Case Studies On Volatility Issues In Financial Markets

Description
A financial market is a market in which people and entities can trade financial securities, commodities, and other fungible items of value at low transaction costs and at prices that reflect supply and demand. Securities include stocks and bonds, and commodities include precious metals or agricultural goods.

CASE STUDIES ON VOLATILITY ISSUES IN FINANCIAL MARKETS
Abstract:Studies of asset returns time-series provide strong evidence that at least two stochastic factors drive volatility. The ?rst essay investigates whether two volatility risks are priced in the stock option market and estimates volatility risk prices in a crosssection of stock option returns. The essay ?nds that the risk of changes in short-term volatility is signi?cantly negatively priced, which agrees with previous studies of the pricing of a single volatility risk. The essay ?nds also that a second volatility risk, embedded in longer-term volatility is signi?cantly positively priced. The di¤erence in the pricing of short- and long-term volatility risks is economically signi?cant - option combinations allowing investors to sell short-term volatility and buy long-term volatility o¤er average pro?ts up to 20% per month. Value-at-Risk measures only the risk of loss at the end of an investment horizon. An alternative measure (MaxVaR) has been proposed recently, which quanti?es the risk of loss at or before the end of an investment horizon. The second essay studies such a risk measure for several jump processes (di¤usions with one- and two-sided jumps and two-sided pure-jump processes with di¤erent structures of jump arrivals). The main tool of analysis is the ?rst passage probability. MaxVaR for jump processes is compared to standard VaR using returns to ?ve major stock indexes over investment horizons up to one month. Typically MaxVaR is 1.5 - 2 times higher than standard VaR, whereby the excess tends to be higher for longer investment horizons and for lower quantiles of the returns distributions. The results of the essay provide one possible justi?cation for the multipliers applied by the Basle Committee to standard VaR for regulatory purposes. Several continuous-time versions of the GARCH model have been proposed in the literature, which typically involve two distinct driving stochastic processes. An interesting alternative is the COGARCH model of Kluppelberg, Lindner and Maller (2004), which is driven by a single Levy process. The third essay derives a backward PIDE for the COGARCH model, in the case when the driving process is VarianceGamma. The PIDE is applied for the calculation of option prices under the COGARCH model.

Contents
Contents List of tables List of graphs 1 Understanding the structure of volatility risks 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Volatility risk prices in models with one and two volatility factors . 1.2.1 The price of a single volatility risk . . . . . . . . . . . . . . . 41.2.2 Models with multiple volatility factors . . . . . . . . . . . . 1.3 Design of the empirical tests . . . . . . . . . . . . . . . . . . . . . . 1.3.1 1.3.2 1.3.3 Construction of option returns . . . . . . . . . . . . . . . . . Data and option-pricing models . . . . . . . . . . . . . . . . Volatility risk factors . . . . . . . . . . . . . . . . . . . . . . 8 10 10 14 20 23 23 37 40 42 42 45 45 48 51 iii v v 1 1 4

1.4 Estimation of volatility risk prices . . . . . . . . . . . . . . . . . . . 1.4.1 1.4.2 Estimation results . . . . . . . . . . . . . . . . . . . . . . . . Evidence from calendar spreads . . . . . . . . . . . . . . . .

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 MaxVar for processes with jumps 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Models and ?rst-passage probabilities . . . . . . . . . . . . . . . . . 2.2.1 2.2.2 2.2.3 CMYD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Double exponential jump-di¤usion . . . . . . . . . . . . . . . CGMY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

2.2.4

Finite-Moment Log-Stable (FMLS) . . . . . . . . . . . . . .

58 60 69 70 70 71 74 88 89 90 96 101

2.3 Empirical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The COGARCH model and option pricing 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The COGARCH process . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Backward PIDE for European options . . . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . ...................... Appendix A. Proof of Proposition 1 Appendix B. Volatility risk prices in a two-factor model Appendix C. Option pricing models References

iv

List of tables Table 1.1. Fixed moneyness and maturity levels Table 1.2. Option data Table 1.3. Estimation errors Table 1.4. VIX and risk-neutral standard deviation Table 1.5. Average option excess returns Table 1.6. Volatility risk prices - one vs. two volatility factors Table 1.7. Volatility risk prices - raw vs. orthogonal volatility risks Table 1.8. Average excess returns on delta-hedged option portfolios Table 1.9. Volatility risk prices - two volatility factors Table 1.10. Volatility risk prices - one volatility factor Table 1.11. Volatility risk prices - abs. market returns and one volatility factor Table 1.12. Average returns to calendar spreads Table 2.1. Goodness-of-?t tests Table 2.2. Levy VaR multiples over Normal VaR - 10 days Table 2.3. Levy VaR multiples over Normal VaR - 20 days Table 2.4. MaxVaR multiples over Normal VaR - 10 days Table 2.5. MaxVaR multiples over Normal VaR - 20 days Table 2.6. Frequency of excessive losses Table 3.1. Put prices - put-call parity vs. PIDE List of graphs Graph 1. Discretization errors - calls Graph 2. Discretization errors - calls Graph 3. COGARCH call option prices Graph 4. COGARCH put option prices v

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1.1

Understanding the structure of volatility risks
Introduction

Recent studies provide evidence that market volatility1 risk is priced in the stock option market (e.g. Chernov and Ghysels (2000), Benzoni (2001), Coval and Shumway (2001), Bakshi and Kapadia (2003), Carr and Wu (2004)). These studies typically ?nd a negative volatility risk price, suggesting that investors are ready to pay a premium for exposure to the risk of changes in volatility. All these studies consider the price of risk, embedded in a single volatility factor. In contrast, time-series studies ?nd that more than one stochastic factor drives asset returns volatility. Engle and Lee (1998) ?nd support for a model with two volatility factors - permanent (trend) and transitory (mean-reverting towards the trend). Gallant, Hsu and Tauchen (1999), Alizadeh, Brandt and Diebold (2002) and Chernov, Gallant, Ghysels and Tauchen (2003) estimate models with one highly persistent and one quickly mean-reverting volatility factor and show that they dominate over one-factor speci?cations for volatility 2. Motivated by the results of the time-series studies, this paper investigates whether the risks in two volatility risks are priced in the stock option market. I construct the volatility factors using implied volatilities from index options with di¤erent maturities (between one month and one year). My main ?nding is that two volatility risks are indeed priced in the stock option market. The risk of changes in short-term volatility is signi?cantly negatively priced. This result is consistent with the previous volatility risk pricing literature, which uses relatively short-term options and ?nds negative price of volatility risk.
This paper only considers stock market volatility and does not touch upon the volatility of individual stocks. For brevity I will refer to market volatility as "volatility". 2 See also Andersen and Bollerslev (1997), Liesenfeld (2001), Jones (2003).
1

1

In addition, the paper reports a novel ?nding - I ?nd that another risk, embedded in longer-term volatility is signi?cantly positively priced. The positive risk price indicates that investors require positive compensation for exposure to long-term volatility risk. This ?nding complements previous results on the pricing of two volatility fac- tors in the stock market: Engle and Lee (1998) ?nd that the permanent (or persis- tent) factor in volatility is signi?cantly positively correlated with the market risk premium, while the transitory factor is not. MacKinlay and Park (2004) con?rm the positive correlation of the permanent volatility factor with the risk premium and also ?nd a time-varying and typically negative correlation of the transitory volatility factor with the risk premium. The di¤erential pricing of volatility risks in the stock option market, found in this paper, is also economically signi?cant, as evidenced by returns on long calendar spreads3. Expected returns on a calendar spread re‡ect mostly the compensations for volatility risks embedded in the two components of the spread. A short position in a negatively priced volatility risk (short-term) combined with a long position in a positively priced volatility risk (long-term) should then have a positive expected return. I calculate returns on calendar spreads written on a number of index and individual options and ?nd that, in full support of the statistical estimations, spreads on puts gain an impressive 20% monthly on average, while spreads on calls gain about 12% on average. Transaction costs would reduce these numbers, but still, a pronounced di¤erence in volatility risk prices can be captured using calendar spreads. To perform the empirical tests I construct time-series of daily returns to options
A long calendar spread is a combination of a short position in an option with short maturity and a long position in an option on the same name, of the same type and with the same strike, but of a longer maturity.
3

2

of several ?xed levels of moneyness and maturity 4. For this construction I use options on six stock indexes and twenty two individual stocks. I estimate volatility risk prices in the cross-section of expected option returns using the Fama-MacBeth approach and Generalized Method of Moments (GMM). These methods have not been applied to a cross-section of option returns in previous studies of volatility risk pricing. The cross-sectional analysis allows easily to decompose implied volatility and to estimate separately the prices of the risks in di¤erent components. This decomposition turns out to be essential for the disentangling of the risks, embedded in long-term implied volatility. What is the economic interpretation of the di¤erent volatility risk prices found in the paper? There is still little theoretical work on the pricing of more than one volatility risks. Tauchen (2004) studies a model with two consumption-related stochastic volatility factors, which generates endogenously two-factor volatility of stock returns. One feature of Tauchen's model is that the risk prices of the two volatility factors are necessarily of the same sign. This paper o¤ers an alternative model, which is able to generate volatility risk prices of di¤erent signs, consistent with the empirical ?ndings. In this model the representative investor's utility function is concave in one source of risk and convex in a second source of risk (for reasonable levels of risk aversion). Such a utility function is closely related to multiplicative habit formation models (e.g. Abel (1990)). Both risk sources exhibit stochastic volatility. The negative volatility risk price is associated to concavity of the utility function, whereas the positively priced volatility risk is associated to convexity of the utility function. The rest of the paper is organized as follows. Section 1.2 discusses the pricing
Such constructs have been used before on a limited scale - e.g. the CBOE used to derive the price of an at-the-money 30-day option to calculate the Volatility Index (VIX) from 1986 to 1993; Buraschi and Jackwerth (2001) use 45-day index options with ?xed moneyness levels close to at-the-money
4

3

of volatility risks in models with one and two volatility factors. Section 1.3 describes the construction of option returns and the volatility risk factors. Section 1.4 presents the estimation results and Section 1.5 concludes.

1.2

Volatility risk prices in models with one and two volatility factors

This section considers ?rst the pricing of volatility risk in a model with a single stochastic volatility factor. It complements previous empirical ?ndings of a nega- tive volatility risk price deriving analytically such a negative price in a stochastic volatility model of the type studied in Heston (1993). A negative price of volatil- ity risk is obtained in this model even if no correlation between asset returns and volatility is assumed. This model is later extended to include a second volatility factor. The predictions of the extended model are consistent with the empirical ?ndings of this paper. Next, the section discusses the evidence for two volatility factors and speci?es the relation that is tested empirically in the rest of the paper.

1.2.1

The price of a single volatility risk

Consider a standard economy with a single volatility risk. The representative investor in this economy holds the market portfolio and has power utility over the terminal value of this portfolio: UT = economy is of the form:
t (S T )1 1

. The pricing kernel process in this

= Et S T

(1.2.1)

Expectation is taken under the statistical measure, S denotes the value of the market portfolio and is the risk aversion coe¢ cient. Assume the following dynamics

4

for S: dSt = DSdt + q St where W tS
t S t

dW S
t

(1.2.2)

is standard Brownian motion. The drift DS is not modeled explicitly, t pS t is

since it does not a¤ect the pricing kernel in the economy. The volatility stochastic. Assume further that
S t

follows a CIR process, which is solution to the

following stochastic di¤erential equation:

d

S t

= k(

t S

+ q

S t

dWt

(1.2.3)

) d t

The model (2)-(3) is Heston's (1993) stochastic volatility model (with possibly time-varying drift DS). The Brownian motions WtS and Wt are assumed here to t be uncorrelated. I discuss below the implication for volatility risk pricing of the correlation between WtS and Wt. Appendix A contains the proof of the following: Proposition 1 The stochastic discount factor for the economy with one sto-

t

chastic volatility factor described in (1.2.1)-(1.2.3) is given by d
t

t

=

t

= dSt + St

d

t

(1.2.4)

where the price of market risk is strictly negative.

is strictly positive and the price of volatility risk

The economic intuition for this negative risk price can be provided by the concavity of the utility function. Higher volatility results in lower expected utility (due to concavity). Then any asset, which is positively correlated with volatility has high payo¤ precisely when expected utility is low. Hence, such an asset acts as

5

insurance and investors are ready to pay a premium for having it in their portfolio. This argument is well known, e.g. from the risk management literature. Given the linear form of of
t

, it is easy to test the model's prediction for the sign

in cross-sectional regressions of the type

E Ri =

M

M i

+

i

+

i

(1.2.5)

where: E [Ri] is expected excess return on test asset i; the betas are obtained in time-series regressions of asset i's returns on proxies for market risk volatility risk (d t); respectively and
i M dSt St

and for

and

are the prices of market risk and volatility risk

is the pricing error. If the two risk factors are normalized to

unit variance, this beta-pricing representation yields risk prices equivalent to those in the stochastic discount factor model (1.2.4) (e.g. Cochrane (2001), Ch. 6). Ang et al. (2004) test a model similar to (1.2.5) and ?nd signi?cantly negative price of volatility risk in the cross-section of expected stock returns. Empirical tests of (1.2.5) involving option returns are reported in Section 1.4 in this paper and also support a negative price of volatility risk. These tests of the simple relation (1.2.5) are consistent with most of the previous studies of volatility risk pricing in the options markets. Benzoni (2001), Pan (2002), Doran and Ronn (2003), among others, use parametric option-pricing models to estimate a negative volatility risk price from option prices and time series of stock market returns. Coval and Shumway (2001) argue that if volatility risk were not priced, then short delta-neutral at-the-money straddles should earn minus the risk free-rate. In contrast, they ?nd 3% average gain per week, which is (tentatively) interpreted as evidence that market volatility risk is negatively priced. Within a general two-dimensional di¤usion model for asset returns, Bakshi and Kapadia

6

(2003) derive that expected returns to delta-hedged options are positive (negative) exactly when the price of volatility risk is positive (negative) and ?nd signi?cantly negative returns. Carr and Wu (2004) construct synthetic variance swap rates from option prices and compare them to realized variance - the variance risk premium obtained in this way is signi?cantly negative. The derivation of (1.2.4) assumes that the two Brownian motions are uncorrelated. If is a non-zero correlation between the Brownian motions, then the and p1 studies have often argued that the negative price of volatility risk they ?nd is due to the negative correlation between changes in volatility and stock returns. So, they focus on the negative term. This is a powerful argument, given that
2

volatility risk price has two components:

. Previous empirical

this negative correlation is among the best-established stylized facts in empirical ?nance (e.g. Black (1976)). Exposure to volatility risk is thus seen as hedging against market downturns and the negative volatility risk price is seen as the pre- mium investors pay for this hedge. However, focusing on the negative correlation leaves aside the second term in volatility risk price ( p1 can be important is indicated e.g. in Carr and Wu (2004) - they ?nd that even after accounting for the correlation between market and volatility risks, there still remains a large unexplained negative component in expected returns to variance swaps. A negative as derived above is consistent with this ?nding. To explore
2

). That such a term

the relative signi?cance of the di¤erent components of volatility risk price, this paper reports empirical tests which include a market risk factor and uncorrelated volatility risks.

7

1.2.2

Models with multiple volatility factors

The study of models with multiple volatility factors has been provoked partly by the observation that the volatility of lower frequency returns is more persistent than the volatility of higher frequency returns. This pattern can be explained by the presence of more than one volatility factors, each with a di¤erent level of persistence. Such factors have been interpreted in several ways in the literature: Andersen and Bollerslev (1997) argue that volatility is driven by heterogeneous information arrival processes with di¤erent persistence. Sudden bursts of volatility are typically dominated by the less persistent processes, which die out as time passes to make the more persistent processes in‡uential. Muller et al. (1997) focus on heterogeneous agents, rather than on heterogeneous information processes. They argue that di¤erent market agents have di¤erent time horizons. The short-term investors evaluate the market more often and perceive the long- term persistent changes in volatility as changes in the average level of volatility at their time scale; in turn, long-term traders perceive short-term changes as random ‡uctuations around a trend. Liesenfeld (2001) argues that investors'sensitivity to new information is not constant but time-varying and is thus a separate source of randomness in the economy. He ?nds that the short-term movements of volatil- ity are primarily driven by the information arrival process, while the long-term movements are driven by the sensitivity to news. MacKinlay and Park (2004) study the correlations between the expected market risk premium and two components of volatility - permanent and transitory. The permanent component is highly persistent and is signi?cantly positively priced in the risk premium, suggesting a positive risk-returns relation. The transitory com- ponent is highly volatile and tends to be negatively priced in the risk premium. This component is related to extreme market movements, transitory market regu8

lations, etc. which can be dominating volatility dynamics over certain periods of time. Tauchen (2004) is the ?rst study to incorporate two stochastic volatility factors in a general equilibrium framework5. The two-factor volatility structure is introduced by assuming consumption growth with stochastic volatility, whereby the volatility process itself exhibits stochastic volatility (this is the second source of randomness in volatility). The model generates endogenously a two-factor conditional volatility of the stock return process. It also generates a negative correlation between stock returns and their conditional volatility as observed empirically in data. One feature of his model is that the risk premia on the two volatility factors are both multiples of the same stochastic process (the volatility of consumption volatility) and are necessarily of the same sign. The model thus imposes a restric- tion on the possible values of volatility risk prices. The model with two volatility factors which is tested in this paper does not impose a'priori restrictions on volatility risk prices. In analogy with (1.2.5) I estimate cross-sectional regressions of the type:

E Ri = where:

M

M i

+

LS i

+

LS i

+

i

:

(1.2.6)

- the test assets are options (unhedged and delta-hedged) on a number of stock indexes and individual stocks and E [Ri] denotes expected excess returns on option i; - the betas are obtained in time-series regressions of option i's returns on proxies
In a related work Bansal and Yaron (2004) model consumption growth as containing a persistent predictable component plus noise. Stochastic volatility is incorporated both in the persistent component and the noise. However, only one source of randomness in volatility is assumed in their model, common to both components of consumption growth.
5

9

for market risk

dSt St

and for two volatility risks d

S t

and d

L t

; in particular, I

consider a short- and a and long-term volatility, denoted by superscripts S and L resp. M

,

S

and
i

L

are the prices of market risk and the two volatility risks

respectively, and

is the pricing error.

Equation (1.2.6) presents a three-factor model with a linear stochastic discount factor. Similar speci?cations have been widely and successfully employed in empirical asset-pricing tests. It would be interesting, though, to search for an economic justi?cation for equation (1.2.6). One possibility would be to formally extend the model in (1.2.1)-(1.2.3) by adding a second source of randomness with stochastic volatility to (1.2.2). However, this approach would come up with the prediction that both volatility risk prices are negative. An alternative economic model consistent with (1.2.6) is presented in Appendix B. This model is related to habit-formation models and is less restrictive - it predicts that one volatility risk is always negatively priced, while the price of the second risk can have both signs. Such a model is consistent with the empirical ?nding of this paper that one volatility risk is signi?cantly positively priced.

1.3

Design of the empirical tests

This section describes the construction of the option returns and volatility risk factors, used in testing (1.2.6).

1.3.1

Construction of option returns

I construct daily returns on hypothetical options with ?xed levels of moneyness and maturity. In particular, the ?xed maturity levels allow to focus on possible

10

maturity e¤ects in option returns, which would be blurred if, for example, only options held until expiration are considered. The convenience of working with such constructs is well known and has been exploited in di¤erent contexts. From 1993 till 2004 the Chicago Board of Options Exchange was calculating the Volatility Index (VIX) as the implied volatility of a 30-day option, struck at-the money forward. In a research context, Buraschi and Jackwerth (2001) construct 45-day options with ?xed moneyness levels close to at-the-money. I follow this approach, and extend it to a number of ?xed moneyness levels and maturities, ranging from one month to one year. Options with predetermined strikes and maturities, most likely, were not actu- ally traded on the exchange on any day in the sample. To ?nd their prices, I apply the following two-step procedure. First, I calibrate an option-pricing model to ex- tract the information contained in the available option prices. Next, this estimated model is used to obtain the prices of the speci?c options I need. This approach has only recently been made feasible by the advent of models, which are capable of accurately calibrating options in the strike and the maturity dimensions together. Section 3.2 presents three models of this type. Extracting information from avail- able options to price other options is a standard procedure. This is how prices are quoted in over-the-counter option markets - if not currently observed, the option price is derived from other available prices by interpolation or a similar procedure. Also, options that have not been traded on a given day are marked-to-market in traders'books in a similar way 6. Once the model is estimated, any option price can be obtained o¤ it. I consider
It is possible to avoid the use of an option-pricing model and apply instead some polynomial smoothing on the implied volatilities of observed options. While an obvious advantage of this approach is that observed prices are ?tted exactly, the downside is that when we need to extrapolate to strikes beyond the range of the observed strikes, this procedure is known to be very inaccurate.
6

11

three levels of moneyness for puts and for calls, and for each one of ?ve maturities, as given in the Table 1.1. (The table shows the ratios between the actual strikes employed and the at-the-money forward price.) The maturities are one, three,

six, nine and twelve months and all strikes are at- or out-of-the-money. To cap- ture the fact that variance increases with maturity, the range of strikes increases accordingly. In this way, for each name I construct 30 time-series of option returns. I estimate volatility risk prices both using all separate time-series and using portfolios, constructed from these series. Returns to unhedged options are constructed as follows. On each day I calculate option prices on the grid of ?xed strikes and maturities. Then I calculate the prices of the same options on the following day - i.e. I keep the strikes but use the following day's parameters and spot price and decrease the time to maturity accordingly. I also take into account the cost of carrying the hedge position to the next day. The daily return on a long zero-cost position in the option is the di¤erence between the second day's option price and the ?rst day's option price with interest:

R = O ( 2 ; S 2 ; K1 ; T2 )

O( 1; S1; K1; T1)(1 + r )

(1.3.1)

The indexes 1 and 2 refer to the ?rst and second day and O denotes an option price. S, K, r and T are spot, strike, interest rate and time to maturity respectively, is an estimated set of parameters and = T1 T2. Note that as the spot price

changes from day to day, the strikes used also change, since the grid of moneyness levels is kept constant. Finally, to make the dollar returns obtained in this way comparable across maturities and names I scale these returns by the option price in the ?rst day. To calculate returns to delta-hedged options, the delta-hedge ratio is needed

12

Table 1.1. Fixed moneyness and maturity levels Moneyness levels for ?ve ?xed maturites at which option returns are calculated as for each name. Moneyness is the ratio between option strike and spot.

Maturity 1 m. 3 m. 6 m. 9 m. 12m. 0.90 0.85 0.80 0.80 0.75

Puts 0.95 0.90 0.90 0.90 0.85 1 1 1 1 1 1 1 1 1 1

Calls 1.05 1.10 1.10 1.10 1.15 1.10 1.15 1.20 1.20 1.25

13

as well. I obtain it numerically in the following way: on each day I move up the spot price by a small amount epsilon, calculate the option price at the new spot (everything else kept the same) and divide the di¤erence between the new and old option prices by epsilon. The daily return on a long zero-cost position in the deltahedged option is the di¤erence between the second day's option price and the ?rst day's option price with interest less delta times the di¤erence between the second day's spot price and the ?rst day's spot price with interest:

R = O ( 2 ; S 2 ; K 1 ; T2 )

O( 1; S1; K1; T1)(1 + r )

( S2

S1(1 + r )) (1.3.2)

where

is the delta-hedge ratio and all other parameters are as before.

To make dollar returns comparable across maturities and names, I scale them by the price of the underlying asset. Scaling by the option price is possible, but it disregards the hedging component, which can be much higher than the option one. Of course, the convenient collection of time-series of option returns comes at the price of daily rebalancing - closing the option position at the previous actual strike and maturity and entering into a new position at a new actual strike (but at the ?xed moneyness) and same maturity. By constructing option returns in this way, I assume away the thorny issue of transaction costs. However, I provide an alternative check for the statistical results by considering monthly returns to calendar spreads.

1.3.2

Data and option-pricing models

All the data I use come from OptionMetrics, a ?nancial research ?rm specializing in the analysis of option markets. The "Ivy DB" data set from OptionMetrics 14

contains daily closing option prices (bid and ask) for all US listed index and equity options, starting in 1996 and updated quarterly. Besides option prices, it also contains daily time-series of the underlying spot prices, dividend payments and projections, stock splits, historical daily interest rate curves and option volumes. Implied volatilities and sensitivities (delta, gamma, vega and theta) for each option are calculated as well. The comprehensive nature of the database makes it most suitable for empirical work on option markets. The data sample includes daily option prices of six stock indexes and twentytwo major individual stocks for six full years: 1997 - 20027. Table 1.2 displays their names and ticker symbols. The 1997-2002 period o¤ers the additional bene?t that it can be split roughly in half to obtain a period of steeply rising stock prices (from January 1997 till mid-2000), and a subsequent period of mostly declining stock prices. As a robustness check, results are presented both for the entire period and for the two sub-periods. Table 1.2 presents also the proportion of three maturity groups in the average daily open interest for at- and out-of-the-money options for each name. It is clear that the longer maturities are well represented, even though the shortmaturity group (up to two months) has a somewhat higher proportion in total open interest. To obtain the option prices needed, I ?t a model to the available option prices on each day. The choice among the numerous models that can accurately ?t the whole set of options on a given day is of secondary importance in this study. I perform below a limited comparison between three candidate models and pick the one, which is slightly more suitable for my purpose. My main consideration is accuracy of the ?t, and I avoid any arguments involving the speci?cs of the modeled price process. So, both a di¤usion-based and a pure-jump model are acceptable,
7

1996 was dropped, since there were much fewer option prices available for this year.

15

Table 1.2. Option data The table displays the names and ticker symbols of the options, used in the estimations. The third column shows average implied at-the-money volatility over 1997 - 2002 for each name. Names are later sorted according to implied volatility in forming portfolios of option returns. The last three columns show the proportion of three maturity groups in the average daily open interest for at- and out-of-the-money options over 1997 - 2002 for each name. Average implied vol. 0.43 0.37 0.56 0.34 0.30 0.36 0.52 0.26 0.32 0.45 0.35 0.45 0.48 0.43 0.36 0.29 0.39 0.65 0.63 0.57 0.34 0.24 0.23 0.55 0.53 0.35 0.45 0.24 Maturity < 2 m. 2-7 m. > 7m. 0.37 0.37 0.35 0.24 0.70 0.34 0.38 0.78 0.33 0.38 0.39 0.35 0.35 0.35 0.28 0.31 0.30 0.38 0.32 0.38 0.31 0.68 0.35 0.37 0.31 0.38 0.57 0.84 0.50 0.48 0.49 0.52 0.29 0.49 0.49 0.22 0.50 0.51 0.46 0.48 0.53 0.49 0.55 0.54 0.48 0.48 0.56 0.50 0.54 0.32 0.47 0.48 0.45 0.48 0.40 0.16 0.14 0.14 0.16 0.24 0.01 0.17 0.13 0.00 0.16 0.11 0.16 0.17 0.12 0.17 0.17 0.16 0.22 0.14 0.11 0.12 0.15 0.01 0.18 0.15 0.24 0.14 0.04 0.00

Company name

Ticker

Amgen AMGN American Express AXP AOL AOL Boeing BA Bank Index BKX Citibank C Cisco Systems CSCO Pharmaceutical Index DRG General Electric GE Hewlett-Packard HWP IBM IBM Intel INTC Lehman Brothers LEH Merryll Lynch MER Phillip Morris MO Merck MRK Microsoft MSFT National Semicond. NSM Nextel Communic. NXTL Oracle ORCL P?zer PFE Russel 2000 RUT S&P 500 SPX Sun Microsystems SUNW Texas Instruments TXN Wal-mart Stores WMT Gold Index XAU Oil Index XOI

16

both models with jumps in volatility and in the price process can be used, etc. It turns out that models, which are conceptually quite di¤erent, perform equally well for my purpose. I focus on the following three models: A stochastic volatility with jumps (SVJ) model studied by Bates (1996) and Bakshi et al. (1997), a double-jump (DJ) model, developed in Du¢ e et al. (2000) and a pure-jump model with stochastic arrival rate of the jumps (VGSA), as in Carr et al. (2003). Appendix C presents some details on the three models. A full-scale comparison between the models is not my purpose here (see Bakshi and Cao (2003) for a recent detailed study). I only compare their pricing accuracy. To do this, I estimate the three models on each day in the sample of S&P500 options (1509 days for the six years). I employ all out-of-themoney options with strike to spot ratio down to 65% for puts and up to 135% for calls, and maturity between one month and one year (140 options per day on average). The main tool for estimation is the characteristic function of the risk-neutral return density, which is available in closed form for all three models (see Appendix B). Following Carr and Madan (1998), I obtain call prices for any parameter set, by inverting the generalized characteristic function of the call price, using the Fast Fourier Transform8. I obtain put prices by put-call parity. I then search for the set of parameters, which minimizes the sum of squared di¤erences between model prices and actual prices. The estimation results are as follows: For the DJ model - 47 days with average % error (A.P.E.) above 5% and average A.P.E. of 2.28% in the remaining days. For the SVJ model - 55 days and 2.42% resp. For the VGSA model - 72 days and 2.48% resp. It is reasonable to apply some ?lter, when working with estimated, not actual option prices. The above
This procedure is strictly valid only for European-style options. Using only at- and outof-the-money options mitigates the bias introduced from applying this procedure to options on individual stocks which are traded American-style.
8

17

estimations show that by discarding days when average A.P.E. is above 5%, fewer than 5% of the data for S&P500 options are lost. I apply this cut-o¤ in all future estimations. As expected, the richest model (DJ) performs best, but the di¤erences are modest. So, when choosing among the models the advantages of using a more parsimonious model should also be considered. As discussed in Bakshi and Cao (2003), the estimation of large option pricing models on individual names is hindered by data limitations. In their sample, the majority of the 100 most actively traded names on the CBOE have on average less than ten out-of-the-money options per day. That's why they need to pool together options from all days in a week to perform estimations. Such an approach is not feasible in this study, since pooling across the days in a week may hinder the construction of returns to options with ?xed moneyness. So I employ only the names with largest number of options per day and further discard days with insu¢ cient number of options. On average, for all names in the sample, fewer than 12 out-of-the-money options are available on 2.4% of the days, fewer than 15 - on 6.8% of the days and fewer than 18 on 11.3% of the days. Obviously, increasing the degrees of freedom would come at the price of giving up an increasing amount of the data. That is why, mostly a data-related consideration leads me to choose VGSA (which has the fewest parameters), while sacri?cing some accuracy. An additional bene?t of such a choice is gain in computational speed. I discard all days with fewer than 12 out-of-the-money options. Table 1.3 presents the average number of options, used in the estimations for each name, and the percentage errors achieved. I discard days with errors above 5% (usually not more than 4-5% of all days). The errors in the remaining days are around 3% and often less, which is quite satisfactory. This is often within

18

Table 1.3. Estimation errors The table displays, for each name, the average number of options per day, used in the estimations (after discarding days w it h le s s t h a n 1 2 o p t io n s ) , t h e p r o p o r t io n o f e s t im a t io n s with average percentage error (A.P.E.) greater than 5% (also discarded), and the average A.P.E.. in the remaining days.

Ticker

Aver. daily options 28 30 35 26 161 30 28 58 35 27 42 36 26 31 33 28 39 19 23 23 34 73 121 34 31 30 35 53

Days with A.P.E.>5% 0.041 0.020 0.066 0.042 0.015 0.044 0.019 0.056 0.040 0.039 0.019 0.009 0.045 0.017 0.079 0.050 0.017 0.036 0.031 0.016 0.071 0.040 0.029 0.007 0.004 0.080 0.021 0.092

Rem aining A.P.E. 0.028 0.029 0.026 0.033 0.024 0.032 0.025 0.035 0.031 0.030 0.025 0.025 0.027 0.027 0.035 0.030 0.025 0.028 0.025 0.026 0.030 0.031 0.025 0.024 0.025 0.030 0.033 0.035

AM GN AXP AOL BA BKX C CSCO DRG GE HW P IB M IN T C LEH M ER MO M RK M SFT NSM NXTL ORCL PFE RUT SPX SUN W TXN WMT XAU XOI

19

the bid-ask spread, in particular for out-of-the-money options. Armed with the estimated parameters for each day, it is easy to generate model prices and returns at the required strikes and maturities.

1.3.3

Volatility risk factors

Cross-sectional estimations of risk prices involve regressions of excess returns on measures of respective risks. I construct these risk measures in two steps. First, I calculate, on each day in the sample, proxies for the market's best estimate of market volatility, realized over di¤erent subsequent periods (from one month to one year). Second, I calculate the daily changes in these volatility factors to obtain the volatility risk measures (or "volatility risk factors"). The market's estimate of volatility, realized over a given future period is taken to be the price of the volatility swap with the respective maturity. When "the market" is de?ned to be the S&P500 index and the length of the period is one month, this best estimate is precisely the CBOE's Volatility Index (VIX). VIX is currently calculated via a non-parametric procedure9 employing all current atand out-of-the-money short-term options on S&P500. One way to obtain the market's volatility estimates for longer future periods would be to extend this procedure, using options with longer maturities. Alternatively, one can use the estimated model parameters for S&P500 and calculate the standard deviations of the S&P500 risk-neutral distribution at the respective horizons10. I apply the second alternative, mostly for computational convenience. I verify that the two approaches produce very similar results. First, the correSee e.g. Carr and Madan (2001). risk-neutral variance is obtained by evaluating at zero the ?rst and second derivatives of the characteristic function of the risk-neutral distribution at di¤erent horizons. The exact form of the second derivative is quite lengthy, but is readily given by any package, implementing symbolic calculations. The volatility proxy is then the square root of this risk-neutral variance.
10 The 9

20

lation between VIX and the one-month risk-neutral standard deviation over 1997 - 2002 is 99.1%. Next, I compare the predictive power of the two volatility timeseries for realized S&P500 volatility. Following Christensen and Prabhala (1998), I regress realized daily return volatility over 30-day non-overlapping intervals on the two implied volatilities at the beginning of the respective 30-day intervals. Table 1.4 shows the results for 1997 - 2002 and two sub-periods. The estimates involving the VIX and the risk-neutral standard deviation are almost identical. The similar- ity is observed both in the entire period and the two sub-periods. This comparison justi?es the use of the risk-neutral standard deviation. It also provides an indirect check of the quality of the model-based option prices used in constructing option returns. I de?ne the volatility risk factors to be the daily changes of the volatility proxies at the respective horizons. I also normalize the volatility risk factors to unit stan- dard deviation, which helps to avoid scaling problems and allows for comparing the prices of di¤erent volatility risks. The calculation of volatility risk factors from option prices is motivated by previous ?ndings that option-implied volatilities at di¤erent horizons exhibit quite di¤erent behavior, indicating that possibly di¤erent risks are embedded in these volatilities: Poterba and Summers (1986) ?nd that the changes in forward short- term implied volatility (which is approximately the di¤erence between short-and long-term volatility) are of the same sign but of consistently smaller absolute value than the changes in current short-term implied volatility. Engle and Mustafa (1992) and Xu and Taylor (1994) ?nd that the volatility of short-term implied volatility is larger and mean-reverts faster than that of long-term implied volatility.

21

Table 1.4. VIX and risk-neutral standard deviation Panel A shows regression output for ln Rt+30 = + ln V IXt + "t. Rt+30 is the realized daily return volatility of S&P500 over a 30-day period starting at time t. Only non-overlapping intervals are involved. V IXt is the CBOE's Volatility Index calculated at the beginning of each 30-day interval. t-statistics are in parentheses. The two sub-periods are 1/1/1997 - 6/30/2000 and 7/1/2000 - 12/31/2002. Panel B shows regression output for ln Rt+1 = + ln SDt + "t. SDt is risk-neutral standard deviation of S&P500 at 30-day horizon, calculated at the beginning of each 30-day interval.

Panel A. VIX 1997 - 2002 -0.52 .82 0.24 (- 1.17) (4 .9 4 ) 1997 - 2000 -0.62 0.79 0.21 ( -0.99) ( 3.42) 2000 - 2002 -0.52 0.81 0.24 (-0 .7 6 ) 0 ( 3.12)

R2

Panel B. Risk-neutral standard-deviation 1997 - 2002 -0.54 .80 0.24 (- 1.23) (4 .9 2 ) 1997 - 2000 -0.62 0.77 0.21 ( -0.97) ( 3.33) 2000 - 2002 -0.57 0.78 0.24 (-0 .8 6 ) 0 ( 3.14)

R2

22

1.4

Estimation of volatility risk prices

This section presents results on the estimation of market volatility risk prices in the cross-section of expected option returns, as in equation (1.2.6). It also presents evidence on the economic signi?cance of the di¤erence between the estimated risk prices. For this purpose I employ returns on calendar spreads. Table 1.5 shows summary statistics of the excess option returns used in estimations. Panel A shows average excess returns to unhedged options across the twenty-eight names, for each strike level and each maturity. Observe that there is a wide variation in these average returns to be explained. There is also a clear pattern across maturities - returns to puts invariably increase with maturity, while those to calls decrease with maturity. Overall, average returns to calls are sig- ni?cantly positive, while returns to puts are mostly negative and sometimes not signi?cantly di¤erent from zero. Panel B shows average returns to delta-hedged options. The variation in these returns is still considerable. The maturity pattern for puts is preserved and is even more signi?cant than for unhedged options. In- terestingly, returns to longer-term calls now tend to be higher than to short-term ones, in contrast to the unhedged case.

1.4.1

Estimation results

I ?rst estimate the prices of two volatility risk factors with two-step cross-sectional regressions on all individual time-series of excess returns to unhedged options (total of 840 series). I apply the standard procedure of ?nding the betas on the risk factors at the ?rst step, then regressing, for each day in the sample, excess returns on betas, and ?nally averaging the second-step regression coe¢ cients and calculating their standard errors. All regressions involve the market risk, the one-month volatility risk factor 23

Table 1.5. Average option excess returns Panel A shows the average of expected excess returns to unhedged option across all names, in each of the strike and maturity groups. Daily returns are multiplied by 30 (monthly basis). E.g. -0.21 stands for -21% of the option price monthly. Each row refers to one of the ?ve maturity groups. O-T-M columns refer to the most out-of-the-money puts / calls; A-T-M columns refer to at-the-money puts / calls. MID columns refer to puts / calls with intermediate moneyness (as in Table 1). Panel B shows the average of expected excess returns to deltahedged options in the same moneyness and maturity groups. Daily returns are multiplied by 30 (monthly basis) and are now given in % of the spot price. E.g. -0.38 stands for -0.38% of spot monthly. The bottom part of each panel shows the respective t-statistics (average returns divided by standard deviation square root of the number of names). Averages are given for the entire 1997 - 2002 period.

Panel A. Average returns to unhedged options Puts Maturity 1 m. m. m. m. 12 m . O -T -M -0.21 -0.05 -0.01 0.02 0.15 M ID -0.16 -0.05 -0.01 -0.00 0.03 A -T -M -0.04 -0.02 -0.02 -0.01 0.00 C a ls A -T -M 0.26 0.15 0.11 0.10 0.10 M ID 0.58 0.28 0.17 0.16 0.29 O -T -M 0.90 0.46 0.40 0.38 0.67

Panel B. Puts O -T -M -0.38 -0.09 0.01 0.07 0.11 M ID -0.37 -0.09 -0.01 0.08 0.12

Average returns to delta-hedged options C a ls A -T -M -0.16 -0.06 -0.01 0.07 0.14 A -T -M -0.04 0.04 0.08 0.13 0.19 M ID 0.19 0.09 0.07 0.12 0.15 O -T -M -0.013 0.066 0.079 0.14 0.15

t -s t a t is t ic s ( u n h e d g e d o p t io n s ) Maturity 1 m. m. m. m. 12 m . O -T -M -4.29 -2.44 -0.71 1.20 2.65 M ID -3.70 -2.86 -0.80 -0.04 1.90 A -T -M -1.38 -1.61 -1.93 -1.28 0.33 A -T -M 5.88 6.91 7.79 8.36 8.20 M ID 11.43 10.66 8.92 7.42 4.39 O -T -M 9.28 6.99 4.67 4.00 3.70

t -s t a t is t ic s ( d e lt a - h e d g e d o p t io n s ) O -T -M -7.83 -1.82 0.18 1.75 2.37 M ID -5.25 -1.49 -0.08 1.65 2.16 A -T -M -1.68 -0.83 -0.16 1.32 2.20 A -T -M -0.48 0.52 1.47 2.21 2.81 M ID 2.48 1.56 1.29 1.91 2.17 O -T -M -0.153 1.156 1.569 2.25 2.35

24

and one of the longer term volatility risk factors. In this set-up the one-month risk factor is proxy for short-term volatility risk. This choice is consistent with MacKinlay and Park (2004) who ?nd that monthly volatility exhibits features typical for short-term (transitory) volatility, while three- to six-month volatility represents well long-term (permanent) volatility. Panel A in Table 1.6 shows the results of these regressions. Market risk is always positively priced (signi?cant at 5%). One-month volatility risk is always signi?cantly negatively priced,while longer-term volatility risks are always signif- icantly positively priced. These results are to a large extend supported by the crosssectional regressions involving a single volatility factor. Panel B in Table 1.6 presents the results for such single-volatility regressions and show that only the onemonth volatility risk has a negative price (insigni?cant), while all other volatility risk prices are positive and mostly signi?cant. Table 1.6 also shows the importance of a second volatility factor for the explanatory power of the regres- sions. The numbers in parentheses show, for each combination of risk factors, the proportion of timeseries regressions (?rst pass) with signi?cant alphas. While practically all regressions with one volatility factor have signi?cant alphas (96% in all cases), this proportion dramatically falls to about 15% when a second volatility factor is included. Table 1.6 also shows the adjusted R2 in regressing average excess returns on betas. For any combination of two volatility factors, R2 increases by 45%. These results strongly indicate that, ?rst, two volatility risks are indeed priced in the option market and second, that these risks are of di¤erent nature, as evi- denced by the di¤erent sign of the risk prices. All previous studies of volatility risk pricing in the option market use relatively short-term options (maturity about one month) and mostly ?nd a negative risk price. So, my ?nding of a negatively priced short-term volatility risk is consistent with previous empirical studies. However, a

25

Table 1.6. Volatility risk prices - one vs. two volatility factors ( a ll u n h e d g e d o p t io n r e t u r n s t im e - s e r ie s ) The table shows volatility risk prices estimated with two-step cross-sectional regressions on all 840 t im e - s e r ie s o f u n h e d g e d d a ily o p t io n r e t u r n s fo r 1 9 9 7 - 2 0 0 2 . T h e e s t im a t e d re la t io n s a r e :

Ri =

i

+

i

M

M KT +
M M i

m m i1 V OL1

+
LL

i

L

V OLL + "i +
i

E Ri =

+

1m 1m i

+

i

At the second step regessions are run separately for each day and the estimates are then averaged. M KT denotes daily returns on S&P 500, V OL1m denotes daily changes in one-month S&P 500 volatility and V OLL denotes daily changes in one of the 3, 6, 9 or 12-month volatilities (i.e riskneutral standard deviations). The -s are estimated risk prices for each of the risk factors. Shanken corrected t-statistics are shown for each risk price estimate. In parenthesis is the proportion of alphas in the ?rst-pass regression, estimated to be signi?cant at 5%. In square brackets is the adj. R2 in regressing average returns on betas. The two panels show results for two and one volatility factors resp.

Panel A. Two volatility factors R is k p r ic e MKT VOL 1m VOL 3m 0.07 -0.09 0.09 ( 0.17) 0.06 -0.12 0.17 ( 0.16) 0.06 -0.14 0.24 ( 0.15) 0.05 -0.14 0.29 ( 0.16) t-sta t. 2.41 -1.99 1.83 [0.3 9] 2.17 -2.61 3.20 [0.4 0] 1.99 -2.88 3.85 [0.4 2] 1.90 -2.90 4.22 [0.4 2] MKT VOL 1m (0 .9 6 ) MKT VOL 3m (0 .9 6 ) MKT VOL 6m

Panel B. One volatility factor R is k p r ic e 0.07 -0.03 [0.3 4] 0.07 0.05 [0.3 5] 0.07 0.12 ( 0.96) 0.07 0.16 ( 0.96) 0.06 0.21 ( 0.95) t-sta t. 2.44 -0.62

MKT VOL 1m VOL 6m

2.46 1.04

MKT VOL 1m VOL 9m

2.34 2.28 [0.3 6] 2.22 2.84 [0.3 6] 2.14 3.26 [0.3 8]

MKT VOL 9m

MKT VOL 1m VOL 12m

MKT VOL 12m

26

positively priced long-term volatility risk has not been identi?ed before. Table 6 also demonstrates the ability of the three-factor model to capture the variance in option excess returns. One possible concern with the results in Table 1.6 relates to the correlations between the market and the volatility factors. This correlation is well known to be negative and typically high in magnitude and is discussed in Section 1.2.1. For the ?ve volatility risk factors used in the estimations the correlation is -0.60% or less. Besides, the volatility risk factors are highly positively correlated (50-90% in the sample). To eliminate the possible e¤ect of these correlations, I run the regressions also with orthogonal volatility risk factors - I use the component of the one-month volatility risk which is orthogonal to market returns and the component of the longer-term volatility risk which is orthogonal to both the market and the one-month orthogonal volatility risk. Table 1.7 compares the results involving the original (raw) volatility risks with these "orthogonal"volatility risks. When orthogonal volatility risk factors are used (Panel B), the results are qualitatively the same. The signi?cance of the negative price of the one-month factor is sometimes lower. Note that the price of risk in the orthogonal long-term factor is much higher in magnitude compared to the raw factor. The estimations with volatility risk factors, orthogonal to market returns also indicate that speci?c volatility risks are priced in the option market. Investors recognize volatility risks beyond those, due to the negative correlation between changes in volatility and market returns. The above test can be subject to several concerns. First, a more precise evaluation of the explanatory power of the model can be performed using the joint distribution of the errors. However, given the large amount of time series involved, this task is computationally quite demanding, as it involves the computation and

27

Table 1.7. Volatility risk prices - raw vs. orthogonal volatility risks ( a ll u n h e d g e d o p t io n r e t u r n s t im e - s e r ie s ) The table shows volatility risk prices estimated with two-step cross-sectional regressions on all 840 time-series of unhedged daily option returns for 1997 - 2002. The estimated relations are:

Ri =

i

+

i

M

M KT +

m 1 V i

OL1m +

L i

V OLL + "i

E Ri =

M

M i

+

1m 1m i

+

LL i

+

i

At the second step regessions are run separately for each day and the estimates are then averaged. M KT denotes daily returns on S&P 500, V OL1m denotes daily changes in one-month S&P 500 volatility and V OLL denotes daily changes in one of the 3, 6, 9 or 12-month volatilities (volatilities here are risk-neutral standard deviations). The -s are estimated risk prices for each of the three risk factors. Shanken corrected t-statistics are shown for each risk price estimate. Panel A shows regressions with the volatility risk factors V OL1m and V OLL. Regressions in panel B use the component of V OL1m orthogonal to M KT , and the component of V OLL orthogonal to each of the other two factors.

Panel A. Raw vol. R is k p r ic e MKT VOL 1m VOL 3m MKT VOL 1m VOL 6m MKT VOL 1m VOL 9m MKT VOL 1m VOL 12m 0.07 -0.09 0.09 0.06 -0.12 0.17 0.06 -0.14 0.24 0.05 -0.14 0.29 t-sta t. 2.41 -1.99 1.83 2.17 -2.61 3.20 1.99 -2.88 3.85 1.90 -2.90 4.22

Panel B. Orthogonal vol. R is k p r ic e 0.07 -0.06 0.35 0.06 -0.10 0.39 0.06 -0.13 0.42 0.05 -0.13 0.43 t-sta t. 2.42 -0.95 3.74 2.18 -1.84 4.57 2.00 -2.02 4.88 1.92 -2.09 4.92

28

inversion of the covariance matrix of the residuals in the time-series regressions. Second, the standard errors of the estimated parameters are not corrected for heteroskedasticity. This may be an issue when dealing with option data and volatility risks, given the persistence in volatility. Third, unhedged options mix the exposure to the risk in the underlying asset and to volatility risk. It can be argued that investors, seeking exposure to volatility risk will hedge away the risk in the underlyer. So, the price of volatility risk may be better re‡ected in returns to hedged option. Fourth, the di¤erent names are given equal weight, even though their relative importance is highly unequal - for example options on the S&P500 amount to almost half the value of all options in the sample. To address these concerns I apply a second test where I consider delta-hedged instead of unhedged options, apply GMM for the estimation and reduce the number of asset-return series by forming option portfolios. Delta-hedging allows for more precise exposure to volatility risk. GMM handles the heteroskedasticity of errors and allows to test for all errors being jointly equal to zero. The portfolios allow for an e¢ cient implementation of GMM and account to an extent for the relative importance of di¤erent options. Forming portfolios addresses one de?ciency of the returns data as well. Because of insu¢ cient out-of-the-money options on certain days and because sometimes the error of estimation has been too high, there are missing observations for certain days for each returns time-series. Since the omis- sions are relatively few and they come at di¤erent days for di¤erent names, having several names in a portfolio leaves no missing data in the aggregated returns series. To form the portfolios I sort the names in the sample according to their average implied volatility (see Table 1.2). Each portfolio belongs to one of the ?ve maturity groups and one of ?ve volatility quintiles (a total of twenty-?ve portfolios). The di¤erent strikes for each name are weighted by the average open interest for the

29

closest available strikes in the data. For example, I ?nd the proportion of the closest to at-the-money puts on each date and assign the average of these proportions across all days to be the weight of at-the-money puts. I proceed in the same way with the other moneyness levels, both for puts and for calls. The di¤erent names within a portfolio are weighted by the average option value for the name, where the average is taken again across all days in the six-year period. Table 1.8 shows the average excess returns on the portfolios (a total of twenty ?ve) over 1997 - 2002. As in Table 1.3 the numbers are in percent and on a monthly basis. The columns show portfolios arranged from the lowest to the highest average implied volatility of the components. Compared to Table 1.5, the weighted returns (i.e. the portfolio returns) tend to be much lower. There are still portfolios with positive returns, but fewer and with smaller absolute returns. Obviously the larger and less volatile names (in particular the indexes) tend to have lower returns. What is preserved however is the maturity e¤ect - returns to longer-maturity options tend to be higher. Tables 1.9 and 1.10 contain the main result of this paper. Table 1.9 shows volatility risk prices from estimations with two volatility risk factor. GMM estimations with ten Newey-West lags are reported11. Results are reported both for the entire six-year period and separately for 1997 to mid-2000 and from mid-2000 to 2002. The one-month volatility factor is always included; the longer-term factors are included both in their raw form, and only with their component orthogonal to the onemonth volatility risk factor. Using the orthogonal component does not change the remaining estimates. In all cases the market risk is not priced. This can be expected given delta-hedging. The price of one-month volatility risk is negative and
11 Five and twenty lags were also used, producing very similar results which are not reported for brevity.

30

Table 1.8. Average excess returns on delta-hedged option portfolios Five portfolios are formed at each maturity by sorting names according to average implied volatility (see Table 1). Volatility quintiles are numbered from 1 (lowest volatility) to 5 (highest volatility). Average daily excess returns for 1997 - 2002 are multiplied by 30 (monthly basis) and given in %; e.g. -0.29 stands for -0.29% of spot monthly.

Maturity

1 Low vol. -0.29 -0.17 -0.10 -0.07 -0.03

2

3

4

5 High vol. 0.02 0.06 0.07 0.05 0.07

1m 3m 6m 9m 12 m

-0.04 -0.03 -0.03 -0.02 -0.02

-0.03 -0.03 0.00 0.02 0.04

0.00 0.00 0.02 0.06 0.07

31

Table 1.9. Volatility risk prices - two volatility factors (twenty ?ve portfolios) The table shows volatility risk prices estimated with GMM on twenty ?ve portfolios of delta-hedged options. The moment conditions are:

2 6 E h(R

E(R

M

M KT

m 1 V

OL1m

L

V OLL)

3

M

M KT

m 1 V

OL1m

L

V OLL)M K T i7

6h
M

M KT

m 1 V

OL1m
L

6 V OLL)V OL1m 7 = 0 i7 7 g = 6 E h(R 6 6
M

M KT
M M

m 1 V

OL1m

L

V OLL)V OLL i7 7 ) 5 7

6 E (R 4 E(R

1m 1m

L

L

M KT denotes daily returns on S&P 500, V OL1m denotes daily changes in one-month volatility, V OLL
denotes daily changes in one of 3, 6 9 or 12 month volatility (volatilities here are risk-neutral standard deviations). The -s are estimated risk prices for each of the three risk factors. z-statistics are distributed standard normal. Tilded factors (e.g. VgL 3m.) are the components of the respective raw O factors, orthogonal to V OL1m. p-values for the chi-squared test for pricing errors jointly equal to zero are in parenthesis. In square brackets is the adj. R2 in regressing average returns on betas.

1997 - 2002 R is k p r ic e MKT VOL 1m. VOL 3m. 0.00 -0.27 0.01 0.42 ( 0.42) 0.02 -0.31 0.12 z -s ta t. 0.06 -3.77 0.16 2.11 [0.6 9] 0.32 -4.07 1.57

1997 - 2000 R is k p r ic e 0.03 -0.33 -0.02 0.46 ( 0.72) 0.08 -0.46 0.19 z -s ta t 0.47 -3.51 -0.25 2.36 [0.6 8] 1.22 -3.67 1.40

2000 - 2002 R is k p ric e -0.01 -0.11 0.07 0.22 (0 .8 5 ) -0.010.120 .09 z -s ta t -0.07 -1.44 0.78 1.31 [0.2 0] -0.151.601 .39

VgL 3m. O

MKT VOL 1m. VOL 6m.

VgL 6m. O

0.46 ( 0.85) 0.00 -0.29 0.10

3.15 [0.8 6] 0.00 -4.37 1.81

0.68 ( 0.97) 0.03 -0.43 1.15

2.79 [0.9 1] 0.47 -3.83 1.53

0.26 (0 .9 0 ) -0.020.120 .08

1.96 [0.2 3] -0.261.731 .53

MKT VOL 1m. VOL 9m.

VgL 9m. O

0.35 ( 0.90) -0.010.280 .11

3.59 [0.8 6] -0.214.43 2.01

0.47 ( 0.88) 0.01 -0.40 0.16

2.91 [0.9 2] 0.12 -3.88 1.79

0.22 (0 .9 2 ) -0.030.120 .08

2.30 [0.1 9] -0.341.811 .46

MKT VOL 1m. VOL 12m.

VgL 12m. O

0.30 ( 0.85)

3.53 [0.8 5]

0.39 ( 0.80)

2.92 [0.9 0]

0.19 (0 .9 2 )

2.40 [0.1 6]

32

Table 10. Volatility risk prices - one volatility factor (twenty ?ve portfolios) The table shows volatility risk prices estimated with GMM on twenty ?ve portfolios of delta-hedged options. The moment conditions are:

2

E(R
M

M

M KT
V

V

V OL)

3 i7

M KT

V OL)M K T

6 E h(R g=6 6
M

i 7=0 7 M KT
V

V OL)V OL

4 6 E h ( R

E(R

M

M

V

V

) 5 7

M KT denotes daily returns on S&P 500, V OL denotes daily changes in volatility (volatilities are
risk-neutral standard deviations at 1, 3, 6, 9 or 12-month horizons). The -s are estimated risk prices for each of the two risk factors. z-statistics are distributed standard normal. p-values for the chi-squared test for the pricing errors jointly equal to zero are in parenthesis. In square brackets is the adj. R2 in regressing average returns on betas.The sub-periods are 1/1/1997 - 6/30/2000 and 7/1/2000 - 12/31/2002

1997 - 2002 R is k p r ic e MKT VOL 1m. -0.04 -0.21 ( 0.24) -0.06 -0.20 ( 0.11) -0.02 -0.12 ( 0.00) z -s ta t. -0.72 -4.31 [0.6 3] -0.95 -3.87 [0.5 2] -0.35 -2.23 [0.4 2]

1997 - 2000 R is k p r ic e -0.03 -0.26 ( 0.30) -0.06 -0.26 ( 0.05) 0.01 -0.13 ( 0.05) z -s ta t. -0.39 -4.09 [0.6 5] -0.70 -3.84 [0.6 3] 0.16 -1.87 [0.5 7]

2000 - 2002 R is k p r ic e -0.03 -0.09 ( 0.88) -0.01 -0.03 ( 0.79) 0.01 0.01 ( 0.63) z -sta t. -0.43 -1.54 [0.1 0] -0.12 -0.53 [0.0 1] 0.11 0.19 [0.0 1]

MKT VOL 3m.

MKT VOL 6m.

MKT VOL 9m.

0.01 -0.06 ( 0.00) 0.02 -0.03 ( 0.00)

0.24 -1.29 [0.4 5] 0.51 -0.75 [0.4 7]

0.06 -0.05 ( 0.02) 0.07 -0.01 ( 0.02)

0.95 -0.81 [0.6 0] 1.24 -0.25 [0.6 1]

0.01 0.01 ( 0.67) 0.00 0.00 ( 0.78)

0.10 0.16 [0.0 0] 0.06 0.02 [0.0 0]

MKT VOL 12m.

33

highly signi?cant, except for the second sub-period. However, all other volatility risk prices are all positive. The prices of the raw factors are marginally signi?cant (zstatistics about 1.5 - 1.6). Note that the signi?cance is higher for the longer maturity raw factors. However, the risk prices of the orthogonal components are all signi?cant (with two exceptions in the second sub-period). The p-values in all cases are very high (typically 80% or more). The adjusted R2 in regressing average returns on betas is typically high (0.80 or more), except for the second sub-period. We have thus strong indication that two volatility factors explain most of the variation in expected option returns in the data sample. Table 1.10 shows volatility risk prices from estimations with one volatility risk factors, which markedly contrast with the one-factor case. The price of market risk is again insigni?cant. The one-month and three-month volatility risks for the whole period are signi?cantly negatively priced. The prices of longer-term volatility risks are all negative, but not signi?cant. Note that both the signi?cance levels and the absolute magnitude of the volatility risk prices steadily decrease as maturity increases. The same pattern is exactly repeated in the ?rst sub-period. The second sub-period presents mostly insigni?cant estimates. The table also shows p-values for the chi-squared test for all pricing errors being jointly zero. The entire period and the ?rst sub-period have high pvalues for the estimation with the one-month factor (24% and 30% resp.). For longer maturities the p-values decrease sharply. The second sub-period is again di¤erent, showing very high p-values (above 60%) for all maturities. The relation between average returns and betas is now weaker (adjusted R2 about 0.60 or less, and even negligible in the second sub-period). The results in Tables 1.9 and 1.10 clearly show that long-term volatilities con- tain two separate risk components with di¤erent prices. Including both short- and long-term volatility in the regressions helps disentangle these two risk components.

34

The second risk, additional to short-term volatility risk is positively priced. The magnitude of the price of this second risk is typically equal or higher than that of the short-term risk price12. What is the actual maturity of the short-term volatility risk? So far, the onemonth volatility factor is assumed to be the short-term one. However, previous studies have considered various frequencies, sometimes much shorter than a month. It is then possible that the true short-term factor is of much lower maturity, and the onemonth factor also mixes two risks. I address this issue by assuming that the absolute value of market returns is proxy for very short-term (one-day) volatility. I run regressions with absolute market returns and each of the former volatility risk factors. If the true maturity of short-term risk is well below one month, this will be re‡ected in signi?cant estimates of the price of one-day volatility risk. In this case the oneday volatility risk can also be expected to help disentangle the risks in the longer term volatilities. The high explanatory power of two volatility factors for option returns (Table 1.9) should also be preserved. On the other hand, if the true maturity of short-term risk is well above one day, the regressions results should resemble those of the one-volatility factor case (Table 1.10). Table 1.11 presents the results of regressions involving absolute market returns. These regressions do not support the hypothesis of a very short maturity of the short term-risk. The estimated risk prices for one- to twelve-month volatilities and their signi?cance levels are very close to the one-factor case (Table 1.10). The explanatory power of the regressions is also similar to that in the one-factor case. The absolute value of market returns indeed has a negative and often signi?cant price, but given the other estimates this is not likely to re‡ect a separate very-short
12 The tests reported in Tables 8 and 9 were also run with the components of short- and long-term volatilities, orthogonal to market returns. The results were very close, which can be expected, given the insigni?cant estimates for market risk price.

35

Table 1.11. Volatility risk prices - absolute market returns and one volatility factor (twenty ?ve portfolios) The table shows volatility risk prices estimated with GMM on twenty ?ve portfolios of delta-hedged options. The moment conditions are analogous to those in Table 9. M KT denotes daily returns on S&P 500, jM KT j is the absolute value of M KT , V OL denotes daily changes in one of 1, 3, 6, 9 or 12 month volatility The -s are estimated risk prices for each of the three risk factors. z-statistics

O
are distributed standard normal. Tilded factors (e.g. VgL 1m.) are the components of the respective raw factors, orthogonal to jM KT j. p-values for the chi-square test for pricing errors jointly equal to zero are in parenthesis. In square brackets is the adj. R2 in regressing average returns on betas. The two sub-periods are as in Table 1.9.

1997 - 2002 R is k p r ic e MKT jM K T j VOL 1m. -0.08 -0.16 -0.20 -0.32 ( 0.36) -0.120.190.19 z -s ta t. -1.30 -1.78 -3.80 -3.27 [0.7 4] -1.682.253.16

1997 - 2000 R is k p r ic e -0.03 -0.03 -0.26 -0.38 ( 0.37) -0.04 0.04 -0.25 z -sta t. -0.48 -0.25 -4.40 -3.71 [0.7 1] -0.53 0.39 -4.27

2000 - 2002 R is k p r ic e -0.05 -0.16 -0.04 -0.07 ( 0.88) -0.050.180 .01 z -s ta t. -0.74 -1.72 -0.68 -0.56 [0.2 1] -0.581.940 .12

VgL 1m. O

MKT jM K T j VOL 3m.

VgL 3m. O

-0.39 ( 0.31) -0.090.200.13

-2.84 [0.6 9] -1.362.382.07

-0.39 ( 0.07) 0.11 0.22 -0.09

-3.35 [0.6 6] 1.83 1.68 -1.45

-0.03 ( 0.83) -0.040.170 .03

-0.18 [0.2 1] -0.511.900 .47

MKT jM K T j VOL 6m.

VgL 6m. O

-0.26 ( 0.10) -0.040.150.07

-2.00 [0.6 6] -0.741.981.28

-0.04 ( 0.01) 0.16 0.28 -0.01

-0.43 [0.6 0] 2.80 1.99 -0.20

0.00 ( 0.82) -0.040.170 .03

0.00 [0.2 3] -0.541.890 .55

MKT jM K T j VOL 9m.

VgL 9m. O

VOL 12m.

-0.11 ( 0.04) -0.02 -0.13 -0.04

-1.28 [0.6 5] -0.351.71-0 .76

0.07 ( 0.01) 0.17 0.28 0.02

0.89 [0.6 1] 2.94 2.01 0.35

0.01 ( 0.85) -0.04-0.1 70.03

MKT jM K T j

0.04 [0.2 3]

-0 -0.05 ( 0.03)

. 5 -0.75 [0.6 6]

81 0.08 ( 0.01)

. 9 1.26 [0.6 1]

0 0 0.01 ( 0.86)

.51 0.05 [0.2 1]

VgL 12m. O

36

term source of volatility risk. In summary, I ?nd that two market volatility risks are signi?cantly priced in the sample of excess option returns. The model with two volatility risk factors has little pricing error. While the short-term factor, embedded in close to one- month implied volatility is negatively priced, the long-term factor which can be extracted from longer term-volatilities is positively priced. Next I consider returns to calendar spreads to investigate whether the di¤erence between the prices of short- and longterm volatility risks is also economically signi?cant.

1.4.2

Evidence from calendar spreads

A long calendar spread is a combination of a short position in an option with short maturity and a long position in an option on the same name, of the same type and with the same strike, but of a longer maturity. These spreads are similar to options, in the sense that the possible loss is limited to the amount of the initial net outlay. The results from cross-sectional regressions on unhedged options reported in Table 1.6 showed a positively priced market risk and long-term volatility risk and a negatively priced short-term risk. Expected returns to calendar spreads then have two components - one re‡ecting the market risks in the two options in the spread and another related to the two volatility risks. The sign and magnitude of the ?rst component should di¤er across types of options (calls or puts) and moneyness (see e.g. Coval and Shumway (2001)). Results in the previous section imply that the sign of the second component should be unambiguously positive (the position is short the short-term risk and long the long-term one). I do not derive here a formal relation between the two components, but verify that for all moneyness ranges and for both option types (put and calls) the expected returns to calendar spreads are positive. This demonstrates that the component related

37

to volatility risk is always positive and dominates the market risk component. Since the results so far only concern market volatility risks, I should strictly focus here only on calendar spreads written on the market. However, given the high explanatory power of market volatility risk factors for option returns (Table 1.9), it can also be expected that calendar spreads written on individual names re‡ect the di¤erence in the pricing of market volatility risks. So, I consider spreads written on all names in the sample as well. I use all options in the data set, which allow to calculate the gain of a position in calendar spread. For each name I record, at the beginning of each month all couples of options of the same type and strike and with di¤erent maturities, for which prices are available at the end of the month as well. In each spread I use a short-term option of the ?rst available maturity above 50 days. Possible liquidity problems when reversing the position at the end of the month are thus avoided. In this way I replicate a strategy, which trades only twice every month - on opening and closing the spread position. While transaction costs are still present, such trades are de?nitely feasible. I calculate returns on spreads where the long term option is of the second or third available maturities above 50 days. The results are very similar and I only report results for the second maturity. Table 1.12 shows average returns to calendar spreads written on the market (S&P500) and on all names in the sample. Separately are shown average returns for di¤erent ranges of moneyness for puts and calls, and for di¤erent periods. On average, spreads on puts gain an impressive 20% monthly, while those on calls about 12%. Average returns on individual names are slightly lower than those on the market alone. Spread returns in di¤erent moneyness ranges vary considerably, but are all positive. Table 1.12 also show that the Sharpe ratios of calendar spreads are typically about 30-40%, going as high as 100% in one case. Transaction costs

38

Table 1.12. Average returns to calendar spreads Panel A shows average returns to calendar spreads formed from short options with maturity at least 50 days and long options with the next available maturity and of the same type and strike. The positions are held for non-overlapping 30-day periods. The strikes of A-T-M options are within 5 % o f th e sp o t at the beginning of each 30-day period. O-T-M (I-T-M) options are at least 5% out-of-the-money (in-themoney) at the beginning of each 30-day period. Average spread returns for S&P500 alone and for all 28 names in the sample are shown. The sub-periods are 1/1/1997 - 6/30/2000 and 7/1/2000 - 12/31/2002. Panel B shows Sharpe ratios for calendar spreads in the same moneyness groups and periods.

Panel A.

Average one-month returns to calendar spreads 1997 - 2002 S& P 500 A l n ames 0.20 0.12 0.09 0.05 0.15 0.20 0.38 0.09 1997 - 2000 S& P 500 0.16 0.13 0.20 0.09 0.02 0.37 0.70 0.03 A l n ames 0.21 0.12 0.10 0.07 0.13 0.25 0.46 0.07 2000 - 2002 S& P 500 0.26 0.14 0.06 -0.03 0.33 0.15 0.41 0.27 A l n ames 0.17 0.10 0.06 0.03 0.17 0.14 0.26 0.11

A ll p u ts A ll c a lls A -T -M p u ts A - T - M c a lls O -T -M p u ts O - T -M c a lls I-T -M p u ts I -T -M c a lls Panel B.

0.20 0.13 0.15 0.05 0.16 0.27 0.56 0.13

S h ar p e r a t ios for c a le n d a r s pr e a ds 1997 - 2002 S& P 500 A l n ames 0.34 0.31 0.29 0.24 0.30 0.43 0.50 0.27 1997 - 2000 S& P 500 0.37 0.26 0.64 0.44 0.05 0.41 1.15 0.13 A l n ames 0.36 0.35 0.32 0.33 0.25 0.57 0.63 0.22 2000 - 2002 S& P 500 0.46 0.24 0.21 -0.12 0.53 0.19 0.68 0.51 A l n ames 0.37 0.29 0.24 0.12 0.43 0.31 0.43 0.38

A ll p u ts A ll c a lls A -T -M p u ts A - T - M c a lls O -T -M p u ts O - T -M c a lls I-T -M p u ts I -T -M c a lls

0.41 0.25 0.49 0.20 0.31 0.31 0.89 0.31

39

would reduce these numbers, but still the di¤erential pricing of volatility risks is very pronounced. Spread returns thus show that the di¤erent prices of short- and long-term volatility risks are not only statistically signi?cant, but economically signi?cant as well.

1.5

Conclusion

A number of volatility-related ?nancial products have been introduced in recent years. Derivatives on realized variance and volatility have been actively traded over the counter. In 2004 the CBOE Futures Exchange introduced futures on the VIX and on the realized three-month variance of the S&P500 index. The practitioners' interest in volatility products has been paralleled by academic research of volatility risk, mostly focused on the risk embedded in a single volatility factor. This paper complements previous studies of volatility risk by presenting evidence that two implied-volatility risks are priced in a cross-section of expected option returns. I ?nd that the risk in short-term volatility is signi?cantly nega- tively priced, while another source of risk, orthogonal to the short-term one and embedded in longer-term volatility is signi?cantly positively priced. I show further that the di¤erence in the pricing of short- and long-term volatility risks is also eco- nomically signi?cant: I examine returns on long calendar spreads and ?nd that, on average, spreads gain up to 20% monthly. The estimations for the two sub-periods reveal considerable di¤erences in the parameters, indicating that an extension to time-varying betas and risk prices is justi?ed. The robustness of the ?ndings in this paper to the choice of an option data set and an option-pricing model in constructing option returns can also be examined.

40

The di¤erential pricing of volatility risks has implication for the modeling of investors' utility. Previous research has found evidence for utility functions over wealth which have both concave and convex sections (Jackwerth (2000), Carr et al. (2002)). It is interesting to explore their results by employing utility functions with more than one arguments and possibly di¤erent volatility risks. Another implication of the ?ndings in this paper relates to the use of options in risk management. It has been argued that ?rms sometimes face risks which are bundled together in a single asset or liability (e.g. Schrand and Unal (1998)). In this case they can use derivatives to allocate their total risk exposure among multiple sources of risk. This paper suggests that derivatives themselves re‡ect multiple risks. How do ?rms chose among derivatives incorporating multiple risks is still to be studied.

41

2
2.1

MaxVar for processes with jumps
Introduction

The Basle Capital Accord was amended in 1996 to include capital charge for market risk. The Amendment gave the banks the option to use their own internal models for measuring market risk in calculating the capital charge. Among other quantitative requirements, the Amendment required the banks to multiply their VaR estimate by a factor of at least three in calculating the charge13. Many market participants expressed the view that the multiplication factors are too high and will possibly undermine the internal models approach. In "Overview of the Amendment" (BIS (1996)) the Basle Committee recognized the controversy, but still defended the multipliers by arguing that they accounts for potential weaknesses in the modelling process. Among the weaknesses mentioned explicitly in the document were the following: - the distributions of asset returns often display fatter tails than the normal distribution; - VaR estimates are typically based on end-of-period positions and generally do not take account of intra-period trading risk14. This paper addresses the following question: How much of the multipliers can be explained, ?rst, by non-normality of returns distributions and, second, by the risk
13 More precisely, the banks should compute VaR using a horizon of 10 days and a 99% con?dence interval. At least one year of historical data should be used, updated at least once every quarter. The capital charge is the product of 1.) the higher of the previous day's VaR and the average VaR over the preceding 60 days and 2.) a multiplicative factor not smaller than three. This multiplicative factor can be increased to up to four if backtesting reveals that the bank's internal model underpredicts losses too often. 14 Other acknowledged weaknesses were that volatilities and correlations can change abruptly, thus rendering the past unreliable approximation to the future, that models cannot adequately capture event risk arising from exceptional market circumstances, and that many models rely on simplifying assumptions, particularly in the case of complex instruments such as options.

42

of loss within the trading period (or interim risk of loss)? To answer this question the paper considers several Levy processes for the underlying assets, all involving jumps. The interim risk of loss is calculated using ?rst passage probabilities for these processes. In particular, the paper considers di¤usions with one- and two-sided jumps and one- and two-sided pure-jump processes. The one-sided pure-jump process employed is the Finite-moment log-stable process (Carr and Wu (2003)). The left tail of the distribution of asset returns in this case declines as a power law and can potentially account for the largest multipliers over normal VaR. The paper has two main ?ndings: First, Levy models can account for multipliers of 1.05 to 1.5. Second, when the interim risk of loss is also taken into account, the multipliers can increase further - 1.5 to 2.1. Typically, the multipliers for longer periods and for lower loss quantiles are slightly higher. Multipliers higher than 2.1 are not observed during the 5-year period 1998 - 2002 for any of the models employed and for any of the underlying time-series. While the multipliers obtained in this paper remain well below the factors of three to four, stipulated in the Amendment, the results are still not conclusive. First, the VaR estimates for Levy processes with interim risk are still violated in some cases more often than the respective quantile levels. Second, stochastic changes in volatility are not taken into account. Third, the calculations are based on time-series of daily returns for several major indexes and are thus only illus- trative typical trading books may exhibit pro?t and loss patterns which greatly di¤er from index returns. In view of these concerns, the paper contributes mostly to our understanding of the importance of employing Levy processes with jumps and of interim risk consideration in VaR estimations, and should not be construed as an evaluation of whether the Basle multipliers are set at appropriate levels.

43

Previous studies have demonstrated the ability of VaR estimates based on Levy models to predict more accurately trading losses (e.g. Eberlein et al. (2003) and references therein). The relation to the multipliers, however has not been considered explicitly. VaR with interim risk has also been studied previously. Kritzman and Rich (2002) introduce "Continuous VaR" as a measure of interim risk and show that over long horizons (up to 10 years) hedging based on this measure improves dramatically the performance of portfolio returns, compared to standard VaR. Bodoukh, Richardson, Stanton and Whitelaw (2004) denote the new risk measure as "MaxVar" and calculate ratios between this measure and standard VaR (analogous to the multipliers calculated in this paper)15. They show that ratios of up to 1.75 can be obtained for certain model parameters and con?dence levels, whereby the ratios are increasing in the drift of returns and the length of horizon and are decreasing in the volatility of returns. Both studies employ Brownian motion as the model for asset returns, which is a special case since the ?rst-passage probability in this case is well-known in closed form. The Brownian motion case is also special in that the ratio of MaxVaR to VaR is mainly driven by the drift of returns (Bodoukh et al. (2004)). If we suppose that the drift is zero, which is a reasonable assumption over the short 10-day regulatory period, then the ratio does not depend on the length of horizon or volatility, but only on the VaR con?dence level. (An easy application of the re‡ection principle for Brownian motion with no drift shows, for example, that the ratio is 1.19 for 5% VaR's and 1.11 for 1% VaR's). The contribution of this paper to the study of Continuous VaR / MaxVaR is in applying the concept to a number of models for asset returns involving jumps. For these, more realistic models, signi?cantly higher multipliers are obtained.
15 Interim risk in VaR estimations has been considered earlier in Bodoukh et al. (1995), Stulz (1996), among others.

44

The rest of the paper is organized as follows: Section 2.2 presents the Levy models and the numerical procedures for calculating ?rst-passage probabilities. Section 2.3 presents the empirical results and Section 2.4 concludes.

2.2

Models and ?rst-passage probabilities

This section presents the models of asset returns considered in the paper. The characteristic functions of log-returns for all models are given in closed form - these functions are used in estimating the model parameters on time-series of returns. The section presents also the numerical procedures for calculating ?rst-passage probability for each model.

2.2.1

CMYD

The CMYD process combines standard Brownian motion and negative jumps. It belongs to the class of spectrally negative processes. The name stands for "CMY plus Di¤usion", whereby the "CMY" part comes from the parameters describing the jump component. This process is closely related to the CGMY process studied in Carr, Geman, Madan and Yor (2002). The CMYD process is given by:

Xt = W t

Zt

(2.2.1)

where Wt is a standard Brownian motion, Zt has a Levy measure

is volatility and the jump component

k(x) = C exp(1+M x) xY

for x > 0.

(2.2.2)

45

Note that Zt has only positive jumps, and so Xt has only negative jumps. The CMYD characteristic function is:
1 2 2 ut

'X (u)
t

E eiuXt 0Z
0

e

iux

= exp @t

1 C exp(1+M x) dx xY
2 2 ut

2 1 A (2.2.3)

= exp tC ( Y ) (M

iu)Y

MY

2

When the uncertainty in asset returns is described by the CMYD process Xt, the asset price dynamics is given by:

St = S0 exp( t + X)]) t E [exp(X t The mean rate of return for the stock (under the statistical measure) is

(2.2.4)

and the

denominator ensures that E [St] = S0 exp( t) The characteristic function of the log price is:
2

E

eiu log(St)

= exp iu log(St) + t( exp tC ( Y ) (M

C ( Y ) (M + 1)Y
2 2 ut

MY

2 (2.2.5)

iu)Y

MY

2

Having the characteristic function in closed form allows for e¢ cient estimation of the parameters of the model using FFT (see Carr and Madan (1999)). Models of stock returns with one-sided jumps have been considered previously, for example in Heston (1993), Carr and Wu (2003). We employ the CMYD model for two main reasons - ?rst, it allows to evaluate the relative performance of models with one- and two-sided jumps in the context of measuring down-jump risk and, second, it o¤ers a signi?cant computational advantage - a technique developed recently by Rogers (2000)

provides an e¢ cient method to calculate ?rst-passage 46

probabilities for Levy processes with one-sided negative jumps16. Denote by f (t; x) the probability that a process Xt with only down-sided jumps and starting at zero does not reach the level x < 0 before time t. Rogers (2000) suggests the following procedure for calculating f (t; x): First, the double Laplace transform of f (t; x) is shown to be:
1 1

Z0 fe( ; z)

Z0

e

t zxf

(t; x)dtdx

=(

() z (z)) ( )z

(2.2.6)

where (z) is the characteristic exponent of Xt :

E [exp(zXt)] = exp(t (z))

(2.2.7)

and

( ) is its inverse, i.e. it is the solution of :

( ( ))) =

(2.2.8)

Then the ?rst-passage probability f (t; x) can be found by standard Fourier inversion, but for the di¢ culty in evaluating ( ) during the inversion. This di¢ culty can be avoided by suitably transforming the contour of integration. The transformation is given by g
1 0

, where p b2 + 2 2 z
2

1 0

(z ) =

b

(2.2.9)

Here b and
16 See

2

are the mean rate and variance of the di¤usion component of Xt:

Khanna and Madan (2003) for an application of the technique to option pricing.

47

Now the inversion formula is: d Z
1

f (t; x) =

2 i(

2

1 0

( ) + b) Z

2

dz 2i

0

(

1 0

( ))exp(tg( ) + xz)fe(g( ); z)

(2.2.10) There is no longer a problem in evaluating the integrand and techniques for twodimensional Laplace inversion (e.g. Choudhury, Lucantoni, Whitt (1994)) can be e¢ ciently applied to obtainf . In the particular case of the CMYD model :
2 2 z

(z) = bz + C ( Y ) (M + z)Y

MY +
2

2

b=

C ( Y ) (M +

1)Y

MY

2

2.2.2

Double exponential jump-di¤usion

The double-exponential jump-di¤usion process (DEJD) process is studied in Kou and Wang (2002) and di¤ers from CMYD in its structure of jumps. The DEJD process is given by: Xt = W t + X Nt
i=1

Yi ,

(2.2.11)

where

and Wt are as before, and Nt is a Poisson process. Nt models the arrival of and is independent of the Brownian motion. Yi are random

jumps, has intensity

variables de?ning the sizes of the jumps. Yi have a common two-sided exponential distribution:

fY ( y ) = p

1

exp(

1

y)1fy 0g + (1

p)

2

exp( 2y)1fy<0g.

(2.2.12)

48

p is the probability of an up-jump given that a jump occurs, and 1= 1and 1= the means of the exponential distributions for the up- and down-jumps respectively. Since the di¤usion and jump components of Xt are independent, the characteristic function of Xt is easily given by:

2

are

'X (u) = E eiuXt
t

= exp

u2 2 t + t 2

p
1

1

iu + (1 +p) 2 2 iu

1

,

(2.2.13)

and the characteristic exponent is: u2 2
2

G ( u) =

+

p
1

1

+ (1 +p) 2 2 iu iu

1.

(2.2.14)

As before, we model the dynamics of asset prices as

St = S0 exp( t + X)]) .
t

(2.2.15)

E [exp(X The characteristic function of the log price is:

t

E eiu log(St) = exp iu log(St) + t + t exp u2 2 t + t 2 u2 2 + 2
1

1

p 1 + (1 p) 2 u +u 2

1 (2.2.16)

p 1 + (1 p) 2 1 iu + iu 2

DEJD has a rare advantage when ?rst passage probabilities are concerned - the Laplace transform of the ?rst passage time to a ?xed level can be calculated analytically. Such explicit solution are possible for processes whose jumps are of the phase type (Assmussen et al. (2002)). The double exponential jumps turn out to

49

be the simplest of this type, hence the calculations are relatively easy to perform. The following theorem (Kou and Wang (2002)) gives the Laplace transform: be the ?rst passage time to level

Theorem 2 Let Xt be the DEJD process and b < 0 for Xt started from 0. For any

b

and 4; be the only 3; 2 (0; 1) let negative roots of the Cramer - Lindberg equation G( ) = , such that 0 < 3; <
2

<

4;

Then

E[exp( =
2

b

)]
4; 4; 3;

+
2

3;

exp(b

3;

)

4; 2

2 4;

3; 3;

exp(b

4;

)

(2.2.17)

For numerical Laplace inversion it is convenient to use the following:
1 1

Z0

e

t

P(

b < t)dt =

1Z

e
0

t

dP (

b

< t) = 1 E[exp(

b

)]

(2.2.18)

Kou and Wang (2002) suggest further the use of the Gaver-Stehfest algorithm for inversion.
>1

where Given the Laplace transform fbof f , we have the approximation f (t) = limn
n

fg)

n

(t ln(2)

fg) = ln(2) n!((2n)!1)!
n

(t

t

n X
k=0

(2.2.19) 0 1

( 1)k B nCb k A f (n + k ) @

t To speed up the convergence Richardson extrapolation can be used, whereby f (t)

is approximated by fn(t) for large t, where
k

fn(t) = Xn
k=1

kn k!(n k)!

(2.2.20)

w(k; n)fg) for w(k; n) = ( 1) n k (t

50

2.2.3

CGMY

The CGMY process was introduced in Carr et al. (2002). The CGMY process is a pure-jump Levy process with the following Levy density:

8

> C exp( Gjxj)
jxj1+Y

for x < 0

< for x > 0 k X(x) = > exp( Mx) : C (2.2.21)

x 1+Y

This is obviously a process with two-sided jumps. The C parameter can be considered as a measure of the arrival rate of jumps - both positive and negative. M and G control the rate of exponential decay of the probability of up- and down- jumps of di¤erent sizes. The Y parameter allows for a ?ne distinction between di¤erent classes of processes: depending on the value of Y , the process may or may not be completely monotone, and may exhibit ?nite or in?nite activity. The CGMY characteristic function is:

'X (u)
t

E eiuXt = exp tC ( Y ) (M

iu)Y

M Y + (G + iu)Y

GY (2.2.22)

The characteristic function of the log price, when the uncertainty is given by the CGMY process is:

Ee

iu log(St)

= exp iu log(St) + t( exp tC ( Y ) (M

C ( Y ) (M + 1)Y iu)Y M Y + (G + iu)Y

M Y + (G GY

1)Y

GY (2.2.23)

Carr et al. (2002) show that CGMY performs well in calibrating both time-series of stock returns and option prices. Since no closed-form expression is available for the ?rst-passage probability 51

of the CGMY process, we apply here a numerical procedure. We note the close relation between the calculation of ?rst-passage probability and the valuation of certain exotic options. In particular, the value of an option paying $1 if the price of the underlying assets hits certain level within a given time period is equal to the ?rstpassage probability of the price to this level (under the risk-neutral measure). Such an option has been considered within the pure-jump context for example in Hirsa (1999). The value function for this option is solution to a partial integro- di¤erential equation (PIDE). Hirsa provides an e¢ cient numerical solution to this equation and we follow closely his approach. (See also Madan and Hirsa (2003)). Note however, that Valueat-Risk calculations are typically performed under the statistical measure, so we also employ this measure throughout. Denote by G(S; t; T ) = E[1(S(u)
H

;

0 < u < T]

(2.2.24)

the conditional expectation at time t < T of the stock price hitting the level H < S0 within the time interval [0; T ]: If at any u < t we have S(u) < H, then G(S; t; T ) = 1. Otherwise 0 < G(S; t; T ) < 1: By construction G(S; t; T ) is a martingale, since it is conditional expectation of a terminal random variable. Then G(S; t; T ) is solution to the following PIDE17:
+1

Gt + S G S +

Z

(G(Sex; t; T )
1

G(S; t; T )

SGS(S; t; T )(ex

1))k(x)dx = 0 (2.2.25)

with boundary conditions

G(S; t; T ) = 1 if S(u) < H for some u < t
17 See Essay 3 in this thesis for a detailed derivation of the PIDE in a more general, twodimensional case.

52

G(S; T; T ) = 0 if S(u)

H for all u

(2.2.26)

After changing variables to s = ln(S) and g(s; t; T ) = G(S; t; T ) the PIDE (2.2.25) becomes:
+1

gt + gs +

Z

(g(s + x; t; T )
1

g(s; t; T )

gs(s; t; T )(ex

1))k(x)dx = 0 (2.2.27)

Using the Levy measure for the CGMY process and writing g(s) for g(s; t; T ):
0

0 = gt + gs + C

Z

( g ( s + x)
1

g( s)

gs(s)(ex

1)) e 1+Y dx
Gjxj

(2.2.28)

j xj +C
0 1

( g ( s + x)

g( s)

gs(ex

Z

1)) e 1+Y dx Mx x

Note that, because of the boundary condition, for any down-jump which brings the price below the level H, g takes the value of 1. For any given s, these are the jumps such that x ln(H) s. [0; T ]

Now (2.2.28) is solved using ?nite di¤erences on the mesh [smin; smax]

si = smin + ih; tj = j ; h = (s max

i = 0; 1; :::N; j = 0; 1; :::M; smin)=N and = T =M

where s denotes the log of the stock price, smin is the log of the ?rst-passage level H < S0, h is the step in the log-price direction and direction. The sum of the two integrals in (2.2.28) is approximated by: is the step in the time

I

I 1 + I 2 + I 3 + I 4 + I5 + I 6

(2.2.29)

53

where
ln(H) s

I1 = C

Z

[1
1

g(s) + gs(s)] e 1+Y dx; Gjxj j xj

I2 = C Zln(H) s

[ g ( s + x)

g(s) + gs(s)] e 1+Y dx; Gjxj j xj
x Gjxj

I3 =

C
1

Z I4 = C Z
0

gs(s) e e1+Y dx; j xj g( s) gs(s)(1 + x + 1 x2 2

g(s) + gs(s)x + 1 gssx2 2

1) e 1+Y dx;
Gjxj

j xj I5 = C Z
0 U

g(s) + gs(s)x + 1 gssx2 2

g( s)

gs(s)(1 + x + 1 x2 2

1)

e

Mx

dx;

I6 =

g( s) Z [ g ( s + x)

gs(s)(ex 1)] e 1+Y dx:
Mx

jxj1+Y

x The discretization for each of the above six integrals is next written down separately. Here s takes the discrete values sj for j = 2 : N gj denotes g(sj; ti; T ). Case 1: 0 < Y < 1: The following integral is essential in what follows: 1 , sj+1 sj = and we

C =

Z

e x dx x1+Y CY Y e ( )Y e ( ) Y + (1 Y )[
inc

(

;1

Y)

inc(

;1

Y )] ; (2.2.30)

where 0

< , and

inc is

the lower incomplete gamma function. Using (2.2.30),

54

for any ti and any j = 2 : N

1:

I1 =

[1

gj + gj+1
(j 1)

gj ] CGY Y Y ) [1
inc(G

eG (G ( j

1))Y + (1

(j

1); 1

Y )] ;

I3 = gj+1 = gj+1 e

e gj C Z
1

(G+1)jxj

1

dx = gj+1

e

(G+1)x

jxj1+Y gj C(G + 1)Y Y + (1 Y ) [1
inc((G

gj C Z

x1+Y dx

(G+1)

+ 1) ; 1

Y )] ;

[(G + 1) ]Y

I4 = C gj +1 2 = CG2
Y 2

2gj + gj
2

1

gj+1 gj Z
1

0

x2 e

Gjxj

dx
inc(G

2gj + gj
2

gj+1

gj

gj+1

jxj1+Y Y) (2

;2

Y );

I5 = 2 C gj +1 2gj + gj 1 gj+1 gj Z x2e Mx dx
Y 2

2 0

x1+Y
2 inc(M

g j+1

= CM2

2gj + gj 1 gj +1 gj (2 Y )

;2

Y ):

For the remaining two integrals I2 and I6 note that there is an I2 term only for j > 2, since for j = 2 down-jumps are small in absolute value and are given only

55

by I4. Equivalently, there is an I6 term only for j < N
s k+1 s j

1. gj e
Gjxj

I2

C XZ
k=1 j2 s k sj

gk+1 + gk+1

gk (x

sk+1 + sj)

gj + gj+1

dx

jxj1+Y =C X j2
k=1

gk+1 + gk+1

gk (s j

sk+1)

gj + gj+1

s k+1 s j

e

Gjxj

dx

jxj1+Y gj Z s k s j +C X j2
k=1

gk+1 gk Z

s k+1 s j

e

Gjxj

dx:

(2.2.31)

s k sj

j xjY The integral in the ?rst summand in (2.2.31) is equal to:
Y

I1 = G
2

e G (j k )

e G (j k 1) (G (j k 1)) Y
Y inc(G

Y (G (j k)) Y

+ G (1 Y ) [ inc(G (j k); 1 Y ) Y

(j

k

1); 1

Y )]

and the integral in the second summand in (2.2.31) is equal to:

I 2 = G1
2

Y

[

inc(G

(j

k ); 1

Y)

inc(G

(j

k

1); 1

Y )] :

In a similar way:

X I6 C
N1 s k+1 s j

gk + gk+1

gk (x

sk + sj )

gj

gj+1

gj (ex

k=j+1 N1

Z

s k sj

1) e 1+Y dx
Mx

x =C

gk +
k=j+1

gk (s

j

s
k

gj + gj+
1

s k+1 sj

e

X gk+1

)

gj Z

sk s
j

M x

d x x
1 + Y N1

C

gj+1

s k+1 s j s k sj

e

(M 1)x

x1+Y

dx

k=j+1

gj Z

X

X +C
N1

gk+1

s k+1 s j

k=j+1

gk Z sk sj

e Mx xY dx

(2.2.32)

56

The integral in the ?rst summand in (2.2.32) is equal to: e M (k j ) I 1 = (M e ( k + 1 M
(k+1 j) 6

j))Y + (1 Y )[
inc(M

(M (k j)) Y (k + 1 j ); 1 Y)
inc(M

(k

j ); 1

Y )] ,

the integral in the second summand in (2.2.32) is equal to: e I2 = ((M e 1) (k + 1
1) (k+1 j) 6 (M (M 1) (k j)

j))Y + (1 Y )[
inc((M

((M

1) (k j ); 1 Y)

j))Y
inc((M

1) (k + 1

1) (k

j ); 1

Y )] ;

and the integral in the third summand is equal to:

I3 = M 1 6

Y

[

inc(M

(k

j ); 1

Y)

inc(M

(k + 1

j ); 1

Y )] :

Case 2: 1 < Y < 2: The only di¤erence in this case is that, for 0 density in (2.2.30) becomes: < , the integral of the Levy

C

Z

e x dx x1+Y
Y

C = Y (1 C + Y (1

e Y ) ( )Y (1 Y )[

Y+

)

e ( )Y (1 Y)
inc(

Y+

)

Y

Y ) (2

inc(

;2

;2

Y )]

All other approximations and calculations follow exactly the previous case. Finally, with this approximation to the summands in (2.2.28) an explicit scheme is applied for solving the PIDE. Assuming that at time ti the values of g are known,

57

the values of g at an earlier time ti

1

are found by using standard ?nite di¤erence

approximations for gt and gs and solving a tri-diagonal linear system.

2.2.4

Finite-Moment Log-Stable (FMLS)

The FMLS process, introduced in Carr and Wu (2003) can be considered as a special case of the CGMY process. It only has down-sided jumps and its Levy density is: C k X ( x) = for x < 0 jxj1+Y (2.2.33)

For this model 1 < Y < 2. Since the numerator in (2.2.33) lacks the exponential damping factor of the CGMY of the Levy density, the left tail is fatter. Actually, the left tail is so fat, that FMLS is only made a feasible model for asset returns by disallowing any up-jumps. Only this restriction ensures that all moments of asset returns are ?nite (unlike the stable processes with two-sided jumps). The FMLS characteristic function is: Y sign(u) tan( 2 )

'X (u)
t

E eiuXt = exp t

Y

uY

1

i

(2.2.34)

where C " =
2 2 Y

1

Y

2 (1 + Y ) #

(2.2.35)

Y1

and

(x) is the gamma function. The PIDE for the value of a claim in the FMLS

case is analogous to (2.2.28), but only has an integral corresponding to down-jumps: ex 58 ( g ( s + x)
1

0

0 = gt + gs + C

Z

g( s)

gs( s)(

1))

1

jxj1+Y

d ( x 2 : . 2 . 3 6 )

The integral is approximated by:

I

I1 + I 2 + I 3 + I 4

(2.2.37)

where
ln(H) s

I1 = C

Z

[1
1

g(s) + gs(s)]

1 jxj1+Y

dx 1

I2 = C Zln(H) s I3 =

[ g ( s + x)

g(s) + gs(s)]

dx

jxj1+Y C Z
1

gs(s) 1+Y ex dx j xj 1 jxj1+Y

0

I4 = C Z

g(s) + gs(s)x + 1 gssx2 2

g( s)

gs(s)(1 + x + 1 x2 2

1)

dx

The integral of the Levy measure is: 1 Y
Y

C

1 dx = C Z jxj1+Y

1
Y

(2.2.38)

where 0 <

< . Using (2.2.38), for any ti and any j = 2 : N C Y ( (j 1)) Y

1:

I1 = [1

gj + gj+1

gj ]

I3 =

gj+1 gj C Z gj
1

e j xj C Y (Y 1)

j xj 1+ Y

dx = (1

gj+1 gj C Z Y+ ) + (2

1

e x dx x1+Y Y ) [1
inc

= gj+1

e
Y

(

;2

Y )]

59

I4 = C gj +1 2 = C gj +1 2 For j = 3 : N
j2

2gj + gj
2

1

gj+1 gj Z gj+1 gj 2

0

x2 dx jxj1+Y
2Y

2gj + gj
2

1

Y

1:
s k+1 s j s k sj

gj gk+1 + gk+1 gk (x sk+1 + sj) gj + gj+1 1
s k+1 s j

1 dx jxj1+Y

I2

C
k=1

XZ =C
k=1

j2

gk+1 + gk+1

gk (s j

sk+1)

gj + gj+1 gj Z

s k sj

dx jxj1+Y

+C X

k=1

gk+1 X j2 gk Z

s k+1 s j s k sj

1 dx j xjY

The integral in the ?rst summand is equal to 1 I1
2

=Y

( (j

1 k

1 1))Y ( (j k))Y

and the integral in the second summand in (44) is equal to 1 k 1))Y 1
1

I = Y 1 1 ( (j
2

2

( (j k)) Y

1

:

2.3

Empirical results

This section presents the results of the time-series estimation of the Levy models

and the Levy and MaxVaR multipliers obtained for each of the models. Results are reported for the original four models plus an additional modi?cation of the CGMY model. The modi?cation (named CGMYLIM) has the G and M parameters ?xed, providing an essentially two-parameter model. By ?xing G and

60

M at very low values (around one18, when typical estimates in the full CGMY model are from 50 to 150), this approximates the stable model with two-sided jumps, while still preserving the ?nite moments. So, CGMYLIM is an interesting twoparameter alternative to FMLS. The estimations are performed for ?ve international stock indexes on daily returns over the ?ve-year period January 1998 - December 2002. The indexes are FTSE, DAX, NIKKEI, Hang Seng and S&P500. Once every week each model is estimated using the 1000 preceding days (about 4 years of mean-adjusted daily returns). The characteristic function, available in closed form for each model, is the main tool of the estimation. Using the fast Fourier transform the characteristic function is inverted once for each parameter setting to obtain the density at a pre-speci?ed set of values for returns (see Carr et al. (2002)). With the density evaluated at these values, the return series are binned by counting the number of observations at each pre-speci?ed return value, assigning data observations to the closest return value. Then a maximization algorithm searches for the parameter estimates that maximize the likelihood of the binned data. Table 2.1 shows one measure of the performance of the models in these estimations - the proportion of estimations for each model where the chi-squared goodness-of-?t test cannot reject the model at 5% (1%). As expected, the normal model performs the worst and can be rejected in more than half of the cases for most underlyers. The models with one-sided jumps are only slightly better, which can also be expected given the approximate symmetry of the density of daily re- turns. The two-sided models are much better and typically can be rejected at most in 1-2% of the cases. It is interesting to note that even with only two parameters,
18 The value of 1.01 for both parameters was chosen to ensure stability of the particular discretization scheme for the PIDE. Other schemes may allow for even lower ?xed values of the G and M parameters.

61

Table 2.1 Goodness-of-?t tests ( t im e- s e r ies e s t im a t io n s ) Proportion of estimations for each model where the chi-squared goodness-of-?t test cannot reject the model at 5% (1%).

5%

1%

5%

1%

FTSE DAX HA N G SEN G N IK K E I S& P 500

N orm al 0 .3 6 0 .5 5 0 .3 7 0 .4 6 0 .0 8 0 .1 6 0 .2 6 0 .4 6 0 .1 9 0 .3 9

FM LS 0 .5 6 0 .6 8 0 .4 8 0 .5 8 0 .1 6 0 .2 4 0 .4 1 0 .5 9 0 .4 7 0 .6 2

FTSE DAX HA N G SEN G N IK K E I S& P 500

CM YD 0 .6 0 0 .7 2 0 .4 9 0 .6 3 0 .1 7 0 .2 6 0 .4 1 0 .5 8 0 .4 2 0 .6 8

D E JD 0 .9 9 0 .9 9 0 .9 8 1 .0 0 0 .8 2 0 .9 8 1 .0 0 1 .0 0 0 .9 1 0 .9 9

FTSE DAX HA N G SEN G N IK K E I S& P 500

CGM Y 0 .9 8 1 .0 0 0 .9 7 1 .0 0 0 .7 7 0 .9 7 1 .0 0 1 .0 0 0 .8 9 1 .0 0

C G M Y L IM 0 .9 4 1 .0 0 0 .6 8 1 .0 0 0 .5 2 0 .6 2 1 .0 0 1 .0 0 0 .8 9 1 .0 0

62

CGMYLIM performs as well as the larger models on four out of the ?ve indexes. Tables 2.2 and 2.3 show the maximum and mean ratios between 5% (resp 1%) VaR's obtained from the Levy models with jumps and standard VaR, obtained from a Normal (Gaussian) model for returns. The ratios (or multiples) are modest (rarely exceeding 1.1) for CMYD, DEJD and CGMY. The two models with heav- iest left tails - FMLS and CGMYLIM exhibit slightly higher multiples at the 5% con?dence level, but signi?cantly higher multiples (up to 1.5 - 1.6) at the 1% level. The multiples for the two horizons - 10 and 20 days are very similar. Tables 2.4 and 2.5 show the maximum and mean multiples of MaxVaR for the Levy models over standard VaR. The multiples for CMYD and CGMY are the lowest - typically below 1.3. The remaining three models show multiples in the range of 1.4 - 1.7, and going slightly above 2 in a few cases. No multiple above 2.1 is recorded. The multiples for the two horizons are very similar. Since all estimations are performed on demeaned returns, the e¤ect of the drift as observed in Bodoukh et al. (2004) does not appear in our data. The above results show clearly that taking jumps and interim risk into account can produce signi?cantly higher VaR values compared to Normal VaR. How well do these higher VaR's perform in predicting future losses? To answer this question we calculate the proportion of non-overlapping 10- and 20-day periods during which the actual maximum loss exceeds the MaxVaR estimated at the beginning of the period. For comparison, the proportion of cases when end-of-period loss exceeds Normal VaR is calculated as well. For a good VaR measure the frequency of ex- cessive losses should correspond to the VaR con?dence level. Table 2.6 reports the frequencies p-values of the chi-squared test for these frequencies to be signi?cantly di¤erent form the respective con?dence levels. Normal VaR performs poorly in this test - in half the cases the test does not

63

Table 2.2 Levy VaR multiples over Normal VaR - 10 days 10-day VaR's are calculated once every 10 days (between 1/1/1998 and 12/31/2002). Each VaR calculation uses daily returns over the preceding 1000 days. For each Levy model the table shows the maximum (mean) ratio of the 5% (1%) VaR's to the respective normal VaR's.

5%VaR's m ax m ean CM YD FTSE DAX H AN G SENG N IK K E I S& P 500 1.06 1.08 1.04 1.05 1.06 1.04 1.05 0.99 1.04 1.04

1%VaR's m ax m ean

5%VaR's m ax m ean FM LS

1%VaR's m ax m ean

1.12 1.17 1.09 1.11 1.19

1.06 1.10 1.04 1.07 1.10

1.06 1.07 1.01 1.03 1.03

1.04 1.05 0.94 0.98 0.99

1.31 1.31 1.21 1.25 1.25

1.28 1.29 1.13 1.18 1.20

D E JD FTSE DAX H AN G SENG N IK K E I S& P 500 1.04 1.05 1.01 1.02 1.03 1.03 1.04 0.96 1.02 1.02 1.08 1.11 1.02 1.05 1.09 1.05 1.07 0.99 1.03 1.05

CGM Y FTSE DAX H AN G SENG N IK K E I S& P 500 1.04 1.06 1.01 1.02 1.03 1.03 1.04 0.96 1.02 1.02 1.07 1.14 1.03 1.06 1.13 1.05 1.07 1.00 1.04 1.07 1.18 1.17 1.18 1.13 1.13

C G M Y L IM 1.09 1.09 1.10 1.10 1.08 1.56 1.52 1.61 1.40 1.49 1.22 1.26 1.38 1.28 1.27

64

Table 2.3. Levy VaR multiples over Normal VaR - 20 days 20-day VaR's are calculated once every 20 days (between 1/1/1998 and 12/31/2002). Each VaR calculation uses daily returns over the preceding 1000 days. For each Levy model the table shows the maximum (mean) ratio of the 5% (1%) VaR's to the respective normal VaR's.

5%VaR's m ax m ean CM YD FTSE DAX H AN G SENG N IK K E I S& P 500 1.06 1.08 1.05 1.05 1.06 1.04 1.05 0.99 1.04 1.04

1%VaR's m ax m ean

5%VaR's m ax m ean FM LS

1%VaR's m ax m ean

1.09 1.13 1.08 1.09 1.14

1.05 1.08 1.02 1.06 1.08

1.08 1.09 1.03 1.05 1.05

1.06 1.08 0.96 1.00 1.01

1.33 1.32 1.21 1.26 1.27

1.29 1.29 1.12 1.17 1.20

D E JD FTSE DAX H AN G SENG N IK K E I S& P 500 1.04 1.05 1.02 1.03 1.03 1.03 1.04 0.97 1.02 1.02 1.06 1.08 1.02 1.03 1.06 1.04 1.05 0.98 1.03 1.03

CGM Y FTSE DAX H AN G SENG N IK K E I S& P 500 1.04 1.06 1.02 1.03 1.03 1.03 1.04 0.97 1.02 1.03 1.05 1.11 1.02 1.04 1.09 1.04 1.06 0.98 1.03 1.05 1.25 1.24 1.26 1.19 1.19

C G M Y L IM 1.12 1.13 1.16 1.14 1.12 1.60 1.53 1.54 1.41 1.52 1.24 1.28 1.34 1.29 1.29

65

Table 2.4 MaxVaR multiples over Normal VaR - 10 days 10-day MaxVaR's are calculated once every 10 days (between 1/1/1998 and 12/31/2002). Each MaxVaR calculation uses daily returns over the preceding 1000 days. For each Levy model the table shows the maximum (mean) ratio of the 5% (1%) MaxVaR's to the respective normal VaR's.

5%VaR's m ax m ean CM YD FTSE DAX HA N G SEN G N IK K E I S& P 500 1.27 1.28 1.23 1.24 1.26 1.23 1.25 1.17 1.23 1.23

1%VaR's m ax m ean

5%VaR's m ax m ean FM LS

1%VaR's m ax m ean

1.25 1.31 1.21 1.23 1.32

1.18 1.22 1.15 1.18 1.23

1.76 1.75 1.51 1.54 1.70

1.59 1.64 1.40 1.46 1.53

1.63 1.59 1.37 1.40 1.57

1.46 1.50 1.27 1.34 1.41

D E JD FTSE DAX HA N G SEN G N IK K E I S& P 500 1.66 1.64 1.41 1.48 1.53 1.43 1.52 1.26 1.35 1.46 1.52 1.59 1.35 1.41 1.49 1.34 1.45 1.22 1.29 1.40

CGM Y FTSE DAX HA N G SEN G N IK K E I S& P 500 1.35 1.33 1.23 1.27 1.35 1.30 1.28 1.16 1.25 1.29 1.29 1.33 1.17 1.23 1.36 1.24 1.23 1.13 1.19 1.26 1.51 1.44 1.41 1.39 1.50

C G M Y L IM 1.36 1.34 1.32 1.34 1.36 1.82 1.76 1.94 1.62 1.76 1.43 1.46 1.62 1.47 1.49

66

Table 2.5. MaxVaR multiples over Normal VaR - 20 days 20-day MaxVaR's are calculated once every 20 days (between 1/1/1998 and 12/31/2002). Each MaxVaR calculation uses daily returns over the preceding 1000 days. For each Levy model the table shows the maximum (mean) ratio of the 5% (1%) MaxVaR's to the respective normal VaR's.

5%VaR's m ax m ean CM YD FTSE DAX HA N G SEN G N IK K E I S& P 500 1.25 1.27 1.25 1.25 1.26 1.23 1.24 1.17 1.23 1.23

1%VaR's m ax m ean

5%VaR's m ax m ean FM LS

1%VaR's m ax m ean

1.19 1.22 1.18 1.18 1.23

1.15 1.18 1.11 1.16 1.18

2.09 2.02 1.65 1.80 2.01

1.88 1.87 1.57 1.71 1.81

1.90 1.83 1.50 1.64 1.84

1.71 1.70 1.44 1.56 1.65

D E JD FTSE DAX HA N G SEN G N IK K E I S& P 500 1.76 1.66 1.40 1.48 1.54 1.47 1.58 1.28 1.36 1.47 1.58 1.57 1.33 1.40 1.47 1.36 1.47 1.21 1.28 1.39

CGM Y FTSE DAX HA N G SEN G N IK K E I S& P 500 1.29 1.30 1.22 1.24 1.29 1.26 1.25 1.15 1.23 1.25 1.22 1.26 1.14 1.18 1.27 1.18 1.19 1.09 1.16 1.20 1.53 1.49 1.51 1.43 1.49

C G M Y L IM 1.36 1.36 1.38 1.36 1.36 1.85 1.78 1.98 1.64 1.77 1.42 1.46 1.64 1.48 1.49

67

Table 2.6 Frequency of excessive losses Proportion of 10- and 20-day periods when actual loss exceeds the 5% Normal VaR and MaxVaR's (resp. the 1% VaR's) calculated at the beginning of the period. The periods are non-overlapping, from 1/1/1998 to 12/31/2002. * (**) denote p-values of the chisquared test for the observed loss to exceed VaR / MaxVaR signi ?cant at 1% (5%).

10 days 5% 1%

20 days 5% 1%

10 days 5% 1%

20 days 5% 1%

FTSE

0.1 4* * 0.1 3* * 0.03 0.00 0.00

0.03 0.0 9* * 0.00 0.00 0.02

0.11 0.05 0.00 0.00 0.00

N orm al 0.05* 0.02 0.02 0.02 0.00 0.05 0.03H AN G SENG 0.02 0.00N IK K E I 0.03 0.00S&P 500 0.08* 0.03

FM LS 0.02DAX 0.0 9* * 0.00 0.00 0.00 0.05 0.1 1* * 0.03

0.0 8* * 0.08* 0.00 0.02

CM YD FTSE DAX H AN G SENG N IK K E I S& P 500 0.05 0.08* 0.00 0.00 0.05 0.02 0.03 0.00 0.00 0.00 0.02 0.06 0.00 0.00 0.02 0.00 0.02 0.00 0.00 0.00 0.05 0.08* 0.00 0.00 0.03 0.02 0.03 0.00 0.00 0.00

D E JD 0.03 0.08* 0.05 0.02 0.03 0.00 0.02 0.00 0.00 0.00

CGM Y FTSE DAX H AN G SENG N IK K E I S& P 500 0.05 0.08* 0.00 0.00 0.05 0.02 0.02 0.00 0.00 0.00 0.06 0.09* 0.05 0.03 0.03 0.02 0.06* 0.02 0.00 0.02 0.05 0.08* 0.00 0.00 0.03

C G M Y L IM 0.02 0.06 0.00 0.09* 0.00 0.03 0.00 0.02 0.00 0.03

0.02 0.02 0.00 0.00 0.00

68

reject (at 5 % con?dence level) the hypothesis that Normal VaR underpredicts future losses. MaxVar preforms much better - typically only in one or two cases out of twenty we see signi?cant underprediction of losses. From this perspective the best model is FMLS, for which all p-values are above 5%.

2.4

Conclusion

It is widely recognized that Normal VaR estimates do not capture properly the magnitude of potential losses in banks' portfolios. The 1996 Amendment to the Basle Capital Accord speci?es that these estimateds should be multiplied by a factor of three or more, as one improvement on the risk measure. This paper examines the multiples over normal VaR which can be obtained when VaR is estimated using processes with jumps as models of asset returns and when taking the risk of iterim losses into account. Typical multiples around 1.5 - 1.7 are found (and as high as 2 in a few cases). One limitation of the paper is that it only considers time-homogeneous (Levy) models. However, it has been shown that incorporating stochastic volatility in models for asset returns improves signi?cantly VaR measures. Second, banks' trading books typically di¤er from the stock indexes used in the estimations in this paper. It is left for future research to examine the question whether richer models (with stochastic volatility) and more realistic portfolios (e.g. including derivatives) can produce even higher VaR multiples.

69

3
3.1

The COGARCH model and option pricing
Introduction

The success of GARCH models in capturing important features of asset returns has provoked interest in GARCH option pricing. Duan (1995) studies the measure change when returns of the underlying asset are modeled by a GARCH process. Garcia and Renault (1998) consider hedging in a GARCH option pricing model and Rithcken and Trevor (1999) construct trinomial trees to price American options under GARCH. Di¤erent GARCH option pricing models have been suggested and tested empirically in Duan (1996b), Heston and Nandi (2000), Hardle and Hafner (2000), Christo¤ersen and Jacobs (2004). Duan, Ritchken and Sun (2004) develop a GARCH option pricing model with jumps. This paper begins the study of option pricing for a new type of a GARCH model - the COGARCH model of Kluppelberg, Lindner and Maller (2004). Like most stochastic volatility models, COGARCH is a continuous-time model. Yet, like GARCH models, it is driven by a single random process and volatility un- der COGARCH exhibits the feedback and autoregressive properties of volatility in GARCH models. One distinctive feature of COGARCH is that the driving ran- dom process is a Levy process with jumps, as in a number of recent models of asset returns19. A second distinctive feature of COGARCH is that volatility is a pure-jump process, as in Barndor¤-Nielsen and Shephard (2001). COGARCH also provides an alternative to the continuous-time limits of certain GARCH processes, which have been previously derived (e.g. Nelson (1990), Duan (1996a)). In con19 For example Madan, Carr and Chang (1998) employ the variance-gamma process as a model of asset returns, Shoutens and Teugels (1998) use the Mexiner process, Barndor¤-Nielsen and Shephard (2001) use the Normal inverse Gaussian process and Eberlein (2001) applies the hyperbolic process.

70

trast to these two-dimensional limits, COGARCH preserves the essential GARCH feature that changes in volatility are only caused by moves in the underlying asset and not by changes in a second, latent random variable20. In the best case options are priced using a closed-form formula or an approxima- tion. Another widely used approach is based on the inversion of the characteristic function of the log-price (e.g. Carr and Madan (1999)). This approach is compu- tationally very e¢ cient, but requires a closed-form expression for the characteristic function. An alternative approach, which is less e¢ cient but is widely used, for example, in pricing exotics, employs the numerical solution of partial di¤erential / integro-di¤erential equations (PDE's / PIDE's). Since for the COGARCH model there is no closed-form option pricing formula or characteristic function currently available, this paper takes the PIDE approach. We exploit the fact that asset returns and volatility under COGARCH are jointly Markov. This Markovian property allows to derive a PIDE for the value of a claim under COGARCH We employ the PIDE for pricing European calls and puts, but the approach can be easily extended to pricing American options and exotics.

3.2

The COGARCH process

The COGARCH process, introduced in Kluppelberg et al. (2004) is a continuoustime version of the GARCH(1,1) process. The innovations of the COGARCH process are given by the jumps of a Levy process. Let (Xt)t
0

be a Levy process with jumps

X t = Xt

Xt , Levy measure

20 For other problems with continuous-time limits of GARCH processes see Corradi (2000), Wang (2002), Brown, Wang and Zhao (2002). But see also Heston and Nandi (2000) and Jeanthau (2004) for discrete time GARCH models which have di¤usion limits driven by a single Brownian motion.

71

kX and let 0 <

< 1 and

0. De?ne a process (Yt)t

0

by:

Yt =

t log( )
0<s<t

log 1 + ( X s)2 :

(3.2.1)

X Proposition 3 (Kluppelberg et al. (2004)) (Yt)t
0

is a spectrally negative Levy

process of bounded variation with drift log( ); with no Gaussian component and with Levy measure kY given by

kY ([0; 1]) = 0 and

kY ([ 1; y]) = k X

(z 2 R : jz j

r

(ey

1) ) , for y > 0:

De?ne further a variance process ( 2)t t
2 t

0

=

0

t

exp(Ys)ds +

2 0

exp( Yt )

(3.2.2)

Z where > 0 and
2 0

is a ?nite random variable, independent of (Xt)t 0. Then the
0

COGARCH process (Gt)t

is given by

dGt =

t

dXt:

(3.2.3)

The logarithmic asset returns over a time period r are modeled as Grt = Gt+r It follows from (3.2.1)-(3.2.3) that both G and jump sizes of G are Gt =
t

Gt .

jump only when X jumps. The

Xt

Proposition 4 (Kluppelberg et al. (2004)) The process ( 2)t t lowing stochastic di¤erential equation:

0

satis?es the fol-

d

2 t+

= dt +

2 t

exp(Yt )d(exp( Yt)

7 2

( 3.2.4)

and it follows that
t 2 t

= t + log( )

Z0

s 2

+
0<s<t

s 2

Xs ) 2 +

2 0

(3.2.5)

d s
2 t

(

and that
2 t+

X =
2 t

(

Xt )2 :

(3.2.6)

Proposition 5 (Kluppelberg et al. (2004)) Markovian.

The bivariate process ( 2; Gt) is t

In what follows we assume that the Levy process Xt is the Variance-Gamma (VG) process (Madan, Carr and Chang (1998)). VG is obtained by evaluating Brownian motion with drift at a random time, given by a gamma process. Let b(t; ; ) = t + W (t) be a Brownian motion with drift (t; ; ) be a gamma process with mean rate and volatility and let

and variance : The VG process

Xt is de?ned in terms of the Brownian motion and the gamma process as:

Xt = b( (t; 1; ); ; ):

Due to the gamma time change VG is a pure-jump process. Its Levy density k(x) is :

>
2 n n

exp( jxj

n j xj ) n

for x < 0

8 <
2 p

exp( x

p p

x)

where k ( x) = >:

for x > 0

(3.2.7)

p

2 2

n

=1

2 =1

p

2 r
2

+ 22 + r + 2 2

2

73

n

=

2 n

and

p

=

2 p

:

We further change the variables in the Levy density to:
2 p p 2 n n

C=

=

;

M=

p p

and

G=

n n

and the Levy density becomes

> C exp( Gjxj) for x < 0 8

k(x) = > exp(jxjMx)
x

for x > 0

:

(3.2.8)

<

:C

3.3

Backward PIDE for European options

Assume that the risk-neutral dynamics of the log price is given by the COGARCH process. Then
Rt

S t = S0 e

0

RtR1 audu+
0

1

ux

(dx;du)

(3.3.1)

where (dx; du) is a random jump measure and Rt

audu is the convexity correction,

0

guaranteeing that the stock price has the proper risk-neutral expectation. To ?nd au we use the fact that the discounted stock price is a positive martingale under the risk neutral measure, and therefore it can be represented as a stochastic exponential of a compensated martingale. This compensated martingale has the form:

1

Zt =
1

(e Z tZ 0

ux

1)( (dx; du)

(dx; du))

(3.3.2)

74

where (dx; du) is the compensator for (dx; du) (see e.g. Shiryaev (1999)). For the discounted stock price we obtain:

Zt

Zs

e

rt S t

= S0 e

X st
st Y
ux

(1 +

Zs )
RtR1

RtR1

= S0 e

0

(e 1

1) (dx;du)+

0

1

ux

) (dx;du :

(3.3.3)

By comparing (3.3.1) and (3.3.3) we obtain
t 1

rt +

Z0

audu =

Z tZ 0

(e
1

ux

1) (dx; du)

and so
1

at = r Z
1

(e

tx

1) (dx):
tx

(3.3.4)

Let st denote the log price at time t and !( t) = (3.3.1) and (3.3.4) we obtain:
1

1

t

x) (dx). From

R1 1 (e
1

st = s0 + r t
1

(e Z tZ 0

ux

1) (dx; du) +
1

u

x (dx; du)

Z tZ 0
u

= s0 + Zt
0

(r + !( u))du + Z tZ 1
0 1

x( (dx; du)

(dx; du)): (3.3.5)

75

Denoting ! = R1 1
2 t

x2 (dx) we rewrite (3.2.5) as:
t 1 u 2

=

2 0

+ t + log( ) Z
0

+ Z tZ 0
1

u 2

(dx; du)

d u

x 2

+ Z tZ 1

u 2

(dx; du))

x 2 ( ( d x ; d u )
1

0

t

1 2

=

2 0

+ t + (log( ) + !)

Z0

du +

u

Z tZ 0

1

u 2

(dx; du)

(dx; du)): (3.3.6)

x 2 (

We can now derive the PIDE for any test function f of the two variables s and For this purpose we use (3.3.5) and (3.3.6), and the fact that the bivariate process (st;
2 t

2

.

) is jointly Markov (Proposition 3). For this Markov process the in?nitesimal
2)

generator I(s;

satis?es the relation
t

f (st; 2) t = f ( s 0; 2 ) +
0

Z0

I(s;

2)

f (su;

2 u

)du + martingale:

(3.3.7)

To obtain the generator we apply Ito's formula for semi-martingales (see e.g. Shiryaev (1999)) and represent f (st; sated martingales:
t t 2

as the sum of drift terms and compen-

)

f (st; 2) t

=

f ( s 0; 2 ) +
0 t 0

(r + ! ( Z

u

))fs(su; u)du 2

+
0

+ log( ) + ! Z

u 2

u;

u

)du 2

f 2 ( s

+ Z t Z 1 f (s u + u x; 2(1 + x2 )) f (s u; 2)
u

2
0 1

3 fs ( s u ; u u x f 2 ( s u ; u

6 4 +martingale:

2

)

2

)

2 2 x u

u7

5 k(x)dxdu (3.3.8)

76

It follows from (3.3.7) and (3.3.8) that:

I(s;

2)

f (st;

t 2

= (r + !( t))fs(st; 2) +

t

+ (log( ) + !) 2 f 2(st; 2)
t t

)

+ 3 Z 1 f (st + t x; 2 (1 + x2)) f (st; 2 ) 2
1 t

6 4

f s ( s t ; 2 ) t x f 2 ( s t ; 2 ) 2 x2
t t t

t7

5 k(x)dx: (3.3.9) By taking expectation and di¤erentiating in (3.3.7) with respect to t we also obtain that for any test function:

E ft(st;

t 2

= E I(s;

2)

f (st;

t

2

)

) which gives the PIDE: ft(st;
t 2

= I(s;

2)

f (st;

t 2

(3.3.10)

)

) :

We now consider the PIDE for a claim in the speci?c case where the COGARCH process is driven by VG and so k(x) is given by (3.2.7). Let g := g(st;
t 2

denote the value of this claim. We further change t to

) and for brevity omit the time subscript . It follows from (3.3.10) that:

rg = g + rgs + ( + (log( ) + !) 2)g 2 +C Z0 1 g(s + x; 2(1 + x2)) g (e
x

1)gs

2 2 xg

e
2

Gjxj

jxj dx e

Mx

+C Z1 0

g(s + x; 2(1 + x2)) g

(e

x

1)gs

2 2 xg

2

x dx:

77

We also change x to

x

to obtain:

rg = g + rgs + ( + (log( ) + !) 2)g 2 e Z0 +C +C
+1 1
G j xj

g(s + x; 2 + x2 ) g (e x

1)gs

x2 g x2 g

2

jxj dx
2

g(s + x;

2

+

x2 )

g

(ex

1)gs

e

M

x

Z
0

x dx: (3.3.11)

To solve numerically this PIDE we use ?nite di¤erences. We break down the integrals in (3.3.11) into four parts:
s

0

s

U

I

Z

( +
D

Z

( +
s

Z0

( +

Z

) = I1 + I2 + I3 + I4
s

(3.3.12)

where D and U are the lower and upper bounds for the discretization in the logprice dimension. For the integrals I1 and I4 we apply the approximation developed in Madan and Hirsa (2003). We index the discretization grid by subscripts (k and j) in the log-price direction and by superscripts in the volatility direction. At

78

the node (j; n) on the grid the approximations are:
s k+1 s j

I1

C XZ
j2 k=1 s k s j

gn+1 +
k

gn+1
k s

gn
k

(x

e Gjxj dx

sk+1 + sj ) jxj +C X Z sk+1 sj j2 C
k=1 s k s j

gn+1
k+1

gn+1
k
2

( x2

(sk+1

sj )2 )

e

G j xj

jxj dx

k=1

gn + gn+1
j

s

gn (ex
j

1) + gn+1 j

2

G j xj

X Z sk+1 sj j2
sk s j

"

j

gn 2 e jxj dx
j

x #

s k+1 s j

=C X j2

"

A1
k

e

G j xj

s k+1 s j

Zsk sj

A2 jxj dx
k

Zsk s j

e

G j xj

dx #

k=1

+C X

s k+1 s j

G j xj

s k+1 s j

e

( G +1)jxj

s k sj

j xj e

dx

A4
k

s k sj

j xj

dx #

(3.3.13)

j

" A3 2

Z

Z

kk=1

where gn+1
k s

A1 =
k

gn+1
k

+

gn
k

( sj

sk+1)

gn+1
k+1

gn+1
k
2

( sj

sk+1

)2

gn +
j

gn+1
j
s

gn
j

A2 =
k

gn+1
k s

gn k
k k+1

A3 =

gn+1

gn+1 k

g
n + 1

+ g
n
2

A4 =
k

gn+1
j
s

gn
j

j

j

:

79

For the integral I4 involving up-jumps we obtain in a similar way:
s k+1 s j

I4

C
k=j+1 s k sj

gn +
k

gn+1
k s

gn
k

(x

sk + sj )

e M x dx x

X N1Z +C X N1 Z
k=j+1 s k sj
2

s k+1 s j

gn+1
k

g n ( x2
k

( sk

sj )2 )

e

M

x

x dx

X C
N1 s k+1 s j

gn + j gn+1 "
j

s

gn (ex
j

1) + gn+1 j

2

gn
j

Mx

k=j+1

Zsk sj

x2 e x dx #
N1

=C X
k=j+1

s k+1 s j

e

M

x

" B1
k s k sj

x dx + Bk 2 Z
s k+1 s j
M

s k+1 s j

e

M

x

dx + B3
k 1)x

s k+1 s j

xe

M

x

dx #

Zsk sj xe
x

Zsk sj dx # (3.3.14)

+C

B3 X N1 " k

dx

B4
k

s k+1 s j

e

(M

Zsk sj

Zsk sj

x

where
k=j+1

B 1 = gn +
k k

gn+1
k s

gn
k

gn+1 ( sj sk )
k
2

g n (s
k j

gn+1
j

gn j
s

gn+1
k

gn
k

sk

)2

gn + j

B2 =
k

s

B3 = 2

k
2

gn+1
k

gn
k

gn+1 + gn
j j

B4 =
k

gn+1
j
s

gn
j

:

The integrals in (3.3.13) and (3.3.14) are given by exponentials and in terms of the exponential integral.

80

For I2 we have:
0 2
s

e 1)gs x2 g
2

G j xj

I2

C Z = gs 2

g + xgs +

x2 g
jxj

2

g (1 + x + x2

jxj dx

0
s

G

x2 e

jxj dx

CZ so the approximation at the (j; n) node is gn+1
j s 0

I2

G j xj s

gn C Z j 2

j xj e

dx:

(3.3.15)

In the same way I3 gn+1
j s
s

xe gn C Z j 20

M

x

dx:

(3.3.16)

When the claim g is European call option we solve (3.3.11) with with initial conditions: g(s;
2

; 0) = (es K)+

and boundary conditions in the log-price direction:

g(s; g(s;

2

;

) = 0 for s ! 0 es
q

2

; )=

Ke

r

+

for s ! 1

where K is the strike price and r and q are the risk-free rate and the dividend yield respectively. (Similar boundary conditions apply for an European put option). In the volatility direction we impose no boundary conditions. Instead, we use one- sided derivatives of g at the volatility end-points of the grid. Having the required

derivatives at each grid point at time

(starting at

= 0), we apply an explicit

scheme to solve the PIDE and ?nd the value of g at each each grid point at time

81

+ 1 (including all points with boundary volatility values). One issue remaining is the discretization error. We do not derive here analytically an expression for the error, but perform a small numerical experiment - we calculate option prices at several di¤erent values of the grid steps and try to infer from them the order of convergence of the scheme. First, it turns out that the solution depends very little on the step in the volatility direction - the option prices change at most by fraction of a percent when doubling or quadrupling the number of volatility grid points. In what follows we use 30 volatility grid points. However, the sensitivity to the step in the log-price direction is much larger. Graphs 1 and 2 show, for a number of steps, the prices (large dots) of options with di¤erent strikes. The spot price is 100, the grid is limited between 20 and 300 in the log-price direction, and we use 50 to 300 uniform log-price steps. The lines on the graphs are quadratic functions ?tted through the prices with the three largest steps (i.e. the three right-most dots). In most cases these lines pass exactly through the prices calculated with smaller steps. We extrapolate these quadratics and take the value at zero to be the option price. (Fitting linear functions through any two of the three right-most dots proved to be less precise.) As a check on the option prices obtained in the above way, we compare put prices calculated directly with the PIDE and put prices obtained from put-call parity using the PIDE to calculate the respective call prices. Table 3.1 shows the prices of puts with strikes from 70 to 130 (where the spot is 100, time to maturity is 0.5, interest and dividends are zero and the COGARCH parameters are: = 0 :1 , = 0:3, = 0 :9 , = 0:1 and = 0:1). = 0:25,

82

Table 3.1. Put prices - put-call parity vs. PIDE

put price via strike put-call parity

put price via PIDE % di¤erence

70 80 90 100 110 120 130

0.3262 1.1689 3.2939 7.2444 13.3758 21.3859 30.4951

0.3274 1.1701 3.2950 7.2456 13.3770 21.3871 30.4963

0.3639 0.1023 0.0361 0.0163 0.0089 0.0056 0.0039

The di¤erences are very small - about 0.1 cent and not more than a third of a percent even for the most out-of-the-money option. Graphs 3 and 4 show COGARCH option prices for a range of strikes, maturities and for several parameter sets. The parameters are chosen to vary widely across parameter sets. At the lowest maturity (0.25 years) there is little di¤erence between the prices obtained at di¤erent parameter sets. For longer maturities (up to one year) the di¤erences become larger with one notable exception - it appears that at all maturities option prices are extremely insensitive to the value of the parameter . The prices obtained for with = 10 (the large dots) hardly di¤er from those obtained

= 0:1 (the thicker solid line). Of course, only the calibration of the model to

actual option prices will reveal what are the appropriate values of the parameters

85

Graph 1. Discretization errors ? Calls
K =70

Call prices are calculated on grids with log?price steps from 0.054 to 0.009 and denoted by large dots. The lines are quadratics through the three right?most dots. Spot is 100, maturity is 0.5, strikes are between 70 and 130.

30.33

30.25 call price 30.16 30.08 0 0.01 0.02 0.03 log?price step 0.04 0.05

K =80
21.17 13.29

K =90

21.06 call price call price 20.94

13.12

12.94

20.83 0 0.01 0.02 0.03 log?price step 0.04 0.05

12.76 0 0.01 0.02 0.03 0.04 0.05 log?price step

K =100
7.24 3.38

K =110

7.03 call price call price 6.81

3.20

3.03

6.59 0 0.01 0.02 0.03 log?price step 0.04 0.05

2.85 0 0.01 0.02 0.03 0.04 0.05 log?price step

K =120
1.39 0.50

K =130

1.30 call price call price 1.21

0.46

0.43

1.12 0 0.01 0.02 0.03 log?price step 0.04 0.05

0.40 0 0.01 0.02 0.03 0.04 0.05 log?price step

Graph 2. Discretization errors ? Puts

K =70
0.33

Put prices are calculated on grids with log?price steps from 0.054 to 0.009 and denoted by large dots. The lines are quadratics through the three right?most dots. Spot is 100, maturity is 0.5, strikes are between 70 and 130.

0.31 put price 0.30 0.28 0 0.01 0.02 0.03 log?price step 0.04 0.05

K =80
1.17 3.30

K =90

1.12 put price put price 1.08

3.18

3.07

1.03 0 0.01 0.02 0.03 log?price step 0.04 0.05

2.96 0 0.01 0.02 0.03 0.04 0.05 log?price step

K =100
7.25 13.38

K =110

7.09 put price put price 6.94

13.27

13.16

6.79 0 0.01 0.02 0.03 log?price step 0.04 0.05

13.05 0 0.01 0.02 0.03 0.04 0.05 log?price step

K =120
21.39 30.60

K =130

21.36 put price put price 21.34

30.56

30.53

21.31 0 0.01 0.02 0.03 log?price step 0.04 0.05

30.50 0 0.01 0.02 0.03 0.04 0.05 log?price step

Graph 3. COGARCH call option prices
spot=100, strikes from 70 to 130
maturity =0.25 35 35 maturity =0.5

30

30

25

25

20

20

15

15

10

10

5

5

0 70

80

90

100

110

120

130 delta=0.9, delta=0.1, delta=0.9, delta=0.9,

0 70 lambda=0.1, lambda=0.1, lambda=10, lambda=0.1,

80 beta=0.05 beta=0.05 beta=0.05 beta=1.00

90

100

110

120

130

maturity =0.75 35 35

maturity =1

30

30

25

25

20

20

15

15

10

10

5

5

0 70

80

90

100

110

120

130

0 70

80

90

100

110

120

130

VG parameters: sigma = 0.25, nu = 0.1, theta = ?0.3

Graph 4. COGARCH put option prices
spot=100, strikes from 70 to 130
maturity =0.25 30 30 maturity =0.5

25

25

20

20

15

15

10

10

5

5

0 70

80

90

100

110

120

130 delta=0.9, delta=0.1, delta=0.9, delta=0.9,

0 70 lambda=0.1, lambda=0.1, lambda=10, lambda=0.1,

80 beta=0.05 beta=0.05 beta=0.05 beta=1.00

90

100

110

120

130

maturity =0.75 30 30

maturity =1

25

25

20

20

15

15

10

10

5

5

0 70

80

90

100

110

120

130

0 70

80

90

100

110

120

130

VG parameters: sigma = 0.25, nu = 0.1, theta = ?0.3

and the price sensitivity.

3.4

Conclusion

This paper studies option pricing under the COGARCH model of Kluppelberg et al. (2004). The paper derives a backward PIDE for the value of a claim written on an asset, following COGARCH. Some properties of European option prices under COGARCH are demonstrated. The backward PIDE derived, however, has one obvious drawback if one intends to use it for calibration of the model to option prices: it needs to be solved separately for each option (or at least for each strike), which may be very costly in computation time. An alternative is the forward PIDE. By solving a single forward equation one can calculate simultaneously the prices of options of all strikes and maturities. Forward PIDE's for models of the log price involving jumps have been derived in Andersen and Andreasen (1999), Carr and Andreasen (2002), Carr and Hirsa (2002), Madan (2005). The derivation of a forward equation for the COGARCH process and its use for calibration of the model is left for future research.

88

Appendix A. Proof of Proposition 1
Expectations are taken under the statistical measure. The pricing kernel process
t

is obtained by conditioning on the realizations of the volatility

S

. Let Et [ ]

denote expectation, conditional on the volatility path.
T

0ZT

DSds + s

T

Z p
S

dW C

ST = St exp @
s s s

(1) 1

1Z

2
t t t S

dsA

t

2

0

ZT DSds
s T S

dW S + 1 s

= Et 4St exp @

Z p
s

2

s

13
t S

Z

dsA5

T

Ds 0 Z 1 2 t 0 ZT

T t

2

s

s

s

1 0

Z
T

1

3

t

p 3

2

t t t S t S

= St exp @

S

dsA E 4exp @ 1

dsA E 4exp @

dW S A55

0

Z DSdsA E 4exp @ ( 2+ 1) t

T t s

(2) 12 0 Z
s

= St exp @

13

Tt

S

dsA5

To evaluate the expectation in (2) we use the fact that the characteristic function =T of theZ integral of a CIR variable is known in closed form. Let Y=
t S s T

t and

ds. Then the

c h a

r a c t e r i s t i c f u n c t i o n o f Y i s :

S

( u ) = E
t

e
i u Y

Y

= A( ; u) exp B( ; u)

t

(3)

A( ; u ) =

exp( cosh( )+ 2iu + coth(

2 2

)
2 2

sinh( )22 ) and = p
2

B ( ; u) =

2 2iu

2

89

On evaluating this characteristic function at u = the stochastic discount factor is: d
t

i

( +1) 2

it follows from (2) that

t

=

t

= dSt St

( + 1) d + coth( 2 )
t

S t

(4)

whereby terms of order dt in the di¤erentiation of

are ignored.

Appendix B. Volatility risk prices in a two-factor model
This appendix presents a stylized model which is consistent with the empirical ?ndings of the paper. The model is an extension of the basic model in (1)-(3), which includes a second stochastic volatility. The stochastic discount factor derived for the extended model corresponds closely to the two-factor model (1.2.6) tested empirically in this paper. For the extended model I show that one volatility risk has always a negative price, while the price of the second volatility risk can be of both signs. In particular, this risk price is positive (as found empirically above) when the utility function is convex in the second volatility factor. In the spirit of habit-formation models, I assume an economy where utility is of the form:
CT HT 1

UT = and the pricing kernel process is:

1

(5)

CT "
t

= Et 90

HT #

(6)

where C denotes an aggregate consumption good, and

is the risk aversion co-

e¢ cient. H is interpreted as a time-varying habit in a multiplicative form, as introduced by Abel (1990) and Gali (1994). In models of this type utility does not depend on the absolute level of consumption, but on the level of consumption relative to a benchmark (habit). The habit is usually related to past consumption. Here I also allow for randomness in habit. Assume the following dynamics for H and C and their respective volatility processes:

H

dW H t
t

(7)

dHt = Ht

DHdt
t

+q

dCt =

DCdt

+ q Ct

t C

+ q

H

dW H t
t

(8)

d W
C t C

t

d

C t

= kC ( C

t C

+

C

dW 1t
t

(9)

) d t d
H t

q +
H H

= kH ( H

t H

dW 2

) d t

q
t t

(10)

All four Brownian motions WtH, WtC, Wt1 and Wt2 are assumed to be independent. The drift of habit growth (DH) can be a function of past consumption as in the t "catching up with the Joneses"versions of habit models; it is not modeled explicitly as before. What is essential is the separate source of randomness in habit - WtH, with stochastic volatility
t

(" abit volatility" . Such a source of randomness h )

re‡ects the notion that habit also depends on some current variables, similar to pH the "keeping up with the Joneses" versions of habit-formation models. The drift DC is not modeled explicitly as before. A key feature of the the model is that
t

it allows for random changes in habit to a¤ect the dynamics of the consumption good ( is a sensitivity parameter, so it should take values between zero and one).

We now prove the following:

91

Proposition 1 The stochastic discount factor factors described in (5)-(10) is given by d
t

t

in the economy with two volatility

t

=

t

= dCt Ct

dHt Ht

B C ( ; uC )d

C t

B H ( ; uH )d

H t

(11)

where BC> 0 BH> 0 if and 1< +1 <1 (12)

Proof. Expectations are taken under the statistical measure. For brevity in notation let:

0ZT

C = exp @

DCdsA
s

and 1

s

1

t

0ZT
t

H = exp @ Condition on the realizations of ing:
H H

DHdsA

and

C

, and obtain for HT and CT the follow-

0ZT HT = HtH exp @ 2
t

dW H
s

p

s

1ZT

s

1 (13)

t

H

dsA

0ZT

C

dW C

CT = CtC exp @

p
s s

+

T

Z p
t t

H s

dW H A
s

0
s C

1 1 2
2 H

(14) Z dsA

exp @ 1 Z

d s

T

T

2 The pricing kernel is:
t t s

1

t

= Ct C

Ht H Et LEt [M ]

(15)

92

where 0
C

1

ds +(

2

1)

s

11

s

L = exp @ @ 2 0 ZT
t H

0

0ZT

Z

dsAA

Tt C s

dW C +(
s

1)

T

M = exp @

@
t

p

Z p
t

H s

dW H AA
s

Take ?rst the conditional expectation in (29):

11

= Ct C
t

2

0
2 C s

ds +(

1)2 Z
H

Ht H Et 4L exp @1 2

0ZT @

dsAA5

T

= Ct C where

Ht H Et [N ]

t

t

s

113 (16) 0
C s

ds

N = exp @1 ( + 1) 2 Z

1 (1 2

) [1 +

(1

)] Z

s

1

T t

T t H

dsA

Using the independence of the two volatility processes and (3), the pricing kernel process is:

t

= Ct C

Ht H AC( ; uC) exp(BC( ; uC)

t C

; uH) exp(BH( ; uH)

t H

) A H ( where superscripts C and H denote parameters related to consumption and habit respectively, and uC = uH = i 1 (1 2 B C ( ; uC ) =
C

) (17)

i ( 2+ 1) ) [1 + ( + 1)
C

(1

)] >0 ) (18)

+

C

coth(

2

93

B H ( ; uH ) = The stochastic discount factor is then: d
t

(1
H

) [1 + +
H

(1
H

)] )

coth(

2

t

=

t

= dCt Ct
1 +1

B C ( ; uC )d

C t

dHt Ht

B H ( ; uH )d

H t

(19)

and BH( ; uH) > 0 if

<

<1

The ?rst two terms are exactly as in the case with one volatility factor and without habit. The third term re‡ects a negative habit risk price. Given that an increase in habit decreases utility, any asset positively correlated with the change in habit has high payo¤s in low-utility states, which provides the intuition for the negative price of habit risk. The last term re‡ects the price of "habit volatility" risk. Assume further that consumption equals dividends and that the market price- todividend ratio is constant. So, the market risk price is positive and the prices of the two market volatility risks have the signs of the risk prices of respectively. Then the stochastic discount factor can also be written as: d
t C t

and

H t

t

=

t

= dSt St

dHt + Ht

d

t

+

H

d

H t

(20)

< 0 as follows from (18) and (19) and the sign of

H

is determined from (12).

What is the intuition for condition (12)? We now show that this condition is closely related to the concavity or convexity of the utility function (5) in habit. Denote for brevity
T

1

CT = CtC exp @
C

dW C

f

0ZT
t

p 94
s

s

1Z s
C t

2

d

s

A

Re-write (14) as:
T

1 0 ZT e CT = CT exp @ p
s H

2

2

s H

s

d W
H s t

Z

d s

21Z

1

T

) 0 exp @1 (1 2
T t

t

t H

= C T Ht H e

Z

H

dsA H

dsA

Then utility at time T is:

s

T

(21)

1

UT = 1 1

2

0

2

)

Z

s

131

T t H

H( dsA5
T

1)(1

)

(22)

4CT Ht H e

exp @1 (1
2 1

This function is convex in HT when

<

< 1. While this condition is not > 3 the positive price of habit

equivalent to (12), it can be easily checked that for

volatility risk follows from the convexity in habit of the utility function. Empirical studies show that > 3 is a plausible range for the values of the risk aversion

parameter. So, intuition for the the price of "habit volatility"risk can be provided, as in the single volatility risk case by the shape of the utility function - concavity in habit re‡ects a decrease in expected utility when "habit volatility" increases, hence the desirability of assets correlated with changes in "habit volatility" and the

negative price of "habit volatility"risk. Exactly the opposite argument applies when the utility function is convex in habit, so the price of "habit volatility" risk is positive in this case. The model presented above is illustrative in nature. It leaves unspeci?ed some important components, in particular the form of the two drifts DH and DC. It also
t t

assumes a constant price-dividend ratio. Still, it points to the essential role that concavity / convexity of utility functions can have in modeling volatility risks. Utility functions which exhibit both concavity and convexity have been found 95

empirically in di¤erent contexts. From results in Jackwerth (2000) it follows that investors'utility from wealth (proxied by a market index) has been changing over time. In particular after 1987 it has exhibited a convex shape over certain ranges of market moves. In a similar spirit, Carr et al. (2002) estimate marginal utility over instantaneous market moves (jumps) of di¤erent size and show that it has both decreasing and increasing sections over di¤erent ranges of jumps. The increasing sections correspond to convex utility. It is interesting to explore their results in the context of utility functions with two arguments and possibly two volatility risks. Now compare (20) with (1.2.6), estimated above. Note that (1.2.6) lacks a term corresponding to
dHt : Ht

However, this lack is re‡ected only in the pricing errors, but

not in the prices of volatility risks (for uncorrelated risk factors). Besides, with excess returns and normalized risk factors the risk prices in the two models are equivalent, as discussed above. It follows that the extended model is consistent with the empirical ?ndings of this paper

Appendix C. Option pricing models
This appendix presents details on the three option pricing models compared in Section 3: 1. Stochastic volatility and jumps model Bates (1996) and Bakshi et al. (1997) consider an eight-parameter model for the log price with stochastic volatility and jumps (SVJ):

dSt = (r dVt = (

)Stdt + pVtStdWt1 + JyStdqy t kVt)dt + 96 pVtdWt2 (23)

Wt1 and W t2 are standard Brownian motions with correlation ; Jy is log-normal with mean
y

and variance
y

2 y

qy is a Poisson process with arrival rate t r is interest rate and Let: a = u2 + iu + 2iu b = iu k

is the jump-compensator
y u2 2 y 2

yy

2 y(eiu

1)

= b +a 2 p2 c = ( + b)t + 2 log(1 h d=2 B= ( + b)(1
a(1 e d
t)

(1 2
+b t

e

t

)) i

e

)

The characteristic function of the log-price at horizon t under the SVJ process is: EQ eiu log(St) = eiu log(S0)+iurt
c 2

+B V

0

(24)

where V0 is instantaneous variance. The expectation is taken here under the riskneutral measure Q. 2. Double jumps model Du¢ e et al. (2000) develop a ‡exible model speci?cation, which also adds jumps in volatility. I use here one of their double-jump (DJ) models:

dSt = (r dVt = (

)Stdt + pVtStdWt1 + JyStdqy t kVt)dt + pVtdWt2 + Jvdqv t (25)

All variables, except for Jv and qv have the same meaning as above. qv is a second
t t

Poisson process with arrival rate

: v The sizes of the jumps are exponentially 97

distributed with mean Let: a = u2 + iu b = iu k

v

= b2 + a 2 p c = ( + b)t + 2 log(1 h d=2
ye y

2 +b t

(1

e

t

)) i

( 2 + b)(1
+ 2y y+ v y u2 2 y 2

e 1

)

+
v

f= g = teiu h= A= B=

b+ t a b

2

2 va ( b v a )2

log(1

+b 2

va

(1

e

t

))

( y+
a(1 e d

v
t)

)(1 + iuf )t +

y

g+

v

h

The characteristic function of the log-price at horizon t under the DJ process is: EQ eiu log(St) = eiu log(S0)+iurt
c

+ A +B V 20

(26)

3. VGSA model Carr and Wu (2003) provide a general study of the ?nancial applications of time-changed Levy processes. They show that most of the stochastic processes employed as models of asset returns, including SVJ and DJ, belong to this class. The third model I consider here is also based on a process of this class. The VGSA process (Carr et al. (2003)) is a six-parameter pure-jump process with stochastic arrival rate of jumps. (Stochastic arrival is an analogue of stochastic volatility for pure-jump processes.) VGSA is introduced in two steps, each step being an explicit timechange of a Levy process.

98

At the ?rst step a Brownian motion with drift, denoted as

b(t; ; ) = t + W t

(27)

where Wt is standard Brownian motion, is evaluated at a time given by an independent gamma process (t; 1; ) with unit mean rate and variance rate . The process, obtained in this way is the Variance Gamma (VG) process (Carr and Madan (1998)): V G(t; ; ; ) = b( (t; 1; ); ; ) VG is a Levy process, like the Brownian motion, but unlike it, is a pure-jump process, due to the gamma time-change. Its characteristic function is: 1 u+(
t= 2

(28)

V Gt

(u) = E e

iuV Gt

=

1

i

=2)u2

(29)

and its characteristic exponent is: = 1 ln 1

VG

(u)

i

u+(

2

=2)u2

(30)

At the second step, the VG process itself is time-changed. The time-change is independent of the VG process and is given by the integral of the mean-reverting CIR process. The CIR process is solution to the following stochastic di¤erential equation: dyt = k(
t

yt)dt + pytdW t

(31)

Denote Yt =

Z

ysds , then
0

V GSA(t; ; ; ; k; ; ) = V G(Yt; ; ; )

(32)

99

The characteristic function of Yt is:

Yt (u)

= E eiuYt = A(t; u) exp B(t; u)y0

where A(t; u), B(t; u) and

are as given in Section 2.2. Since Yt is independent of

the VG process, the characteristic function of the VGSA process can be obtained via conditional expectation:

V GSAt

(u) = E eiuV GSAt =

Yt (i

VG

(u)

)

(33)

Then the characteristic function of the log-price at horizon t under the VGSA process is:
V GSAt

(u) i) (34)

EQ

eiu log(St)

=

eiu log(S0)+iurt

V GSAt

(

100

References
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