Options futures and other derivatives 5th

F I F T H

E D I T I O N

JOHN C.HULL


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

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

Graduate Investments/Portfolio Management
Fischer/Jordan, Security Analysis and Portfolio Management, Sixth Edition
Francis/Ibbotson. Investments: A Global Perspective
Haugen, The Inefficient Stock Market, Second Edition
Holden, Spreadsheet Modeling: A Book and CD-ROM Series

(Available in Graduate and Undergraduate Versions)
Nofsinger, The Psychology of Investing
Sharpe/Alexander/Bailey. Investments, Sixth Edition

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

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

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

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

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

Shapiro/Balbirer, Modern Corporate Finance: A Multidiseiplinary Approach to Value Creation
Van Horne/Wachowicz, Fundamentals of Financial Management, Eleventh Edition
Mastering Finance CD-ROM


Fifth Edition

OPTIONS, FUTURES,
& OTHER DERIVATIVES
John C. Hull
Maple Financial Group Professor of Derivatives and Risk Management
Director, Bonham Center for Finance
Joseph L. Rotman School of Management
University of Toronto

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


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

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

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

1
1
2
2

5
6
10
14
15
16
17

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

19
19
20
23
24

27
31
31
32
33
35
36
37
38
38
40

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

;

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


Contents

3.13
3.14
3.15

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

rates are constant

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

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

6. Swaps
6.1
6.2
6.3
6.4
6.5
6.6
6.7

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

60
60
61
63
64
65
67
68
70
70
72
75
78
82
86
87
88
88

90
92
93
93
94
94
96
98
100
102
102
103
104
110
Ill
112
116
118
119
120
123
125
125
131
134
136
140
143
145
146

147
147
149


Contents

xi

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

options markets
Underlying assets
Specification of stock options
Newspaper quotes
Trading
Commissions
Margins
The options clearing corporation

Regulation
Taxation
Warrants, executive stock options, and convertibles
Over-the-counter markets
Summary
Suggestions for further reading
Questions and problems
Assignment questions

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

8. Properties of
8.1
8.2
8.3

8.4
8.5
8.6
8.7
8.8

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

167
167
170
171
174
175
177
178
179
180

181
182
183

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

185
185
187
194
197
197
198
198
199

10. Introduction
10.1
10.2
10.3
10.4
10.5

10.6
10.7
10.8

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

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

Assignment questions


xii

Contents

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

216
216
217
222
223
225
226
227
228

229
229
230
232

12. The Black-Scholes model
234
12.1
Lognormal property of stock prices
234
12.2 The distribution of the rate of return
236
12.3 The expected return
237
12.4 Volatility
238
12.5
Concepts underlying the Black-Scholes-Merton differential equation
241
12.6
Derivation of the Black-Scholes-Merton differential equation
242
12.7 Risk-neutral valuation
244
12.8 Black-Scholes pricing formulas
246
12.9 Cumulative normal distribution function
248
12.10 Warrants issued by a company on its own stock
249

12.11 Implied volatilities
250
12.12 The causes of volatility
251
12.13 Dividends
252
Summary
256
Suggestions for further reading
257
Questions and problems
258
Assignment questions
261
Appendix 12A: Proof of Black-Scholes-Merton formula
262
Appendix 12B: Exact procedure for calculating the values of American calls on
dividend-paying stocks
265
Appendix 12C: Calculation of cumulative probability in bivariate normal
distribution
266
13. Options on
13.1
13.2
13.3
13.4
13.5
13.6
13.7

13.8
13.9

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

267
267
268
270
276
278
284
286
287
288

289
290
291
294
295


Contents

xiii

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

297

14. The Greek letters
14.1 Illustration
14.2 Naked and covered positions
14.3 A stop-loss strategy
14.4 Delta hedging
14.5 Theta
14.6 Gamma
14.7 Relationship between delta, theta, and gamma
14.8 Vega
14.9 Rho
14.10 Hedging in practice
14.11 Scenario analysis
14.12 Portfolio insurance
14.13 Stock market volatility

Summary
Suggestions for further reading
Questions and problems
:
Assignment questions
Appendix 14A: Taylor series expansions and hedge parameters

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

15. Volatility smiles
15.1 Put-call parity revisited
15.2 Foreign currency options

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

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

16. Value at risk
16.1 The VaR measure
16.2 Historical simulation
16.3 Model-building approach
16.4 Linear model

16.5 Quadratic model
16.6 Monte Carlo simulation
16.7 Comparison of approaches
16.8 Stress testing and back testing
16.9 Principal components analysis
Summary
Suggestions for further reading
Questions and problems
Assignment questions
Appendix 16A: Cash-flow mapping
Appendix 16B: Use of the Cornish-Fisher expansion to estimate VaR

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

345

xiv

Contents

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

372
372
374
376
377
378
382
385
388
388
389

391

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

392
392

19. Exotic options
19.1 Packages
19.2 Nonstandard American options
19.3 Forward start options
19.4 Compound options
19.5 Chooser options

19.6 Barrier options
19.7 Binary options
19.8 Lookback options
19.9 Shout options
19.10 Asian options
19.11
Options to exchange one asset for another
19.12 Basket options
19.13 Hedging issues
19.14 Static options replication
Summary
Suggestions for further reading
Questions and problems
Assignment questions
Appendix 19A: Calculation of the first two moments of arithmetic averages
and baskets

435
435
436
437
437
438
439
441
441
443
443
445
446

447
447
449
449
451
452

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

456
456
457

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

454

Contents

xv

20.3
20.4
20.5
20.6
20.7
20.8
20.9

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

458
460
461
465

467
472
474
478
479
480
481

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

and measures
The market price of risk
Several state variables
•
Martingales
Alternative choices for the numeraire
Extension to multiple independent factors
Applications
Change of numeraire
Quantos
Siegel's paradox

Summary
Suggestions for further reading
Questions and problems
Assignment questions
Appendix 21 A: Generalizations of Ito's lemma
Appendix 2IB: Expected excess return when there are multiple sources of
uncertainty

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

22. Interest rate
22.1
22.2
22.3
22.4
22.5

22.6
22.7
22.8
22.9

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

508
508
511
515
520
524
524
527
529

530
531
531
532
534
536

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

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

537

537
538
538
539
542
543
543
544
546

506


xvi

Contents

23.10
23.11
23.12
23.13
23.14
23.15
23.16

24. Interest rate
24.1
24.2
24.3
24.4

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

549
550
552
563
564
565
566
566
567
568
570

derivatives: more advanced models
571
Two-factor models of the short rate
571
The Heath, Jarrow, and Morton model

574
The LIBOR market model
577
Mortgage-backed securities
586
Summary
588
Suggestions for further reading
589
Questions and problems
590
Assignment questions
591
Appendix 24A: The A(t, T), aP, and 0(t) functions in the two-factor Hull-White
model
593

25. Swaps revisited
25.1 Variations on the vanilla deal
25.2 Compounding swaps
25.3 Currency swaps
25.4 More complex swaps
25.5 Equity swaps
25.6 Swaps with embedded options
25.7 Other swaps
25.8 Bizarre deals
Summary
Suggestions for further reading
Questions and problems
Assignment questions

Appendix 25A: Valuation of an equity swap between payment dates

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

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

610
610

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

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

Contents

xvn

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

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

29. Insurance, weather, and energy derivatives
29.1
Review of pricing issues
29.2 Weather derivatives
29.3 Energy derivatives
29.4 Insurance derivatives

Summary
Suggestions for further reading
Questions and problems
Assignment questions

678
678
679
680
682
683
684
684
685

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

686
686
690
693
694

695
697

Glossary of terms

700

DerivaGem software

:

715

Major exchanges trading futures and options

720

Table for N{x) when x sj 0

722

Table for N(x) when x ^ 0

723

Author index

725

Subject index

729


PREFACE
It is sometimes hard for me to believe that the first edition of this book was only 330 pages and
13 chapters long! There have been many developments in derivatives markets over the last 15 years
and the book has grown to keep up with them. The fifth edition has seven new chapters that cover
new derivatives instruments and recent research advances.
Like earlier editions, the book serves several markets. It is appropriate for graduate courses in
business, economics, and financial engineering. It can be used on advanced undergraduate courses
when students have good quantitative skills. Also, many practitioners who want to acquire a
working knowledge of how derivatives can be analyzed find the book useful.
One of the key decisions that must be made by an author who is writing in the area of derivatives
concerns the use of mathematics. If the level of mathematical sophistication is too high, the
material is likely to be inaccessible to many students and practitioners. If it is too low, some
important issues will inevitably be treated in a rather superficial way. I have tried to be particularly
careful about the way I use both mathematics and notation in the book. Nonessential mathematical material has been either eliminated or included in end-of-chapter appendices. Concepts that
are likely to be new to many readers have been explained carefully, and many numerical examples
have been included.
The book covers both derivatives markets and risk management. It assumes that the reader has
taken an introductory course in finance and an introductory course in probability and statistics.
No prior knowledge of options, futures contracts, swaps, and so on is assumed. It is not therefore
necessary for students to take an elective course in investments prior to taking a course based on
this book. There are many different ways the book can be used in the classroom. Instructors
teaching a first course in derivatives may wish to spend most time on the first half of the book.
Instructors teaching a more advanced course will find that many different combinations of the
chapters in the second half of the book can be used. I find that the material in Chapters 29 and 30
works well at the end of either an introductory or an advanced course.

What's New?
Material has been updated and improved throughout the book. The changes in this edition
include:
1. A new chapter on the use of futures for hedging (Chapter 4). Part of this material was
previously in Chapters 2 and 3. The change results in the first three chapters being less
intense and allows hedging to be covered in more depth.
2. A new chapter on models and numerical procedures (Chapter 20). Much of this material is
new, but some has been transferred from the chapter on exotic options in the fourth edition.
xix


xx

Preface

3. A new chapter on swaps (Chapter 25). This gives the reader an appreciation of the range of
nonstandard swap products that are traded in the over-the-counter market and discusses how
they can be valued.
4. There is an extra chapter on credit risk. Chapter 26 discusses the measurement of credit risk
and credit value at risk while Chapter 27 covers credit derivatives.
5. There is a new chapter on real options (Chapter 28).
6. There is a new chapter on insurance, weather, and energy derivatives (Chapter 29).
7. There is a new chapter on derivatives mishaps and what we can learn from them (Chapter 30).
8. The chapter on martingales and measures has been improved so that the material flows
better (Chapter 21).
9. The chapter on value at risk has been rewritten so that it provides a better balance between
the historical simulation approach and the model-building approach (Chapter 16).
10. The chapter on volatility smiles has been improved and appears earlier in the book.
(Chapter 15).
11. The coverage of the LIBOR market model has been expanded (Chapter 24).

12. One or two changes have been made to the notation. The most significant is that the strike
price is now denoted by K rather than X.
13. Many new end-of-chapter problems have been added.
Software
A new version of DerivaGem (Version 1.50) is released with this book. This consists of two Excel
applications: the Options Calculator and the Applications Builder. The Options Calculator consists
of the software in the previous release (with minor improvements). The Applications Builder
consists of a number of Excel functions from which users can build their own applications. It
includes a number of sample applications and enables students to explore the properties of options
and numerical procedures more easily. It also allows more interesting assignments to be designed.
The software is described more fully at the end of the book. Updates to the software can be
downloaded from my website:
www.rotman.utoronto.ca/~hull
Slides
Several hundred PowerPoint slides can be downloaded from my website. Instructors who adopt the
text are welcome to adapt the slides to meet their own needs.
Answers to Questions
As in the fourth edition, end-of-chapter problems are divided into two groups: "Questions and
Problems" and "Assignment Questions". Solutions to the Questions and Problems are in Options,
Futures, and Other Derivatives: Solutions Manual, which is published by Prentice Hall and can be
purchased by students. Solutions to Assignment Questions are available only in the Instructors
Manual.


Preface

xxi

A cknowledgments
Many people have played a part in the production of this book. Academics, students, and

practitioners who have made excellent and useful suggestions include Farhang Aslani, Jas Badyal,
Emilio Barone, Giovanni Barone-Adesi, Alex Bergier, George Blazenko, Laurence Booth, Phelim
Boyle, Peter Carr, Don Chance, J.-P. Chateau, Ren-Raw Chen, George Constantinides, Michel
Crouhy, Emanuel Derman, Brian Donaldson, Dieter Dorp, Scott Drabin, Jerome Duncan, Steinar
Ekern, David Fowler, Louis Gagnon, Dajiang Guo, Jrgen Hallbeck, Ian Hawkins, Michael
Hemler, Steve Heston, Bernie Hildebrandt, Michelle Hull, Kiyoshi Kato, Kevin Kneafsy, Tibor
Kucs, Iain MacDonald, Bill Margrabe, Izzy Nelkin, Neil Pearson, Paul Potvin, Shailendra Pandit,
Eric Reiner, Richard Rendleman, Gordon Roberts, Chris Robinson, Cheryl Rosen, John Rumsey,
Ani Sanyal, Klaus Schurger, Eduardo Schwartz, Michael Selby, Piet Sercu, Duane Stock, Edward
Thorpe, Yisong Tian, P. V. Viswanath, George Wang, Jason Wei, Bob Whaley, Alan White,
Hailiang Yang, Victor Zak, and Jozef Zemek. Huafen (Florence) Wu and Matthew Merkley
provided excellent research assistance.
I am particularly grateful to Eduardo Schwartz, who read the original manuscript for the first
edition and made many comments that led to significant improvements, and to Richard Rendleman and George Constantinides, who made specific suggestions that led to improvements in more
recent editions.
The first four editions of this book were very popular with practitioners and their comments and
suggestions have led to many improvements in the book. The students in my elective courses on
derivatives at the University of Toronto have also influenced the evolution of the book.
Alan White, a colleague at the University of Toronto, deserves a special acknowledgment. Alan
and I have been carrying out joint research in the area of derivatives for the last 18 years. During
that time we have spent countless hours discussing different issues concerning derivatives. Many of
the new ideas in this book, and many of the new ways used to explain old ideas, are as much Alan's
as mine. Alan read the original version of this book very carefully and made many excellent
suggestions for improvement. Alan has also done most of the development work on the DerivaGem software.
Special thanks are due to many people at Prentice Hall for their enthusiasm, advice, and
encouragement. I would particularly like to thank Mickey Cox (my editor), P. J. Boardman (the
editor-in-chief) and Kerri Limpert (the production editor). I am also grateful to Scott Barr, Leah
Jewell, Paul Donnelly, and Maureen Riopelle, who at different times have played key roles in the
development of the book.
I welcome comments on the book from readers. My email address is:

John C. Hull
University of Toronto


C H A P T E R

1

INTRODUCTION
In the last 20 years derivatives have become increasingly important in the world of finance. Futures
and options are now traded actively on many exchanges throughout the world. Forward contracts,
swaps, and many different types of options are regularly traded outside exchanges by financial
institutions, fund managers, and corporate treasurers in what is termed the over-the-counter
market. Derivatives are also sometimes added to a bond or stock issue.
A derivative can be defined as a financial instrument whose value depends on (or derives from)
the values of other, more basic underlying variables. Very often the variables underlying derivatives are the prices of traded assets. A stock option, for example, is a derivative whose value is
dependent on the price of a stock. However, derivatives can be dependent on almost any variable,
from the price of hogs to the amount of snow falling at a certain ski resort.
Since the first edition of this book was published in 1988, there have been many developments in
derivatives markets. There is now active trading in credit derivatives, electricity derivatives, weather
derivatives, and insurance derivatives. Many new types of interest rate, foreign exchange, and
equity derivative products have been created. There have been many new ideas in risk management
and risk measurement. Analysts have also become more aware of the need to analyze what are
known as real options. (These are the options acquired by a company when it invests in real assets
such as real estate, plant, and equipment.) This edition of the book reflects all these developments.
In this opening chapter we take a first look at forward, futures, and options markets and provide
an overview of how they are used by hedgers, speculators, and arbitrageurs. Later chapters will give
more details and elaborate on many of the points made here.

1.1

EXCHANGE-TRADED MARKETS

A derivatives exchange is a market where individuals trade standardized contracts that have been
defined by the exchange. Derivatives exchanges have existed for a long time. The Chicago Board of
Trade (CBOT, www.cbot.com) was established in 1848 to bring farmers and merchants together.
Initially its main task was to standardize the quantities and qualities of the grains that were traded.
Within a few years the first futures-type contract was developed. It was known as a to-arrive
contract. Speculators soon became interested in the contract and found trading the contract to be
an attractive alternative to trading the grain itself. A rival futures exchange, the Chicago
Mercantile Exchange (CME, www.cme.com), was established in 1919. Now futures exchanges
exist all over the world.
The Chicago Board Options Exchange (CBOET www.cboe.com) started trading call option


CHAPTER 1

contracts on 16 stocks in 1973. Options had traded prior to 1973 but the CBOE succeeded in
creating an orderly market with well-defined contracts. Put option contracts started trading on the
exchange in 1977. The CBOE now trades options on over 1200 stocks and many different stock
indices. Like futures, options have proved to be very popular contracts. Many other exchanges
throughout the world now trade options. The underlying assets include foreign currencies and
futures contracts as well as stocks and stock indices.
Traditionally derivatives traders have met on the floor of an exchange and used shouting and a
complicated set of hand signals to indicate the trades they would like to carry out. This is known as
the open outcry system. In recent years exchanges have increasingly moved from the open outcry
system to electronic trading. The latter involves traders entering their desired trades at a keyboard
and a computer being used to match buyers and sellers. There seems little doubt that eventually all
exchanges will use electronic trading.

1.2

OVER-THE-COUNTER MARKETS

Not all trading is done on exchanges. The over-the-counter market is an important alternative to
exchanges and, measured in terms of the total volume of trading, has become much larger than the
exchange-traded market. It is a telephone- and computer-linked network of dealers, who do not
physically meet. Trades are done over the phone and are usually between two financial institutions
or between a financial institution and one of its corporate clients. Financial institutions often act as
market makers for the more commonly traded instruments. This means that they are always
prepared to quote both a bid price (a price at which they are prepared to buy) and an offer price
(a price at which they are prepared to sell).
Telephone conversations in the over-the-counter market are usually taped. If there is a dispute
about what was agreed, the tapes are replayed to resolve the issue. Trades in the over-the-counter
market are typically much larger than trades in the exchange-traded market. A key advantage of
the over-the-counter market is that the terms of a contract do not have to be those specified by an
exchange. Market participants are free to negotiate any mutually attractive deal. A disadvantage is
that there is usually some credit risk in an over-the-counter trade (i.e., there is a small risk that the
contract will not be honored). As mentioned earlier, exchanges have organized themselves to
eliminate virtually all credit risk.

1.3

FORWARD CONTRACTS

A forward contract is a particularly simple derivative. It is an agreement to buy or sell an asset at a
certain future time for a certain price. It can be contrasted with a spot contract, which is an
agreement to buy or sell an asset today. A forward contract is traded in the over-the-counter
market—usually between two financial institutions or between a financial institution and one of its

clients.
One of the parties to a forward contract assumes a long position and agrees to buy the underlying
asset on a certain specified future date for a certain specified price. The other party assumes a short
position and agrees to sell the asset on the same date for the same price.
Forward contracts on foreign exchange are very popular. Most large banks have a "forward
desk" within their foreign exchange trading room that is devoted to the trading of forward


Introduction

Table 1.1 Spot and forward quotes for the USD-GBP exchange
rate, August 16, 2001 (GBP = British pound; USD = U.S. dollar)

Spot
1-month forward
3-month forward
6-month forward
1-year forward

Bid

Offer

1.4452
1.4435
1.4402
1.4353
1.4262

1.4456

1.4440
1.4407
1.4359
1.4268

contracts. Table 1.1 provides the quotes on the exchange rate between the British pound (GBP) and
the U.S. dollar (USD) that might be made by a large international bank on August 16, 2001. The
quote is for the number of USD per GBP. The first quote indicates that the bank is prepared to buy
GBP (i.e., sterling) in the spot market (i.e., for virtually immediate delivery) at the rate of $1.4452
per GBP and sell sterling in the spot market at $1.4456 per GBP. The second quote indicates that
the bank is prepared to buy sterling in one month at $1.4435 per GBP and sell sterling in one month
at $1.4440 per GBP; the third quote indicates that it is prepared to buy sterling in three months at
$1.4402 per GBP and sell sterling in three months at $1.4407 per GBP; and so on. These quotes are
for very large transactions. (As anyone who has traveled abroad knows, retail customers face much
larger spreads between bid and offer quotes than those in given Table 1.1.)
Forward contracts can be used to hedge foreign currency risk. Suppose that on August 16, 2001,
the treasurer of a U.S. corporation knows that the corporation will pay £1 million in six months (on
February 16, 2002) and wants to hedge against exchange rate moves. Using the quotes in Table 1.1,
the treasurer can agree to buy £1 million six months forward at an exchange rate of 1.4359. The
corporation then has a long forward contract on GBP. It has agreed that on February 16, 2002, it
will buy £1 million from the bank for $1.4359 million. The bank has a short forward contract on
GBP. It has agreed that on February 16, 2002, it will sell £1 million for $1.4359 million. Both sides
have made a binding commitment.
Payoffs from Forward Contracts
Consider the position of the corporation in the trade we have just described. What are the possible
outcomes? The forward contract obligates the corporation to buy £1 million for $1,435,900. If the
spot exchange rate rose to, say, 1.5000, at the end of the six months the forward contract would be
worth $64,100 (= $1,500,000 - $1,435,900) to the corporation. It would enable £1 million to be
purchased at 1.4359 rather than 1.5000. Similarly, if the spot exchange rate fell to 1.4000 at the end of
the six months, the forward contract would have a negative value to the corporation of $35,900

because it would lead to the corporation paying $35,900 more than the market price for the sterling.
In general, the payoff from a long position in a forward contract on one unit of an asset is
ST-K
where K is the delivery price and ST is the spot price of the asset at maturity of the contract. This is
because the holder of the contract is obligated to buy an asset worth ST for K. Similarly, the payoff
from a short position in a forward contract on one unit of an asset is
K-ST


CHAPTER 1

Figure 1.1

Payoffs from forward contracts: (a) long position, (b) short position.
Delivery price = K; price of asset at maturity = SV

These payoffs can be positive or negative. They are illustrated in Figure 1.1. Because it costs
nothing to enter into a forward contract, the payoff from the contract is also the trader's total gain
or loss from the contract.
Forward Price and Delivery Price
It is important to distinguish between the forward price and delivery price. The forward price is the
market price that would be agreed to today for delivery of the asset at a specified maturity date.
The forward price is usually different from the spot price and varies with the maturity date
(see Table 1.1).
In the example we considered earlier, the forward price on August 16, 2001, is 1.4359 for a
contract maturing on February 16, 2002. The corporation enters into a contract and 1.4359
becomes the delivery price for the contract. As we move through time the delivery price for the
corporation's contract does not change, but the forward price for a contract maturing on February
16, 2002, is likely to do so. For example, if GBP strengthens relative to USD in the second half of
August the forward price could rise to 1.4500 by September 1, 2001.

Forward Prices and Spot Prices
We will be discussing in some detail the relationship between spot and forward prices in Chapter 3.
In this section we illustrate the reason why the two are related by considering forward contracts on
gold. We assume that there are no storage costs associated with gold and that gold earns no income.1
Suppose that the spot price of gold is $300 per ounce and the risk-free interest rate for
investments lasting one year is 5% per annum. What is a reasonable value for the one-year
forward price of gold?
1
This is not totally realistic. In practice, storage costs are close to zero, but an income of 1 to 2% per annum can be
earned by lending gold.


Introduction

Suppose first that the one-year forward price is $340 per ounce. A trader can immediately take
the following actions:
1. Borrow $300 at 5% for one year.
2. Buy one ounce of gold.
3. Enter into a short forward contract to sell the gold for $340 in one year.
The interest on the $300 that is borrowed (assuming annual compounding) is $15. The trader can,
therefore, use $315 of the $340 that is obtained for the gold in one year to repay the loan. The
remaining $25 is profit. Any one-year forward price greater than $315 will lead to this arbitrage
trading strategy being profitable.
Suppose next that the forward price is $300. An investor who has a portfolio that includes gold can
1. Sell the gold for $300 per ounce.
2. Invest the proceeds at 5%.
3. Enter into a long forward contract to repurchase the gold in one year for $300 per ounce.
When this strategy is compared with the alternative strategy of keeping the gold in the portfolio for
one year, we see that the investor is better off by $15 per ounce. In any situation where the forward
price is less than $315, investors holding gold have an incentive to sell the gold and enter into a

long forward contract in the way that has been described.
The first strategy is profitable when the one-year forward price of gold is greater than $315. As
more traders attempt to take advantage of this strategy, the demand for short forward contracts
will increase and the one-year forward price of gold will fall. The second strategy is profitable for
all investors who hold gold in their portfolios when the one-year forward price of gold is less than
$315. As these investors attempt to take advantage of this strategy, the demand for long forward
contracts will increase and the one-year forward price of gold will rise. Assuming that individuals
are always willing to take advantage of arbitrage opportunities when they arise, we can conclude
that the activities of traders should cause the one-year forward price of gold to be exactly $315.
Any other price leads to an arbitrage opportunity.2

1.4

FUTURES CONTRACTS

Like a forward contract, a futures contract is an agreement between two parties to buy or sell an
asset at a certain time in the future for a certain price. Unlike forward contracts, futures contracts
are normally traded on an exchange. To make trading possible, the exchange specifies certain
standardized features of the contract. As the two parties to the contract do not necessarily know
each other, the exchange also provides a mechanism that gives the two parties a guarantee that the
contract will be honored.
The largest exchanges on which futures contracts are traded are the Chicago Board of Trade
(CBOT) and the Chicago Mercantile Exchange (CME). On these and other exchanges throughout
the world, a very wide range of commodities and financial assets form the underlying assets in the
various contracts. The commodities include pork bellies, live cattle, sugar, wool, lumber, copper,
aluminum, gold, and tin. The financial assets include stock indices, currencies, and Treasury bonds.
2

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

CHAPTER 1

One way in which a futures contract is different from a forward contract is that an exact delivery
date is usually not specified. The contract is referred to by its delivery month, and the exchange
specifies the period during the month when delivery must be made. For commodities, the delivery
period is often the entire month. The holder of the short position has the right to choose the time
during the delivery period when it will make delivery. Usually, contracts with several different
delivery months are traded at any one time. The exchange specifies the amount of the asset to be
delivered for one contract and how the futures price is to be quoted. In the case of a commodity,
the exchange also specifies the product quality and the delivery location. Consider, for example, the
wheat futures contract currently traded on the Chicago Board of Trade. The size of the contract is
5,000 bushels. Contracts for five delivery months (March, May, July, September, and December)
are available for up to 18 months into the future. The exchange specifies the grades of wheat that
can be delivered and the places where delivery can be made.
Futures prices are regularly reported in the financial press. Suppose that on September 1, the
December futures price of gold is quoted as $300. This is the price, exclusive of commissions, at
which traders can agree to buy or sell gold for December delivery. It is determined on the floor of the
exchange in the same way as other prices (i.e., by the laws of supply and demand). If more traders
want to go long than to go short, the price goes up; if the reverse is true, the price goes down.3
Further details on issues such as margin requirements, daily settlement procedures, delivery
procedures, bid-offer spreads, and the role of the exchange clearinghouse are given in Chapter 2.

1.5

OPTIONS

Options are traded both on exchanges and in the over-the-counter market. There are two basic
types of options. A call option gives the holder the right to buy the underlying asset by a certain date

for a certain price. A put option gives the holder the right to sell the underlying asset by a certain
date for a certain price. The price in the contract is known as the exercise price or strike price; the
date in the contract is known as the expiration date or maturity. American options can be exercised at
any time up to the expiration date. European options can be exercised only on the expiration date
itself.4 Most of the options that are traded on exchanges are American. In the exchange-traded
equity options market, one contract is usually an agreement to buy or sell 100 shares. European
options are generally easier to analyze than American options, and some of the properties of an
American option are frequently deduced from those of its European counterpart.
It should be emphasized that an option gives the holder the right to do something. The holder
does not have to exercise this right. This is what distinguishes options from forwards and futures,
where the holder is obligated to buy or sell the underlying asset. Note that whereas it costs nothing
to enter into a forward or futures contract, there is a cost to acquiring an option.
Call Options
Consider the situation of an investor who buys a European call option with a strike price of $60 to
purchase 100 Microsoft shares. Suppose that the current stock price is $58, the expiration date of
3

In Chapter 3 we discuss the relationship between a futures price and the spot price of the underlying asset (gold, in
this case).
4

Note that the terms American and European do not refer to the location of the option or the exchange. Some
options trading on North American exchanges are European.


Introduction

Profit ($)
30

20

10
Terminal
stock price ($)
0

30

40

50

60

70

80

90

-5

Figure 1.2 Profit from buying a European call option on one Microsoft share.
Option price = $5; strike price = $60
the option is in four months, and the price of an option to purchase one share is $5. The initial
investment is $500. Because the option is European, the investor can exercise only on the expiration
date. If the stock price on this date is less than $60, the investor will clearly choose not to exercise.
(There is no point in buying, for $60, a share that has a market value of less than $60.) In these
circumstances, the investor loses the whole of the initial investment of $500. If the stock price is

above $60 on the expiration date, the option will be exercised. Suppose, for example, that the stock
price is $75. By exercising the option, the investor is able to buy 100 shares for $60 per share. If the
shares are sold immediately, the investor makes a gain of $15 per share, or $1,500, ignoring
transactions costs. When the initial cost of the option is taken into account, the net profit to the
investor is $1,000.
Figure 1.2 shows how the investor's net profit or loss on an option to purchase one share varies
with the final stock price in the example. (We ignore the time value of money in calculating the
profit.) It is important to realize that an investor sometimes exercises an option and makes a loss
overall. Suppose that in the example Microsoft's stock price is $62 at the expiration of the option.
The investor would exercise the option for a gain of 100 x ($62 — $60) = $200 and realize a loss
overall of $300 when the initial cost of the option is taken into account. It is tempting to argue that
the investor should not exercise the option in these circumstances. However, not exercising would
lead to an overall loss of $500, which is worse than the $300 loss when the investor exercises. In
general, call options should always be exercised at the expiration date if the stock price is above the
strike price.
Put Options

Whereas the purchaser of a call option is hoping that the stock price will increase, the purchaser of a
put option is hoping that it will decrease. Consider an investor who buys a European put option to
sell 100 shares in IBM with a strike price of $90. Suppose that the current stock price is $85, the
expiration date of the option is in three months, and the price of an option to sell one share is $7. The
initial investment is $700. Because the option is European, it will be exercised only if the stock price
is below $90 at the expiration date. Suppose that the stock price is $75 on this date. The investor can


CHAPTER 1

Profit (S)
30

20

10
Terminal
stock price ($)
0

—V

60

70

80

90

100

110

120

-7

Figure 1.3 Profit from buying a European put option on one IBM share.
Option price = $7; strike price = $90
buy 100 shares for $75 per share and, under the terms of the put option, sell the same shares for $90
to realize a gain of $15 per share, or $1,500 (again transactions costs are ignored). When the $700
initial cost of the option is taken into account, the investor's net profit is $800. There is no guarantee

that the investor will make a gain. If the final stock price is above $90, the put option expires
worthless, and the investor loses $700. Figure 1.3 shows the way in which the investor's profit or loss
on an option to sell one share varies with the terminal stock price in this example.
Early Exercise
As already mentioned, exchange-traded stock options are usually American rather than European.
That is, the investor in the foregoing examples would not have to wait until the expiration date before
exercising the option. We will see in later chapters that there are some circumstances under which it is
optimal to exercise American options prior to maturity.
Option Positions
There are two sides to every option contract. On one side is the investor who has taken the long
position (i.e., has bought the option). On the other side is the investor who has taken a short
position (i.e., has sold or written the option). The writer of an option receives cash up front, but
has potential liabilities later. The writer's profit or loss is the reverse of that for the purchaser of the
option. Figures 1.4 and 1.5 show the variation of the profit or loss with the final stock price for
writers of the options considered in Figures 1.2 and 1.3.
There are four types of option positions:
1.
2.
3.
4.

A
A
A
A

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

Introduction

• • Profit ($)

30

X

40

50

70

80

90

i

60

Terminal
stock price ($)

-10

-20

-30

Figure 1.4 Profit from writing a European call option on one Microsoft share.
Option price = $5; strike price = $60
It is often useful to characterize European option positions in terms of the terminal value or payoff
to the investor at maturity. The initial cost of the option is then not included in the calculation. If
K is the strike price and S? is the final price of the underlying asset, the payoff from a long position
in a European call option is
max(5 r - K, 0)
This reflects the fact that the option will be exercised if ST > K and will not be exercised if ST < K.
The payoff to the holder of a short position in the European call option is
- max(S r - K, 0) = min(K - ST, 0)
.. Profit I
7
60
0

70

Terminal
stock price ($)

80
90

100

110

120

-10

-20

-30

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


10

CHAPTER 1

,, Payoff

| Payoff

,, Payoff

Figure 1.6 Payoffs from positions in European options: (a) long call, (b) short call, (c) long put,
(d) short put. Strike price = K; price of asset at maturity = ST
The payoff to the holder of a long position in a European put option is

max(K-ST, 0)
and the payoff from a short position in a European put option is
- ma\{K -ST,0) = min (S r - K, 0)
Figure 1.6 shows these payoffs.

1.6

TYPES OF TRADERS

Derivatives markets have been outstandingly successful. The main reason is that they have
attracted many different types of traders and have a great deal of liquidity. When an investor
wants to take one side of a contract, there is usually no problem in finding someone that is
prepared to take the other side.
Three broad categories of traders can be identified: hedgers, speculators, and arbitrageurs.
Hedgers use futures, forwards, and options to reduce the risk that they face from potential future
movements in a market variable. Speculators use them to bet on the future direction of a market
variable. Arbitrageurs take offsetting positions in two or more instruments to lock in a profit. In
the next few sections, we consider the activities of each type of trader in more detail.