acted as a price taker in the permit market in the sense that it set its
marginal abatement costs equal to the permit price.
b. If the dominant firm’s allocation was greater (smaller) than
0
1
x its
marginal abatement costs were below (above) the permit price and it
manipulated the permit price upwards (downwards) by over- (under-)
purchasing permits.
c. The threshold allocation
0
1
x is smaller than the firm’s emissions and
necessarily makes the firm a net buyer of permits.
46
This caveat applies to some extent also to Hahn’s results. Only if the firm’s cost function
is known can the regulator compute its efficient emissions and thus determine
H
x
1
. The
difference is that in my setup, the regulator also needs to know the firm’s degree of market
power and find a closed-form or numerical solution for ) (
1
*
1
x x . I will leave the proof for
the existence and uniqueness of such a solution for future work.
Chapter 4: Market Power
60
Result 1 is the core finding of this paper and states that even if the dominant
firm is a net buyer of permits it can find it in its interest to manipulate the permit price
upwards, provided that its allocation is sufficiently high.
Note that this is a generalization of Hahn?s result, which I will denominate as
1 1
x x
H
= : A dominant firm will only abstain from manipulating the price if it
receives exactly the number of allowances necessary to cover its emissions and
therefore does not trade. To see this, simply set 0 /
1
= x p ? ? in (4.6) or (4.7), thus
eliminating the link between output and permit markets. Also note that if the second
term on the RHS on (4.7) is sufficiently large (i.e. if the impact of the firm’s permit
purchases on output and permit price is sufficiently strong) then 0
0
1
< x . In this case,
even full auctioning would lead the firm to choose a permit price that is greater than
its abatement costs.
On the other hand, if a firm has been observed to emit more (less) than its
initial permit allocation, Hahn's model would imply that the firms marginal abatement
costs are below (above) the market price. My model therefore shows that this
conclusion is premature if output markets are taken into account.
So far I have focused on the effect of permit allocation on the permit price.
However, as is clear from (4.3) and (4.4), the dominant firm?s allocation also has an
impact on the output price. I start by re-writing (4a) as
(4.8)
1
1
*
1
*
1
1
1
) ( ) ( ) (
q
x x q
q
p
C p
q
?
??
?
?
? + ? ? = ?
Chapter 4: Market Power
61
With neither market power nor a permit market there would be the standard
outcome that price equals marginal production cost, i.e.
1
q
C p = . Market power in the
output market increases the output price by the second term on the RHS, which is also
a familiar result. The last term describes the effect of linking a permit market to the
output market. Because 0 /
1
< q ? ?? , this term decreases (increases) the output price
if the firm is a net buyer (seller) of permits. Substituting Hahn’s result of
*
1 1
x x
H
=
would cancel this third term, but it would not remove the output price distortion
introduced by the second term. To see how my generalized threshold
0
1
x performs in
this case, I solve (4.5) for
1
*
1
x x ? and substitute into (4.8) to get
(4.9) ) (
/
/
/
/
) ( ) (
1
1
1 *
1
1 1
1 *
1
1
1
?
? ??
? ??
?
??
? ??
? ?
?
?
? ? + + ? ? = ?
e q
C
x
q
q
q x
x p
q
q
p
C p
By construction, allocating
0
1
x to the dominant firm eliminates the last term,
as in this case the marginal abatement costs are equal to the permit price. The third
term on the RHS is negative and thus decreases output price distortion. However, the
price distortion is not fully removed because it can be shown that
(4.10)
1 1 1
1
/
/
q
p
q x
x p
?
?
?
??
? ??
? ?
> (Proof: Appendix B)
It follows immediately that the output price can be brought to its efficient
level only by allocating less than
0
1
x to the dominant firm, because in this case the
Chapter 4: Market Power
62
last term will be negative. The threshold allocation to the dominant firm that yields
1
q
C p = can be computed using (4.8) and is
(4.11)
*
1
1
1 *
1
00
1
/
/
q
q
q p
x x
? ??
? ?
? =
The fact that
0
1
00
1
x x < can easily be verified by using the inequality in (4.10).
As before, due to the dependence of
*
1
x on
1
x , this threshold can be evaluated ex-
ante only under very simple functional forms. This leads to the following result:
Result 2:
After the market has been instituted and firms’ allocation, emissions and output
decisions have been observed, we can infer that:
a. If the dominant firm received an allocation of
00
1
x , its marginal
production costs were equal to the output price.
b. If the firm received more (less) than
00
1
x , marginal production costs
were greater (smaller) than the output price.
c. The threshold allocation x
1
00
is smaller than
0
1
x .
Chapter 4: Market Power
63
The fact that marginal costs are equal to price in the two markets at two
different levels means that efficiency cannot be restored completely. Either the
permit price is distorted, or the output price, or both:
Result 3:
a. The first-best solution in the sense that both the output and the permit
price are at their competitive levels cannot be achieved by means of
permit allocation alone, because
0
1
00
1
x x < .
b. If the firm received more than
0
1
x (less than
00
1
x ), both output and
permit price were distorted upwards (downwards) relative to
marginal costs. If the firm’s allocation was
0
1 1
00
1
x x x < < , the output
price was above and the permit price below marginal costs.
Results 1-3 imply that under the assumption of market power in both markets,
the amount of free allocation is crucial for price distortion, and that Hahn?s
“neutralizing” allocation prescription will result in an inflation of both output and
permit price. In the following section I will empirically address the relevance of
these findings in the context of the EU ETS.
Chapter 4: Market Power
64
4.3. Application to the EU ETS
4.3.a.) General applicability
Although the EU ETS covers six broad industry sectors, the main players both
in terms of allocation and emissions are firms within the power & heat sector. There
is evidence that this sector was subject to significant windfall profits (Grubb and
Neuhoff, 2006, Hepburn, et al., 2006, Neuhoff, et al., 2006, Sijm, et al., 2006), which
sets the stage for price manipulation as analyzed in the previous section. According to
market observers (e.g. Point Carbon), it was the sustained allowance purchases from
power & heat, combined with a relatively short allowance supply from the other
sectors, that drove the price to the –in hindsight-very high level. There are a number
of very large power producers for which the assumption of some market power seems
at least possible.
On aggregate, firms in the power & heat sector were net demanders of
allowances, whereas the other sectors covered by the EU ETS were over-allocated as
a whole (Figs. 2.2-3). Hahn?s results imply that in this case, price manipulation by
dominant firms within this sector could not have been behind the allowance price
increase, but as I show in section 4.2, this does not hold if firms are able to influence
both the output and the allowance price. According to Result 1, a dominant firm
would have found it optimal to use its market power to increase the allowance price
even if it was a net buyer of allowances, as long as its free allocation exceeded x
1
0
.
Chapter 4: Market Power
65
I will now address the question whether there is any evidence relating firms’
actual allocation to x
1
0
. To do this, it will be convenient to introduce a substitution in
notation. At x
1
= x
1
0
the emissions constraint will be binding such that x
1
= e
1
.
Defining y
1
= x
1
/e
1
to be the proportion of actual emissions that the dominant firm
receives allocated for free, I can re-state (4.7) in terms of the corresponding y
1
0
:
(4.7’)
1
1
1
1
1
1
0
1
1
/
/
1
q
e x
x p
y ? < ? = ?
?
? ??
? ?
The threshold in (4.7’) is exactly equivalent to
0
1
x as defined by (4.7), but
instead of depending on the dominant firm’s output
1
q it now contains its emission
intensity (average emissions per unit of output), denoted
1
? .
The main difficulty to empirical assessment of (4.7) is to determine the effect
of a dominant firm’s permit purchase decisions on the output and the permit price.
To get around this problem, I will use the fact that the numerator on the RHS of (4.7’)
is equivalent to the impact of the permit price on the output price if there is no
demand response:
47
47
Totally differentiating output and permit price and dividing yields
1 1 1 1
1 1 1 1
* / * /
* / * /
dx x dq q
dx x p dq q p
d
dp
? ?? ? ??
? ? ? ?
? +
+
=
Chapter 4: Market Power
66
(4.12)
0 1
1
1
/
/
=
=
dq
d
dp
x
x p
? ? ??
? ?
I will argue that the short-term demand response for electricity by households
is very small, as the most efficient means to reduce demand is to make changes in the
portfolio of household appliances towards more energy-efficient items, which takes
time. Assuming no consumer demand response during the first 18 months of the EU
ETS, I can substitute (4.12) into (4.7’). The effect of the allowance price on the
electricity price ( ? d dp/ ) can be estimated by looking at the electricity spreads
before and after the institution of the permit market relative to the allowance price. I
will do this separately using market data from the UK and Germany, two of the
largest players in the EU ETS.
4.3.b.) The UK power market
Gas-fired power plants are at the margin during medium and peak hours in the
UK and are therefore price setting during these loads (Grubb and Newbery, 2007).
48
Figure 4.1 shows year-ahead spark spreads and clean spark spreads in the UK. The
spark spread is the theoretical gross profit of a gas-fired power plant from selling a
unit of electricity, having bought the fuel necessary to generate it:
48
During base loads, the marginal generator is coal and nuclear for most hours, as they are
generally ranked lower in the merit order than gas plants.
Chapter 4: Market Power
67
(4.13)
) / ( ) / ( ) / ( ) / (
*
e MWh g MWh g MWh Euro
gas
e MWh Euro e MWh Euro
p p spread Spark ? ? =
Here, p refers to the power price as before,
gas
p to the price for natural gas
and ? to the heat rate (or efficiency). The clean spark spread, also called green spark
spread, further adjusts the revenue stream by the CO
2
costs embedded in power
generation:
(4.14)
gas
spread spark spread spark Clean ? ? * ? =
where
gas
? denotes the emission intensity of a typical gas plant. Solving (4.14) for
the electricity price and its derivative with respect to the permit price yields
(4.15)
gas gas gas
d
dp
p spread spark clean p ?
?
? ? ? = ? ? + = *
At first sight, this seems to imply that the extent to which the electricity price
is increased due to CO
2
costs is simply the emissions intensity used to compute the
clean dark spread. However, these spreads are created as benchmarks and don’t
imply that the emission intensity used in their calculation is necessarily the average
emission intensity of the marginal generator in the market. In theory, one could
calculate the clean spark spread for any type of generator.
Chapter 4: Market Power
68
To see how
gas
? relates to ? d dp / it is useful to compare pre-market spark
spreads with post-market clean spark spreads. With full cost pass-through,
49
constant
demand and stable technology, the clean spark spread will be equal to the pre-market
spread if the “correct” emission factor is used in (4.14), i.e. if ? ? d dp
gas
/ = . In
other words, theoretical gross profits
50
of a power producer will not change if the
power price is increased by an amount exactly equivalent to the embedded carbon
cost.
Figure 4.1 shows almost precisely such a market. Before the carbon market,
UK spark spreads fluctuated around a level of £6-8. In January 2004, the year-ahead
spreads immediately incorporated the carbon cost of producing electricity based on
the year-ahead forward price for CO
2
, with the spark spread increasing and the clean
spark spread taking the place of the spark spread. The equalization of pre-market
spark spreads and post-market clean spark spreads implies that the CO
2
costs
embedded in the production of peak electricity are those that correspond to the
allowance price multiplied by ?
g
used to calculate the clean dark spread, which is
0.41 tCO
2
/MWh, the emission intensity of a typical Combined Cycle Gas Turbine
(CCGT) (PointCarbon, 2007). This means that in the UK,
e MWh tCO d dp
g
/ 41 . 0 /
2
= ? ? ? . Naturally, this is an approximation as the clean
spark spread has its own variation over time, and factors other than the allowance
49
Assuming a completely inelastic demand, profit-maximizing firms will pass their costs
fully through to consumers.
50
As discussed above, there may well be windfall profits due to the institution of the market
because of free allocation.
Chapter 4: Market Power
69
price could potentially have affected the spreads. However, the figure and other
evidence (Grubb and Newbery, 2007, Vorspools, 2006) implies that electricity prices
increased roughly by the amount that corresponds to carbon costs incurred by an
average CCGT.
I will now do the following back-of-the envelope calculation: Firms in the UK
power sector on average received 77.1% of their emissions allocated for free.
51
Substituting a value of 0.77 for the UK, setting 41 . 0 = ? d dp and solving (4.7’) for
the emission intensity, this means that it would have been profitable for dominant UK
power firms to use their market power to increase the allowance price (and with it the
electricity price) by over-purchasing allowances if
(4.16) e MWh tCO / 78 . 1 ) 77 . 0 1 ( 41 . 0
2
= ? < ?
In comparison, the emission intensity of an anthracite coal power plant (the
most emission-intensive method of power generation used today in Europe) with a
heat rate of 33.3%,
52
is 1.06 tCO
2
per MWh of output. In other words, even the least
efficient of all power production companies received an allocation in excess of
0
1
y (or
0
1
x ) and would have found it profitable to manipulate the permit price upwards,
provided that it had some market power.
51
Power & heat generators with an allocation of at least 100,000 allowances, based on 2005
numbers. This list is a subset of installations with activity code 1 (combustion) in the
Community Independent Transaction Log (CITL). Many code-1 installations produce process
power & heat only and are much smaller than power plants. For all code 1 installations
combined, the fraction of free allocation is even larger.
52
Power plants with such a low heat rate are most likely not allowed to operate in the UK.
Chapter 4: Market Power
70
Note that this result is quite robust to what cost pass-through rate I substitute
into the numerator of (4.7?). The fact that companies received a free allocation
covering around 77 % of their emissions means that the inequality in (4.16) holds as
long as a firm?s emission intensity is smaller than (1/0.23=) 4.3 times the cost pass-
through rate. Also note that power companies in the UK received the least generous
allowance allocation in the EU ETS.
4.3.c.) The German power market
Large power producers in Germany received an average of 99.5% of their
actual emissions allocated for free, much more than their UK counterparts.
Differences in allocation for firms in the same sector but different countries have
raised discussions about the ability to achieve economic efficiency in a system where
each member country is able to independently define its own National Allocation
Plan (Boehringer and Lange, 2005a).
In Germany, the marginal generator is a coal-fired power plant during most
hours of the year, including the entire base load (Grubb and Newbery, 2007, Sijm, et
al., 2006). German dark spreads and clean dark spreads are presented in Figure 4.2,
along with their average (middle line). The dark spread is equivalent to the spark
spread described above, but instead of gas it is applied to coal generation. Likewise,
the clean dark spread is the dark spread adjusted for the CO
2
emission costs inherent
in coal generation. The heat rate and emission intensity used to calculate dark spreads
Chapter 4: Market Power
71
and clean dark spreads are 35% and 0.96 tCO2/MWh of output, respectively
(PointCarbon, 2007).
Cost pass-through was not immediate as in the UK, but 50% of carbon costs
were passed through after one year, as indicated by the center line reaching the pre-
market level of €10/MWh. By January 2006, the clean dark spread reached the pre-
market dark spread level, implying that by then the carbon costs embedded in coal
generation had been fully passed through to consumers. Again, the movement of
clean dark spreads towards pre-market dark spread levels could include reasons other
than CO2 cost, but the figure implies that some cost pass-through was very likely.
Suppose now that cost pass-through on average was 0.5*
c
? , where
c
? is the
carbon intensity of coal generation used for the calculation of the green dark spread.
Substituting this value along with an allocation of 99.5 % into (4.7’) reveals that
German power firms with market power would have found it profitable to inflate the
allowance price if their emission intensity was below
e MWh tCO / 96 ) 995 . 0 1 ( 96 . 0 * 5 . 0
2
= ? < ?
which is trivially the case for any power company. The choice of pass-through rate
(and the assumption of zero demand response) is even less important here than for the
UK, because almost any positive cost pass-through rate multiplied by (1/0.005=) 200
will lead to a number that exceeds real-world emission intensities.
Chapter 4: Market Power
72
Naturally, these calculations do not show in any way that electricity producers
in the UK or in Germany actually had market power in both the output and permit
market, and used this power to inflate the allowance price. But they do show that
assuming that some firms had market power, 1.) they would have found it profitable
to over-purchase allowances and under-abate emissions in order to inflate the permit
price, because the ensuing increase in the electricity price would have more than
compensated them for increased allowance purchase costs, and 2.) this would have
lead to an electricity price increase relative to a situation with less free allocation.
4.4. Conclusions
There is a large literature about market power in permit markets, but, to my
knowledge, no paper has directly addressed the effect of free allocation on price
manipulation in the presence of market power in both permit market as well as the
linked output market. Besides being of general economic interest, this particular
question is motivated by a very high (in hindsight too high) allowance price during
the first phase of the EU ETS, which reportedly led to large windfall profits
especially for firms in the power & heat sector. These firms received most of their
allowances for free but were able to pass through a large part of the opportunity costs
to consumers. The reason for the apparent price inflation is not clear to date, but the
presence of windfall profits (which are increasing in the permit price) and the history
of imperfect competition in the power & heat sector raises the question whether
Chapter 4: Market Power
73
dominant power producers could have used their market weight in order to increase
the permit price.
According to Hahn?s (1984) well-known results, the answer to this question is
clearly negative, because power & heat is the only sector that was under-allocated
with permits and thus was a net allowance buyer. In Hahn?s framework, any dominant
permit buyer would depress rather than inflate the permit price, and would act
competitively only when given the exact amount of free allocation that covers its
emissions.
In this paper, I show that Hahn?s results no longer hold with market power in
both markets. I derive the threshold of free allocation above which the dominant firm
finds it profitable to under-abate and over-purchase allowances in order to push up
the permit price. This threshold is a function of cost pass-through and firms? average
emission intensity and is always less than a firm?s emissions were it to set its marginal
abatement costs equal to the permit price.
These findings are not subject to stringent assumptions about relative
efficiency in production and/or abatement among firms, as is typically the case in the
raising rivals? costs literature that discusses market manipulation in input and output
markets. Firms do not profit at the expense of their industry rivals but that of
consumers via the increased output price. In fact, the industry fringe profits from
market manipulation on behalf of the dominant firm, as its revenue increases as well.
I apply my theoretical results to the UK and German power market. Using
market evidence, I show that power generators in these countries received an
Chapter 4: Market Power
74
allocation in excess of x
1
0
and would therefore have been interested in increasing the
allowance price, provided they had the ability to do so. In the UK, this result is due to
an almost complete and immediate cost pass-through, whereas in Germany costs were
passed through more slowly and/or less completely. However, because of the very
generous allocation to German power generators, almost any positive pass-through
rate would have made it profitable for them to manipulate the permit price upwards.
Chapter 5: Options Pricing
75
Chapter 5: An Options Pricing Approach to CO
2
Allowances in
the EU ETS
Abstract
The EU ETS was set up very quickly, which could have made it impossible
for firms to adjust their production technology before the end of Phase I. I derive an
allowance pricing formula under the assumption that abatement was infeasible, which
renders the allowance price a function of the penalty for noncompliance and the
probability that the cap turns out to be binding. This is the pricing formula of a
binary option, with the underlying process being CO
2
emissions. The options pricing
formula depends on the mean and variance of future emissions.
I define the processes driving (stochastic) emissions and estimate their
parameters using market data. I then incorporate these parameter estimates into the
options pricing formula and estimate the remaining free parameters. The results
make economic sense, and the model fits the data reasonably well. This implies that
allowance prices may indeed not have been determined by marginal abatement costs
during the first market phase.
Keywords: Emissions permit markets, air pollution, CO
2
, climate change, options
pricing, asset pricing, EU ETS.
JEL classification: G12, G14, G18, Q52, Q53, Q54
Chapter 5: Options Pricing
76
5.1 Introduction
The centerpiece of emissions permit market theory is that firms equate their
marginal abatement costs to the permit price. Intuitively, if a firm finds it cheaper to
abate an additional unit of emissions than what a permit is worth, it will make a profit
from abating and either buy one fewer or sell one more permit on the market.
Likewise, if the firm finds that purchasing a permit on the market is cheaper than to
abate another unit of emissions, it will not abate but use the market to reach
compliance. The efficient solution of this arbitrage game is that all firms abate exactly
to the point where their marginal abatement costs are equal to the permit price.
However, this leaves out two important possibilities: For one, firms don’t
know exactly what their emissions are going to be, even if they are engaging in
abatement, if abatement consists of reducing emissions per unit of (stochastic) output.
And second, abatement may not be feasible, or at least not practical, for the involved
firms in the short run. Most permit markets to date, including the EU ETS, impose a
penalty for noncompliance: For every unit (usually a ton) of emissions for which the
firm cannot surrender a permit, it is fined a penalty; in addition, the missing permits
have to be surrendered in the following year.
In this paper I will address precisely this question: Was the allowance price in
the EU ETS determined predominantly by firms looking with one eye to the penalty
Chapter 5: Options Pricing
77
and with the other on the realized emissions to date and the expected emissions to
come?
I develop an allowance pricing formula using options pricing techniques that
does not incorporate abatement, but instead relies on the penalty and expected overall
emissions levels as price drivers. While eliminating abatement from the problem is
mainly intended to simplify the calculations, one can also argue that this assumption
is quite realistic for the following reasons:
First, the timely construction of cleaner production technology (e.g. more
efficient power plants) was largely infeasible before the end of the first phase. The
market was set up at a very rapid speed, with only a little over year between its legal
conception and the start of the first phase. Some EU member countries did not
finalize their national allocation plans (NAPs) until mid 2005, which created
considerable uncertainty about the total cap and the resulting allowance price.
Uncertainty over the return of irreversible investment delays such investment.
53
Besides, even under complete certainty, many new plants simply take longer than 4
years (the time between the inception of the market and the end of the first phase) to
plan and construct.
In the absence of cost-effective technology that filters CO
2
from exhaust
gases, this leaves essentially only fuel switching as a method of abatement (Alberola,
et al., 2008, Bunn and Fezzi, 2007, Mansanet-Bataller, et al., 2007, Rickels, et al.,
53
For a thorough treatment of investment under uncertainty and the ensuing option value of
waiting, see Dixit and Pindyck (1994).
Chapter 5: Options Pricing
78
2007, Sijm, et al., 2006). However, energy-intensive industries are typically locked
into long-term contracts. It is questionable whether firms were able to adjust these
contracts in time (Chesney and Taschini, 2008), and/or whether they were willing to
do so, considering the large uncertainty about future caps.
Second, firms probably anticipated that their first-phase emissions were going
to be used to guide the distribution of second-phase allowances. The EC vehemently
argued against this and repeatedly promised that this was not going to be the case, but
it happened nevertheless: Most EU member countries based their second-phase NAPs
on verified 2005 emissions. One possible reason for this –from an economic point of
view highly inefficient- choice is the scarcity of information about historic emissions
in the EU, which made it almost impossible for the EU member countries to fight the
temptation to use the information gained from the first round of emissions
verification. Basing future allocation on current emissions creates a disincentive to
abate, because every unit of abatement comes causes not only costs in the current
period (e.g. due to fuel switching) but also a reduction in future free allocation
(Boehringer and Lange, 2005b).
Third, aggregate emissions were below the total cap for each individual year
of the market.
54
An allowance surplus by itself does not automatically mean that there
was no abatement, especially during the first 16 months of the market when the
allowance price was very high and actual emissions had yet to be verified (Ellerman
54
During the first phase as a whole, the cap and total emissions were 6.250 and 6.081 billion
tons, respectively. The allowance surplus of 168 million tons corresponds to about 2.7 % of
the cap.
Chapter 5: Options Pricing
79
and Buchner, 2006). On the other hand, it is precisely the beginning of the market
that was especially constrained in terms of building new infrastructure and adjusting
long-term fuel contracts. And last but not least, I found no consistent correlation
between allowance prices and commonly identified abatement price drivers (see
Chapter 3 of this Dissertation). For all these reasons, the assumption of no abatement
during the first phase may not be far from the truth.
The purchase of an allowance gives the bearer the option to use it for
compliance at the end of the period, or, equivalently, to sell it. However, if the cap
turns out to be not binding, the bearer can retire the allowance. In other words, the
holder has the right, but not the obligation to use the allowance. This makes an
allowance a financial option. Specifically, the payoff function is that of a type of
binary option called a cash-or-nothing call option. There exist well-established
pricing formulae for binary options if the underlying asset follows a lognormal
distribution. I develop a pricing formula based on two normally distributed
underlying processes, namely electricity consumption and precipitation.
The fact that allowances can be viewed as financial options is neither
remarkable in itself nor new. A handful of studies have used financial methods to
either predict allowance prices themselves, or to price options on allowances.
However, to my knowledge, no one has used observed allowance prices and then
used options pricing formulas to back out the underlying parameters.
In Section 5.2 I give some more background and derive an options pricing
formula for EU ETS allowances as a function of past and future emissions in the
Chapter 5: Options Pricing
80
power & heat sector, the cap, the penalty for noncompliance and a set of free
parameters. This pricing formula contains the mean and variance of expected future
emissions between the current day and the end of the market, which I derive in
Section 5.3 as a function of exogenous stochastic processes. Section 5.4 contains
empirical estimates for these underlying processes as well as estimates for the free
parameters in the options pricing formula. Section 5.5 concludes.
5.2. Emission allowance pricing in the absence of abatement
5.2.a.) Literature on permit pricing
Historically, permit pricing formulas were derived by solving an optimal
control problem, originating with Montgomery (1972). This was later extended to the
dynamic case (Leiby and Rubin, 2001), to incorporate banking and borrowing
(Cronshaw and Brown-Kruse, 1996, Rubin, 1996) and to address uncertainty
(Schennach, 2000, Zhao, 2003) and to some extent volatility (Newell et al., 2005).
But it was not until recently that financial methods have been employed to derive
emissions permit prices.
Kosobud et al. (2005) introduced financial tools to the analysis of SO
2
permits
in the US Acid Rain program. Other contributions that approach permit markets and
the efficient price of an allowance from a financial perspective include Benz and
Trueck (2006) and Fehr and Hinz (2006). Seifert et al (2008) explicitly mention the
Chapter 5: Options Pricing
81
option value of a permit when compared with the alternative of irreversible
investment in emissions abatement. Chesney and Taschini (2008) go one step further
and define EU ETS allowances to be financial (as opposed to “real” options) and
derive a pricing formula that comes close to options pricing. However, all of these
approaches start with the definition of underlying pollution processes, and then derive
a market-clearing permit price by method of simulation. This allows for valuable
insights in terms of price volatility and hedging strategies, but it does not address the
question of what actually drives permit prices. My paper intends to fill this gap by
taking the allowance price series in the EU ETS as given and test whether it is
consistent with a model that is based on options pricing techniques, and, in particular,
whether it was driven by the penalty of noncompliance and the probability that the
cap would turn out to be binding.
5.2.b.) Derivation of option pricing formula
Let
t
P be the closing price for an allowance on day t, with the day index
) ,..., 2 , 1 ( T t = starting on January 1, 2005, and ending at t =T on December 31, 2007.
Let
t
g represent CO
2
emissions on day t,
?
=
?
t
k
k
t
g G
1
1
cumulative emissions
since the beginning of the market, and
?
+ =
?
T
t k
k
T
t
g G
1
cumulative future emissions
until the end of the market. It follows from these definitions that at time t,
t
G
1
is
observed and
T
t
G is stochastic. Furthermore, let P be the penalty for noncompliance
Chapter 5: Options Pricing
82
and
0
S be the total emissions cap over the entire market period imposed by the
regulator. Finally, it will also be useful to define
t
t
G S S
1 0
? = to be the “remaining
cap” until the end of the market.
At time T, the price of an allowance is zero if emissions are below the cap, or
equal to the penalty if the cap turned out to be binding. Naturally, if the cap is
already exceeded at time t<T, the probability that the allowance price is equal to the
penalty is one:
(5.1)
0
0 0
?
>
¹
´
¦
=
T
T
T
T
S
S
if
if
P
P
The penalty is the sum of the per-unit penalty and the cost of buying an
additional permit for the second phase, for which I use the forward price of second-
phase allowances
II Phase
t
P :
=
T
P €
II Phase
T
P + 40
At t<T, it is not known with certainty whether the cap will be exceeded,
provided that it has not been exceeded already. The expected price is
(5.2)
] [ ] 0 | [
) ( * ] [ ] 0 | [
T t t T t
S
T
t
T
t t T t t T t
P E S P E
dG G P E S P E
t
= ?
= >
?
?
?
Chapter 5: Options Pricing
83
where ) (
T
t t
G ? denotes the probability density function over cumulative future CO
2
emissions and ] [?
t
E stands for the expectation taken using all information available at
time t. Note that equation (5.2) is very similar to an equation derived by Chesney and
Taschini (2008).
I will assume that daily emissions g
t
are normally distributed.
55
As I will
argue below, emissions are a linear function of underlying normally distributed AR(1)
processes, meaning that the assumption of normally distributed emissions is really an
assumption about normally distributed underlying processes. Because
T
t
G represents
the summation of random events
k
g for t k > , it follows that
T
t
G is normally
distributed as well. I will denote the mean and standard deviation as
t
µ and
t
s ,
respectively. It follows that
) 1 , 0 ( ~ N
s
G
Q
t
t
T
t
t
µ ?
?
has a standard normal distribution. Let ) (? ? and ) (? ? be the probability density
function (pdf) and cumulative probability density function (cdf) of the standard
55
In theory, the choice of a normal distribution makes a truncation at zero necessary since
negative emissions are not defined. But because CO
2
emissions in the EU are many standard
deviations away from zero, the correction implied by the truncation is very small, such that
for the remainder of this paper I will neglect the truncation issue and assume that emissions
are normally distributed. Note that this assumption is similar to that of demographers who
assume that people’s height is normally distributed.
Chapter 5: Options Pricing
84
normal distribution, respectively. I now convert the integral in (5.3) into an integral
over
t
Q :
(5.4)
?
?
?
= >
t t t
s S
t t T t t T t
dQ Q P E S P E
/ ) (
) ( * ] [ ] 0 | [
µ
?
which is equivalent to
(5.5)
|
|
¹
|
\
| ?
? = >
t
t t
T t t T t
s
S
P E S P E
µ
* ] [ ] 0 | [
Arbitrage considerations dictate that the price at time t be equal to the
expected price at T, discounted by the risk-free rate of interest r.
56
The same
reasoning applies to the forward price for Phase II allowances, such that
) (
* ] [
t T r II Phase
t
II Phase
T t
e P P E
?
= . This means that the allowance price is a martingale
defined by
(5.6)
] 40 [ 0 |
* ] 40 [ 0 |
) (
) (
II Phase
t
t T r
t t
t
t t II Phase
t
t T r
t t
P e S P
s
S
P e S P
+ = ?
|
|
¹
|
\
| ?
? + = >
? ?
? ?
µ
56
Real-world markets are typically not risk-neutral, but option prices based on risk neutrality
nevertheless yield the correct (meaning no-arbitrage) solution for traded assets (Hull, 2002).
Risk aversion may be more important for the pricing of non-traded assets such as the weather
or electricity demand, but the price of market risk can never be measured with a sufficient
degree of confidence in order to make its inclusion in a pricing formula worthwhile, due to
measurement and identification issues (e.g. a greater market fundamental and a higher price
of risk have the same effect on the price). In absence of a convincing prior for market risk, I
will omit the latter in my analysis.
Chapter 5: Options Pricing
85
This is the option pricing formula for a cash-or-nothing call option based on a
normally distributed underlying asset or process.
Given knowledge of past emissions
t
G
0
, the overall cap
0
S , the forward price
penalty
II Phase
t
P and the interest rate r, and estimates for the mean
t
µ and standard
deviation
t
s of future emissions at every point in time, an estimate for the allowance
price
t
P
ˆ
can be computed for reach day, and the resulting time series compared with
the observed series
t
P for T t ..., , 2 , 1 = . A situation where the two series correspond to
one another would be interpreted as evidence that the allowance price was driven by
emissions and the penalty, but not abatement.
What remains to be determined in order to evaluate (5.6) are past emissions
and the mean and standard deviation of cumulative future emissions. These are not
directly observed, but have to be derived from underlying processes and ultimately
estimated using market data. This is the subject of the following section.
5.3. Deriving the mean and standard deviation of future emissions
5.3.a.) CO
2
emissions as a function of exogenous stochastic processes
Emissions are verified only once a year, and there exists no direct data about
daily emissions. However, for the power and heat sector (which is by far the most
Chapter 5: Options Pricing
86
important in the EU ETS) here is something that comes close: Daily electricity
consumption.
Electricity is special in the sense that demand has to be met with a matching
supply at all times in order for the grid not to crash. I will make the simplifying
assumption of zero short-term demand elasticity of electricity. This makes electricity
demand an exogenous process driven by stochastic elements such as the weather.
57
More precisely, it is not generation of electricity in general that drives CO
2
emissions, but generation of electricity using conventional thermal generators by
burning fossil fuels like coal and natural gas. Thus, daily emissions are a function of
consumer demand, as well as the availability of “clean” (i.e. non-CO
2
-emitting)
sources of energy, mainly hydroelectric and nuclear power.
58
Hydroelectric
generation depends on rainfall and varies within and between years, but nuclear
generation is largely constant due to prohibitively high start-up costs.
Let c
t
represent overall electricity consumption; c
t
c
consumption of
conventional fossil-fueled generation; n nuclear power generation (all in Giga-Watt-
hours (GWh) per day); and h
t
rainfall in the EU in millimeters (mm) per day,
weighted by installed hydroelectric capacity. Assuming that all available
57
In the long run, consumers will react to higher electricity prices by changing their
consumption habits and appliance portfolio, such that electricity demand is also a function of
the electricity price. But regardless of the time horizon, exogenous weather shocks will
always drive short-term consumption.
58
Although wind generation has increased rapidly during the past few years, it still accounts
for a relatively small fraction of total power production.
Chapter 5: Options Pricing
87
hydroelectric and nuclear power is used (i.e. that they are lowest in the merit order
59
),
demand for conventional generation can be expressed as
(5.7) n h c c
t t
c
t
? ? = ?
where ? is a fixed coefficient translating precipitation into hydroelectric power.
Since precipitation can be stored to some extent, either in reservoirs or as
snow in the mountains, there is no immediate relationship between precipitation and
hydro generation on any given day, which makes a regression of hydro output on
daily precipitation impractical. On the long run, however, all net hydro generation is
ultimately due to precipitation, and even though rainfall today may not translate into
more generation today, it nevertheless reduces expected conventional generation
needed to satisfy consumer electricity demand until the end of the market. I compute
the precipitation-to-rainfall conversion factor ? by dividing the EU’s total hydro
generation
60
in 1990-2005 of 4,852,339 GWh by cumulative weighted precipitation
over the same period of 9,775.28 mm, using installed hydroelectric capacity per
country as weights. This results in a conversion factor of ?=496.389 GWh/mm.
In the EU, 12 member countries have nuclear power plants (BE, CZ, DE, ES,
FI, FR, HU, NL, SK, SL, SW, UK). Their average total output in the years 2003-
2005 was 2,679 GWh per day, which I will use as a measure for n.
59
The merit order is the sequence by which individual generators are brought online and is
usually based on marginal cost.
60
From World Development Indicators database, World Bank.
Chapter 5: Options Pricing
88
The emission intensity (in CO
2
/GWh) of the marginal generator varies with
demand. The correct way to express emissions in Europe’s power & heat sector is
(5.8)
?
? =
c
t
c
t
dy y g
0
) (
where ) (
c
t
c ? is an emission intensity function transforming conventional thermal
power generation into CO
2
emissions. To compute the integral in (5.8) I would need
to know the exact dispatch order and the marginal emission intensity of all generators
involved, which is information that is not readily available. Instead, I assume that the
emission intensity of the marginal generators (i.e. all generators that are not running
all the time) follow a quadratic function. This allows me to express (5.8) as
(5.9)
) min( *
*
min c
t
c
t t
c g
c g
?
?
? ? ?
+ ? ?
The parameter ? translates fossil-fueled electricity generation into CO
2
emissions. ? is a constant defined as the difference between CO
2
emissions
associated with minimum thermal generation
min
g and the (theoretical) emissions if
the emission intensity ? were applicable to the inframarginal generation.
61
61
In theory, the average emission intensity of inframarginal generation could be greater or
smaller than the emission intensity of marginal generation. For example, if inframarginal
generation consists to a large part of lignite or anthracite coal power plants, then K>0, but if it
exists largely of efficient generators such as combined cycle gas turbines (CCGTs) that are
low in the dispatch order due to their small marginal costs, then K<0. In the EU, the former
Chapter 5: Options Pricing
89
Combining (5.7) and (5.9), daily CO
2
emissions can be expressed as
(5.10) ) ( * n h c g
t t t
? ? + ? = ? ?
In this specification, emissions are a function of a set of parameters and the
two stochastic and exogenous processes
t
c and
t
h . The properties of
t
g , and thus of
t
µ and
t
s , are therefore a function of the properties of
t
c and
t
h . At time s, the
mean of future CO
2
emissions at s t ? is defined by
(5.11)
( )
?
?
?
+ =
+ =
+ =
? ? + ? =
(
¸
(
¸
? ? + =
(
¸
(
¸
= =
T
t k
t s t s
T
t k
t t s
T
t k
k s
T
t s t
n h E c E K t T
n h c K E
g E G E
1
1
1
] [ ] [ * ) (
) ( *
] [
? ?
? ?
µ
The calculation of the variance of future emissions is a little more
complicated. In Appendix C (Result 1), I show that at time u t s ? ? , the variance is
(5.12)
( )
( )
? ?
?
? ? ?
+ = + =
+ =
+ = + = + =
? ? + +
+ ? =
+ = =
T
t k
T
k u
u k s u k s u k s u k s
T
t k
k s k k s k s
T
t k
T
k u
u k s
T
t k
k s
T
t s t
c h Cov h c Cov h h Cov c c Cov
h Var h c Cov c Var
g g Cov g Var G Var s
1 1
2 2
1
2 2
1 1 1
2
] , [ ] , [ ] , [ ] , [ 2
] [ ] , [ 2 ] [
] , [ 2 ] [ ] [
? ? ? ?
? ? ?
is much more likely given the large number of lignite plants in Germany and the new EU
member countries from Eastern Europe.
Chapter 5: Options Pricing
90
Both expressions are functions of the constants ? and n, the parameters K
and ? , the mean and variance of electricity consumption and precipitation, and the
covariance of electricity consumption and precipitation between different days.
I defined the constants ? and n above and will treat them as known; K and
? will enter the estimation as free parameters. The derivation of the mean, variance
and covariance of electricity consumption and rainfall is the subject of the next
subsection.
5.3.b.) Properties of the stochastic processes c
t
and h
t
For the definition of the stochastic processes of electricity demand and
precipitation, I will draw extensively from a paper by Peter Alaton, Boualem
Djehiche and David Stillberger (2002). Although their analysis focuses on pricing a
weather option over heating-degree days with the underlying process being
temperature, it is very similar in principle to both electricity demand and
precipitation, as both are exogenously driven stochastic processes that contain
deterministic annual fluctuation and long-term trends. The contribution of my paper
is not the derivation of the property of such processes, but the application of these
methods to model CO
2
emissions and, ultimately, to CO
2
allowance pricing.
Chapter 5: Options Pricing
91
I will model both electricity consumption and precipitation diffusion
processes
62
consisting of a deterministic mean and a stochastic part, and which exhibit
mean-reversion.
63
For mathematical tractability, I include the stochastic element in
the form of a generalized Wiener process. Combining the processes in the index x,
they can be described as
(5.13) h c x dW t i dt x x a
dt
dx
dx
x
t x t
m
t x
m
t
t
, ; )] ( [ ) ( * = +
(
¸
(
¸
? + = ?
This is known as an Ornstein-Uhlenbeck process with a non-zero mean and
time-varying volatility.
64
The term in brackets represents the drift of the processes,
followed by the diffusion term defined by the standard Wiener process
x
t
dW times
the corresponding volatility. The first element of the drift term in (5.13) is due to the
fact that mean consumption and precipitation change throughout the year. The mean
reversion parameters a
x
measure the speed at which the processes revert back to their
long-term mean.
I constrain the volatility to be constant within each calendar month, but allow
it to differ across months. The index i labels the month to which the time index t
62
A diffusion process is the solution to a stochastic differential equation. In particular, it is a
continuous-time Markov-process with a continuous sample path. This is a realistic
description for electricity consumption and precipitation and makes the derivation and
exposition easier, but the market and weather data discrete are naturally only available for
discrete points in time.
63
Mean reversion is a commonly observed characteristic in many naturally occurring
processes, as they generally do not grow without bounds and eventually return to their long-
term mean.
64
See, for example, Bibby and Sorensen (1995).
Chapter 5: Options Pricing
92
refers. For reasons of data availability (see below) I will start this index at 1 in
January 1976 and finish at 384 in December 2007. Thus,
2007 Dec 384
1976 Feb 2
1976 Jan 1 ) (
? =
? =
? =
t if
t if
t if t i
?
Because I assume that the volatility is the same for each calendar month, it
must be that ] 12 * [ ] [ k i i + =? ? for any integer k .
I define the long-term mean of electricity consumption and precipitation as
(5.14)
] 365 / 2 sin[ * *
* ] 365 / 2 sin[ * *
2 1 0
2 1 0
h h h h m
t
t
c c c c c m
t
t t h
WD D t t c
? ? ? ? ?
? ? ? ? ?
+ + + =
+ + + + =
The parameters
x
0
? and
x
1
? ( h c x , = ) describe the level and trend of the two
process, respectively, whereas
x
2
? describe the amplitudes of the respective sine
waves. The phase angles
x
? shift the oscillation of the two processes to their correct
position. Lastly, the vector of coefficients
c
D (not applicable to rainfall) accounts
differences in electricity consumption across different weekdays, and
t
WD is a vector
of weekday dummies.
Equation (5.13) describes two stochastic differential equations. At time t s ? ,
their solution is
65
65
See, for example, Øksendahl (2007) Chapter 2.
Chapter 5: Options Pricing
93
(5.15) h c x dW i e x e x x x
x
x
t
s
t a m
t
s t a m
s s t
x x
, ; )] ( [ ) (
) ( ) (
= + + ? =
?
? ? ? ?
?
?
? ?
The first term on the RHS is the deviation of actual consumption/precipitation
at the present time s from its mean. As time goes on, the impact of this deviation will
diminish due to the mean-reversion property of both processes, measured by the
exponent. If one of the processes is at its average at time s, or if s t >> , then the first
term will drop out, and the expectation at time t simply becomes the mean
expectation
m
t
x defined by (5.14).
The mean and variance of electricity demand and precipitation can be
computed as
(5.16) h c x x e x x x E
m
t
s t a m
s s t s
x
, ; ] [ ] [
) (
= + ? =
? ?
(5.17)
( ) | |
( )
h c x dy e y i
dW e i E
x E x E x Var
t
s
y t a
x
t
s
t a
x s
t s t s t s
x
x
, ; )] ( [
)] ( [
] [ ] [
) *( 2 2
2
2 ) *( 2 2
2
= =
(
¸
(
¸
=
? =
?
?
? ?
? ?
?
? ?
?
?
The second equality follows from the fact that | | dt dW E
x
t
=
2
) ( . If the
volatility does not change between s and time t, (5.17) can be solved to
(5.18) ( ) ) ( ) ( ; , ; 1
2
)] ( [
] [
) ( 2
2
t i s i h c x e
a
t i
x Var
s t a
x
x
t s
x
= = ? =
?
?
Chapter 5: Options Pricing
94
If s and t are not within the same month, the expression becomes more
complicated. I will denote the first day of each month as { } i t i t t i t = = ) ( : min )] ( [
min
.
In Appendix C (Result 2) I show that for h c x , = and ) ( ) ( t i s i ? , the general
expression for the variance is
(5.19) ( )
)
`
¹
¹
´
¦
? + + ? =
? ? + ? ?
?
=
?
)] ( [ )] ( [ ] 1 [ ] [
2
1
] [
2 ) ( 2 2 ]) 1 [ ( 2
1 ) (
) (
2 2
min
s i e t i e k k
a
x Var
x
s t a
x
k t t a
t i
s i k
x x
x
t s
x x
? ? ? ?
It is easy to verify that if the volatility is the same for each month, (5.19)
collapses to (5.18).
To calculate the covariance between electricity consumption and rainfall on
the same day, note that dt dW dW E
ch h
t
c
t
? = ] [ , where
] [ * ] [ / ] , [
t t t t
ch
h Var c Var h c Cov ? ? is the correlation coefficient between the two
processes. Thus,
( )( ) | |
?
?
? + ?
? + ?
=
(
¸
(
¸
=
? ? =
t
s
y t a a
h c
ch
t
s
h c t a a
h c s
t s t t s t s t t s
dy e y i y i
dW dW e i i E
h E h c E c E h c Cov
h c
h c
) )*( (
) )*( (
)] ( [ )] ( [
)] ( [ )] ( [
] [ ] [ ] , [
? ? ?
? ? ? ?
? ?
?
Analogous to the procedure used for the variance, this can be solved to
Chapter 5: Options Pricing
95
(5.20)
( )
¦
)
¦
`
¹
¦
¹
¦
´
¦
? +
+ + ?
+
=
? + ?
+ ? + ?
?
=
?
)] ( [ )] ( [ )] ( [ )] ( [
] 1 [ ] 1 [ ] [ ] [
] , [
) )( (
]) 1 [ )( (
1 ) (
) (
min
s i s i e t i t i
e k k k k
a a
h c Cov
h c
s t a a
h c
k t t a a
t i
s i k
h c h c
h c
ch
t t s
h c
h c
? ? ? ?
? ? ? ?
?
Lastly, the covariance between electricity consumption/precipitation on day t
and u for s ? t ? u is defined by (see Appendix C, Result 3):
(5.21)
] , [ * ] , [
] , [ * ] , [
, ]; [ * ] , [
) *(
) *(
) *(
t t s
t u a
u t s
t t s
t u a
u t s
t s
t u a
u t s
h c Cov e c h Cov
h c Cov e h c Cov
h c x x Var e x x Cov
c
h
x
? ?
? ?
? ?
=
=
= =
Expressions (5.16) and (5.19)-(5.21) can now be substituted into (5.11) and
(5.12). In the following section I obtain empirical parameter estimates for
ch
x
c x x x x
a D ? ? ? ? ? , , , , , ,
2 1 0
and ) (i
x
? .
5.4. Estimation
There are two different steps in the estimation. The final goal is to express
equation (5.6) as a function of data, known constants, and a set of free parameters,
and calculate these free parameters using market data. Since most countries lack
daily electricity consumption data prior to 2006, I will evaluate (5.6) for the period
between January 1, 2006 and December 31, 2007.
Because in January 2006, realizations of electricity consumption and
precipitation for later days were not yet known, I estimate
t
µ and
t
s using data
Chapter 5: Options Pricing
96
through 2005 only, with some exceptions where necessary. In theory I could update
the estimates for every day, but for simplicity I will use pre-2006 data.
5.4.a.) Data
Daily data about electricity consumption is available from the Union for the
Coordination of Transmission of Electricity (UCTE)
66
for continental European
countries, including all EU member states except for the Nordic countries,
67
the UK,
Ireland, the Baltic States, Malta and Cyprus. Electricity consumption has been
measured on every third Wednesday of each month
68
since 1994 for 9 EU countries,
since 1996 for Germany and since 1999 for another 5 EU countries. Weekend
consumption is available for every Weekend following the third Wednesday of each
month in the year 2000. Starting in January 2006, electricity consumption is
available on a daily basis for all UCTE countries. To supplement the UCTE data I
obtained all available historic electricity consumption data directly from the
transmission system operators (TSOs) in the UK, Ireland and the Nordic countries.
69
66
Available at www.ucte.org, last accessed in September 2008.
67
Sweden, Denmark and Finland. Note that Norway is not part of the EU, and although it is
now linked to the EU ETS, this was not the case during the first phase of the market.
68
Wednesdays are supposed to be the most typical weekdays (as opposed to Mondays and
Fridays, which may be slightly different), and the third week is supposed to be the typical
week of a month.
69
UK: Daily data since 2001 from the National grid, available at
http://www.nationalgrid.com/uk/Electricity/Data/; Ireland: Daily data since 2002 from
Eirgrid, available at http://www.eirgrid.com; Denmark: Daily data since 2000 from
Energinet, available at http://www.energinet.dk; Finland: Daily data since 2004 from Fingrid,
Chapter 5: Options Pricing
97
I exclude Malta, Cyprus and the Baltic States from the analysis, because the former
are not integrated into Europe’s electricity grid and no daily electricity consumption
data for the latter is available. In terms of annual electricity production, the 20
countries included account for 99% of total production in the EU-25.
70
The EU produces nearly all of the electricity it consumes, with net
imports/exports accounting for less than 0.1 percent overall consumption. I therefore
exclude imports/exports in my calculations and set consumption equal to production.
In order to accommodate the variation in type and provenance of the data I
will carry out the analyses separately for each group of countries for which the
available data is of the same type (e.g. daily vs. monthly) and covers the same time
period. The six groups are listed in Table 5.1. Figures 5.1a-f show the available pre-
2006 electricity consumption data by group. All countries have daily data for the
years 2006 and 2007.
For precipitation, I use the European Climate Assessment and Dataset.
71
This
dataset contains daily data for 1,048 monitoring stations located in 42 countries. The
length of the series varies from a few years to >150 years, with most series spanning
available at http://www.fingrid.fi; Sweden: Daily data since 2000 from Svenska Kraftnät,
available at http://www.svk.se/web/Page.aspx?id=5794.
70
In 2007, Romania and Bulgaria joined the Union to make it the EU-27. However, because
they were not part of the market during the first two years, and their registries were not ready
until the end of 2007, they can be excluded from Phase I.
71
Klein Tank et al. (2007): “Daily Dataset of 20
th
-Century Surface Air Temperature and
Precipitation Series for the European Climate Assessment”, available at eca.knmi.nl, last
accessed in September 2008.
Chapter 5: Options Pricing
98
several decades. To model the stochastic process underlying precipitation, I use data
covering the years 1976-2005.
The conversion of precipitation into hydroelectric power is location-specific.
For example, rainfall in the Netherlands or in Denmark is largely irrelevant for power
generation because these countries have very little installed hydroelectric generation
capacity, whereas hydro generation constitutes a large share of total power production
in Alpine and Scandinavian countries. I average station entries by country,
72
and then
create a weighted European average using installed hydroelectric capacity in 2006 as
weights.
73
Installed hydro generation is given in the last column of Table 5.1.
Weighted precipitation in millimeters (mm) is shown in Figure 5.2 for a
subset of the sample period. Whereas it is difficult to visually discern a pattern in the
raw data (Fig. 5.2a), using moving 7-day-average (Fig. 5.2b) reveals a clear
seasonality.
72
For low-lying countries such as Belgium and Luxembourg, I simply take an average of all
monitoring stations. However, since hydro generation in the Alps and in Scandinavia is
highly location-specific, I take an average of the subset of monitoring stations that are located
in or near mountains. A full list of the selected stations is available from the author upon
request.
73
This data comes from UCTE (www.ucte.org) for continental Europe; from Nordpool
(www.nordel.org) for Scandinavia; from the Austrian Energy Agency
(www.energyagency.at/enercee/) for the Baltic States; from Harrison (2005) for the UK; and
from the Electricity Supply Board (ESB, available at
http://www.esb.ie/main/about_esb/power_stations_intro.jsp) for Ireland; all accessed in
September 2008.
Chapter 5: Options Pricing
99
5.4.b.) Parameter estimation for electricity consumption and precipitation
I estimate the parameters
x x x x x
D , , , ,
2 1 0
? ? ? ? and ] [i
x
? with a model that
features an autoregressive error to account for mean-reversion and multiplicative
heteroskedasticity to allow the variance to differ across months:
(5.22)
h c c c x Nov Jan t i
t i N u
u
WD D t t t x
t
x
t
x x
x
x
x
t
x
t
x
t x
x
t
x
t t
x x x x x
t
, ..., , , ; } * ... * exp{ )] ( [
)] ( [ , 0 ( ~
*
* ) 365 / 2 cos( * ) 365 / 2 sin( * *
6 2 1
11 1 0
2
2
1
2 1 1 0
= + + + =
+ =
+ + + + + =
?
? ? ? ?
?
? ? ?
? ? ? ? ? ? ?
Note that the index x now covers six different electricity consumption series,
plus the (weighted) precipitation series, all of which are estimated separately by
maximum likelihood.
The parameters
x x
1 0
, ? ? and
x
D are the same as in (5.14) and are estimated
directly. The transformation of the sine wave plus the phase angle into a sine and
cosine wave is based on a Fourier transform and serves to linearize the equation. The
parameters
x
2
? and
x
? can be computed using the estimates of
x
1
? and
x
2
? :
74
(5.23)
] / arctan[
) ( ) (
1 2
2
2
2
1 2
x x x
x x x
? ? ?
? ? ?
=
+ =
; h c c c x , ..., , ,
6 2 1
=
The t-statistics and confidence intervals have to be calculated using the delta
method.
74
See, for example, Beckwith et al. (1995), p. 131.
Chapter 5: Options Pricing
100
I estimate the daily variance ] [
2
i
x
? from the autocorrelation parameters
x
?
and the variance of the white noise ] [
2
i
x
? .
75
For a stationary AR(1) process, the
variance is given as
(5.24) ( ) | |
x
x
t t t x
i
E x E x E i
?
?
? ?
?
= = ? =
1
] [
] [ ] [ ] [
2
2 2 2
The mean-reversion parameters
x
a measure the speed at which a shock to
t
x
is felt at later times. From (5.16), the expectation of future electricity consumption or
precipitation is
h c x x e
x e x x x E
m
t
s t a x
s
m
t
s t a m
s s t s
x
x
, *
] [ ] [
) (
) (
= + =
+ ? =
? ?
? ?
?
This makes it clear that the term
) ( s t a
x
e
? ?
is equivalent to the impulse-response
function of the AR(1) process defined by
76
s t
s t
?
= ? ? ) , (
75
Because I cannot estimate an AR(1) parameter with data that only contains entries for
every 3
rd
Wednesday per month, I use the 2006-7 data to estimate this parameter for Series 1-
3. Likewise, the estimate of the variance is sensitive to the frequency of measurement
(Hayashi and Yoshida, 2005) and generally improves with greater frequency. I therefore also
use the 2006-7 data to estimate the variance and the correlation coefficients (see below).
Note that for all other parameters, I use pre-2006 data only. Note that the daily variance and
mean reversion parameter for Series 4-6 are not significantly different between pre- and post-
2006 data.
76
See Hamilton (1994) p. 53-54.
Chapter 5: Options Pricing
101
which measures the impact of an exogenous shock occurring in period s on the
variable in period t. Equating the two and solving yields
(5.25) ) ln(
x x
a ? ? =
All parameter estimates are given in Table 5.2.
I compute the correlation coefficients among the different series
kl
?
) ..., , , , (
1 1
h c c l k = by using the data from 2006-2007, for which all series have daily
entries.
77
The results in Table 5.3 show that electricity consumption across the six
different regions is highly correlated, but that precipitation weighted by available
hydroelectric power and electricity consumption is not. Because the correlation
coefficient between precipitation and all six electricity consumption series is zero, I
will set j h c Cov h c Cov
t t s t
j
t s
? = = 0 ] , [ ] , [ .
I derived the expressions for the variance and covariance in (5.16), (5.19) and
(5.21) for total electricity consumption. Due to the six different data groups the parts
77
Hayashi and Yoshida (2005) developed an unbiased estimator to compute the correlation
coefficient between time series of different measuring intervals, but that estimator is not
bounded by unity in magnitude, relying on truncation instead. Also, this would only address
the problem of differing frequencies within the same time period, but not that of different
time periods.
Using the much higher-frequency data for 2006-2007 is equivalent to assuming that the
covariance between electricity consumption in the six different regions and EU-wide
weighted precipitation is the same before and after January 1, 2006. For the two groups for
which ample data is available (groups 4-6), this assumption appears to hold. Note that the
market participants very likely have much better information about these covariances than
what would be gleaned based on a few monthly data points from pre-2006 data available to
the researcher.
Chapter 5: Options Pricing
102
involving electricity consumption that are based on overall consumption have to be
adjusted to
(5.16’)
?
=
=
6
1
] [ ] [
j
j
t s t s
c E c E
(5.19’)
( )
¦
)
¦
`
¹
¦
¹
¦
´
¦
? +
+ + ?
+
=
+ =
? + ?
+ ? + ?
?
=
= + = =
?
? ? ?
)] ( [ )] ( [ )] ( [ )] ( [
] 1 [ ] 1 [ ] [ ] [
] , [
] , [ 2 ] [ ] [
) )( (
]) 1 [ )( (
1 ) (
) (
6
1
6
1
6
1
min
s i s i e t i t i
e k k k k
a a
c c Cov
c c Cov c Var c Var
l j
l
c
j
c
l j
l
c
j
c
l j l j
l j
c c
s t a a
c c
k t t a a
t i
s i k
c c c c
c c
l j
l
t
j
t s
j j l
l
t
j
t s
j
j
t s t s
? ? ? ?
? ? ? ?
?
(5.21’) ( )
? ? ?
= + =
? ? ? ?
=
? ?
+ + =
6
1
6
1
) *( ) *(
6
1
) *(
] , [ ] [ * ] , [
j j l
l
t
j
t s
t u a t u a
j
j
t s
t u a
u t s
c c Cov e e c Var e c c Cov
l
c
j
c
j
c
5.4.c.) Evaluation of the options pricing formula
With these parameter estimates, I can now proceed to evaluating the options
pricing formula. Because emissions were below the total cap at the end of the
market, as well as for each year individually, I will disregard the second line of
equation (5.6). I use first- and second-phase over-the-counter (OTC) allowance
prices from Point Carbon.
The mean and standard deviation of future emissions are a function of free
parameters and estimates of the mean, variance and covariance of the processes for
Chapter 5: Options Pricing
103
electricity consumption and precipitation. Substituting (5.11) and (5.12) into (5.6)
and simplifying gives
(5.26)
( ) ( )
2 / 1
1
2
1
2
1
1 1
0 ) (
] , [ ] , [ 2 ] [ ] [
] [ ] [
(
* ] 40 [
|
¹
|
\
|
+ + + ?
? ? + ? ? ?
|
|
¹
|
\
| ? +
? + =
? ? ?
? ?
+ = + = + =
= + =
? ?
T
t k
u k s
T
k u
u k s k s k
T
t k
s t
t
k
k k
T
t k
k s k s t
t
t II Phase
t
t T r
t
h h Cov c c Cov h Var c Var B
n h c n h E c E A
with
B
S A TK
P e P
? ?
? ?
?
?
where A
t
and B
t
are known functions of the parameters of the diffusion processes
for electricity consumption and precipitation.
To account for the price crash after the first round of emissions verifications, I
add dummy variable to allow for the updating of firms’ expectation about emissions
from other sectors (and emissions of other firms in the power & heat sector for that
matter, but these parameters cannot be individually identified). Simplifying leads to
(5.27)
V S TK K
A
P e
P
B Y with
S D K
Y
t
II Phase
t
t T r
t
t t
EV EV
t
t
+ ? ?
?
|
|
¹
|
\
|
+
? ?
?
=
? ?
?
0
) (
1
40
*
?
Chapter 5: Options Pricing
104
where
1 ?
? refers to the c.d.f of the standard normal distribution and
EV
t
D is a
dummy that takes on the value of zero before the first round of emissions
verifications, and of one thereafter. Given the price crash in April 2006,
EV
S has to
be positive, implying that firms updated their expectation of the total number of
remaining available permits upwards. I use an annual interest rate of 10% for the
calculation of the discounted penalty.
78
Total emissions from other sectors or emissions during 2005 are contained in
V S TK K + ? ?
0
, where V stands for any time-invariant parameter that shifts the
amount of permits available to the power & heat sector for the years 2006-2007.
Note that K has to be negative, since
0
S is the total cap and the number of permits
available to power generators (net of the correction associated with minimum
generation TK ) has to be positive.
Estimates for the free parameters K , ? and
EV
S can be computed by taking
averages of
t
Y for the period before and after the allowance price crash. Note that the
parameters are not individually identified, such that one of them has to be held fixed
in order to calculate the others. For example, when holding
EV
S fixed, K
ˆ
and ?ˆ are
defined as
78
Use of 0% and 20% did not alter the results significantly.
Chapter 5: Options Pricing
105
(5.28)
?
?
?
?
EV
EV
t t
EV
t t
S K
D Y
K
D Y
?
= = ?
= = ?
) 1 ( |
) 0 ( |
2
1
2 1 1 2
ˆ ;
/ 1
ˆ
? ?
?
? ? ?
=
?
= ?
EV EV
S S
K
where ) ( | ?
t
Y refers to the conditional sample average. Results for fixed values of
EV
S are shown in the left panel of Table 5.4. The shaded areas are the parameter
combinations that make economic sense. The emission intensity ? has to be
somewhere between 600 and 900 tCO
2
/GWh,
79
which is the case for
MT S MT
EV
150 100 < < , and for MT K MT 770 , 1 660 , 2 ? < < ? . Both ranges are
plausible: The range for
EV
S implies that firms expected 2005 emissions to exceed the
annual cap by 6-56 MT, but instead they turned out to be 94 MT below.
80
The range
for K means that of the about 4,200 MT of permits issued for the years 2006 and
2007, between 42-63 % were used to cover emissions of power generators.
81
The
goodness of fit (defined by the model sum of squares divided by total sum of squares)
is 0.81, much larger than the corresponding values for the model presented in Chapter
3 (see Tables 3.1-2).
79
The average emissions intensity of the marginal generators in the EU will not exceed that
of a coal-fired power plant, which emits about 920 tCO2/GWh. The emission intensity of
Combined Cycle Gas Turbines (CCGTs) is about half of this, but coal is at the margin for the
majority of the load in the EU.
80
The first round of emissions verifications found emissions to be 94 MT below the total
2005 allocation.
81
Note that the sector defined as “Power & Heat” accounts for about 70% of total emissions,
but this includes production of heat as well as industrial process combustion, not just
electricity producers.
Chapter 5: Options Pricing
106
Using the estimates
EV
S K | ˆ ,
ˆ
? I compute the estimated price series
t
P
ˆ
:
(5.29)
|
|
¹
|
\
|
+
?
? + =
? ?
t
t
t
EV EV
t II Phase
t
t T r
t
B
A
B
S D K
P e P
?ˆ
ˆ
(
* ] 40 [
ˆ
) (
Figure 5.3 shows the predicted price series, along with the actual allowance
price and the forward price for second-phase allowances. The estimated series
follows the data quite well until the April 2006 price crash, after which it falls below
the actual price, before crossing it and finishing the period slightly too high.
Importantly, the prediction shows the post-crash stabilization followed by a gradual
decline to zero observed in the real price series, which has puzzled market observers.
This is the result of the probability of a binding cap slowly approaching zero, as time
progresses and actual emissions are observed.
A striking difference between prediction and actual price is the volatility. The
allowance price series appears to fluctuate much more than the prediction, which of
course could be due to shocks in unobserved variables that drive emissions that end
up in the model residual. However, there is an alternative explanation: If power
generators had a better idea about the variance of future demand than the researcher
(because they have access to better data, especially for the area which they have
exclusively serviced for decades before market liberalization), then I would have
overestimated the standard deviation of future generation denominated by B
t
in
Chapter 5: Options Pricing
107
(5.27). A too large standard deviation would attenuate demand shocks, leading to a
prediction that is too smooth and too inert to electricity demand shocks.
To evaluate this possibility, I test the results for their sensitivity to
t
B . I
divide
t
B by 10 and 10, corresponding to a factor of 10 resp. 100 by which my
estimate for the variance exceeds the estimate used by the market participants.
82
The
predictions are shown in Figure 5.4. It is clear that the results are greatly influenced
by the uncertainty embedded in future emissions. Whereas the larger correction
appears to overshoot and lead to excessive price volatility, the more modest
correction by one order of magnitude fits the data much better, with a goodness of fit
of 0.92. The corresponding estimates for K , ? and
EV
S are shown in the right panel
of Table 5.4. Again, for sensible emission intensities (shaded region), the associated
values for
EV
S and K are plausible.
5.5 Conclusions
In this paper I derive an allowance pricing formula based on the assumption
that firms were not able to engage in significant abatement during the first three years
of the EU ETS, and therefore had no control over their emissions. In this case, the
value of an allowance can be characterized by an options pricing formula for a cash-
82
Note that when computing the standard deviation of future consumption using the 2006-7
data and adding all the series up, I get a result that is about 1.4 times lower than the standard
deviation calculated using separate series.
Chapter 5: Options Pricing
108
or-nothing call option. This formula contains the mean and standard deviation of CO
2
emissions for the remainder of the market.
I calculate daily emissions based on daily demand for electricity generated by
conventional thermal combustion. This is a function of total electricity consumption
and the availability of non-emitting sources for electricity such as hydro and nuclear.
I assume that these processes are characterized by diffusions and estimate the
diffusion parameters using market data. This allows me to express the allowance
price as a function of data, estimates for future emissions and three free parameters.
The parameter estimates are highly significant and make economic sense. The
predicted allowance price series fits the actually observed prices quite well, especially
when adjusting the estimate for future electricity demand volatility downwards, based
on the hypothesis that power generators had a better idea about it. Importantly, the
model is able to explain the price stabilization after the price crash, followed by a
long and steady decline towards zero, which can be explained by a declining
probability that the cap was going to be binding. A model based on abatement
parameters would only be able to explain such a movement if the price of
fundamentals related to abatement also expressed such a steady decline, which was
not the case (see Chapter 3).
I conclude that the allowance price during the first phase of the EU ETS was
to a large extent driven by the penalty for noncompliance and the probability of a
binding cap, at least for the years 2006 and 2007. This could be due to the speed of
market-setup, which did not allow the involved firms to adjust their emissions in
Chapter 5: Options Pricing
109
time, and/or the realization after the April 2006 price crash that the market was likely
to be oversupplied with permits and that therefore abatement measures would not be
profitable.
Chapter 6: Conclusions
110
Chapter 6: Conclusions
In my dissertation I examine the relationship between first-phase allowances
in the EU ETS and various price drivers. In Chapter 3 I start with the most
commonly cited market fundamentals that are assumed to drive marginal abatement
costs, but find them to have little explanatory power for the allowance price path,
despite using the best available data. I then go on to test the allowance market for
the presence of a price bubble. The results are consistent with the presence of a
bubble (or a series of bubbles), although a conclusive proof of the existence of a
bubble is impossible on theoretical grounds.
In my analysis I skip the issue of how the bubble(s), if any, got started, and
focus exclusively on their presence. But something must drive the price up initially
before self-fulfilling expectations can form and take over. In Chapter 4 I focus on the
group that profited most from the high allowance price: Power producers, due to free
allocation and cost pass-through to electricity prices. Although no reason for price
inflation in competitive markets, the presence of market power can change this. The
problem with this hypothesis is that power producers were net buyers of allowances,
and existing economic theory predicts that they would decrease, rather than increase,
the allowance price, provided that they had market power. I show that when taking
the interaction between output and permit market into account, this prescription no
Chapter 6: Conclusions
111
longer holds true. Actual allocation amounts and cost pass-through rates in the EU
ETS indicate that dominant power producers would indeed have found it profitable to
inflate the allowance price.
Chapter 5 is to some extent complementary to Chapter 3: If allowance prices
were not driven by marginal abatement costs, is there an alternative explanation to a
bubble? There is some evidence supporting the hypothesis that firms simply did not
have enough time to adjust their process emissions in time for the first phase. In this
case, the allowance price would be driven by the penalty for noncompliance and the
probability that the overall cap turns out to be binding. I set up an options pricing
model that has these characteristics and find that it fits the data well, better indeed
than the market model in Chapter 3. One caveat for the findings of the options
pricing paper is that they only apply to the years 2006-7.
Taken together, my findings imply the following:
First, the allowance price during the first phase of the EU ETS was not equal
to marginal abatement costs. This could be due to the formation of a price bubble,
market power in the power sector or firms’ inability to engage in timely and
significant abatement. It is quite possible that all three of these reasons were involved
simultaneously or sequentially: A bubble needs to get started somehow, which would
reconcile the first two reasons. Also, the bubbles results are much stronger for the
period before the crash, whereas the options pricing story may apply to the latter part
of the market only. In this case, it would not have been so much inability, but rather
Chapter 6: Conclusions
112
unwillingness to abate on the part of power producers, once they were fairly certain
that the market was going to be long after the price crash. Thus, my findings are
consistent with market power starting a bubble, which then sustained itself for a while
until it was popped by the first round of emissions verifications. From then on, firms
were no longer concentrating on abatement but rather on optimizing their allowance
portfolio, taking the stochastic nature of emissions into account.
Second, regardless of the exact combination of reasons, the fact that first-
phase allowance prices were not equal to marginal abatement costs means that the
first phase of the EU ETS failed from the perspective of reaching a given emissions
goal at least cost. It is possible that the prime goal of the EU was not the design of an
efficient policy instrument for 2005-2007, but instead to prepare the EU for the Kyoto
compliance period of 2008-2012. But this could arguably have been achieved at a
lower cost to consumers and the economy as a whole, considering the large increase
in output prices without the benefit of an emissions reduction.
Future cap-and-trade markets should be set up such that they avoid the
problems encountered during the first phase of the EU ETS. Specifically, regulators
of future markets should consider auctioning most if not all allowances, which would
avoid the problem of market manipulation while giving the regulator the opportunity
to reimburse consumers for higher output prices from carbon cost pass-through.
Further, more frequent rounds of emissions verifications would prick any price
bubble sooner by breaking the cycle of self-fulfilling expectations and bring the price
back to its fundamental value. Lastly, companies should be given sufficient time and
Chapter 6: Conclusions
113
regulatory certainty to engage in large-scale abatement decisions. If the time is too
short, or the price signal to uncertain, the allowance market may deteriorate into a
betting game where firms aim to reach compliance exclusively by buying allowances
on the market, rather than abating emissions.
Future research is needed in this area. The second phase of the EU ETS, as
well as upcoming carbon markets in the eastern USA, Japan, Australia and Canada
should be examined as to whether the permit price is truly driven by marginal
abatement costs. Another promising area of research would be the design of
sophisticated auctioning schemes that would allow the regulator to gather more
information about firms’ marginal abatement costs. This knowledge would aid in
setting the cap as well as in determining whether there is a discrepancy between
permit price and marginal costs, once the market is under way.
114
Tables
Table 2.1: Summary results for Phase I of the EU ETS
2005 2006 2007
Total Phase I
Price (time average) € 18.40 € 18.05 € 0.72
€ 12.39
Trading volume
a
262 Mt 817 Mt 1,364 Mt
2,443 Mt
Trading value
a
€ 5.4 billion € 14.6 billion € 28.0 billion
€ 48.0 billion
Allocation 2,099 Mt 2,072 Mt 2,079 Mt
6,250 Mt
Emissions 2,010 Mt 2,031 Mt 2,041 Mt
6,081 Mt
Surplus (volume) 89 Mt 41 Mt 39 Mt
168 Mt
Surplus (%) 4.22 % 1.98 % 1.85 %
2.69 %
a: OTC and exchange trading for phase I and II, but excluding bilateral trades
115
Table 3.1: Results from estimating Equation (3.10)
Full period Pre-crash Post-crash
D.Gas
F
26.0812*** 37.5707*** 20.0823***
p <0.0001 <0.0001 <0.0001
D.Coal -26.1478 333.9588 -22.2437
p 0.8549 0.2534 0.7444
Temp
5d
W -3.7832 -7.2270 -1.7798
p 0.5922 0.6950 0.6378
Temp
5d
S 11.2833 65.2161* 13.3100**
p 0.3355 0.0606 <0.0001
D.Res 0.4370 -29.7230 0.2881
p 0.9744 0.4210 0.9652
Crash -4.1946***
p <0.0001
N 609 333 272
Chi
2
2143.59 29.95 54.23
p <0.0001 <0.0001 <0.0001
LL -415.91 -244.37 -143.59
AIC 847.81 502.74 301.17
BIC 883.11 529.39 326.41
Goodness of fit
#
0.2481 0.0376 0.0105
*: p<0.1; **: p<0.05; ***
<0.01; all variables defined in text #: Model sum of squares/total sum of squares
116
Table 3.2: Results from estimating equation (3.11)
(1) (2) (3) (4)
Pre-crash Post-crash Pre-crash Post-crash
D.GasF 53.1555*** 28.7834*** 42.2153*** 32.4172***
p <0.0001 <0.0001 0.0001 <0.0001
L.D.GasF 30.1149*** 20.3575***
p <0.0001 0.0100
L2.D.GasF 9.0277 -10.7088
p 0.4483 0.4630
D.GasS -17.1810*** -7.4000**
p <0.0001 0.0182
L.D.Coal 268.0765 93.2723
p 0.2671 0.7679
D20.Coal -16.3700 4.3739
p 0.1694 0.8020
Temp1M W -0.0008 -0.0010** -0.0007 0.0002
p 0.2149 0.0171 0.3869 0.7611
Temp1M S 0.0074*** -0.0015*** -0.0025* -0.0003
p <0.0001 0.0010 0.0696 0.5886
D5.Res -18.2342*** 2.2228
p <0.0001 0.6021
D20.Res -0.1130 18.6926*** -0.0460 -2.2647
p 0.2287 <0.0001 0.7251 0.6056
L.D.EUA 0.1724*** 0.4123***
p 0.0085 <0.0001
L2.D.EUA -0.0827* -0.1305***
p 0.0756 0.0043
L3.D.EUA 0.1140*** 0.0973***
p 0.0008 0.0001
L4.D.EUA 0.1699*** 0.0379**
p <0.0001 0.0358
L5.D.EUA -0.0224 0.1149***
p 0.6284 <0.0001
N 329 272 318 253
Chi2 152.08 293.00 260.52 188.26
p <0.0001 <0.0001 <0.0002 <0.0003
AIC 469.88 234.24 439.52 162.35
BIC 504.04 266.69 492.18 211.82
Goodness of fit 0.1092 0.0278 0.1325 0.2374
*: p<0.1; **: p<0.05; ***
<0.01; all variables defined in text 117
Table 3.3: Cointegration test results
Full period Pre-crash Post-crash
A(1) -0.09517 -0.49985 0.38262
p 0.247 <0.001 0.006
B1(1) (Gas) -0.00016 0.00146 0.00225
p 0.971 0.822 0.868
B2(1) (Coal) -0.03021 -0.31725 0.10131
p 0.764 0.018 0.749
B3(1) (Reservoirs) 0.00210 -0.03683 0.00839
p 0.497 0.005 0.350
B4(1) (DAX) -0.00002 0.00032 -0.00043
p 0.671 0.024 0.107
_const 0.3711 1.1521 2.3260
p 0.589 0.152 0.412
Unit Root test on z
t
-0.391 -0.974 -0.543
Unit Root test on
(
z
t
-2.241 -1.915 -2.751
Table 3.4: Results from regime-switching tests
Full period pre-crash post-crash
p 0.979 0.984 0.995
q 0.994 0.931 1.000
LR statistic 118.320 38.940 60.360
p <0.0001 0.0007 <0.0001
118
Table 5.1: Data availability and installed hydroelectric capacity by country
Country per Start of data series
a
Hydro capacity
data series Type Year Source
b
in 2006 (MW)
Series 1
Austria 3rd Wed. 1994 UCTE 11,811
Belgium 3rd Wed. 1994 UCTE 1,411
France 3rd Wed. 1994 UCTE 25,457
Greece 3rd Wed. 1994 UCTE 3,133
Italy 3rd Wed. 1994 UCTE 21,070
Luxembourg 3rd Wed. 1994 UCTE 1,128
Netherlands 3rd Wed. 1994 UCTE 37
Portugal 3rd Wed. 1994 UCTE 4,948
Spain 3rd Wed. 1994 UCTE 20,714
Series 2
Germany 3rd Wed. 1996 UCTE 9,100
Series 3
Czech Republic 3rd Wed. 1999 UCTE 2,175
Hungary 3rd Wed. 1999 UCTE 46
Poland 3rd Wed. 1999 UCTE 2,324
Slovak Republic 3rd Wed. 1999 UCTE 2,429
Slovenia 3rd Wed. 1999 UCTE 873
Series 4
UK daily 2002 Country TSO 4,256
Ireland Daily 2002 Country TSO 512
Series 5
Sweden daily 2001 Country TSO 16,180
Denmark daily 2000 Country TSO 10
Series 6
Finland daily 2004 Country TSO 3,044
a: All countries have daily data starting in 2006
b: UCTE: Union for the Coordination of transmission of electricity;
TSO: Transmission system operator
119
Table 5.2: Parameter estimates for diffusion processes
c1 c2 c3 c4 c5 c6 h
N 168 144 108 1,460 2,190 730 10,950
Const. 1486.06 1248.56 654.25 763.47 569.68 207.54 23.45
z 22.73 36.44 17.47 16.92 25.12 1.95 44.10
Trend 86.98 5.33 4.84 9.06 -2.07 1.20 -0.01
z 32.44 3.92 3.42 5.57 -2.44 0.33 -0.28
Mo n/a n/a n/a -20.84 -3.51 0.66 n/a
z n/a n/a n/a -22.31 -5.98 1.60 n/a
Fr n/a n/a n/a -20.31 -13.98 1.01 n/a
z n/a n/a n/a -20.31 -22.31 2.31 n/a
Sa -416.47 -207.72 -72.13 -128.22 -67.15 -15.71 n/a
z 32.44 3.92 3.42 -101.66 -97.18 -28.56 n/a
Su -750.21 -328.49 -128.43 -157.64 -72.43 -21.45 n/a
z -13.80 -26.70 -12.23 -133.32 -103.87 -43.08 n/a
XNY n/a n/a n/a -86.72 -37.25 -11.54 n/a
z n/a n/a n/a -20.08 -12.89 -4.85 n/a
?
2
x
(sine)
375.85 145.36 116.96 134.10 104.99 36.98 3.00
z 18.99 25.19 32.22 35.09 41.12 10.91 7.06
?
x
(phase) 1.33 1.39 1.41 1.23 1.34 1.35 -0.40
z 42.06 49.94 46.71 38.54 47.09 14.26 -2.96
AR(1)* 0.58 0.39 0.59 0.84 0.86 0.91 0.52
z 18.95 11.32 21.52 95.98 87.02 74.92 103.92
a* 0.54 0.94 0.53 0.18 0.15 0.09 0.65
z 10.24 10.68 11.38 17.15 13.20 6.94 68.02
?*
Jan 499.71 133.17 88.72 65.21 46.42 23.69 17.21
Feb 316.67 94.92 53.10 45.35 42.27 23.84 16.15
Mar 366.30 119.39 64.96 67.47 41.32 21.19 19.15
Apr 453.41 142.96 79.08 79.32 51.52 31.36 14.49
May 400.48 135.97 55.56 92.75 48.67 33.66 16.28
Jun 387.02 132.80 59.94 45.64 46.51 34.84 16.54
Jul 427.82 116.79 55.50 20.17 30.71 12.12 18.07
Aug 305.51 97.78 50.71 69.65 12.10 7.77 20.91
Sep 389.43 122.02 61.45 23.50 18.56 7.65 20.21
Oct 387.15 120.30 64.95 31.66 27.65 11.17 22.63
Nov 432.38 108.56 73.35 39.00 35.50 19.57 21.50
Dec 414.69 163.85 85.40 96.82 55.91 42.84 17.68
*For series 1-3, based on 2006-7 data; all other estimates based on pre-2006 data
120
Table 5.3: Correlation coefficients
a
among different series
c1 c2 c3 c4 c5 c6 h
c1 1.000
c2 0.8814* 1.000
c3 0.9016* 0.8730* 1.000
c4 0.4554* 0.2976* 0.4927* 1.000
c5 0.5170* 0.3897* 0.6032* 0.9231* 1.000
c6 0.4588* 0.3672* 0.5573* 0.8496* 0.9418* 1.000
h -0.067 0.014 -0.036 -0.038 -0.033 -0.020 1.000
*p<0.05; all coefficients based on 2006-7 data
a: The correlation coefficient between series x
t
i
and x
t
j
and the corresponding p-value
are computed as
ˆ ? =
(x
t
i
? x
i
)(x
t
j
? x
j
)
t=1
T
i , j
?
(x
t
i
? x
i
)
2
t=1
T
i, j
?
(x
t
j
? x
j
)
2
t=1
T
i, j
?
; p = 2*ttail T
i, j
?2, ˆ ? T
i, j
?2 / 1? ˆ ?
2
( )
where T
i, j
refers to the number of days for which both series have valid entries.
121
Table 5.4: Parameter estimates from options pricing formula
Original Model using B
t
Model using B
t
/ 10
1
?
-2,967,479 1
?
-2,955,860
2
?
-3,134,951 2
?
-2,991,328
Good. of fit:
a
0.8055 Good. of fit:
a
0.9232
EV
S (MT) K
ˆ
(MT)
?ˆ
EV
S (MT) K
ˆ
(MT)
?ˆ
10 -177 60 10 -833 282
20 -354 119 20 -1,667 564
30 -532 179 30 -2,500 846
40 -709 239 40 -3,334 1,128
50 -886 299 50 -4,167 1,410
60 -1,063 358 60 -5,000 1,692
70 -1,240 418 70 -5,834 1,974
80 -1,418 478 80 -6,667 2,256
90 -1,595 537 90 -7,500 2,537
100 -1,772 597 100 -8,334 2,819
110 -1,949 657 110 -9,167 3,101
120 -2,126 717 120 -10,001 3,383
130 -2,304 776 130 -10,834 3,665
140 -2,481 836 140 -11,667 3,947
150 -2,658 896 150 -12,501 4,229
160 -2,835 955 160 -13,334 4,511
170 -3,012 1,015 170 -14,168 4,793
180 -3,189 1,075 180 -15,001 5,075
190 -3,367 1,135 190 -15,834 5,357
200 -3,544 1,194 200 -16,668 5,639
Shaded areas: Economically plausible parameter ranges for ?
a: Goodness of fit=
?
=
? ?
T
t
t t t t
P E P P E P
1
2 2
]) [ /( ])
ˆ
[
ˆ
(
Figures
Figure 1.1: EUA price and trading volumes, Phase I
122
Figure 1.1: EUA price and trading volumes, Phase I
Figure 2.1: Allowance allocation and emissions by EU member
Figure 2.2: Allowance allocation and emissions by sector (total values)
0
50
100
150
200
250
300
350
400
450
500
D
E
P
L
G
B
I
T
E
S
M
i
l
l
i
o
n
t
o
n
s
C
O
2
0
200
400
600
800
1000
1200
1400
1600
Power & Heat
M
i
l
l
i
o
n
t
o
n
s
C
O
2
123
Figure 2.1: Allowance allocation and emissions by EU member country
Figure 2.2: Allowance allocation and emissions by sector (total values)
E
S
F
R
C
Z
N
L
G
R
B
E
F
I
P
T
A
T
S
K
H
U
D
K
S
E
I
E
E
E
L
T
S
I
Country
Power & Heat Metals Cement &
lime
Oil & Gas Glass &
ceramics
C
Y
L
V
L
U
M
T
Allocation
Emissions
Pulp & Paper
Allocation
Emissions
Figure 2.3: Allowance allocation and emissions by sector (percent of total)
9.6
9.0
7.6
a.) Allocation by sector (%)
8.3
8.9
7.3
124
Figure 2.3: Allowance allocation and emissions by sector (percent of total)
69.9
2.0 1.8
a.) Allocation by sector (%)
Power & Heat
Metals
Cement & lime
Oil & Gas
Glass & ceramics
Pulp & Paper
72.3
1.7 1.5
b.) Emissions by sector (%)
Power & Heat
Metals
Cement & lime
Oil & Gas
Glass & ceramics
Pulp & Paper
Figure 2.3: Allowance allocation and emissions by sector (percent of total)
Power & Heat
Cement & lime
Glass & ceramics
Pulp & Paper
Power & Heat
Cement & lime
Glass & ceramics
Pulp & Paper
125
Figure 3.1: EUA, coal, gas, DAX and reservoir levels
Figure 4.1: Spark spread
Figure 4.2: Dark spreads and green dark spreads in Germany
126
Figure 4.1: Spark spread and green (clean) spark spread in the UK
Figure 4.2: Dark spreads and green dark spreads in Germany
127
Figure 5.1: Available electricity consumption data, pre-2006
128
Figure 5.2: Weighted average precipitation in the EU
129
Figure 5.3: EUA price, prediction and forward price for Phase II
Figure 5.4: Sensitivity of prediction to variance of future electricity demand
Appendix A
130
Appendices
Appendix A: Proof of Equation (3.8)
Derivation of Equation (3.8)
I start by restating equation (3.7):
(3.7) ( ) ( ) ( )
? ? ? ?
+ =
?
+ =
?
+ =
?
+ =
? ? ? + + ? + =
T
t k
k k k
T
t k
k k k
T
t k
k k
T
t k
k
E b g N F E F d E
1
1
1
1
1
1
1
] [ * ] [ ] [ ? ? ?
If allowance and fuel prices have the Markov property such that
t t t
P P E ? =
+
] [
1
where
t
P represents any price, r + = 1 ? the discount factor and r the interest rate, at k=T-1
it must be that
(A1) ( ) ( ) ] [
1 1 1 T T T T T T T
E h F F d ? ? ? + ? + =
? ? ?
? ?? ?
where ? ) ( b g N h + ? . Now I move one period back to k=T-2:
( ) ( ) ] [ ] [ ) (
1 1 2 1 1 2 1 2
2
1 T T T T T T T T T T t T T
E E h F F F F d ? ? ? + ? ? ? + ? + ? + + = +
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ?
Substituting (A1) for ?
T
and rearranging yields
(A2) ( ) ( ) ] [ ) ( ) 1 (
1 2 1 2 1 2
2
1 ? ? ? ? ? ? ?
? ? ? + ? + + = +
T T T T T T T
E h F F d ? ? ? ? ? ?
Appendix A
131
Moving another period back to k=T-3:
( )
( ) ] [ ] [ ] [
) (
1 1 2 1 2 3 2
1 2 1 3 2 3
2 3
1 2
T T T T T T T T T
T T T T T T T T T T
E E E h
F F F F F F d
? ? ? + ? ? ? + ? ? ? +
? + ? + ? + + + = + +
? ? ? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
As before, I substitute (A1) for
T
? and simplify:
( )
( ) ] [ ] [
) ( ) 1 (
1 2 1 2 3 2
2 1 3 2 3
2 3
1 2
? ? ? ? ? ?
? ? ? ? ? ? ?
? ? ? + ? ? ? +
? + ? + + + = + +
T T T T T T
T T T T T T T
E E h
F F F F d ? ? ? ? ? ? ? ? ?
Now I further substitute (A2) for
1
) 1 (
?
+
T
? ? to get
(A3) ( ) ( ) ] [ ) ( ) 1 (
2 3 2 3 2 3
2 3
2
2
? ? ? ? ? ? ?
? ? ? + ? + + + = + +
T T T T T T T
E h F F d ? ? ? ? ? ? ? ?
The next step would be to move to period k=T-4 and successively substituting (A1),
(A2) and (A3). However, the general solution is apparent:
(A4) ( ) ] [ ) (
1 1
1
1 t t t
GC
t
GC
t
T
t k
k T
t
T
t k
k T
t
E h F F d ? ? ? + ? + =
? ?
=
+ ?
?
=
?
? ?
? ? ? ?
The first term on the RHS can be re-written as
?
=
?
?
T
t k
k T
t
? ? ? * *
1
. Dividing (A4)
by the summation term on the LHS yields the result:
(3.8) ( )
( )
? ?
?
?
?
?
?
? ? ?
+ +
?
+ =
T
t
k T
t t t
T
t
k T
t t
t t
E
b g N
F F
d
?
?
?
?
?? ?
] [
1 1
1
Appendix B
132
Appendix B: Proof of Lemma 1 and Eq. (4.9)
Proof of Lemma 1
Differentiating (4.2) w.r.t. p and rearranging gives
(
¸
(
¸
=
(
¸
(
¸
(
(
¸
(
¸
0
1
/
/
p e
p q
C C
C C
i
i
i
ee
i
qe
i
qe
i
? ?
? ?
Solving for the effect of a price change on output and emissions yields
(B1)
( ) 0 0
0
2
> ? ? ? >
?
?
=
>
?
=
i
qe
i
ee
i
i
i
i
qe
i
i
i
ee i
C C C
C
p
e
C
p
q
?
?
?
?
Similarly, differentiating (4.2) w.r.t. the permit price and solving yields
(B2)
0
0
<
?
?
=
<
?
=
i
i
i
i
i
qe
i
C
e
C
q
??
?
??
?
To analyze the effect of the dominant firm’s output on output price p and
permit price ?, differentiate (4.3) w.r.t. q
1
and rearrange:
Appendix B
133
(B3)
(
¸
(
¸
? ?
=
(
(
(
(
¸
(
¸
(
(
(
(
¸
(
¸
|
|
¹
|
\
|
? ? ?
? ?
? ?
= =
= =
0
1
1
1
2 2
2 2
P
q
p
q
p
x x
p
q
P
q
P
N
i
i
N
i
i
N
i
i
N
i
i
?
?
?
??
?
?
??
?
?
?
??
?
Solving for the effect of q
1
on the permit price:
(B4)
(
¸
(
¸
? ? +
? ?
=
? ? ? ? ?
?
= = = = =
=
N
i
N
i
N
i
i i
N
i
i i
N
i
i
N
i
i
x
p
q
p
x q
P
x
p
x
P
q
2 2 2 2 2
2
1
??
?
?
?
?
?
??
?
??
?
?
?
?
??
Because P’<0 and e
i
=x
i
for i=(2,…,N), it follows immediately from (B1) that
the numerator is positive. As for the denominator, the fist term is negative from (B2).
In order to show that (B4) is negative I have to show that the term in the brackets is
positive, i.e. that
(B5) 0
?
2 2 2 2
> ? ? ?
? ? ? ?
= = = =
N
i
N
i
i i
N
i
N
i
i i
x
p
q
p
x q
??
?
?
?
?
?
??
?
Substituting (B1) and (B2), this is equivalent to showing that
(B6) 0
?
2 2 2 2
>
?
?
?
?
?
?
?
? ? ? ?
= = = =
N
i
i
i
N
i
i
i
ee
N
i
i
i
qe
N
i
i
i
qe
C
C
C C
to prove the inequality in (B5). Separating out the a single firm, it is clear that
Appendix B
134
( )
( )
( )
0
1
2
2
2
>
?
=
?
?
?
i
i
i
qe
i
i
ee
i
C C C
which enables me to express (B6) as
0
1
?
2
>
?
?
+
?
?
?
=
N
j i
i
i
j
qe
i
qe
j
ee
i
i
C C C C
Noting the symmetry between i/j and j/i multiplications and dropping the first
(positive) term, I can express this as
(B7) 0
2
?
2
>
?
? +
?
< ?
N
j i
i
j
qe
i
qe
i
ee
j
j
ee
i
C C C C C C
Squaring both sides of the numerator in (B7) yields
( ) ( ) ( )
2
?
2 2
4 4 2
j
qe
i
qe
j
ee
j
i
ee
i
i
ee
j
i
ee
j
j
ee
i
j
ee
i
C C C C C C C C C C C C C C > > + +
where the second inequality comes from the fact that
( ) ( )
2 2
j
qe
i
qe
j
ee
j
i
ee
i
i
qe
i
ee
i
C C C C C C i C C C > ? ? >
Subtracting the RHS of the first inequality completes the proof:
(B8) ( ) ( ) ( ) 0 0 2
! !
2 2 2
> ? ? > ? = + ?
i
ee
j
j
ee
i
i
ee
j
i
ee
j
j
ee
i
j
ee
i
C C C C C C C C C C C C
Appendix B
135
Now I derive the sign of the other three expressions in Lemma 1 by solving
(B3) for the effect of firm 1’s output on the output price and using (B2) and (B5):
(B9) 0
/
2
2
1
<
? ? +
?
=
?
?
=
=
P
x
x P
q
p
N
i
i
N
i
i
??
?
?? ?
?
?
because the numerator is positive. Finally, differentiating (4.3) w.r.t. x
1
gives
(
¸
(
¸
?
=
(
(
(
(
¸
(
¸
(
(
(
(
¸
(
¸
? ?
? ?
? ?
= =
= =
1
0
1
1
2 2
2 2
x
p
x
p
x x
p
q
P
q
P
M
i
i
M
i
i
N
i
i
N
i
i
?
?
?
??
?
?
??
?
?
?
??
?
Solving this yields
(B10) 0
/
2
1
>
?
=
?
=
N
i
i
p q
x
? ?
?
??
(B11) 0
/
2
1
>
?
?
=
?
=
N
i
i
q
x
p
?? ?
?
?
Proof of equation (4.9)
Keeping in mind that 0 /
1
< q ? ?? , I re-write (4.9) as
1
1
?
1
1
/
/
/
/
q
q p
x
x p
? ??
? ?
? ??
? ?
<
Appendix B
136
Substituting (B4) and (B9)-(B10) into this expression and simplifying yields
(B12)
?
?
?
?
=
=
=
=
?
<
?
N
i
i
N
i
i
N
i
i
N
i
i
p x
x
p q
q
2
2
?
2
2
/
/
/
/
? ?
?? ?
? ?
?? ?
Multiplying both sides by the two denominators (again reversing the
inequality) and bringing both terms to the left hand side gives
(B13) 0
!
2 2 2 2
> ? = ?
? ? ? ?
= = = =
N
i
N
i
i i
N
i
N
i
i i
x
p
q
p
x q
??
?
?
?
?
?
??
?
which I prove above.
Appendix C
137
Appendix C: Derivation of variance and covariance of future emissions
Result 1: Variance of future CO
2
emissions
The variance of
T
t
G is defined by
(C1)
? ? ?
+ = + = + =
+ = =
T
t k
T
k u
u k s
T
t k
k s
T
t s t
g g Cov g Var G Var s
1 1 1
2
] , [ 2 ] [ ] [
The variance of g
t
and the covariance between g
t
and g
u
are
(C2)
| |
| |
( ) ] [ ] , [ 2 ] [
] [
2 2
1
2
1
1
t s t t s t s
c
t s
c
t s t s
h Var h c Cov c Var
c Var
c K Var g Var
? ? ?
?
?
+ ? =
=
+ =
(C3)
( )( ) | |
( )( ) | |
{ }{ } | |
( )( ) | | ( )( ) | |
( )( ) | | ( )( ) | |
( ) ] , [ ] , [ ] , [ ] , [
] [ ] [ ] [ ] [
] [ ] [ ] [ ] [
]) [ ( ] [ ]) [ ( ] [
] [ ] [
] [ ] [ ] , [
2 2
2 2
2 2 2
2
2
u t s u t s u t s u t s
u s u t s t s u s u t s t s
u s u t s t s u s u t s t s
u s u u s u t s t t s t s
c
u s
c
u
c
t s
c
t s
u s u t s t s u t s
c h Cov h c Cov h h Cov c c Cov
c E c h E h E h E h c E c E
h E h h E h E c E c c E c E
h E h c E c h E h c E c E
c E c c E c E
g E g g E g E g g Cov
? ? ? ?
?? ??
? ? ?
? ? ?
?
? ? + =
? ? ? ? ? ?
? ? + ? ? =
? ? ? ? ? ? =
? ? =
? ? =
Combining (C2) and (C3) establishes the result shown in equation (5.12)
Appendix C
138
(5.12)
( )
( )
? ?
?
+ = + =
+ =
? ? + +
+ =
T
t k
T
k u
u t s u t s u t s u t s
T
t k
t s t s t
c h Cov h c Cov h h Cov c c Cov
h Var c Var s
1 1
2 2
1
2 2 2
] , [ ] , [ ] , [ ] , [ 2
] [ ] [
? ? ? ?
? ?
Result 2: Generalization of the variance for different volatilities
I start by restating the equation (5.17): The variance of
t
c and
t
h for t s ? ? 0
is
(5.17) h c x dy y i e x Var
x
t
s
t a
t s
x
, )] ( [ ] [
2 ) ( 2
= =
?
? ?
?
?
Suppose that at time s, we’re in month 5 and want to calculate the variance of
consumption/precipitation in month 8. Using the notation defined in the text that
{ } i t i t t i t = = ) ( : min )] ( [
min
, we have that ] 9 [ ] 8 [ ] 7 [ ] 6 [
min min min min
t t t t t s < < < < < . I
now split up the integral in (17) into four integrals with constant volatility:
dy e dy e
dy e dy e x Var
x
t
t
y t a
x
t
t
y t a
x
t
t
y t a
x
t
s
y t a
t s
x x
x x
] 8 [ ] 7 [
] 6 [ ] 5 [ ] [
2
] 8 [
) ( 2 2
] 8 [
] 7 [
) ( 2
2
] 7 [
] 6 [
) ( 2 2
] 6 [
) ( 2
min
min
min
min
min
min
? ?
? ?
? ?
? ?
? ? ? ?
? ? ? ?
+ +
+ =
Next, I split the exponents such that they match with the new upper limits of
the integrals and move the remainder (a constant) in front:
Appendix C
139
dy e dy e e
dy e e dy e e x Var
x
t
t
y t a
x
t
t
y t a t t a
x
t
t
y t a t t a
x
t
s
y t a t t a
t s
x x x
x x x x
] 8 [ ] 7 [
] 6 [ ] 5 [ ] [
2
] 8 [
) ( 2 2
] 8 [
] 7 [
) ] 8 [ ( 2 ]) 8 [ ( 2
2
] 7 [
] 6 [
) ] 7 [ ( 2 ]) 7 [ ( 2 2
] 6 [
) ] 6 [ ( 2 ]) 6 [ ( 2
min
min
min
min min
min
min
min min
min
min min
? ?
? ?
? ?
? ?
? ? ? ? ? ?
? ? ? ? ? ? ? ?
+ +
+ =
Because the volatilities are constant within each integral, each of them can be
easily solved:
( ) ( )
( ) ( )
]) 7 [ ( 2
2
]) 7 [ ] 8 [ ( 2
2
]) 8 [ ( 2
]) 6 [ ] 7 [ ( 2
2
]) 7 [ ( 2 ) ] 6 [ ( 2
2
]) 6 [ ( 2
min min min min
min min min min min
1 *
2
] 8 [
1 *
2
] 7 [
*
1 *
2
] 6 [
* 1 *
2
] 5 [
* ] [
t t a
x
x t t a
x
x t t a
t t a
x
x t t a s t a
x
x t t a
t s
x x x
x x x x
e e e
e e e e x Var
? ? ? ? ? ?
? ? ? ? ? ? ? ?
? + ? +
? + ? =
?
?
?
?
?
?
?
?
Multiplying out and some rearranging gives
( ) ( )
( ) ¦
)
¦
`
¹
¦
¹
¦
´
¦
? + ? +
? + ?
=
? ? ? ?
? ? ? ?
) ( 2 2 2 ]) 8 [ ( 2 2 2
]) 7 [ ( 2 2 2 ]) 6 [ ( 2 2 2
] 5 [ ] 8 [ ] 8 [ ] 7 [
] 7 [ ] 6 [ ] 6 [ ] 5 [
2
1
] [
min
min min
s t a
x x
t t a
x x
t t a
x x
t t a
x x
x
t s
x x
x x
e e
e e
a
x Var
? ? ? ?
? ? ? ?
which can be generalized to
(5.19) ( )
)
`
¹
¹
´
¦
? + + ? =
? ? + ? ?
?
=
?
)] ( [ )] ( [ ] 1 [ ] [
2
1
] [
2 ) ( 2 2 ]) 1 [ ( 2
1 ) (
) (
2 2
min
s i e t i e k k
a
x Var
x
s t a
x
k t t a
t i
s i k
x x
x
t s
x x
? ? ? ?
Result 3: Covariance of x on two different days
The covariance between
t
x and
u
x , for h c x , = and u t s ? ? is given by
Appendix C
140
( )( ) | |
(
¸
(
¸
=
? ? =
? ?
? ? ? ?
?
? ?
?
? ?
? ? ? ? dW i e dW i e E
x E x x E x E x x Cov
x
u
s
u
x
t
s
t
s
u s u t s t s u t s
x x
)] ( [ * )] ( [
] [ ] [ ] , [
) ( ) (
I split up the second integral into two parts and pull out the constant term:
(
(
¸
(
¸
|
|
¹
|
\
|
+ =
? ? ?
? ? ? ? ? ? ? ?
?
? ?
?
? ? ?
?
? ?
? ? ? ? ? ? dW i e dW i e e dW i e E x x Cov
x
u
t
u
x
t
s
t t u
x
t
s
t
s u t s
x x x x
)] ( [ )] ( [ * )] ( [ ] , [
) ( ) ( ) ( ) (
Multiplying out gives
(
¸
(
¸
+
(
¸
(
¸
=
? ?
? ?
? ? ? ?
? ? ? ? ? ?
?
? ?
?
? ?
?
? ?
?
? ? ?
? ? ? ?
? ? ? ?
dW i e dW i e E
dW i e dW i e E e x x Cov
x
u
t
u
x
t
s
t
s
x
t
s
t
x
t
s
t
s
t u
u t s
x x
x x x
)] ( [ * )] ( [
)] ( [ * )] ( [ ] , [
) ( ) (
) ( ) ( ) (
The second term is the expectation of the product of two stochastic processes
occurring during non-overlapping time periods. Because a Wiener process is iid, this
term drops out. Using the fact that dt dW =
2
) ( establishes the result:
(5.20)
] [ *
) )]( ( [
)] ( [ * )] ( [ ] , [
) (
2 2 ) ( 2 ) (
) ( ) ( ) (
t s
t u
x
t
s
t
s
t u
x
t
s
t
x
t
s
t
s
t u
u t s
x Var e
dW i e E e
dW i e dW i e E e x x Cov
x
x x
x x x
? ?
? ? ? ?
? ? ? ? ? ?
=
(
¸
(
¸
=
(
¸
(
¸
=
?
? ?
?
?
? ? ?
?
? ?
?
? ? ?
? ?
? ? ? ?
141
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