Description
The document about analyzes the various derivative models like Black Model, Factors Models, HJM and Libor Market Model used for pricing the interest rate derivatives, which include both plain vanilla as well as exotic interest rate derivatives.
ANALYSIS
AND PRICING
OF INTEREST
RATE
DERIVATIVES
For: Capital Metrics and Risk Solutions, Pune
By: Biswadeep Ghosh
31262, MBA Finance
SIBM Pune
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CONTENTS
ACKNOWLEDGEMENTS ................................................................................................................................................. 3
1. OBJECTIVE ................................................................................................................................................................. 4
2. ROAD MAP ................................................................................................................................................................. 4
3. METHODOLGY .......................................................................................................................................................... 5
4. COMPANY ANALYSIS: AN OVERVIEW ........................................................................................................... 6
4.1 Introduction .................................................................................................................................................... 6
4.2 Services .............................................................................................................................................................. 6
4.2.1 Equity Research.................................................................................................................................... 6
4.2.2 Financial Analytics .............................................................................................................................. 6
4.2.3 Research Process ................................................................................................................................. 7
4.2.4 People ....................................................................................................................................................... 7
4.2.5 Clientele ................................................................................................................................................... 7
5. INDUSTRIAL ANALYSIS: FINANCIAL DERIVATIVES MARKET ............................................................ 8
5.1 Development of exchange-traded derivatives .................................................................................. 8
5.2 The need for a derivatives market ......................................................................................................... 8
5.3 The participants in a derivatives market ............................................................................................ 8
5.4 Development of derivatives market in India ..................................................................................... 9
5.5 Exchange-traded vs. OTC derivatives markets .............................................................................. 10
5.6 Comparative Analysis ............................................................................................................................... 11
5.6.1 World Exchanges: .................................................................................................................................. 11
5.6.2 Business Growth in NSE Derivatives segment ..................................................................... 11
5.6.3 Month wise Product wise Traded Value Analysis ............................................................... 11
5.7 Indian Forex and Interest Rate Derivative market ...................................................................... 12
5.7.1 Currency Futures .............................................................................................................................. 12
5.7.2 Interest Rate Derivatives ............................................................................................................... 13
5.8 Final Note ...................................................................................................................................................... 14
6. STUDY OF DERIVATIVES .................................................................................................................................. 15
6.1 Future Contracts ......................................................................................................................................... 15
6.2 Forward Contracts ..................................................................................................................................... 16
6.3 Options ........................................................................................................................................................... 16
6.4 Swaps .............................................................................................................................................................. 17
6.5 Exotics ............................................................................................................................................................. 18
7. STUDY OF YIELD CURVE ................................................................................................................................... 20
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7.1 Yield to Maturity ......................................................................................................................................... 20
7.2 MATLAB conversion functions ............................................................................................................. 21
7.3 Inferences ...................................................................................................................................................... 22
8. DERIVATIVE MODELS ........................................................................................................................................ 23
8.1 Weiner Process and Ito’s Lemma ........................................................................................................ 23
8.2 Markov property ........................................................................................................................................ 23
8.3 Ito Process ..................................................................................................................................................... 24
8.4 The Black-Scholes-Merton Model ........................................................................................................ 25
8.5 Black’s-76 Model ........................................................................................................................................ 26
8.6 One Factor Models ..................................................................................................................................... 27
8.6.1 Merton/Ho-Lee Model .................................................................................................................... 28
8.6.2 Vasicek Model .................................................................................................................................... 28
8.6.3 The CIR Model .................................................................................................................................... 29
8.7 Multi-Factor Models .................................................................................................................................. 31
8.8 Heath-Jarrow-Morton (HJM) Model ................................................................................................... 33
8.9 LIBOR Market Model ................................................................................................................................. 34
9. PRICING USING MATLAB AND OCTAVE ..................................................................................................... 38
9.1 Black-Scholes ............................................................................................................................................... 38
9.2 Black-76 ......................................................................................................................................................... 38
9.3 HJM ................................................................................................................................................................... 39
9.4 LIBOR Market Model ................................................................................................................................. 39
10. VALUE ADDITION ........................................................................................................................................... 41
10.1 To the Organization ................................................................................................................................... 41
10.2 To “me” as a student of management ................................................................................................ 41
11. TEAM PLAY IN THE ORGANIZATION ..................................................................................................... 42
12. LIMITATIONS OF THE STUDY.................................................................................................................... 42
13. IMPLICATION AND CONCLUSION ............................................................................................................ 43
BIBLIOGRAPHY .............................................................................................................................................................. 45
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ACKNOWLEDGEMENTS
First of all, I would like to express my gratitude to Dr. Arun Mudbidri, Director, SIBM
who provided me the platform to enable me to work on this project. I take this
opportunity to thank Prof. Kaustubh Medhekar, Head of Department, Finance, SIBM for
his nurturing and relentless support. I also thank Prof. S. Kalidas, College Guide for his
valuable inputs from time to time. I express my gratitude to Mr. Rahul Rathi, CEO,
Capital Metrics & Risk Solutions for giving me the opportunity to work in this esteemed
organization and making the whole learning experience memorable. I express my
gratitude to Mr. Ravi Kumar, Research Analyst, CMRS and Company Guide for his help
and persistent support without which this project would not have been possible. I am
also very thankful to Mr. Jeevan., Mr. Sawan, Research Analysts, CMRS and Mr. Rajesh
Shah, Head of Quantitave Analytics Team, CMRS for their valuable inputs. I am also
thankful to Placement Advisory Team, which made sure I get the best summer
placement. I also thank all my classmates and specially my colleagues at CMRS, Rahul
Kumar and Mangesh Bhagat for their suggestions and inputs.
-Biswadeep Ghosh
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1. OBJECTIVE
The project will analyze the various derivative models like Black Model, Factors Models,
HJM and Libor Market Model used for pricing the interest rate derivatives, which
include both plain vanilla as well as exotic interest rate derivatives.
The most important concept in derivative pricing is arbitrage. Arbitrage becomes
possible when two identical or very similar assets trade at two different prices. If and
when such a situation occurs, a trader may have an opportunity to realize virtually
riskless profit because there would be no need to commit any actual money to the trade.
After studying these models, the interest rate derivatives like caps, floors and callable
and non-callable bonds are priced according to the best fitting models.
2. ROAD MAP
DATE ACTIVITY
14/04/2009 IN-DEPTH COMPANY AND INDUSTRY ANALYSIS
01/05/2009 UNDERSTANDING THE BASIC CONCEPTS OF OPTIONS, FUTURES AND
OTHER DERIVATIVES
13/05/2009 UNDERSTANDING THE YIELD CURVE AND VARIOUS PRICING
MODELS
27/05/2009 QUANTITATIVE ASPECTS OF THE INTEREST RATE DERIVATIVE
MODEL AND BUILDING UPON THE EXISTING MODEL BASED ON
ADDITIONAL PARAMETERS
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3. METHODOLGY
Pricing using MATLAB and OCTAVE
Running the basic models in MATLAB
Writing algorithm to implement HJM and
LMM
Studying various Interest Rate Derivative Models
Wiener
Process
Black Scholes
Model
Black -76
Model
Factor
Models
HJM
Libor Market
Model
Study of Yield Curve
Study of ytm, spot curve and forward curves
Running the various curves in MATLAB and
drawing inferences.
Study of Derivatives
Study of futures, forwards, options, swaps and exotics
Company and Industry Analysis
Study of Indian Derivative Market
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4. COMPANY ANALYSIS: AN OVERVIEW
4.1 Introduction
Capital Metrics & Risk Solutions is an independent equity research and financial analytics
company based in India. It started in 2002 exclusively as a financial research company. It is
a financial services firm with three areas of focus:
1) Risk Management Consultancy – Aimed at customized product development including
pricing calculators & hedging strategies
2) Quantitative Trading Strategies – Client specific and proprietary
3) Fundamental Equity Research – Proprietary and customized to suit client needs.
4.2 Services
Exclusively a financial research company, Capital Metrics & Risk Solutions is focused on two
main areas of service:
4.2.1 Equity Research
Capital Metrics & Risk Solutions is an independent research company with no
investment banking relationships. It offers objective and thorough equity research to
multiply client investment returns. Using qualitative and quantitative techniques and
proprietary and secondary databases, it offers analyses of different asset classes like
REITs and MLPs. It covers emerging as well as developed markets.
4.2.2 Financial Analytics
Capital Metrics & Risk Solutions has a rich and diverse experience in financial and
quantitative risk analytics. Here are some needs it has met:
It developed a quantitative model for assessing the credit and market risk of a leading
bank for their expanded loan portfolio.
It helped a mutual fund optimize its portfolio in a volatile market.
It developed a quantitative trading strategy for an institutional investor to buy a large
chunk of stock with minimized volatility risk and impact costs.
To generate its analytic reports, it makes use of sophisticated statistical and modeling
tools like Monte Carlo Simulations, Fat Tails Analysis and Risk Budgeting. It makes
conventional models more robust to meet particular requirements.
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4.2.3 Research Process
Capital Metrics & Risk Solutions works closely with clients at all stages of a project. Its
research process is customized in every case.
Arriving At a Shared Platform of Understanding
It spends a lot of time and resources in understanding the client's objectives. Quite often,
a new engagement starts with a short term pilot project. This helps in understanding the
client's needs and the client gets a fair chance to assess its capabilities.
Building a Report
It forms a research team selected carefully to suit the client's need. In each team, there is
a one-person client contact for easy accountability and quick, two-way flow of
communication.
Quality Check
Every report is scrutinized by a quality assurance team for comprehensiveness and
reliability of data, soundness of analysis and accuracy of results.
Internal Assessment
The QA-approved report is presented before a committee comprising experts and fund
managers. A report is submitted to a client only after it has satisfied the committee.
Follow Up
The relationship does not end with the submission of a report. Throughout the year, it
updates the client with information that could impact our analysis. It also works with
clients to implement a strategy.
4.2.4 People
The people in company are bright, young and highly capable. Over four-fifths of
its research team members hold a masters and/or MBA degrees from top US, UK
and Indian institutions. They have wide industry exposure, across geographies
and business sectors.
4.2.5 Clientele
It focuses on equities as an asset class. It is targeting a growth of 2X in clientele. Its book
includes esteemed clients like IDBI Capital, David Landes of Bondsonline.com, HDFC
Bank and ICICI Bank. It has established a relationship with CNBC TV18 in the recent past
and has recently added SBI Capital, a potentially large business.
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5. INDUSTRIAL ANALYSIS: FINANCIAL DERIVATIVES
MARKET
Financial markets are, by nature, extremely volatile and hence the risk factor is an important
concern for financial agents. To reduce this risk, the concept of derivatives comes into the
picture. Derivatives are products whose values are derived from one or more basic variables
called bases. These bases can be underlying assets (for example forex, equity, etc), bases or
reference rates. For example, wheat farmers may wish to sell their harvest at a future date to
eliminate the risk of a change in prices by that date. The transaction in this case would be the
derivative, while the spot price of wheat would be the underlying asset.
5.1 Development of exchange-traded derivatives
Derivatives have probably been around for as long as people have been trading with one
another. Forward contracting dates back at least to the 12th century and well have been
around before then. Merchants entered into contracts with one another for future delivery
of specified amount of commodities at specified price. A primary motivation for pre-
arranging a buyer or seller for a stock of commodities in early forward contracts was to
lessen the possibility that large swings would inhibit marketing the commodity after a
harvest.
5.2 The need for a derivatives market
The derivatives market performs a number of economic functions:
- They help in transferring risks from risk averse to risk oriented people
- They help in the discovery of future as well as current prices
- They catalyze entrepreneurial activity
- They increase the volume traded in markets because of participation of risk
averse people in greater numbers
- They increase savings and investment in the long run
5.3 The participants in a derivatives market
- Hedgers use futures or options markets to reduce or eliminate the risk
associated with price of an asset.
- Speculators use futures and options contracts to get extra leverage in betting on
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future movements in the price of an asset. They can increase both the potential
gains and potential losses by usage of derivatives in a speculative venture.
- Arbitrageurs are in business to take advantage of a discrepancy between prices
in two different markets. If, for example, they see the futures price of an asset
getting out of line with the cash price, they will take offsetting positions in the
two markets to lock in a profit.
5.4 Development of derivatives market in India
1. The first step towards introduction of derivatives trading in India was the
promulgation of the Securities Laws (Amendment) Ordinance, 1995, which
withdrew the prohibition on options in securities. The market for derivatives,
however, did not take off, as there was no regulatory framework to govern
trading of derivatives. SEBI set up a 24–member committee under the
Chairmanship of Dr.L.C.Gupta on November 18, 1996 to develop appropriate
regulatory framework for derivatives trading in India. The committee
submitted its report on March 17, 1998 prescribing necessary pre–conditions
for introduction of derivatives trading in India. The committee recommended
that derivatives should be declared as ‘securities’ so that regulatory
framework applicable to trading of ‘securities’ could also govern trading of
securities. SEBI also set up a group in June 1998 under the Chairmanship of
Prof.J.R.Varma, to recommend measures for risk containment in derivatives
market in India. The report, which was submitted in October 1998, worked
out the operational details of margining system, methodology for charging
initial margins, broker net worth, deposit requirement and real–time
monitoring requirements.
2. The Securities Contract Regulation Act (SCRA) was amended in December
1999 to include derivatives within the ambit of ‘securities’ and the regulatory
framework were developed for governing derivatives trading. The act also
made it clear that derivatives shall be legal and valid only if such contracts
are traded on a recognized stock exchange, thus precluding OTC derivatives.
The government also rescinded in March 2000, the three– decade old
notification, which prohibited forward trading in securities.
3. Derivatives trading commenced in India in June 2000 after SEBI granted the
final approval to this effect in May 2001. SEBI permitted the derivative
segments of two stock exchanges, NSE and BSE, and their clearing
house/corporation to commence trading and settlement in approved
derivatives contracts.
4. To begin with, SEBI approved trading in index futures contracts based on S&P
CNX Nifty and BSE–30(Sensex) index. This was followed by approval for
trading in options based on these two indexes and options on individual
securities.
5. The trading in BSE Sensex options commenced on June 4, 2001 and the
trading in options on individual securities commenced in July 2001. Futures
contracts on individual stocks were launched in November 2001.
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6. The derivatives trading on NSE commenced with S&P CNX Nifty Index
futures on June 12, 2000. The trading in index options commenced on June 4,
2001 and trading in options on individual securities commenced on July 2,
2001.
7. Single stock futures were launched on November 9, 2001. The index futures
and options contract on NSE are based on S&P CNX
8. Trading and settlement in derivative contracts is done in accordance with the
rules, byelaws, and regulations of the respective exchanges and their clearing
house/corporation duly approved by SEBI and notified in the official gazette.
Foreign Institutional Investors (FIIs) are permitted to trade in all Exchange
traded derivative products.
5.5 Exchange-traded vs. OTC derivatives markets
The OTC derivatives markets have witnessed rather sharp growth over the last few
years, which have accompanied the modernization of commercial and investment
banking and globalization of financial activities. The recent developments in
information technology have contributed to a great extent to these developments.
While both exchange-traded and OTC derivative contracts offer many benefits, the
former have rigid structures compared to the latter. It has been widely discussed
that the highly leveraged institutions and their OTC derivative positions were the
main cause of turbulence in financial markets in 1998. These episodes of turbulence
revealed the risks posed to market stability originating in features of OTC derivative
instruments and markets.
The OTC derivatives markets have the following features compared to exchange-
traded derivatives:
- The management of counter-party (credit) risk is decentralized and located
within individual institutions,
- There are no formal centralized limits on individual positions, leverage, or
margining,
- There are no formal rules for risk and burden-sharing,
- There are no formal rules or mechanisms for ensuring market stability and
integrity, and for safeguarding the collective interests of market participants
- The OTC contracts are generally not regulated by a regulatory authority and the
Exchanges self-regulatory organization, although they are affected indirectly by
national legal systems, banking supervision and market surveillance.
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5.6 Comparative Analysis
5.6.1 World Exchanges:
Top 10 Derivatives Exchanges ranked by Number of Contracts Traded
Source: www.futuresindustry.org
5.6.2 Business Growth in NSE Derivatives segment
Source: National Stock Exchange of India
5.6.3 Month wise Product wise Traded Value Analysis
A graphical representation of the month wise product wise turnover in the F&O
Segment for the period October 2008 to March 2009 is as below:
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Source: National Stock Exchange of India
5.7 Indian Forex and Interest Rate Derivative market
Indian forex and derivative markets have also developed significantly over the years.
As per the BIS global survey the percentage share of the rupee in total turnover
covering all currencies increased from 0.3 percent in 2004 to 0.7 percent in 2007. As
per geographical distribution of foreign exchange market turnover, the share of
India at $34 billion per day increased from 0.4 in 2004 to 0.9 percent in 2007.
5.7.1 Currency Futures
Currently only Currency Futures are allowed to trade by SEBI. Since the launch of
the first currency futures exchange in September 2008, currency futures
contracts are being traded in three recognized exchanges.
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The average daily volume of currency futures contracts traded on all the
exchanges increased from Rs.260 crore in September 2008 to Rs.2,181 crore in
December 2008 and further to Rs.5,235 crore in March 2009.
The functioning of the exchanges continues to be reviewed by the RBI-SEBI
Standing Technical Committee. On the recommendation of the Committee, the
position limits on the clients and trading members have been doubled from US $
5 million and US $ 25 million respectively to US $ 10 million and US $ 50 million.
However, the upper limits of 6 per cent and 15 per cent of the total open interest
on the clients and trading members remain unchanged. The position limit for
banks continues at 15 per cent of total open interest or US $ 100 million,
whichever is higher.
5.7.2 Interest Rate Derivatives
- Rupee derivatives in India were introduced in July 1999 when RBI permitted
banks/FIs/PDs to undertake Interest rate swaps and Forward rate
agreements.
- The rupee interest rate derivatives presently permissible are Forward Rate
Agreements (FRA), Interest Rate Swaps (IRS) and Interest Rate Futures (IRF).
- As regards interest rate derivatives, the inter-bank Rupee swap market
turnover, as reported on the CCIL platform, has averaged around USD 4
billion (Rs. 16,000 crores) per day in notional terms.
- The outstanding Rupee swap contracts in banks’ balance sheet, as on
Avg. daily Volume( in Rs crore)
0
1000
2000
3000
4000
5000
6000
Sep-
08
Oct-
08
Nov-
08
Dec-
08
Jan-
09
Feb-
09
Mar-
09
Avg. daily Volume( in
Rs crore)
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August 31, 2007, amounted to nearly USD 1600 billion (Rs. 64,00,000 crore)
in notional terms.
- Outstanding notional amounts in respect of cross currency interest rate
swaps in the banks’ books as on August 31, 2007, amounted to USD 57 billion
(Rs. 2,24,000 crore).
- Interest Rate Futures Contracts are contracts based on the list of underlying
as may be specified by the Exchange and approved by SEBI from time to time.
To begin with, interest rate futures contracts on the following underlying
shall be available for trading on the F&O Segment of the National Stock
Exchange:
o Notional T – Bills
o Notional 10 year bonds (coupon bearing and non-coupon bearing)
5.8 Final Note
In terms of the growth of derivatives markets, and the variety of derivatives users, the
Indian market has equaled or exceeded many other regional markets. While the growth is
being spearheaded mainly by retail investors, private sector institutions and large
corporations, smaller companies and state-owned institutions are gradually getting into the
act. Foreign brokers such as JP Morgan Chase are boosting their presence in India in
reaction to the growth in derivatives. The variety of derivatives instruments available for
trading is also expanding.
There remain major areas of concern for Indian derivatives users. Large gaps exist in the
range of derivatives products that are traded actively. In equity derivatives, NSE figures
show that almost 90% of activity is due to stock futures or index futures, whereas trading in
options is limited to a few stocks, partly because they are settled in cash and not the
underlying stocks.
As Indian derivatives markets grow more sophisticated, greater investor awareness will
become essential. NSE has programs to inform and educate brokers, dealers, traders, and
market personnel. In addition, institutions will need to devote more resources to develop
the business processes and technology necessary for derivatives trading.
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6. STUDY OF DERIVATIVES
The Merriam-Webster dictionary defines a derivative in the field of chemistry as “a
substance that can be made from another substance.” Derivatives in finance work on
the same principle.
Derivatives are the financial instruments which promise payoffs that are derived from
the value of something else, which is called the “underlying.” The underlying is often a
financial asset or rate, but it does not have to be. For example, derivatives exist with
payments linked to the S&P 500 stock index, the temperature at IGI Airport, and the
number of bankruptcies among a group of selected companies. Some estimates of the
size of the market for derivatives are in excess of $270trillion – more than 100 times
larger than 30 years ago. When derivative contracts lead to large financial losses, they
can make headlines. Derivatives had a role in the fall of Enron. Just a few years ago,
Warren Buffett concluded that “derivatives are financial weapons of mass destruction,
carrying dangers that, while now latent, are potentially lethal.” But there are two sides
to this coin. Although some serious dangers are associated with derivatives, handled
with care they have proved to be immensely valuable to modern economies, and will
surely remain so.
Derivatives come in flavors from plain vanilla to mint chocolate-chip. The plain vanilla
include contracts to buy or sell something for future delivery (forward and futures
contracts), contracts involving an option to buy or sell something at a fixed price in the
future (options) and contracts to exchange one cash flow for another (swaps), along
with simple combinations of forward, futures and options contracts. (Futures contracts
are similar to forward contracts, but they are standardized contracts that trade on
exchanges.) At the mint chocolate-chip end of the spectrum, however, the sky is the
limit.
6.1 Future Contracts
In finance, a futures contract is a standardized contract, traded on a futures
exchange, to buy or sell a certain underlying instrument at a certain date in the
future, at a pre-set price. The future date is called the delivery date or final
settlement date. The pre-set price is called the futures price. The price of the
underlying asset on the delivery date is called the settlement price. The settlement
price, normally, converges towards the futures price on the delivery date.
A futures contract gives the holder the right and the obligation to buy or sell, which
differs from an options contract, which gives the buyer the right, but not the
obligation, and the option writer (seller) the obligation, but not the right. To exit the
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commitment, the holder of a futures position has to sell his long position or buy back
his short position, effectively closing out the futures position and its contract
obligations. Futures contracts are exchange traded derivatives. The exchange acts as
counterparty on all contracts, sets margin requirements, etc.
6.2 Forward Contracts
A forward contract obligates one party to buy the underlying at a fixed price at a
certain future date (called the maturity) from a counterparty, who is obligated to sell
the underlying at that fixed price. Consider a U.S. exporter who expects to receive a
€100 million payment for goods in six months. Suppose that the price of the euro is
$1.20 today. If the euro were to fall by 10 percent over the next six months, the
exporter would lose $12 million. But by selling euros forward, the exporter locks in
the current forward exchange rate. If the forward rate is $1.18 (less than $1.20
because the market apparently expects the euro to depreciate a bit), the exporter is
guaranteed to receive $118 million at maturity. Hedging consists of taking a financial
position to reduce exposure to a risk. In this example, the financial position is a
forward contract, the risk is depreciation of the euro, and the exposure is €100
million in six months, which is perfectly hedged with the forward contract. Since no
money changes hands when the exporter buys euros forward, the market value of
the contract must be zero when it is initiated, since otherwise the exporter would
get something for nothing.
6.3 Options
Options Contract is a type of Derivatives Contract which gives the buyer/holder of
the contract the right (but not the obligation) to buy/sell the underlying asset at a
predetermined price within or at end of a specified period. The buyer / holder of the
option purchase the right from the seller/writer for a consideration which is called
the premium. The seller/writer of an option is obligated to settle the option as per
the terms of the contract when the buyer/holder exercises his right. The underlying
asset could include securities, an index of prices of securities etc.
A call option on a stock gives its holder the right to buy a fixed number of shares at a
given price by some future date, while a put option gives its holder the right to sell a
fixed number of shares on the same terms. The specified price is called the exercise
price. When the holder of an option takes advantage of her right, she is said to
exercise the option. The purchase price of an option – the money that changes hands
on day one – is called the option premium. Options enable their holders to lever
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their resources, while at the same time limiting their risk. Suppose Smith believes
that the current price of $50 for Upside Inc. stock is too low. Let’s assume that the
premium on a call option that confers the right to buy shares at $50 each for six
months is $10 per share. Smith can buy call options to purchase 100 shares for
$1,000. She will gain from stock price increases as if she had invested in 100 shares,
even though she invested an amount equal to the value of 20 shares. With only
$1,000 to invest, Smith could have borrowed $4,000 to buy 100 shares. At maturity,
she would then have to repay the loan. The gain made upon exercising the option is
therefore similar to the gain from a levered position in the stock – a position
consisting of purchasing shares with one’s own money plus money that’s borrowed.
However, if Smith borrowed $4,000, she could lose up to $5,000 plus interest if the
stock price fell to zero. With the call option, the most she can lose is $1,000. But
there’s no free lunch here; she’ll lose the entire $1,000 if the stock price does not rise
above $50.
6.4 Swaps
Swaps are private agreements between two parties to exchange cash flows in the
future according to a prearranged formula. A swap is a contract to exchange cash
flows over a specific period. The principal used to compute the flows is the “notional
amount.” Suppose you have an adjustable-rate mortgage with principal of $200,000
and current payments of $11,000 per year. If interest rates doubled, your payments
would increase dramatically. You could eliminate this risk by refinancing with a
fixed-rate mortgage, but the transaction could be expensive. A swap contract, by
contrast, would not entail renegotiating the mortgage. You would agree to make
payments to a counter party – say a bank – equal to a fixed interest rate applied to
$200,000. In exchange, the bank would pay you a floating rate applied to $200,000.
With this interest-rate swap, you would use the floating-rate payments received
from the bank to make your mortgage payments. The only payments you would
make out of your own pocket would be the fixed interest payments to the bank, as if
you had a fixed-rate mortgage. Therefore, a doubling of interest rates would no
longer affect your out-of pocket costs. Nor, for that matter, would a halving of
interest rates.
The two commonly used swaps are:
• Interest rate swaps:
These entail swapping only the interest related cash flows between the parties
in the same currency.
• Currency swaps:
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These entail swapping both principal and interest between the parties, with the
cash flows in one direction being in a different currency than those in the
opposite direction.
Swaps are usually entered into at-the-money (i.e. with minimal initial cash payments
because fair value is zero), through brokers or dealers who take an up-front cash
payment or who adjust the rate to bear default risk. The two most prevalent swaps
are interest rate swaps and foreign currency swaps, while others include equity
swaps, commodity swaps, and swaptions.
6.5 Exotics
An exotic derivative is one that cannot be created by mixing and matching option
and forward contracts. Instead, the payoff is a complicated function of one or many
underlings’. When P&G lost $160 million on derivatives in 1994, the main culprit
was an exotic swap. The amount it had to pay on the swap depended on the five-year
Treasury note yield and the price of the 30-year Treasury bond. Another example of
an exotic derivative is a binary option, which pays a fixed amount if some condition
is met. For instance, a binary option might pay $10 million if, before a specified date,
one of the three largest banks in Indonesia has defaulted on its debt.
A few common exotics are
- Swaptions
Swaptions are an option granting its owner the right but not the obligation to enter
into an underlying swap. Although options can be traded on a variety of swaps, the
term "swaptions" typically refers to options on interest rate swaps.
- Caps
An cap or commonly called as a interest rate cap is a derivative in which the buyer
receives payments at the end of each period in which the interest rate exceeds the
agreed strike price. An example of a cap would be an agreement to receive a
payment for each month the LIBOR rate exceeds 2.5%.
The interest rate cap can be analyzed as a series of European call options or caplets
which exist for each period the cap agreement is in existence.
- Floors
P a g e | 19 SIBM Pune
An interest rate floor is a series of European put options or floorlets on a specified
reference rate, usually LIBOR. The buyer of the floor receives money if on the
maturity of any of the floorlets, the reference rate fixed is below the agreed strike
price of the floor.
P a g e | 20 SIBM Pune
7. STUDY OF YIELD CURVE
7.1 Yield to Maturity
The yield-to-maturity (ytm) is the internal rate of return of a bond. It is that number y such
that when all the cash flows from the bond are discounted at the rate y and added up, we
obtain the current price of the bond.
¿
=
+
=
n
i
t
i
i
i
y
c
P
1
) 1 (
In order to value large portfolios of bonds, there are three methods:
Discount function:
The discount function measures, for each t, the present value of $1 receivable in t years.
¿
=
=
n
i
i i
t d c P
1
) (
Zero-coupon rate:
The zero-coupon rate or the spot rate is the ytm of a zero-coupon bond with maturity t
years from the present. The price d(t) of the t-maturity zero is related to its ytm r(t) as
kt
k t r
t d
) / ) ( 1 (
1
) (
+
=
The curve {r (t) |t ? 0} is called the spot curve, zero-coupon yield curve or simply, the yield
curve. In practice, the yield curve is typically upward sloping, i.e, zcr’s increase as maturity
increases.
Forward rates:
The forward rate for the period (t1,t2) refers to the rate we can lock in today for the
borrowing or lending over this time period. It is denoted by f(t1,t2).
1 2
2 1
2 1
) ( ln ) ( ln
) , (
t t
t d t d
t t f
÷
÷
=
P a g e | 21 SIBM Pune
7.2 MATLAB conversion functions
rate2disc -Discounting factors from interest rates
Syntax
[Disc, EndTimes, StartTimes] = rate2disc(Compounding, Rates,
EndDates, StartDates, ValuationDate)
disc2rate -Interest rates from cash flow discounting factors
Syntax
[Rates, EndTimes, StartTimes] = disc2rate(Compounding, Disc,
EndDates, StartDates, ValuationDate)
fwd2zero -Zero curve given forward curve
Syntax
[ZeroRates, CurveDates] = fwd2zero(ForwardRates, CurveDates, Settle,
Compounding, Basis)
zero2fwd -Forward curve given zero curve
Syntax
[ForwardRates, CurveDates] = zero2fwd(ZeroRates, CurveDates, Settle,
Compounding, Basis)
P a g e | 22 SIBM Pune
Fig. : Treasury Spot and Forward Curves
7.3 Inferences
- If the zcr curve is increasing, the ytm curve will also be increasing and will lie below the zcr
curve.
- If the zcr curve is increasing, the forward curve must lie above the zcr curve.
- If the zcr curve is decreasing, the ytm curve will also be decreasing and will lie above the zcr
curve.
- If the zcr curve is decreasing, the forward curve must lie below the zcr curve.
P a g e | 23 SIBM Pune
8. DERIVATIVE MODELS
Derivatives are priced on the assumption that financial markets are frictionless. One can
then find an asset-buying-and-selling strategy that only requires an initial investment
that ensures that the portfolio generates the same payoff as the derivative. This is called
a “replicating portfolio.” The value of the derivative must be the same as that of the
replicating portfolio; otherwise there would be a way to make a risk-free profit by
buying the portfolio and selling the derivative.
8.1 Weiner Process and Ito’s Lemma
Any variable whose value changes over time in an uncertain way is said to follow a
stochastic process. Stochastic processes can be classified as discrete time or
continuous time. A discrete-time stochastic process is one where the value of the
variable can change only at certain fixed points in time, whereas a continuous-time
stochastic process is one where changes can take place at any time. Stochastic
processes can also be classified as continuous variable or discrete variable. In a
continuous-variable process, the underlying variable can take any value within a
certain range, whereas in a discrete-variable process, only certain discrete values
are possible.
Wiener process is a particular type of Markov stochastic process with a mean
change of zero and a variance rate of 1.0 per year. It has been used in physics
to describe the motion of a particle that is subject to a large number of small
molecular shocks and is sometimes referred to as Brownian motion.
8.2 Markov property
A Markov process is a particular type of stochastic process where only the present
value of a variable is relevant for predicting the future. The past history of the
variable and the way that the present has emerged from the past are irrelevant.
Stock prices are usually assumed to follow a Markov process.
A generalized Wiener process for a variable x can be defined in terms of dz as follows:
dx = adt + bdz
The mean change per unit time of a stochastic process is known as the drift rate and
the variance per unit time is known as the variance rate.
P a g e | 24 SIBM Pune
8.3 Ito Process
A further type of stochastic process can be defined. This is known as an Ito process.
This is a generalized Wiener process in which the parameters a and b are
functions of the value of the underlying variable x and time t. Algebraically, an Ito
process can be written
dx = a(x,t)dt + b(x,t)dz
Ito’s lemma
The price of a stock option is a function of the underlying stock's price and time.
More generally, we can say that the price of any derivative is a function of the
stochastic variables underlying the derivative and time. An important result in this
area was discovered by the mathematician Kiyosi Ito in 1951. It is known as Ito's
lemma.
Suppose that the value of a variable x follows the Ito process
dx = a(x,t)dt + b(x,i)dz
where dz is a Wiener process and a and b are functions of x and t. The variable x has
a drift rate of a and a variance rate of b2. Ito's lemma shows that a function G of x
and t follows the process
P a g e | 25 SIBM Pune
z b
x
G
dt b
x
G
t
G
a
x
G
dG c
c
c
+
|
|
.
|
\
|
c
c
+
c
c
+
c
c
=
2
2
2
2
1
So based on the equation mentioned above G follows a Ito process with a drift rate of
2
2
2
2
1
b
x
G
t
G
a
x
G
c
c
+
c
c
+
c
c
and a variance rate of
2
2
b
x
G
|
.
|
\
|
c
c
. The above equation is Ito’s
lemma.
8.4 The Black-Scholes-Merton Model
Lognormal property of Stock Prices
We define S G ln =
S S
G 1
=
c
c
2 2
2
1
S S
G
÷ =
c
c
0 =
c
c
t
G
Substituting the value in the equation derived above in
Sdz
S
G
dt S
S
G
t
G
S
S
G
dG o o µ
c
c
+
|
|
.
|
\
|
c
c
+
c
c
+
c
c
=
2 2
2
2
2
1
Or
dz dt dG o
o
µ +
|
|
.
|
\
|
÷ =
2
2
Since µ and ? are constant, the above equation indicates that our definition
S G ln = follows a generalized Wiener process. It has constant drift rate of
2
2
o
µ ÷ and constant variance
2
o .
The concept
The Black-Scholes-Merton differential equation is an equation that must be satisfied
by the price of any derivative dependent on a non-dividend-paying stock. The
arguments involve setting up a riskless portfolio consisting of a position in the
derivative and a position in the stock. In absence of arbitrage opportunities, the
P a g e | 26 SIBM Pune
return from the portfolio must be the risk-free interest rate, r. This leads to the
Black-Scholes-Merton differential equation.
The reason a riskless portfolio can be set up is that the stock price and the derivative
price are both affected by the same underlying source of uncertainty: stock price
movements. In any short period of time, the price of the derivative is perfectly
correlated with the price of the underlying stock. When an appropriate portfolio of
the stock and the derivative is established, the gain or loss from the stock position
always offsets the gain or loss from the derivative position so that the overall value
of the portfolio at the end of the short period of time is known with certainty.
The difference between the Black-Scholes-Merton analysis and the analysis using
the binomial model is, in the former the position in the stock and the derivative is
riskless for only a very short period of time. (In theory, it remains riskless only for
an instantaneously short period of time.) To remain riskless, it must be adjusted, or
rebalanced, frequently. It is nevertheless true that the return from the riskless
portfolio in any very short period of time must be the risk-free interest rate. This is
the key element in the Black-Scholes analysis and leads to their pricing formulas.
The Black-Scholes formulas for the prices at time zero of a European call option on a
non dividend paying stock and a European put option on a non-dividend paying stock
are
) ( ) (
2 1 0
d N Ke d N S c
rt ÷
÷ = and ) ( ) (
1 0 2
d N S d N Ke p
rt
÷ ÷ ÷ =
÷
where,
( ) ( )
( ) ( )
T d
T
T r K S
d
T
T r K S
d
o
o
o
o
o
÷ =
÷ +
=
+ +
=
1
2
0
2
2
0
1
2 ln
2 ln
8.5 Black’s-76 Model
This model is used to value European futures options. Fischer Black was the first to
show this in a paper published in 1976. The underlying assumption is that futures
prices follow a lognormal property. The European call price, c, and the European put
price, p, for a futures option are given by equations
P a g e | 27 SIBM Pune
( ) | | ) (
2 1 0
d KN d N F e c
rT
÷ =
÷
and
( ) | | ) (
1 0 2
d N F d KN e p
rT
÷ ÷ ÷ =
÷
where
( )
( )
T d
T
T K F
d
T
T K F
d
o
o
o
o
o
÷ =
÷
=
+
=
1
2
0
2
2
0
1
2 ln
2 ln
and ? is the volatility of the futures price. When the cost of carry and the
convenience yield are functions only of time, it can be shown that the volatility of the
futures price is the same as the volatility of the underlying asset. Therefore the
Black's model does not require the options contract and the futures contract to
mature at the same time.
8.6 One Factor Models
In a one-factor model, a single stochastic factor drives all the changes in the yield
curve. In one simple and versatile model of interest rates, all security prices and
rates depends on only one factor-the short rate. In the context of interest
rate derivatives, a short rate model is a mathematical model that describes the
future evolution of interest rates by describing the future evolution of the short rate.
The short rate, usually written rt is the (annualized) interest rate at which an entity
can borrow money for an infinitesimally short period of time from time t. Specifying
the current short rate does not specify the entire yield curve. However no-arbitrage
arguments show that, under some fairly relaxed technical conditions, if we model
the evolution of rt as a stochastic process under a risk-neutral measure Q then the
price at time t of a zero-coupon bond maturing at time T is given by
P a g e | 28 SIBM Pune
where is the natural filtration for the process. Thus specifying a model for the
short rate specifies future bond prices. This means that instantaneous forward rates
are also specified by the usual formula.
And its third equivalent, the yields are given as well.
A general one-factor model is one in which the short rate rt evolves according to the
process
t t t t
dW r t dt r t dr ) , ( ) , ( | o + =
where (.) o and (.) | denote, respectively, the drift and diffusion component of the
short rate, and
t
W is a standard Brownian motion process. Three special cases of this
process of interest are the Merton/Ho-Lee, Vasicek, and CIR models.
8.6.1 Merton/Ho-Lee Model
The Merton/Ho-Lee model is a short rate model to predict future interest rates.
It is the simplest model that can be calibrated to market data, by implying the
form of
÷
o from market prices. Ho and Lee does not allow for mean reversion.
) , (
t
r t o =
÷
o and ) , (
t
r t | =
÷
|
8.6.2 Vasicek Model
This is the first detailed no-arbitrage modeling in finance of the term-structure of
interest rates. Here,
) , (
t
r t o = ) (
t
r ÷ u k and ) , (
t
r t | =
÷
|
where, k , u , and
÷
| are all positive constants.
The drift term ) (
t
r ÷ u k in the Vasicek model exhibits mean-reversion. The short
rates have a normal distribution with mean and variance given by,
P a g e | 29 SIBM Pune
) 1 ( ). 2 / 1 ( ) | (
) ( ] | [
2
2
0
0 0
t
t
t
t
e r r Var
r e r r E
k
k
| k
u u
÷
÷
÷
÷ =
÷ + =
8.6.3 The CIR Model
Cox, Ingersoll, and Ross introduced a mean-reverting square-root diffusion
process for the short rate rt :
) , (
t
r t o = ) (
t
r ÷ u k and ) , (
t
r t | =
t
r
÷
|
The form of the diffusion component
t
r
÷
| implies that short rates follow a non-
centric chi-square distribution with
2 2 2 2
0 0
0 0
) 1 ( ) 2 / 1 ( ) ( ). / 1 ( ) | (
) ( ] | [
t t t
t
t
t
e e e r r r Var
r e r r E
k k k
k
| u k | k
u u
÷
÷
÷ ÷
÷
÷
÷ + ÷ =
÷ + =
One-Factor Interest Rate Models
Model Drift ) , (
t
r t o Diffusion ) , (
t
r t |
Merton (1973)
÷
o
÷
|
Vasicek (1977) ) (
t
r ÷ u k
÷
|
Dothan (1978) 0
÷
|
Brennan and Schwartz
(1979)
÷ ÷
+
t
r
1
0 o o
t
r
÷
|
Cox et al. (1985) ) (
t
r ÷ u k
t
r
÷
|
Chan et al. (1992)
÷ ÷
+
t
r
1
0 o o
5 . 1
t
r
÷
|
Geometric Brownian motion
÷
t
r o
t
r
÷
|
Constant elasticity of
variance (CEV)
÷
t
r o
¸
|
t
r
÷
P a g e | 30 SIBM Pune
Solving One Factor Models
- PDE Approach
The “one factor” assumption in a one-factor model is that the evolution of all the
bond prices in the model depends only on the evolution of the model’s single
factor. That is, if P(t,T) denotes the time-t price of a zero-coupon bond maturing
at T, then
P(t,T) = P(t,T,
t
r )
Let the short rate process be given by
t t t t
dW r t dt r t dr ) , ( ) , ( | o + =
By Ito’s lemma
dP(t,T) =
t
t r rr r t
rr r t
PdW Pdt
dW P dt P P P
dt P dr P dt P
o µ
| | o
|
+
+ + +
+ +
]
2
1
[
2
1
2
2
In the absence of arbitrage, bond prices must satisfy the “fundamental partial
differential equation (pde)”,
0
2
1
) (
2
= ÷ + + ÷ rP P P P
rr t r
| |ì o
where, ì is called the “market price of risk” associated with the model’s single
factor.
ì is defined through the drift and volatility of the unknown bond prices.
Introducing a functional form for ì is necessary to be able to solve this pde. Two
of the specific case-solutions are:
- Bond Prices in the Vasicek Model
P a g e | 31 SIBM Pune
Vasicek makes the assumption that ) , (
t
r t ì is a constant ì . The bond prices in
this model is given by
P (t, T,
t
r ) = exp )] ( ) ( [ t t B r A
t
÷
where, t =T-t is the time left to the bond maturity
)] exp( 1 [
1
) ( kt
k
t ÷ ÷ = B
2
3
2
)] exp( 1 [
4
) ( ] ) ( [ ) ( kt
k
o
t t t ÷ ÷ ÷ · ÷ = R B A
2
2
2
) (
k
o
k
ìo
u ÷ + = · R
- Bond Prices in the CIR Model
In this model the market price of risk, ) , (
t
r t ì =
t
r ì . The bond prices in this
model is given by
P (t, T,
t
r ) = exp )] ( ) ( [ t t B r A
t
÷
¦
)
¦
`
¹
¦
¹
¦
´
¦
(
¸
(
¸
+ ÷ + +
=
+ +
2
/ 2
2 / ) (
2 ) 1 )( (
2
ln ) (
o ku
¸t
t ¸ ì k
¸ ì k ¸
¸
t
e
e
A
¸ ì k ¸
t
¸t
¸t
2 ) 1 )( (
) 1 ( 2
) (
+ ÷ + +
÷
=
e
e
B
2 2
2 ) ( o ì k ¸ + + =
8.7 Multi-Factor Models
Any model in which there are two or more uncertain parameters in the option price
(one-factor models incorporate only one cause of uncertainty: the future price).
Multi-factor models are useful for two main reasons. Firstly, they permit more
realistic modeling, particularly of interest rates, although they are very difficult to
compute. Secondly, multi-factor options (for example, spread options) have several
P a g e | 32 SIBM Pune
parameters, each with independent volatilities, and also the correlation between the
underlying must be dealt with separately.
In one-factor models, the single factor was the short rate,
t
r . In multi-factor models,
the factors may be
- The long rate
- The rate of inflation
- The mean of the short rate
- The volatility of the short rate; and many others
In an n-factor model, there are n factors which affect changes in the yield curve. The
regular flow of modeling is:
1. Model each of the factors separately.
2. Solve for the market prices of risk associated with each factor.
3. Price the bonds using this and the n-factor version of the Partial
Differentiation Equation (PDE) approach.
The n-factor pde is
2
2
1
o µ
XX X t
P P P rP + + =
where, X=(X1, …….Xn) are the n factors driving the term structure.
Two-factor version of CIR model
The bond price where the short-rate is the sum of the two independent square-
root diffusions is as follows:
|
|
.
|
\
|
(
¸
(
¸
÷ ×
|
|
.
|
\
|
=
¿ [
= = 2 , 1 2 , 1
2 1
) ( exp ) ( ) , , , (
i
i it
i
i t t
B x A x x T t P t t
where, t =T-t
¦
)
¦
`
¹
¦
¹
¦
´
¦
(
¸
(
¸
+ ÷ + +
=
+ +
2
/ 2
2 / ) (
2 ) 1 )( (
2
ln ) (
i i i
i
i i i
i i i i
i
i
e
e
A
o u k
t ¸
t ¸ ì k
¸ ì k ¸
¸
t
P a g e | 33 SIBM Pune
i i i i
i
i
i
e
e
B
¸ ì k ¸
t
t ¸
t ¸
2 ) 1 )( (
) 1 ( 2
) (
+ ÷ + +
÷
=
2 2
2 ) (
i i i i
o ì k ¸ + + =
} 2 , 1 { e i
8.8 Heath-Jarrow-Morton (HJM) Model
HJM marks a significant advance to the modeling of the term- structure movements and the
pricing of the interest rate derivatives. HJM framework works directly with the entire yield
curve and models simultaneously the changes in the rates of all maturities. It is consistent with
any initial yield curve. Implementation only requires two pieces of information: the initial yield
curve and the volatilities of the forward rates of different maturities.
The difference of HJM with the factors models are as follows:
HJM Factor models
HJM is a general framework where the
choice of the factor is left to the
modeler.
Factor models are specific.
HJM framework works directly with the
entire yield curve.
Factor models works by modeling certain
points on the yield curve (e.g. Short rate)
HJM model the forward rate curve. Factor models model the spot yield curve.
The risk-neutral drifts of the forwards
are functions of the volatilities, not the
market price of the risk.
The risk-neutral drift of the short-rate
depends on the market price of the risk
ì
One-Factor HJM Model:
Let P(t,s) denote the time-t price of a zero-coupon bond maturing at time s and with a face value
of $1.
)
`
¹
¹
´
¦
÷ =
¿
÷
÷
1
). , ( exp ) , (
s
t i
h i t f s t P
where, f(t,i) means $1 invested at time I at the rate f(t,i) will grow by time i+1 to
exp{f(t,s).h}
h=length of each period in years
P a g e | 34 SIBM Pune
The forward curve f (t.T) can move “up“ or move “down” in the forward curve with probability q
and 1-q respectively.
(
(
(
(
¸
(
¸
÷ +
+ +
+ +
= +
) 2 , 1 (
) 2 , 1 (
) 1 , 1 (
) , 1 (
n t f
t t f
t t f
T t f
u
u
u
u
?
(
(
(
(
¸
(
¸
÷ +
+ +
+ +
= +
) 2 , 1 (
) 2 , 1 (
) 1 , 1 (
) , 1 (
n t f
t t f
t t f
T t f
d
d
d
d
?
8.9 LIBOR Market Model
Introduction
LIBOR Market Model, as the name suggests, models the LIBOR rates. It is also
known as the BGM Model (Brace Gatarek Musiela Model). It models the
forward LIBOR rates which is directly observable in the market. One of the
advantages of this model is that it is consistent with the Black’s pricing of
Swaptions and Caplets.
There are many variations of LIBOR Market Models. Here we are going to discuss
following variations:
- LIBOR Market Model with non-stochastic volatility.
- LIBOR Market Model with Zhu stochastic volatility model.
- LIBOR Market Model with Heston stochastic variance model.
Dynamics of LIBOR Market Model
Forward LIBOR rates at any time t can be calculated using corresponding yield
curve as follows
i i
i
i i i
T t P
T t P
T T t f
t
1
1
) , (
) , (
) , , (
1
1
×
|
|
.
|
\
|
÷ =
÷
÷
Here, ) , , (
1 i i i
T T t f
÷
is the forward LIBOR rate at time t corresponding to time
period
1 ÷ i
T to
i
T for i = 1, 2, 3… m. ) , (
i
T t P is the discount factor (bond price) of
maturity
i
T calculated at time t.
i
t is the year fraction between
1 ÷ i
T and
i
T .
LIBOR Market Model with non-stochastic volatility
P a g e | 35 SIBM Pune
The dynamics of LIBOR Market Model is given by following equation:
¿
=
+ =
m
k
k k i i i
i
i
dZ b dt
f
df
1
,
o µ
Here
i
µ is the drift part which depends upon the forward measure in which this
dynamic equation is written. Drift term can be written in self, spot and terminal
measures as follows
? Self Measure – 0 =
i
µ
? Spot Measure –
¿
=
+
÷ =
i
j j j
j j j i j
i i
f
f
1
,
1 t
t µ o
o µ
? Terminal Measure –
¿
+ =
+
÷ =
m
i j j j
j j j i j
i i
f
f
1
,
1 t
t µ o
o µ
Here
i
o is the volatility of forward rate ) , , (
1 i i i
T T t f
÷
.
k i
b
,
is the loading factor
which depends upon the correlation between forward LIBOR rates. The
relationship between loading factor
k i
b
,
and correlation is given by following
matrix equation.
] [ ] ][ [ µ =
T
b b
Here is m×n matrix. m is the total number of LIBOR rates while n is the total
number of independent wiener process dZk required for calculating the dynamics
of the all these LIBOR rates.
LIBOR Market Model with Zhu stochastic volatility model
In this model, unlike previous one, volatility of the LIBOR rate is stochastic in
nature. Therefore stochastic equation of LIBOR rate remains the same but the
parametric equation for volatility changes to mean reverting stochastic equation.
Dynamics of this model is written as
¿
=
+ =
n
k
k k i i i
i
i
dZ b v dt
f
df
1
,
µ
Here
k i
b
,
is loading factor which depends upon the correlation between forward
LIBOR rates.
i
v is the stochastic volatility term which under forward measure
Q
i+1
is written as
dW dt v v
i i i i i
o u k + ÷ = ) (
P a g e | 36 SIBM Pune
Here,
i
k is mean reverting rate,
i
u is long term volatility and
i
o is volatility of
volatility of respective forward LIBOR rate. dW is the stochastic part of the
volatility equation. Each of these
i
k ,
i
u ,
i
o and
i
v (0) is written as following
parametric equations
) exp(
i i
aT k k =
) exp( ) (
3 2 1 i i i
T b b T b + + =u u
) exp( ) ( ) 0 (
3 2 1 i i i
T c c T c v v + + = , initial volatility.
) exp(
i i
dT o o =
i
µ is the drift part which depends upon the forward measure in which this
dynamic equation is written. Drift term in self, spot and terminal measures can
be written as follows
? Self Measure – 0 =
i
µ
? Spot Measure –
¿
=
+
÷ =
i
j j j
j j j i j
i i
f
f Corr
1
,
1 t
t o
o µ
? Terminal Measure –
¿
+ =
+
÷ =
m
i j j j
j j j i j
i i
f
f Corr
1
,
1 t
t o
o µ
Followings are some of its form that is frequently used in LMM.
? Rebonatho one parameter correlation equation
( ) | | exp
0 , j i j i
T T Corr ÷ ÷ = |
? Rebonatho two parameters correlation equation
( ) ( ) | | ) , min( exp exp
0 , j i j i j i
T T T T gamma Corr ÷ × × ÷ ÷ = |
LIBOR Market Model with Heston stochastic variance model
? In this model forward LIBOR rate dynamics in risk neutral world (spot
measure) is written as follows
(
¸
(
¸
÷ =
¿ ¿
= =
+
n
k
n
k
k i k i k k i i i
dt t t t V dZ t t V t f t df
1 1
, , 1 ,
) ( ) ( ) ( ) ( ) ( ) ( ) ( ¸ o ¸
? Here
k i , 1 +
o is the volatility of bond price of maturity Ti+1 at time t, while
V(t) is the stochastic variance term and in risk neutral world it is written
as follows
P a g e | 37 SIBM Pune
) ( ) ( )) ( ( ) ( t dW t V dt t V t dV c u k + ÷ =
Here k is the mean reverting rate, u is long term variance value and c is
volatility of stochastic part of the stochastic variance equation. Also here
‘n’ is the total number of independent wiener process required to write
dynamics of all the ‘m’ LIBOR rates. Correlations between different
forward LIBOR rates are associated with vectors
i
¸ for i = 1, 2, 3…… m.
Followings are some of its form that is frequently used in LMM.
? Rebonatho one parameter correlation equation
( ) | | exp
0 , j i j i
T T Corr ÷ ÷ = |
? Rebonatho two parameters correlation equation
( ) ( ) | | ) , min( exp exp
0 , j i j i j i
T T T T gamma Corr ÷ × × ÷ ÷ = |
P a g e | 38 SIBM Pune
9. PRICING USING MATLAB AND OCTAVE
9.1 Black-Scholes
blsprice -Black-Scholes put and call option pricing
Syntax
[Call, Put] = blsprice(Price, Strike, Rate, Time, Volatility, Yield)
blsprice can handle other types of underlies like Futures and Currencies. When
pricing Futures (Black model), enter the input argument Yield as:
Yield = Rate
When pricing currencies (Garman-Kohlhagen model), enter the input
argument Yield as:
Yield = ForeignRate
where ForeignRate is the continuously compounded, annualized risk free interest
rate in the foreign country.
Example:
Consider European stock options that expire in three months with an exercise price
of $95. Assume that the underlying stock pays no dividend, trades at $100, and has a
volatility of 50% per annum. The risk-free rate is 10% per annum. Using this data
[Call, Put] = blsprice(100, 95, 0.1, 0.25, 0.5)
returns call and put prices of $13.70 and $6.35, respectively.
9.2 Black-76
blkprice -Black's model for pricing futures options
Syntax
[Call, Put] = blkprice(Price, Strike, Rate, Time, Volatility)
Examples
P a g e | 39 SIBM Pune
Consider European futures options with exercise prices of $20 that expire in four
months. Assume that the current underlying futures price is also $20 with a
volatility of 25% per annum. The risk-free rate is 9% per annum. Using this data
[Call, Put] = blkprice(20, 20, 0.09, 4/12, 0.25)
returns equal call and put prices of $1.1166.
9.3 HJM
Following is a program implemented in OCTAVE for creating HJM price tree.
function u = hjm(f0,sig0,h);
n = length(f0);
m = n-1;
fu = f0(2:n);
fd = f0(2:n);
sigma = sig0(2:n);
alpha = zeros(m:1);
for j=[1:m];
if (j==1);
alpha(j) = log(0.5*(exp(-sigma(j)*h*sqrt(h)) + ...
exp(sigma(j)*h*sqrt(h))))/h^2;
end;
if (j>1);
alpha(j) = log(0.5*(exp(-sigma(1:j)*h*sqrt(h)) + ...
exp(sum(sigma(1:j))*h*sqrt(h))))/h^2 - sum(alpha(1:j-1));
end;
end;
fu = fu + alpha*h + sigma*sqrt(h);
fd = fd + alpha*h - sigma*sqrt(h);
u = [fu fd];
9.4 LIBOR Market Model
Following is a program implemented in OCTAVE for one-factor LMM
%Initial LIBOR curve
libor0 = [0.06 0.07 0.08 0.09 0.10 0.11]';
%Initial volatility curve (could be matrix)
sig = [0.15 0.14 0.13 0.12 0.11 0.10]';
P a g e | 40 SIBM Pune
n = length(libor0)-1; %Denotes the final LIBOR rate time
delta = 0.25; %Time interval in years
h = delta;
libor = zeros(n+1,n+1);
libor
,1) = libor0;
nsims = 10000;
%START SIMULATION
for i = 1:nsims;
for t = 1:n;
e = randn*sqrt(h);
drift = 0;
for k = [n:-1:t];
if k==n; %Do for numeraire
libor(k+1,t+1) = libor(k+1,t)*exp(-
0.5*sig(k+1)^2*h...
+ sig(k+1)*e);
else
drift = drift + delta*sig(k+2)*libor(k+2,t)/...
(1+delta*libor(k+2,t));
libor(k+1,t+1) = libor(k+1,t) * exp(-
0.5*sig(k+1)^2*h...
- drift*sig(k+1)*h + sig(k+1)*e);
end
end
end
price(i) = 100/prod(1+delta*diag(libor));
end;
%CROSS CHECK RESULTS
cprice = 100/prod(1+delta*libor0); %Price off initial curve
sprice = mean(price); %Simulated price
fprintf('Price off curve = %10.6f, Price off simulation = %10.6f \n',...
cprice,sprice);
P a g e | 41 SIBM Pune
10. VALUE ADDITION
10.1 To the Organization
The Analysis of the available interest rate derivative models and building upon them
based on additional parameters which would help in better analysis and
approximation of the LIBOR rates.
This project involves a lot of quantitative techniques and C++ coding which is out of
the scope due to limited time frame of 2 months.
- Running the pricing models in MATLAB and OCTAVE.
- Summarizing research reports
- Proof reading for a book on Derivative under publication.
- Fixing a process for a continuous refresh of data from Bloomberg.
- Evaluate the various ETL tools like Pentaho, Talend etc. to manage and
analyze Hedge Fund data.
10.2 To “me” as a student of management
The undertaking of this project has been very useful in multiple ways. It has helped
in the following ways
- To develop a broad understanding of the derivative market.
- To get a detailed understanding of the various derivative instruments, their
working.
- To price the interest rate derivatives based on different models using
MATLAB and OCTAVE.
- To get a insights on the hedge funds related data and it’s analysis using data
warehousing tools like Pentaho and Talend
- To work on Bloomberg terminals
P a g e | 42 SIBM Pune
11. TEAM PLAY IN THE ORGANIZATION
The quantitative analytics team at Capital Metrics and Risk Solutions is headed by Mr.
Rajesh Shah (PhD, Cornell), and operates under the overall guidance of Mr. Raghu
Sundaram. The team provides customized high-end derivatives valuation and risk-
management solutions for the clients.
As a part of this team, my role was to run the various derivative models in MATLAB and
price the interest rate derivatives.
Also, I implemented the HJM and LMM models in OCTAVE by writing the algorithm and
then running them to get the HJM tree and one-factor LMM.
Finally, I was asked to study a few research papers on LMM and summarize the findings
so that the team members can build upon them in calibrating the LMM.
12. LIMITATIONS OF THE STUDY
- This Project Involves A Lot Of Quantitative Techniques And C++ Coding Which Is Out
Of The Scope Due To Limited Time Frame Of 2 Months.
- The Libor Market Model (LMM) is very exhaustive and complex. Research is still in progress
to optimize the calibration process. Hence in this project only the basic of this model is
touched and further implementation is beyond the scope of the project.
P a g e | 43 SIBM Pune
13. IMPLICATION AND CONCLUSION
The derivative markets have been growing rapidly over the past few decades and
have been accused lately for their alleged role in the financial crisis. The leveraged
operations are said to have generate an “irrational appeal” for risk taking, and the
lack of clearing obligations also appeared as very damaging for the balance of the
market.
The countries all over the world are trying to extend its oversight of the financial
system to include the shadowy market of derivatives, the kind of complex financial
instruments that helped catapult the world into an economic crisis.
Countries all over the world, want to create a central electronic-based system that
would track the buying and selling of derivatives. They want to ensure that financial
firms selling the instruments have enough capital on hand in case they default and
subject them to stringent standards of conduct and new reporting requirements.
All (over-the-counter) derivatives dealers and all other firms whose activities in the
markets create large exposures to counterparties should be subject to a robust
regime of prudential supervision and regulation. Key elements of that robust
regulatory regime must include conservative capital requirements, business conduct
standards, reporting requirements and conservative requirements relating to initial
margins on counterparty credit exposures. New rules should be in place to deter
financial firms from taking undue risk, prevent fraud and ensure they are marketed
appropriately.
Current law largely excludes regulation of the instruments, which are referred to as
"over-the-counter" derivatives because they are traded privately rather than
through commodity exchanges. It was unclear how the rules would affect hedge
funds, which are large, mostly unregulated entities that use complex trading tactics
to earn big returns for high-dollar investors. Many hedge funds use derivatives
contracts to offset risk on other transactions.
New laws should be formed to check such activities and regulators should be always
on a vigil. There should a plan which talks to establish an "audit trail" for the
derivatives and have "clear unimpeded authority to police fraud, market
manipulation and other market abuse
Even after the above mentioned regulations and their implication, the derivatives
volumes are soaring, and at the same time instruments are becoming more and
more complex. This has created a great demand for pricing models to handle these
sophisticated structures, with a short supply of people capable of creating these
models.
P a g e | 44 SIBM Pune
And hence implementing Derivatives Models, particularly the over-the-counter
market in complex or exotic options, are continuing to expand rapidly on a global
scale. However, the availability of information regarding the theory and applications
of the numerical techniques required to succeed in these markets is limited. This
lack of information is extremely damaging to all kinds of financial institutions and
consequently there is enormous demand for a source of sound numerical methods
for pricing and hedging.
P a g e | 45 SIBM Pune
BIBLIOGRAPHY
1. Financial Derivatives Toolbox-User’s Guide, Version 2, The Math Works, Inc.
2. Lixin Wu and Fan Zhang(May 2006), Libor market model with stochastic volatility,
Journal Of Industrial And Management Optimization, Volume 2, Number 2, May
2006
3. Black, Fischer; Derman, Emanuel; Toy, William(1990), A One-Factor Model Of
Interest Rates And Its Application, Financial Analysts Journal; Jan/Feb 1990; 46, 1;
ABI/INFORM Global, pg. 33
4. Paul Wilmott (1995), The Mathematics of Financial Derivatives
5. Hull, John C. (2005), Options, Futures and Other Derivatives, Sixth Edition. Prentice
Hall
6. http://www.nseindia.com/ , National Stock Exchange of India
7. http://www.bseindia.com/ Bombay Stock Exchange of India
doc_871367592.docx
The document about analyzes the various derivative models like Black Model, Factors Models, HJM and Libor Market Model used for pricing the interest rate derivatives, which include both plain vanilla as well as exotic interest rate derivatives.
ANALYSIS
AND PRICING
OF INTEREST
RATE
DERIVATIVES
For: Capital Metrics and Risk Solutions, Pune
By: Biswadeep Ghosh
31262, MBA Finance
SIBM Pune
P a g e | 1 SIBM Pune
CONTENTS
ACKNOWLEDGEMENTS ................................................................................................................................................. 3
1. OBJECTIVE ................................................................................................................................................................. 4
2. ROAD MAP ................................................................................................................................................................. 4
3. METHODOLGY .......................................................................................................................................................... 5
4. COMPANY ANALYSIS: AN OVERVIEW ........................................................................................................... 6
4.1 Introduction .................................................................................................................................................... 6
4.2 Services .............................................................................................................................................................. 6
4.2.1 Equity Research.................................................................................................................................... 6
4.2.2 Financial Analytics .............................................................................................................................. 6
4.2.3 Research Process ................................................................................................................................. 7
4.2.4 People ....................................................................................................................................................... 7
4.2.5 Clientele ................................................................................................................................................... 7
5. INDUSTRIAL ANALYSIS: FINANCIAL DERIVATIVES MARKET ............................................................ 8
5.1 Development of exchange-traded derivatives .................................................................................. 8
5.2 The need for a derivatives market ......................................................................................................... 8
5.3 The participants in a derivatives market ............................................................................................ 8
5.4 Development of derivatives market in India ..................................................................................... 9
5.5 Exchange-traded vs. OTC derivatives markets .............................................................................. 10
5.6 Comparative Analysis ............................................................................................................................... 11
5.6.1 World Exchanges: .................................................................................................................................. 11
5.6.2 Business Growth in NSE Derivatives segment ..................................................................... 11
5.6.3 Month wise Product wise Traded Value Analysis ............................................................... 11
5.7 Indian Forex and Interest Rate Derivative market ...................................................................... 12
5.7.1 Currency Futures .............................................................................................................................. 12
5.7.2 Interest Rate Derivatives ............................................................................................................... 13
5.8 Final Note ...................................................................................................................................................... 14
6. STUDY OF DERIVATIVES .................................................................................................................................. 15
6.1 Future Contracts ......................................................................................................................................... 15
6.2 Forward Contracts ..................................................................................................................................... 16
6.3 Options ........................................................................................................................................................... 16
6.4 Swaps .............................................................................................................................................................. 17
6.5 Exotics ............................................................................................................................................................. 18
7. STUDY OF YIELD CURVE ................................................................................................................................... 20
P a g e | 2 SIBM Pune
7.1 Yield to Maturity ......................................................................................................................................... 20
7.2 MATLAB conversion functions ............................................................................................................. 21
7.3 Inferences ...................................................................................................................................................... 22
8. DERIVATIVE MODELS ........................................................................................................................................ 23
8.1 Weiner Process and Ito’s Lemma ........................................................................................................ 23
8.2 Markov property ........................................................................................................................................ 23
8.3 Ito Process ..................................................................................................................................................... 24
8.4 The Black-Scholes-Merton Model ........................................................................................................ 25
8.5 Black’s-76 Model ........................................................................................................................................ 26
8.6 One Factor Models ..................................................................................................................................... 27
8.6.1 Merton/Ho-Lee Model .................................................................................................................... 28
8.6.2 Vasicek Model .................................................................................................................................... 28
8.6.3 The CIR Model .................................................................................................................................... 29
8.7 Multi-Factor Models .................................................................................................................................. 31
8.8 Heath-Jarrow-Morton (HJM) Model ................................................................................................... 33
8.9 LIBOR Market Model ................................................................................................................................. 34
9. PRICING USING MATLAB AND OCTAVE ..................................................................................................... 38
9.1 Black-Scholes ............................................................................................................................................... 38
9.2 Black-76 ......................................................................................................................................................... 38
9.3 HJM ................................................................................................................................................................... 39
9.4 LIBOR Market Model ................................................................................................................................. 39
10. VALUE ADDITION ........................................................................................................................................... 41
10.1 To the Organization ................................................................................................................................... 41
10.2 To “me” as a student of management ................................................................................................ 41
11. TEAM PLAY IN THE ORGANIZATION ..................................................................................................... 42
12. LIMITATIONS OF THE STUDY.................................................................................................................... 42
13. IMPLICATION AND CONCLUSION ............................................................................................................ 43
BIBLIOGRAPHY .............................................................................................................................................................. 45
P a g e | 3 SIBM Pune
ACKNOWLEDGEMENTS
First of all, I would like to express my gratitude to Dr. Arun Mudbidri, Director, SIBM
who provided me the platform to enable me to work on this project. I take this
opportunity to thank Prof. Kaustubh Medhekar, Head of Department, Finance, SIBM for
his nurturing and relentless support. I also thank Prof. S. Kalidas, College Guide for his
valuable inputs from time to time. I express my gratitude to Mr. Rahul Rathi, CEO,
Capital Metrics & Risk Solutions for giving me the opportunity to work in this esteemed
organization and making the whole learning experience memorable. I express my
gratitude to Mr. Ravi Kumar, Research Analyst, CMRS and Company Guide for his help
and persistent support without which this project would not have been possible. I am
also very thankful to Mr. Jeevan., Mr. Sawan, Research Analysts, CMRS and Mr. Rajesh
Shah, Head of Quantitave Analytics Team, CMRS for their valuable inputs. I am also
thankful to Placement Advisory Team, which made sure I get the best summer
placement. I also thank all my classmates and specially my colleagues at CMRS, Rahul
Kumar and Mangesh Bhagat for their suggestions and inputs.
-Biswadeep Ghosh
P a g e | 4 SIBM Pune
1. OBJECTIVE
The project will analyze the various derivative models like Black Model, Factors Models,
HJM and Libor Market Model used for pricing the interest rate derivatives, which
include both plain vanilla as well as exotic interest rate derivatives.
The most important concept in derivative pricing is arbitrage. Arbitrage becomes
possible when two identical or very similar assets trade at two different prices. If and
when such a situation occurs, a trader may have an opportunity to realize virtually
riskless profit because there would be no need to commit any actual money to the trade.
After studying these models, the interest rate derivatives like caps, floors and callable
and non-callable bonds are priced according to the best fitting models.
2. ROAD MAP
DATE ACTIVITY
14/04/2009 IN-DEPTH COMPANY AND INDUSTRY ANALYSIS
01/05/2009 UNDERSTANDING THE BASIC CONCEPTS OF OPTIONS, FUTURES AND
OTHER DERIVATIVES
13/05/2009 UNDERSTANDING THE YIELD CURVE AND VARIOUS PRICING
MODELS
27/05/2009 QUANTITATIVE ASPECTS OF THE INTEREST RATE DERIVATIVE
MODEL AND BUILDING UPON THE EXISTING MODEL BASED ON
ADDITIONAL PARAMETERS
P a g e | 5 SIBM Pune
3. METHODOLGY
Pricing using MATLAB and OCTAVE
Running the basic models in MATLAB
Writing algorithm to implement HJM and
LMM
Studying various Interest Rate Derivative Models
Wiener
Process
Black Scholes
Model
Black -76
Model
Factor
Models
HJM
Libor Market
Model
Study of Yield Curve
Study of ytm, spot curve and forward curves
Running the various curves in MATLAB and
drawing inferences.
Study of Derivatives
Study of futures, forwards, options, swaps and exotics
Company and Industry Analysis
Study of Indian Derivative Market
P a g e | 6 SIBM Pune
4. COMPANY ANALYSIS: AN OVERVIEW
4.1 Introduction
Capital Metrics & Risk Solutions is an independent equity research and financial analytics
company based in India. It started in 2002 exclusively as a financial research company. It is
a financial services firm with three areas of focus:
1) Risk Management Consultancy – Aimed at customized product development including
pricing calculators & hedging strategies
2) Quantitative Trading Strategies – Client specific and proprietary
3) Fundamental Equity Research – Proprietary and customized to suit client needs.
4.2 Services
Exclusively a financial research company, Capital Metrics & Risk Solutions is focused on two
main areas of service:
4.2.1 Equity Research
Capital Metrics & Risk Solutions is an independent research company with no
investment banking relationships. It offers objective and thorough equity research to
multiply client investment returns. Using qualitative and quantitative techniques and
proprietary and secondary databases, it offers analyses of different asset classes like
REITs and MLPs. It covers emerging as well as developed markets.
4.2.2 Financial Analytics
Capital Metrics & Risk Solutions has a rich and diverse experience in financial and
quantitative risk analytics. Here are some needs it has met:
It developed a quantitative model for assessing the credit and market risk of a leading
bank for their expanded loan portfolio.
It helped a mutual fund optimize its portfolio in a volatile market.
It developed a quantitative trading strategy for an institutional investor to buy a large
chunk of stock with minimized volatility risk and impact costs.
To generate its analytic reports, it makes use of sophisticated statistical and modeling
tools like Monte Carlo Simulations, Fat Tails Analysis and Risk Budgeting. It makes
conventional models more robust to meet particular requirements.
P a g e | 7 SIBM Pune
4.2.3 Research Process
Capital Metrics & Risk Solutions works closely with clients at all stages of a project. Its
research process is customized in every case.
Arriving At a Shared Platform of Understanding
It spends a lot of time and resources in understanding the client's objectives. Quite often,
a new engagement starts with a short term pilot project. This helps in understanding the
client's needs and the client gets a fair chance to assess its capabilities.
Building a Report
It forms a research team selected carefully to suit the client's need. In each team, there is
a one-person client contact for easy accountability and quick, two-way flow of
communication.
Quality Check
Every report is scrutinized by a quality assurance team for comprehensiveness and
reliability of data, soundness of analysis and accuracy of results.
Internal Assessment
The QA-approved report is presented before a committee comprising experts and fund
managers. A report is submitted to a client only after it has satisfied the committee.
Follow Up
The relationship does not end with the submission of a report. Throughout the year, it
updates the client with information that could impact our analysis. It also works with
clients to implement a strategy.
4.2.4 People
The people in company are bright, young and highly capable. Over four-fifths of
its research team members hold a masters and/or MBA degrees from top US, UK
and Indian institutions. They have wide industry exposure, across geographies
and business sectors.
4.2.5 Clientele
It focuses on equities as an asset class. It is targeting a growth of 2X in clientele. Its book
includes esteemed clients like IDBI Capital, David Landes of Bondsonline.com, HDFC
Bank and ICICI Bank. It has established a relationship with CNBC TV18 in the recent past
and has recently added SBI Capital, a potentially large business.
P a g e | 8 SIBM Pune
5. INDUSTRIAL ANALYSIS: FINANCIAL DERIVATIVES
MARKET
Financial markets are, by nature, extremely volatile and hence the risk factor is an important
concern for financial agents. To reduce this risk, the concept of derivatives comes into the
picture. Derivatives are products whose values are derived from one or more basic variables
called bases. These bases can be underlying assets (for example forex, equity, etc), bases or
reference rates. For example, wheat farmers may wish to sell their harvest at a future date to
eliminate the risk of a change in prices by that date. The transaction in this case would be the
derivative, while the spot price of wheat would be the underlying asset.
5.1 Development of exchange-traded derivatives
Derivatives have probably been around for as long as people have been trading with one
another. Forward contracting dates back at least to the 12th century and well have been
around before then. Merchants entered into contracts with one another for future delivery
of specified amount of commodities at specified price. A primary motivation for pre-
arranging a buyer or seller for a stock of commodities in early forward contracts was to
lessen the possibility that large swings would inhibit marketing the commodity after a
harvest.
5.2 The need for a derivatives market
The derivatives market performs a number of economic functions:
- They help in transferring risks from risk averse to risk oriented people
- They help in the discovery of future as well as current prices
- They catalyze entrepreneurial activity
- They increase the volume traded in markets because of participation of risk
averse people in greater numbers
- They increase savings and investment in the long run
5.3 The participants in a derivatives market
- Hedgers use futures or options markets to reduce or eliminate the risk
associated with price of an asset.
- Speculators use futures and options contracts to get extra leverage in betting on
P a g e | 9 SIBM Pune
future movements in the price of an asset. They can increase both the potential
gains and potential losses by usage of derivatives in a speculative venture.
- Arbitrageurs are in business to take advantage of a discrepancy between prices
in two different markets. If, for example, they see the futures price of an asset
getting out of line with the cash price, they will take offsetting positions in the
two markets to lock in a profit.
5.4 Development of derivatives market in India
1. The first step towards introduction of derivatives trading in India was the
promulgation of the Securities Laws (Amendment) Ordinance, 1995, which
withdrew the prohibition on options in securities. The market for derivatives,
however, did not take off, as there was no regulatory framework to govern
trading of derivatives. SEBI set up a 24–member committee under the
Chairmanship of Dr.L.C.Gupta on November 18, 1996 to develop appropriate
regulatory framework for derivatives trading in India. The committee
submitted its report on March 17, 1998 prescribing necessary pre–conditions
for introduction of derivatives trading in India. The committee recommended
that derivatives should be declared as ‘securities’ so that regulatory
framework applicable to trading of ‘securities’ could also govern trading of
securities. SEBI also set up a group in June 1998 under the Chairmanship of
Prof.J.R.Varma, to recommend measures for risk containment in derivatives
market in India. The report, which was submitted in October 1998, worked
out the operational details of margining system, methodology for charging
initial margins, broker net worth, deposit requirement and real–time
monitoring requirements.
2. The Securities Contract Regulation Act (SCRA) was amended in December
1999 to include derivatives within the ambit of ‘securities’ and the regulatory
framework were developed for governing derivatives trading. The act also
made it clear that derivatives shall be legal and valid only if such contracts
are traded on a recognized stock exchange, thus precluding OTC derivatives.
The government also rescinded in March 2000, the three– decade old
notification, which prohibited forward trading in securities.
3. Derivatives trading commenced in India in June 2000 after SEBI granted the
final approval to this effect in May 2001. SEBI permitted the derivative
segments of two stock exchanges, NSE and BSE, and their clearing
house/corporation to commence trading and settlement in approved
derivatives contracts.
4. To begin with, SEBI approved trading in index futures contracts based on S&P
CNX Nifty and BSE–30(Sensex) index. This was followed by approval for
trading in options based on these two indexes and options on individual
securities.
5. The trading in BSE Sensex options commenced on June 4, 2001 and the
trading in options on individual securities commenced in July 2001. Futures
contracts on individual stocks were launched in November 2001.
P a g e | 10 SIBM Pune
6. The derivatives trading on NSE commenced with S&P CNX Nifty Index
futures on June 12, 2000. The trading in index options commenced on June 4,
2001 and trading in options on individual securities commenced on July 2,
2001.
7. Single stock futures were launched on November 9, 2001. The index futures
and options contract on NSE are based on S&P CNX
8. Trading and settlement in derivative contracts is done in accordance with the
rules, byelaws, and regulations of the respective exchanges and their clearing
house/corporation duly approved by SEBI and notified in the official gazette.
Foreign Institutional Investors (FIIs) are permitted to trade in all Exchange
traded derivative products.
5.5 Exchange-traded vs. OTC derivatives markets
The OTC derivatives markets have witnessed rather sharp growth over the last few
years, which have accompanied the modernization of commercial and investment
banking and globalization of financial activities. The recent developments in
information technology have contributed to a great extent to these developments.
While both exchange-traded and OTC derivative contracts offer many benefits, the
former have rigid structures compared to the latter. It has been widely discussed
that the highly leveraged institutions and their OTC derivative positions were the
main cause of turbulence in financial markets in 1998. These episodes of turbulence
revealed the risks posed to market stability originating in features of OTC derivative
instruments and markets.
The OTC derivatives markets have the following features compared to exchange-
traded derivatives:
- The management of counter-party (credit) risk is decentralized and located
within individual institutions,
- There are no formal centralized limits on individual positions, leverage, or
margining,
- There are no formal rules for risk and burden-sharing,
- There are no formal rules or mechanisms for ensuring market stability and
integrity, and for safeguarding the collective interests of market participants
- The OTC contracts are generally not regulated by a regulatory authority and the
Exchanges self-regulatory organization, although they are affected indirectly by
national legal systems, banking supervision and market surveillance.
P a g e | 11 SIBM Pune
5.6 Comparative Analysis
5.6.1 World Exchanges:
Top 10 Derivatives Exchanges ranked by Number of Contracts Traded
Source: www.futuresindustry.org
5.6.2 Business Growth in NSE Derivatives segment
Source: National Stock Exchange of India
5.6.3 Month wise Product wise Traded Value Analysis
A graphical representation of the month wise product wise turnover in the F&O
Segment for the period October 2008 to March 2009 is as below:
P a g e | 12 SIBM Pune
Source: National Stock Exchange of India
5.7 Indian Forex and Interest Rate Derivative market
Indian forex and derivative markets have also developed significantly over the years.
As per the BIS global survey the percentage share of the rupee in total turnover
covering all currencies increased from 0.3 percent in 2004 to 0.7 percent in 2007. As
per geographical distribution of foreign exchange market turnover, the share of
India at $34 billion per day increased from 0.4 in 2004 to 0.9 percent in 2007.
5.7.1 Currency Futures
Currently only Currency Futures are allowed to trade by SEBI. Since the launch of
the first currency futures exchange in September 2008, currency futures
contracts are being traded in three recognized exchanges.
P a g e | 13 SIBM Pune
The average daily volume of currency futures contracts traded on all the
exchanges increased from Rs.260 crore in September 2008 to Rs.2,181 crore in
December 2008 and further to Rs.5,235 crore in March 2009.
The functioning of the exchanges continues to be reviewed by the RBI-SEBI
Standing Technical Committee. On the recommendation of the Committee, the
position limits on the clients and trading members have been doubled from US $
5 million and US $ 25 million respectively to US $ 10 million and US $ 50 million.
However, the upper limits of 6 per cent and 15 per cent of the total open interest
on the clients and trading members remain unchanged. The position limit for
banks continues at 15 per cent of total open interest or US $ 100 million,
whichever is higher.
5.7.2 Interest Rate Derivatives
- Rupee derivatives in India were introduced in July 1999 when RBI permitted
banks/FIs/PDs to undertake Interest rate swaps and Forward rate
agreements.
- The rupee interest rate derivatives presently permissible are Forward Rate
Agreements (FRA), Interest Rate Swaps (IRS) and Interest Rate Futures (IRF).
- As regards interest rate derivatives, the inter-bank Rupee swap market
turnover, as reported on the CCIL platform, has averaged around USD 4
billion (Rs. 16,000 crores) per day in notional terms.
- The outstanding Rupee swap contracts in banks’ balance sheet, as on
Avg. daily Volume( in Rs crore)
0
1000
2000
3000
4000
5000
6000
Sep-
08
Oct-
08
Nov-
08
Dec-
08
Jan-
09
Feb-
09
Mar-
09
Avg. daily Volume( in
Rs crore)
P a g e | 14 SIBM Pune
August 31, 2007, amounted to nearly USD 1600 billion (Rs. 64,00,000 crore)
in notional terms.
- Outstanding notional amounts in respect of cross currency interest rate
swaps in the banks’ books as on August 31, 2007, amounted to USD 57 billion
(Rs. 2,24,000 crore).
- Interest Rate Futures Contracts are contracts based on the list of underlying
as may be specified by the Exchange and approved by SEBI from time to time.
To begin with, interest rate futures contracts on the following underlying
shall be available for trading on the F&O Segment of the National Stock
Exchange:
o Notional T – Bills
o Notional 10 year bonds (coupon bearing and non-coupon bearing)
5.8 Final Note
In terms of the growth of derivatives markets, and the variety of derivatives users, the
Indian market has equaled or exceeded many other regional markets. While the growth is
being spearheaded mainly by retail investors, private sector institutions and large
corporations, smaller companies and state-owned institutions are gradually getting into the
act. Foreign brokers such as JP Morgan Chase are boosting their presence in India in
reaction to the growth in derivatives. The variety of derivatives instruments available for
trading is also expanding.
There remain major areas of concern for Indian derivatives users. Large gaps exist in the
range of derivatives products that are traded actively. In equity derivatives, NSE figures
show that almost 90% of activity is due to stock futures or index futures, whereas trading in
options is limited to a few stocks, partly because they are settled in cash and not the
underlying stocks.
As Indian derivatives markets grow more sophisticated, greater investor awareness will
become essential. NSE has programs to inform and educate brokers, dealers, traders, and
market personnel. In addition, institutions will need to devote more resources to develop
the business processes and technology necessary for derivatives trading.
P a g e | 15 SIBM Pune
6. STUDY OF DERIVATIVES
The Merriam-Webster dictionary defines a derivative in the field of chemistry as “a
substance that can be made from another substance.” Derivatives in finance work on
the same principle.
Derivatives are the financial instruments which promise payoffs that are derived from
the value of something else, which is called the “underlying.” The underlying is often a
financial asset or rate, but it does not have to be. For example, derivatives exist with
payments linked to the S&P 500 stock index, the temperature at IGI Airport, and the
number of bankruptcies among a group of selected companies. Some estimates of the
size of the market for derivatives are in excess of $270trillion – more than 100 times
larger than 30 years ago. When derivative contracts lead to large financial losses, they
can make headlines. Derivatives had a role in the fall of Enron. Just a few years ago,
Warren Buffett concluded that “derivatives are financial weapons of mass destruction,
carrying dangers that, while now latent, are potentially lethal.” But there are two sides
to this coin. Although some serious dangers are associated with derivatives, handled
with care they have proved to be immensely valuable to modern economies, and will
surely remain so.
Derivatives come in flavors from plain vanilla to mint chocolate-chip. The plain vanilla
include contracts to buy or sell something for future delivery (forward and futures
contracts), contracts involving an option to buy or sell something at a fixed price in the
future (options) and contracts to exchange one cash flow for another (swaps), along
with simple combinations of forward, futures and options contracts. (Futures contracts
are similar to forward contracts, but they are standardized contracts that trade on
exchanges.) At the mint chocolate-chip end of the spectrum, however, the sky is the
limit.
6.1 Future Contracts
In finance, a futures contract is a standardized contract, traded on a futures
exchange, to buy or sell a certain underlying instrument at a certain date in the
future, at a pre-set price. The future date is called the delivery date or final
settlement date. The pre-set price is called the futures price. The price of the
underlying asset on the delivery date is called the settlement price. The settlement
price, normally, converges towards the futures price on the delivery date.
A futures contract gives the holder the right and the obligation to buy or sell, which
differs from an options contract, which gives the buyer the right, but not the
obligation, and the option writer (seller) the obligation, but not the right. To exit the
P a g e | 16 SIBM Pune
commitment, the holder of a futures position has to sell his long position or buy back
his short position, effectively closing out the futures position and its contract
obligations. Futures contracts are exchange traded derivatives. The exchange acts as
counterparty on all contracts, sets margin requirements, etc.
6.2 Forward Contracts
A forward contract obligates one party to buy the underlying at a fixed price at a
certain future date (called the maturity) from a counterparty, who is obligated to sell
the underlying at that fixed price. Consider a U.S. exporter who expects to receive a
€100 million payment for goods in six months. Suppose that the price of the euro is
$1.20 today. If the euro were to fall by 10 percent over the next six months, the
exporter would lose $12 million. But by selling euros forward, the exporter locks in
the current forward exchange rate. If the forward rate is $1.18 (less than $1.20
because the market apparently expects the euro to depreciate a bit), the exporter is
guaranteed to receive $118 million at maturity. Hedging consists of taking a financial
position to reduce exposure to a risk. In this example, the financial position is a
forward contract, the risk is depreciation of the euro, and the exposure is €100
million in six months, which is perfectly hedged with the forward contract. Since no
money changes hands when the exporter buys euros forward, the market value of
the contract must be zero when it is initiated, since otherwise the exporter would
get something for nothing.
6.3 Options
Options Contract is a type of Derivatives Contract which gives the buyer/holder of
the contract the right (but not the obligation) to buy/sell the underlying asset at a
predetermined price within or at end of a specified period. The buyer / holder of the
option purchase the right from the seller/writer for a consideration which is called
the premium. The seller/writer of an option is obligated to settle the option as per
the terms of the contract when the buyer/holder exercises his right. The underlying
asset could include securities, an index of prices of securities etc.
A call option on a stock gives its holder the right to buy a fixed number of shares at a
given price by some future date, while a put option gives its holder the right to sell a
fixed number of shares on the same terms. The specified price is called the exercise
price. When the holder of an option takes advantage of her right, she is said to
exercise the option. The purchase price of an option – the money that changes hands
on day one – is called the option premium. Options enable their holders to lever
P a g e | 17 SIBM Pune
their resources, while at the same time limiting their risk. Suppose Smith believes
that the current price of $50 for Upside Inc. stock is too low. Let’s assume that the
premium on a call option that confers the right to buy shares at $50 each for six
months is $10 per share. Smith can buy call options to purchase 100 shares for
$1,000. She will gain from stock price increases as if she had invested in 100 shares,
even though she invested an amount equal to the value of 20 shares. With only
$1,000 to invest, Smith could have borrowed $4,000 to buy 100 shares. At maturity,
she would then have to repay the loan. The gain made upon exercising the option is
therefore similar to the gain from a levered position in the stock – a position
consisting of purchasing shares with one’s own money plus money that’s borrowed.
However, if Smith borrowed $4,000, she could lose up to $5,000 plus interest if the
stock price fell to zero. With the call option, the most she can lose is $1,000. But
there’s no free lunch here; she’ll lose the entire $1,000 if the stock price does not rise
above $50.
6.4 Swaps
Swaps are private agreements between two parties to exchange cash flows in the
future according to a prearranged formula. A swap is a contract to exchange cash
flows over a specific period. The principal used to compute the flows is the “notional
amount.” Suppose you have an adjustable-rate mortgage with principal of $200,000
and current payments of $11,000 per year. If interest rates doubled, your payments
would increase dramatically. You could eliminate this risk by refinancing with a
fixed-rate mortgage, but the transaction could be expensive. A swap contract, by
contrast, would not entail renegotiating the mortgage. You would agree to make
payments to a counter party – say a bank – equal to a fixed interest rate applied to
$200,000. In exchange, the bank would pay you a floating rate applied to $200,000.
With this interest-rate swap, you would use the floating-rate payments received
from the bank to make your mortgage payments. The only payments you would
make out of your own pocket would be the fixed interest payments to the bank, as if
you had a fixed-rate mortgage. Therefore, a doubling of interest rates would no
longer affect your out-of pocket costs. Nor, for that matter, would a halving of
interest rates.
The two commonly used swaps are:
• Interest rate swaps:
These entail swapping only the interest related cash flows between the parties
in the same currency.
• Currency swaps:
P a g e | 18 SIBM Pune
These entail swapping both principal and interest between the parties, with the
cash flows in one direction being in a different currency than those in the
opposite direction.
Swaps are usually entered into at-the-money (i.e. with minimal initial cash payments
because fair value is zero), through brokers or dealers who take an up-front cash
payment or who adjust the rate to bear default risk. The two most prevalent swaps
are interest rate swaps and foreign currency swaps, while others include equity
swaps, commodity swaps, and swaptions.
6.5 Exotics
An exotic derivative is one that cannot be created by mixing and matching option
and forward contracts. Instead, the payoff is a complicated function of one or many
underlings’. When P&G lost $160 million on derivatives in 1994, the main culprit
was an exotic swap. The amount it had to pay on the swap depended on the five-year
Treasury note yield and the price of the 30-year Treasury bond. Another example of
an exotic derivative is a binary option, which pays a fixed amount if some condition
is met. For instance, a binary option might pay $10 million if, before a specified date,
one of the three largest banks in Indonesia has defaulted on its debt.
A few common exotics are
- Swaptions
Swaptions are an option granting its owner the right but not the obligation to enter
into an underlying swap. Although options can be traded on a variety of swaps, the
term "swaptions" typically refers to options on interest rate swaps.
- Caps
An cap or commonly called as a interest rate cap is a derivative in which the buyer
receives payments at the end of each period in which the interest rate exceeds the
agreed strike price. An example of a cap would be an agreement to receive a
payment for each month the LIBOR rate exceeds 2.5%.
The interest rate cap can be analyzed as a series of European call options or caplets
which exist for each period the cap agreement is in existence.
- Floors
P a g e | 19 SIBM Pune
An interest rate floor is a series of European put options or floorlets on a specified
reference rate, usually LIBOR. The buyer of the floor receives money if on the
maturity of any of the floorlets, the reference rate fixed is below the agreed strike
price of the floor.
P a g e | 20 SIBM Pune
7. STUDY OF YIELD CURVE
7.1 Yield to Maturity
The yield-to-maturity (ytm) is the internal rate of return of a bond. It is that number y such
that when all the cash flows from the bond are discounted at the rate y and added up, we
obtain the current price of the bond.
¿
=
+
=
n
i
t
i
i
i
y
c
P
1
) 1 (
In order to value large portfolios of bonds, there are three methods:
Discount function:
The discount function measures, for each t, the present value of $1 receivable in t years.
¿
=
=
n
i
i i
t d c P
1
) (
Zero-coupon rate:
The zero-coupon rate or the spot rate is the ytm of a zero-coupon bond with maturity t
years from the present. The price d(t) of the t-maturity zero is related to its ytm r(t) as
kt
k t r
t d
) / ) ( 1 (
1
) (
+
=
The curve {r (t) |t ? 0} is called the spot curve, zero-coupon yield curve or simply, the yield
curve. In practice, the yield curve is typically upward sloping, i.e, zcr’s increase as maturity
increases.
Forward rates:
The forward rate for the period (t1,t2) refers to the rate we can lock in today for the
borrowing or lending over this time period. It is denoted by f(t1,t2).
1 2
2 1
2 1
) ( ln ) ( ln
) , (
t t
t d t d
t t f
÷
÷
=
P a g e | 21 SIBM Pune
7.2 MATLAB conversion functions
rate2disc -Discounting factors from interest rates
Syntax
[Disc, EndTimes, StartTimes] = rate2disc(Compounding, Rates,
EndDates, StartDates, ValuationDate)
disc2rate -Interest rates from cash flow discounting factors
Syntax
[Rates, EndTimes, StartTimes] = disc2rate(Compounding, Disc,
EndDates, StartDates, ValuationDate)
fwd2zero -Zero curve given forward curve
Syntax
[ZeroRates, CurveDates] = fwd2zero(ForwardRates, CurveDates, Settle,
Compounding, Basis)
zero2fwd -Forward curve given zero curve
Syntax
[ForwardRates, CurveDates] = zero2fwd(ZeroRates, CurveDates, Settle,
Compounding, Basis)
P a g e | 22 SIBM Pune
Fig. : Treasury Spot and Forward Curves
7.3 Inferences
- If the zcr curve is increasing, the ytm curve will also be increasing and will lie below the zcr
curve.
- If the zcr curve is increasing, the forward curve must lie above the zcr curve.
- If the zcr curve is decreasing, the ytm curve will also be decreasing and will lie above the zcr
curve.
- If the zcr curve is decreasing, the forward curve must lie below the zcr curve.
P a g e | 23 SIBM Pune
8. DERIVATIVE MODELS
Derivatives are priced on the assumption that financial markets are frictionless. One can
then find an asset-buying-and-selling strategy that only requires an initial investment
that ensures that the portfolio generates the same payoff as the derivative. This is called
a “replicating portfolio.” The value of the derivative must be the same as that of the
replicating portfolio; otherwise there would be a way to make a risk-free profit by
buying the portfolio and selling the derivative.
8.1 Weiner Process and Ito’s Lemma
Any variable whose value changes over time in an uncertain way is said to follow a
stochastic process. Stochastic processes can be classified as discrete time or
continuous time. A discrete-time stochastic process is one where the value of the
variable can change only at certain fixed points in time, whereas a continuous-time
stochastic process is one where changes can take place at any time. Stochastic
processes can also be classified as continuous variable or discrete variable. In a
continuous-variable process, the underlying variable can take any value within a
certain range, whereas in a discrete-variable process, only certain discrete values
are possible.
Wiener process is a particular type of Markov stochastic process with a mean
change of zero and a variance rate of 1.0 per year. It has been used in physics
to describe the motion of a particle that is subject to a large number of small
molecular shocks and is sometimes referred to as Brownian motion.
8.2 Markov property
A Markov process is a particular type of stochastic process where only the present
value of a variable is relevant for predicting the future. The past history of the
variable and the way that the present has emerged from the past are irrelevant.
Stock prices are usually assumed to follow a Markov process.
A generalized Wiener process for a variable x can be defined in terms of dz as follows:
dx = adt + bdz
The mean change per unit time of a stochastic process is known as the drift rate and
the variance per unit time is known as the variance rate.
P a g e | 24 SIBM Pune
8.3 Ito Process
A further type of stochastic process can be defined. This is known as an Ito process.
This is a generalized Wiener process in which the parameters a and b are
functions of the value of the underlying variable x and time t. Algebraically, an Ito
process can be written
dx = a(x,t)dt + b(x,t)dz
Ito’s lemma
The price of a stock option is a function of the underlying stock's price and time.
More generally, we can say that the price of any derivative is a function of the
stochastic variables underlying the derivative and time. An important result in this
area was discovered by the mathematician Kiyosi Ito in 1951. It is known as Ito's
lemma.
Suppose that the value of a variable x follows the Ito process
dx = a(x,t)dt + b(x,i)dz
where dz is a Wiener process and a and b are functions of x and t. The variable x has
a drift rate of a and a variance rate of b2. Ito's lemma shows that a function G of x
and t follows the process
P a g e | 25 SIBM Pune
z b
x
G
dt b
x
G
t
G
a
x
G
dG c
c
c
+
|
|
.
|
\
|
c
c
+
c
c
+
c
c
=
2
2
2
2
1
So based on the equation mentioned above G follows a Ito process with a drift rate of
2
2
2
2
1
b
x
G
t
G
a
x
G
c
c
+
c
c
+
c
c
and a variance rate of
2
2
b
x
G
|
.
|
\
|
c
c
. The above equation is Ito’s
lemma.
8.4 The Black-Scholes-Merton Model
Lognormal property of Stock Prices
We define S G ln =
S S
G 1
=
c
c
2 2
2
1
S S
G
÷ =
c
c
0 =
c
c
t
G
Substituting the value in the equation derived above in
Sdz
S
G
dt S
S
G
t
G
S
S
G
dG o o µ
c
c
+
|
|
.
|
\
|
c
c
+
c
c
+
c
c
=
2 2
2
2
2
1
Or
dz dt dG o
o
µ +
|
|
.
|
\
|
÷ =
2
2
Since µ and ? are constant, the above equation indicates that our definition
S G ln = follows a generalized Wiener process. It has constant drift rate of
2
2
o
µ ÷ and constant variance
2
o .
The concept
The Black-Scholes-Merton differential equation is an equation that must be satisfied
by the price of any derivative dependent on a non-dividend-paying stock. The
arguments involve setting up a riskless portfolio consisting of a position in the
derivative and a position in the stock. In absence of arbitrage opportunities, the
P a g e | 26 SIBM Pune
return from the portfolio must be the risk-free interest rate, r. This leads to the
Black-Scholes-Merton differential equation.
The reason a riskless portfolio can be set up is that the stock price and the derivative
price are both affected by the same underlying source of uncertainty: stock price
movements. In any short period of time, the price of the derivative is perfectly
correlated with the price of the underlying stock. When an appropriate portfolio of
the stock and the derivative is established, the gain or loss from the stock position
always offsets the gain or loss from the derivative position so that the overall value
of the portfolio at the end of the short period of time is known with certainty.
The difference between the Black-Scholes-Merton analysis and the analysis using
the binomial model is, in the former the position in the stock and the derivative is
riskless for only a very short period of time. (In theory, it remains riskless only for
an instantaneously short period of time.) To remain riskless, it must be adjusted, or
rebalanced, frequently. It is nevertheless true that the return from the riskless
portfolio in any very short period of time must be the risk-free interest rate. This is
the key element in the Black-Scholes analysis and leads to their pricing formulas.
The Black-Scholes formulas for the prices at time zero of a European call option on a
non dividend paying stock and a European put option on a non-dividend paying stock
are
) ( ) (
2 1 0
d N Ke d N S c
rt ÷
÷ = and ) ( ) (
1 0 2
d N S d N Ke p
rt
÷ ÷ ÷ =
÷
where,
( ) ( )
( ) ( )
T d
T
T r K S
d
T
T r K S
d
o
o
o
o
o
÷ =
÷ +
=
+ +
=
1
2
0
2
2
0
1
2 ln
2 ln
8.5 Black’s-76 Model
This model is used to value European futures options. Fischer Black was the first to
show this in a paper published in 1976. The underlying assumption is that futures
prices follow a lognormal property. The European call price, c, and the European put
price, p, for a futures option are given by equations
P a g e | 27 SIBM Pune
( ) | | ) (
2 1 0
d KN d N F e c
rT
÷ =
÷
and
( ) | | ) (
1 0 2
d N F d KN e p
rT
÷ ÷ ÷ =
÷
where
( )
( )
T d
T
T K F
d
T
T K F
d
o
o
o
o
o
÷ =
÷
=
+
=
1
2
0
2
2
0
1
2 ln
2 ln
and ? is the volatility of the futures price. When the cost of carry and the
convenience yield are functions only of time, it can be shown that the volatility of the
futures price is the same as the volatility of the underlying asset. Therefore the
Black's model does not require the options contract and the futures contract to
mature at the same time.
8.6 One Factor Models
In a one-factor model, a single stochastic factor drives all the changes in the yield
curve. In one simple and versatile model of interest rates, all security prices and
rates depends on only one factor-the short rate. In the context of interest
rate derivatives, a short rate model is a mathematical model that describes the
future evolution of interest rates by describing the future evolution of the short rate.
The short rate, usually written rt is the (annualized) interest rate at which an entity
can borrow money for an infinitesimally short period of time from time t. Specifying
the current short rate does not specify the entire yield curve. However no-arbitrage
arguments show that, under some fairly relaxed technical conditions, if we model
the evolution of rt as a stochastic process under a risk-neutral measure Q then the
price at time t of a zero-coupon bond maturing at time T is given by
P a g e | 28 SIBM Pune
where is the natural filtration for the process. Thus specifying a model for the
short rate specifies future bond prices. This means that instantaneous forward rates
are also specified by the usual formula.
And its third equivalent, the yields are given as well.
A general one-factor model is one in which the short rate rt evolves according to the
process
t t t t
dW r t dt r t dr ) , ( ) , ( | o + =
where (.) o and (.) | denote, respectively, the drift and diffusion component of the
short rate, and
t
W is a standard Brownian motion process. Three special cases of this
process of interest are the Merton/Ho-Lee, Vasicek, and CIR models.
8.6.1 Merton/Ho-Lee Model
The Merton/Ho-Lee model is a short rate model to predict future interest rates.
It is the simplest model that can be calibrated to market data, by implying the
form of
÷
o from market prices. Ho and Lee does not allow for mean reversion.
) , (
t
r t o =
÷
o and ) , (
t
r t | =
÷
|
8.6.2 Vasicek Model
This is the first detailed no-arbitrage modeling in finance of the term-structure of
interest rates. Here,
) , (
t
r t o = ) (
t
r ÷ u k and ) , (
t
r t | =
÷
|
where, k , u , and
÷
| are all positive constants.
The drift term ) (
t
r ÷ u k in the Vasicek model exhibits mean-reversion. The short
rates have a normal distribution with mean and variance given by,
P a g e | 29 SIBM Pune
) 1 ( ). 2 / 1 ( ) | (
) ( ] | [
2
2
0
0 0
t
t
t
t
e r r Var
r e r r E
k
k
| k
u u
÷
÷
÷
÷ =
÷ + =
8.6.3 The CIR Model
Cox, Ingersoll, and Ross introduced a mean-reverting square-root diffusion
process for the short rate rt :
) , (
t
r t o = ) (
t
r ÷ u k and ) , (
t
r t | =
t
r
÷
|
The form of the diffusion component
t
r
÷
| implies that short rates follow a non-
centric chi-square distribution with
2 2 2 2
0 0
0 0
) 1 ( ) 2 / 1 ( ) ( ). / 1 ( ) | (
) ( ] | [
t t t
t
t
t
e e e r r r Var
r e r r E
k k k
k
| u k | k
u u
÷
÷
÷ ÷
÷
÷
÷ + ÷ =
÷ + =
One-Factor Interest Rate Models
Model Drift ) , (
t
r t o Diffusion ) , (
t
r t |
Merton (1973)
÷
o
÷
|
Vasicek (1977) ) (
t
r ÷ u k
÷
|
Dothan (1978) 0
÷
|
Brennan and Schwartz
(1979)
÷ ÷
+
t
r
1
0 o o
t
r
÷
|
Cox et al. (1985) ) (
t
r ÷ u k
t
r
÷
|
Chan et al. (1992)
÷ ÷
+
t
r
1
0 o o
5 . 1
t
r
÷
|
Geometric Brownian motion
÷
t
r o
t
r
÷
|
Constant elasticity of
variance (CEV)
÷
t
r o
¸
|
t
r
÷
P a g e | 30 SIBM Pune
Solving One Factor Models
- PDE Approach
The “one factor” assumption in a one-factor model is that the evolution of all the
bond prices in the model depends only on the evolution of the model’s single
factor. That is, if P(t,T) denotes the time-t price of a zero-coupon bond maturing
at T, then
P(t,T) = P(t,T,
t
r )
Let the short rate process be given by
t t t t
dW r t dt r t dr ) , ( ) , ( | o + =
By Ito’s lemma
dP(t,T) =
t
t r rr r t
rr r t
PdW Pdt
dW P dt P P P
dt P dr P dt P
o µ
| | o
|
+
+ + +
+ +
]
2
1
[
2
1
2
2
In the absence of arbitrage, bond prices must satisfy the “fundamental partial
differential equation (pde)”,
0
2
1
) (
2
= ÷ + + ÷ rP P P P
rr t r
| |ì o
where, ì is called the “market price of risk” associated with the model’s single
factor.
ì is defined through the drift and volatility of the unknown bond prices.
Introducing a functional form for ì is necessary to be able to solve this pde. Two
of the specific case-solutions are:
- Bond Prices in the Vasicek Model
P a g e | 31 SIBM Pune
Vasicek makes the assumption that ) , (
t
r t ì is a constant ì . The bond prices in
this model is given by
P (t, T,
t
r ) = exp )] ( ) ( [ t t B r A
t
÷
where, t =T-t is the time left to the bond maturity
)] exp( 1 [
1
) ( kt
k
t ÷ ÷ = B
2
3
2
)] exp( 1 [
4
) ( ] ) ( [ ) ( kt
k
o
t t t ÷ ÷ ÷ · ÷ = R B A
2
2
2
) (
k
o
k
ìo
u ÷ + = · R
- Bond Prices in the CIR Model
In this model the market price of risk, ) , (
t
r t ì =
t
r ì . The bond prices in this
model is given by
P (t, T,
t
r ) = exp )] ( ) ( [ t t B r A
t
÷
¦
)
¦
`
¹
¦
¹
¦
´
¦
(
¸
(
¸
+ ÷ + +
=
+ +
2
/ 2
2 / ) (
2 ) 1 )( (
2
ln ) (
o ku
¸t
t ¸ ì k
¸ ì k ¸
¸
t
e
e
A
¸ ì k ¸
t
¸t
¸t
2 ) 1 )( (
) 1 ( 2
) (
+ ÷ + +
÷
=
e
e
B
2 2
2 ) ( o ì k ¸ + + =
8.7 Multi-Factor Models
Any model in which there are two or more uncertain parameters in the option price
(one-factor models incorporate only one cause of uncertainty: the future price).
Multi-factor models are useful for two main reasons. Firstly, they permit more
realistic modeling, particularly of interest rates, although they are very difficult to
compute. Secondly, multi-factor options (for example, spread options) have several
P a g e | 32 SIBM Pune
parameters, each with independent volatilities, and also the correlation between the
underlying must be dealt with separately.
In one-factor models, the single factor was the short rate,
t
r . In multi-factor models,
the factors may be
- The long rate
- The rate of inflation
- The mean of the short rate
- The volatility of the short rate; and many others
In an n-factor model, there are n factors which affect changes in the yield curve. The
regular flow of modeling is:
1. Model each of the factors separately.
2. Solve for the market prices of risk associated with each factor.
3. Price the bonds using this and the n-factor version of the Partial
Differentiation Equation (PDE) approach.
The n-factor pde is
2
2
1
o µ
XX X t
P P P rP + + =
where, X=(X1, …….Xn) are the n factors driving the term structure.
Two-factor version of CIR model
The bond price where the short-rate is the sum of the two independent square-
root diffusions is as follows:
|
|
.
|
\
|
(
¸
(
¸
÷ ×
|
|
.
|
\
|
=
¿ [
= = 2 , 1 2 , 1
2 1
) ( exp ) ( ) , , , (
i
i it
i
i t t
B x A x x T t P t t
where, t =T-t
¦
)
¦
`
¹
¦
¹
¦
´
¦
(
¸
(
¸
+ ÷ + +
=
+ +
2
/ 2
2 / ) (
2 ) 1 )( (
2
ln ) (
i i i
i
i i i
i i i i
i
i
e
e
A
o u k
t ¸
t ¸ ì k
¸ ì k ¸
¸
t
P a g e | 33 SIBM Pune
i i i i
i
i
i
e
e
B
¸ ì k ¸
t
t ¸
t ¸
2 ) 1 )( (
) 1 ( 2
) (
+ ÷ + +
÷
=
2 2
2 ) (
i i i i
o ì k ¸ + + =
} 2 , 1 { e i
8.8 Heath-Jarrow-Morton (HJM) Model
HJM marks a significant advance to the modeling of the term- structure movements and the
pricing of the interest rate derivatives. HJM framework works directly with the entire yield
curve and models simultaneously the changes in the rates of all maturities. It is consistent with
any initial yield curve. Implementation only requires two pieces of information: the initial yield
curve and the volatilities of the forward rates of different maturities.
The difference of HJM with the factors models are as follows:
HJM Factor models
HJM is a general framework where the
choice of the factor is left to the
modeler.
Factor models are specific.
HJM framework works directly with the
entire yield curve.
Factor models works by modeling certain
points on the yield curve (e.g. Short rate)
HJM model the forward rate curve. Factor models model the spot yield curve.
The risk-neutral drifts of the forwards
are functions of the volatilities, not the
market price of the risk.
The risk-neutral drift of the short-rate
depends on the market price of the risk
ì
One-Factor HJM Model:
Let P(t,s) denote the time-t price of a zero-coupon bond maturing at time s and with a face value
of $1.
)
`
¹
¹
´
¦
÷ =
¿
÷
÷
1
). , ( exp ) , (
s
t i
h i t f s t P
where, f(t,i) means $1 invested at time I at the rate f(t,i) will grow by time i+1 to
exp{f(t,s).h}
h=length of each period in years
P a g e | 34 SIBM Pune
The forward curve f (t.T) can move “up“ or move “down” in the forward curve with probability q
and 1-q respectively.
(
(
(
(
¸
(
¸
÷ +
+ +
+ +
= +
) 2 , 1 (
) 2 , 1 (
) 1 , 1 (
) , 1 (
n t f
t t f
t t f
T t f
u
u
u
u
?
(
(
(
(
¸
(
¸
÷ +
+ +
+ +
= +
) 2 , 1 (
) 2 , 1 (
) 1 , 1 (
) , 1 (
n t f
t t f
t t f
T t f
d
d
d
d
?
8.9 LIBOR Market Model
Introduction
LIBOR Market Model, as the name suggests, models the LIBOR rates. It is also
known as the BGM Model (Brace Gatarek Musiela Model). It models the
forward LIBOR rates which is directly observable in the market. One of the
advantages of this model is that it is consistent with the Black’s pricing of
Swaptions and Caplets.
There are many variations of LIBOR Market Models. Here we are going to discuss
following variations:
- LIBOR Market Model with non-stochastic volatility.
- LIBOR Market Model with Zhu stochastic volatility model.
- LIBOR Market Model with Heston stochastic variance model.
Dynamics of LIBOR Market Model
Forward LIBOR rates at any time t can be calculated using corresponding yield
curve as follows
i i
i
i i i
T t P
T t P
T T t f
t
1
1
) , (
) , (
) , , (
1
1
×
|
|
.
|
\
|
÷ =
÷
÷
Here, ) , , (
1 i i i
T T t f
÷
is the forward LIBOR rate at time t corresponding to time
period
1 ÷ i
T to
i
T for i = 1, 2, 3… m. ) , (
i
T t P is the discount factor (bond price) of
maturity
i
T calculated at time t.
i
t is the year fraction between
1 ÷ i
T and
i
T .
LIBOR Market Model with non-stochastic volatility
P a g e | 35 SIBM Pune
The dynamics of LIBOR Market Model is given by following equation:
¿
=
+ =
m
k
k k i i i
i
i
dZ b dt
f
df
1
,
o µ
Here
i
µ is the drift part which depends upon the forward measure in which this
dynamic equation is written. Drift term can be written in self, spot and terminal
measures as follows
? Self Measure – 0 =
i
µ
? Spot Measure –
¿
=
+
÷ =
i
j j j
j j j i j
i i
f
f
1
,
1 t
t µ o
o µ
? Terminal Measure –
¿
+ =
+
÷ =
m
i j j j
j j j i j
i i
f
f
1
,
1 t
t µ o
o µ
Here
i
o is the volatility of forward rate ) , , (
1 i i i
T T t f
÷
.
k i
b
,
is the loading factor
which depends upon the correlation between forward LIBOR rates. The
relationship between loading factor
k i
b
,
and correlation is given by following
matrix equation.
] [ ] ][ [ µ =
T
b b
Here is m×n matrix. m is the total number of LIBOR rates while n is the total
number of independent wiener process dZk required for calculating the dynamics
of the all these LIBOR rates.
LIBOR Market Model with Zhu stochastic volatility model
In this model, unlike previous one, volatility of the LIBOR rate is stochastic in
nature. Therefore stochastic equation of LIBOR rate remains the same but the
parametric equation for volatility changes to mean reverting stochastic equation.
Dynamics of this model is written as
¿
=
+ =
n
k
k k i i i
i
i
dZ b v dt
f
df
1
,
µ
Here
k i
b
,
is loading factor which depends upon the correlation between forward
LIBOR rates.
i
v is the stochastic volatility term which under forward measure
Q
i+1
is written as
dW dt v v
i i i i i
o u k + ÷ = ) (
P a g e | 36 SIBM Pune
Here,
i
k is mean reverting rate,
i
u is long term volatility and
i
o is volatility of
volatility of respective forward LIBOR rate. dW is the stochastic part of the
volatility equation. Each of these
i
k ,
i
u ,
i
o and
i
v (0) is written as following
parametric equations
) exp(
i i
aT k k =
) exp( ) (
3 2 1 i i i
T b b T b + + =u u
) exp( ) ( ) 0 (
3 2 1 i i i
T c c T c v v + + = , initial volatility.
) exp(
i i
dT o o =
i
µ is the drift part which depends upon the forward measure in which this
dynamic equation is written. Drift term in self, spot and terminal measures can
be written as follows
? Self Measure – 0 =
i
µ
? Spot Measure –
¿
=
+
÷ =
i
j j j
j j j i j
i i
f
f Corr
1
,
1 t
t o
o µ
? Terminal Measure –
¿
+ =
+
÷ =
m
i j j j
j j j i j
i i
f
f Corr
1
,
1 t
t o
o µ
Followings are some of its form that is frequently used in LMM.
? Rebonatho one parameter correlation equation
( ) | | exp
0 , j i j i
T T Corr ÷ ÷ = |
? Rebonatho two parameters correlation equation
( ) ( ) | | ) , min( exp exp
0 , j i j i j i
T T T T gamma Corr ÷ × × ÷ ÷ = |
LIBOR Market Model with Heston stochastic variance model
? In this model forward LIBOR rate dynamics in risk neutral world (spot
measure) is written as follows
(
¸
(
¸
÷ =
¿ ¿
= =
+
n
k
n
k
k i k i k k i i i
dt t t t V dZ t t V t f t df
1 1
, , 1 ,
) ( ) ( ) ( ) ( ) ( ) ( ) ( ¸ o ¸
? Here
k i , 1 +
o is the volatility of bond price of maturity Ti+1 at time t, while
V(t) is the stochastic variance term and in risk neutral world it is written
as follows
P a g e | 37 SIBM Pune
) ( ) ( )) ( ( ) ( t dW t V dt t V t dV c u k + ÷ =
Here k is the mean reverting rate, u is long term variance value and c is
volatility of stochastic part of the stochastic variance equation. Also here
‘n’ is the total number of independent wiener process required to write
dynamics of all the ‘m’ LIBOR rates. Correlations between different
forward LIBOR rates are associated with vectors
i
¸ for i = 1, 2, 3…… m.
Followings are some of its form that is frequently used in LMM.
? Rebonatho one parameter correlation equation
( ) | | exp
0 , j i j i
T T Corr ÷ ÷ = |
? Rebonatho two parameters correlation equation
( ) ( ) | | ) , min( exp exp
0 , j i j i j i
T T T T gamma Corr ÷ × × ÷ ÷ = |
P a g e | 38 SIBM Pune
9. PRICING USING MATLAB AND OCTAVE
9.1 Black-Scholes
blsprice -Black-Scholes put and call option pricing
Syntax
[Call, Put] = blsprice(Price, Strike, Rate, Time, Volatility, Yield)
blsprice can handle other types of underlies like Futures and Currencies. When
pricing Futures (Black model), enter the input argument Yield as:
Yield = Rate
When pricing currencies (Garman-Kohlhagen model), enter the input
argument Yield as:
Yield = ForeignRate
where ForeignRate is the continuously compounded, annualized risk free interest
rate in the foreign country.
Example:
Consider European stock options that expire in three months with an exercise price
of $95. Assume that the underlying stock pays no dividend, trades at $100, and has a
volatility of 50% per annum. The risk-free rate is 10% per annum. Using this data
[Call, Put] = blsprice(100, 95, 0.1, 0.25, 0.5)
returns call and put prices of $13.70 and $6.35, respectively.
9.2 Black-76
blkprice -Black's model for pricing futures options
Syntax
[Call, Put] = blkprice(Price, Strike, Rate, Time, Volatility)
Examples
P a g e | 39 SIBM Pune
Consider European futures options with exercise prices of $20 that expire in four
months. Assume that the current underlying futures price is also $20 with a
volatility of 25% per annum. The risk-free rate is 9% per annum. Using this data
[Call, Put] = blkprice(20, 20, 0.09, 4/12, 0.25)
returns equal call and put prices of $1.1166.
9.3 HJM
Following is a program implemented in OCTAVE for creating HJM price tree.
function u = hjm(f0,sig0,h);
n = length(f0);
m = n-1;
fu = f0(2:n);
fd = f0(2:n);
sigma = sig0(2:n);
alpha = zeros(m:1);
for j=[1:m];
if (j==1);
alpha(j) = log(0.5*(exp(-sigma(j)*h*sqrt(h)) + ...
exp(sigma(j)*h*sqrt(h))))/h^2;
end;
if (j>1);
alpha(j) = log(0.5*(exp(-sigma(1:j)*h*sqrt(h)) + ...
exp(sum(sigma(1:j))*h*sqrt(h))))/h^2 - sum(alpha(1:j-1));
end;
end;
fu = fu + alpha*h + sigma*sqrt(h);
fd = fd + alpha*h - sigma*sqrt(h);
u = [fu fd];
9.4 LIBOR Market Model
Following is a program implemented in OCTAVE for one-factor LMM
%Initial LIBOR curve
libor0 = [0.06 0.07 0.08 0.09 0.10 0.11]';
%Initial volatility curve (could be matrix)
sig = [0.15 0.14 0.13 0.12 0.11 0.10]';
P a g e | 40 SIBM Pune
n = length(libor0)-1; %Denotes the final LIBOR rate time
delta = 0.25; %Time interval in years
h = delta;
libor = zeros(n+1,n+1);
libor
,1) = libor0; nsims = 10000;
%START SIMULATION
for i = 1:nsims;
for t = 1:n;
e = randn*sqrt(h);
drift = 0;
for k = [n:-1:t];
if k==n; %Do for numeraire
libor(k+1,t+1) = libor(k+1,t)*exp(-
0.5*sig(k+1)^2*h...
+ sig(k+1)*e);
else
drift = drift + delta*sig(k+2)*libor(k+2,t)/...
(1+delta*libor(k+2,t));
libor(k+1,t+1) = libor(k+1,t) * exp(-
0.5*sig(k+1)^2*h...
- drift*sig(k+1)*h + sig(k+1)*e);
end
end
end
price(i) = 100/prod(1+delta*diag(libor));
end;
%CROSS CHECK RESULTS
cprice = 100/prod(1+delta*libor0); %Price off initial curve
sprice = mean(price); %Simulated price
fprintf('Price off curve = %10.6f, Price off simulation = %10.6f \n',...
cprice,sprice);
P a g e | 41 SIBM Pune
10. VALUE ADDITION
10.1 To the Organization
The Analysis of the available interest rate derivative models and building upon them
based on additional parameters which would help in better analysis and
approximation of the LIBOR rates.
This project involves a lot of quantitative techniques and C++ coding which is out of
the scope due to limited time frame of 2 months.
- Running the pricing models in MATLAB and OCTAVE.
- Summarizing research reports
- Proof reading for a book on Derivative under publication.
- Fixing a process for a continuous refresh of data from Bloomberg.
- Evaluate the various ETL tools like Pentaho, Talend etc. to manage and
analyze Hedge Fund data.
10.2 To “me” as a student of management
The undertaking of this project has been very useful in multiple ways. It has helped
in the following ways
- To develop a broad understanding of the derivative market.
- To get a detailed understanding of the various derivative instruments, their
working.
- To price the interest rate derivatives based on different models using
MATLAB and OCTAVE.
- To get a insights on the hedge funds related data and it’s analysis using data
warehousing tools like Pentaho and Talend
- To work on Bloomberg terminals
P a g e | 42 SIBM Pune
11. TEAM PLAY IN THE ORGANIZATION
The quantitative analytics team at Capital Metrics and Risk Solutions is headed by Mr.
Rajesh Shah (PhD, Cornell), and operates under the overall guidance of Mr. Raghu
Sundaram. The team provides customized high-end derivatives valuation and risk-
management solutions for the clients.
As a part of this team, my role was to run the various derivative models in MATLAB and
price the interest rate derivatives.
Also, I implemented the HJM and LMM models in OCTAVE by writing the algorithm and
then running them to get the HJM tree and one-factor LMM.
Finally, I was asked to study a few research papers on LMM and summarize the findings
so that the team members can build upon them in calibrating the LMM.
12. LIMITATIONS OF THE STUDY
- This Project Involves A Lot Of Quantitative Techniques And C++ Coding Which Is Out
Of The Scope Due To Limited Time Frame Of 2 Months.
- The Libor Market Model (LMM) is very exhaustive and complex. Research is still in progress
to optimize the calibration process. Hence in this project only the basic of this model is
touched and further implementation is beyond the scope of the project.
P a g e | 43 SIBM Pune
13. IMPLICATION AND CONCLUSION
The derivative markets have been growing rapidly over the past few decades and
have been accused lately for their alleged role in the financial crisis. The leveraged
operations are said to have generate an “irrational appeal” for risk taking, and the
lack of clearing obligations also appeared as very damaging for the balance of the
market.
The countries all over the world are trying to extend its oversight of the financial
system to include the shadowy market of derivatives, the kind of complex financial
instruments that helped catapult the world into an economic crisis.
Countries all over the world, want to create a central electronic-based system that
would track the buying and selling of derivatives. They want to ensure that financial
firms selling the instruments have enough capital on hand in case they default and
subject them to stringent standards of conduct and new reporting requirements.
All (over-the-counter) derivatives dealers and all other firms whose activities in the
markets create large exposures to counterparties should be subject to a robust
regime of prudential supervision and regulation. Key elements of that robust
regulatory regime must include conservative capital requirements, business conduct
standards, reporting requirements and conservative requirements relating to initial
margins on counterparty credit exposures. New rules should be in place to deter
financial firms from taking undue risk, prevent fraud and ensure they are marketed
appropriately.
Current law largely excludes regulation of the instruments, which are referred to as
"over-the-counter" derivatives because they are traded privately rather than
through commodity exchanges. It was unclear how the rules would affect hedge
funds, which are large, mostly unregulated entities that use complex trading tactics
to earn big returns for high-dollar investors. Many hedge funds use derivatives
contracts to offset risk on other transactions.
New laws should be formed to check such activities and regulators should be always
on a vigil. There should a plan which talks to establish an "audit trail" for the
derivatives and have "clear unimpeded authority to police fraud, market
manipulation and other market abuse
Even after the above mentioned regulations and their implication, the derivatives
volumes are soaring, and at the same time instruments are becoming more and
more complex. This has created a great demand for pricing models to handle these
sophisticated structures, with a short supply of people capable of creating these
models.
P a g e | 44 SIBM Pune
And hence implementing Derivatives Models, particularly the over-the-counter
market in complex or exotic options, are continuing to expand rapidly on a global
scale. However, the availability of information regarding the theory and applications
of the numerical techniques required to succeed in these markets is limited. This
lack of information is extremely damaging to all kinds of financial institutions and
consequently there is enormous demand for a source of sound numerical methods
for pricing and hedging.
P a g e | 45 SIBM Pune
BIBLIOGRAPHY
1. Financial Derivatives Toolbox-User’s Guide, Version 2, The Math Works, Inc.
2. Lixin Wu and Fan Zhang(May 2006), Libor market model with stochastic volatility,
Journal Of Industrial And Management Optimization, Volume 2, Number 2, May
2006
3. Black, Fischer; Derman, Emanuel; Toy, William(1990), A One-Factor Model Of
Interest Rates And Its Application, Financial Analysts Journal; Jan/Feb 1990; 46, 1;
ABI/INFORM Global, pg. 33
4. Paul Wilmott (1995), The Mathematics of Financial Derivatives
5. Hull, John C. (2005), Options, Futures and Other Derivatives, Sixth Edition. Prentice
Hall
6. http://www.nseindia.com/ , National Stock Exchange of India
7. http://www.bseindia.com/ Bombay Stock Exchange of India
doc_871367592.docx