Options futures and other derivatives 5th

3.13 Cost of carry 60 3.14 Delivery options 60 3.15 Futures prices and the expected future spot price 61 Summary 63 Suggestions for further reading 64 Questions and problems 65 Assignmen

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

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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

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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

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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

OPTIONS, FUTURES,

& OTHER DERIVATIVES

John C Hull

Maple Financial Group Professor of Derivatives and Risk Management

Director, Bonham Center for Finance Joseph L Rotman School of Management

University of Toronto

Prentice Hall

P R E N T I C E H A L L , U P P E R S A D D L E R I V E R , N E W J E R S E Y 0 7 4 5 8

Preface xix

1 Introduction 1

1.1 Exchange-traded markets 1 1.2 Over-the-counter markets 2 1.3 Forward contracts 2 1.4 Futures contracts 5 1.5 Options 6 1.6 Types of traders 10 1.7 Other derivatives 14 Summary 15 Questions and problems 16 Assignment questions 17

2 Mechanics of futures markets 19

2.1 Trading futures contracts 19 2.2 Specification of the futures contract 20 2.3 Convergence of futures price to spot price 23 2.4 Operation of margins 24 2.5 Newspaper quotes 27 2.6 Keynes and Hicks 31 2.7 Delivery 31 2.8 Types of traders 32

2.10 Accounting and tax 35 2.11 Forward contracts vs futures contracts 36 Summary 37 Suggestions for further reading 38 Questions and problems 38 Assignment questions 40

3 Determination of forward and futures prices 41

3.1 Investment assets vs consumption assets 41 3.2 Short selling 41 3.3 Measuring interest rates 42 3.4 Assumptions and notation 44 3.5 Forward price for an investment asset 45 3.6 Known income 47 3.7 Known yield 49 3.8 Valuing forward contracts 49 3.9 Are forward prices and futures prices equal? 51 3.10 Stock index futures 52 3.11 Forward and futures contracts on currencies 55 3.12 Futures on commodities 58

ix

3.13 Cost of carry 60 3.14 Delivery options 60 3.15 Futures prices and the expected future spot price 61 Summary 63 Suggestions for further reading 64 Questions and problems 65 Assignment questions 67 Appendix 3A: Proof that forward and futures prices are equal when interest

rates are constant 68

4 Hedging strategies using futures 70

4.1 Basic principles 70 4.2 Arguments for and against hedging 72 4.3 Basis risk 75 4.4 Minimum variance hedge ratio 78 4.5 Stock index futures 82 4.6 Rolling the hedge forward 86 Summary 87 Suggestions for further reading 88 Questions and problems 88 Assignment questions 90 Appendix 4A: Proof of the minimum variance hedge ratio formula 92

5 Interest rate markets 93

5.1 Types of rates 93 5.2 Zero rates 94 5.3 Bond pricing 94 5.4 Determining zero rates 96 5.5 Forward rates 98 5.6 Forward rate agreements 100 5.7 Theories of the term structure 102 5.8 Day count conventions 102 5.9 Quotations 103 5.10 Treasury bond futures 104 5.11 Eurodollar futures 110

5.12 The LIBOR zero curve I l l

5.13 Duration 112 5.14 Duration-based hedging strategies 116 Summary 118 Suggestions for further reading 119 Questions and problems 120 Assignment questions 123

6 Swaps 125

6.1 Mechanics of interest rate swaps 125 6.2 The comparative-advantage argument 131 6.3 Swap quotes and LIBOR zero rates 134 6.4 Valuation of interest rate swaps 136 6.5 Currency swaps 140 6.6 Valuation of currency swaps 143 6.7 Credit risk 145 Summary 146 Suggestions for further reading 147 Questions and problems 147 Assignment questions 149

Contents xi

7 Mechanics of options markets 151

7.1 Underlying assets 151 7.2 Specification of stock options 152 7.3 Newspaper quotes 155 7.4 Trading 157 7.5 Commissions 157 7.6 Margins 158 7.7 The options clearing corporation 160 7.8 Regulation 161 7.9 Taxation 161 7.10 Warrants, executive stock options, and convertibles 162 7.11 Over-the-counter markets 163 Summary 163 Suggestions for further reading 164 Questions and problems 164 Assignment questions 165

8 Properties of stock options 167

8.1 Factors affecting option prices 167

8.2 Assumptions and notation 170 8.3 Upper and lower bounds for option prices 171 8.4 Put-call parity 174 8.5 Early exercise: calls on a non-dividend-paying stock 175 8.6 Early exercise: puts on a non-dividend-paying stock 177 8.7 Effect of dividends 178 8.8 Empirical research 179 Summary 180 Suggestions for further reading 181 Questions and problems 182 Assignment questions 183

9 Trading strategies involving options 185

9.1 Strategies- involving a single option and a stock 185 9.2 Spreads 187 9.3 Combinations 194 9.4 Other payoffs 197 Summary 197 Suggestions for further reading 198 Questions and problems 198 Assignment questions 199

10 Introduction to binomial trees 200

10.1 A one-step binomial model 200 10.2 Risk-neutral valuation 203 10.3 Two-step binomial trees 205 10.4 A put example 208 10.5 American options 209 10.6 Delta 210

10.7 Matching volatility with u and d 211

10.8 Binomial trees in practice 212 Summary 213 Suggestions for further reading 214 Questions and problems 214 Assignment questions 215

11 A model of the behavior of stock prices 216

11.1 The Markov property 216 11.2 Continuous-time stochastic processes 217 11.3 The process for stock prices 222 11.4 Review of the model 223 11.5 The parameters 225 11.6 Ito's lemma 226 11.7 The lognormal property 227 Summary 228 Suggestions for further reading 229 Questions and problems 229 Assignment questions 230 Appendix 11 A: Derivation of Ito's lemma 232

12 The Black-Scholes model 234

12.1 Lognormal property of stock prices 234 12.2 The distribution of the rate of return 236 12.3 The expected return 237 12.4 Volatility 238 12.5 Concepts underlying the Black-Scholes-Merton differential equation 241 12.6 Derivation of the Black-Scholes-Merton differential equation 242 12.7 Risk-neutral valuation 244 12.8 Black-Scholes pricing formulas 246 12.9 Cumulative normal distribution function 248 12.10 Warrants issued by a company on its own stock 249 12.11 Implied volatilities 250 12.12 The causes of volatility 251 12.13 Dividends 252 Summary 256 Suggestions for further reading 257 Questions and problems 258 Assignment questions 261 Appendix 12A: Proof of Black-Scholes-Merton formula 262 Appendix 12B: Exact procedure for calculating the values of American calls on

dividend-paying stocks 265 Appendix 12C: Calculation of cumulative probability in bivariate normal

distribution 266

13 Options on stock indices, currencies, and futures 267

13.1 Results for a stock paying a known dividend yield 267 13.2 Option pricing formulas 268 13.3 Options on stock indices 270 13.4 Currency options 276 13.5 Futures options 278 13.6 Valuation of futures options using binomial trees 284 13.7 Futures price analogy 286 13.8 Black's model for valuing futures options 287 13.9 Futures options vs spot options 288 Summary 289 Suggestions for further reading 290 Questions and problems 291 Assignment questions 294 Appendix 13 A: Derivation of differential equation satisfied by a derivative

dependent on a stock providing a dividend yield 295

Contents xiii

Appendix 13B: Derivation of differential equation satisfied by a derivative

dependent on a futures price 297

14 The Greek letters 299

14.1 Illustration 299 14.2 Naked and covered positions 300 14.3 A stop-loss strategy 300 14.4 Delta hedging 302 14.5 Theta 309 14.6 Gamma 312 14.7 Relationship between delta, theta, and gamma 315 14.8 Vega 316 14.9 Rho 318 14.10 Hedging in practice 319 14.11 Scenario analysis 319 14.12 Portfolio insurance 320 14.13 Stock market volatility 323 Summary 323 Suggestions for further reading 324 Questions and problems : 326 Assignment questions 327 Appendix 14A: Taylor series expansions and hedge parameters 329

15 Volatility smiles 330

15.1 Put-call parity revisited 330 15.2 Foreign currency options 331 15.3 Equity options 334 15.4 The volatility term structure and volatility surfaces 336 15.5 Greek letters 337 15.6 When a single large jump is anticipated 338 15.7 Empirical research 339 Summary 341 Suggestions for further reading 341 Questions and problems 343 Assignment questions 344 Appendix 15A: Determining implied risk-neutral distributions from volatility

smiles 345

16 Value at risk 346

16.1 The VaR measure 346 16.2 Historical simulation 348 16.3 Model-building approach 350 16.4 Linear model 352 16.5 Quadratic model 356 16.6 Monte Carlo simulation 359 16.7 Comparison of approaches 359 16.8 Stress testing and back testing 360 16.9 Principal components analysis 360 Summary 364 Suggestions for further reading 364 Questions and problems 365 Assignment questions 366 Appendix 16A: Cash-flow mapping 368 Appendix 16B: Use of the Cornish-Fisher expansion to estimate VaR 370

17 Estimating volatilities and correlations 372

17.1 Estimating volatility 372 17.2 The exponentially weighted moving average model 374 17.3 The GARCH(1,1) model 376 17.4 Choosing between the models 377 17.5 Maximum likelihood methods 378 17.6 Using GARCHfl, 1) to forecast future volatility 382 17.7 Correlations 385 Summary 388 Suggestions for further reading 388 Questions and problems 389 Assignment questions 391

18 Numerical procedures 392

18.1 Binomial trees 392 18.2 Using the binomial tree for options on indices, currencies, and futures

contracts 399

18.3 Binomial model for a dividend-paying stock ' 402

18.4 Extensions to the basic tree approach 405 18.5 Alternative procedures for constructing trees 406 18.6 Monte Carlo simulation 410 18.7 Variance reduction procedures 414 18.8 Finite difference methods 418 18.9 Analytic approximation to American option prices 427 Summary 427 Suggestions for further reading 428 Questions and problems 430 Assignment questions 432 Appendix 18A: Analytic approximation to American option prices of

MacMillan and of Barone-Adesi and Whaley 433

19 Exotic options 435

19.1 Packages 435 19.2 Nonstandard American options 436 19.3 Forward start options 437 19.4 Compound options 437 19.5 Chooser options 438 19.6 Barrier options 439 19.7 Binary options 441 19.8 Lookback options 441 19.9 Shout options 443 19.10 Asian options 443 19.11 Options to exchange one asset for another 445 19.12 Basket options 446 19.13 Hedging issues 447 19.14 Static options replication 447 Summary 449 Suggestions for further reading 449 Questions and problems 451 Assignment questions 452 Appendix 19A: Calculation of the first two moments of arithmetic averages

and baskets 454

20 More on models and numerical procedures 456

20.1 The CEV model 456 20.2 The jump diffusion model 457

Contents xv

20.3 Stochastic volatility models 458 20.4 The IVF model 460 20.5 Path-dependent derivatives 461 20.6 Lookback options 465 20.7 Barrier options 467 20.8 Options on two correlated assets 472 20.9 Monte Carlo simulation and American options 474 Summary 478 Suggestions for further reading 479 Questions and problems 480

Assignment questions 481

21 Martingales and measures 483

21.1 The market price of risk 484 21.2 Several state variables • 487 21.3 Martingales 488 21.4 Alternative choices for the numeraire 489 21.5 Extension to multiple independent factors 492 21.6 Applications 493 21.7 Change of numeraire 495 21.8 Quantos 497 21.9 Siegel's paradox 499 Summary 500 Suggestions for further reading 500 Questions and problems 501 Assignment questions 502 Appendix 21 A: Generalizations of Ito's lemma 504 Appendix 2IB: Expected excess return when there are multiple sources of

uncertainty 506

22 Interest rate derivatives: the standard market models 508

22.1 Black's model 508 22.2 Bond options 511 22.3 Interest rate caps 515 22.4 European swap options 520 22.5 Generalizations 524 22.6 Convexity adjustments 524 22.7 Timing adjustments 527 22.8 Natural time lags 529 22.9 Hedging interest rate derivatives 530 Summary 531 Suggestions for further reading 531 Questions and problems 532 Assignment questions 534 Appendix 22A: Proof of the convexity adjustment formula 536

23 Interest rate derivatives: models of the short rate 537

23.1 Equilibrium models 537 23.2 One-factor equilibrium models 538 23.3 The Rendleman and Bartter model 538 23.4 The Vasicek model 539 23.5 The Cox, Ingersoll, and Ross model 542 23.6 Two-factor equilibrium models 543 23.7 No-arbitrage models 543 23.8 The Ho and Lee model 544 23.9 The Hull and White model 546

23.10 Options on coupon-bearing bonds 549 23.11 Interest rate trees 550 23.12 A general tree-building procedure 552 23.13 Nonstationary models 563 23.14 Calibration 564 23.15 Hedging using a one-factor model 565 23.16 Forward rates and futures rates 566 Summary 566 Suggestions for further reading 567 Questions and problems 568 Assignment questions 570

24 Interest rate derivatives: more advanced models 571

24.1 Two-factor models of the short rate 571 24.2 The Heath, Jarrow, and Morton model 574 24.3 The LIBOR market model 577 24.4 Mortgage-backed securities 586 Summary 588 Suggestions for further reading 589 Questions and problems 590 Assignment questions 591

Appendix 24A: The A(t, T), a P , and 0(t) functions in the two-factor Hull-White

model 593

25 Swaps revisited 594

25.1 Variations on the vanilla deal 594

25.2 Compounding swaps 595 25.3 Currency swaps 598 25.4 More complex swaps 598 25.5 Equity swaps 601 25.6 Swaps with embedded options 602 25.7 Other swaps 605 25.8 Bizarre deals 605 Summary 606 Suggestions for further reading 606 Questions and problems 607 Assignment questions 607 Appendix 25A: Valuation of an equity swap between payment dates 609

26 Credit risk 610

23.1 Bond prices and the probability of default 610 26.2 Historical data 619 26.3 Bond prices vs historical default experience 619 26.4 Risk-neutral vs real-world estimates 620 26.5 Using equity prices to estimate default probabilities 621 26.6 The loss given default 623 26.7 Credit ratings migration 626 26.8 Default correlations 627 26.9 Credit value at risk 630 Summary 633 Suggestions for further reading 633 Questions and problems 634 Assignment questions 635 Appendix 26A: Manipulation of the matrices of credit rating changes 636

Contents xvn

27 Credit derivatives 637

27.1 Credit default swaps 637 27.2 Total return swaps 644 27.3 Credit spread options 645 27.4 Collateralized debt obligations 646 27.5 Adjusting derivative prices for default risk 647 27.6 Convertible bonds 652 Summary 655 Suggestions for further reading 655 Questions and problems 656 Assignment questions 658

28 Real options 660

28.1 Capital investment appraisal 660 28.2 Extension of the risk-neutral valuation framework 661 28.3 Estimating the market price of risk 665 28.4 Application to the valuation of a new business 666 28.5 Commodity prices 667 28.6 Evaluating options in an investment opportunity 670 Summary 675 Suggestions for further reading 676 Questions and problems 676 Assignment questions 677

29 Insurance, weather, and energy derivatives 678

29.1 Review of pricing issues 678 29.2 Weather derivatives 679 29.3 Energy derivatives 680 29.4 Insurance derivatives 682 Summary 683 Suggestions for further reading 684 Questions and problems 684 Assignment questions 685

30 Derivatives mishaps and what we can learn from them 686

30.1 Lessons for all users of derivatives 686

30.2 Lessons for financial institutions 690 30.3 Lessons for nonfinancial corporations 693 Summary 694 Suggestions for further reading 695

Glossary of notation 697 Glossary of terms 700 DerivaGem software : 715 Major exchanges trading futures and options 720 Table for N{x) when x sj 0 722 Table for N(x) when x ^ 0 723 Author index 725 Subject index 729

It is sometimes hard for me to believe that the first edition of this book was only 330 pages and

13 chapters long! There have been many developments in derivatives markets over the last 15 yearsand the book has grown to keep up with them The fifth edition has seven new chapters that covernew derivatives instruments and recent research advances

Like earlier editions, the book serves several markets It is appropriate for graduate courses inbusiness, economics, and financial engineering It can be used on advanced undergraduate courseswhen students have good quantitative skills Also, many practitioners who want to acquire aworking knowledge of how derivatives can be analyzed find the book useful

One of the key decisions that must be made by an author who is writing in the area of derivativesconcerns the use of mathematics If the level of mathematical sophistication is too high, thematerial is likely to be inaccessible to many students and practitioners If it is too low, someimportant issues will inevitably be treated in a rather superficial way I have tried to be particularlycareful about the way I use both mathematics and notation in the book Nonessential mathema-tical material has been either eliminated or included in end-of-chapter appendices Concepts thatare likely to be new to many readers have been explained carefully, and many numerical exampleshave been included

The book covers both derivatives markets and risk management It assumes that the reader hastaken an introductory course in finance and an introductory course in probability and statistics

No prior knowledge of options, futures contracts, swaps, and so on is assumed It is not thereforenecessary for students to take an elective course in investments prior to taking a course based onthis book There are many different ways the book can be used in the classroom Instructorsteaching a first course in derivatives may wish to spend most time on the first half of the book.Instructors teaching a more advanced course will find that many different combinations of thechapters in the second half of the book can be used I find that the material in Chapters 29 and 30works well at the end of either an introductory or an advanced course

2 A new chapter on models and numerical procedures (Chapter 20) Much of this material isnew, but some has been transferred from the chapter on exotic options in the fourth edition

xix

xx Preface

3 A new chapter on swaps (Chapter 25) This gives the reader an appreciation of the range ofnonstandard swap products that are traded in the over-the-counter market and discusses howthey can be valued

4 There is an extra chapter on credit risk Chapter 26 discusses the measurement of credit riskand credit value at risk while Chapter 27 covers credit derivatives

5 There is a new chapter on real options (Chapter 28)

6 There is a new chapter on insurance, weather, and energy derivatives (Chapter 29)

7 There is a new chapter on derivatives mishaps and what we can learn from them (Chapter 30)

8 The chapter on martingales and measures has been improved so that the material flowsbetter (Chapter 21)

9 The chapter on value at risk has been rewritten so that it provides a better balance betweenthe historical simulation approach and the model-building approach (Chapter 16)

10 The chapter on volatility smiles has been improved and appears earlier in the book.(Chapter 15)

11 The coverage of the LIBOR market model has been expanded (Chapter 24)

12 One or two changes have been made to the notation The most significant is that the strike

price is now denoted by K rather than X.

13 Many new end-of-chapter problems have been added

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 Builderconsists of a number of Excel functions from which users can build their own applications Itincludes a number of sample applications and enables students to explore the properties of optionsand numerical procedures more easily It also allows more interesting assignments to be designed.The software is described more fully at the end of the book Updates to the software can bedownloaded from my website:

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 InstructorsManual

A cknowledgments

Many people have played a part in the production of this book Academics, students, andpractitioners who have made excellent and useful suggestions include Farhang Aslani, Jas Badyal,Emilio Barone, Giovanni Barone-Adesi, Alex Bergier, George Blazenko, Laurence Booth, PhelimBoyle, Peter Carr, Don Chance, J.-P Chateau, Ren-Raw Chen, George Constantinides, MichelCrouhy, Emanuel Derman, Brian Donaldson, Dieter Dorp, Scott Drabin, Jerome Duncan, SteinarEkern, David Fowler, Louis Gagnon, Dajiang Guo, Jrgen Hallbeck, Ian Hawkins, MichaelHemler, Steve Heston, Bernie Hildebrandt, Michelle Hull, Kiyoshi Kato, Kevin Kneafsy, TiborKucs, Iain MacDonald, Bill Margrabe, Izzy Nelkin, Neil Pearson, Paul Potvin, Shailendra Pandit,Eric Reiner, Richard Rendleman, Gordon Roberts, Chris Robinson, Cheryl Rosen, John Rumsey,Ani Sanyal, Klaus Schurger, Eduardo Schwartz, Michael Selby, Piet Sercu, Duane Stock, EdwardThorpe, Yisong Tian, P V Viswanath, George Wang, Jason Wei, Bob Whaley, Alan White,Hailiang Yang, Victor Zak, and Jozef Zemek Huafen (Florence) Wu and Matthew Merkleyprovided excellent research assistance

I am particularly grateful to Eduardo Schwartz, who read the original manuscript for the firstedition and made many comments that led to significant improvements, and to Richard Rendle-man and George Constantinides, who made specific suggestions that led to improvements in morerecent editions

The first four editions of this book were very popular with practitioners and their comments andsuggestions have led to many improvements in the book The students in my elective courses onderivatives at the University of Toronto have also influenced the evolution of the book

Alan White, a colleague at the University of Toronto, deserves a special acknowledgment Alanand I have been carrying out joint research in the area of derivatives for the last 18 years Duringthat time we have spent countless hours discussing different issues concerning derivatives Many ofthe new ideas in this book, and many of the new ways used to explain old ideas, are as much Alan's

as mine Alan read the original version of this book very carefully and made many excellentsuggestions for improvement Alan has also done most of the development work on the Deriva-Gem software

Special thanks are due to many people at Prentice Hall for their enthusiasm, advice, andencouragement I would particularly like to thank Mickey Cox (my editor), P J Boardman (theeditor-in-chief) and Kerri Limpert (the production editor) I am also grateful to Scott Barr, LeahJewell, Paul Donnelly, and Maureen Riopelle, who at different times have played key roles in thedevelopment of the book

I welcome comments on the book from readers My email address is:

hull@rotman.utoronto.ca

John C HullUniversity 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 Futuresand options are now traded actively on many exchanges throughout the world Forward contracts,swaps, and many different types of options are regularly traded outside exchanges by 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 tives are the prices of traded assets A stock option, for example, is a derivative whose value isdependent on the price of a stock However, derivatives can be dependent on almost any variable,from the price of hogs to the amount of snow falling at a certain ski resort

deriva-Since the first edition of this book was published in 1988, there have been many developments inderivatives markets There is now active trading in credit derivatives, electricity derivatives, weatherderivatives, and insurance derivatives Many new types of interest rate, foreign exchange, andequity derivative products have been created There have been many new ideas in risk managementand risk measurement Analysts have also become more aware of the need to analyze what 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 givemore 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 beendefined by the exchange Derivatives exchanges have existed for a long time The Chicago Board ofTrade (CBOT, www.cbot.com) was established in 1848 to bring farmers and merchants together.Initially its main task was to standardize the quantities and qualities of the grains that were traded

Within a few years the first futures-type contract was developed It was known as a to-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 ChicagoMercantile Exchange (CME, www.cme.com), was established in 1919 Now futures exchangesexist all over the world

The Chicago Board Options Exchange (CBOET www.cboe.com) started trading call option

contracts on 16 stocks in 1973 Options had traded prior to 1973 but the CBOE succeeded increating an orderly market with well-defined contracts Put option contracts started trading on theexchange in 1977 The CBOE now trades options on over 1200 stocks and many different stockindices Like futures, options have proved to be very popular contracts Many other exchangesthroughout the world now trade options The underlying assets include foreign currencies andfutures contracts as well as stocks and stock indices.

Traditionally derivatives traders have met on the floor of an exchange and used shouting and acomplicated set of hand signals to indicate the trades they would like to carry out This is known 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 allexchanges 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 theexchange-traded market It is a telephone- and computer-linked network of dealers, who do notphysically meet Trades are done over the phone and are usually between two financial institutions

or between a financial institution and one of its corporate clients Financial institutions often act asmarket makers for the more commonly traded instruments This means that they are alwaysprepared to quote both a bid price (a price at which they are prepared to buy) and an offer price(a price at which they are prepared to sell)

Telephone conversations in the over-the-counter market are usually taped If there is a disputeabout what was agreed, the tapes are replayed to resolve the issue Trades in the over-the-countermarket are typically much larger than trades in the exchange-traded market A key advantage ofthe over-the-counter market is that the terms of a contract do not have to be those specified by anexchange Market participants are free to negotiate any mutually attractive deal A disadvantage isthat there is usually some credit risk in an over-the-counter trade (i.e., there is a small risk that thecontract will not be honored) As mentioned earlier, exchanges have organized themselves toeliminate virtually all credit risk

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 "forwarddesk" within their foreign exchange trading room that is devoted to the trading of forward

Table 1.1 Spot and forward quotes for the USD-GBP exchange

rate, August 16, 2001 (GBP = British pound; USD = U.S dollar)

Spot1-month forward3-month forward6-month forward1-year forward

Bid

1.44521.44351.44021.43531.4262

Offer

1.44561.44401.44071.43591.4268

contracts Table 1.1 provides the quotes on the exchange rate between the British pound (GBP) andthe U.S dollar (USD) that might be made by a large international bank on August 16, 2001 Thequote is for the number of USD per GBP The first quote indicates that the bank is prepared to buyGBP (i.e., sterling) in the spot market (i.e., for virtually immediate delivery) at the rate of $1.4452

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 muchlarger spreads between bid and offer quotes than those in given Table 1.1.)

Forward contracts can be used to hedge foreign currency risk Suppose that on August 16, 2001,the treasurer of a U.S corporation knows that the corporation will pay £1 million in six months (onFebruary 16, 2002) and wants to hedge against exchange rate moves Using the quotes in Table 1.1,the treasurer can agree to buy £1 million six months forward at an exchange rate of 1.4359 Thecorporation then has a long forward contract on GBP It has agreed that on February 16, 2002, itwill buy £1 million from the bank for $1.4359 million The bank has a short forward contract onGBP It has agreed that on February 16, 2002, it will sell £1 million for $1.4359 million Both sideshave made a binding commitment

Payoffs from Forward Contracts

Consider the position of the corporation in the trade we have just described What are the possibleoutcomes? The forward contract obligates the corporation to buy £1 million for $1,435,900 If thespot exchange rate rose to, say, 1.5000, at the end of the six months the forward contract would beworth $64,100 (= $1,500,000 - $1,435,900) to the corporation It would enable £1 million to bepurchased at 1.4359 rather than 1.5000 Similarly, if the spot exchange rate fell to 1.4000 at the end ofthe six months, the forward contract would have a negative value to the corporation of $35,900because it would lead to the corporation paying $35,900 more than the market price for the sterling

In general, the payoff from a long position in a forward contract on one unit of an asset is

S T -K

where K is the delivery price and S T 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 S T for K Similarly, the payoff

from a short position in a forward contract on one unit of an asset is

K-S

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 costsnothing 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 acontract maturing on February 16, 2002 The corporation enters into a contract and 1.4359becomes the delivery price for the contract As we move through time the delivery price for thecorporation'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 ofAugust 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 ongold We assume that there are no storage costs associated with gold and that gold earns no income.1

Suppose that the spot price of gold is $300 per ounce and the risk-free interest rate forinvestments lasting one year is 5% per annum What is a reasonable value for the one-yearforward price of gold?

1 This is not totally realistic In practice, storage costs are close to zero, but an income of 1 to 2% per annum can be earned by lending gold.

Suppose first that the one-year forward price is $340 per ounce A trader can immediately takethe following actions:

1 Borrow $300 at 5% for one year

2 Buy one ounce of gold

3 Enter into a short forward contract to sell the gold for $340 in one year

The interest on the $300 that is borrowed (assuming annual compounding) is $15 The trader can,therefore, use $315 of the $340 that is obtained for the gold in one year to repay the loan Theremaining $25 is profit Any one-year forward price greater than $315 will lead to this arbitragetrading strategy being profitable

Suppose next that the forward price is $300 An investor who has a portfolio that includes gold 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 forone year, we see that the investor is better off by $15 per ounce In any situation where the forwardprice is less than $315, investors holding gold have an incentive to sell the gold and enter into along forward contract in the way that has been described

The first strategy is profitable when the one-year forward price of gold is greater than $315 Asmore traders attempt to take advantage of this strategy, the demand for short forward contractswill increase and the one-year forward price of gold will fall The second strategy is profitable forall investors who hold gold in their portfolios when the one-year forward price of gold is less than

$315 As these investors attempt to take advantage of this strategy, the demand for long forwardcontracts will increase and the one-year forward price of gold will rise Assuming that individualsare always willing to take advantage of arbitrage opportunities when they arise, we can concludethat the activities of traders should cause the one-year forward price of gold to be exactly $315.Any other price leads to an arbitrage opportunity.2

1.4 FUTURES CONTRACTS

Like a forward contract, a futures contract is an agreement between two parties to buy or sell anasset at a certain time in the future for a certain price Unlike forward contracts, futures contractsare normally traded on an exchange To make trading possible, the exchange specifies certainstandardized features of the contract As the two parties to the contract do not necessarily knoweach other, the exchange also provides a mechanism that gives the two parties a guarantee that thecontract will be honored

The largest exchanges on which futures contracts are traded are the Chicago Board of Trade(CBOT) and the Chicago Mercantile Exchange (CME) On these and other exchanges throughoutthe world, a very wide range of commodities and financial assets form the underlying assets in thevarious contracts The commodities include pork bellies, live cattle, sugar, wool, lumber, copper,aluminum, gold, and tin The financial assets include stock indices, currencies, and Treasury bonds

2 Our arguments make the simplifying assumption that the rate of interest on borrowed funds is the same as the rate

of interest on invested funds.

One way in which a futures contract is different from a forward contract is that an exact deliverydate is usually not specified The contract is referred to by its delivery month, and the exchangespecifies the period during the month when delivery must be made For commodities, the deliveryperiod is often the entire month The holder of the short position has the right to choose the timeduring the delivery period when it will make delivery Usually, contracts with several differentdelivery months are traded at any one time The exchange specifies the amount of the asset to bedelivered for one contract and how the futures price is to be quoted In the case of a commodity,the exchange also specifies the product quality and the delivery location Consider, for example, thewheat futures contract currently traded on the Chicago Board of Trade The size of the contract is5,000 bushels Contracts for five delivery months (March, May, July, September, and December)are available for up to 18 months into the future The exchange specifies the grades of wheat thatcan be delivered and the places where delivery can be made.

Futures prices are regularly reported in the financial press Suppose that on September 1, theDecember futures price of gold is quoted as $300 This is the price, exclusive of commissions, atwhich traders can agree to buy or sell gold for December delivery It is determined on the floor of theexchange in the same way as other prices (i.e., by the laws of supply and demand) If more traderswant to go long than to go short, the price goes up; if the reverse is true, the price goes down.3

Further details on issues such as margin requirements, daily settlement procedures, deliveryprocedures, bid-offer spreads, and the role of the exchange clearinghouse are given in Chapter 2

1.5 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-tradedequity options market, one contract is usually an agreement to buy or sell 100 shares Europeanoptions are generally easier to analyze than American options, and some of the properties of anAmerican option are frequently deduced from those of its European counterpart

It should be emphasized that an option gives the holder the right to do something The holderdoes not have to exercise this right This is what distinguishes options from forwards and futures,where the holder is obligated to buy or sell the underlying asset Note that whereas it costs nothing

to enter into a forward or futures contract, there is a cost to acquiring an option

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.

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 initialinvestment is $500 Because the option is European, the investor can exercise only on the expirationdate If the stock price on this date is less than $60, the investor will clearly choose not to exercise.(There is no point in buying, for $60, a share that has a market value of less than $60.) In thesecircumstances, the investor loses the whole of the initial investment of $500 If the stock price isabove $60 on the expiration date, the option will be exercised Suppose, for example, that the stockprice is $75 By exercising the option, the investor is able to buy 100 shares for $60 per share If theshares are sold immediately, the investor makes a gain of $15 per share, or $1,500, ignoringtransactions costs When the initial cost of the option is taken into account, the net profit to theinvestor is $1,000

Figure 1.2 shows how the investor's net profit or loss on an option to purchase one share varieswith the final stock price in the example (We ignore the time value of money in calculating theprofit.) It is important to realize that an investor sometimes exercises an option and makes a lossoverall Suppose that in the example 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 lossoverall of $300 when the initial cost of the option is taken into account It is tempting to argue thatthe investor should not exercise the option in these circumstances However, not exercising wouldlead to an overall loss of $500, which is worse than the $300 loss when the investor exercises Ingeneral, call options should always be exercised at the expiration date if the stock price is above thestrike price

Put Options

Whereas the purchaser of a call option is hoping that the stock price will increase, the purchaser of aput option is hoping that it will decrease Consider an investor who buys a European put option tosell 100 shares in IBM with a strike price of $90 Suppose that the current stock price is $85, theexpiration date of the option is in three months, and the price of an option to sell one share is $7 Theinitial investment is $700 Because the option is European, it will be exercised only if the stock price

is below $90 at the expiration date Suppose that the stock price is $75 on this date The investor can

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 $700initial cost of the option is taken into account, the investor's net profit is $800 There is no guaranteethat the investor will make a gain If the final stock price is above $90, the put option expiresworthless, and the investor loses $700 Figure 1.3 shows the way in which the 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 beforeexercising the option We will see in later chapters that there are some circumstances under which it isoptimal to exercise American options prior to maturity

Option Positions

There are two sides to every option contract On one side is the investor who has taken the longposition (i.e., has bought the option) On the other side is the investor who has taken a 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 theoption Figures 1.4 and 1.5 show the variation of the profit or loss with the final stock price forwriters of the options considered in Figures 1.2 and 1.3

There are four types of option positions:

1 A long position in a call option

2 A long position in a put option

3 A short position in a call option

4 A short position in a put option

-30

Figure 1.4 Profit from writing a European call option on one Microsoft share.

Option price = $5; strike price = $60

It is often useful to characterize European option positions in terms of the terminal value or payoff

to the investor at maturity The initial cost of the option is then not included in the calculation If

K is the strike price and S? is the final price of the underlying asset, the payoff from a long position

in a European call option is

max(5r - K, 0) This reflects the fact that the option will be exercised if S T > K and will not be exercised if S T < K.

The payoff to the holder of a short position in the European call option is

Figure 1.5 Profit from writing a European put option on one IBM share.

Option price = $7; strike price = $90

,, Payoff | Payoff

,, Payoff

Figure 1.6 Payoffs from positions in European options: (a) long call, (b) short call, (c) long put,

(d) short put Strike price = K; price of asset at maturity = S T

The payoff to the holder of a long position in a European put option is

Three broad categories of traders can be identified: hedgers, speculators, and arbitrageurs.Hedgers use futures, forwards, and options to reduce the risk that they face from potential futuremovements in a market variable Speculators use them to bet on the future direction of a marketvariable Arbitrageurs take offsetting positions in two or more instruments to lock in a profit Inthe next few sections, we consider the activities of each type of trader in more detail

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, knowsthat it will pay £ 10 million on November 16,2001, for goods it has purchased from a British supplier.The USD-GBP exchange rate quotes made by a financial institution are given in Table 1.1.ImportCo could hedge its foreign exchange risk by buying pounds (GBP) from the financialinstitution in the three-month forward market at 1.4407 This would have the effect of fixing theprice to be paid to the British exporter at $14,407,000

Consider next another U.S company, which we will refer to as ExportCo, that is exportinggoods to the United Kingdom and on August 16, 2001, knows that it will receive £30 million threemonths later ExportCo can hedge its foreign exchange risk by selling £30 million in the three-month forward market at an exchange rate of 1.4402 This would have the effect of locking in theU.S dollars to be realized for the sterling at $43,206,000

Note that if the companies choose not to hedge they might do better than if they do hedge.Alternatively, they might do worse Consider ImportCo If the exchange rate is 1.4000 on November

16 and the company has not hedged, the £10 million that it has to pay will cost $14,000,000, which isless than $14,407,000 On the other hand, if the exchange rate is 1.5000, the £10 million will cost

$15,000,000—and the company will wish it had hedged! The position of ExportCo if it does nothedge is the reverse If the exchange rate in September proves to be less than 1.4402, the company willwish it had hedged; if the rate is greater than 1.4402, it will be pleased it had not done so.This example illustrates a key aspect of hedging The cost of, or price received for, the underlyingasset is ensured However, there is no assurance that the outcome with hedging will be better thanthe outcome without hedging

Options can also be used for hedging Consider an investor who in May 2000 owns 1,000Microsoft shares The current share price is $73 per share The investor is concerned that thedevelopments in Microsoft's antitrust case may cause the share price to decline sharply in the nexttwo months and wants protection The investor could buy 10 July put option contracts with a strikeprice of $65 on the Chicago Board Options Exchange This would give the investor the right to sell1,000 shares for $65 per share If the quoted option price is $2.50, each option contract would cost

100 x $2.50 = $250, and the total cost of the hedging strategy would be 10 x $250 = $2,500.The strategy costs $2,500 but guarantees that the shares can be sold for at least $65 per shareduring the life of the option If the market price of Microsoft falls below $65, the options can beexercised so that $65,000 is realized for the entire holding When the cost of the options is takeninto account, the amount realized is $62,500 If the market price stays above $65, the options arenot exercised and expire worthless However, in this case the value of the holding is always above

$65,000 (or above $62,500 when the cost of the options is taken into account)

There is a fundamental difference between the use of forward contracts and options for hedging.Forward contracts are designed to neutralize risk by fixing the price that the hedger will pay orreceive for the underlying asset Option contracts, by contrast, provide insurance They offer a wayfor investors to protect themselves against adverse price movements in the future while stillallowing them to benefit from favorable price movements Unlike forwards, options involve thepayment of an up-front fee

Speculators

We now move on to consider how futures and options markets can be used by speculators.Whereas hedgers want to avoid an exposure to adverse movements in the price of an asset,

speculators wish to take a position in the market Either they are betting that the price will go up

or they are betting that it will go down

Consider a U.S speculator who in February thinks that the British pound will strengthenrelative to the U.S dollar over the next two months and is prepared to back that hunch to thetune of £250,000 One thing the speculator can do is simply purchase £250,000 in the hope thatthe sterling can be sold later at a profit The sterling once purchased would be kept in aninterest-bearing account Another possibility is to take a long position in four CME Aprilfutures contracts on sterling (Each futures contract is for the purchase of £62,500.) Supposethat the current exchange rate is 1.6470 and the April futures price is 1.6410 If the exchangerate turns out to be 1.7000 in April, the futures contract alternative enables the speculator torealize a profit of (1.7000 - 1.6410) x 250,000 = $14,750 The cash market alternative leads 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, thefutures contract gives rise to a (1.6410 - 1.6000) x 250,000 = $10,250 loss, whereas the cashmarket alternative gives rise to a loss of (1.6470 - 1.6000) x 250,000 = $11,750 The alternativesappear to give rise to slightly different profits and losses But these calculations do not reflectthe interest that is earned or paid It will be shown in Chapter 3 that when the interest earned

in sterling and the interest paid in dollars are taken into account, the profit or loss from thetwo alternatives is the same

What then is the difference between the two alternatives? The first alternative of buying sterlingrequires an up-front investment of $411,750 As we will see in Chapter 2, the second alternativerequires only a small amount of cash—perhaps $25,000—to be deposited by the speculator inwhat is termed a margin account The futures market allows the speculator to obtain leverage.With a relatively small initial outlay, the investor is able to take a large speculative position

We consider next an example of how a speculator could use options Suppose that it is Octoberand a speculator considers that Cisco is likely to increase in value over the next two months Thestock price is currently $20, and a two-month call option with a $25 strike price is currently sellingfor $1 Table 1.2 illustrates two possible alternatives assuming that the speculator is willing toinvest $4,000 The first alternative involves the purchase of 200 shares The second involves thepurchase of 4,000 call options (i.e., 20 call option contracts)

Suppose that the speculator's hunch is correct and the price of Cisco's shares rises to $35 byDecember The first alternative of buying the stock yields a profit of

200 x ($35 - $20) = $3,000 '

However, the second alternative is far more profitable A call option on Cisco with a strike price

Table 1.2 Comparison of profits (losses) from two alternative

strategies for using $4,000 to speculate on Cisco stock in October

Investor's strategy

Buy shares Buy call options

December

$15

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

stock price

$35

$3,000

$36,000

Introduction 13

of $25 gives a payoff of $10, because it enables something worth $35 to be bought for $25 Thetotal payoff from the 4,000 options that are purchased under the second alternative is

4,000 x $10 = $40,000Subtracting the original cost of the options yields a net profit of

$40,000 - $4,000 = $36,000The options strategy is, therefore, 12 times as profitable as the strategy of buying the stock.Options also give rise to a greater potential loss Suppose the stock price falls to $15 byDecember The first alternative of buying stock yields a loss of

200 x ($20-$15) = $1,000Because the call options expire without being exercised, the options strategy would lead to a loss of

$4,000—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 giveninvestment, the use of options magnifies the financial consequences Good outcomes become verygood, while bad outcomes become very bad!

Futures and options are similar instruments for speculators in that they both provide a way inwhich a type of leverage can be obtained However, there is an important difference between the two.With futures the speculator's potential loss as well as the potential gain is very large With options nomatter how bad things get, the speculator's loss is limited to the amount paid for the options

100 x [($1.55 x 100)-$152]

or $300 in the absence of transactions costs Transactions costs would probably eliminate the profitfor a small investor However, a large investment house faces very low transactions costs in boththe stock market and the foreign exchange market It would find the arbitrage opportunity veryattractive and would try to take as much advantage of it as possible

Arbitrage opportunities such as the one just described cannot last for long As arbitrageurs buythe stock in New York, the forces of supply and demand will cause the dollar price to rise.Similarly, as they sell the stock in London, the sterling price will be driven down Very quickly thetwo prices will become equivalent at the current exchange rate Indeed, the existence of profit-hungry arbitrageurs makes it unlikely that a major disparity between the sterling price and thedollar price could ever exist in the first place Generalizing from this example, we can say that the

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

1.7 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 veryimaginative in designing nonstandard derivatives to meet the needs of clients Sometimes these aresold by financial institutions to their corporate clients in the over-the-counter market On otheroccasions, they are added to bond or stock issues to make these issues more attractive to investors.Some nonstandard derivatives are simply portfolios of two or more "plain vanilla" call and putoptions Others are far more complex The possibilities for designing new interesting 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 bedecomposed 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 anadditional amount based on the price of oil at that time The additional amount was equal to theproduct of 170 and the excess (if any) of the price of a barrel of oil at maturity over $25 Themaximum additional amount paid was $2,550 (which corresponds to a price of $40 per barrel) Thesebonds provided holders with a stake in a commodity that was critically important to the fortunes ofthe company If the price of the commodity went up, the company was in a good position to providethe 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, K x and K 2 , are specified with K x > K 2 If the exchange rate at the

bond's maturity is above K u the bondholder receives the full face value If it is less than K 2 , the

bondholder receives nothing Between K 2 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, S T , 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 ofthe bond is

1,000maxk l , 0 0 0 ( ^

-L \ S 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

for-wards) are popular in foreign exchange markets Suppose that on August 16, 2001, a U.S companyfinds 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 thederivatives 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 onthe 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 anotheroption; some have payoffs dependent on the square of a future interest rate; and so on

Traditionally, the variables underlying options and other derivatives have been stock prices, stockindices, interest rates, exchange rates, and commodity prices However, other underlying 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 ofderivatives markets In many situations, both hedgers and speculators find it more attractive totrade a derivative on an asset than to trade the asset itself Some derivatives are traded onexchanges Others are traded by financial institutions, fund managers, and corporations in theover-the-counter market, or added to new issues of debt and equity securities Much of this book isconcerned with the valuation of derivatives The aim is to present a unifying framework withinwhich all 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 acertain 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 traderscan be identified: hedgers, speculators, and arbitrageurs Hedgers are in the position where theyface risk associated with the price of an asset They use derivatives to reduce or eliminate this risk.Speculators wish to bet on future movements in the price of an asset They use derivatives to getextra leverage Arbitrageurs are in business to take advantage of a discrepancy between prices intwo different markets If, for example, they see the futures price of an asset getting out of line withthe cash price, they will take offsetting positions in the two markets to lock in a profit

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.

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?

This chapter covers the details of how futures markets work We examine issues such as thespecification of contracts, the operation of margin accounts, the organization of exchanges, theregulation of markets, the way in which quotes are made, and the treatment of futures transactionsfor accounting and tax purposes We also examine forward contracts and explain the differencebetween the pattern of payoffs realized from futures and forward contracts.

2.1 TRADING FUTURES CONTRACTS

As mentioned in Chapter 1, futures contracts are now traded very actively all over the world Thetwo largest futures exchanges in the United States are the Chicago Board of Trade (CBOT,www.cbot.com) and the Chicago Mercantile Exchange (CME, www.cme.com) The two largestexchanges in Europe are the London International Financial Futures and Options Exchange(www.liffe.com) and Eurex (www.eurexchange.com) Other large exchanges include Bolsa deMercadorias y Futuros (www.bmf.com.br) in Sao Paulo, the Tokyo International Financial FuturesExchange (www.tiffe.or.jp), the Singapore International Monetary Exchange (www.simex.com.sg),and the Sydney Futures Exchange (www.sfe.com.au) For a more complete list, see the table at theend of this book

We examine how a futures contract comes into existence by considering the corn futurescontract traded on the Chicago Board of Trade (CBOT) On March 5, an investor in New Yorkmight call a broker with instructions to buy 5,000 bushels of corn for delivery in July of the sameyear The broker would immediately pass these instructions on to a trader on the floor of theCBOT The broker would request a long position in one contract because each corn contract onthe CBOT is for the delivery of exactly 5,000 bushels At about the same time, another investor

in Kansas might instruct a broker to sell 5,000 bushels of corn for July delivery This brokerwould then pass instructions to short one contract to a trader on the floor of the CBOT The two

19

floor traders would meet, agree on a price to be paid for the corn in July, and the deal would bedone.

The investor in New York who agreed to buy has a long futures position in one contract; 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 ofsupply and demand If at a particular time more traders wish to sell July corn than buy Julycorn, the price will go down New buyers then enter the market so that a balance between buyersand sellers is maintained If more traders wish to buy July corn than to sell July corn, the pricegoes up New sellers then enter the market and a balance between buyers and sellers ismaintained

Closing Out Positions

The vast majority of futures contracts do not lead to delivery The reason is that most traderschoose to close out their positions prior to the delivery period specified in the contract Closing out

a position means entering into the opposite type of trade from the original one For example theNew York investor who bought a July corn futures contract on March 5 can close out the 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 futuresprice between March 5 and April 20

It is important to realize that there is no particular significance to the party on the other side of atrade in a futures transaction Consider trader A who initiates a long futures position by tradingone contract Suppose that trader B is on the other side of the transaction At a later stage trader Amight close out the position by entering into a short contract The trader on the other side of thissecond transaction does not have to be, and usually is not, trader B

2.2 THE SPECIFICATION OF THE FUTURES CONTRACT

When developing a new contract, the exchange must specify in some detail the exact nature of theagreement between the two parties In particular, it must specify the asset, the contract size (exactlyhow much of the asset will be delivered under one contract), where delivery will be made, and whendelivery will be made

Sometimes alternatives are specified for the grade of the asset that will be delivered or for thedelivery locations As a general rule, it is the party with the short position (the party that hasagreed to sell the asset) that chooses what will happen when alternatives are specified by 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 inthe 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 hasspecified the asset in its orange juice futures contract as

U.S Grade A, with Brix value of not less than 57 degrees, having a Brix value to acid ratio of not lessthan 13 to 1 nor more than 19 to 1, with factors of color and flavor each scoring 37 points or higherand 19 for defects, with a minimum score 94

The Chicago Mercantile Exchange in its random-length lumber futures contract has specified thatEach delivery unit shall consist of nominal 2x4s of random lengths from 8 feet to 20 feet, grade-stamped Construction and Standard, Standard and Better, or #1 and #2; however, in no case maythe quantity of Standard grade or #2 exceed 50% Each delivery unit shall be manufactured inCalifornia, Idaho, Montana, Nevada, Oregon, Washington, Wyoming, or Alberta or BritishColumbia, Canada, and contain lumber produced from grade-stamped Alpine fir, Englemannspruce, hem-fir, lodgepole pine, and/or spruce pine fir

For some commodities a range of grades can be delivered, but the price received depends the gradechosen For example, in the Chicago Board of Trade corn futures contract, the standard grade is

"No 2 Yellow", but substitutions are allowed with the price being adjusted in a way established bythe exchange

The financial assets in futures contracts are generally well defined and unambiguous Forexample, there is no need to specify the grade of a Japanese yen However, there are someinteresting features of the Treasury bond and Treasury note futures contracts traded on theChicago Board of Trade The underlying asset in the Treasury bond contract is any long-termU.S Treasury bond that has a maturity of greater than 15 years and is not callable within 15 years

In the Treasury note futures contract, the underlying asset is any long-term Treasury note with amaturity of no less than 6.5 years and no more than 10 years from the date of delivery In bothcases, the exchange has a formula for adjusting the price received according to the coupon andmaturity date of the bond delivered This is discussed in Chapter 5

The Contract 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 beunable to use the exchange On the other hand, if the contract size is too small, trading may beexpensive as there is a cost associated with each contract traded

The correct size for a contract clearly depends on the likely user Whereas the value of 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 futurescontract traded on the Chicago Board of Trade, instruments with a face value of $100,000 aredelivered

In some cases exchanges have introduced "mini" contracts to attract smaller investors Forexample, the CME's Mini Nasdaq 100 contract is on 20 times the Nasdaq 100 index whereas theregular contract is on 100 times the index

Delivery Arrangements

The place where delivery will be made must be specified by the exchange This is particularlyimportant for commodities that involve significant transportation costs In the case of the Chicago

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 inCalifornia, Idaho, Montana, Nevada, Oregon, and Washington, and in the province of BritishColumbia

When alternative delivery locations are specified, the price received by the party with the shortposition is sometimes adjusted according to the location chosen by that party For example, in thecase of the corn futures contract traded by the Chicago Board of Trade, delivery can be made atChicago, Burns Harbor, Toledo, or St Louis However, deliveries at Toledo and St Louis aremade at a discount of 4 cents per bushel from the Chicago contract price

Delivery Months

A futures contract is referred to by its delivery month The exchange must specify the preciseperiod during the month when delivery can be made For many futures contracts, the deliveryperiod is the whole month

The delivery months vary from contract to contract and are chosen by the exchange to meet theneeds of market participants For example, the main delivery months for currency futures on theChicago Mercantile Exchange are March, June, September, and December; corn futures traded onthe Chicago Board of Trade have delivery months of January, March, May, July, September,November, and December At any given time, contracts trade for the closest delivery month and anumber of subsequent delivery months The exchange specifies when trading in a particularmonth's contract will begin The exchange also specifies the last day on which trading can takeplace for a given contract Trading generally ceases a few days before the last day on which deliverycan be made

Price Quotes

The futures price is quoted in a way that is convenient and easy to understand For example, crudeoil futures prices on the New York Mercantile Exchange are quoted in dollars per barrel to twodecimal places (i.e., to the nearest cent) Treasury bond and Treasury note futures prices on theChicago Board of Trade are quoted in dollars and thirty-seconds of a dollar The minimum pricemovement that can occur in trading is consistent with the way in which the price is quoted Thus, it

is SO.01 per barrel for the oil futures and one thirty-second of a dollar for the Treasury bond andTreasury note futures

Daily Price Movement 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 ofspeculative excesses However, limits can become an artificial barrier to trading when the price ofthe underlying commodity is increasing or decreasing rapidly Whether price limits are, on balance,good for futures markets is controversial

Mechanics of Futures Markets 23

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 spotprice of the underlying asset When the delivery period is reached, the futures price equals—or isvery close to—the spot price

To see why this is so, we first suppose that the futures price is above the spot price during thedelivery period Traders then have a clear arbitrage opportunity:

1 Short a futures contract

2 Buy the asset

3 Make delivery

These steps are certain to lead to a profit equal to the amount by which the futures price exceedsthe spot price As traders exploit this arbitrage opportunity, the futures price will fall Suppose nextthat the futures price is below the spot price during the delivery period Companies interested inacquiring the asset will find it attractive to enter into a long futures contract and then wait fordelivery to be made As they do so, the futures price will tend to rise

The result is that the futures price is very close to the spot price during the delivery period.Figure 2.1 illustrates the convergence of the futures price to the spot price In Figure 2.1a thefutures price is above the spot price prior to the delivery month, and in Figure 2.1b the futures

Figure 2.1 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

price is below the spot price prior to the delivery month The circumstances under which these twopatterns 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 acertain 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 onThursday, June 5 to buy two December gold futures contracts on the New York CommodityExchange (COMEX) We suppose that the current futures price is $400 per ounce Because thecontract 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 themargin account would therefore be reduced by $600 to $3,400 Similarly, if the price of Decembergold 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 Whenthere is a decrease in the futures price so that the margin account of an investor with a longposition is reduced by $600, the investor's broker has to pay the exchange $600 and the exchangepasses the money on to the broker of an investor with a short position Similarly, when there is anincrease in the futures price, brokers for parties with short positions pay money to the exchangeand brokers for parties with long positions receive money from the exchange Later we willexamine 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, closingout the position would involve neutralizing the existing contract by selling 200 ounces of gold fordelivery 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

<$)

400.00 397.00 396.10 398.20 397.10 396.70 395.40 393.30 393.60 391.80 392.70 387.00 387.00 388.10 388.70 391.00 392.30

Daily gain (loss) ($)

(600) (180) 420 (220) (80) (260) (420) 60 (360) 180 (1,140) 0 220 120 460 260

Cumulative gain (loss) ($)

(600) (780) (360) (580) (660) (920) (1,340) (1,280) (1,640) (1,460) (2,600) (2,600) (2,380) (2,260) (1,800) (1,540)

Margin account balance ($)

4,000 3,400 3,220 3,640 3,420 3,340 3,080 2,660 4,060 3,700 3,880 2,740 4,000 4,220 4,340 4,800 5,060

Margin call ($)

1,340

1,260

prices in the case of the investor considered earlier The maintenance margin is assumed for the purpose of the illustration to be $1,500 per contract, or $3,000 in total On June 13 the balance in the margin account falls $340 below the maintenance margin level This drop triggers a margin call from the broker for additional $1,340 Table 2.1 assumes that the investor does in fact provide this margin by the close of trading on June 16 On June 19 the balance in the margin account again falls below the maintenance margin level, and a margin call for $1,260 is sent out The investor provides this margin by the close of trading on June 20 On June 26 the investor decides to close out the position by selling two contracts The futures price on that day is $392.30, and the investor has a cumulative loss of $1,540 Note that the investor has excess margin on June 16, 23, 24, and 25 Table 2.1 assumes that the excess is not withdrawn.

Further Details

Many brokers allow an investor to earn interest on the balance in a margin account The balance in the account does not therefore represent a true cost, provided that the interest rate is competitive with what could be earned elsewhere To satisfy the initial margin requirements (but not sub- sequent margin calls), an investor can sometimes deposit securities with the broker Treasury bills are usually accepted in lieu of cash at about 90% of their face value Shares are also sometimes accepted in lieu of cash—but at about 50% of their face value.