JOHN C.HULL
Personal Finance
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Mastering Finance CD-ROM
OPTIONS, FUTURES,
& OTHER DERIVATIVES
John C. Hull
Maple Financial Group Professor of Derivatives and Risk Management Director, Bonham Center for Finance
Joseph L. Rotman School of Management University of Toronto
Prentice Hall
P 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 J E R S E Y 0 7 4 5 8
Preface xix 1. Introduction 1 1.1 Exchange-traded markets 1 1.2 Over-the-counter markets 2 1.3 Forward contracts 2 1.4 Futures contracts 5 1.5 Options 6 1.6 Types of traders 10 1.7 Other derivatives 14 Summary 15 Questions and problems 16 Assignment questions 17 2. Mechanics of futures markets 19 2.1 Trading futures contracts 19 2.2 Specification of the futures contract 20 2.3 Convergence of futures price to spot price 23 2.4 Operation of margins 24 2.5 Newspaper quotes 27 2.6 Keynes and Hicks 31 2.7 Delivery 31 2.8 Types of traders 32
2.9 Regulation ; 33
2.10 Accounting and tax 35 2.11 Forward contracts vs. futures contracts 36 Summary 37 Suggestions for further reading 38 Questions and problems 38 Assignment questions 40 3. Determination of forward and futures prices 41 3.1 Investment assets vs. consumption assets 41 3.2 Short selling 41 3.3 Measuring interest rates 42 3.4 Assumptions and notation 44 3.5 Forward price for an investment asset 45 3.6 Known income 47 3.7 Known yield 49 3.8 Valuing forward contracts 49 3.9 Are forward prices and futures prices equal? 51 3.10 Stock index futures 52 3.11 Forward and futures contracts on currencies 55 3.12 Futures on commodities 58 ix
3.13 Cost of carry 60 3.14 Delivery options 60 3.15 Futures prices and the expected future spot price 61 Summary 63 Suggestions for further reading 64 Questions and problems 65 Assignment questions 67 Appendix 3A: Proof that forward and futures prices are equal when interest
rates are constant 68 4. Hedging strategies using futures 70 4.1 Basic principles 70 4.2 Arguments for and against hedging 72 4.3 Basis risk 75 4.4 Minimum variance hedge ratio 78 4.5 Stock index futures 82 4.6 Rolling the hedge forward 86 Summary 87 Suggestions for further reading 88 Questions and problems 88 Assignment questions 90 Appendix 4A: Proof of the minimum variance hedge ratio formula 92 5. Interest rate markets 93 5.1 Types of rates 93 5.2 Zero rates 94 5.3 Bond pricing 94 5.4 Determining zero rates 96 5.5 Forward rates 98 5.6 Forward rate agreements 100 5.7 Theories of the term structure 102 5.8 Day count conventions 102 5.9 Quotations 103 5.10 Treasury bond futures 104 5.11 Eurodollar futures 110 5.12 The LIBOR zero curve I l l 5.13 Duration 112 5.14 Duration-based hedging strategies 116 Summary 118 Suggestions for further reading 119 Questions and problems 120 Assignment questions 123 6. Swaps 125 6.1 Mechanics of interest rate swaps 125 6.2 The comparative-advantage argument 131 6.3 Swap quotes and LIBOR zero rates 134 6.4 Valuation of interest rate swaps 136 6.5 Currency swaps 140 6.6 Valuation of currency swaps 143 6.7 Credit risk 145 Summary 146 Suggestions for further reading 147 Questions and problems 147 Assignment questions 149
7. Mechanics of options markets 151 7.1 Underlying assets 151 7.2 Specification of stock options 152 7.3 Newspaper quotes 155 7.4 Trading 157 7.5 Commissions 157 7.6 Margins 158 7.7 The options clearing corporation 160 7.8 Regulation 161 7.9 Taxation 161 7.10 Warrants, executive stock options, and convertibles 162 7.11 Over-the-counter markets 163 Summary 163 Suggestions for further reading 164 Questions and problems 164 Assignment questions 165 8. Properties of stock options 167 8.1 Factors affecting option prices 167 8.2 Assumptions and notation 170 8.3 Upper and lower bounds for option prices 171 8.4 Put-call parity 174 8.5 Early exercise: calls on a non-dividend-paying stock 175 8.6 Early exercise: puts on a non-dividend-paying stock 177 8.7 Effect of dividends 178 8.8 Empirical research 179 Summary 180 Suggestions for further reading 181 Questions and problems 182 Assignment questions 183 9. Trading strategies involving options 185 9.1 Strategies- involving a single option and a stock 185 9.2 Spreads 187 9.3 Combinations 194 9.4 Other payoffs 197 Summary 197 Suggestions for further reading 198 Questions and problems 198 Assignment questions 199 10. Introduction to binomial trees 200 10.1 A one-step binomial model 200 10.2 Risk-neutral valuation 203 10.3 Two-step binomial trees 205 10.4 A put example 208 10.5 American options 209 10.6 Delta 210 10.7 Matching volatility with u and d 211 10.8 Binomial trees in practice 212 Summary 213 Suggestions for further reading 214 Questions and problems 214 Assignment questions 215
11. A model of the behavior of stock prices 216 11.1 The Markov property 216 11.2 Continuous-time stochastic processes 217 11.3 The process for stock prices 222 11.4 Review of the model 223 11.5 The parameters 225 11.6 Ito's lemma 226 11.7 The lognormal property 227 Summary 228 Suggestions for further reading 229 Questions and problems 229 Assignment questions 230 Appendix 11 A: Derivation of Ito's lemma 232 12. The Black-Scholes model 234 12.1 Lognormal property of stock prices 234 12.2 The distribution of the rate of return 236 12.3 The expected return 237 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 function 248 12.10 Warrants issued by a company on its own stock 249 12.11 Implied volatilities 250 12.12 The causes of volatility 251 12.13 Dividends 252 Summary 256 Suggestions for further reading 257 Questions and problems 258 Assignment questions 261 Appendix 12A: Proof of Black-Scholes-Merton formula 262 Appendix 12B: Exact procedure for calculating the values of American calls on
dividend-paying stocks 265 Appendix 12C: Calculation of cumulative probability in bivariate normal
distribution 266 13. Options on stock indices, currencies, and futures 267 13.1 Results for a stock paying a known dividend yield 267 13.2 Option pricing formulas 268 13.3 Options on stock indices 270 13.4 Currency options 276 13.5 Futures options 278 13.6 Valuation of futures options using binomial trees 284 13.7 Futures price analogy 286 13.8 Black's model for valuing futures options 287 13.9 Futures options vs. spot options 288 Summary 289 Suggestions for further reading 290 Questions and problems 291 Assignment questions 294 Appendix 13 A: Derivation of differential equation satisfied by a derivative
dependent on a stock providing a dividend yield 295
Appendix 13B: Derivation of differential equation satisfied by a derivative
dependent on a futures price 297 14. The Greek letters 299 14.1 Illustration 299 14.2 Naked and covered positions 300 14.3 A stop-loss strategy 300 14.4 Delta hedging 302 14.5 Theta 309 14.6 Gamma 312 14.7 Relationship between delta, theta, and gamma 315 14.8 Vega 316 14.9 Rho 318 14.10 Hedging in practice 319 14.11 Scenario analysis 319 14.12 Portfolio insurance 320 14.13 Stock market volatility 323 Summary 323 Suggestions for further reading 324 Questions and problems : 326 Assignment questions 327 Appendix 14A: Taylor series expansions and hedge parameters 329 15. Volatility smiles 330 15.1 Put-call parity revisited 330 15.2 Foreign currency options 331 15.3 Equity options 334 15.4 The volatility term structure and volatility surfaces 336 15.5 Greek letters 337 15.6 When a single large jump is anticipated 338 15.7 Empirical research 339 Summary 341 Suggestions for further reading 341 Questions and problems 343 Assignment questions 344 Appendix 15A: Determining implied risk-neutral distributions from volatility
smiles 345 16. Value at risk 346 16.1 The VaR measure 346 16.2 Historical simulation 348 16.3 Model-building approach 350 16.4 Linear model 352 16.5 Quadratic model 356 16.6 Monte Carlo simulation 359 16.7 Comparison of approaches 359 16.8 Stress testing and back testing 360 16.9 Principal components analysis 360 Summary 364 Suggestions for further reading 364 Questions and problems 365 Assignment questions 366 Appendix 16A: Cash-flow mapping 368 Appendix 16B: Use of the Cornish-Fisher expansion to estimate VaR 370
17. Estimating volatilities and correlations 372 17.1 Estimating volatility 372 17.2 The exponentially weighted moving average model 374 17.3 The GARCH(1,1) model 376 17.4 Choosing between the models 377 17.5 Maximum likelihood methods 378 17.6 Using GARCHfl, 1) to forecast future volatility 382 17.7 Correlations 385 Summary 388 Suggestions for further reading 388 Questions and problems 389 Assignment questions 391 18. Numerical procedures 392 18.1 Binomial trees 392 18.2 Using the binomial tree for options on indices, currencies, and futures
contracts 399 18.3 Binomial model for a dividend-paying stock '. 402 18.4 Extensions to the basic tree approach 405 18.5 Alternative procedures for constructing trees 406 18.6 Monte Carlo simulation 410 18.7 Variance reduction procedures 414 18.8 Finite difference methods 418 18.9 Analytic approximation to American option prices 427 Summary 427 Suggestions for further reading 428 Questions and problems 430 Assignment questions 432 Appendix 18A: Analytic approximation to American option prices of
MacMillan and of Barone-Adesi and Whaley 433 19. Exotic options 435 19.1 Packages 435 19.2 Nonstandard American options 436 19.3 Forward start options 437 19.4 Compound options 437 19.5 Chooser options 438 19.6 Barrier options 439 19.7 Binary options 441 19.8 Lookback options 441 19.9 Shout options 443 19.10 Asian options 443 19.11 Options to exchange one asset for another 445 19.12 Basket options 446 19.13 Hedging issues 447 19.14 Static options replication 447 Summary 449 Suggestions for further reading 449 Questions and problems 451 Assignment questions 452 Appendix 19A: Calculation of the first two moments of arithmetic averages
and baskets 454 20. More on models and numerical procedures 456 20.1 The CEV model 456 20.2 The jump diffusion model 457
20.3 Stochastic volatility models 458 20.4 The IVF model 460 20.5 Path-dependent derivatives 461 20.6 Lookback options 465 20.7 Barrier options 467 20.8 Options on two correlated assets 472 20.9 Monte Carlo simulation and American options 474 Summary 478 Suggestions for further reading 479 Questions and problems 480 Assignment questions 481 21. Martingales and measures 483 21.1 The market price of risk 484 21.2 Several state variables • 487 21.3 Martingales 488 21.4 Alternative choices for the numeraire 489 21.5 Extension to multiple independent factors 492 21.6 Applications 493 21.7 Change of numeraire 495 21.8 Quantos 497 21.9 Siegel's paradox 499 Summary 500 Suggestions for further reading 500 Questions and problems 501 Assignment questions 502 Appendix 21 A: Generalizations of Ito's lemma 504 Appendix 2IB: Expected excess return when there are multiple sources of
uncertainty 506 22. Interest rate derivatives: the standard market models 508 22.1 Black's model 508 22.2 Bond options 511 22.3 Interest rate caps 515 22.4 European swap options 520 22.5 Generalizations 524 22.6 Convexity adjustments 524 22.7 Timing adjustments 527 22.8 Natural time lags 529 22.9 Hedging interest rate derivatives 530 Summary 531 Suggestions for further reading 531 Questions and problems 532 Assignment questions 534 Appendix 22A: Proof of the convexity adjustment formula 536 23. Interest rate derivatives: models of the short rate 537 23.1 Equilibrium models 537 23.2 One-factor equilibrium models 538 23.3 The Rendleman and Bartter model 538 23.4 The Vasicek model 539 23.5 The Cox, Ingersoll, and Ross model 542 23.6 Two-factor equilibrium models 543 23.7 No-arbitrage models 543 23.8 The Ho and Lee model 544 23.9 The Hull and White model 546
23.10 Options on coupon-bearing bonds 549 23.11 Interest rate trees 550 23.12 A general tree-building procedure 552 23.13 Nonstationary models 563 23.14 Calibration 564 23.15 Hedging using a one-factor model 565 23.16 Forward rates and futures rates 566 Summary 566 Suggestions for further reading 567 Questions and problems 568 Assignment questions 570 24. Interest rate derivatives: more advanced models 571 24.1 Two-factor models of the short rate 571 24.2 The Heath, Jarrow, and Morton model 574 24.3 The LIBOR market model 577 24.4 Mortgage-backed securities 586 Summary 588 Suggestions for further reading 589 Questions and problems 590 Assignment questions 591 Appendix 24A: The A(t, T), aP, and 0(t) functions in the two-factor Hull-White
model 593 25. Swaps revisited 594 25.1 Variations on the vanilla deal 594 25.2 Compounding swaps 595 25.3 Currency swaps 598 25.4 More complex swaps 598 25.5 Equity swaps 601 25.6 Swaps with embedded options 602 25.7 Other swaps 605 25.8 Bizarre deals 605 Summary 606 Suggestions for further reading 606 Questions and problems 607 Assignment questions 607 Appendix 25A: Valuation of an equity swap between payment dates 609 26. Credit risk 610 23.1 Bond prices and the probability of default 610 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 default 623 26.7 Credit ratings migration 626 26.8 Default correlations 627 26.9 Credit value at risk 630 Summary 633 Suggestions for further reading 633 Questions and problems 634 Assignment questions 635 Appendix 26A: Manipulation of the matrices of credit rating changes 636
27. Credit derivatives 637 27.1 Credit default swaps 637 27.2 Total return swaps 644 27.3 Credit spread options 645 27.4 Collateralized debt obligations 646 27.5 Adjusting derivative prices for default risk 647 27.6 Convertible bonds 652 Summary 655 Suggestions for further reading 655 Questions and problems 656 Assignment questions 658 28. Real options 660 28.1 Capital investment appraisal 660 28.2 Extension of the risk-neutral valuation framework 661 28.3 Estimating the market price of risk 665 28.4 Application to the valuation of a new business 666 28.5 Commodity prices 667 28.6 Evaluating options in an investment opportunity 670 Summary 675 Suggestions for further reading 676 Questions and problems 676 Assignment questions 677 29. Insurance, weather, and energy derivatives 678 29.1 Review of pricing issues 678 29.2 Weather derivatives 679 29.3 Energy derivatives 680 29.4 Insurance derivatives 682 Summary 683 Suggestions for further reading 684 Questions and problems 684 Assignment questions 685 30. Derivatives mishaps and what we can learn from them 686 30.1 Lessons for all users of derivatives 686 30.2 Lessons for financial institutions 690 30.3 Lessons for nonfinancial corporations 693 Summary 694 Suggestions for further reading 695 Glossary of notation 697 Glossary of terms 700 DerivaGem software : 715 Major exchanges trading futures and options 720 Table for N{x) when x sj 0 722 Table for N(x) when x ^ 0 723 Author index 725 Subject index 729
It is sometimes hard for me to believe that the first edition of this book was only 330 pages and 13 chapters long! There have been many developments in derivatives markets over the last 15 years and the book has grown to keep up with them. The fifth edition has seven new chapters that cover new derivatives instruments and recent research advances.
Like earlier editions, the book serves several markets. It is appropriate for graduate courses in business, economics, and financial engineering. It can be used on advanced undergraduate courses when students have good quantitative skills. Also, many practitioners who want to acquire a working 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 derivatives concerns the use of mathematics. If the level of mathematical sophistication is too high, the material is likely to be inaccessible to many students and practitioners. If it is too low, some important issues will inevitably be treated in a rather superficial way. I have tried to be particularly careful 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 that are likely to be new to many readers have been explained carefully, and many numerical examples have been included.
The book covers both derivatives markets and risk management. It assumes that the reader has taken 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 therefore necessary for students to take an elective course in investments prior to taking a course based on this book. There are many different ways the book can be used in the classroom. Instructors teaching 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 the chapters in the second half of the book can be used. I find that the material in Chapters 29 and 30 works well at the end of either an introductory or an advanced course.
What's New?
Material has been updated and improved throughout the book. The changes in this edition include:
1. A new chapter on the use of futures for hedging (Chapter 4). Part of this material was previously in Chapters 2 and 3. The change results in the first three chapters being less intense and allows hedging to be covered in more depth.
2. A new chapter on models and numerical procedures (Chapter 20). Much of this material is new, but some has been transferred from the chapter on exotic options in the fourth edition.
xix
3. A new chapter on swaps (Chapter 25). This gives the reader an appreciation of the range of nonstandard swap products that are traded in the over-the-counter market and discusses how they can be valued.
4. There is an extra chapter on credit risk. Chapter 26 discusses the measurement of credit risk and 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 flows better (Chapter 21).
9. The chapter on value at risk has been rewritten so that it provides a better balance between the 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.
Software
A new version of DerivaGem (Version 1.50) is released with this book. This consists of two Excel applications: the Options Calculator and the Applications Builder. The Options Calculator consists of the software in the previous release (with minor improvements). The Applications Builder consists of a number of Excel functions from which users can build their own applications. It includes a number of sample applications and enables students to explore the properties of options and 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 be downloaded from my website:
www.rotman.utoronto.ca/~hull Slides
Several hundred PowerPoint slides can be downloaded from my website. Instructors who adopt the text are welcome to adapt the slides to meet their own needs.
Answers to Questions
As in the fourth edition, end-of-chapter problems are divided into two groups: "Questions and Problems" 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 be purchased by students. Solutions to Assignment Questions are available only in the Instructors Manual.
A cknowledgments
Many people have played a part in the production of this book. Academics, students, and practitioners who have made excellent and useful suggestions include Farhang Aslani, Jas Badyal, Emilio Barone, Giovanni Barone-Adesi, Alex Bergier, George Blazenko, Laurence Booth, Phelim Boyle, Peter Carr, Don Chance, J.-P. Chateau, Ren-Raw Chen, George Constantinides, Michel Crouhy, Emanuel Derman, Brian Donaldson, Dieter Dorp, Scott Drabin, Jerome Duncan, Steinar Ekern, David Fowler, Louis Gagnon, Dajiang Guo, Jrgen Hallbeck, Ian Hawkins, Michael Hemler, Steve Heston, Bernie Hildebrandt, Michelle Hull, Kiyoshi Kato, Kevin Kneafsy, Tibor Kucs, 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, Edward Thorpe, 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 Merkley provided excellent research assistance.
I am particularly grateful to Eduardo Schwartz, who read the original manuscript for the first edition 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 more recent editions.
The first four editions of this book were very popular with practitioners and their comments and suggestions have led to many improvements in the book. The students in my elective courses on derivatives 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. Alan and I have been carrying out joint research in the area of derivatives for the last 18 years. During that time we have spent countless hours discussing different issues concerning derivatives. Many of the new ideas in this book, and many of the new ways used to explain old ideas, are as much Alan's as mine. Alan read the original version of this book very carefully and made many excellent suggestions 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, and encouragement. I would particularly like to thank Mickey Cox (my editor), P. J. Boardman (the editor-in-chief) and Kerri Limpert (the production editor). I am also grateful to Scott Barr, Leah Jewell, Paul Donnelly, and Maureen Riopelle, who at different times have played key roles in the development of the book.
I welcome comments on the book from readers. My email address is:
hull@rotman.utoronto.ca
John C. Hull University of Toronto
INTRODUCTION
In the last 20 years derivatives have become increasingly important in the world of finance. Futures and 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 financial institutions, fund managers, and corporate treasurers in what is termed the over-the-counter market. 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 is dependent 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 in derivatives markets. There is now active trading in credit derivatives, electricity derivatives, weather derivatives, and insurance derivatives. Many new types of interest rate, foreign exchange, and equity derivative products have been created. There have been many new ideas in risk management and risk measurement. Analysts have also become more aware of the need to analyze what are known as real options. (These are the options acquired by a company when it invests in real assets such 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 provide an overview of how they are used by hedgers, speculators, and arbitrageurs. Later chapters will give more details and elaborate on many of the points made here.
1.1 EXCHANGE-TRADED MARKETS
A derivatives exchange is a market where individuals trade standardized contracts that have been defined by the exchange. Derivatives exchanges have existed for a long time. The Chicago Board of Trade (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-arrive contract. Speculators soon became interested in the contract and found trading the contract to be an attractive alternative to trading the grain itself. A rival futures exchange, the Chicago Mercantile Exchange (CME, www.cme.com), was established in 1919. Now futures exchanges exist all over the world.
The Chicago Board Options Exchange (CBOET www.cboe.com) started trading call option
contracts on 16 stocks in 1973. Options had traded prior to 1973 but the CBOE succeeded in creating an orderly market with well-defined contracts. Put option contracts started trading on the exchange in 1977. The CBOE now trades options on over 1200 stocks and many different stock indices. Like futures, options have proved to be very popular contracts. Many other exchanges throughout the world now trade options. The underlying assets include foreign currencies and futures contracts as well as stocks and stock indices.
Traditionally derivatives traders have met on the floor of an exchange and used shouting and a complicated set of hand signals to indicate the trades they would like to carry out. This is known as the open outcry system. In recent years exchanges have increasingly moved from the open outcry system to electronic trading. The latter involves traders entering their desired trades at a keyboard and a computer being used to match buyers and sellers. There seems little doubt that eventually all exchanges will use electronic trading.
1.2 OVER-THE-COUNTER MARKETS
Not all trading is done on exchanges. The over-the-counter market is an important alternative to exchanges and, measured in terms of the total volume of trading, has become much larger than the exchange-traded market. It is a telephone- and computer-linked network of dealers, who do not physically meet. Trades are done over the phone and are usually between two financial institutions or between a financial institution and one of its corporate clients. Financial institutions often act as market makers for the more commonly traded instruments. This means that they are always prepared 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 dispute about what was agreed, the tapes are replayed to resolve the issue. Trades in the over-the-counter market are typically much larger than trades in the exchange-traded market. A key advantage of the over-the-counter market is that the terms of a contract do not have to be those specified by an exchange. Market participants are free to negotiate any mutually attractive deal. A disadvantage is that there is usually some credit risk in an over-the-counter trade (i.e., there is a small risk that the contract will not be honored). As mentioned earlier, exchanges have organized themselves to eliminate virtually all credit risk.
1.3 FORWARD CONTRACTS
A forward contract is a particularly simple derivative. It is an agreement to buy or sell an asset at a certain future time for a certain price. It can be contrasted with a spot contract, which is an agreement to buy or sell an asset today. A forward contract is traded in the over-the-counter market—usually between two financial institutions or between a financial institution and one of its clients.
One of the parties to a forward contract assumes a long position and agrees to buy the underlying asset on a certain specified future date for a certain specified price. The other party assumes a short position 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 "forward desk" within their foreign exchange trading room that is devoted to the trading of forward
Table 1.1 Spot and forward quotes for the USD-GBP exchange rate, August 16, 2001 (GBP = British pound; USD = U.S. dollar)
Spot
1-month forward 3-month forward 6-month forward 1-year forward
Bid 1.4452 1.4435 1.4402 1.4353 1.4262
Offer 1.4456 1.4440 1.4407 1.4359 1.4268
contracts. Table 1.1 provides the quotes on the exchange rate between the British pound (GBP) and the U.S. dollar (USD) that might be made by a large international bank on August 16, 2001. The quote is for the number of USD per GBP. The first quote indicates that the bank is prepared to buy GBP (i.e., sterling) in the spot market (i.e., for virtually immediate delivery) at the rate of $1.4452 per GBP and sell sterling in the spot market at $1.4456 per GBP. The second quote indicates that the bank is prepared to buy sterling in one month at $1.4435 per GBP and sell sterling in one month at $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 are for very large transactions. (As anyone who has traveled abroad knows, retail customers face much larger 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 (on February 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. The corporation then has a long forward contract on GBP. It has agreed that on February 16, 2002, it will buy £1 million from the bank for $1.4359 million. The bank has a short forward contract on GBP. It has agreed that on February 16, 2002, it will sell £1 million for $1.4359 million. Both sides have made a binding commitment.
Payoffs from Forward Contracts
Consider the position of the corporation in the trade we have just described. What are the possible outcomes? The forward contract obligates the corporation to buy £1 million for $1,435,900. If the spot exchange rate rose to, say, 1.5000, at the end of the six months the forward contract would be worth $64,100 (= $1,500,000 - $1,435,900) to the corporation. It would enable £1 million to be purchased at 1.4359 rather than 1.5000. Similarly, if the spot exchange rate fell to 1.4000 at the end of the six months, the forward contract would have a negative value to the corporation of $35,900 because 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 is ST-K
where K is the delivery price and ST is the spot price of the asset at maturity of the contract. This is because the holder of the contract is obligated to buy an asset worth ST for K. Similarly, the payoff from a short position in a forward contract on one unit of an asset is
K-ST
Figure 1.1 Payoffs from forward contracts: (a) long position, (b) short position.
Delivery price = K; price of asset at maturity = SV
These payoffs can be positive or negative. They are illustrated in Figure 1.1. Because it costs nothing to enter into a forward contract, the payoff from the contract is also the trader's total gain or loss from the contract.
Forward Price and Delivery Price
It is important to distinguish between the forward price and delivery price. The forward price is the market 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 a contract maturing on February 16, 2002. The corporation enters into a contract and 1.4359 becomes the delivery price for the contract. As we move through time the delivery price for the corporation's contract does not change, but the forward price for a contract maturing on February 16, 2002, is likely to do so. For example, if GBP strengthens relative to USD in the second half of August the forward price could rise to 1.4500 by September 1, 2001.
Forward Prices and Spot Prices
We 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 on gold. 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 for investments lasting one year is 5% per annum. What is a reasonable value for the one-year forward price of gold?
1 This is not totally realistic. In practice, storage costs are close to zero, but an income of 1 to 2% per annum can be earned by lending gold.
Suppose first that the one-year forward price is $340 per ounce. A trader can immediately take the 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. The remaining $25 is profit. Any one-year forward price greater than $315 will lead to this arbitrage trading strategy being profitable.
Suppose next that the forward price is $300. An investor who has a portfolio that includes gold can 1. 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 for one year, we see that the investor is better off by $15 per ounce. In any situation where the forward price is less than $315, investors holding gold have an incentive to sell the gold and enter into a long 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. As more traders attempt to take advantage of this strategy, the demand for short forward contracts will increase and the one-year forward price of gold will fall. The second strategy is profitable for all 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 forward contracts will increase and the one-year forward price of gold will rise. Assuming that individuals are always willing to take advantage of arbitrage opportunities when they arise, we can conclude that 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.2
1.4 FUTURES CONTRACTS
Like a forward contract, a futures contract is an agreement between two parties to buy or sell an asset at a certain time in the future for a certain price. Unlike forward contracts, futures contracts are normally traded on an exchange. To make trading possible, the exchange specifies certain standardized features of the contract. As the two parties to the contract do not necessarily know each other, the exchange also provides a mechanism that gives the two parties a guarantee that the contract 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 throughout the world, a very wide range of commodities and financial assets form the underlying assets in the various 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.
2 Our arguments make the simplifying assumption that the rate of interest on borrowed funds is the same as the rate of interest on invested funds.
One way in which a futures contract is different from a forward contract is that an exact delivery date is usually not specified. The contract is referred to by its delivery month, and the exchange specifies the period during the month when delivery must be made. For commodities, the delivery period is often the entire month. The holder of the short position has the right to choose the time during the delivery period when it will make delivery. Usually, contracts with several different delivery months are traded at any one time. The exchange specifies the amount of the asset to be delivered 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, the wheat futures contract currently traded on the Chicago Board of Trade. The size of the contract is 5,000 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 that can be delivered and the places where delivery can be made.
Futures prices are regularly reported in the financial press. Suppose that on September 1, the December futures price of gold is quoted as $300. This is the price, exclusive of commissions, at which traders can agree to buy or sell gold for December delivery. It is determined on the floor of the exchange in the same way as other prices (i.e., by the laws of supply and demand). If more traders want 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, delivery procedures, bid-offer spreads, and the role of the exchange clearinghouse are given in Chapter 2.
1.5 OPTIONS
Options are traded both on exchanges and in the over-the-counter market. There are two basic types of options. A call option gives the holder the right to buy the underlying asset by a certain date for a certain price. A put option gives the holder the right to sell the underlying asset by a certain date for a certain price. The price in the contract is known as the exercise price or strike price; the date in the contract is known as the expiration date or maturity. American options can be exercised at any time up to the expiration date. European options can be exercised only on the expiration date itself.4 Most of the options that are traded on exchanges are American. In the exchange-traded equity options market, one contract is usually an agreement to buy or sell 100 shares. European options are generally easier to analyze than American options, and some of the properties of an American 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 holder does 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 nothing to enter into a forward or futures contract, there is a cost to acquiring an option.
Call Options
Consider the situation of an investor who buys a European call option with a strike price of $60 to purchase 100 Microsoft shares. Suppose that the current stock price is $58, the expiration date of
3 In Chapter 3 we discuss the relationship between a futures price and the spot price of the underlying asset (gold, in this case).
4 Note that the terms American and European do not refer to the location of the option or the exchange. Some options trading on North American exchanges are European.
30
20
10
0 - 5
Profit ($)
Terminal stock price ($)
30 40 50 60 70 80 90
Figure 1.2 Profit from buying a European call option on one Microsoft share.
Option price = $5; strike price = $60
the option is in four months, and the price of an option to purchase one share is $5. The initial investment is $500. Because the option is European, the investor can exercise only on the expiration date. 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 these circumstances, the investor loses the whole of the initial investment of $500. If the stock price is above $60 on the expiration date, the option will be exercised. Suppose, for example, that the stock price is $75. By exercising the option, the investor is able to buy 100 shares for $60 per share. If the shares are sold immediately, the investor makes a gain of $15 per share, or $1,500, ignoring transactions costs. When the initial cost of the option is taken into account, the net profit to the investor is $1,000.
Figure 1.2 shows how the investor's net profit or loss on an option to purchase one share varies with the final stock price in the example. (We ignore the time value of money in calculating the profit.) It is important to realize that an investor sometimes exercises an option and makes a loss overall. Suppose that in the example Microsoft's 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 loss overall of $300 when the initial cost of the option is taken into account. It is tempting to argue that the investor should not exercise the option in these circumstances. However, not exercising would lead to an overall loss of $500, which is worse than the $300 loss when the investor exercises. In general, call options should always be exercised at the expiration date if the stock price is above the strike price.
Put Options
Whereas the purchaser of a call option is hoping that the stock price will increase, the purchaser of a put option is hoping that it will decrease. Consider an investor who buys a European put option to sell 100 shares in IBM with a strike price of $90. Suppose that the current stock price is $85, the expiration date of the option is in three months, and the price of an option to sell one share is $7. The initial investment is $700. Because the option is European, it will be exercised only if the stock price is below $90 at the expiration date. Suppose that the stock price is $75 on this date. The investor can
30
20
10
0
- 7
Profit (S)
—V
Terminal stock price ($)
60 70 80 90 100 110 120
Figure 1.3 Profit from buying a European put option on one IBM share.
Option price = $7; strike price = $90
buy 100 shares for $75 per share and, under the terms of the put option, sell the same shares for $90 to realize a gain of $15 per share, or $1,500 (again transactions costs are ignored). When the $700 initial cost of the option is taken into account, the investor's net profit is $800. There is no guarantee that the investor will make a gain. If the final stock price is above $90, the put option expires worthless, and the investor loses $700. Figure 1.3 shows the way in which the investor's profit or loss on an option to sell one share varies with the terminal stock price in this example.
Early Exercise
As already mentioned, exchange-traded stock options are usually American rather than European.
That is, the investor in the foregoing examples would not have to wait until the expiration date before exercising the option. We will see in later chapters that there are some circumstances under which it is optimal to exercise American options prior to maturity.
Option Positions
There are two sides to every option contract. On one side is the investor who has taken the long position (i.e., has bought the option). On the other side is the investor who has taken a short position (i.e., has sold or written the option). The writer of an option receives cash up front, but has potential liabilities later. The writer's profit or loss is the reverse of that for the purchaser of the option. Figures 1.4 and 1.5 show the variation of the profit or loss with the final stock price for writers of the options considered in Figures 1.2 and 1.3.
There are four types of option positions:
1. A long position in a call option.
2. A long position in a put option.
3. A short position in a call option.
4. A short position in a put option.
• • Profit ($)
X
70 80i 9030 40 50 60 Terminal
stock price ($) -10
-20
-30
Figure 1.4 Profit from writing a European call option on one Microsoft share.
Option price = $5; strike price = $60
It is often useful to characterize European option positions in terms of the terminal value or payoff to the investor at maturity. The initial cost of the option is then not included in the calculation. If K is the strike price and S? is the final price of the underlying asset, the payoff from a long position in a European call option is
max(5r - K, 0)
This reflects the fact that the option will be exercised if ST > K and will not be exercised if ST < K.
The payoff to the holder of a short position in the European call option is - max(Sr - K, 0) = min(K - ST, 0)
.. Profit I 7
0
-10
-20
-30
60 70 80
Terminal stock price ($)
90 100 110 120
Figure 1.5 Profit from writing a European put option on one IBM share.
Option price = $7; strike price = $90
,, 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 = ST
The 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 (Sr - K, 0) Figure 1.6 shows these payoffs.
1.6 TYPES OF TRADERS
Derivatives markets have been outstandingly successful. The main reason is that they have attracted many different types of traders and have a great deal of liquidity. When an investor wants to take one side of a contract, there is usually no problem in finding someone that is prepared 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 future movements in a market variable. Speculators use them to bet on the future direction of a market variable. Arbitrageurs take offsetting positions in two or more instruments to lock in a profit. In the next few sections, we consider the activities of each type of trader in more detail.
Hedgers
We 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, knows that 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 financial institution in the three-month forward market at 1.4407. This would have the effect of fixing the price 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 exporting goods to the United Kingdom and on August 16, 2001, knows that it will receive £30 million three months 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 the U.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 is less than $14,407,000. On the other hand, if the exchange rate is 1.5000, the £10 million will cost
$15,000,000—and the company will wish it had hedged! The position of ExportCo if it does not hedge is the reverse. If the exchange rate in September proves to be less than 1.4402, the company will wish 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 underlying asset is ensured. However, there is no assurance that the outcome with hedging will be better than the outcome without hedging.
Options can also be used for hedging. Consider an investor who in May 2000 owns 1,000 Microsoft shares. The current share price is $73 per share. The investor is concerned that the developments in Microsoft's antitrust case may cause the share price to decline sharply in the next two months and wants protection. The investor could buy 10 July put option contracts with a strike price 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 share during the life of the option. If the market price of Microsoft falls below $65, the options can be exercised so that $65,000 is realized for the entire holding. When the cost of the options is taken into account, the amount realized is $62,500. If the market price stays above $65, the options are not 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 or receive for the underlying asset. Option contracts, by contrast, provide insurance. They offer a way for investors to protect themselves against adverse price movements in the future while still allowing them to benefit from favorable price movements. Unlike forwards, options involve the payment of an up-front fee.
Speculators
We 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,
speculators wish to take a position in the market. Either they are betting that the price will go up or they are betting that it will go down.
Consider a U.S. speculator who in February thinks that the British pound will strengthen relative to the U.S. dollar over the next two months and is prepared to back that hunch to the tune of £250,000. One thing the speculator can do is simply purchase £250,000 in the hope that the sterling can be sold later at a profit. The sterling once purchased would be kept in an interest-bearing account. Another possibility is to take a long position in four CME April futures contracts on sterling. (Each futures contract is for the purchase of £62,500.) Suppose that the current exchange rate is 1.6470 and the April futures price is 1.6410. If the exchange rate turns out to be 1.7000 in April, the futures contract alternative enables the speculator to realize a profit of (1.7000 - 1.6410) x 250,000 = $14,750. The cash market alternative leads to an asset being purchased for 1.6470 in February and sold for 1.7000 in April, so that a profit of (1.7000- 1.6470) x 250,000 = $13,250 is made. If the exchange rate falls to 1.6000, the futures contract gives rise to a (1.6410 - 1.6000) x 250,000 = $10,250 loss, whereas the cash market alternative gives rise to a loss of (1.6470 - 1.6000) x 250,000 = $11,750. The alternatives appear to give rise to slightly different profits and losses. But these calculations do not reflect the interest that is earned or paid. It will be shown in Chapter 3 that when the interest earned in sterling and the interest paid in dollars are taken into account, the profit or loss from the two alternatives is the same.
What then is the difference between the two alternatives? The first alternative of buying sterling requires an up-front investment of $411,750. As we will see in Chapter 2, the second alternative requires only a small amount of cash—perhaps $25,000—to be deposited by the speculator in what 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 October and a speculator considers that Cisco is likely to increase in value over the next two months. The stock price is currently $20, and a two-month call option with a $25 strike price is currently selling for $1. Table 1.2 illustrates two possible alternatives assuming that the speculator is willing to invest $4,000. The first alternative involves the purchase of 200 shares. The second involves the purchase of 4,000 call options (i.e., 20 call option contracts).
Suppose that the speculator's hunch is correct and the price of Cisco's shares rises to $35 by December. The first alternative of buying the stock yields a profit of
200 x ($35 - $20) = $3,000 '
However, the second alternative is far more profitable. A call option on Cisco with a strike price Table 1.2 Comparison of profits (losses) from two alternative
strategies for using $4,000 to speculate on Cisco stock in October
Investor's strategy Buy shares Buy call options
December
$15 ($1,000) ($4,000)
stock price
$35
$3,000
$36,000
of $25 gives a payoff of $10, because it enables something worth $35 to be bought for $25. The total payoff from the 4,000 options that are purchased under the second alternative is
4,000 x $10 = $40,000 Subtracting the original cost of the options yields a net profit of
$40,000 - $4,000 = $36,000
The 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 by December. The first alternative of buying stock yields a loss of
200 x ($20-$15) = $1,000
Because the call options expire without being exercised, the options strategy would lead to a loss of
$4,000—the original amount paid for the options.
It is clear from Table 1.2 that options like futures provide a form of leverage. For a given investment, the use of options magnifies the financial consequences. Good outcomes become very good, while bad outcomes become very bad!
Futures and options are similar instruments for speculators in that they both provide a way in which a type of leverage can be obtained. However, there is an important difference between the two.
With futures the speculator's potential loss as well as the potential gain is very large. With options no matter how bad things get, the speculator's loss is limited to the amount paid for the options.
Arbitrageurs
Arbitrageurs 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 two or more markets. In later chapters we will see how arbitrage is sometimes possible when the futures price of an asset gets out of line with its cash price. We will also examine how arbitrage can be used in 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 the London Stock Exchange (www.stockex.co.uk). Suppose that the stock price is $152 in New York and £100 in London at a time when the exchange rate is $1.5500 per pound. An arbitrageur could simultaneously 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 profit for a small investor. However, a large investment house faces very low transactions costs in both the stock market and the foreign exchange market. It would find the arbitrage opportunity very attractive 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 buy the 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 the two 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 the dollar price could ever exist in the first place. Generalizing from this example, we can say that the
