9.6 Barrier options... 43919.7 Binary options ... 44119.8 Lookback options.. 44119.9 Shout options.... 44319.10 Asian options ............ cc eeee cece eeee eee ee ee 44319.11 Options to exchange one asset for another 44519.12 Basket options ............000. cc ceeeeeeeece neces 44619.13 Hedging issues 2.00.00... cece cece e cece erent teeta tr reese nnn teeeernibeeeetneeeeeen nies 44719.14 Static options replication ............00 cocci cece cece cece ttn te tenenebenebeteeceeeeeeeeel 447Summary ......00...0 ee - 449Suggestions for further reading. 449Questions and problems ....... 451Assignment QUueSLIONS 2.0.0.0... ce eee ee ene EEE Ete ere terete ete 452Appendix 19A: Calculation of the first two moments of arithmetic averagesand baskets ...00..... ce ccceeeee sete eeeeeeeeeeesete ane teeseeeeeeeeaetaees 45420. More on models and numerical procedures ............:sccssecsssevecsecessccecsscacerseseescoesoneeeeseseeees 45620.1 The CEV model20.2. The jump diffusion model
 Contents
20.320.420.520.620.720.820.9
21. Martingales and measures
21.121.221.321.421.521.621.721.821.9
22. Interest rate derivatives: the standard market models
xvStochastic volatility models ............::cecccccaseeeecececeeeeceeeeeeeeeesenaaeseseeacauaenees 458The IVF model................ ..460
            
Path-dependent derivatives...Lookback options..........Barrier options...............0006Options on two correlated assets..............0065
Monte Carlo simulation and American options ..........0..cccececseeeceeeecee ener renters 474STU 00 10 478Suggestions for further reading . 479Questions and problems ........ :
Assignment questions
The market price of risk..........000 00 ceee cece eeeeee ener eee reece cece tet eae eee ee ete ngeaneaaeeee 484Several state variables 0.0.0.0... eee eaneeeneeeeseeccenevessanaseuescesseeseeasaes ses 487Martingales ............... ieceeeeeee teers ..488
                          
Alternative choices for the numeraire ....
Extension to multiple independent factors, 492Applications ......... 493Change of numeraire . 1 A495Quantos.......... 497Siegel’s paradox . 499Summary ......0..-. eee .-500Suggestions for further reading. ..500Questions and problems ..... .-501Assignment questions +... 902Appendix 21A: Generalizations of Il6’s lemma ......0..0... cece eeee cee eee terre eee tere 504Appendix 21B: Expected excess return when there are multiple sources ofUNCETAINLY «6 eee cece eect ee ese eer etter tne beeseee stan aeertee
       
22.1 Black’s model ...........c eens22.2 Bond options...22.3 Interest rate caps ......0... ee22.4 European swap options........0........0022.5 Generalizations ...........22.6 Convexity adjustments ..22.7 Timing adjustments...22.8 Natural time lags.............22.9 Hedging interest rate derivativesSUMMALY 2.0.0... cece eee cere eeeSuggestions for further reading.Questions and problems ........Assignment QuesliONs .........:cccccecee cence ecseeeeeeneceeneneeeenens ”Appendix 22A: Proof of the convexity adjustment formula......0.00.....ccccceeeeeee 53623. Interest rate derivatives: models of the short rate ................::sceeeeeeeceececeeseece eeu eeceeceeeeneeeeeee 53723.1 Equilibrium models ............0000., .-53723.2. One-factor equilibrium models..... 53823.3. The Rendleman and Bartter model.. 53823.4 The Vasicek model.............0000000- 53923.5 The Cox, Ingersoll, and Ross model ..54223.6 Two-factor equilibrium models .......23.7. No-arbitrage models.............23.8 The Ho and Lee model...23.9 The Hull and White mode
      
 
 
   
 
       
xvi Contents23,10 Options on cowpon-bearing bonds........ 0. ee cccee cnc eeceecee nese ee eeeaneeteueeetnaees 54923.11  Imterest rate trees oo... cece cee c cee reece reece ea eee ede e tesa eee Eet eee e rete eet ee tee 55023.12 A general tree-building procedure ...... 00... .cecccceceeee ee tee ee ee cen eeeeeettaneeeeaeeeenees 55223.13. Nonstationary models............... ... 56323.14 Calibration...0....0.. eee . 56423.15 Hedging using a one-factor model..............ccccccce nsec seeeteeeeeee seen eteeetaes .-- 56523.16 Forward rates and futures rates 566
SUMIMALY oe. cece eeee eee neues 566Suggestions for further reading 567Questions and problems . 568Assignment questions 000... ener center rn eete es cece een ee ee ete eneeee ees 57024, Interest rate derivatives: more advanced MOdEIS...,......sccesssescevsseecserevssceeeeeseessseeeeneesecueees24.1 Two-factor models of the short rate .24.2 The Heath, Jarrow, and Morton model ...........0...0 0c ccceececceceaceneeeete ee eteaeteeaee 57424.3. The LIBOR market model.........0.... ccc cceccc cece ce eee ce te ce eeee cere tat tesa eseeaeeteeesn S7724.4 Mortgage-backed securities -- 586SUMMary .0.... eee eee - 588Suggestions for further reading .......00. eee ccc e cece sc eeeeteereeeeteenenne eee 589Questions and problemsAssignment questionsAppendix 24A: The A(t, T), op, and 0(t) functions in the two-factor Hull-Whitemodel
25, Swaps revisited ........cccsecsecccseseccsesssccnssnsecaesnsseasseeeeeeseeneeesneceeeseessesseeseeeeeegeeesens .
25.1 Variations on the vanilla deal 59425.2 Compounding swaps.......... ... 59525.3 Currency swaps......... 59825.4 More complex swaps... 59825.5 Equity swaps ...........0 ee .. 60125.6 Swaps with embedded options.
25.7. Other swaps
25.8 Bizarre deals
SUMMALY 00.0... eee eee
26. Credit risk23.126,226.326.426.526.626.726.826.9
   
Suggestions for further reading .Questions and problems ........00....... cece ceeeeeteeecatececueeeedeeeeeescne essa seetanerenaeee 607Assignment questions 0.000.000... eccceececec cere tener tern n rence eeeeeeeseeteeeete nant eninge 607Appendix 25A: Valuation of an equity swap between payment dates ................ 609deen aeeeeeesteaeeetececstsneeesteseneseeeaeneseseceeee ne eeeeeeseenecsseeneeeseneaeneeseareerseeeeeeneseenees 610
Bond prices and the probability of default .Historical data
    
  
Bond prices vs. historical default experience 619Risk-ueutral vs. real-world esiimates 0... . 620Using equity prices to estimate default probabilities .......00.... eee 621The loss given default 0.0.00... ccc ccecceceeeceeeceeeere ea eeeneeseaeeetaa esse eeeeaeeeeaa eee 623Credit ratings migration . 626Default correlations... --627Credit value at risk 0.0... ce cece neee ence eer eece eee eea een eta een ee sect eeadeeaeeennegs 630SUMIMATY 200. ieee cece cee ee ene e een eee eee eee een ee eee eet esse eaneeeeenesaeteneeeeeeeaeen tens 633Suggestions for further reading ...633Questions and problems......... .. 634
   
Assignment questions ......000000.c0cececesceeeeeceee eee ee sete nee teeeeeeererereesAppendix 26A: Manipulation of the matrices of credit rating changes............... 636Contents xvii
27. Credit derivatives ..............ccssscenseseesesceseceeenseccessenserenserees deeeneteneeresceeereeeeenes tee eeaeseneee 63727.1 Credit default swaps... cccccecccccctseeeeeeneeeeeeceeeaaeereeenaaneeeneeeeaaaeeeseseqaaanees 63727.2 Total return swaps27.3 Credit spread options .
 
     
27.4 Collateralized debt obligations.............. . 646
27.5 Adjusting derivative prices for default risk . 647
27.6 Convertible bonds... eee 652Summary .0......c cece 655
Suggestions for further reading .............0eccceececceeeeeeseeee eee set estes tetesaeeeesaeenene 655
Questions and problems .................ceceeseceeeeeeccceceeceeeeceeeeeeeeeeeeeenertatnsteaeeeea 656
Assignment Questions ...........00ccccccceeeeeeeeeeeceeseeeeeeeeeeeeeeeeeeaeeeeeeeneteeeeaneeerees 658
28. Real options ..........scesessceseseesseeeee saevecessceevecscnesrecceesseecseeseeceseseaceesteccsssecesssssseensees 12000
                 
28.1 Capital investment appraisal.28.2 Extension of the risk-neutral valuation framework ...28.3. Estimating the market price of risk28.4 Application to the valuation of a new business,
28.5 Commodity prices......0..0000. ee .28.6 Evaluating options in an investment opportumity ........ eee cece cent eeeSUMIMALY 000. cece eee cece eect rere ete eaten nnn Dn DEE EDD re DE terete teteeeeeeeneeeets®"Suggestions for further readingQuestions and problems..............Assignment questions ..............29. Insurance, weather, and energy derivatives....... veebeoarsesconsenscceseneccnersesceetsesscessooesse seseverees 07829.1 Review of pricing issues ............ cece cceeeceeeeeeeeeeeeeeeeeeeeeeaeeaeereeeenaenaeereeseneenaes 67829.2 Weather derivatives .... --679
29.3. Energy derivatives...
29.4 Insurance derivatives .. 682SUEMUIMALY . 0... eeeeeee eee ee ee 683Suggestions for further reading - 684Questions and problems ...............ccceceecceeneeeecenaeeesceeenacaeeceeeesaeaaeeeeeeeeaenaees 684ASSIQMMENE QUESTIONS 0.00... cece cecccccceeeeneeeeen ese caeeeneeeeeaeeeesueeeesenseeaersenaaeeeeaes 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 . --693SUMMALY 2.0... eee cee ee ence 694Suggestions for further reading 0.0.00... cece cceeceenteeceee ena eteeteeeenaeeeeeeeeeenaaa 695
Glossary of notation
Glossary of terms...
 
DerivaGem softwareMajor exchanges trading futures ANd OPVIONS 0.0.00. .ccccce ccc teceee eee ee een eeteeee nee teeet cess neteeeeeeeenetnesTable for N(x) When X <0 ciicciccecccceceec nce ee nee ee ee ee tear ee tee ene eegea ee eet eeaenenseeneenesernetasepeeanetersTable for N(x) when x 20 vo ccccccccccccctecccce nee cece nent eter ee nneee ee setaeeeeeeeesaeeeteeseeeaueeeeeeeeseseaaeanesAuthor Index o..ccccecccececetecee nee neeeneeeenneens
 
Subject 10dex ccc cec ccc eee nee EEE EE EE eet EL e en EE Sct abba cna det ned esaaeteeseaaeneeae eesPREFACE
 
 
It is sometimes hard for me to believe that the first edition of this book was only 330 pages and13 chapters long! There have been many developments in derivatives markets over the last 15 yearsand the book has grown to keep up with them. The fifth edition has seven new chapters that covernew derivatives instruments and recent research advances.
Like earlier editions, the book serves several markets. It is appropriate for graduate courses inbusiness, economics, and financial engineering. It can be used on advanced undergraduate courseswhen siudents have good quantitative skiils. Aliso, many practitioners who wani 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. | 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-chapier 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.
What’s New?
Material has been updated and improved throughout the book. The changes in this editioninclude:
1. A new chapter on the use of futures for hedging (Chapter 4). Part of this material waspreviously in Chapters 2 and 3. The change results in the first three chapters being lessintense and allows hedging to be covered in more depth.
2. A new chapter on models and numerical procedures (Chapter 20). Much of this materia! isnew, but some has been transferred from the chapter on exotic options in the fourth edition.
xi
mexx 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.
. There is a new chapter on real options (Chapter 28).. There is a new chapter on insurance, weather, and energy derivatives (Chapter 29).
. There is a new chapter on derivatives mishaps and what we can learn from them (Chapter 30).
oO s DN M
. 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 heen made to the notation. The most significant is that the strikeprice 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 Excelapplications: the Options Calculator and the Applications Ruilder. The Options Calculator consistsof the software in the previous release (with minor improvements). The Applications Builderconsists of a number of Excel functions from which users can build their own applications. Itincludes a number of sample applications and enables students to explore the properties of optionsand numerical procedures more easily. It also allows more interesting assignments to be designed.
The software is described more fully at the end of the book. Updates to the software can bedownloaded from my website:
www.rotman.utoronto.ca/"hull
Slides
Several hundred PowerPoint slides can be downloaded from my website. Instructors who adopt thetext are welcome to adapt the slides to meet their own needs.
Answers to Questions
As in the fourth edition, end-of-chapter problems are divided into two groups: “Questions andProblems” and “Assignment Questions”. Solutions to the Questions and Problems are in Options,Futures, and Other Derivatives: Solutions Manual, which is published by Prentice Hall and can bepurchased by students. Solutions to Assignment Questions are available only in the InstructorsManual.Preface xxi
Acknowledgments
Many people have played a part in the production of this book. Academics, students, andpractitioners who have made excellent and useful suggestions include Farhang Astlani, 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, MichaelHemiler, 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 Rendieman, Gordon Roberts, Chris Robinson, Cheryl Rosen, John Rumsey,Ani Sanyal, Kiaus 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’sas 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. Hull
University of TorontoCHAPTER 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 financialinstitutions, fund managers, and corporate treasurers in what is termed the over-the-countermarket. Derivatives are also sometimes added to a bond or stock issue.
A derivative can be defined as a financial instrument whose value depends on (or derives from)the values of other, more basic underlying variables. Very often the variables underlying deriva-tives are the prices of traded assets. A stock option, for example, is a derivative whose value isdependent on the price of a stock. However, derivatives can be dependent on almost any variable,from the price of hogs to the amount of snow falling at a certain ski resort.
Since the first edition of this book was published in 1988, there have been many developments inderivatives markets. There is now active trading in credit derivatives, electricity derivatives, weatherderivatives, and insurance derivatives. Many new types of interest rate, foreign exchange, andequity derivative products have been created. There have been many new ideas in risk management
and risk measurement, Analvsts have also become more aware of the need to analyze what are
and risk Measurement, Anatysts have aiso Decome more aware the need analyze what ar
known as real options, (These are the options acquired by a company when it invests in real assetssuch as real estate, plant, and equipment.) This edition of the book reflects all these developments.
In this opening chapter we take a first look at forward, futures, and options markets and providean overview of how they are used by hedgers, speculators, and arbitrageurs. Later chapters will givemore details and elaborate on many of the points made here.
1.1 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 fo-arrivecontract. Speculators soon became interested in the contract and found trading the contract to bean attractive alternative to trading the grain itself. A rival futures exchange, the ChicagoMercantile Exchange (CME, www.cme.com), was established in 1919. Now futures exchangesexist all over the world.
The Chicago Board Options Exchange (CBOE; www.cboe.com) started trading call option
1
 2 CHAPTER 1
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 asthe open outcry system. In recent years exchanges have increasingly moved from the open outcrysystem to electronic trading. The latter involves traders entering their desired trades at a keyboardand a computer being used to match buyers and sellers. There seems little doubt that eventually allexchanges will use electronic trading.
1.2 | OVER-THE-COUNTER 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 mutch larcer than theexchanges anc, measured in terms of ine tota: vOume of tracing, as oecome muecn iarger tnan tne
exchange-traded market. It is a telephone- and computer-linked network of dealers, who do notphysically meet. Trades are done over the phone and are usually between two financial institutionsor between a financial institution and one of its corporate clients. Financial institutions often act asmarket makers for the more commonly traded instruments. This means that they are alwaysprepared to quote both a bid price (a price at which they are prepared to buy) and an offer price(a price at which they are prepared to sell).
Telephone conversations in the over-the-counter market are usually taped. If there is a disputeabout what was agreed, the tapes are replayed to resolve the issue. Trades in the over-the-countermarket are typically much larger than trades in the exchange-traded market. A key advantage ofthe over-the-counter market is that the terms of a contract do not have to be those specified by anexchange. Market participants are free to negotiate any mutually attractive deal. A disadvantage isthat there is usually some credit risk in an over-the-counter trade (i.e., there is a small risk that thecontract will not be honored). As mentioned earlier, exchanges have organized themselves toeliminate virtually all credit risk.
1.3| FORWARD CONTRACTS
 
 
A forward contract is a particularly simple derivative. It is an agreement to buy or sell an asset at acertain future time for a certain price. It can be contrasted with a spot contract, which is anagreement to buy or sell an asset today. A forward contract is traded in the over-the-countermarket—usually between two financial institutions or between a financial institution and one of itsclients.
One of the parties to a forward contract assumes a Jong position and agrees to buy the underlyingasset on a certain specified future date for a certain specified price. The other party assumes a shortposition and agrees to sell the asset on the same date for the same price.
Forward contracts on foreign exchange are very popular. Most large banks have a “forwarddesk” within their foreign exchange trading room that is devoted to the trading of forwardIntroduction 3
Table1.1 Spot and forward quotes for the USD-GBP exchangerate, August 16, 2001 (GBP = British pound; USD = U.S. dollar)
 
Bid OfferSpot 1.4452 1.4456{-month forward 1.4435 1.44403-month forward {.4402 1.44076-month forward (.4353 1.4359
1-year forward 1.4262 1.4268
contracts. Table 1.1 provides the quotes on the exchange rate between the British pound (GBP) andthe U.S. doilar (USD) that might be made by a large international bank on August 16, 2001. hequote is for the number of USD per GBP. The first quote indicates that the bank is prepared to buyGBP (i.e., sterling) in the spot market (i.e., for virtually immediate delivery) at the rate of $1.4452per GBP and sell sterling in the spot market at $1.4456 per GBP. The second quote indicates thatthe bank is prepared to buy sterling in one month at $1.4435 per GBP and sell sterling in one monthat $1.4440 per GBP; the third quote indicates that it is prepared to buy sterling in three months at$1.4402 per GBP and sell sterling in three months at $1.4407 per GBP; and so on. These quotes arefor very large transactions. (As anyone who has traveled abroad knows, retail customers face muchlarger spreads between bid and offer quotes than those in given Table 1.1.)
Forward contracts can be used to hedge foreign currency risk. Suppose that on August 16, 2001,the treasurer of a U.S. corporation knows that the corporation will pay £1 million in six months (onFebruary 16, 2002) and wants to hedge against exchange rate moves. Using the quotes in Table 1.1,the treasurer can agree to buy £1 million six months forward at an exchange rate of 1.4359. Thecorporation then has a long forward contract on GBP. It has agreed that on February 16, 2002, itwill buy £1 million from the bank for $1.4359 million. The bank has a short forward contract onGBP. It has agreed that on February 16, 2002, it will sell £1 million for $1.4359 million. Both sideshave made a binding commitment.
Payoffs from Forward Contract