1.F I F T H E D I T I O NJOHN C.HULL

2. PRENTICE HALL FINANCE SERIESPersonal FinanceKeown, Personal Finance: Turning Monev into Wealth, Second EditionTrivoli, Personal Portfolio Management: Fundamentals & StrategiesWinger/Frasca, Personal Finance: An Integrated Planning Approach, Sixth EditionUndergraduate Investments/Portfolio ManagementAlexander/Sharpe/Bailey, Fundamentals of Investments, Third EditionFabozzi, Investment Management, Second EditionHaugen, Modern Investment Theory, Fifth EditionHaugen, The New Finance, Second EditionHaugen, The Beast on Wall StreetHaugen, The Inefficient Stock Market, Second EditionHolden, Spreadsheet Modeling: A Book and CD-ROM Series (Available in Graduate and Undergraduate Versions)Nofsinger. The Psychology of InvestingTaggart, Quantitative Analysis for Investment ManagementWinger/Frasca, Investments, Third EditionGraduate Investments/Portfolio ManagementFischer/Jordan, Security Analysis and Portfolio Management, Sixth EditionFrancis/Ibbotson. 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Fifth Edition OPTIONS, FUTURES,& OTHER DERIVATIVES John C. HullMaple Financial Group Professor of Derivatives and Risk ManagementDirector, Bonham Center for Finance Joseph L. Rotman School of ManagementUniversity of Toronto PrenticeHallP R E N T I C E H A L L , U P P E R S A D D L E R I V E R , N E W JERSEY 0 7 4 5 8 4. CONTENTSPreface xix1. Introduction1 1.1Exchange-traded markets1 1.2Over-the-counter markets 2 1.3Forward contracts2 1.4Futures contracts5 1.5Options6 1.6Types of traders10 1.7Other derivatives 14Summary 15Questions and problems16Assignment questions172. Mechanics of futures markets 19 2.1 Trading futures contracts19 2.2 Specification of the futures contract20 2.3 Convergence of futures price to spot price 23 2.4 Operation of margins 24 2.5 Newspaper quotes 27 2.6Keynes and Hicks31 2.7Delivery31 2.8Types of traders32 2.9Regulation; 332.10Accounting and tax352.11 Forward contracts vs. futures contracts36Summary 37Suggestions for further reading 38Questions and problems38Assignment questions403. Determination of forward and futures prices413.1 Investment assets vs. consumption assets413.2 Short selling 413.3 Measuring interest rates423.4 Assumptions and notation443.5Forward price for an investment asset453.6 Known income473.7 Known yield 493.8 Valuing forward contracts 493.9 Are forward prices and futures prices equal?51 3.10 Stock index futures 52 3.11Forward and futures contracts on currencies55 3.12 Futures on commodities58 ix 5. Contents 3.13 Cost of carry 60 3.14 Delivery options60 3.15 Futures prices and the expected future spot price 61Summary 63Suggestions for further reading 64Questions and problems65Assignment questions67Appendix 3A: Proof that forward and futures prices are equal when interest rates are constant684. Hedging strategies using futures704.1 Basic principles 704.2 Arguments for and against hedging724.3 Basisrisk754.4 Minimum variance hedge ratio 784.5 Stock index futures824.6 Rolling the hedge forward86 Summary 87 Suggestions for further reading 88 Questions and problems88 Assignment questions90 Appendix 4A: Proof of the minimum variance hedge ratio formula925. Interest rate markets 935.1 Types of rates 935.2 Zero rates 945.3 Bond pricing 945.4 Determining zero rates 965.5 Forward rates985.6 Forward rate agreements 1005.7 Theories of the term structure1025.8 Day count conventions 1025.9 Quotations103 5.10 Treasury bond futures 104 5.11 Eurodollar futures110 5.12 The LIBOR zero curveIll 5.13 Duration112 5.14 Duration-based hedging strategies 116Summary 118Suggestions for further reading 119Questions and problems120Assignment questions1236. Swaps1256.1 Mechanics of interest rate swaps1256.2 The comparative-advantage argument1316.3 Swap quotes and LIBOR zero rates1346.4 Valuation of interest rate swaps1366.5 Currency swaps1406.6 Valuation of currency swaps 1436.7 Credit risk 145Summary 146Suggestions for further reading 147Questions and problems147Assignment questions149 6. Contentsxi7. Mechanics ofoptions markets 151 7.1 Underlying assets 151 7.2Specification of stock options 152 7.3 Newspaper quotes155 7.4 Trading 157 7.5 Commissions 157 7.6 Margins 158 7.7 The options clearing corporation160 7.8 Regulation161 7.9 Taxation1617.10 Warrants, executive stock options, and convertibles 1627.11 Over-the-counter markets163 Summary 163 Suggestions for further reading 164 Questions and problems164 Assignment questions1658. Properties of stock options 167 8.1Factors affecting option prices167 8.2Assumptions and notation 170 8.3Upper and lower bounds for option prices 171 8.4Put-call parity174 8.5Early exercise: calls on a non-dividend-paying stock 175 8.6Early exercise: puts on a non-dividend-paying stock177 8.7Effect of dividends178 8.8Empirical research 179Summary180Suggestions for further reading181Questions and problems 182Assignment questions 1839. Trading strategies involving options185 9.1 Strategies- involving a single option and a stock 185 9.2 Spreads 187 9.3 Combinations194 9.4 Other payoffs 197 Summary 197 Suggestions for further reading 198 Questions and problems198 Assignment questions19910. Introduction to binomial trees 20010.1A one-step binomial model20010.2 Risk-neutral valuation20310.3 Two-step binomial trees 20510.4 A put example 20810.5 American options20910.6 Delta 21010.7 Matching volatility with u and d21110.8 Binomial trees in practice212 Summary 213 Suggestions for further reading 214 Questions and problems214 Assignment questions215 7. xii Contents11. A model of the behavior of stock prices 21611.1 The Markov property21611.2 Continuous-time stochastic processes 21711.3 The process for stock prices 22211.4 Review of the model22311.5 The parameters 22511.6 Itos lemma22611.7 The lognormal property 227Summary 228Suggestions for further reading 229Questions and problems229Assignment questions230Appendix 11 A: Derivation of Itos lemma23212. The Black-Scholes model234 12.1 Lognormal property of stock prices 234 12.2 The distribution of the rate of return 236 12.3 The expected return237 12.4 Volatility 238 12.5 Concepts underlying the Black-Scholes-Merton differential equation 241 12.6 Derivation of the Black-Scholes-Merton differential equation 242 12.7 Risk-neutral valuation 244 12.8 Black-Scholes pricing formulas 246 12.9 Cumulative normal distribution function24812.10 Warrants issued by a company on its own stock24912.11 Implied volatilities 25012.12 The causes of volatility 25112.13 Dividends252Summary256Suggestions for further reading257Questions and problems 258 Assignment questions261Appendix 12A: Proof of Black-Scholes-Merton formula262 Appendix 12B: Exact procedure for calculating the values of American calls ondividend-paying stocks 265 Appendix 12C: Calculation of cumulative probability in bivariate normaldistribution 26613. Options on stock indices, currencies, and futures267 13.1Results for a stock paying a known dividend yield 267 13.2Option pricing formulas 268 13.3Options on stock indices270 13.4Currency options276 13.5Futures options 278 13.6Valuation of futures options using binomial trees 284 13.7Futures price analogy 286 13.8Blacks model for valuing futures options 287 13.9Futures options vs. spot options288 Summary 289 Suggestions for further reading 290 Questions and problems291 Assignment questions294 Appendix 13 A: Derivation of differential equation satisfied by a derivativedependent on a stock providing a dividend yield295 8. Contents xiiiAppendix 13B: Derivation of differential equation satisfied by a derivativedependent on a futures price 29714. The Greek letters299 14.1 Illustration 299 14.2 Naked and covered positions300 14.3 A stop-loss strategy 300 14.4 Delta hedging302 14.5 Theta309 14.6 Gamma312 14.7 Relationship between delta, theta, and gamma 315 14.8 Vega 316 14.9 Rho31814.10 Hedging in practice31914.11 Scenario analysis31914.12 Portfolio insurance32014.13 Stock market volatility323Summary323Suggestions for further reading324Questions and problems : 326Assignment questions 327Appendix 14A: Taylor series expansions and hedge parameters32915. Volatility smiles33015.1 Put-call parity revisited 33015.2 Foreign currency options33115.3 Equity options33415.4 The volatility term structure and volatility surfaces 33615.5 Greek letters 33715.6 When a single large jump is anticipated 33815.7 Empirical research339Summary341Suggestions for further reading341Questions and problems 343Assignment questions 344Appendix 15A: Determining implied risk-neutral distributions from volatilitysmiles 34516. Value at risk346 16.1 The VaR measure346 16.2 Historical simulation348 16.3 Model-building approach350 16.4 Linear model 352 16.5 Quadratic model356 16.6 Monte Carlo simulation 359 16.7 Comparison of approaches 359 16.8 Stress testing and back testing360 16.9 Principal components analysis360Summary364Suggestions for further reading364Questions and problems 365Assignment questions 366Appendix 16A: Cash-flow mapping368Appendix 16B: Use of the Cornish-Fisher expansion to estimate VaR370 9. xivContents17. Estimating volatilities and correlations 372 17.1 Estimating volatility372 17.2 The exponentially weighted moving average model374 17.3 The GARCH(1,1) model 376 17.4 Choosing between the models377 17.5 Maximum likelihood methods 378 17.6 Using GARCHfl, 1) to forecast future volatility382 17.7 Correlations 385 Summary 388 Suggestions for further reading 388 Questions and problems389 Assignment questions39118. Numerical procedures 39218.1 Binomial trees39218.2 Using the binomial tree for options on indices, currencies, and futurescontracts39918.3 Binomial model for a dividend-paying stock . 40218.4 Extensions to the basic tree approach 40518.5 Alternative procedures for constructing trees 40618.6 Monte Carlo simulation41018.7 Variance reduction procedures 41418.8 Finite difference methods 41818.9 Analytic approximation to American option prices427Summary427Suggestions for further reading428Questions and problems 430Assignment questions 432Appendix 18A: Analytic approximation to American option prices ofMacMillan and of Barone-Adesi and Whaley 43319. Exotic options 43519.1 Packages43519.2 Nonstandard American options43619.3 Forward start options 43719.4 Compound options43719.5 Chooser options 43819.6 Barrier options 43919.7 Binary options44119.8 Lookback options44119.9 Shout options 44319.10 Asian options44319.11Options to exchange one asset for another 44519.12 Basket options 44619.13 Hedging issues 44719.14 Static options replication 447 Summary 449 Suggestions for further reading 449 Questions and problems451 Assignment questions452 Appendix 19A: Calculation of the first two moments of arithmetic averages and baskets 45420. More on models and numerical procedures456 20.1 The CEV model456 20.2 The jump diffusion model 457 10. Contents xv 20.3 Stochastic volatility models458 20.4 The IVF model 460 20.5 Path-dependent derivatives461 20.6 Lookback options465 20.7 Barrier options 467 20.8 Options on two correlated assets472 20.9 Monte Carlo simulation and American options 474Summary 478Suggestions for further reading 479Questions and problems480Assignment questions48121. Martingales and measures483 21.1The market price of risk 484 21.2 Several state variables487 21.3 Martingales 488 21.4 Alternative choices for the numeraire 489 21.5Extension to multiple independent factors492 21.6 Applications493 21.7 Change of numeraire 495 21.8 Quantos 497 21.9 Siegels paradox499Summary 500Suggestions for further reading 500Questions and problems501Assignment questions502Appendix 21 A: Generalizations of Itos lemma 504Appendix 2IB: Expected excess return when there are multiple sources of uncertainty50622. Interest rate derivatives: the standard market models 50822.1Blacks model 50822.2Bond options51122.3Interest rate caps51522.4European swap options 52022.5Generalizations 52422.6Convexity adjustments 52422.7Timing adjustments52722.8Natural time lags 52922.9Hedging interest rate derivatives 530Summary 531Suggestions for further reading 531Questions and problems532Assignment questions534Appendix 22A: Proof of the convexity adjustment formula 53623. Interest rate derivatives: models of the short rate 53723.1Equilibrium models53723.2One-factor equilibrium models 53823.3The Rendleman and Bartter model 53823.4The Vasicek model 53923.5The Cox, Ingersoll, and Ross model54223.6Two-factor equilibrium models 54323.7No-arbitrage models 54323.8The Ho and Lee model54423.9The Hull and White model546 11. xviContents 23.10Options on coupon-bearing bonds 549 23.11Interest rate trees 550 23.12A general tree-building procedure 552 23.13Nonstationary models563 23.14Calibration 564 23.15Hedging using a one-factor model565 23.16Forward rates and futures rates 566Summary 566Suggestions for further reading 567Questions and problems568Assignment questions57024. Interest rate derivatives: more advanced models57124.1 Two-factor models of the short rate 57124.2The Heath, Jarrow, and Morton model57424.3The LIBOR market model 57724.4 Mortgage-backed securities586Summary588Suggestions for further reading589Questions and problems 590Assignment questions 591Appendix 24A: The A(t, T), aP, and 0(t) functions in the two-factor Hull-Whitemodel59325. Swaps revisited 594 25.1 Variations on the vanilla deal594 25.2 Compounding swaps 595 25.3 Currency swaps598 25.4 More complex swaps598 25.5 Equity swaps601 25.6 Swaps with embedded options 602 25.7 Other swaps 605 25.8 Bizarre deals 605 Summary606 Suggestions for further reading606 Questions and problems 607 Assignment questions 607 Appendix 25A: Valuation of an equity swap between payment dates60926. Credit risk 610 23.1 Bond prices and the probability of default610 26.2 Historical data 619 26.3 Bond prices vs. historical default experience 619 26.4 Risk-neutral vs. real-world estimates 620 26.5 Using equity prices to estimate default probabilities 621 26.6 The loss given default623 26.7 Credit ratings migration626 26.8 Default correlations627 26.9 Credit value at risk630Summary 633Suggestions for further reading 633Questions and problems634Assignment questions635Appendix 26A: Manipulation of the matrices of credit rating changes 636 12. Contentsxvn27. Credit derivatives63727.1Credit default swaps63727.2 Total return swaps 64427.3 Credit spread options64527.4 Collateralized debt obligations64627.5 Adjusting derivative prices for default risk 64727.6 Convertible bonds652 Summary655 Suggestions for further reading655 Questions and problems 656 Assignment questions 65828. Real options66028.1Capital investment appraisal66028.2Extension of the risk-neutral valuation framework 66128.3 Estimating the market price of risk66528.4 Application to the valuation of a new business 66628.5 Commodity prices 66728.6 Evaluating options in an investment opportunity670Summary 675Suggestions for further reading 676 Questions and problems 676 Assignment questions 67729. Insurance, weather, and energy derivatives67829.1 Review of pricing issues 678 29.2 Weather derivatives 679 29.3 Energy derivatives680 29.4 Insurance derivatives 682 Summary683 Suggestions for further reading684 Questions and problems 684Assignment questions68530. Derivatives mishaps and what we can learn from them 68630.1Lessons for all users of derivatives68630.2 Lessons for financial institutions 69030.3Lessons for nonfinancial corporations 693Summary 694Suggestions for further reading 695Glossary of notation697Glossary of terms 700DerivaGem software: 715Major exchanges trading futures and options 720Table for N{x) when x sj 0722Table for N(x) when x ^ 0 723Author index725Subject index 729 13. PREFACEIt is sometimes hard for me to believe that the first edition of this book was only 330 pages and13 chapters long! There have been many developments in derivatives markets over the last 15 yearsand the book has grown to keep up with them. The fifth edition has seven new chapters that covernew derivatives instruments and recent research advances. Like earlier editions, the book serves several markets. It is appropriate for graduate courses inbusiness, economics, and financial engineering. It can be used on advanced undergraduate courseswhen students have good quantitative skills. Also, many practitioners who want to acquire aworking knowledge of how derivatives can be analyzed find the book useful. One of the key decisions that must be made by an author who is writing in the area of derivativesconcerns the use of mathematics. If the level of mathematical sophistication is too high, thematerial is likely to be inaccessible to many students and practitioners. If it is too low, someimportant issues will inevitably be treated in a rather superficial way. I have tried to be particularlycareful about the way I use both mathematics and notation in the book. Nonessential mathema-tical material has been either eliminated or included in end-of-chapter appendices. Concepts thatare likely to be new to many readers have been explained carefully, and many numerical exampleshave been included. The book covers both derivatives markets and risk management. It assumes that the reader hastaken an introductory course in finance and an introductory course in probability and statistics.No prior knowledge of options, futures contracts, swaps, and so on is assumed. It is not thereforenecessary for students to take an elective course in investments prior to taking a course based onthis book. There are many different ways the book can be used in the classroom. Instructorsteaching a first course in derivatives may wish to spend most time on the first half of the book.Instructors teaching a more advanced course will find that many different combinations of thechapters in the second half of the book can be used. I find that the material in Chapters 29 and 30works well at the end of either an introductory or an advanced course.Whats New?Material has been updated and improved throughout the book. The changes in this editioninclude: 1. A new chapter on the use of futures for hedging (Chapter 4). Part of this material waspreviously in Chapters 2 and 3. The change results in the first three chapters being lessintense and allows hedging to be covered in more depth. 2. A new chapter on models and numerical procedures (Chapter 20). Much of this material isnew, but some has been transferred from the chapter on exotic options in the fourth edition.xix 14. xxPreface 3. A new chapter on swaps (Chapter 25). This gives the reader an appreciation of the range ofnonstandard swap products that are traded in the over-the-counter market and discusses howthey can be valued. 4. There is an extra chapter on credit risk. Chapter 26 discusses the measurement of credit riskand credit value at risk while Chapter 27 covers credit derivatives. 5. There is a new chapter on real options (Chapter 28). 6. There is a new chapter on insurance, weather, and energy derivatives (Chapter 29). 7. There is a new chapter on derivatives mishaps and what we can learn from them (Chapter 30). 8. The chapter on martingales and measures has been improved so that the material flowsbetter (Chapter 21). 9. The chapter on value at risk has been rewritten so that it provides a better balance betweenthe historical simulation approach and the model-building approach (Chapter 16). 10. The chapter on volatility smiles has been improved and appears earlier in the book. (Chapter 15). 11. The coverage of the LIBOR market model has been expanded (Chapter 24). 12. One or two changes have been made to the notation. The most significant is that the strike price is now denoted by K rather than X. 13. Many new end-of-chapter problems have been added.SoftwareA new version of DerivaGem (Version 1.50) is released with this book. This consists of two Excelapplications: the Options Calculator and the Applications Builder. The Options Calculator consistsof the software in the previous release (with minor improvements). The Applications Builderconsists of a number of Excel functions from which users can build their own applications. Itincludes a number of sample applications and enables students to explore the properties of optionsand numerical procedures more easily. It also allows more interesting assignments to be designed.The software is described more fully at the end of the book. Updates to the software can bedownloaded from my website: www.rotman.utoronto.ca/~hullSlidesSeveral hundred PowerPoint slides can be downloaded from my website. Instructors who adopt thetext are welcome to adapt the slides to meet their own needs.Answers to QuestionsAs in the fourth edition, end-of-chapter problems are divided into two groups: "Questions andProblems" and "Assignment Questions". Solutions to the Questions and Problems are in Options,Futures, and Other Derivatives: Solutions Manual, which is published by Prentice Hall and can bepurchased by students. Solutions to Assignment Questions are available only in the InstructorsManual. 15. PrefacexxiA cknowledgmentsMany people have played a part in the production of this book. Academics, students, andpractitioners who have made excellent and useful suggestions include Farhang Aslani, Jas Badyal,Emilio Barone, Giovanni Barone-Adesi, Alex Bergier, George Blazenko, Laurence Booth, PhelimBoyle, Peter Carr, Don Chance, J.-P. Chateau, Ren-Raw Chen, George Constantinides, MichelCrouhy, Emanuel Derman, Brian Donaldson, Dieter Dorp, Scott Drabin, Jerome Duncan, SteinarEkern, David Fowler, Louis Gagnon, Dajiang Guo, Jrgen Hallbeck, Ian Hawkins, MichaelHemler, Steve Heston, Bernie Hildebrandt, Michelle Hull, Kiyoshi Kato, Kevin Kneafsy, TiborKucs, Iain MacDonald, Bill Margrabe, Izzy Nelkin, Neil Pearson, Paul Potvin, Shailendra Pandit,Eric Reiner, Richard Rendleman, Gordon Roberts, Chris Robinson, Cheryl Rosen, John Rumsey,Ani Sanyal, Klaus Schurger, Eduardo Schwartz, Michael Selby, Piet Sercu, Duane Stock, EdwardThorpe, Yisong Tian, P. V. Viswanath, George Wang, Jason Wei, Bob Whaley, Alan White,Hailiang Yang, Victor Zak, and Jozef Zemek. Huafen (Florence) Wu and Matthew Merkleyprovided excellent research assistance.I am particularly grateful to Eduardo Schwartz, who read the original manuscript for the firstedition and made many comments that led to significant improvements, and to Richard Rendle-man and George Constantinides, who made specific suggestions that led to improvements in morerecent editions.The first four editions of this book were very popular with practitioners and their comments andsuggestions have led to many improvements in the book. The students in my elective courses onderivatives at the University of Toronto have also influenced the evolution of the book.Alan White, a colleague at the University of Toronto, deserves a special acknowledgment. Alanand I have been carrying out joint research in the area of derivatives for the last 18 years. Duringthat time we have spent countless hours discussing different issues concerning derivatives. Many ofthe new ideas in this book, and many of the new ways used to explain old ideas, are as much Alansas mine. Alan read the original version of this book very carefully and made many excellentsuggestions for improvement. Alan has also done most of the development work on the Deriva-Gem software. Special thanks are due to many people at Prentice Hall for their enthusiasm, advice, andencouragement. I would particularly like to thank Mickey Cox (my editor), P. J. Boardman (theeditor-in-chief) and Kerri Limpert (the production editor). I am also grateful to Scott Barr, LeahJewell, Paul Donnelly, and Maureen Riopelle, who at different times have played key roles in thedevelopment of the book.I welcome comments on the book from readers. My email address is:hull@rotman.utoronto.caJohn C. Hull University of Toronto 16. C H A P T E R 1INTRODUCTIONIn the last 20 years derivatives have become increasingly important in the world of finance. Futuresand options are now traded actively on many exchanges throughout the world. Forward contracts,swaps, and many different types of options are regularly traded outside exchanges by financialinstitutions, fund managers, and corporate treasurers in what is termed the over-the-countermarket. Derivatives are also sometimes added to a bond or stock issue. A derivative can be defined as a financial instrument whose value depends on (or derives from)the values of other, more basic underlying variables. Very often the variables underlying deriva-tives are the prices of traded assets. A stock option, for example, is a derivative whose value isdependent on the price of a stock. However, derivatives can be dependent on almost any variable,from the price of hogs to the amount of snow falling at a certain ski resort. Since the first edition of this book was published in 1988, there have been many developments inderivatives markets. There is now active trading in credit derivatives, electricity derivatives, weatherderivatives, and insurance derivatives. Many new types of interest rate, foreign exchange, andequity derivative products have been created. There have been many new ideas in risk managementand risk measurement. Analysts have also become more aware of the need to analyze what areknown as real options. (These are the options acquired by a company when it invests in real assetssuch as real estate, plant, and equipment.) This edition of the book reflects all these developments. In this opening chapter we take a first look at forward, futures, and options markets and providean overview of how they are used by hedgers, speculators, and arbitrageurs. Later chapters will givemore details and elaborate on many of the points made here.1.1EXCHANGE-TRADED MARKETSA derivatives exchange is a market where individuals trade standardized contracts that have beendefined by the exchange. Derivatives exchanges have existed for a long time. The Chicago Board ofTrade (CBOT, www.cbot.com) was established in 1848 to bring farmers and merchants together.Initially its main task was to standardize the quantities and qualities of the grains that were traded.Within a few years the first futures-type contract was developed. It was known as a to-arrivecontract. Speculators soon became interested in the contract and found trading the contract to bean attractive alternative to trading the grain itself. A rival futures exchange, the ChicagoMercantile Exchange (CME, www.cme.com), was established in 1919. Now futures exchangesexist all over the world.The Chicago Board Options Exchange (CBOET www.cboe.com) started trading call option 17. CHAPTER 1contracts on 16 stocks in 1973. Options had traded prior to 1973 but the CBOE succeeded increating an orderly market with well-defined contracts. Put option contracts started trading on theexchange in 1977. The CBOE now trades options on over 1200 stocks and many different stockindices. Like futures, options have proved to be very popular contracts. Many other exchangesthroughout the world now trade options. The underlying assets include foreign currencies andfutures contracts as well as stocks and stock indices.Traditionally derivatives traders have met on the floor of an exchange and used shouting and acomplicated set of hand signals to indicate the trades they would like to carry out. This is known asthe open outcry system. In recent years exchanges have increasingly moved from the open outcrysystem to electronic trading. The latter involves traders entering their desired trades at a keyboardand a computer being used to match buyers and sellers. There seems little doubt that eventually allexchanges will use electronic trading.1.2 OVER-THE-COUNTER MARKETSNot all trading is done on exchanges. The over-the-counter market is an important alternative toexchanges and, measured in terms of the total volume of trading, has become much larger than theexchange-traded market. It is a telephone- and computer-linked network of dealers, who do notphysically meet. Trades are done over the phone and are usually between two financial institutionsor between a financial institution and one of its corporate clients. Financial institutions often act asmarket makers for the more commonly traded instruments. This means that they are alwaysprepared to quote both a bid price (a price at which they are prepared to buy) and an offer price(a price at which they are prepared to sell). Telephone conversations in the over-the-counter market are usually taped. If there is a disputeabout what was agreed, the tapes are replayed to resolve the issue. Trades in the over-the-countermarket are typically much larger than trades in the exchange-traded market. A key advantage ofthe over-the-counter market is that the terms of a contract do not have to be those specified by anexchange. Market participants are free to negotiate any mutually attractive deal. A disadvantage isthat there is usually some credit risk in an over-the-counter trade (i.e., there is a small risk that thecontract will not be honored). As mentioned earlier, exchanges have organized themselves toeliminate virtually all credit risk.1.3 FORWARD CONTRACTSA forward contract is a particularly simple derivative. It is an agreement to buy or sell an asset at acertain future time for a certain price. It can be contrasted with a spot contract, which is anagreement to buy or sell an asset today. A forward contract is traded in the over-the-countermarketusually between two financial institutions or between a financial institution and one of itsclients. One of the parties to a forward contract assumes a long position and agrees to buy the underlyingasset on a certain specified future date for a certain specified price. The other party assumes a shortposition and agrees to sell the asset on the same date for the same price. Forward contracts on foreign exchange are very popular. Most large banks have a "forwarddesk" within their foreign exchange trading room that is devoted to the trading of forward 18. IntroductionTable 1.1 Spot and forward quotes for the USD-GBP exchangerate, August 16, 2001 (GBP = British pound; USD = U.S. dollar)Bid OfferSpot 1.44521.44561-month forward1.44351.44403-month forward1.44021.44076-month forward1.43531.43591-year forward 1.42621.4268contracts. Table 1.1 provides the quotes on the exchange rate between the British pound (GBP) andthe U.S. dollar (USD) that might be made by a large international bank on August 16, 2001. Thequote is for the number of USD per GBP. The first quote indicates that the bank is prepared to buyGBP (i.e., sterling) in the spot market (i.e., for virtually immediate delivery) at the rate of $1.4452per GBP and sell sterling in the spot market at $1.4456 per GBP. The second quote indicates thatthe bank is prepared to buy sterling in one month at $1.4435 per GBP and sell sterling in one monthat $1.4440 per GBP; the third quote indicates that it is prepared to buy sterling in three months at$1.4402 per GBP and sell sterling in three months at $1.4407 per GBP; and so on. These quotes arefor very large transactions. (As anyone who has traveled abroad knows, retail customers face muchlarger spreads between bid and offer quotes than those in given Table 1.1.) Forward contracts can be used to hedge foreign currency risk. Suppose that on August 16, 2001,the treasurer of a U.S. corporation knows that the corporation will pay 1 million in six months (onFebruary 16, 2002) and wants to hedge against exchange rate moves. Using the quotes in Table 1.1,the treasurer can agree to buy 1 million six months forward at an exchange rate of 1.4359. Thecorporation then has a long forward contract on GBP. It has agreed that on February 16, 2002, itwill buy 1 million from the bank for $1.4359 million. The bank has a short forward contract onGBP. It has agreed that on February 16, 2002, it will sell 1 million for $1.4359 million. Both sideshave made a binding commitment.Payoffs from Forward ContractsConsider the position of the corporation in the trade we have just described. What are the possibleoutcomes? The forward contract obligates the corporation to buy 1 million for $1,435,900. If thespot exchange rate rose to, say, 1.5000, at the end of the six months the forward contract would beworth $64,100 (= $1,500,000 - $1,435,900) to the corporation. It would enable 1 million to bepurchased at 1.4359 rather than 1.5000. Similarly, if the spot exchange rate fell to 1.4000 at the end ofthe six months, the forward contract would have a negative value to the corporation of $35,900because it would lead to the corporation paying $35,900 more than the market price for the sterling.In general, the payoff from a long position in a forward contract on one unit of an asset isST-Kwhere K is the delivery price and ST is the spot price of the asset at maturity of the contract. This isbecause the holder of the contract is obligated to buy an asset worth ST for K. Similarly, the payofffrom a short position in a forward contract on one unit of an asset isK-ST 19. CHAPTER 1Figure 1.1 Payoffs from forward contracts: (a) long position, (b) short position. Delivery price = K; price of asset at maturity = SVThese payoffs can be positive or negative. They are illustrated in Figure 1.1. Because it costsnothing to enter into a forward contract, the payoff from the contract is also the traders total gainor loss from the contract.Forward Price and Delivery PriceIt is important to distinguish between the forward price and delivery price. The forward price is themarket price that would be agreed to today for delivery of the asset at a specified maturity date.The forward price is usually different from the spot price and varies with the maturity date(see Table 1.1). In the example we considered earlier, the forward price on August 16, 2001, is 1.4359 for acontract maturing on February 16, 2002. The corporation enters into a contract and 1.4359becomes the delivery price for the contract. As we move through time the delivery price for thecorporations contract does not change, but the forward price for a contract maturing on February16, 2002, is likely to do so. For example, if GBP strengthens relative to USD in the second half ofAugust the forward price could rise to 1.4500 by September 1, 2001.Forward Prices and Spot PricesWe will be discussing in some detail the relationship between spot and forward prices in Chapter 3.In this section we illustrate the reason why the two are related by considering forward contracts ongold. We assume that there are no storage costs associated with gold and that gold earns no income.1 Suppose that the spot price of gold is $300 per ounce and the risk-free interest rate forinvestments lasting one year is 5% per annum. What is a reasonable value for the one-yearforward price of gold?1This is not totally realistic. In practice, storage costs are close to zero, but an income of 1 to 2% per annum can beearned by lending gold. 20. IntroductionSuppose first that the one-year forward price is $340 per ounce. A trader can immediately takethe following actions:1. Borrow $300 at 5% for one year.2. Buy one ounce of gold.3. Enter into a short forward contract to sell the gold for $340 in one year.The interest on the $300 that is borrowed (assuming annual compounding) is $15. The trader can,therefore, use $315 of the $340 that is obtained for the gold in one year to repay the loan. Theremaining $25 is profit. Any one-year forward price greater than $315 will lead to this arbitragetrading strategy being profitable. Suppose next that the forward price is $300. An investor who has a portfolio that includes gold can1. Sell the gold for $300 per ounce.2. Invest the proceeds at 5%.3. Enter into a long forward contract to repurchase the gold in one year for $300 per ounce.When this strategy is compared with the alternative strategy of keeping the gold in the portfolio forone year, we see that the investor is better off by $15 per ounce. In any situation where the forwardprice is less than $315, investors holding gold have an incentive to sell the gold and enter into along forward contract in the way that has been described. The first strategy is profitable when the one-year forward price of gold is greater than $315. Asmore traders attempt to take advantage of this strategy, the demand for short forward contractswill increase and the one-year forward price of gold will fall. The second strategy is profitable forall investors who hold gold in their portfolios when the one-year forward price of gold is less than$315. As these investors attempt to take advantage of this strategy, the demand for long forwardcontracts will increase and the one-year forward price of gold will rise. Assuming that individualsare always willing to take advantage of arbitrage opportunities when they arise, we can concludethat the activities of traders should cause the one-year forward price of gold to be exactly $315.Any other price leads to an arbitrage opportunity.21.4 FUTURES CONTRACTSLike a forward contract, a futures contract is an agreement between two parties to buy or sell anasset at a certain time in the future for a certain price. Unlike forward contracts, futures contractsare normally traded on an exchange. To make trading possible, the exchange specifies certainstandardized features of the contract. As the two parties to the contract do not necessarily knoweach other, the exchange also provides a mechanism that gives the two parties a guarantee that thecontract will be honored.The largest exchanges on which futures contracts are traded are the Chicago Board of Trade(CBOT) and the Chicago Mercantile Exchange (CME). On these and other exchanges throughoutthe world, a very wide range of commodities and financial assets form the underlying assets in thevarious contracts. The commodities include pork bellies, live cattle, sugar, wool, lumber, copper,aluminum, gold, and tin. The financial assets include stock indices, currencies, and Treasury bonds.2Our arguments make the simplifying assumption that the rate of interest on borrowed funds is the same as the rateof interest on invested funds. 21. CHAPTER 1 One way in which a futures contract is different from a forward contract is that an exact deliverydate is usually not specified. The contract is referred to by its delivery month, and the exchangespecifies the period during the month when delivery must be made. For commodities, the deliveryperiod is often the entire month. The holder of the short position has the right to choose the timeduring the delivery period when it will make delivery. Usually, contracts with several differentdelivery months are traded at any one time. The exchange specifies the amount of the asset to bedelivered for one contract and how the futures price is to be quoted. In the case of a commodity,the exchange also specifies the product quality and the delivery location. Consider, for example, thewheat futures contract currently traded on the Chicago Board of Trade. The size of the contract is5,000 bushels. Contracts for five delivery months (March, May, July, September, and December)are available for up to 18 months into the future. The exchange specifies the grades of wheat thatcan be delivered and the places where delivery can be made. Futures prices are regularly reported in the financial press. Suppose that on September 1, theDecember futures price of gold is quoted as $300. This is the price, exclusive of commissions, atwhich traders can agree to buy or sell gold for December delivery. It is determined on the floor of theexchange in the same way as other prices (i.e., by the laws of supply and demand). If more traderswant to go long than to go short, the price goes up; if the reverse is true, the price goes down.3 Further details on issues such as margin requirements, daily settlement procedures, deliveryprocedures, bid-offer spreads, and the role of the exchange clearinghouse are given in Chapter 2.1.5 OPTIONSOptions are traded both on exchanges and in the over-the-counter market. There are two basictypes of options. A call option gives the holder the right to buy the underlying asset by a certain datefor a certain price. A put option gives the holder the right to sell the underlying asset by a certaindate for a certain price. The price in the contract is known as the exercise price or strike price; thedate in the contract is known as the expiration date or maturity. American options can be exercised atany time up to the expiration date. European options can be exercised only on the expiration dateitself.4 Most of the options that are traded on exchanges are American. In the exchange-tradedequity options market, one contract is usually an agreement to buy or sell 100 shares. Europeanoptions are generally easier to analyze than American options, and some of the properties of anAmerican option are frequently deduced from those of its European counterpart. It should be emphasized that an option gives the holder the right to do something. The holderdoes not have to exercise this right. This is what distinguishes options from forwards and futures,where the holder is obligated to buy or sell the underlying asset. Note that whereas it costs nothingto enter into a forward or futures contract, there is a cost to acquiring an option.Call OptionsConsider the situation of an investor who buys a European call option with a strike price of $60 topurchase 100 Microsoft shares. Suppose that the current stock price is $58, the expiration date of3In Chapter 3 we discuss the relationship between a futures price and the spot price of the underlying asset (gold, inthis case).4Note that the terms American and European do not refer to the location of the option or the exchange. Someoptions trading on North American exchanges are European. 22. IntroductionProfit ($)3020 10Terminal stock price ($) 0304050 607080 90-5Figure 1.2 Profit from buying a European call option on one Microsoft share. Option price = $5; strike price = $60the option is in four months, and the price of an option to purchase one share is $5. The initialinvestment is $500. Because the option is European, the investor can exercise only on the expirationdate. If the stock price on this date is less than $60, the investor will clearly choose not to exercise.(There is no point in buying, for $60, a share that has a market value of less than $60.) In thesecircumstances, the investor loses the whole of the initial investment of $500. If the stock price isabove $60 on the expiration date, the option will be exercised. Suppose, for example, that the stockprice is $75. By exercising the option, the investor is able to buy 100 shares for $60 per share. If theshares are sold immediately, the investor makes a gain of $15 per share, or $1,500, ignoringtransactions costs. When the initial cost of the option is taken into account, the net profit to theinvestor is $1,000. Figure 1.2 shows how the investors net profit or loss on an option to purchase one share varieswith the final stock price in the example. (We ignore the time value of money in calculating theprofit.) It is important to realize that an investor sometimes exercises an option and makes a lossoverall. Suppose that in the example Microsofts stock price is $62 at the expiration of the option.The investor would exercise the option for a gain of 100 x ($62 $60) = $200 and realize a lossoverall of $300 when the initial cost of the option is taken into account. It is tempting to argue thatthe investor should not exercise the option in these circumstances. However, not exercising wouldlead to an overall loss of $500, which is worse than the $300 loss when the investor exercises. Ingeneral, call options should always be exercised at the expiration date if the stock price is above thestrike price.Put OptionsWhereas the purchaser of a call option is hoping that the stock price will increase, the purchaser of aput option is hoping that it will decrease. Consider an investor who buys a European put option tosell 100 shares in IBM with a strike price of $90. Suppose that the current stock price is $85, theexpiration date of the option is in three months, and the price of an option to sell one share is $7. Theinitial investment is $700. Because the option is European, it will be exercised only if the stock priceis below $90 at the expiration date. Suppose that the stock price is $75 on this date. The investor can 23. CHAPTER 1 Profit (S)302010 Terminalstock price ($) 0 V 607080 90100 110 120-7 Figure 1.3 Profit from buying a European put option on one IBM share. Option price = $7; strike price = $90buy 100 shares for $75 per share and, under the terms of the put option, sell the same shares for $90to realize a gain of $15 per share, or $1,500 (again transactions costs are ignored). When the $700initial cost of the option is taken into account, the investors net profit is $800. There is no guaranteethat the investor will make a gain. If the final stock price is above $90, the put option expiresworthless, and the investor loses $700. Figure 1.3 shows the way in which the investors profit or losson an option to sell one share varies with the terminal stock price in this example.Early ExerciseAs already mentioned, exchange-traded stock options are usually American rather than European.That is, the investor in the foregoing examples would not have to wait until the expiration date beforeexercising the option. We will see in later chapters that there are some circumstances under which it isoptimal to exercise American options prior to maturity.Option PositionsThere are two sides to every option contract. On one side is the investor who has taken the longposition (i.e., has bought the option). On the other side is the investor who has taken a shortposition (i.e., has sold or written the option). The writer of an option receives cash up front, buthas potential liabilities later. The writers profit or loss is the reverse of that for the purchaser of theoption. Figures 1.4 and 1.5 show the variation of the profit or loss with the final stock price forwriters of the options considered in Figures 1.2 and 1.3.There are four types of option positions: 1. A long position in a call option. 2. A long position in a put option. 3. A short position in a call option. 4. A short position in a put option. 24. Introduction Profit ($) 304050 X 607080 i 90Terminal stock price ($) -10 -20 -30 Figure 1.4 Profit from writing a European call option on one Microsoft share.Option price = $5; strike price = $60It is often useful to characterize European option positions in terms of the terminal value or payoffto the investor at maturity. The initial cost of the option is then not included in the calculation. IfK is the strike price and S? is the final price of the underlying asset, the payoff from a long positionin a European call option is max(5 r - K, 0)This reflects the fact that the option will be exercised if ST > K and will not be exercised if ST < K.The payoff to the holder of a short position in the European call option is - max(S r - K, 0) = min(K - ST, 0) .. Profit I 7Terminal 60 7080 stock price ($) 090100110 120 -10 -20 -30Figure 1.5 Profit from writing a European put option on one IBM share.Option price = $7; strike price = $90 25. 10 CHAPTER 1 ,, Payoff | Payoff ,, Payoff Figure 1.6 Payoffs from positions in European options: (a) long call, (b) short call, (c) long put,(d) short put. Strike price = K; price of asset at maturity = STThe payoff to the holder of a long position in a European put option is max(K-ST, 0)and the payoff from a short position in a European put option is- ma{K -ST,0) = min (S r - K, 0)Figure 1.6 shows these payoffs.1.6 TYPES OF TRADERSDerivatives markets have been outstandingly successful. The main reason is that they haveattracted many different types of traders and have a great deal of liquidity. When an investorwants to take one side of a contract, there is usually no problem in finding someone that isprepared to take the other side. Three broad categories of traders can be identified: hedgers, speculators, and arbitrageurs.Hedgers use futures, forwards, and options to reduce the risk that they face from potential futuremovements in a market variable. Speculators use them to bet on the future direction of a marketvariable. Arbitrageurs take offsetting positions in two or more instruments to lock in a profit. Inthe next few sections, we consider the activities of each type of trader in more detail. 26. Introduction 11HedgersWe now illustrate how hedgers can reduce their risks with forward contracts and options.Suppose that it is August 16, 2001, and ImportCo, a company based in the United States, knowsthat it will pay 10 million on November 16,2001, for goods it has purchased from a British supplier.The USD-GBP exchange rate quotes made by a financial institution are given in Table 1.1.ImportCo could hedge its foreign exchange risk by buying pounds (GBP) from the financialinstitution in the three-month forward market at 1.4407. This would have the effect of fixing theprice to be paid to the British exporter at $14,407,000.Consider next another U.S. company, which we will refer to as ExportCo, that is exportinggoods to the United Kingdom and on August 16, 2001, knows that it will receive 30 million threemonths later. ExportCo can hedge its foreign exchange risk by selling 30 million in the three-month forward market at an exchange rate of 1.4402. This would have the effect of locking in theU.S. dollars to be realized for the sterling at $43,206,000.Note that if the companies choose not to hedge they might do better than if they do hedge.Alternatively, they might do worse. Consider ImportCo. If the exchange rate is 1.4000 on November 16 and the company has not hedged, the 10 million that it has to pay will cost $14,000,000, which isless than $14,407,000. On the other hand, if the exchange rate is 1.5000, the 10 million will cost$15,000,000and the company will wish it had hedged! The position of ExportCo if it does nothedge is the reverse. If the exchange rate in September proves to be less than 1.4402, the company willwish it had hedged; if the rate is greater than 1.4402, it will be pleased it had not done so.This example illustrates a key aspect of hedging. The cost of, or price received for, the underlyingasset is ensured. However, there is no assurance that the outcome with hedging will be better thanthe outcome without hedging.Options can also be used for hedging. Consider an investor who in May 2000 owns 1,000Microsoft shares. The current share price is $73 per share. The investor is concerned that thedevelopments in Microsofts antitrust case may cause the share price to decline sharply in the nexttwo months and wants protection. The investor could buy 10 July put option contracts with a strikeprice of $65 on the Chicago Board Options Exchange. This would give the investor the right to sell 1,000 shares for $65 per share. If the quoted option price is $2.50, each option contract would cost 100 x $2.50 = $250, and the total cost of the hedging strategy would be 10 x $250 = $2,500. The strategy costs $2,500 but guarantees that the shares can be sold for at least $65 per shareduring the life of the option. If the market price of Microsoft falls below $65, the options can beexercised so that $65,000 is realized for the entire holding. When the cost of the options is takeninto account, the amount realized is $62,500. If the market price stays above $65, the options arenot exercised and expire worthless. However, in this case the value of the holding is always above$65,000 (or above $62,500 when the cost of the options is taken into account). There is a fundamental difference between the use of forward contracts and options for hedging.Forward contracts are designed to neutralize risk by fixing the price that the hedger will pay orreceive for the underlying asset. Option contracts, by contrast, provide insurance. They offer a wayfor investors to protect themselves against adverse price movements in the future while stillallowing them to benefit from favorable price movements. Unlike forwards, options involve thepayment of an up-front fee.SpeculatorsWe now move on to consider how futures and options markets can be used by speculators.Whereas hedgers want to avoid an exposure to adverse movements in the price of an asset, 27. 12 CHAPTER 1speculators wish to take a position in the market. Either they are betting that the price will go upor they are betting that it will go down. Consider a U.S. speculator who in February thinks that the British pound will strengthenrelative to the U.S. dollar over the next two months and is prepared to back that hunch to thetune of 250,000. One thing the speculator can do is simply purchase 250,000 in the hope thatthe sterling can be sold later at a profit. The sterling once purchased would be kept in aninterest-bearing account. Another possibility is to take a long position in four CME Aprilfutures contracts on sterling. (Each futures contract is for the purchase of 62,500.) Supposethat the current exchange rate is 1.6470 and the April futures price is 1.6410. If the exchangerate turns out to be 1.7000 in April, the futures contract alternative enables the speculator torealize a profit of (1.7000 - 1.6410) x 250,000 = $14,750. The cash market alternative leads toan asset being purchased for 1.6470 in February and sold for 1.7000 in April, so that a profitof (1.7000- 1.6470) x 250,000 = $13,250 is made. If the exchange rate falls to 1.6000, thefutures contract gives rise to a (1.6410 - 1.6000) x 250,000 = $10,250 loss, whereas the cashmarket alternative gives rise to a loss of (1.6470 - 1.6000) x 250,000 = $11,750. The alternativesappear to give rise to slightly different profits and losses. But these calculations do not reflectthe interest that is earned or paid. It will be shown in Chapter 3 that when the interest earnedin sterling and the interest paid in dollars are taken into account, the profit or loss from thetwo alternatives is the same. What then is the difference between the two alternatives? The first alternative of buying sterlingrequires an up-front investment of $411,750. As we will see in Chapter 2, the second alternativerequires only a small amount of cashperhaps $25,000to be deposited by the speculator inwhat is termed a margin account. The futures market allows the speculator to obtain leverage.With a relatively small initial outlay, the investor is able to take a large speculative position. We consider next an example of how a speculator could use options. Suppose that it is Octoberand a speculator considers that Cisco is likely to increase in value over the next two months. Thestock price is currently $20, and a two-month call option with a $25 strike price is currently sellingfor $1. Table 1.2 illustrates two possible alternatives assuming that the speculator is willing toinvest $4,000. The first alternative involves the purchase of 200 shares. The second involves thepurchase of 4,000 call options (i.e., 20 call option contracts). Suppose that the speculators hunch is correct and the price of Ciscos shares rises to $35 byDecember. The first alternative of buying the stock yields a profit of200 x ($35 - $20) = $3,000 However, the second alternative is far more profitable. A call option on Cisco with a strike priceTable 1.2 Comparison of profits (losses) from two alternativestrategies for using $4,000 to speculate on Cisco stock in OctoberDecember stock priceInvestors strategy$15 $35Buy shares ($1,000) $3,000Buy call options ($4,000)$36,000 28. Introduction13of $25 gives a payoff of $10, because it enables something worth $35 to be bought for $25. Thetotal payoff from the 4,000 options that are purchased under the second alternative is4,000 x $10 = $40,000Subtracting the original cost of the options yields a net profit of $40,000 - $4,000 = $36,000The options strategy is, therefore, 12 times as profitable as the strategy of buying the stock.Options also give rise to a greater potential loss. Suppose the stock price falls to $15 byDecember. The first alternative of buying stock yields a loss of 200 x ($20-$15) = $1,000Because the call options expire without being exercised, the options strategy would lead to a loss of$4,000the original amount paid for the options.It is clear from Table 1.2 that options like futures provide a form of leverage. For a giveninvestment, the use of options magnifies the financial consequences. Good outcomes become verygood, while bad outcomes become very bad!Futures and options are similar instruments for speculators in that they both provide a way inwhich a type of leverage can be obtained. However, there is an important difference between the two.With futures the speculators potential loss as well as the potential gain is very large. With options nomatter how bad things get, the speculators loss is limited to the amount paid for the options.ArbitrageursArbitrageurs are a third important group of participants in futures, forward, and options markets.Arbitrage involves locking in a riskless profit by simultaneously entering into transactions in twoor more markets. In later chapters we will see how arbitrage is sometimes possible when the futuresprice of an asset gets out of line with its cash price. We will also examine how arbitrage can be usedin options markets. This section illustrates the concept of arbitrage with a very simple example. Consider a stock that is traded on both the New York Stock Exchange (www.nyse.com) and theLondon Stock Exchange (www.stockex.co.uk). Suppose that the stock price is $152 in New Yorkand 100 in London at a time when the exchange rate is $1.5500 per pound. An arbitrageur couldsimultaneously buy 100 shares of the stock in New York and sell them in London to obtain a risk-free profit of 100 x [($1.55 x 100)-$152]or $300 in the absence of transactions costs. Transactions costs would probably eliminate the profitfor a small investor. However, a large investment house faces very low transactions costs in boththe stock market and the foreign exchange market. It would find the arbitrage opportunity veryattractive and would try to take as much advantage of it as possible. Arbitrage opportunities such as the one just described cannot last for long. As arbitrageurs buythe stock in New York, the forces of supply and demand will cause the dollar price to rise.Similarly, as they sell the stock in London, the sterling price will be driven down. Very quickly thetwo prices will become equivalent at the current exchange rate. Indeed, the existence of profit-hungry arbitrageurs makes it unlikely that a major disparity between the sterling price and thedollar price could ever exist in the first place. Generalizing from this example, we can say that the 29. 14CHAPTER 1very existence of arbitrageurs means that in practice only very small arbitrage opportunities areobserved in the prices that are quoted in most financial markets. In this book most of thearguments concerning futures prices, forward prices, and the values of option contracts will bebased on the assumption that there are no arbitrage opportunities.1.7OTHER DERIVATIVESThe call and put options we have considered so far are sometimes termed "plain vanilla" or"standard" derivatives. Since the early 1980s, banks and other financial institutions have been veryimaginative in designing nonstandard derivatives to meet the needs of clients. Sometimes these aresold by financial institutions to their corporate clients in the over-the-counter market. On otheroccasions, they are added to bond or stock issues to make these issues more attractive to investors.Some nonstandard derivatives are simply portfolios of two or more "plain vanilla" call and putoptions. Others are far more complex. The possibilities for designing new interesting nonstandardderivatives seem to be almost limitless. Nonstandard derivatives are sometimes termed exoticoptions or just exotics. In Chapter 19 we discuss different types of exotics and consider how theycan be valued.We now give examples of three derivatives that, although they appear to be complex, can bedecomposed into portfolios of plain vanilla call and put options.5 Example 1.1: Standard Oils Bond Issue A bond issue by Standard Oil worked as follows. The holder received no interest. At the bonds maturity the company promised to pay $1,000 plus an additional amount based on the price of oil at that time. The additional amount was equal to the product of 170 and the excess (if any) of the price of a barrel of oil at maturity over $25. The maximum additional amount paid was $2,550 (which corresponds to a price of $40 per barrel). These bonds provided holders with a stake in a commodity that was critically important to the fortunes of the company. If the price of the commodity went up, the company was in a good position to provide the bondholder with the additional payment. Example 1.2: ICON In the 1980s, Bankers Trust developed index currency option notes (ICONs). These are bonds in which the amount received by the holder at maturity varies with a foreign exchange rate. Two exchange rates, Kx and K2, are specified with Kx > K2. If the exchange rate at the bonds maturity is above Ku the bondholder receives the full face value. If it is less than K2, the bondholder receives nothing. Between K2 and K, a portion of the full face value is received. Bankers Trusts first issue of an ICON was for the Long Term Credit Bank of Japan. The ICON specified that if the yen-USD exchange rate, ST, is greater than 169 yen per dollar at maturity (in 1995), the holder of the bond receives SI ,000. If it is less than 169 yen per dollar, the amount received by the holder of the bond is 1,000-maxkl , 0 0 0S ( ^ -LT When the exchange rate is below 84.5, nothing is received by the holder at maturity. Example 1.3: Range Forward Contract Range forward contracts (also known as flexible for- wards) are popular in foreign exchange markets. Suppose that on August 16, 2001, a U.S. company finds that it will require sterling in three months and faces the exchange rates given in Table 1.1. It See Problems 1.24, 1.25, and 1.30 at the end of this chapter for how the decomposition is accomplished. 30. Introduction15 could enter into a three-month forward contract to buy at 1.4407. A range forward contract is an alternative. Under this contract an exchange rate band straddling 1.4407 is set. Suppose that the chosen band runs from 1.4200 to 1.4600. The range forward contract is then designed to ensure that if the spot rate in three months is less than 1.4200, the company pays 1.4200; if it is between 1.4200 and 1.4600, the company pays the spot rate; if it is greater than 1.4600, the company pays 1.4600.Other, More Complex ExamplesAs mentioned earlier, there is virtually no limit to the innovations that are possible in thederivatives area. Some of the options traded in the over-the-counter market have payoffs dependenton maximum value attained by a variable during a period of time; some have payoffs dependent onthe average value of a variable during a period of time; some have exercise prices that are functionsof time; some have features where exercising one option automatically gives the holder anotheroption; some have payoffs dependent on the square of a future interest rate; and so on. Traditionally, the variables underlying options and other derivatives have been stock prices, stockindices, interest rates, exchange rates, and commodity prices. However, other underlying variablesare becoming increasingly common. For example, the payoffs from credit derivatives, which arediscussed in Chapter 27, depend on the creditworthiness of one or more companies; weatherderivatives have payoffs dependent on the average temperature at particular locations; insurancederivatives have payoffs dependent on the dollar amount of insurance claims of a specified type madeduring a specified period; electricity derivatives have payoffs dependent on the spot price ofelectricity; and so on. Chapter 29 discusses weather, insurance, and energy derivatives. SUMMARYOne of the exciting developments in finance over the last 25 years has been the growth ofderivatives markets. In many situations, both hedgers and speculators find it more attractive totrade a derivative on an asset than to trade the asset itself. Some derivatives are traded onexchanges. Others are traded by financial institutions, fund managers, and corporations in theover-the-counter market, or added to new issues of debt and equity securities. Much of this book isconcerned with the valuation of derivatives. The aim is to present a unifying framework withinwhich all derivativesnot just options or futurescan be valued. In this chapter we have taken a first look at forward, futures, and options contracts. A forwardor futures contract involves an obligation to buy or sell an asset at a certain time in the future for acertain price. There are two types of options: calls and puts. A call option gives the holder the rightto buy an asset by a certain date for a certain price. A put option gives the holder the right to sellan asset by a certain date for a certain price. Forwards, futures, and options trade on a wide rangeof different underlying assets. Derivatives have been very successful innovations in capital markets. Three main types of traderscan be identified: hedgers, speculators, and arbitrageurs. Hedgers are in the position where theyface risk associated with the price of an asset. They use derivatives to reduce or eliminate this risk.Speculators wish to bet on future movements in the price of an asset. They use derivatives to getextra leverage. Arbitrageurs are in business to take advantage of a discrepancy between prices intwo different markets. If, for example, they see the futures price of an asset getting out of line withthe cash price, they will take offsetting positions in the two markets to lock in a profit. 31. 16 CHAPTER 1 QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL) 1.1. What is the difference between a long forward position and a short forward position? 1.2. Explain carefully the difference between hedging, speculation, and arbitrage."j 1.3. What is the difference between entering into a long forward contract when the forward price is $50 and taking a long position in a call option with a strike price of $50? 1.4. Explain carefully the difference between writing a call option and buying a put option. 1.5. A trader enters into a short forward contract on 100 million yen. The forward exchange rate is$0.0080 per yen. How much does the trader gain or lose if the exchange rate at the end of thecontract is (a) $0.0074 per yen; (b) $0^00? 1 per yen? 1.6. A trader enters into* a "snort cotton futures contract when the futures price is 50 cents per pound.The contract is for the delivery of 50,000 pounds. How much does the trader gain or lose if thecotton price at the end of the contract is (a) 48.20 cents per pound; (b) 51.30 cents per pound? 1.7. Suppose that you write a put contract on AOL Time Warner with a strike price of $40 and anexpiration date in three months. The current stock price of AOL Time Warner is $41 and thecontract is on 100 shares. What have you committed yourself to? How much could you gain or lose? 1.8. You would like to speculate on a rise in the price of a certain stock. The current stock price is $29,and a three-month call with a strike of $30 costs $2.90. You have $5,800 to invest. Identify twoalternative strategies, one involving an investment in the stock and the other involving investmentin the option. What are the potential gains and losses from each? 1.9. Suppose that you own 5,000 shares worth $25 each. How can put options be used to provide youwith insurance against a decline in the value of your holding over the next four months?1.10. A trader buys a European put on a share for $3. The stock price is $42 and the strike price is $40.Under what circumstances does the trader make a profit? Under what circumstances will theoption be exercised? Draw a diagram showing the variation of the traders profit with the stockprice at the maturity of the option.1.11. A trader sells a European call on a share for $4. The stock price is $47 and the strike price is $50.Under what circumstances does the trader make a profit? Under what circumstances will theoption be exercised? Draw a diagram showing the variation of the traders profit with the stockprice at the maturity of the option.1.12. A trader buys a call option with a strike price of $45 and a put option with a strike price of $40.Both options have the same maturity. The call costs $3 and the put costs $4. Draw a diagramshowing the variation of the traders profit with the asset price.1.13. When first issued, a stock provides funds for a company. Is the same true of a stock option? Discuss.1.14. Explain why a forward contract can be used for either speculation or hedging.1.15. Suppose that a March call option to buy a share for $50 costs $2.50 and is held until March.Under what circumstances will the holder of the option make a profit? Under what circumstanceswill the option be exercised? Draw a diagram illustrating how the profit from a long position inthe option depends on the stock price at maturity of the option.1.16. Suppose that a June put option to sell a share for $60 costs $4 and is held until June. Under whatcircumstances will the seller of the option (i.e., the party with the short position) make a profit?Under what circumstances will the option be exercised? Draw a diagram illustrating how theprofit from a short position in the option depends on the stock price at maturity of the option. 32. Introduction 171.17. A trader writes a September call option with a strike price of $20. It is now May, the stock price is$18, and the option price is $2. Describe the traders cash flows if the option is held untilSeptember and the stock price is $25 at that time.1.18. A trader writes a December put option with a strike price of S30. The price of the option is S4.Under what circumstances does the trader make a gain?1.19. A company knows that it is due to receive a certain amount of a foreign currency in four months.What type of option contract is appropriate for hedging?1.20. A United States company expects to have to pay 1 million Canadian dollars in six months.Explain how the exchange rate risk can be hedged using (a) a forward contract; (b) an option.1.21. The Chicago Board of Trade offers a futures contract on long-term Treasury bonds. Characterizethe traders likely to use this contract.1.22. "Options and futures are zero-sum games." What do you think is meant by this statement?1.23. Describe the profit from the following portfolio: a long forward contract on an asset and a longEuropean put option on the asset with the same maturity as the forward contract and a strikeprice that is equal to the forward price of the asset at the time the portfolio is set up.1.24. Show that an ICON such as the one described in Section 1.7 is a combination of a regular bondand two options.1.25. Show that a range forward contract such as the one described in Section 1.7 is a combination oftwo options. How can a range forward contract be constructed so that it has zero value?1.26. On July 1, 2002, a company enters into a forward contract to buy 10 million Japanese yen onJanuary 1, 2003. On September 1, 2002, it enters into a forward contract to sell 10 millionJapanese yen on January 1, 2003. Describe the payoff from this strategy.1.27. Suppose that sterling-USD spot and forward exchange rates are as follows:Spot1.608090-day forward 1.6056180-day forward 1.6018What opportunities are open to an arbitrageur in the following situations?a. A 180-day European call option to buy 1 for $1.57 costs 2 cents.b. A 90-day European put option to sell 1 for $1.64 costs 2 cents. ASSIGNMENT QUESTIONS1.28. The price of gold is currently $500 per ounce. The forward price for delivery in one year is $700.An arbitrageur can borrow money at 10% per annum. What should the arbitrageur do? Assumethat the cost of storing gold is zero and that gold provides no income.1.29. The current price of a stock is $94, and three-month call options with a strike price of $95currently sell for $4.70. An investor who feels that the price of the stock will increase is trying todecide between buying 100 shares and buying 2,000 call options (= 20 contracts). Both strategiesinvolve an investment of $9,400. What advice would you give? How high does the stock price haveto rise for the option strategy to be more profitable?1.30. Show that the Standard Oil bond described in Section 1.7 is a combination of a regular bond, along position in call options on oil with a strike price of $25, and a short position in call optionson oil with a strike price of $40. 33. 18 CHAPTER 11.31. Use the DerivaGem software to calculate the value of the range forward contract considered inSection 1.7 on the assumption that the exchange rate volatility is 15% per annum. Adjust theupper end of the band so that the contract has zero value initially. Assume that the dollar andsterling risk-free rates are 5.0% and 6.4% per annum, respectively.1.32. A trader owns gold as part of a long-term investment portfolio. The trader can buy gold for $250per ounce and sell gold for $249 per ounce. The trader can borrow funds at 6% per year andinvest funds at 5.5% per year. (Both interest rates are expressed with annual compounding.) Forwhat range of one-year forward prices of gold does the trader have no arbitrage opportunities?Assume there is no bid-offer spread for forward prices.1.33. Describe how foreign currency options can be used for hedging in the situation considered inSection 1.6 so that (a) ImportCo is guaranteed that its exchange rate will be less than 1.4600, and(b) ExportCo is guaranteed that its exchange rate will be at least 1.4200. Use DerivaGem tocalculate the cost of setting up the hedge in each case assuming that the exchange rate volatility is12%, interest rates in the United States are 3% and interest rates in Britain are 4.4%. Assume thatthe current exchange rate is the average of the bid and offer in Table 1.1.1.34. A trader buys a European call option and sells a European put option. The options have the sameunderlying asset, strike price and maturity. Describe the traders position. Under whatcircumstances does the price of the call equal the price of the put? 34. CHAPTER 2MECHANICS OF FUTURESMARKETSIn Chapter 1 we explained that both futures and forward contracts are agreements to buy or sell anasset at a future time for a certain price. Futures contracts are traded on an organized exchange,and the contract terms are standardized by that exchange. By contrast, forward contracts areprivate agreements between two financial institutions or between a financial institution and one ofits corporate clients. This chapter covers the details of how futures markets work. We examine issues such as thespecification of contracts, the operation of margin accounts, the organization of exchanges, theregulation of markets, the way in which quotes are made, and the treatment of futures transactionsfor accounting and tax purposes. We also examine forward contracts and explain the differencebetween the pattern of payoffs realized from futures and forward contracts.2.1TRADING FUTURES CONTRACTSAs mentioned in Chapter 1, futures contracts are now traded very actively all over the world. Thetwo largest futures exchanges in the United States are the Chicago Board of Trade (CBOT,www.cbot.com) and the Chicago Mercantile Exchange (CME, www.cme.com). The two largestexchanges in Europe are the London International Financial Futures and Options Exchange(www.liffe.com) and Eurex (www.eurexchange.com). Other large exchanges include Bolsa deMercadorias y Futuros (www.bmf.com.br) in Sao Paulo, the Tokyo International Financial FuturesExchange (www.tiffe.or.jp), the Singapore International Monetary Exchange (www.simex.com.sg),and the Sydney Futures Exchange (www.sfe.com.au). For a more complete list, see the table at theend of this book.We examine how a futures contract comes into existence by considering the corn futurescontract traded on the Chicago Board of Trade (CBOT). On March 5, an investor in New Yorkmight call a broker with instructions to buy 5,000 bushels of corn for delivery in July of the sameyear. The broker would immediately pass these instructions on to a trader on the floor of theCBOT. The broker would request a long position in one contract because each corn contract onthe CBOT is for the delivery of exactly 5,000 bushels. At about the same time, another investorin Kansas might instruct a broker to sell 5,000 bushels of corn for July delivery. This brokerwould then pass instructions to short one contract to a trader on the floor of the CBOT. The two19 35. 20 CHAPTER 2floor traders would meet, agree on a price to be paid for the corn in July, and the deal would bedone. The investor in New York who agreed to buy has a long futures position in one contract; theinvestor in Kansas who agreed to sell has a short futures position in one contract. The priceagreed to on the floor of the exchange is the current futures price for July corn. We will supposethe price is 170 cents per bushel. This price, like any other price, is determined by the laws ofsupply and demand. If at a particular time more traders wish to sell July corn than buy Julycorn, the price will go down. New buyers then enter the market so that a balance between buyersand sellers is maintained. If more traders wish to buy July corn than to sell July corn, the pricegoes up. New sellers then enter the market and a balance between buyers and sellers ismaintained.Closing Out PositionsThe vast majority of futures contracts do not lead to delivery. The reason is that most traderschoose to close out their positions prior to the delivery period specified in the contract. Closing outa position means entering into the opposite type of trade from the original one. For example theNew York investor who bought a July corn futures contract on March 5 can close out the positionby selling (i.e., shorting) one July corn futures contract on April 20. The Kansas investor who sold(i.e., shorted) a July contract on March 5 can close out the position on by buying one July contracton April 20. In each case, the investors total gain or loss is determined by the change in the futuresprice between March 5 and April 20. It is important to realize that there is no particular significance to the party on the other side of atrade in a futures transaction. Consider trader A who initiates a long futures position by tradingone contract. Suppose that trader B is on the other side of the transaction. At a later stage trader Amight close out the position by entering into a short contract. The trader on the other side of thissecond transaction does not have to be, and usually is not, trader B.2.2THE SPECIFICATION OF THE FUTURES CONTRACTWhen developing a new contract, the exchange must specify in some detail the exact nature of theagreement between the two parties. In particular, it must specify the asset, the contract size (exactlyhow much of the asset will be delivered under one contract), where delivery will be made, and whendelivery will be made.Sometimes alternatives are specified for the grade of the asset that will be delivered or for thedelivery locations. As a general rule, it is the party with the short position (the party that hasagreed to sell the asset) that chooses what will happen when alternatives are specified by theexchange. When the party with the short position is ready to deliver, it files a notice of intention todeliver with the exchange. This notice indicates selections it has made with respect to the grade ofasset that will be delivered and the delivery location.The AssetWhen the asset is a commodity, there may be quite a variation in the quality of what is available inthe marketplace. When the asset is specified, it is therefore important that the exchange stipulate 36. Mechanics of Futures Markets21the grade or grades of the commodity that are acceptable. The New York Cotton Exchange hasspecified the asset in its orange juice futures contract as U.S. Grade A, with Brix value of not less than 57 degrees, having a Brix value to acid ratio of not less than 13 to 1 nor more than 19 to 1, with factors of color and flavor each scoring 37 points or higher and 19 for defects, with a minimum score 94.The Chicago Mercantile Exchange in its random-length lumber futures contract has specified that Each delivery unit shall consist of nominal 2x4s of random lengths from 8 feet to 20 feet, grade- stamped Construction and Standard, Standard and Better, or #1 and #2; however, in no case may the quantity of Standard grade or #2 exceed 50%. Each delivery unit shall be manufactured in California, Idaho, Montana, Nevada, Oregon, Washington, Wyoming, or Alberta or British Columbia, Canada, and contain lumber produced from grade-stamped Alpine fir, Englemann spruce, hem-fir, lodgepole pine, and/or spruce pine fir.For some commodities a range of grades can be delivered, but the price received depends the gradechosen. For example, in the Chicago Board of Trade corn futures contract, the standard grade is"No. 2 Yellow", but substitutions are allowed with the price being adjusted in a way established bythe exchange.The financial assets in futures contracts are generally well defined and unambiguous. Forexample, there is no need to specify the grade of a Japanese yen. However, there are someinteresting features of the Treasury bond and Treasury note futures contracts traded on theChicago Board of Trade. The underlying asset in the Treasury bond contract is any long-termU.S. Treasury bond that has a maturity of greater than 15 years and is not callable within 15 years.In the Treasury note futures contract, the underlying asset is any long-term Treasury note with amaturity of no less than 6.5 years and no more than 10 years from the date of delivery. In bothcases, the exchange has a formula for adjusting the price received according to the coupon andmaturity date of the bond delivered. This is discussed in Chapter 5.The Contract SizeThe contract size specifies the amount of the asset that has to be delivered under one contract. Thisis an important decision for the exchange. If the contract size is too large, many investors who wishto hedge relatively small exposures or who wish to take relatively small speculative positions will beunable to use the exchange. On the other hand, if the contract size is too small, trading may beexpensive as there is a cost associated with each contract traded. The correct size for a contract clearly depends on the likely user. Whereas the value of whatis delivered under a futures contract on an agricultural product might be $10,000 to $20,000,it is much higher for some financial futures. For example, under the Treasury bond futurescontract traded on the Chicago Board of Trade, instruments with a face value of $100,000 aredelivered. In some cases exchanges have introduced "mini" contracts to attract smaller investors. Forexample, the CMEs Mini Nasdaq 100 contract is on 20 times the Nasdaq 100 index whereas theregular contract is on 100 times the index.Delivery ArrangementsThe place where delivery will be made must be specified by the exchange. This is particularlyimportant for commodities that involve significant transportation costs. In the case of the Chicago 37. 22CHAPTER 2Mercantile Exchanges random-length lumber contract, the delivery location is specified as On track and shall either be unitized in double-door boxcars or, at no additional cost to the buyer, each unit shall be individually paper-wrapped and loaded on flatcars. Par delivery of hem-fir in California, Idaho, Montana, Nevada, Oregon, and Washington, and in the province of British Columbia.When alternative delivery locations are specified, the price received by the party with the shortposition is sometimes adjusted according to the location chosen by that party. For example, in thecase of the corn futures contract traded by the Chicago Board of Trade, delivery can be made atChicago, Burns Harbor, Toledo, or St. Louis. However, deliveries at Toledo and St. Louis aremade at a discount of 4 cents per bushel from the Chicago contract price.Delivery MonthsA futures contract is referred to by its delivery month. The exchange must specify the preciseperiod during the month when delivery can be made. For many futures contracts, the deliveryperiod is the whole month.The delivery months vary from contract to contract and are chosen by the exchange to meet theneeds of market participants. For example, the main delivery months for currency futures on theChicago Mercantile Exchange are March, June, September, and December; corn futures traded onthe Chicago Board of Trade have delivery months of January, March, May, July, September,November, and December. At any given time, contracts trade for the closest delivery month and anumber of subsequent delivery months. The exchange specifies when trading in a particularmonths contract will begin. The exchange also specifies the last day on which trading can takeplace for a given contract. Trading generally ceases a few days before the last day on which deliverycan be made.Price QuotesThe futures price is quoted in a way that is convenient and easy to understand. For example, crudeoil futures prices on the New York Mercantile Exchange are quoted in dollars per barrel to twodecimal places (i.e., to the nearest cent). Treasury bond and Treasury note futures prices on theChicago Board of Trade are quoted in dollars and thirty-seconds of a dollar. The minimum pricemovement that can occur in trading is consistent with the way in which the price is quoted. Thus, itis SO.01 per barrel for the oil futures and one thirty-second of a dollar for the Treasury bond andTreasury note futures.Daily Price Movement LimitsFor most contracts, daily price movement limits are specified by the exchange. If the price movesdown by an amount equal to the daily price limit, the contract is said to be limit down. If it movesup by the limit, it is said to be limit up. A limit move is a move in either direction equal to the dailyprice limit. Normally, trading ceases for the day once the contract is limit up or limit down.However, in some instances the exchange has the authority to step in and change the limits. The purpose of daily price limits is to prevent large price movements from occurring because ofspeculative excesses. However, limits can become an artificial barrier to trading when the price ofthe underlying commodity is increasing or decreasing rapidly. Whether price limits are, on balance,good for futures markets is controversial. 38. Mechanics of Futures Markets23Position LimitsPosition limits are the maximum number of contracts that a speculator may hold. In the ChicagoMercantile Exchanges random-length lumber contract, for example, the position limit at the timeof writing is 1,000 contracts with no more than 300 in any one delivery month. Bona fide hedgersare not affected by position limits. The purpose of the limits is to prevent speculators fromexercising undue influence on the market.2.3 CONVERGENCE OF FUTURES PRICE TO SPOT PRICEAs the delivery month of a futures contract is approached, the futures price converges to the spotprice of the underlying asset. When the delivery period is reached, the futures price equalsor isvery close tothe spot price. To see why this is so, we first suppose that the futures price is above the spot price during thedelivery period. Traders then have a clear arbitrage opportunity:1. Short a futures contract.2. Buy the asset.3. Make delivery.These steps are certain to lead to a profit equal to the amount by which the futures price exceedsthe spot price. As traders exploit this arbitrage opportunity, the futures price will fall. Suppose nextthat the futures price is below the spot price during the delivery period. Companies interested inacquiring the asset will find it attractive to enter into a long futures contract and then wait fordelivery to be made. As they do so, the futures price will tend to rise.The result is that the futures price is very close to the spot price during the delivery period.Figure 2.1 illustrat