Options, Futures, and Other Derivatives, 5th edition

F I F T H

E D I T I O N

JOHN C.HULL

PRENTICE HALL FINANCE SERIES Personal Finance Keown, Personal Finance: Turning Monev into Wealth, Second Edition Trivoli, Personal Portfolio Management: Fundamentals & Strategies Winger/Frasca, Personal Finance: An Integrated Planning Approach, Sixth Edition

Undergraduate Investments/Portfolio Management Alexander/Sharpe/Bailey, Fundamentals of Investments, Third Edition Fabozzi, Investment Management, Second Edition Haugen, Modern Investment Theory, Fifth Edition Haugen, The New Finance, Second Edition Haugen, The Beast on Wall Street Haugen, The Inefficient Stock Market, Second Edition Holden, Spreadsheet Modeling: A Book and CD-ROM Series (Available in Graduate and Undergraduate Versions) Nofsinger. The Psychology of Investing Taggart, Quantitative Analysis for Investment Management Winger/Frasca, Investments, Third Edition

Graduate Investments/Portfolio Management Fischer/Jordan, Security Analysis and Portfolio Management, Sixth Edition Francis/Ibbotson. Investments: A Global Perspective Haugen, The Inefficient Stock Market, Second Edition Holden, Spreadsheet Modeling: A Book and CD-ROM Series (Available in Graduate and Undergraduate Versions) Nofsinger, The Psychology of Investing Sharpe/Alexander/Bailey. Investments, Sixth Edition

Options/Futures/Derivatives Hull, Fundamentals of Futures and Options Markets, Fourth Edition Hull, Options, Futures, and Other Derivatives, Fifth Edition

Risk Management/Financial Engineering Mason/Merton/Perold/Tufano, Cases in Financial Engineering

Fixed Income Securities Handa, FinCoach: Fixed Income (software)

Bond Markets Fabozzi, Bond Markets, Analysis and Strategies, Fourth Edition

Undergraduate Corporate Finance Bodie/Merton, Finance Emery/Finnerty/Stowe, Principles of Financial Management Emery/Finnerty, Corporate Financial Management Gallagher/Andrew, Financial Management: Principles and Practices, Third Edition Handa, FinCoach 2.0 Holden, Spreadsheet Modeling: A Book and CD-ROM Series (Available in Graduate and Undergraduate Versions) Keown/Martin/Petty/Scott, Financial Management, Ninth Edition Keown/Martin/Petty/Scott, Financial Management, 9/e activehook M Keown/Martin/Petty/Scott, Foundations of Finance: The Logic and Practice of Financial Management, Third Edition Keown/Martin/Petty/Scott, Foundations of Finance, 3je activebook ' Mathis, Corporate Finance Live: A Web-based Math Tutorial Shapiro/Balbirer, Modern Corporate Finance: A Multidiseiplinary Approach to Value Creation Van Horne/Wachowicz, Fundamentals of Financial Management, Eleventh Edition Mastering Finance CD-ROM

Fifth Edition

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 JERSEY 0 7 4 5 8

CONTENTS Preface 1. Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7

2. Mechanics of 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

xix Exchange-traded markets Over-the-counter markets Forward contracts Futures contracts Options Types of traders Other derivatives Summary Questions and problems Assignment questions

1 1 2 2 5 6 10 14 15 16 17

futures markets Trading futures contracts Specification of the futures contract Convergence of futures price to spot price Operation of margins Newspaper quotes Keynes and Hicks Delivery Types of traders Regulation Accounting and tax Forward contracts vs. futures contracts Summary Suggestions for further reading Questions and problems Assignment questions

19 19 20 23 24 27 31 31 32 33 35 36 37 38 38 40

3. Determination of forward and futures prices 3.1 Investment assets vs. consumption assets 3.2 Short selling 3.3 Measuring interest rates 3.4 Assumptions and notation 3.5 Forward price for an investment asset 3.6 Known income 3.7 Known yield 3.8 Valuing forward contracts 3.9 Are forward prices and futures prices equal? 3.10 Stock index futures 3.11 Forward and futures contracts on currencies 3.12 Futures on commodities

;

41 41 41 42 44 45 47 49 49 51 52 55 58 ix

Contents

3.13 3.14 3.15

Cost of carry Delivery options Futures prices and the expected future spot price Summary Suggestions for further reading Questions and problems Assignment questions Appendix 3A: Proof that forward and futures prices are equal when interest rates are constant

4. Hedging strategies using futures 4.1 Basic principles 4.2 Arguments for and against hedging 4.3 Basis risk 4.4 Minimum variance hedge ratio 4.5 Stock index futures 4.6 Rolling the hedge forward Summary Suggestions for further reading Questions and problems Assignment questions Appendix 4A: Proof of the minimum variance hedge ratio formula 5. Interest rate 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14

markets Types of rates Zero rates Bond pricing Determining zero rates Forward rates Forward rate agreements Theories of the term structure Day count conventions Quotations Treasury bond futures Eurodollar futures The LIBOR zero curve Duration Duration-based hedging strategies Summary Suggestions for further reading Questions and problems Assignment questions

6. Swaps 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Mechanics of interest rate swaps The comparative-advantage argument Swap quotes and LIBOR zero rates Valuation of interest rate swaps Currency swaps Valuation of currency swaps Credit risk Summary Suggestions for further reading Questions and problems Assignment questions

60 60 61 63 64 65 67 68 70 70 72 75 78 82 86 87 88 88 90 92 93 93 94 94 96 98 100 102 102 103 104 110 Ill 112 116 118 119 120 123 125 125 131 134 136 140 143 145 146 147 147 149

Contents

xi

7. Mechanics of 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11

options markets Underlying assets Specification of stock options Newspaper quotes Trading Commissions Margins The options clearing corporation Regulation Taxation Warrants, executive stock options, and convertibles Over-the-counter markets Summary Suggestions for further reading Questions and problems Assignment questions

151 151 152 155 157 157 158 160 161 161 162 163 163 164 164 165

8. Properties of 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

stock options Factors affecting option prices Assumptions and notation Upper and lower bounds for option prices Put-call parity Early exercise: calls on a non-dividend-paying stock Early exercise: puts on a non-dividend-paying stock Effect of dividends Empirical research Summary Suggestions for further reading Questions and problems Assignment questions

167 167 170 171 174 175 177 178 179 180 181 182 183

9. Trading strategies involving options 9.1 Strategies- involving a single option and a stock 9.2 Spreads 9.3 Combinations 9.4 Other payoffs Summary Suggestions for further reading Questions and problems Assignment questions

185 185 187 194 197 197 198 198 199

10. Introduction 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

200 200 203 205 208 209 210 211 212 213 214 214 215

to binomial trees A one-step binomial model Risk-neutral valuation Two-step binomial trees A put example American options Delta Matching volatility with u and d Binomial trees in practice Summary Suggestions for further reading Questions and problems Assignment questions

xii

Contents

11. A model of the behavior of stock prices 11.1 The Markov property 11.2 Continuous-time stochastic processes 11.3 The process for stock prices 11.4 Review of the model 11.5 The parameters 11.6 Ito's lemma 11.7 The lognormal property Summary Suggestions for further reading Questions and problems Assignment questions Appendix 11 A: Derivation of Ito's lemma

216 216 217 222 223 225 226 227 228 229 229 230 232

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 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9

stock indices, currencies, and futures Results for a stock paying a known dividend yield Option pricing formulas Options on stock indices Currency options Futures options Valuation of futures options using binomial trees Futures price analogy Black's model for valuing futures options Futures options vs. spot options Summary Suggestions for further reading Questions and problems Assignment questions Appendix 13 A: Derivation of differential equation satisfied by a derivative dependent on a stock providing a dividend yield

267 267 268 270 276 278 284 286 287 288 289 290 291 294 295

Contents

xiii

Appendix 13B: Derivation of differential equation satisfied by a derivative dependent on a futures price

297

14. The Greek letters 14.1 Illustration 14.2 Naked and covered positions 14.3 A stop-loss strategy 14.4 Delta hedging 14.5 Theta 14.6 Gamma 14.7 Relationship between delta, theta, and gamma 14.8 Vega 14.9 Rho 14.10 Hedging in practice 14.11 Scenario analysis 14.12 Portfolio insurance 14.13 Stock market volatility Summary Suggestions for further reading Questions and problems : Assignment questions Appendix 14A: Taylor series expansions and hedge parameters

299 299 300 300 302 309 312 315 316 318 319 319 320 323 323 324 326 327 329

15. Volatility smiles 15.1 Put-call parity revisited 15.2 Foreign currency options 15.3 Equity options 15.4 The volatility term structure and volatility surfaces 15.5 Greek letters 15.6 When a single large jump is anticipated 15.7 Empirical research Summary Suggestions for further reading Questions and problems Assignment questions Appendix 15A: Determining implied risk-neutral distributions from volatility smiles

330 330 331 334 336 337 338 339 341 341 343 344

16. Value at risk 16.1 The VaR measure 16.2 Historical simulation 16.3 Model-building approach 16.4 Linear model 16.5 Quadratic model 16.6 Monte Carlo simulation 16.7 Comparison of approaches 16.8 Stress testing and back testing 16.9 Principal components analysis Summary Suggestions for further reading Questions and problems Assignment questions Appendix 16A: Cash-flow mapping Appendix 16B: Use of the Cornish-Fisher expansion to estimate VaR

346 346 348 350 352 356 359 359 360 360 364 364 365 366 368 370

345

xiv

Contents

17. Estimating volatilities and correlations 17.1 Estimating volatility 17.2 The exponentially weighted moving average model 17.3 The GARCH(1,1) model 17.4 Choosing between the models 17.5 Maximum likelihood methods 17.6 Using GARCHfl, 1) to forecast future volatility 17.7 Correlations Summary Suggestions for further reading Questions and problems Assignment questions

372 372 374 376 377 378 382 385 388 388 389 391

18. Numerical procedures 18.1 Binomial trees 18.2 Using the binomial tree for options on indices, currencies, and futures contracts 18.3 Binomial model for a dividend-paying stock '. 18.4 Extensions to the basic tree approach 18.5 Alternative procedures for constructing trees 18.6 Monte Carlo simulation 18.7 Variance reduction procedures 18.8 Finite difference methods 18.9 Analytic approximation to American option prices Summary Suggestions for further reading Questions and problems Assignment questions Appendix 18A: Analytic approximation to American option prices of MacMillan and of Barone-Adesi and Whaley

392 392

19. Exotic options 19.1 Packages 19.2 Nonstandard American options 19.3 Forward start options 19.4 Compound options 19.5 Chooser options 19.6 Barrier options 19.7 Binary options 19.8 Lookback options 19.9 Shout options 19.10 Asian options 19.11 Options to exchange one asset for another 19.12 Basket options 19.13 Hedging issues 19.14 Static options replication Summary Suggestions for further reading Questions and problems Assignment questions Appendix 19A: Calculation of the first two moments of arithmetic averages and baskets

435 435 436 437 437 438 439 441 441 443 443 445 446 447 447 449 449 451 452

20. More on models and numerical procedures 20.1 The CEV model 20.2 The jump diffusion model

456 456 457

399 402 405 406 410 414 418 427 427 428 430 432 433

454

Contents

xv

20.3 20.4 20.5 20.6 20.7 20.8 20.9

Stochastic volatility models The IVF model Path-dependent derivatives Lookback options Barrier options Options on two correlated assets Monte Carlo simulation and American options Summary Suggestions for further reading Questions and problems Assignment questions

458 460 461 465 467 472 474 478 479 480 481

21. Martingales 21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 21.9

and measures The market price of risk Several state variables • Martingales Alternative choices for the numeraire Extension to multiple independent factors Applications Change of numeraire Quantos Siegel's paradox Summary Suggestions for further reading Questions and problems Assignment questions Appendix 21 A: Generalizations of Ito's lemma Appendix 2IB: Expected excess return when there are multiple sources of uncertainty

483 484 487 488 489 492 493 495 497 499 500 500 501 502 504

22. Interest rate 22.1 22.2 22.3 22.4 22.5 22.6 22.7 22.8 22.9

derivatives: the standard market models Black's model Bond options Interest rate caps European swap options Generalizations Convexity adjustments Timing adjustments Natural time lags Hedging interest rate derivatives Summary Suggestions for further reading Questions and problems Assignment questions Appendix 22A: Proof of the convexity adjustment formula

508 508 511 515 520 524 524 527 529 530 531 531 532 534 536

23. Interest rate 23.1 23.2 23.3 23.4 23.5 23.6 23.7 23.8 23.9

derivatives: models of the short rate Equilibrium models One-factor equilibrium models The Rendleman and Bartter model The Vasicek model The Cox, Ingersoll, and Ross model Two-factor equilibrium models No-arbitrage models The Ho and Lee model The Hull and White model

537 537 538 538 539 542 543 543 544 546

506

xvi

Contents

23.10 23.11 23.12 23.13 23.14 23.15 23.16

24. Interest rate 24.1 24.2 24.3 24.4

Options on coupon-bearing bonds Interest rate trees A general tree-building procedure Nonstationary models Calibration Hedging using a one-factor model Forward rates and futures rates Summary Suggestions for further reading Questions and problems Assignment questions

549 550 552 563 564 565 566 566 567 568 570

derivatives: more advanced models 571 Two-factor models of the short rate 571 The Heath, Jarrow, and Morton model 574 The LIBOR market model 577 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 25.1 Variations on the vanilla deal 25.2 Compounding swaps 25.3 Currency swaps 25.4 More complex swaps 25.5 Equity swaps 25.6 Swaps with embedded options 25.7 Other swaps 25.8 Bizarre deals Summary Suggestions for further reading Questions and problems Assignment questions Appendix 25A: Valuation of an equity swap between payment dates

594 594 595 598 598 601 602 605 605 606 606 607 607 609

26. Credit 23.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 26.9

610 610 619 619 620 621 623 626 627 630 633 633 634 635 636

risk Bond prices and the probability of default Historical data Bond prices vs. historical default experience Risk-neutral vs. real-world estimates Using equity prices to estimate default probabilities The loss given default Credit ratings migration Default correlations Credit value at risk Summary Suggestions for further reading Questions and problems Assignment questions Appendix 26A: Manipulation of the matrices of credit rating changes

Contents

xvn

27. Credit derivatives 27.1 Credit default swaps 27.2 Total return swaps 27.3 Credit spread options 27.4 Collateralized debt obligations 27.5 Adjusting derivative prices for default risk 27.6 Convertible bonds Summary Suggestions for further reading Questions and problems Assignment questions 28. Real options 28.1 Capital investment appraisal 28.2 Extension of the risk-neutral valuation framework 28.3 Estimating the market price of risk 28.4 Application to the valuation of a new business 28.5 Commodity prices 28.6 Evaluating options in an investment opportunity Summary Suggestions for further reading Questions and problems Assignment questions

637 637 644 645 646 647 652 655 655 656 658 660 660 661 665 666 667 670 675 676 676 677

29. Insurance, weather, and energy derivatives 29.1 Review of pricing issues 29.2 Weather derivatives 29.3 Energy derivatives 29.4 Insurance derivatives Summary Suggestions for further reading Questions and problems Assignment questions

678 678 679 680 682 683 684 684 685

30. Derivatives mishaps and what we can learn from them 30.1 Lessons for all users of derivatives 30.2 Lessons for financial institutions 30.3 Lessons for nonfinancial corporations Summary Suggestions for further reading Glossary of notation

686 686 690 693 694 695 697

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

PREFACE 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 mathematical 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

xx

Preface

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.

Preface

xxi

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 Rendleman 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 DerivaGem 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: [email protected] John C. Hull University of Toronto

C H A P T E R

1

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 derivatives 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

CHAPTER 1

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

Introduction

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

Offer

1.4452 1.4435 1.4402 1.4353 1.4262

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

CHAPTER 1

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.

Introduction

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.

CHAPTER 1

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.

Introduction

Profit ($) 30

20

10 Terminal stock price ($) 0

30

40

50

60

70

80

90

-5

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

CHAPTER 1

Profit (S) 30

20

10 Terminal stock price ($) 0

—V

60

70

80

90

100

110

120

-7

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. 2. 3. 4.

A A A A

long position in a call option. long position in a put option. short position in a call option. short position in a put option.

Introduction

• • Profit ($)

30

X

40

50

70

80

90

i

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(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 7 60 0

70

Terminal stock price ($)

80 90

100

110

120

-10

-20

-30

Figure 1.5 Profit from writing a European put option on one IBM share. Option price = $7; strike price = $90

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 = 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 (S r - 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.

Introduction

11

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 threemonth 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,

12

CHAPTER 1

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 December stock price Investor's strategy Buy shares Buy call options

$15

$35

($1,000) ($4,000)

$3,000 $36,000

Introduction

13

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 riskfree 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 profithungry 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

14

CHAPTER 1

very existence of arbitrageurs means that in practice only very small arbitrage opportunities are observed in the prices that are quoted in most financial markets. In this book most of the arguments concerning futures prices, forward prices, and the values of option contracts will be based on the assumption that there are no arbitrage opportunities.

1.7

OTHER DERIVATIVES

The 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 very imaginative in designing nonstandard derivatives to meet the needs of clients. Sometimes these are sold by financial institutions to their corporate clients in the over-the-counter market. On other occasions, 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 put options. Others are far more complex. The possibilities for designing new interesting nonstandard derivatives seem to be almost limitless. Nonstandard derivatives are sometimes termed exotic options or just exotics. In Chapter 19 we discuss different types of exotics and consider how they can be valued. We now give examples of three derivatives that, although they appear to be complex, can be decomposed into portfolios of plain vanilla call and put options.5 Example 1.1: Standard Oil's Bond Issue A bond issue by Standard Oil worked as follows. The holder received no interest. At the bond's 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 bond's 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 Trust's 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-maxk L

l , 0 0 0S ( ^ \T

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 forwards) 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.

Introduction

15

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 Examples As mentioned earlier, there is virtually no limit to the innovations that are possible in the derivatives area. Some of the options traded in the over-the-counter market have payoffs dependent on maximum value attained by a variable during a period of time; some have payoffs dependent on the average value of a variable during a period of time; some have exercise prices that are functions of time; some have features where exercising one option automatically gives the holder another option; 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, stock indices, interest rates, exchange rates, and commodity prices. However, other underlying variables are becoming increasingly common. For example, the payoffs from credit derivatives, which are discussed in Chapter 27, depend on the creditworthiness of one or more companies; weather derivatives have payoffs dependent on the average temperature at particular locations; insurance derivatives have payoffs dependent on the dollar amount of insurance claims of a specified type made during a specified period; electricity derivatives have payoffs dependent on the spot price of electricity; and so on. Chapter 29 discusses weather, insurance, and energy derivatives.

SUMMARY One of the exciting developments in finance over the last 25 years has been the growth of derivatives markets. In many situations, both hedgers and speculators find it more attractive to trade a derivative on an asset than to trade the asset itself. Some derivatives are traded on exchanges. Others are traded by financial institutions, fund managers, and corporations in the over-the-counter market, or added to new issues of debt and equity securities. Much of this book is concerned with the valuation of derivatives. The aim is to present a unifying framework within which all derivatives—not just options or futures—can be valued. In this chapter we have taken a first look at forward, futures, and options contracts. A forward or futures contract involves an obligation to buy or sell an asset at a certain time in the future for a certain price. There are two types of options: calls and puts. A call option gives the holder the right to buy an asset by a certain date for a certain price. A put option gives the holder the right to sell an asset by a certain date for a certain price. Forwards, futures, and options trade on a wide range of different underlying assets. Derivatives have been very successful innovations in capital markets. Three main types of traders can be identified: hedgers, speculators, and arbitrageurs. Hedgers are in the position where they face 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 get extra leverage. Arbitrageurs are in business to take advantage of a discrepancy between prices in two different markets. If, for example, they see the futures price of an asset getting out of line with the cash price, they will take offsetting positions in the two markets to lock in a profit.

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 the contract 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 the cotton 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 an expiration date in three months. The current stock price of AOL Time Warner is $41 and the contract 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 two alternative strategies, one involving an investment in the stock and the other involving investment in 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 you with 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 the option be exercised? Draw a diagram showing the variation of the trader's profit with the stock price 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 the option be exercised? Draw a diagram showing the variation of the trader's profit with the stock price 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 diagram showing the variation of the trader's 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 circumstances will the option be exercised? Draw a diagram illustrating how the profit from a long position in the 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 what circumstances 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 the profit from a short position in the option depends on the stock price at maturity of the option.

Introduction

17

1.17. 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 trader's cash flows if the option is held until September 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. Characterize the 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 long European put option on the asset with the same maturity as the forward contract and a strike price 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 bond and two options. 1.25. Show that a range forward contract such as the one described in Section 1.7 is a combination of two 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 on January 1, 2003. On September 1, 2002, it enters into a forward contract to sell 10 million Japanese 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: Spot 1.6080 90-day forward 1.6056 180-day forward 1.6018 What 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 QUESTIONS 1.28. 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? Assume that 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 $95 currently sell for $4.70. An investor who feels that the price of the stock will increase is trying to decide between buying 100 shares and buying 2,000 call options (= 20 contracts). Both strategies involve an investment of $9,400. What advice would you give? How high does the stock price have to 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, a long position in call options on oil with a strike price of $25, and a short position in call options on oil with a strike price of $40.

18

CHAPTER 1

1.31. Use the DerivaGem software to calculate the value of the range forward contract considered in Section 1.7 on the assumption that the exchange rate volatility is 15% per annum. Adjust the upper end of the band so that the contract has zero value initially. Assume that the dollar and sterling 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 $250 per ounce and sell gold for $249 per ounce. The trader can borrow funds at 6% per year and invest funds at 5.5% per year. (Both interest rates are expressed with annual compounding.) For what 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 in Section 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 to calculate the cost of setting up the hedge in each case assuming that the exchange rate volatility is 12%, interest rates in the United States are 3% and interest rates in Britain are 4.4%. Assume that the 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 same underlying asset, strike price and maturity. Describe the trader's position. Under what circumstances does the price of the call equal the price of the put?

CHAPTER

2

MECHANICS OF FUTURES MARKETS In Chapter 1 we explained that both futures and forward contracts are agreements to buy or sell an asset 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 are private agreements between two financial institutions or between a financial institution and one of its corporate clients. This chapter covers the details of how futures markets work. We examine issues such as the specification of contracts, the operation of margin accounts, the organization of exchanges, the regulation of markets, the way in which quotes are made, and the treatment of futures transactions for accounting and tax purposes. We also examine forward contracts and explain the difference between the pattern of payoffs realized from futures and forward contracts.

2.1

TRADING FUTURES CONTRACTS

As mentioned in Chapter 1, futures contracts are now traded very actively all over the world. The two 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 largest exchanges in Europe are the London International Financial Futures and Options Exchange (www.liffe.com) and Eurex (www.eurexchange.com). Other large exchanges include Bolsa de Mercadorias y Futuros (www.bmf.com.br) in Sao Paulo, the Tokyo International Financial Futures Exchange (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 the end of this book. We examine how a futures contract comes into existence by considering the corn futures contract traded on the Chicago Board of Trade (CBOT). On March 5, an investor in New York might call a broker with instructions to buy 5,000 bushels of corn for delivery in July of the same year. The broker would immediately pass these instructions on to a trader on the floor of the CBOT. The broker would request a long position in one contract because each corn contract on the CBOT is for the delivery of exactly 5,000 bushels. At about the same time, another investor in Kansas might instruct a broker to sell 5,000 bushels of corn for July delivery. This broker would then pass instructions to short one contract to a trader on the floor of the CBOT. The two 19

20

CHAPTER 2

floor traders would meet, agree on a price to be paid for the corn in July, and the deal would be done. The investor in New York who agreed to buy has a long futures position in one contract; the investor in Kansas who agreed to sell has a short futures position in one contract. The price agreed to on the floor of the exchange is the current futures price for July corn. We will suppose the price is 170 cents per bushel. This price, like any other price, is determined by the laws of supply and demand. If at a particular time more traders wish to sell July corn than buy July corn, the price will go down. New buyers then enter the market so that a balance between buyers and sellers is maintained. If more traders wish to buy July corn than to sell July corn, the price goes up. New sellers then enter the market and a balance between buyers and sellers is maintained. Closing Out Positions The vast majority of futures contracts do not lead to delivery. The reason is that most traders choose to close out their positions prior to the delivery period specified in the contract. Closing out a position means entering into the opposite type of trade from the original one. For example the New York investor who bought a July corn futures contract on March 5 can close out the position by 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 contract on April 20. In each case, the investor's total gain or loss is determined by the change in the futures price 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 a trade in a futures transaction. Consider trader A who initiates a long futures position by trading one contract. Suppose that trader B is on the other side of the transaction. At a later stage trader A might close out the position by entering into a short contract. The trader on the other side of this second transaction does not have to be, and usually is not, trader B.

2.2

THE SPECIFICATION OF THE FUTURES CONTRACT

When developing a new contract, the exchange must specify in some detail the exact nature of the agreement between the two parties. In particular, it must specify the asset, the contract size (exactly how much of the asset will be delivered under one contract), where delivery will be made, and when delivery will be made. Sometimes alternatives are specified for the grade of the asset that will be delivered or for the delivery locations. As a general rule, it is the party with the short position (the party that has agreed to sell the asset) that chooses what will happen when alternatives are specified by the exchange. When the party with the short position is ready to deliver, it files a notice of intention to deliver with the exchange. This notice indicates selections it has made with respect to the grade of asset that will be delivered and the delivery location. The Asset When the asset is a commodity, there may be quite a variation in the quality of what is available in the marketplace. When the asset is specified, it is therefore important that the exchange stipulate

Mechanics of Futures Markets

21

the grade or grades of the commodity that are acceptable. The New York Cotton Exchange has specified 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, gradestamped 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 grade chosen. 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 by the exchange. The financial assets in futures contracts are generally well defined and unambiguous. For example, there is no need to specify the grade of a Japanese yen. However, there are some interesting features of the Treasury bond and Treasury note futures contracts traded on the Chicago Board of Trade. The underlying asset in the Treasury bond contract is any long-term U.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 a maturity of no less than 6.5 years and no more than 10 years from the date of delivery. In both cases, the exchange has a formula for adjusting the price received according to the coupon and maturity date of the bond delivered. This is discussed in Chapter 5. The Contract Size The contract size specifies the amount of the asset that has to be delivered under one contract. This is an important decision for the exchange. If the contract size is too large, many investors who wish to hedge relatively small exposures or who wish to take relatively small speculative positions will be unable to use the exchange. On the other hand, if the contract size is too small, trading may be expensive 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 what is 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 futures contract traded on the Chicago Board of Trade, instruments with a face value of $100,000 are delivered. In some cases exchanges have introduced "mini" contracts to attract smaller investors. For example, the CME's Mini Nasdaq 100 contract is on 20 times the Nasdaq 100 index whereas the regular contract is on 100 times the index. Delivery Arrangements The place where delivery will be made must be specified by the exchange. This is particularly important for commodities that involve significant transportation costs. In the case of the Chicago

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CHAPTER 2

Mercantile Exchange's 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 short position is sometimes adjusted according to the location chosen by that party. For example, in the case of the corn futures contract traded by the Chicago Board of Trade, delivery can be made at Chicago, Burns Harbor, Toledo, or St. Louis. However, deliveries at Toledo and St. Louis are made at a discount of 4 cents per bushel from the Chicago contract price. Delivery Months A futures contract is referred to by its delivery month. The exchange must specify the precise period during the month when delivery can be made. For many futures contracts, the delivery period is the whole month. The delivery months vary from contract to contract and are chosen by the exchange to meet the needs of market participants. For example, the main delivery months for currency futures on the Chicago Mercantile Exchange are March, June, September, and December; corn futures traded on the 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 a number of subsequent delivery months. The exchange specifies when trading in a particular month's contract will begin. The exchange also specifies the last day on which trading can take place for a given contract. Trading generally ceases a few days before the last day on which delivery can be made. Price Quotes The futures price is quoted in a way that is convenient and easy to understand. For example, crude oil futures prices on the New York Mercantile Exchange are quoted in dollars per barrel to two decimal places (i.e., to the nearest cent). Treasury bond and Treasury note futures prices on the Chicago Board of Trade are quoted in dollars and thirty-seconds of a dollar. The minimum price movement that can occur in trading is consistent with the way in which the price is quoted. Thus, it is SO.01 per barrel for the oil futures and one thirty-second of a dollar for the Treasury bond and Treasury note futures. Daily Price Movement Limits For most contracts, daily price movement limits are specified by the exchange. If the price moves down by an amount equal to the daily price limit, the contract is said to be limit down. If it moves up by the limit, it is said to be limit up. A limit move is a move in either direction equal to the daily price 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 of speculative excesses. However, limits can become an artificial barrier to trading when the price of the underlying commodity is increasing or decreasing rapidly. Whether price limits are, on balance, good for futures markets is controversial.

23

Mechanics of Futures Markets

Position Limits

Position limits are the maximum number of contracts that a speculator may hold. In the Chicago Mercantile Exchange's random-length lumber contract, for example, the position limit at the time of writing is 1,000 contracts with no more than 300 in any one delivery month. Bona fide hedgers are not affected by position limits. The purpose of the limits is to prevent speculators from exercising undue influence on the market.

2.3 CONVERGENCE OF FUTURES PRICE TO SPOT PRICE As the delivery month of a futures contract is approached, the futures price converges to the spot price of the underlying asset. When the delivery period is reached, the futures price equals—or is very close to—the spot price. To see why this is so, we first suppose that the futures price is above the spot price during the delivery 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 exceeds the spot price. As traders exploit this arbitrage opportunity, the futures price will fall. Suppose next that the futures price is below the spot price during the delivery period. Companies interested in acquiring the asset will find it attractive to enter into a long futures contract and then wait for delivery 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 illustrates the convergence of the futures price to the spot price. In Figure 2.1a the futures price is above the spot price prior to the delivery month, and in Figure 2.1b the futures

Futures price

Spot price

Time (a)

Figure 2.1

Time >(b)

Relationship between futures price and spot price as the delivery month is approached, (a) Futures price above spot price; (b) futures price below spot price

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CHAPTER 2

price is below the spot price prior to the delivery month. The circumstances under which these two patterns are observed are discussed later in this chapter and in Chapter 3.

2.4

OPERATION OF MARGINS

If two investors get in touch with each other directly and agree to trade an asset in the future for a certain price, there are obvious risks. One of the investors may regret the deal and try to back out. Alternatively, the investor simply may not have the financial resources to honor the agreement. One of the key roles of the exchange is to organize trading so that contract defaults are avoided. This is where margins come in. Marking to Market

To illustrate how margins work, we consider an investor who contacts his or her broker on Thursday, June 5 to buy two December gold futures contracts on the New York Commodity Exchange (COMEX). We suppose that the current futures price is $400 per ounce. Because the contract size is 100 ounces, the investor has contracted to buy a total of 200 ounces at this price. The broker will require the investor to deposit funds in a margin account. The amount that must be deposited at the time the contract is entered into is known as the initial margin. We suppose this is $2,000 per contract, or $4,000 in total. At the end of each trading day, the margin account is adjusted to reflect the investor's gain or loss. This practice is referred to as marking to market the account. Suppose, for example, that by the end of June 5 the futures price has dropped from $400 to $397. The investor has a loss of $600 (= 200 x $3), because the 200 ounces of December gold, which the investor contracted to buy at $400, can now be sold for only $397. The balance in the margin account would therefore be reduced by $600 to $3,400. Similarly, if the price of December gold rose to $403 by the end of the first day, the balance in the margin account would be increased by $600 to $4,600. A trade is first marked to market at the close of the day on which it takes place. It is then marked to market at the close of trading on each subsequent day. Note that marking to market is not merely an arrangement between broker and client. When there is a decrease in the futures price so that the margin account of an investor with a long position is reduced by $600, the investor's broker has to pay the exchange $600 and the exchange passes the money on to the broker of an investor with a short position. Similarly, when there is an increase in the futures price, brokers for parties with short positions pay money to the exchange and brokers for parties with long positions receive money from the exchange. Later we will examine in more detail the mechanism by which this happens. The investor is entitled to withdraw any balance in the margin account in excess of the initial margin. To ensure that the balance in the margin account never becomes negative a maintenance margin, which is somewhat lower than the initial margin, is set. If the balance in the margin account falls below the maintenance margin, the investor receives a margin call and is expected to top up the margin account to the initial margin level the next day. The extra funds deposited are known as a variation margin. If the investor does not provide the variation margin, the broker closes out the position by selling the contract. In the case of the investor considered earlier, closing out the position would involve neutralizing the existing contract by selling 200 ounces of gold for delivery in December. Table 2.1 illustrates the operation of the margin account for one possible sequence of futures

Mechanics of Futures Markets

25

Table 2.1 Operation of margins for a long position in two gold futures contracts. The initial margin is $2,000 per contract, or $4,000 in total, and the maintenance margin is $1,500 per contract, or $3,000 in total. The contract is entered into on June 5 at $400 and closed out on June 26 at $392.30. The numbers in the second column, except the first and the last, represent the futures prices at the close of trading

Day

Futures price Soe