Derivatives and Risk Management MBAB 5P44...

Hatem Ben Ameur Derivatives and Risk Management Brock University �MBA

Derivatives and Risk

Management

MBAB 5P44 �MBA

Hatem Ben Ameur

Brock University �Faculty of Business

Winter 2010

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Contents

1. Introduction

1.1 Derivatives and Hedging

1.2 Options

1.3 Forward and Futures Contracts

1.4 Swaps and Other Derivatives

1.5 Arbitrage

1.6 The Role of Derivatives Markets

1.7 Assignment

2. Structure of Options Markets

2.1 The Risk of an Option Position

2.2 Organized versus Over-the-Counter Options Mar-kets: Pros and Cons

2.3 Standardized Options

2.4 The Trading Process

2.5 The Clearing House2


2.6 Transaction Costs

2.7 Types of Options

2.8 Assignment

3. Principles of Options Pricing

3.1 Basic Notation and Concepts

3.2 Bounds for Call Options

3.3 Optimal Exercise of American Call Options

3.4 Bounds for Put Options

3.5 Call-Put Parity

3.6 Assignment

4. Biomial Model

4.1 The One-Period Binomial Tree

4.2 Pricing Formula for a One-Period Binomial Tree

4.3 No-Arbitrage Pricing

4.4 Dividends and Early Exercise

4.5 Assignment

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5. The Black and Scholes Model

5.1 Introduction

5.2 The Black and Scholes Formula for EuropeanOptions

5.3 Sensitivity Analysis

5.4 Dividends

5.5 The Volatility of Stock Returns

5.6 The Black and Scholes Formula for European PutOptions

5.7 Relationship to the Binomial Model

5.8 Assignment

6. Selected Option Strategies

6.1 Introduction

6.2 Basic Strategies

6.3 More Complex Strategies

6.4 Advanced Strategies

6.5 Assignment

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7. The Structure of Forward and Futures Markets

7.1 The Development of Forward and Futures Mar-kets


7.3 The Daily Settlement System

7.4 Assignment

8. Pricing Forwards, Futures, and Options on Futures

8.1 Principles of Carry-Arbitrage Pricing

8.2 Pricing Forward Contracts on Stocks

8.3 Pricing Forward Contracts on Stock Indices

8.4 Pricing Forward Contracts on Foreign Currencies

8.5 Pricing Forward Contracts on Commodities

8.6 Futures Prices and Expected Future Spot Prices

8.7 Call-Put-Futures Parity

8.8 Pricing Options on Futures

8.9 Assignment

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9. Forward and Futures Hedging

9.1 Is Hedging Relevant?

9.2 Hedging Design

9.3 Hedging T-bonds and Stock Portfolios

9.4 Assignment

10. Vanilla Interest-Rate Swaps

10.1 Design

10.2 Pricing

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A derivative is a contract whose return depends on theprice movements of some underlying assets. There arethree main families of derivative contracts: options, fu-tures, and swaps. They all have the ability to reduce risk;thus, are widely used for hedging purposes.

This course covers basic topics on options, futures, andswaps. We focus on their 1) market organization, 2)hedging properties, and 3) evaluation principles. Foreach product, we report on its market rules and speci-�cities, and study some of its associated hedging strate-gies. Models for derivatives pricing are based on the no-arbitrage assumption. We discuss the most popular: theBlack-Scholes-Merton model for options pricing and thecost-of-carry model for forward/futures pricing.

The following textbooks are required. The �rst one willbe used frequently during the term; the others o¤er usefulalternate explanations, and are listed by increasing orderof di¢ culty:

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1. Don M. Chance and Robert Brooks, 2007, An Intro-duction to Derivatives and Risk Management, 7thEdition, Thomson.

2. John C. Hull, 2005, Fundamentals of Futures andOptions, 6th Edition, Prentice Hall.

3. Robert W. Kolb and James A. Overdahl, 2007, Fu-tures, Options, and Swaps, 5th Edition, Blackwell.

4. John C. Hull, 2008, Options, Futures, and OtherDerivative Securities, 7th Edition, Prentice Hall.

All Figures and Tables referred to in the text are takenfrom Chance and Brooks (2007). Monitoring the newson �nancial derivatives is recommended as an importantcomplement to classroom work. Selected articles frombusiness newspapers such us The Financial Times, TheGlobe and Mail, The National Post, and The Wall StreetJournal will be discussed in class.

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The grading policy is based on: four assignments eachworth 5%; two midterm exams each worth 30%; a quizworth 10%; and one presentation worth 10%. The midtermexams and the quiz are cumulative.

The assignments will improve your skills and prepare youfor the exams. For each assignment, you will be given se-lected problems to solve, articles to comment on, and(possibly) empirical and numerical experiments to im-plement. The �nal requirement, due at the end of theterm, consists of a short presentation (20 minutes foreach group) on a complex �nancial derivative of yourchoice.

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1 Introduction

1.1 Derivatives and Hedging

A derivative is a contract whose performance dependson the price movement of some underlying assets. Anunderlying may be either a �nancial asset such as a stockor a real asset such as a commodity. The underlyingassets we consider here are usually traded on cash orspot markets, which, in turn, strongly impact derivativesmarkets.

Exercise 1: Give a de�nition of an asset. Provide exam-ples of �nancial assets and real assets.

Options trading on organized markets goes back to the18th century in the Netherlands. Dutsh horticulturistsused to hold put options on Tulip. They would lock in theTulip sell price before the harvest period, thereby hedgingagainst a potential decrease on the Tulip spot price.

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Derivatives have the ability to reduce risk; thus, are widelyused for hedging purposes.

Example 2: The spot foreign exchange rate for the USdollar is 0.69056 Euros. Your company agrees to pay abank 63,694 Euros in 3 months in exchange for 100,000US dollars. This is a foreign currency forward contract.No cash is exchanged up front. Give the underlying as-set, the maturity date, and the forward rate (for the USdollar). Compare to the spot forward rate. Conclude.

Example 3: A contract promises its holder $1000 if noearthquake hits Tokyo during the next year. This con-tract is known as a digital option, and it�s used here asinsurance against earthquake damage. How much wouldyou pay for it?

The return (in %) is a numerical measure of investmentperformance, and the risk is the uncertainty of futurereturns. There are various types of risk, including mar-ket risk and credit risk. Market risk is associated to the

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movement of asset prices, and credit risk to the failure ofa counterparty to ful�ll his obligations.

Among investment opportunities that have the same ex-pected return, a risk-averse investor would prefer the onethat has the lowest risk, while a risk-neutral investorwould be indi¤erent, as long as the expected return re-mains constant. A risk-averse investor wouldn�t take onadditional risk unless he expects a large enough risk pre-mium.

Hedging is relevant because investors are usually risk-averse, pessimistic, and they support market imperfec-tions, including transaction costs, information asymme-tries, and taxes.

There are three main subfamilies of derivative contracts,namely, options, futures, and swaps. All can be designedto hedge risk. If one party is hedging, then the other isspeculating!

Exercise 4: Why is hedging relevant for a company?(stability, convexity...)

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1.2 Options

An option is a right (privilege), but not an obligation. Acall option is a contract that gives its holder the right,but not the obligation, to buy an underlying asset at aspeci�ed future maturity date for a known strike price. Aput option is a contract that gives its holder the right,but not the obligation, to sell an underlying asset at aspeci�ed future maturity date for a known strike price.

Options are traded both on exchanges and over the counter.

The option holder is also called the buyer or the longparty. The option buyer bene�ts from a privilege; there-fore, pays for it usually up front. The option signer isalso called the writer, the seller, or the short party.

The up-front payment made by the buyer to the seller isthe option price, also known as the option premium.

Example 5: The ABC October 30 call option is quotedat $0.5, the ABC October 30 put option is quoted at

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$1.5, while the ABC spot price is quoted at $28. The calloption is said to be out of the money since its immediateexercise (if possible) has no value. Conversely, the putoption is in the money. Suppose the ABC spot price risesto $31 on October 16, which is the option maturity date.Then, the call holder is better of exercising his right andmaking a pro�t of $1. The call expires in the money,while the put expires out of the money.

1.3 Forward and Futures Contracts

Under a forward contract, one party agrees to buy andthe other to sell an underlying asset at a speci�ed fu-ture maturity date for an agreed-upon forward price. Aforward contract sounds like an option, but the two arefundamentally di¤erent. While a forward contract is anobligation for both parties, an option is a right for itsholder. In addition, an option assumes an up-front pay-ment by the buyer to the seller, while a forward contractdoes not.

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Forward contracts are traded over the counter, and arewritten on foreign currencies, bonds, stocks, stock in-dexes, and commodities.

A futures contract is otherwise similar to a forward con-tract except that it is traded on an exchange and is sub-ject to a daily settlement. At the end of each tradingday, the futures contract is closed and rewritten at thenew closing settlement price. If the di¤erence to the pre-vious settlement price is positive, it is subtracted fromthe short-trader account and added to the long-traderaccount. Else, the reverse is done. The daily-settlementsystem protects both parties against counterparty risk.

Few traders hold their positions on a futures contractuntil the maturity date; thus, deliver or take delivery ofthe underlying asset. They usually reverse their positionsand exit the market before the futures contract maturity.

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1.4 Swaps and Other Derivatives

A swap commits two parties to exchange cash �ows atspeci�ed settlement dates, up to a given maturity date.

Example 6: A company agrees to pay its bank semi-annual interest over 20 years, based on a �xed nominalinterest rate of 5.5% (per year), and to receive interestpayments based on a �oating interest rate, say, the 6-month LIBOR rate. Both interest payments assume aprincipal amount of $1 million. No payment is exchangedup front.

Several swaps are traded over the counter, namely, in-terest rate swaps, currency swaps, equity swaps, and themore recently introduced and controversial credit defaultswaps.

There are several combinations of derivative contracts,such as options on swaps or swaptions, options on fu-tures contracts, and options embedded in bonds, amongothers.

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1.5 Arbitrage

Suppose an investor enters the market without initialwealth, initiates a riskless investment strategy, and leavesthe market at a certain future date with a strictly pos-itive riskless pro�t. This is a free lunch or an arbitrageopportunity.

A short selling of a �nancial asset consists of borrow-ing the asset from a broker for an immediate sell, withthe promise to return it later with its generated revenues.Short selling is usually required to initiate arbitrage strate-gies.

Models for pricing derivatives are built to preclude againstarbitrage opportunities. They are said to be arbitragefree. On the other hand, �nancial markets may allowfor sporadic arbitrage opportunities; however, in e¢ cientmarkets, they will quickly disappear since arbitrageurs willpro�t immediately and keep prices in line.

Example 7: Two assets A1 and A2 are traded in a one-period market model as follows.

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[Please discuss Figure 1.2 on page 10.]

Which asset is over- and which is under-priced? Give anarbitrage opportunity.

Researchers have worked hard to characterize arbitrage-free market models, and have established a strong rela-tionship between the no-arbitrage property and the so-called martingale property of the underlying-asset price,discounted at the risk-free interest rate. The law ofone price, which characterizes no-arbitrage market mod-els, stipulates that two portfolios with identical cash-�owstreams necessarily have the same present value.

Market participants �gure out their investment strate-gies in part by comparing theoretical values to marketprices. One should buy undervalued assets, and sell over-valued assets. Theoretical values of �nancial assets arecomputed using the present value principle, or more elab-orated methods such as the risk-neutral evaluation prin-ciple, which plays a central role for pricing derivatives.

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In all cases, discount and compound factors are used tomove cash �ows backward and forward in time.

Exercise 8: The nominal interest rate is �xed at rnom =4% (per year), and interest is compounded semi-annually.Give the semi-annual interest rate r:5 (in % per six months),which is relevant to compound interest, the compoundfactor c0;1 over one year. Compute the equivalent an-nual interest rate r1 (in % per year), which would applyif interest were compounded annually. This is the e¤ec-tive interest rate. Compute again the compound factorc0;1. Compute the equivalent monthly interest rate r1=12(in % per month) and quarterly interest rate r0:25 (in %per quarter). Compute again the compound factor c0;1.Recognize that r:5, r1, r1=12 and r:25 are all equivalentrates. The continuously compounded interest rate rc (in% per year) is a nominal rate, which would apply if inter-est were compounded at each second (or even a fractionof a second). Compute again the compound factor c0;1.Recall that:�

1 +a

n

�n! ea, when n!1,

where a 2 R and n 2 N.19


1.6 The Role of Derivatives Markets

Hedgers use derivatives to reduce risk; however, theircounterparts, called speculators, use them to increaserisk. This is the dark side of derivatives, as they aresometimes considered legalized gambling tools.

Next, derivatives provide investors with signi�cant andvaluable information about spot markets. For example,the convergence of the forward price to the spot priceat maturity provides useful information to market partic-ipants. In addition, derivative markets o¤er some opera-tional advantages. Transaction costs are often lower andshort-selling positions are usually easier than in cash mar-kets. Economic and �nancial crises have had great im-pact on derivatives markets. Examples include the majorstock-market crash in 1987, and the recent credit-risk cri-sis. Consequently, derivatives markets su¤ered; however,investors�con�dence and growth were usually back.

In sum, derivative markets contribute to making �nancialmarkets more complete and e¢ cient.

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1.7 Assignment

Read the chapter, and give precise and concise de�nitionsfor the keywords. This part is to be prepared for themidterm exam, but should not be handed in as part ofthe assignment.

Answer questions: 7, 11, 13 and 15 on page 19.

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2 Structure of Options Markets

2.1 The Risk of an Option Position

Consider a call option on a stock that pays no dividendover the option�s life.

The stock price at maturuty is indicated by ST and theoption strike price by X.

If ST > X, then the call-option payo¤ is YT = ST �K.Else if ST � K, then the option payo¤ is null. In sum,the payo¤ for the call-option holder at maturity resumesto:

YT = max (0, ST �K) .

The payo¤ for the call-option holder, as a function of theterminal stock price, is as follows.

[Please plot the graph of a call payo¤ function.]

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The call-option seller, who is in charge to pay the promisedcash �ow, may fail to ful�ll his obligation. In addition,the payo¤ for the call-option buyer is unbounded, whichincreases further the seller�s credit risk.

This is also the case for a put option, even though itspayo¤ function is bounded.

[Please plot the graph of a put payo¤ function.]

Both parties support market risk in both options exchangesand over-the-counter options markets. While long partiessupport some credit risk in over-the-counter options mar-kets, they are protected in options exchanges by marginsystems.

2.2 Organized versus Over-the-Counter Op-

tions Markets: Pros and Cons

An exchange is a legal corporate entity that is organizedfor trading securities. The Chicago Board of Trade, the

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�rst large derivatives exchange known worldwide, was cre-ated in 1973. Futures contracts have been traded �rst,followed by standardized options. Since then, derivativesexchanges have been created and promoted all over theworld.

[Please comment on Table 2.1 on page 30.]

Meanwhile, over-the-counter markets had been moder-ately growing up till the early 1980s, when corporationsstarted hedging against interest- and exchange-rate risk.Over-the-counter markets are now larger and deeper.

In the following table, an advantage for an options marketis a disadvantage for the other.

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Over-the-Counter Markets Organized Markets

1�Great design �exibility

2�Easy market access

3�Total con�dentiality

4�Very low counterparty

risk5�High liquidity

6�Low transaction costs

7�Price e¤ectiveness

Dealing over the counter o¤ers great �exibility to �x theoption underlying asset, strike price, and maturity, amongother parameters. In addition, more �exibility was re-cently added for revising minimal trading-volume levels.

The �exibility o¤ered in over-the-counter options mar-kets is e¤ective for designing speci�c hedging strategies.These are usually conducted by corporations and institu-tional investors in total privacy as they are highly strategicand valuable.

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Options buyers support some credit risk in over-the-countermarkets since options sellers may fail to ful�ll their oblig-ations. In an organized market, however, long partiesbene�t from a protective margin system, mastered by aclearing house. Conversely, both parties support marketrisk whether they trade on exchange-listed options or not.

Standardized options, traded in organized markets, areliquid. A market intervenant may invert his position atany time before the maturity date, o¤set his risk, andleave the market. Unfortunately, leaving the market priormaturity is not allowed in over-the-counter markets.

Liquid markets provide investors with e¤ective and valu-able signals, including prices and volumes."

Please comment on the statistics of theBank for International Settlements

#.

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2.3 Standardized Options

Exchanges have made options as marketable as stocks bymeans of standards. Firstly, the exchange speci�es theunderlying assets to be used. Options tend to be writtenon large �rms; however, options on small �rms may bemore useful for hedging purposes.

To initiate an exchange-listed option, the decision whetherto consider an underlying asset or not belongs to the ex-change but not to the underlying �rm.

An option class refers to a given underlying asset, andan option serie to the same class and same strike price.Next, the exchange speci�es the size of any traded optioncontract. For example, on the Chicago Board of OptionsExchange (CBOE), an option on a stock gives exposureto 100 stocks. The option size is adjusted upward for astock split or a stock dividend. Meanwhile, the optionstrike price is revised downward consistently.

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For example, in the case of a two-for-one stock split,any old option contract is exchanged for two new optioncontracts, and, accordingly, the strike price is divided bytwo. In sum, the payo¤ to the option holder must remainthe same before and after the split.

Exercise 9: Consider the case of a 15% stock dividend.The option size is revised upward, and the strike price isrevised downward. Explain.

In addition, exercise prices are also standardized by theexchange to maintain the most attractive options, whichare usually signed at-the-money and nearly at-the-moneyoptions:

X = S0 and X � S0,

where S0 is the current stock price, and X is the optionexercise price.

Exercise prices are not adjusted when a cash-dividendis paid. Traders must accordingly revise options prices,

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downward for call options and upward for put options.Along the same lines, options maturities are standard-ized. For example, in the CBOE, each stock is assignedto one of the following three expiration cycles:

1. January, April, July, and October;

2. February, May, August, and November;

3. March, June, September, and December.

For a given stock, traded maturities are: the currentmonth that de�nes the current cycle, the next monthin the next cycle, and the next two months in the stockcycle.

Example 10: IBM is assigned to the January cycle. Thecurrent date is January 20, 2010. Traded maturities ofoptions on IBM are thus: January, February, April, andJuly.

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Finally, exchanges put constraints on options positionsand exercising to prevent agaist market manipulations.A position on an option class on one side of the market islimited to a certain maximum, say, 50,000 stocks. Thisis a position limit.

Unlike a European option, which can be exercised onlyat maturity, an American option gives its holder the ad-ditional right of early exercise. Most exchange-listed op-tions in the United States allow for early exercise.

The number of option contracts one can exercise on agiven trading day may also be limited. This is an exerciselimit.

Options standards can be seen as limitations and con-straints; nevertheless, they add liquidity. Many large in-stitutional investors, however, left options exchanges totrade over the counter, looking for more �exibility. Mean-while, exchanges have introduced �exibility via some op-tions contracts. Examples include FLEX and LEAPS,both traded on the CBOE. FLEX options allow for var-ious exercise prices, while LEAPS options allow for longmaturities.

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Trading within an exchange is mainly maintained by mar-ket makers. Electronic trading systems, however, havebeen increasing in popularity.

A market maker is responsible for matching options buy-ers with options sellers. If there is a mismatch, the marketmaker may complete the trade. Depending on how thetrading information is disclosed and shared, the marketmaker may be called a specialist. To survive, the mar-ket maker pro�ts by buying at one price and selling at ahigher price. The bid price is the highest price the marketmaker is willing to pay for an option, and the ask priceis the lowest price he is willing to accept for an option.The bid-ask spread, which is the di¤erence between thetwo prices quoted by a market maker, can be seen as atransaction cost for market participants.

Market makers act in various ways:

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1. Scalpers pro�t from bid-ask spreads and tend toclose their positions in the very short term;

2. Position traders may hold positions to keep the trad-ing process going;

3. Spreaders look for quasi-arbitrage opportunities inan attempt to earn small pro�ts at a very low risk.

A �oor broker is a member of the exchange who exe-cutes trading orders on behalf of non-member brokerage�rms. He earns a �at salary or a commission on eachorder. Market makers and �oor brokers are members ofthe exchange. Each membership is referred to as a seat.

To trade on exchange-listed options, one must establishan account with a brokerage �rm, which employs or is inbusiness with a �oor broker.

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A board broker is an employee of the exchange who isin charge to introduce trading orders into the computersystem; thereby keeping the market maker informed.

An electronic-trading system allows for trading from elec-tronic terminals located anywhere. The pertinent infor-mation, including bid and ask prices, is vehiculed andprocessed in the system to keep the trading process going.For example, EUREX, located in Frankfurt, Germany, isa fully automated exchange.

There are several o¤-�oor options traders, including largeinstitutional as well as individual investors.

While options on individual stocks may lead to deliver theunderlying asset at the exercise date, options on stockindexes obey a cash settlement procedure.

Example 11: Consider an American index call option,which has a multiple of 100. The current level of the

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index is It = 1500, and the exercise price is X = 1450.If the call-option is exercised at t, the holder get paid:

100� (It �X) = 500 dollars in cash.

Quotations are reported in business newspapers and onexchanges Web sites.

[Please read the Microsoft price quotation on page 38] .

An investor can place several types of orders on an ex-change. While a market order instructs to obtain the bestavailable market price, a limit order speci�es a maximumprice for buying and a minimum price for selling. Theycan be either good-till-canceled or day orders. Limit or-ders are executed as soon as possible in order of priority.

Example 12: Suppose you issue a limit order, good-till-cancelled, to buy a call option at a maximum price of $3,while the bid-ask spread is quoted by the market maker2.75�3.25. Obviously, your order cannot be completed.

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The board broker will input it into the computer systemtill the ask price, now at $3.25, is revised downward.

A stop order speci�es a price, which is lower than thecurrent price, at which the broker has to sell for the bestavailable price.

Some trading orders are so large that they cannot beachieved in total for the same price. An all-or-none orderinforms to execute the order in total, even at di¤erentprices. On the other hand, an all-or-none, same-priceorder instructs to execute the order in total at a uniqueprice, should it be cancelled.

2.5 The Clearing House

A clearing house, formally known as an Options Clear-ing Corporation, is an organization that guarantees shortparties�performance. Their members are called clearing�rms.

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The protecting system works as follows. The option selleris constrained to deposit a margin amount in a marginaccount held by his broker. The margin amount per op-tion is typically a fraction of the underlying asset price,which depends on whether the option is covered or notand out or in the money. Similarly, the seller�s broker ishimself constrained to deposit a margin amount in a mar-gin account held by his clearing �rm within the clearinghouse.

Now, if the shares are not delivered by an option sellerwhen the option is exercised by an option buyer, the clear-ing �rm of the seller�s broker will appeal to the seller�sbroker, and the seller�s broker will appeal to the seller.

Liquidity is the main concern of exchanges. An optiontrader may o¤set his position at any time he wishes byissuing an o¤setting order, which simply consists on in-verting his position at the best available market price.

[Please discuss Figure 2.2 on page 35.]

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2.6 Transaction Costs

Trading options entails various transaction costs. Thebid-ask spread is a signi�cant transaction cost for optionstraders. Floor trading and clearing fees are included inthe broker�s commission, and are generally expressed asa �xed amount plus a per-contract charge. Internet ratesare about $20 plus $1.25 per contract from major dis-count brokers. They charge less than full-service brokers,but usually provide fewer service (valuable informationand advice). Margins induce an opportunity cost.

2.7 Types of Options

Options are written on various underlying assets, includ-ing stocks, indexes, currencies, bonds, and commodities;however, options on major stocks and indexes are themost widely traded.

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Standard options are called vanilla options. Several oth-ers, with more complex payo¤ functions, are called exoticoptions. For example, the payo¤ of an Asian option isbased on the average price of the underlying asset pricealong its path, and the one of a lookback option is basedon extreme prices. On the other hand, options on swapsand futures are actively traded. Options may also beembedded in securities. For example, government bondsoften contain embedded call and put options, and corpo-rate bonds often contain embedded call and conversionoptions.

An executive stock option is a call option written by acorporation and held by its executives. These optionsare used as incentives for managers. They have to out-perform in an attempt to push upward the corporation�smarket price at higher levels, and in turn to push theircall options deeply in the money.

A real option is an option that is embedded in an invest-ment project. Examples are options to expand, temporaryshut down, abandon, or sell an already started investmentproject. 38


2.8 Assignment

Read the chapter, and give precise and concise de�nitionsfor the keywords listed on page 46. This part is to beprepared for the midterm exam, but should not be handedin as part of the assignment.

Answer questions no 2, 4, 7, 13, and 17 on page 47.

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3 Principles of Options Pricing

3.1 Basic Notation and Concepts

Throughout this material, we assume that arbitrage op-portunities (if any) are quickly eliminated by investors.The following notation is used from now on:

1. St : the stock price at time t 2 [0, T ], where t = 0refers to the current date and t = T to the optionmaturity;

2. X : the option strike price;

3. r : the (e¤ective) risk-free rate (in % per year);

4. v (S0; X; T ) and V (S0; X; T ) : a European optionvalue and the associated American option value, re-spectively, seen as functions of the current underlyingasset price, strike price, and maturity;

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Lowercase and uppercase letters are used to distinguishbetween European and American options, respectively.For example, c refers to the premium of a European calloption, and C to the premium of an American call option.

The exercise value of an American call option, Ce(S0; X,T ) = max (0; S0 �X), is known, and the holding value,Ch (S0; X; T ), is computed based on the future poten-tialities of the contract. The value of the American calloption is:

C = max�Ce; Ch

�,

with the convention that Ch (ST ; X; 0) = 0 and C(ST ,X; 0) = Ce(ST , X; 0) = max (0; ST �X), for all ST .

The exercise value of an American put option, P e(S0; X,T ) = max (0; X � S0), is known, and the holding value,Ph (S0; X; T ), is computed based on the future poten-tialities of the contract. The value of the American putoption is:

P = max�P e; Ph

�,

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with the convention that Ph (ST ; X; 0) = 0 and P (ST ,X; 0) = max (0; X � ST ), for all ST .

The time value of an American option, called also thespeculative value, is de�ned as the di¤erence between itsvalue and its exercise value, that is, V � V e � 0.

The risk-free rate is the return earned on a risk-free in-vestment, which is associated to T-bills for short matu-rities. They are traded in over-the-counter markets, andquoted by special dealers for various maturities up to oneyear.

Instead of earning interest, T-bills are quoted at discount,that is, they are traded at lower prices than their principalamount, set here at $100. For a given maturity, thediscount (in $ per $100 of principal) is:

Discount = Principal amount� Discount rate�Time to maturity,

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where the principal amount is bT = $100, the discountrate rd is expressed in % per year, and the time to ma-turity is expressed as a fraction of a year, with the con-vention that one year equals 360 days.

The current T-bill price for a principal amount of $100comes:

b0 = Principal� Discount= Principal� (1� rd � Time to maturity) .

Suppose that the investment horizon is 7 days, the biddiscount rate is quoted at 4.45% (per year), and the askdiscount rate is quoted 4.37% (per year). The averagediscount rate, 4.41% (per year), is an approximation ofthe discount rate. The T-bill price is then:

b0 = 100��1� 4:41%� 7

360

�= $99:91425.

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The periodic return (in % per 7 days) is:

r7 =bT � b0b0

=100� 99:9142599:91425

= 0:0858%.

Compounding interest annually at the (e¤ective) risk-freerate r = r365 (in % per year) is equivalent to compound-ing interest each 7 days at the periodic rate r7 (in % per7 days). Indeed, their associated compound factors areidentical.

The risk-free rate (in % per year) is then:

r = (1 + r7)3657 � 1

= 4:57% (per year).

44


3.2 Bounds for Call Options

A call option is a right for its holder; therefore, mustgenerally pay for it up front. The value of the call optionhas an obvious lower bound:

C (S0; X; T ) � c (S0; X; T ) � 0.

Since exercising early is not mandatory, the exercise valueof an American call option, called also the intrinsic value,is a lower bound:

C (S0; X; T ) � max (0; S0 �X) .

[Please explain Figure 3.1 on page 59.]

Exercise 13: Check that this lower bound holds for theAmerican call options in Table 3.1 on page 57.

45


An upper bound for a call option is simply the currentstock price, that is,

c (S0; X; T ) � C (S0; X; T ) � S0.

Since a call option is a vehicle to purchase the stock, onecannot pay more for it than for the stock. To make surethat this upper bound holds, suppose the opposite, thatis,

C (S0; X; T ) > S0.

If this were the case, the call option would be over- andthe stock under-priced. An arbitrage strategy is as fol-lows: sell the call option, buy the stock, and make apro�t. In the worst-case scenario, whenever the call op-tion is exercised ahead in time, deliver the stock and getpaid. This is an arbitrage opportunity. Since a free lunchis supposed to disappear very quickly, the assumed strictinequality cannot hold for long. Thus, its opposite holdsdurably:

C (S0; X; T ) � S0.46


Exercise 14: Check that this upper bound holds for theAmerican call options in Table 3.1 on page 57.

At expiry, the value of a call option is:

max (0; ST �X) .


The value of an American call option is an increasingfunction of its time to maturity:

C (S0; X; T1) � C (S0; X; T2) , for T1 � T2.

The two call options are American. The call option (S0; X,T2) may be exercised early whenever (S0; X; T1) is ex-ercised.

Exercise 15: Check this inequality for the American calloptions in Table 3.1 on page 57.

47


Exercise 16: Assume the opposite to hold. Give anarbitrage opportunity, and draw conclusions.


This property also holds for European call options.

The value of a call option is a decreasing function of itsexercise price:

c (S0; X1; T ) � c (S0; X2; T ) , for X1 � X2.

The payment promised by the call option (S0; X1; T ) toits holder is greater than the one promised by (S0; X2; T )in all scenarios. The cost for the call-option holder followsthe same rule.

Exercise 17: Check this inequality for the American calloptions in Table 3.1 on page 57.

Exercise 18: Assume the opposite to hold. Give anarbitrage opportunity, and draw conclusions.

48


For European call options, one has:

0 � c (S0; X1; T )� c (S0; X2; T ) �X2 �X1(1 + r)T

� X2 �X1, for X1 � X2.

The lower bound for the di¤erence in the prices of thetwo call options has already been established.

Exercise 19: To establish the upper bound, suppose theopposite. Give an arbitrage strategy, and draw conclu-sions.

For American call options, one has:

0 � C (S0; X1; T )� C (S0; X2; T ) � X2 �X1.

Exercise 20: Check the upper bound for DCRB Euro-pean call options in Table 3.4 on page 57 (done in Table3.4 on page 65).

49


A lower bound for the European call option (S0; X; T )is:

c (S0; X; T ) � max 0; S0 �

X

(1 + r)T

!.

Exercise 21: Suppose the opposite to hold. Give anarbitrage opportunity, and draw conclusions.


We now consider a European call option on a foreigncurrency. At time t, one unit of the foreign currency isequivalent to St units of the local currency. The risk-freeforeign interest rate is �. In this context, a lower boundfor the European call option (S0; X; T ) is:

c (S0; X; T ) � max 0;

S0

(1 + �)T� X

(1 + r)T

!.


50


The value of a call option is an increasing function of therisk-free interest rate and of the stock-return volatility.

Interest rates have multiple impacts on the value of a calloption. In one hand, higher interest rates reduce presentvalues. This has a negative impact on the value of thecall option. On the other hand, the underlying stock priceincreases more when interest rates are high than it doeswhen interest rates are low. This has a positive impact onthe value of the call option. In this context, the positivee¤ect is stronger.

Under high stock-return volatility, the stock price is morelikely to reach greater highs and lows. This has a positiveimpact on the value of the call option, since only highprices are accounted for.

51


3.3 Optimal Exercise of American Call Op-

tions

An American option costs at least as much as its Euro-pean counterpart with the same parameters since it givesits holder the additional right of early exercise. Thisadditional cost is null for American call options on no-dividend-paying stocks.

Exercising a call option gives:

Ce (S0; X; T ) � C (S0; X; T ) = S0 �X

� S0 �X

(1 + r)T� c � Ch,

which in turn implies:

Ce (S0; X; T ) � Ch (S0; X; T ) .

The conclusion is that it is never optimal to exercise anAmerican call option early when the underlying stock doesnot pay dividends.

52


If the stock pays a cash-dividend amount, the stock pricetends to fall by the same amount on the ex-dividend date.The value of the call option is revised downward accord-ingly. The holder of the call option can avoid such a lossby exercising his right just before the ex-dividend date.Exercising early may be optimal for American call optionson dividend-paying stocks.

3.4 Bounds for Put Options

A put option is a right for its holder; therefore, must payfor it generally up front.

The value of the put option has an obvious lower bound:

p (S0; X; T ) � 0.

A lower bound for an American put option is its exercisevalue, that is,

P (S0; X; T ) � max (0; X � S0) .53



The value of a European put option has the followingupper bound:

p (S0; X; T ) �X

(1 + r)T.

Since an American put option can be exercised early, itsvalue has the following upper bound:

P (S0; X; T ) � X.

Exercise 23: For each contract, suppose the opposite tohold. Give an arbitrage strategy, and draw conclusions.


The value of an American put option is an increasingfunction of its time to maturity:

P (S0; X; T1) � P (S0; X; T2) , for T1 � T2.54


The two put options are American. The put option (S0; X,T2) may be exercised early whenever (S0; X; T1) is ex-ercised.

Exercise 24: Check this property for DCRB Americanput options in Table 3.1 on page 57.


[Please explain Figure 3.8 and Figure 3.9 on page 74.]

This property usually holds for European put options.

The value of a put option is an increasing function of itsexercise price:

p (S0; X1; T ) � p (S0; X2; T ) , for X1 � X2.

The payment promised by the put option (S0; X1; T ) toits holder is lower than the one promised by (S0; X2; T )

55


in all scenarios. The cost for the option holder followsthe same rule.

Exercise 26: Check this inequality for the DCRB Amer-ican put options in Table 3.1 on page 57.


The di¤erence in value of two European put options withthe same characteristics except for their strike prices ver-i�es:

p (S0; X2; T )� p (S0; X1; T ) �X2 �X1(1 + r)T

.

The di¤erence in value of two American put options withthe same characteristics except for their strike prices ver-i�es:

P (S0; X2; T )� P (S0; X1; T ) � X2 �X1.

56


Exercise 28: Check the inequality for the DCRB Amer-ican put options in Table 3.1 on page 57 (done in Table3.9 on page 77).


A lower bound for a European put option is:

p (S0; X; T ) � max 0;

X

(1 + r)T� S0

!.



Exercise: Show that a European put option on a foreigncurrency veri�es:

p (S0; X; T ) � max 0;

X

(1 + r)T� S0

(1 + �)T

!,

57


where S0 is the current foreign exchange rate and � therisk-free foreign interest rate.

The value of a put option is a decreasing function of therisk-free interest rate and an increasing function of thestock-return volatility. The reasons for this are similar tothose evoked for call options.

An American put option is exercised early if, and only if,the stock price is under a certain threshold.

[Explain again Figure 3.8 on page 74.]

3.5 Call-Put Parity

The put-call parity states that a stock plus a Europeanput is equivalent to a European call plus some risk-freebounds:

p+ S0 = c+X

(1 + r)T.

58


No-arbitrage implies a strong relationship between Euro-pean call and put options.

Exercise 31: An equality is equivalent to two inequali-ties. For each one, suppose the opposite to hold. Givean arbitrage opportunity, and draw conclusions.

The call-put parity for European currency options is:

p+S0

(1 + �)T= c+

X

(1 + r)T.

3.6 Assignment

Read the chapter, and give precise and concise de�nitionsfor the keywords listed on page 86. This part is to beprepared for the midterm exam, but should not be handedin as part of the assignment.

Answer questions no 1, 2, 6, 8, 9, 12, 14, 16, 20, and 24.

59


4 Binomial Model

4.1 The One-Period Binomial Tree

We consider a market for a riskless saving account anda risky stock. Trading activities take place only at thecurrent time t0 = 0 and at horizon t1 = T , all positionsare then closed at the horizon. No trading is allowed inbetween.

In addition, the stock price is assumed to move from itscurrent level S0 according to a one-period binomial tree.

p Sup1 = uS0

%S0

&1� p Sdown1 = dS0

t0 = 0 t1 = T

60


The stock price can rise by a factor u or drop by a factord, where u > d.

The probabilities p and 1�p de�ne the physical probabil-ity measure P , under which investors evaluate likelihoodsand take decisions.

The parameters u and d can be seen as volatility para-meters. The greater is u�d, the greater the volatility ofthe stock return.

Example 32: Assume that the stock price is currentlyquoted at $100, and can either increase by 25% or de-crease by 20%. The factors u and d are:

u = 1:25 and d = 0:8.

The one-period binomial tree for the stock is as follows.


61


In this case, the one-period binomial tree is:

Sup1 = uS0 = 1:25� 100 = 125

%S0

&Sdown1 = dS0 = 0:8� 100 = 80

t0 = 0 t1 = T

.

The price can move upward from $100 to $125 or down-ward from $100 tp $80.

Trading is allowed at the current time t0, and all positionsare closed at the horizon t1.

The (periodic) risk-free interest rate is indicated by r,and is expressed in % over the time period [0, T ]. It canbe inferred from the discount rate on T-bills that havealmost the same maturity.

62


The binomial tree is arbitrage free if, and only if, thefollowing property holds:

d < 1 + r < u.

Indeed, in the case of an upward movement S1 = uS0,the rate of return on the stock u � 1 (in % per period)must exceed the risk-free interest rate r (in % per pe-riod). Otherwise, the stock would be overpriced, and anarbitrage opportunity would appear. In addition, the risk-free interest rate r must exceed the rate of return on thestock under of a downward movement d� 1.

We consider the European call option (S0; X; T ).


Example 32 (continued): The risk-free interest rate isr = 7% (per period). Is the model arbitrage free?

The binomial tree is simple but viable. It is used here togo through the fundamentals of options pricing.

63


The goal now is to derive a formula for the value of theEuropean call option c0 in the one-period binomial tree,as a function of the current stock price S0, the optionstrike price X, the volatility parameters u and d, and therisk-free interest rate r. As usual, we use the no-arbitrageprinciple.

Next, we set up a general pricing formula that holds notonly within the binomial tree but also in all arbitrage-freemodels.

4.2 Pricing Formula for a One-Period Bi-

nomial Tree

Consider a European call option on a stock with a matu-rity date T and a strike price X.

The so-called hedge portfolio consists of h shares of stockand a single signed call option with an initial value of:

H0 = hS0 � C0,64


and a terminal value of:

H1 = hS1 � C1.

For a speci�c level of h, the hedge portfolio is riskless,and, by the no-arbitrage principle, must earn the risk-freerate. A formula for the call-option value can therefore bederived.

The value of the hedge portfolio moves along the binomialtree as follows:

Hup1 = hS

up1 � Cup1

%H0

&Hdown1 = hSdown � Cdown1

.

The hedge portfolio is riskless if, and only if,

Hup1 = Hdown1 .

65


Solving for h gives:

h =Cup1 � Cdown1

Sup1 � Sdown1

,

which depends on the known parameters S0, X, u, andd.

Given the hedge parameter h, the hedge portfolio, as ariskless investment, should earn the risk-free rate:

H0 = hS0 � C0

=H11 + r

=Hup1

1 + r=Hdown1

1 + r.

Solving for C0 gives:

C0 =p�Cup1 + (1� p�)Cdown1

1 + r,

where

p� =1 + r � du� d

2 (0; 1) .

66


The probabilities p� and 1� p� de�ne the so-called risk-neutral probability measure P �, which is not related inany way to the physical probability measure P .

In sum, the value of the European call option can beexpressed as a weighted average (an expectation) of itspromised cash �ow that is discounted at the risk-free in-terest rate:

c0 = E��c11 + r

�,

where E� [:] is the expectation sign under the risk-neutralprobability measure P �, c0 is the option value at time t0,and c1 is the option value at time t1.

The pricing formula discounts the risky cash �ow of theoption by the risk-free interest rate as if investors wererisk neutral, while they are not. This was done via a majorcorrection. We move from the real world, seen under thephysical probability measure P , to a risk-neutral world,seen under the risk-neutral probability measure P �.

Example 32 (continued): Consider a European call op-tion on the previously mentioned stock with a strike priceof X = $100. See page 96.

67


1. Compute the risk-neutral probabilities for upward anddownward movements.

2. Draw the one-period binomial tree for the call option.

3. Compute the hedge ratio.

4. Use the formula to compute the value of the calloption.

5. Check that the hedge portfolio earns the risk-freeinterest rate.

6. Compute in two di¤erent ways the value of its asso-ciated European put option.

The risk-neutral probabilities are:

p� =1 + r � du� d

=1 + 7%� 0:81:25� 0:8

= 0:6 and

1� p� = 0:4.

68


The one-period binomial tree for the stock, the call op-tion, and the hedge portfolio is:

S0 = 100

c0 =0:6�25+0:4�0

1+7%= 14:02

h = 25�0125�80 = 0:556

H0 = 0:56� 100� 14:02 = 42:02t0 = 0

,

and

Sup1 = 125

% cup1 = 25

S0 = 100 Hup1 = 0:56� 125� 25 = 45

&Sdown1 = dS0 = 0:8� 100 = 80cdown1 = 0

Hdown1 = 0:56� 80� 0 = 45t0 = 0 t1 = T

.

The hedge portfolio, which is a riskless investment, isequivalent to a long position on the stock and a shortposition on the call option:

H0 = hS0 � C0.69


Equivalently, one has:

C0 = hS0 �H0.

Thus, the call option is equivalent to a long position onthe stock and a short position on T-bills. More precisely,the call option is equivalent to holding h shares of stockand borrowing H0 (in $) at the risk-free interest rate.

In other words, the call option can be replicated (du-plicated) by a hedging strategy based on the underlyingstock and the saving account:

C1 = hS1 �H1.

If the signer decides to fully replicate the option, he wouldinsure the promised payment to the holder at maturity. Amarket in which one can fully replicate derivatives is calleda complete market. The binomial tree is an example.

70


The value of the associated put option is rather obtainedthrough the binomial tree or by the put-call parity:

p0 = c0 +X

1 + r� S0

= $7:48.

Example 32 (continued): Show how the call option canbe fully replicated. Do the same work for the associatedEuropean put option.

4.3 No-Arbitrage Pricing

The extension of the pricing formula is straightforward aslong as the model is arbitrage free.

Consider an American option on a stock with a strike priceof X and a maturity date of T . We use the followingnotation:

71


1. St : the stock price at time t;

2. Vt (s) : the option value at time t, seen as a functionof the stock price St = s;

3. V ht (s) : the option holding value at time t, seen asa function of the stock price St = s;

4. V et (s) : the option exercise value at time t, seen asa function of the stock price St = s.

The holding value summarizes for all the future poten-tialities of the option. The notation proposed here isconsistent with the one introduced in the previous chap-ters, but is more general. For an American call option, forexample, one has Vt = Ct and V et = max (0; St �X).At each decision date, the option holder has to decide

72


whether it is optimal to exercise its right or not. Theoption value becomes:

Vt (s) = max�V et (s) , V

ht (s)

�, for all t and all s,

with the convention that V hT (s) = 0 or VT (s) = VeT (s),

for all s.

A European option can be seen as a special case with:

V et (s) = 0, for all t < T and all s.

The holding value of an option at time tn can be ex-pressed as a conditional expectation of its future poten-tialities discounted back at the risk-free rate. The no-arbitrage pricing formula is:

V hn (s) = E�24Vn+1

�Stn+1

�1 + r

j Stn = s

35 ,where V hn (s) is the option holding value at the currenttime tn when Stn = s, E

� [: j Stn = s] is the conditional

73


expectation symbol taken under the risk-neutral probabil-ity measure P �, Vn+1

�Stn+1

�is the future value of the

option at the next decision date tn+1 when the stockprice is Stn+1, and r is the periodic risk-free rate.

From the perspective of an investor at time tn, the stockprice Stn+1 is random. The pricing formula consists ofcomputing a weighted average on all feasible future valuesof the option, discounted at the risk-free rate, the weightsbeing given by the risk-neutral probabilities.

Example 32 (continued): Consider a two-period bino-mial tree for the stock. Compute the value of the Euro-pean call option. Compute in two ways the value of itsassociated European put option. The two-period bino-mial tree for the stock and the call option are as follows.See Figure 4.4 on page 104. Again, the hedge portfolio

74


can be used to replicate the call option.

Suu2 = 156:25% cuu2 = 56:25

Sup1 = 125

% cu1 = 31:54 &S0 = 100 Sud2 = Sdu2 = 100

c0 = 17:69 & % cud2 = Cdu2 = 0

Sd1 = 80

cd1 = 0 &Sdd2 = 64

cdd2 = 0

t0 = 0 t1 t2 = T

.

4.4 Dividends and Early Exercise

Unlike American put options, American call options can-not be exercised optimally before maturity. If the stockpays dividends over the option�s life, however, an Amer-ican call option can be exercised early just before ex-dividend dates.

75


Consider an American option on a dividend-paying stock.Let D1; : : : ; Dp be the dividends, and �1; : : : ; �p theirassociated ex-dividend dates over the option�s life [0, T ].Pricing an option on a dividend-paying stock requires thefollowing adjustments for the binomial tree.

1. Compute the PV of all dividends to be paid over theoption�s life as of the present time t0 = 0, that is,

D =pXi=1

Die�r� i,

where r is the continuou risk-free rate;

2. Adjust downward the current stock price by the PVof all dividends:

S�0 = S0 �D;

3. Create the binomial tree that starts from S�0 usingthe up and down factors as usual;

76


4. At each node of the binomial tree, add to the stockprice the PV of all future dividends to be paid overthe option�s life, including the one that is about togo ex-dividend (if any);

5. Move backward through the binomial tree, and com-pute the option values as usual from maturity to thestart.

This method applies for European as well as Americanoptions.

Example 33: Consider a stock that promises a divi-dend of $2 in 90 days. The stock is now traded at $50.The risk-free rate is a continuously compounded rate, ex-pressed in % per year. The parameters of the binomial

77


tree are:

�t = 24 days

= 0:065756 (years),

r = 9% (per year),

u = 1:108015,

d = 0:902515,

p� =er�t � du� d

= 0:503262.

The PV of all dividends over the option�s life is:

D = D1e�r�1 = $1:96.

78


The binomial tree in step 3 is as follows:

80:2372:40

65:35 65:3558:98 58:98

53:23 53:23 53:2348:04 48:04 48:04

43:36 43:36 43:3639:13 39:13

35:32 35:3231:87

28:77

79


The adjusted binomial tree in step 4 is as follows:

80:2372:40

67:34 65:3560:96 58:98

55:20 55:22 53:2350:00 50:02 48:04

45:32 45:35 43:3641:11 39:13

37:31 35:3231:87

28:77

Consider an American call option on this stock startingat parity and maturing in 120 days. The binomial tree

80


for this American call option is

30:2322:70

17:34� 15:3511:36 9:27

7:22 5:44 3:234:48 3:12 1:62

1:76 0:81 0:000:41 0:00

0:00 0:000:00

0:00

The value of this American call option is $4:48. Thisoption will be optimally exercised if the stock price willrecord an �up-up-up�movement from the start.

4.5 Assignment

Read the chapter, and give precise and concise de�nitionsfor the keywords listed on page 86. This part is to be

81


prepared for the midterm exam, but should not be handedin as part of the assignment.

Answer questions no 2, 4, 12, 13, and 20.

Exercise 34: Read and comment on the business articleentitled: Dow jones sells index business to CME.

82


5 The Black and Scholes Model

5.1 Introduction

Black and Scholes introduced a frictionless market for anon-dividend paying stock (the risky asset) and a savingaccount (the riskless asset) in which trading takes placecontinuously to time horizon T .

The stock price process fSg is assumed to follow a long-normal process characterized by:

ST = SteR,

where R is the continuously compounded rate of returnon the stock over the period [t, T ]. Under the physicalprobability measure P , the rate of return on the stockover [t, T ] follows a normal distribution in the form:

R =

��

2

2

!(T � t) + �

pT � tZ,

83


where � is a real parameter known as the instantaneousrate of return on the stock (in % per year), � is a positiveparameter known as the volatility of the stock log-returns(in % per year), T � t is the investment�s remaining timeto maturity (in years), and Z is a random variable thatfollows the standard normal distribution N (0, 1) with amean value of 0 and a standard deviation of 1. The stock-price process is said to be lognormal since the naturallogarithm of any future stock price ST , given the currentstock price St, follows the normal distribution:

N ln (St) +

��

2

2

!(T � t) , �

pT � t

!.

Example 35: The rate of return on a stock (in % peryear) follows the normal distribution N (5%, 4%) witha mean value of 5% and a standard deviation of 4%.Draw the curve of its density function n (:). Explain themeaning of its distribution function N (:). Compute theprobability that the stock will record a loss during theupcoming year P (N (5%, 4%) � 0). Use Excel. Un-fortunately, such a manipulation is not so easy to achieve

84


in practice since the parameter � is typically unknown,and is hard to estimate from observed data.

Example 36: Compute the probability that a standardnormal distribution N (0, 1) records values between itsmean value �1, �2,and �3 standard deviations. UseExcel and Table 5.1 on page 135.

Consider a European call option on the risky asset whosecurrent value in the Black and Scholes model is indicatedby c. The hedge portfolio, as for the binomial tree, isriskless for:

h = �

= lim�S!0

�c

�S

=@c

@S,

if it is rebalanced continuously. Therefore, the PV prin-ciple can be extended to the so-called risk-neutral eval-uation principle in the Black and Scholes model, whichturns out to be arbitrage free. Under the risk-neutral

85


probability measure P �, the rate of return on the stockR� is random, and follows the normal distribution:

R� =

r � �

2

2

!(T � t) + �

pT � tZ�,

where r is the continuously compounded risk-free rate (in% per year), and Z� is a random variable that follows thenormal distribution N (0, 1).

In sum, the rate of return on the stock behaves similarlyunder P and P �, except that the instantaneous rate ofreturn on the stock � in the real world is exchanged by therisk-free rate r in the risk-neutral world, where investorsact as if they were risk neutral.

5.2 The Black and Scholes Formula for

European Options

Risk-neutral evaluation holds in the Black and Scholesmarket, for it is arbitrage free. The value of a European

86


call option is:

ct = E�he�r(T�t)max (0, ST �X) j St

i= E�

he�r(T�t) (ST �X) I (ST > X) j St

i= E�

he�r(T�t)ST I (ST > X) j St

i�

E�he�r(T�t)XI (ST > X) j St

i,

where the indicator function I is de�ned as follows:

I (ST > X) =(1, if ST > X0, if ST � X

.

The call-option value consists of two terms. The �rst(second) term is the conditional expectation of the dis-counted stock price (strike price) given that the optionexpires in the money.

Computing this conditional expectation is straightforwardusing integration theory and calculus. The �nal result isthe Black and Scholes formula for a European call optionon a non-dividend paying stock:

ct = N (d1)St �Xe�r(T�t)N (d2) ,87


where

d1 =ln (St=X) +

�r + �2=2

�(T � t)

�pT � t

and

d2 = d1 � �pT � t.

The Black and Scholes formula for a European put optionon a non-dividend paying stock is

pt = Xe�r(T�t)N (�d2)� StN (�d1) .

Example 37: Compute the value of a European call op-tion. The initial stock price is S0 = $125:94, the strikeprice is X = $125, the e¤ective risk-free interest rate isr = 4:56% (per year), the volatility is � = 83% (peryear), and the maturity is T = 0:0959 (in years). Usethe Black and Scholes formula and Excel. See Table 5.2on page 136. Do a sensitivity analysis of c0, as a functionof S0, X, �, and r.

88


5.3 Sensitivity Analysis

The value of a European call option c = c0 at the startis an increasing function of the stock price S0 = S, thevolatility �, and the risk-free rate r, and a decreasingfunction of the strike price X.

The Delta coe¢ cient, indicated by �, is the sensitivityof c to small variations in S0, all other parameters being�xed:

� =@c

@S

= lim�S0!0

�c

�S= N (d1) 2 [0, 1]

� �c

�S, for small �S.

For small changes in S, the �rst-order Taylor approxima-tion of c is:

cnew � cold +��Snew � Sold

�.

89


The Gamma coe¢ cient, indicated by �, is the sensitivityof the � coe¢ cient to small variations in S, all otherparameters being �xed:

� =@�

@S

=@2c

@S2

=e�d

21=2

S�p2�T

> 0, for all S0.

For small changes in S, the second-order Taylor approxi-mation of c is:

cnew � cold +��Snew � Sold

�+1

2��Snew � Sold

�2.


Example 37 (continued): The coe¢ cient Delta is � =

0:5692. Approximate the new call price if the new stockprice were quoted at $127.

90


Example 37 (continued): The coe¢ cient Gamma is0:0123. Approximate the new call value if the new stockprice were quoted at $127. Check that the second-orderapproximation is more precise than the �rst-order approx-imation.

The coe¢ cients � and � are always positive; hence, thecurve of the call option is not only an increasing function,but also convex.


The Vega function, indicated by #, is the sensitivity ofCe to small variations in �, all other parameters being�xed:

# =@Ce

@�

= lim��!0

�Ce

��

=S0pTe�d

21=2

p2�

> 0,

for all �.91


Example 37 (continued): The coe¢ cient Vega is 15:32.If the volatility changes by 1%, the call value would changeby $0.1532 in the same direction. The call value is highlysensitive to the volatility of stock returns.

The Theta function, indicated by �, is the sensitivity ofCe to small variations in the current time t, all otherparameters being �xed:

� =@Ce

@t= � @Ce

@ (T � t)

= lim�t!0

�Ce

�t

= �S0pT � te�d21=2

2q2� (T � t)

�

rcXe�rc(T�t)N (d2) ,

for all T .

The Theta function is always negative.

92


5.4 Dividends

In the case of a known discrete dividend D1 at an ex-dividend date t1, one has to adjust the stock price down-ward by the discounted value of the dividend:

S�0 = S0 �D1e�rct1,

and again apply the Black and Scholes formula with S�0.Now, if the dividends are paid continuously at a rate �c (in% per year), the stock price must be adjusted downwardas follows:

S�0 = S0e��cT .

Example 37 (continued): Compute the call value if 1�a dividend of $2 is paid in 21 days, or 2� the dividendsare paid continuously at a rate of 4% (per year).

93


5.5 The Volatility of Stock Returns

In the Black and Scholes model, there are two methodsfor computing the volatility of stock returns �.

The �rst approach, known as the historical method, con-sists of 1� selecting a time step �t, 2� computing thecontinuous rates of return on the stock Rc1; : : : ; R

cN re-

alized on successive time intervals of length �t, and 3�estimating the volatility � as follows:

b� = 1p�t

vuuutPNn=1 �Rcn �Rc�2N � 1

, (per year).

Example 37 (continued): Table 5.6 on page 156 showsN = 59 realized daily stock returns. For example, thediscrete rate of return on the stock on day 1 is:

S2 � S1S1

= Rd1

= 7:87% (on day 1),94


and the continuous rate of return on the stock on day 1veri�es:

Rc1 = ln�1 +Rd1

�= ln

S2S1

!,

= 7:57% (on day 1).

Since the sumP59n=1

�Rcn �Rc

�2= 0:182817, the vol-

atility estimate is:

b� =p250

s0:182817

58= 0:8877 (per year)

The second approach, known as the implied method, con-sists of approximating the volatility by solving the non-linear equation for �:

CBSe (�) = Cmarkete .

The implied volalatility acts as if the market behavedexactly as the Black and Scholes model, while it doesnot.

[Please explain Figure 5.19 on page 161.]95


A more general approach consists of minimizing the modelerror:

min�

NXn=1

�vBSn � vmarketn

�2,

where the European options are selected to be the mosttraded contracts.

5.6 The Black and Scholes Formula for

European Put Options

The Black and Scholes market is arbitrage free. So, giventhe fair value for a call option, we can deduce the fairvalue of a put option by using the put-call parity.

The Black and Scholes formula for a European put optionis:

Pe = Xe�rcTN (�d2)� S0N (�d1) .

96


Example 37 (continued): The value of the associatedput option is P = $12:08.

[See Table 5.8 on page 165 for a sensitivity analysis.]

As for a European call option, a sensibility analysis canbe done for a European put option.

For a European put option, the Delta coe¢ cient � isnegative. The � hedging strategy, which allows one toreplicate the put option, starts by taking a short positionon j�j shares of stock and saving at the risk-free rate.Next, when the stock price moves, compute the new co-e¢ cient �, adjust the quantity of shares sold short, andsave or withdraw money at the risk-free rate consequently.

5.7 Relatioship to the Binomial Model

The binomial tree can be calibrated to the Black andScholes model by selecting its coe¢ cients as follows:

u = e�p�t and d =

1

u,

97


where � is the volaltility of stock returns (per year), Tthe horizon (in years), and N the number of time steps,and �t = T

N the time step (in years).

The binomial tree remains arbitrage free if, and only if,

d < 1 + r < u.

Selecting the coe¢ cients in that way ensures the con-vergence of the binomial tree to the Black and Scholesmodel, when the number of time steps N converges toin�nity.

5.8 Assignment

Read the chapter, and give precise and concise de�nitionsfor the keywords listed on page 172. This part is to beprepared for the second midterm exam, but should notbe handed in as part of the assignment.

Answer questions no 1, 2, 6, 7, and 12.

98


6 Selected Option Strategies

6.1 Introduction

Combinations of stocks and options are so diverse thatany investor can �nd a strategy that �ts his risk prefer-ence, and marker forescast.

A bull market is a market in which stock prices go up, anda bear market is one in which stock prices go down. Abullish investor believes that a bull market is coming, anda bearish investor believes that a bear market is coming.

We focus on strategies based on European call and putoptions combined with their underlying stocks. The pro�tassociated to a given portfolio strategy is de�ned as thedi¤erence in the value of the portfolio over the option�slife:

� = VT � V0,

99


Vt is the value of the portfolio at time t.

The strategy is then analyzed through its pro�t seen as afunction of the (unknown) terminal stock price, indicatedby ST .

6.2 Basic Strategies

The pro�t from a long position on the stock is:

� (ST ) = ST � S0.

The price level S�T is the terminal stock price, such that��S�T�= 0. The following pairs (ST , � (ST )) belong

to the pro�t line:

(0;�S0) and (S�T = S0; 0) .

The pro�t from a short position on the stock is simplythe opposite of the pro�t from the long position."

Please explain Figure 6.1 and Figure 6.2on page 186.

#100


Holding a stock is a bullish strategy, which has an un-limited potential gain and limited potential loss. Sellingthe stock short is a bearish strategy, which has a limitedpotential gain and unlimited potential loss.

The pro�t from a long position on a European call optionis:

� (ST ) = max (0; ST �X)� Ce.

The pro�t function can be expressed as:

� (ST ) =

(�Ce, for ST � XST �X � Ce for ST > X

,

where ST = X is a breakeven point. The following pairsbelong to the piecewise-linear pro�t curve:

(0, � Ce) , (X, � Ce) , and (S�T = X + Ce, 0) .

A short position on a European call option is also knownas an uncovered call or a naked call."

Please explain Figure 6.3 on page 188 andFigure 6.6 on page 192.

#101


A long position on a European call is a bullish strategy,which has an unlimited potential gain and limited poten-tial loss. Conversely, a short position on a European calloption is a bearish strategy, which has a limited potentialgain and unlimited potential loss. The pro�t from a longposition on a European put option is:

� (ST ) = max (0, X � ST )� Pe.


� (ST ) =

(X � ST � Pe, for ST � X�Pe, for ST > X

.

Again, ST = X is a breakeven point. The following pairsbelong to the piecewise-linear pro�t curve:

(0; X � Pe) , (S�T = X � Pe; 0) , and (X;�Pe) .

The pro�t from a short position on the European putoption is simply the opposite of the pro�t from the longposition."

Please explain Figures 6.9, 6.12 and 6.15on pages 194, 197, and 199.

#102


A long position on a European put option is a bearishstrategy, which has a limited potential loss and a largerbut limited potential gain. Conversely, a short positionon a European put option is a bullish strategy, which hasa limited potential gain and a larger but limited potentialloss.

6.3 More Complex Strategies

A covered call consists of a short position on a call optionand a long position on the underlying stock. A similar butsomewhat di¤erent strategy is the so-called � hedgingstrategy. A short position on a call option and a longposition on � shares of stock de�ne a riskless strategy.The covered call is expected to be a bit riskier, but is stilla safe strategy.

The pro�t from a covered call is:

� (ST ) = � (max (0; ST �X)� Ce) + ST � S0.103



� (ST ) =

(ST � (S0 � Ce), for ST � XX � (S0 � Ce), for ST > X

,

where ST = X is a breakeven point. The following pairsbelong to the piecewise-linear pro�t curve:

(0;� (S0 � Ce)) , (S�T = S0 � Ce; 0) , and

(X;X � (S0 � Ce)) ,

where S0�Ce � 0, S0�Ce � X, andX�(S0 � Ce) �0.


A covered call is a bullish strategy, which has a limitedpotential loss and a limited and small potential gain. Aprotective put consists of a long position on a put optionand a long position on its underlying stock. The pro�tfrom a covered call is:

� (ST ) = (max (0, X � ST )� Pe) + ST � S0.

104



� (ST ) =

(X � S0 � Pe, for ST < XST � S0 � Pe, for ST � X

,

where ST = X is a breakeven point.

The following pairs belong to the piecewise-linear pro�tcurve:

(0; X � S0 � Pe) , (X;X � S0 � Pe) , and

(S�T = S0 + Pe; 0) ,

where X � S0 � Pe is supposed to be negative, andX � S�T .


A protective put is a bullish strategy, which has an un-limited potential gain and a limited and small potentialloss. The protective put acts as an insurance policy onthe stock.

105


The put-call parity is:

Ce (S0; X; T ) = S0 + Pe (S0; X; T )�X

(1 + r)T.

The right-hand expression de�nes a portfolio, called asynthetic call, that is equivalent to a European call op-tion.

Suppose that the call option is overpriced. The put-callparity suggests the actual call option to be sold, and thesynthetic call option to be bought. This strategy is knownas a conversion. A reverse conversion consists of sellingthe actual put option and buying the synthetic put option.This strategy is motivated when the actual put option isunderpriced.

6.4 Advanced Option Strategies

A spread is a portfolio consisting of a long position on oneoption and a short position on another option. Options

106


under money spreads di¤er only by their strike prices, andby their maturity dates under calendar spreads.

A call bull spread is a money spread on call options thatdi¤er only by their strike prices. The long call has thelower strike price X1 < X2. The pro�t from a call bullspread is:

� (ST ) = max (0; ST �X1)� C1 �(max (0, ST �X2)� C2) ,

where C1 > C2 are the call-option prices associated toX1 < X2. The pro�t function from a call bull spread is:

� (ST )

=

8><>:�(C1 � C2), for ST � X1 < X2ST �X1 � (C1 � C2) , for X1 < ST � X2X2 �X1 � (C1 � C2) for X1 < X2 < ST

,

where X1 and X2 are two breakeven points.


(0;� (C1 � C2)) , (X1;� (C1 � C2)) ,(S�T = X1 + (C1 � C2) ; 0) , and(X2; X2 �X1 � (C1 � C2)) ,

107


where C1�C2 > 0, X1 < X1+(C1 � C2) < X2, andX2 �X1 � (C1 � C2) > 0.


A call bull spread is a bullish strategy that has a limitedand small gain and loss.

A put bear spread is a money spread that consists of along position on the put option with the higher exerciseprice X2 and a short position on the put option with thelower exercise price X1.

The pro�t function from a call bull spread is:

� (ST )

=

8><>:X2 �X1 � (P2 � P1), for ST � X1 < X2�ST +X2 � (P2 � P1) , for X1 < ST � X2� (P2 � P1) , for X1 < X2 < ST

,

where X1 and X2 are two breakeven points.

108



(0; X2 �X1 � (P2 � P1)) ,(X1; X2 �X1 � (P2 � P1)) ,(S�T = X2 � (P2 � P1); 0) , and(X2;� (P2 � P1)) ,

whereX2�X1�(P2�P1) � 0,X1 � X2�(P2�P1) �X2, and P2 � P1 � 0.


The put bear spread is a bearish strategy that has a lim-ited and small gain and loss.

6.5 Assignment

Read the chapter, and give precise and concise de�nitionsfor the keywords listed on page 215. This part is to beprepared for the second midterm exam, but should notbe handed in as part of the assignment.

Answer questions no 1, 3, 5, 9, and 10 on pages 215 and216.

109


7 The Structure of Forward and Fu-

tures Markets

7.1 The Development of Forward and Fu-

tures Markets

A forward contract commits two parties one to buy andthe other to sell an underlying asset at a known futurematurity date (delivery date) for an agreed-upon forwardprice (delivery price). No payment is exchanged up front.

Example 38: Several every-day transactions can be in-terpreted as forward contracts. Examples include pizzadelivery and car rentals.

A forward contract is fundamentally di¤erent from anoption. The former commits the long party to buyingthe underlying asset, while the latter does not. Forward

110


contracts are traded over the counter; therefore, forward-market participants are subject to both market and creditrisk.

If the forward price rises (falls), the long party records again (loss) since he committed to buying the underlyingasset at a lower (higher) price. On the other hand, theshort party records a loss (gain). The cumulative lossrecorded by one party at maturity can be so great thathe can fail to ful�ll his obligation.

A futures contract is identical to a forward contract ex-cept that it is traded on an exchange and is subject toa daily settlement of gains and losses. Unlike a forwardcontract, which is settled at maturity, a futures contractis settled at the end of each trading day, thereby keepingthe loss and the default probability at a very low level.

Since investors can invert their positions at any time ina liquid market, futures contracts are usually (but notalways) settled in cash prior to maturity.

111


While forward contracts go back to the beginnings ofcommerce, futures contracts are more recent. Futurescontracts on agricultural products have been traded since1898, when the Chicago Mercantile Exchange was cre-ated. The exchange de�nes a set of terms and conditionsfor each futures contract such as, size, quotation unit,minimum price �uctuation, grade, trading hours, deliveryterms, daily price limits, and delivery procedures. Finan-cial futures contracts started to be traded later, in theearly 1970s, when Western economies allowed foreign ex-change rates to �uctuate. The International MonetaryMarket was created in 1972 to trade futures contracts onforeign currencies. Later on, in the mid-1970s, interest-rate futures contracts started to be traded. Examplesinclude the GNMA, T-bill, T-notes, and T-bond futurescontracts.

Example 39: The T-bond futures contract, traded onthe CBOT, calls for the delivery of $100,000 T-bonds witha minimum remaining life of 15 years at the �rst deliverydate. This contract contains several embedded options

112


held by the short trader, such as the choosing and timingoptions. The short trader can deliver, at his discretion,any bond of a set of eligible long-term T-bonds. Theseller is expected to act in an optimal way, and deliverthe cheapest T-bond. The short trader also has the rightto deliver the underlying asset within a month, followingspeci�c delivery rules.

Stock index futures contracts were the next to be traded,followed by options on futures. For example, a call optionon a futures contract gives its holder the right to take along position on the underlying futures contract at optionmaturity for the agreed-upon option strike price. Theoption holder will exercise his right if, and only if, thefutures price at option maturity is higher than the optionstrike price.


A futures exchange is a corporate entity organized totrade futures contracts. The owners of the exchange are

113


known as the full members. Each one of them owns aseat.

[Please explain Figure 8.1 o page 264.]

Several market participants are involved in trading a fu-tures contract: the buyer (seller), the buyer�s (seller�s)broker, and the commission broker of the buyer�s (seller�s)broker. The latter is a full or a partial member of theexchange, depending on whether he owns or leases a seat.

[Please explain the top of Figure 8.2 on page 267.]

7.3 The Daily Settlement System

The clearinghouse guarantees the performance of eachparticipant in a futures exchange. The oldest one in NorthAmerica, associated to the Chicago Board of Trade, wascreated in 1920. The owners of a clearinghouse are knownas the clearing �rms.

114


To ensure the performance of an investor, a futures con-tract is subject to a daily settlement managed by theclearinghouse.

[Please explain the bottom of Figure 8.2 on page 267.]

First of all, when a futures contract is issued, each partyis invited to open a margin account with his broker, andto deposit an initial margin. The broker is invited to dothe same with his clearing �rm, which in turn, does thesame with the clearinghouse. Next, the trading processcontinues untill the day�s end. The settlement price isthen registered based on the last futures price. To makeit simple, de�ne the settlement price within a tradingday as the last registered futures price. Finally, the fu-tures contract is marked to the market at the end of thetrading day: it is closed and reopened at the new settle-ment price. The di¤erence between the current and thelast settlement prices, if positive, is subtracted from theseller�s margin account, and added to the buyer�s mar-gin account. If the di¤erence is negative, the opposite is

115


done. When a margin account falls below a certain level,known as the maintenance margin, the investor receives amargin call to raise the margin balance to the initial mar-gin level. The daily settlement system drastically reducescounterparty risk, since losses, which are monitored everyday, remain limited.

Example 39 (continued): The detail is in Table 8.2 onpage 269. Consider a short position on the CBOT T-bond futures contract, which was taken on 08/01 at thequoted futures price of 97-27. The futures price was at$(97+27/32) per $100 of principal (of the underlying T-bond), which corresponded to $97,843.75 per $100,000of principal. The initial margin was $2,500, and the main-tenance margin $2,000. The seller then deposited $2,500into his margin account.

The futures price fell to $97,406.25 at the end of the�rst trading day. This was the new settlement price, theold one being $97,843.75. The seller recorded a gain,and a buyer recorded a loss. The (absolute) di¤rence

116


between the two settlement prices, that is, $437.50, wassubstracted from the buyer�s margin account, and addedto the seller�s margin account. The balance of the seller�smargin account was then at $2,937.50.

The futures price rose to $97,781.25 at the end of thesecond trading day. This was the new settlement price,the old one having been $97,406.25. The seller recordeda loss, and the buyer a gain. The (absolute) di¤erencebetween the two settlement prices, that is, $375.50, wassubtracted from the seller�s margin account, and addedto the buyer�s margin account. The balance of the seller�smargin account then was at $2,562.50.

On 08/08, the balance of the seller�s margin account (at$1250) fell under the maintenance margin ($2000) justafter the contract was marked to the market. A margincall of $1250 was issued the same day, and honoured thenext trading day, on 08/11.

On 08/18, the last trading day, the settlement price fellfrom $100,781.25 to $100,500.00, and the seller then

117


recorde a gain of $281.25. The latter closed his posi-tion by inversion, and withdrew the remaining balance of$3,031.25.

7.4 Assignment

Read the chapter, and give precise and concise de�nitionsfor the keywords listed on page 279. This part is to beprepared for the second midterm exam, but should notbe handed in as part of Assignment 2.

Answer questions no 1, 2, 3, 15, 16.

118


8 Pricing Forwards, Futures, and

Options on Futures

8.1 Principles of Carry-Arbitrage Pricing

A forward contract commits two parties one to buy andthe other to sell an underlying asset at a known futurematurity date, for an agreed-upon forward price. No pay-ment is exchanged up front. The maturity date is alsoknown as the delivery date, and the forward price as thedelivery price. The underlying asset may be a stock, stockindex, foreign currency, bond, or commodity.

A futures contract is identical to a forward contract ex-cept that it is traded on an exchange, subject to a marginsystem, and a daily settlement of gains and losses.

An important issue is that the forward (futures) price isdi¤erent from the forward (futures) value. For example,

119


you may commit to buy a share of stock in three monthsfor the forward price of $50; however, the value of yourposition is zero at the start since no up-front payment isexchanged.

The value of both positions on a forward contract is zeroat the start. This is also the case for a futures contractat the start and end of each trading day, as soon as thecontract is marked to market.

We use the following notation:

1. F (t; T ) : the price of a forward contract starting att and maturing at T ;

2. Vt (0; T ) : the value at t of a long position on aforward contract starting at 0 and maturing at T ;

3. ft (T ) : the price at t of a futures contract maturingat T ;

120


4. ft;�1 (T ) : the settlement price on the day prior tot of a futures contract maturing at T ;

5. vt (T ) : the value at t of a long position on a futurescontract maturing at T .

8.2 Pricing Forward Contracts on Stocks

Consider a forward contract on a single stock, which paysno dividends over the contract�s life. Prices and valuesare computed assuming that the market is frictionless andarbitrage-free. The convergence principle is:

F (T; T ) = limt!T

F (t; T ) = ST ,

where ST is the spot price at T of the stock.

The valuation of a forward contract is based on the prop-erty:

V0 (0; T ) = 0.121


So, a forward contract cannot be considered an asset ora liability at the start. The forward price F (0, T ) isnegotiated at 0, such that V0 (0, T ) = 0. The value ofa long position on the forward contract at T is:

VT (0; T ) = ST � F (0; T ) ,

which can be either positive or negative. Thus, lookingahead, the forward contract can be considered an assetor a liability. The value of a short position on a forward(futures) contract is simply the opposite of that of a longposition.

Now, consider a portfolio consisting of a long position onthe underlying asset and a short position on the forwardcontract. Its value at T is:

ST � VT (0, T ) = F (0; T ) ,

which is known in advance. Since this portfolio bears norisk at maturity, its value before maturity at t is:

St � Vt (0; T ) =F (0; T )

(1 + r)T�t,

122


where r is the e¤ective risk-free interest rate (in % peryear), and T�t the remaining time to maturity (in years).This results in the following equation:

Vt (0; T ) = St �F (0; T )

(1 + r)T�t.

Solving this equation at time 0 for F (0, T ) gives thewell-known result:

F (0, T ) = S0 (1 + r)T .

Example 40: You bought a forward contract maturing in45 days for a price of $100, and have held it for 20 days.The risk-free rate is 10% (per year). Now, the spot priceof the underlying stock is $102. Give the stock price thatwould have been observed at the start, and compute thecurrent value of the forward contract.

For a futures contract, the convergence property holds:

fT (T ) = limt!T

ft (T ) = ST .

123


The value of a long position on a futures contract beforebeing marked to market is:

vt (T ) = ft (T )� ft;�1 (T ) .

Since the gain/loss under a futures contract is settledevery day, the value of both positions switches to zero assoon as the contract is marked to market.

Ignoring the margin system, and assuming a constantrisk-free rate, the futures price will be equal to the for-ward price.

8.3 Pricing Forward Contracts on Stock

Indices

If the stock pays sure dividendsD1; : : : ; DJ at t1; : : : ; tJduring the contract�s life, the carry-arbitrage evaluationequation at t becomes:

St � Vt (0; T ) = Dt;T +F (0; T )

(1 + r)T�t,

124


where

Dt;T =Xtj>t

Dj

(1 + r)tj,

is the present value at t of all remaining dividends from tto T . The present value at time 0 of all dividends D0;Tis indicated by D0. This results in:

F (0; T ) = (S0 �D0) (1 + r)T ,

Example 41: The current stock price is at $100, and therisk-free rate at 6%. The stock pays a dividend of $5 in20 days. Compute the forward price with a maturity of45 days.

A stock index is a weighted combination of single stocks,each paying dividends on a regular basis, as if the stockindex were paying dividends in continuous time. Assum-ing that the dividends are paid continuously at a known�xed rate �c, one has:

D0 = S0�1� e��cT

�.

125


The forward price is then:

F (0; T ) = S0e(rc��c)T ,

where rc is the continuous-time risk-free interest rate (in% per year).

Example 42: A stock index is at 50, and its dividend rateat 6%. Compute the futures price when the continuous-time risk-free rate is at 8% and the time to maturity is 60days. Compute the futures price again when the risk-freerate is at 5%.

For a futures contract on a stock index, one can observethe futures price f0 (T ) and solve for the dividend rate�c:

f0 (T ) = S0e(rc��c)T .

The solution is:

�c = rc �1

Tln

f0 (T )

S0

!.

126


Example 43: Consider a futures contract on a stock in-dex with a time to maturity of 60 days. The futures priceis at 50.16, the stock index at 50, and the continuous-time risk-free rate at 8%. Compute the constant dividendrate of this stock index.

8.4 Pricing Forward Contracts on Foreign

Currencies

Consider again the portfolio of a long position on theunderlying asset and a short position on the forward con-tract. Interpret the spot exchange rate S0 as the price attime 0 in local money of one unit of the foreign currency.For this portfolio to be riskless at maturity, it must con-tain one unit of the foreign currency. Less then one unitis required prior to maturity, since holding a foreign cur-rency earns interest. More precisely, the portfolio mustcontain:

1

(1 + �)T�t

127


units of the foreign currency at time t < T , where � isthe foreign risk-free rate (in % per year).

The carry-arbitrage evaluation equation becomes

St

(1 + �)T�t� Vt (0; T ) =

F (0; T )

(1 + r)T�t.

Solving for F (0, T ) at time 0 gives:

F (0; T ) =(1 + r)T

(1 + �)TS0.

Example 44: The spot exchange rate for US dollars was0.7908 euros. The US interest rate was 5.84%, while theeuro interest rate was 3.59%. The time to expiration was90 days. Compute the forward exchange rate for dollars.

From the perspective of a European investor, it may beinteresting to convert euros to dollars to invest at a higherinterest rate. When time comes to convert back dollarsto euros at the agreed-upon forward exchange rate, a lossis recorded. This is the well known interest-rate parity.

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8.5 Pricing Forward Contracts on Com-

modities

Holding a commodity induces a storage cost s per unit,assumed to be known and paid at maturity. Consideragain the portfolio of a long position on the commodityand a short position on the forward contract. The carry-arbitrage evaluation equation at t becomes:

St � Vt (0; T ) =�s+ F (0; T )(1 + r)T�t

.

Solving for F (0, T ) at time 0 gives:

F (0; T ) = S0 (1 + r)T + s.

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8.6 Futures Prices and Expected Future

Spot Prices

Depending on the bene�t/cost of carry, the futures pricemay be higher or lower than the spot price:

f0 (T ) :

8>>><>>>:> S0, then the market is called

Contango< S0, then the market is called

Backwardation

.

On the other hand, futures prices give insights into futurespot prices. Empirical investigations, however, show thatfutures prices are generally biased estimations of expectedfuture spot prices:

f0 (T ) :

8>>><>>>:> E [ST ] , then the market is called

Normal Contango< E [ST ] , then the market is called

Normal Backwardation

,

where the expectation is computed under the physicalprobability measure.

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8.7 Call-Put-Futures Parity

Consider a portfolio of a long position on a European calloption and a short position on a European put option.Both have the same underlying asset, exercise price, andmaturity. The value of this portfolio at T is:

max (0; ST �X)�max (0; X � ST )= ST �X= ST � F (0; T ) + F (0; T )�X.

This results in the call-put-futures parity:

Ce � Pe =F (0; T )�X(1 + r)T

.

Example 45: On May 14, the S&P 500 index was at1337.80, and the June futures price on the S&P 500 indexat 1339.30. The June 1340 call option on this stockindex was at 40, and the June 1340 put option at 39.The maturity date was June 18, and the risk-free rateat 4.56% (per year). Check the call-put-futures parity.Conclude. See the example on page 310.131


8.8 Pricing Options on Futures

A call (put) option on a futures contract gives its holderthe right (but not the obligation) to take a long (short)position in the underlying futures contract at the optionmaturity.

The call (put) option is exercised at maturity if, and onlyif, its strike price is lower (higher) than the futures price.The option is then settled in cash. While the call-optionholder is paid the di¤erence between the futures price andthe option strike price, the put-option holder is paid theopposite amount when the option expires in the money.

Black (1976) developed a variation of the Black and Sc-holes formula (1973) for pricing options on futures. Thefutures contract in the Black model plays the same roleas the stock in the Black and Scholes model.

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To alleviate the notation, the futures price at the optionstart f0 (T2) is indicated by f0, where T2 is the maturityof the futures contract. Black�s formula is as follows:

Ce = e�rcT1 [f0N (d1)�XN (d2)] ,

and

Pe = e�rcT1 [XN (�d2)� f0N (�d1)] ,

with

d1 =ln (f0=X) + �

2T1=2

�pT1

and d2 = d1 � �qT1,

where T1 (prior to T2) is the option maturity, X theoption strike price, rc is the risk-free rate, and � is thevolatility associated to the futures contract.

8.9 Assignment

Read the chapter, and give precise and concise de�ni-tions for the keywords. This part is to be prepared for

133


the exams, but should not be handed in as part of theassignment.

Answer questions no 10, 11, 14, 16, 19.

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9 Forward and Futures Hedging

9.1 Is Hedging Relevant?

Hedging is a trading activity designed to reduce risk. Wehave been discussing hedging strategies, and giving ex-amples such as covered calls and protective puts. Now,we focus on activities based on forward/futures contractsand their underlying assets.

Mainly, a hedger should take opposite positions in thespot and forward/futures markets. Indeed, the underlying-asset spot and forward/futures prices move in the samedirection, so that a loss (gain) recorded in the spot mar-ket is at least partially o¤set by a gain (loss) in the for-ward/futures market.

Hedging activities reduce the possibility of loss, but alsoof gain. They have no impact on the average of thepro�t function, but reduce its �uctuation. In addition,

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the value of a company is independent of any �nancialdecision, including hedging (Modigliani and Miller)! Ishedging relevant?

The answer is obviously, yes. The result by Modiglianiand Miller is true only in a frictionless market. Real activ-ities involve various frictions, including transaction costs,information asymmetries, and taxes.

Firms hedge to save taxes, improve credit terms, andreduce bankruptcy costs. Also, hedging a sporadic trans-action may be extremely e¤ective.

9.2 Hedging Design

A short hedge consists of a short position in the for-ward/futures market, possibly coupled with a long posi-tion in the spot market. A long hedge consists of a longposition in the forward/futures market, possibly coupledwith a short position in the spot market.

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The pro�t from a short position in the futures market anda long position in the spot market is:

�t = St � S0 � (ft � f0)= St � ft � (S0 � f0)= bt � b0= �b,

where bt = St � ft is the basis at time t.

The pro�t turns out to be the change in the basis. In thiscontext, analysing the risk is equivalent to analysing thebasis.


For a hedge to completely eliminate risk, there must bea perfect match between the hedging and the underly-ing asset, maturity, and size. This is a perfect hedge. Aperfect hedge can be achieved only by using a forwardcontract since the underlying asset, maturity, and sizecan be selected by the hedger at his discretion. Hedging

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activities based on forward contracts, however, are of-ten undesirable because forward markets are illiquid, andinvolve large transaction costs.

Hedging activities based on futures contracts are viablealternatives. Hedging strategies with futures contractsalso have their disadvantages.

First of all, the hedging asset may not be traded on thefutures markets; instead, a related underlying asset maybe used. Suppose one holds a corporate bond, and wantsto hedge against a rise in its yield to maturity. Futurescontracts on corporate bonds are not traded; however, fu-tures contracts on T-bonds can be used. Next, the hedg-ing maturity may not be known in advance, or may notmatch the trading maturities. The maturity of the futurescontract should be subsequent to the hedging maturity,so that the hedger can leave the market at the hedgingmaturity, just before the delivery month. This works for ashort-maturity hedge, but is less viable for longer hedgingmaturities. Indeed, the longer the maturity of a futures

138


contract, and the lower its liquidity. For a long-maturityhedge, hedgers often roll forward the hedge. Finally, thehedge size should be selected with care. The hedge ratiois the number of futures contracts used to hedge a givenexposure in the spot market.

To start with, assume that the size of the exposure and ofthe futures contract is one. The pro�t from a long posi-tion on one asset to be hedged and Nf futures contractsis:

�t = St � S0 +Nf (ft � f0)= �S +Nf�f .

The pro�t function is risky. Here, the asset to be hedgedis not necessarily the asset underlying the futures con-tract. The optimal hedge ratio N�f is the one that mini-mizes the variance of the pro�t function, which is:

Var [�]

= Var [�S] + VarhNf�f

i+ 2Cov

h�S, Nf�f

i= N2fVar [�f ] + 2NfCov [�S, �f ] + Var [�S] .

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The variance of the pro�t function can be seen as a func-tion of the unknown parameter nf . Check that the pro�tis a convex function that reaches its minimum at:

N�f = �Cov [�S, �f ]Var [�f ]

= �� [�S, �f ] � [�S]� [�f ]

,

where � [�S;�f ] is the correlation between �S and�f , � [�S] the standard deviation of �S, and � [�f ]the standard deviation of�f . These coe¢ cients are esti-mated using their sample counterparts. This formula maybe expressed in di¤erent ways, depending on the asset tobe hedged and the asset underlying the futures contract.

Now, given the size of the futures contract, the numberof futures contracts required to hedge a given exposureis obtained by computing the quantity:

N�f �Size of the exposure

Size of the futures contract,

and rounding it to the �rst whole number.

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Example 46: A company knows it will buy 1 milliongallons of jet fuel in three months. The standard devia-tion of the three-month change in the price of jet oil is0.032. The company chooses to hedge by buying futurescontracts on heating oil. The standard deviation of thechange in the futures price is 0.04. The correlation be-tween the three-month change in the price of jet fuel andthe three-month change in the futures price is 0.8. Com-pute the optimal hedge ratio and the number of futurescontracts to be used. Each futures contract on heatingoil is on 42,000 gallons.

The correlation � between two variables measures theirlinear association, and indicates how much their realisa-tions lie along the same line. The correlation � belongs tothe interval [�1, 1], and must be interpreted as follows:

� =

8>>>>>><>>>>>>:

around + 1! stronglinear increasing relationshiparound 0! no linear relationshiparound � 1! stronglinear decreasing relationship

.

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The coe¢ cient � [�S;�f ] is expected to be positivesince the asset underlying the futures contract should beclosely related to the asset to be hedged. A negative signof N�f shows that the hedge is achieved by a short po-sition in the futures market in case of a long position inthe spot market. A long hedge is required in case of ashort position in the spot market.

Example 47: Discuss examples for � around 1, 0, and�1.

9.3 Hedging T-Bonds and Stock Portfo-

lios

Consider a position on a T-bond to be hedged by anopposite position on a T-bond futures contract. In thiscontext, a rise (fall) in the term-structure of interest rates

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induces a fall (rise) in the bond�s price. The current T-bond�s price is:

B =NXn=1

CFn(1 + y)tn

,

where y is the bond�s yield to maturity (e¤ective rate in% per year), and CFn (in $) the cash �ow made at timetn (in years). One has:

@B

@y=

NXn=1

CFn(1 + y)tn�1

� (�tn)

= � 1

1 + y

NXn=1

CFn(1 + y)tn

� tn

= � B

1 + y

NXn=1

CFn= (1 + y)tn

B� tn

= � B

1 + y

NXn=1

!ntn,

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where the weights !n, for n = 1; : : : ; N , are non nega-tive and verify

NXn=1

!n = 1.

The duration of the bond, de�ned as:

D =NXn=1

!ntn,

is a weighted average of the time to each cash-�ow pay-ment.

Finally, for a small change in y, one has:

@B

@y= lim

�y!0�B

�y

� �B

�y

= � B

1 + yD.

Said in other words, one has:

�B=B

�y� � D

1 + y= MDB,

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where MDB is known as the modi�ed duration of thebond.

Along the same lines, we de�ne the modi�ed duration ofthe T-bond futures contract as:

MDf =�f=f

�y,

where �y is the change in the yield to maturity of theT-bond to be hedged.

The optimal hedge ratio is:

N�f = �Cov [�B, �f ]Var [�f ]

= �Cov [�B=B, �f=f ]�B � fVar [�f=f ]� f2

= �Bf

CovhMDB�y, MDf�y

iVar

hMDf�y

i= �B

f� MDBMDf

.

This is the price-sensitivity hedge ratio.145


Example 48: A portfolio manager holds $1 million facevalue of a T-bond. The bond is currently priced 101per $100 par value, and has a modi�ed duration of 7.83(years). The manager plans to sell the bond in one monthto meet an obligation, and is concerned by an increasein interest rates. A short position on the T-bond futurescontract maturing in four months is used. Its quotedprice is at 70 16/32, and implied modi�ed duration at7.2 (years). Give the price-sensitivity hedge ratio. SeeFigure 11.7 on page 376.

Now, consider a position on a stock portfolio to be hedgedby an opposite position on a stock index futures. assumethat the stock portfolio is not well diversi�ed, while thestock index underlying the futures contract is diversi�edenough to be considered as a market proxy.

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The optimal hedge ratio is:

�Cov [�S;�f ]Var [�f ]

= �Cov [�S=S;�f=f ]� S � fVar [�f=f ]� f2

= �Sf� Cov [�S=S � r;�f=f � r]

Var [�f=f � r]

� �Sf� Cov [�S=S � r;�I=I � r]

Var [�I=I � r]

= �Sf� �,

where r is the risk-free rate, I the index underlying thefutures contract, and � is the beta of the portfolio. Thelatter parameter indicates the sensitivity of the excessreturn on the asset to be hedged to a change in the excessreturn on the market portfolio.

Example 49: A company wishes to hedge a portfolioworth $2,100,000 over the next three months using theS&P 500 index futures contract maturing in four months.The S&P 500 index is quoted 900, and has a multiplierof 250. The beta of the portfolio is 1.5. Give the optimalhedge ratio. How can the company reduce the beta ofthe portfolio to 0.75? 147


9.4 Assignment

Read the chapter, and give precise and concise de�ni-tions for the keywords. This part is to be prepared forthe exams, but should not be handed in as part of theassignment.

Exercise (worth an extra 5% in the second midtermexam): Consider the variance of the pro�t from a longposition in the spot market and a short position in thefutures market, seen as a function of Nf 2 R:

Var [�] = f�Nf

�.

The parameters � [�S;�f ], � [�S], and � [�f ] are�xed at 0.6, 0.2, and 0.1, respectively. Draw the curveof Var[�]. Describe the shape of the curve. Locate theoptimal hedge ratio, and check its consistency with theformula. Answer the same questions, for � [�S;�f ] 2f0:8; 0:6; 0:4;�0:4;�0:6;�0:8g. Interpret your result.

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10 Vanilla Interest-Rate Swaps

10.1 Design

A swap commits two parties to exchanging series of pay-ments or assets in the future, in accordance with speci�edrules and schedule. There are swaps on interest rates, ex-change rates, equities, and commodities.

Swaps are traded in over-the-counter markets, and de-signed in several ways to match clients� needs. In thefollowing, we focus on swaps on interest rates, which arethe most-traded swap contracts.

A vanilla interest-rate swap commits two parties to ex-changing interest payments on a regular basis. The swappayer commits to paying interest based on a �xed rate,known as the swap rate, and the swap receiver to payinginterest based on a �oating rate. Only the net paymentsare exchanged. No payment is made up front. The swap

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rate is negotiated so that the value of both parties is nullat the start.

A vanilla swap is characterized by its starting date, settle-ment dates (including its maturity), tenor or settlementperiod, notional principal, �oating rate, and �xed rate.

Example 50: A �rm enters a swap as a payer with thefollowing parameters: a notional principal of $50 million,settlement period of 3 months, and maturity of one year.The �oating rate is the 3-month LIBOR rate, and the�xed rate is 7.5% (per year). The swap is settled inarrears, that is, the interest payment at time tn is basedon the �oating rate observed at time tn�1, which coversthe period [tn�1, tn]. The 3-month LIBOR rate and theswap rate are nominal rates, both expressed in % peryear. They assume here that the interest is compoundedquarterly. Draw a �gure for the cash-�ow stream, fromthe perspective of the swap payer.

A swap payer can exchange a �oating-rate loan for a�xed-rate loan, and thus hedge against interest-rate �uc-tuations.

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Example 50 (continued): Explain Table 12.1 on page410.

10.2 Pricing

The position of a swap payer can be seen as a portfo-lio consisting of a long position on a �oating-rate bondand a short position on a �xed-rate bond, both with thesame maturity, settlement dates, and principal as thoseof the swap. The �oating-rate bond pays coupons at the�oating-rate of the swap, and the �xed-rate bond payscoupons at the swap rate.

The discount factor for the horizon tn, B0 (tn), is theprice at time 0 of $1 to be received at time tn. This isthe price at time 0 of a zero-coupon bond maturing attime tn.

In the LIBOR market, the discount factors are related tothe term-structure of interest rates as follows:

B0 (tn) =1

1 + L0 (tn)� tn, for n = 1; : : : ; N ,

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where L0 (tn) (in % per year) is the rate of return asso-ciated to the zero-coupon bond maturing at tn. In thefollowing, we assume that the LIBOR term-structure ofinterest rates is given, as are the discount factors.

The value of the swap at the start is:

v0 = 1�NXn=1

R��t�B0 (tn) +B0 (tN)

= 0

where R is the swap rate (in % per year), t1; : : : ; tN arethe settlement dates (including the swap maturity), and�t is the tenor. Solving this equation for R gives:

R =1

�t� 1�B0 (T )PN

n=1B0 (tn).

Example 51: Explain Table 12.2.

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