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D e r i v a t i v e s IV a r k e t s '

Aladura Perstal

Copeland/Weston F/k'tpp')? and Colvorate Policy

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Library of Congress Cataloging-inublication McDonald, Robert L. (Robert Lynch), 19.54Derivatives markers, 2e / Robert L. McDonald. P. Cm. lncludes index. E ISBN 0-321-28030-X 1. Derivative securities. 1. Titlez 14G6024 A3 113946 2006 E 332 64'.5-dc21 E 0-321-28030-X ISBN 08 07 06 t)5 E

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FINANCIAL ENGINEERINGAND APPLICATIONS 4 71

Exotic Options: I 443 443 lntroduction Asian Optims 444

Barrier Optiops 449

a*

503

Debt qnd Equity as Options 503 Multiple Debt Issues 511 warrants512 co'nvertible Bonds 513 Callable Bonds 516 Bond Valuation Based on the Stock Price

PART FO

Chapter 14

14.3

.

corporate

Applications 503 16.1 Equity Debt, and Warrants

'

Market-Maha-ng as Instumnce 436 437 Sttmman' 438 C/pt7pf'c?. FJ/rl/JD438 Readq p/-o/p/tu/ps 438 Appotdix 13.A: 'Thy/op.Series 44j Approximations Appendix j3.B: Greeks j;; tbe Model 441

413

An Exampl of Delta-l-ledging for 2 D alrs 41# Interpreting the Prolit Calculation 418 Delta-l-ledging for Several Dgys 420 A Self-Financing Portfolio: The Stock Moves One (r 422 The Mathematics of

Delta-l-ledging

Analysis

The Blackcholes Argument 429 Delta-Hedging of American Options What Is the Advantage to Frequent Re-l-ledging? 431 Delta-Hedging in Practice 432 Gamma-Neutraliv 433 lnsurance 436 Mafket-Makers

'

13.4

The Blackcholes

XYZ'S

Option Risk in the Absence of Hedging 414 oi Delta and Gamma as Measures Exposure 416

Delta-Hedgzg

13.6

14.1 14.2

Chapter 13

13.3

424 Delta-Gamma Approximations Theta: Accounting for Time 425 Undersranding the Market-/laker's Prolit 427

395

Profit Diagrams Before Maturity

-$,,77,'77,,3,),

k

colq'rErq'rs

o h1 E rqT s

17.2

Real Options 547 Investment and the NPV Rule

548

Static NPV 548 The Correct Use of NPV 549 The Project as an Option 5.50 lnvestment under Uncertaint'y 551 A Simple DCF Problem 551 Valuing Derivatives on rhe Cash Flow 55l

Evaluating a Project with a z-eear Investment Horizon 54 Evaluating the Project with an lrnite lnvestment Horizon 558

Real Options in Practice

Lognormal Confdence Intervals 600 The Conditional Expected Price 602 The Black-scholes Formula 604 the Parameters of a Estmating 18..$

558

Peak-Load Electricity Generation Research and Development 563

559

Commodity Extraction as an ' Option 565

17.4

Lognormal Distribution How Are Asset Prices Distributed? 608

Single-Barrel Extraction under Certainty 565 Single-Barrel Extraction upder Uncertainty 569 Valuing an Infinite Oil Reserve Commodit'y Extraction with Shut-Down and Restart Options Permanent Shutting Down 574 Investment Whe Shutdown Is Possible 576 578 Restarting roduction Additional Options 578

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.

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lrjjjjkjr'ajjjK.zkjjj/kukr

PART F

...

.

jk.j.;tryqt!(,q()..;;.) !rqqq4j,j;.z;.

. . . ..

.z

587

The Lognormal Distribution 593 A Lognormal Model of Stock Prices 595 Lognormal Probabilit'y Calculations 598 599

Computing Random Numbers

621

Simulating Correlated Stock 643 Prits

Monte Carlo Valuation

P?-o/J/cp/.

19.6

Ecient

Control Variate Method 630 Other Monte Carlo Methods 632 Valuation of American Options

21.1

646

21.2

Geometric Brownian Modpn

The Shqrpe Rqtio 659 The Risk-Neutral Procss It's Lemma 663

about

msk-Neutral Pricing

655

660

Functions of an It Process 663 Multivariate It's Lemma 66.5

Valuing a Claim on Stl 666 667 The Proceqs Followed by

The Black-scholes Equation Verifying the Formula for a

681

Derivative 683 The Black-scholes Equation and Equilibrium Returns 686 What If the Underlying Asset Is Not an Investment Asset? 688

Brownian Motion and It's

Lognormality 65.5 Relative Importance of the Drif't and Noise Terms 6.56 Correlated lt Processes 657 Multiplicaton Rules 658

20.4 20.5 20.6

The Black-scholes 679 Differential Equations and Valuation uder Certainty 679 The Valuation Equation 680 Bonds 680 Dividendaying stocks 681 The General Structure 681

Defnition of Brownian Motion 650 Properties of Bmwnian Motion 651 Arithmetic roWian Motion 6.5'3 The Ornstein-hlenbeck Process 6.54

20.3

675

Equation

Lemma 649 20.1 The Black-scholes Assunptin Stock Prices 649 Brownian Motion 650

624

Monte Carlo Valuation

Pl-ol7/tD//,s

Chapter 21

A pE'a dix 19.A: Fora7i//uz.JorGeometric Average Options 648

Chapter 20

XV

Fitrtbel. Reading

,9f?/7?/;f7?-),

Monte Carlo Vatuation of a European Call 62,5 Accuracy of Monte Carlo 626 Arithmetic Asian Option 627

19.5

Jumps 643

Generating 11 Correlated Lognormal 644 Random Vriables C/7f7p 645 ter Fltrtber Readiltg 645

Using Sums of Uniformly Distributed Random Varia es 622 Using the Inverse Cumulative Normal Distribution 622

Simulating a Sequence of Stock Prices 623

Converting p.NormalRandpm Variable to Standard Normal 590 Sums of Normal Random Variables 591

Probabilities

619

Simulating Lognormal Stock Prices 623

Chapter 18

18.4

Monte Carlp Valuation Computing the Optio Price as a Discounted Expected Value 617 Valuation with Risk-Neutral Probabilities 618 Valuation with True Probabilities

THEORY 585

18.2 18.3

605

Chapter 19 19.1

ADVANCEDPRICING

The Lognormal Distribution 587 18.1 The Normal Distribution

19.9

.

19.2 .

The Poisson Distribution 636 Simulatirlg Jumpswith the Poisson Distribution 639

,sfap/pfp-.)#

Cbapter .Flf?-#pcrReadinz 580 P/-ol7/cpo.$'580 Appendix 17.A: Calcttlation of Opfi??C$'# 583 Tinte to Drill a?: Oil 'T# Appendix 17.Bt T/pcSoltttion tf/ff/; Sbtttting Dorfal and Rcsfflrff/i,g 583 .,

19.7 19.8

Multiple

Histograms 608 Normal Probability Plots 609 6.13 Cbapter lhtrtber Reading 613 Prola/eaos 614 Appendix 18.A: T/lc Expectation ofa fwogaongt'l/Variable 615 a Ntllwffv Appca-x 18 rB: Casfnfco-n,g . .. ( Probabilit Plot 616

jiib

sifp.n//7f/rl,

..

k.

colql-Elq'rs

%.CONTENTS

xiv

21.4 21.5

690

Interpreting the Black-scholes Equation 690 The Backward Equation 691 Derivative Prices as Discounted Expected Cash Flows 692 Changing the Numeraire 693 Option Pricing When the Stok Price

Can

Jump

696

Merton's Solution for Diversifiable 697

Jumps I

Cbapter 5'J/777777f7r, 698 Fl/r#pcr Reading 698 .'I'h-olalEv/z,s 699 Appoldix 21 .A: Mltltivariate Black-scboles Analysis 700 Appendix 21.B: Proof ofpropositiol'

701

.'1

668 Proving the Propositior @ Specifc Examples 669 Q?7 670 Valuing a Claim on

630

.%l

633

20.8

Jumpsin the Stock Price Cbaptel. 67# .9/77777:f7?7,

672

Chapter 22 Exotic Options: 11 703 22.1 M-or-Nothing Options 703 Terminology 703 Cash-or-Nothing Options Asset-or-Nothing Options

704 706

21.1

k

xvi

Cox'rElq'rs CONTENTS

Ordinary Options and Gap Options Delta-l-ledging All-or-Nothing Options 707

706

All-or-Nothing Barrie Options

710

Cash-or-Nothing Barrier Options Asser-or-Noching Barrier Oprions Rebate Options 716

Barrier Options Quanto, 718

22.3 22.4

22.5

22.6

Variance and Volatility Swaps Pricing Volatility 759 xtending the Blackcholes

763

710 715

Currency-t-inlced Options

724

Foreign Euity Call Struck in Foreign Currency 728 Foreign Equity Call Struck in Domestic Currency 729 Fixed Exchange Rate Foreign Equity Call 730 Equity-Linked Foreign Exchange Call 731

Pricing

Measmement and Behvior ' Vol at ility 744

Value at lkisk for One Stock 815 817 VaR for Two or More Stcks Poitfolios for Nonlinear 819 VaR VaR for Bonds 826 Estimating Volatility 830 Bootstrapping Return Distributions 831 Issues with VaR 832 Alternative ltisk Measures 832 VaR and the ltisk-Neutral Distribution

Model 790 A Binomial Interest Rate Model

24.5

The Black-Derman--foy Model Verifying Yields 802 Verihring Volatilities 803 Constructing a Black-Derman--fby Tree 804 Pricing Examples 80.5

PART SlX

APPENDIXES 871

Appendix A

The Greek Alphabet

Continuous Compounding 875 B.1 The Language of lnterest Rates 875 B.2 The Logarithmic and Exponential Functions 876 Changing Interest Rates 877 Symmetry for lncreases and Decreases

878

837

Problellls

Appendix C C.1

Jensen'sInequality

881 Example: The Exponential Function 881 Example: The Price of a Call 882 Proof of Jensen's Inequality 884

C.2 C.3

26.2

Credit Risk 841 Default Concepts and Terminology 841 The Merton Default Model

26.3

Default at Maturity 843 Related Moels 845 and Default Experience Bond Ratings

Appendix D An Introduction to Visual Basic for Applications 885 D.1 Calculations without VBA 885 D.2 How to Learn VBA 886 D.3 Calculations with 'VBA 886

26.1

847

793

Using Ratings to Assess Bankruptcy Probability 847 Recovery Rates 8.50 Reduced Form Bankruptcy Models

796

26.4

Credit Instruments

873

Appendix B

83.5 Subadditive Iisk Measures Cbapter ,Slf?/pplzp-)?838 Rfr//pcr Reading 839 l''?-ol//tDns 839

Chapter 26

The Rendelman-Bartter Model 785 The Vasicek Model 786 The Cox-lngersoll-ltossModel 787 Comparing Vasicek and C1R 788 Bond Options, Caps, and the Black

Zero-coupon Bond Prices 794 Yields and Expected lnterest Rates Option Pricing 797

of

Time-varying Volatility: ARCH 747 The GARCH Model 751 Variation 755 Realized Quadratic

779

Equl'librium Short-Rate Bond Price Models 785

733

Historical Volatility 744 Exponentially Weiglzted Moving Average

746

Chapter 25 Value at Risk 25.1 Value at ltiik 813

The Belavior of Bonds and Inieresr' .. . . . '. . .. . . . .. Rates 780 A.n Impossibl l Prtcing Vdl 780 Etation for Bonds 781 An Equilibrium for Delta-Gamma Approxirtions Bonds 784

24.3

23.2

and

Chapter 24 Interest Rate Models 24.1 Market-Maling and Bond

Other Multivariate Options

Chapter 23 Volatility 741 23.1 lmplied Volatility 741

Model

Evidence 771 77.3 Cbapter &fn;?77'I).)? 773 htrtbel. Rct-ltfia,g Problel'ls 774 777 Appendix 23.4

727

Exchange Options 732 Options on the Best of Two Assets Basket Options 735 Cbaptel. &f7?;/?7t7?''y736 ltrtber Ref.ltfb/g 736 Problellls 737

lhfrl/Jcr

758

xvii

C/gtzplcrSltnul'an' 866 Flf/-#pcl-Readtg 867 f'lrol-l/c/as 867

sfp7/ntp'),808 Reading 808 f-b-o/pkcp/s809 Appendix 24.A: T/JC HeatbJtefrrow-Mb?-fo/zModel 811 Clmpter

757

Implied Volarility 764 Constant Elasticity of Variance 766 The Heston Model 768

JumpRisk

717

The Yen Perspective 720 The Dollar Perspective 721 A Binomial Model for the Dollar-Denominated Investor

-

Hedging and Pricing Volatility

%

852

853

Collateralized Debt Obligations 8.53 Credit Default Swaps and Related Structures 858 Pricing a Default Swap 862 CDS Indices 864

Problons

'

884

Creating a Simple Funcyion 886 A Simple Example of a Subroutine 888 Creating a Button to Invoke a Subroutine 888 Functions Can Call Functions 889 lllegal Function Names 889 Differences between Functions and Subroutines 890

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xviii

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Using a Named Range to Read and Write Numbers from a Spreadsheet 891 Reading and Writing to CellsThat Are Not Named 892 Using the Cetls Functions to Read and Write to Cells 892 Reading from within a Function 893

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envatives have moved to the center of modern corporate hnance, investments. and the mnnagement of financ ial institutions. They have also had a profound impact on other management functions such as business strategy, operations management, and marketina. A malor drawback. however. to makinz the power of derivatives accessible to students and practitioners alike has been the relatively high degree of mathematical sophistication required for understanding the underlying concepts and tools. With Robert McDonald s Derivatives Markets, we fnally have a derivatives text that is a wonderful blend of the economics and mathematics of delivatives plicing and easily accessible to A students and advanced undergraduates. It is a special pleasure for me to introduce this new edition, since I have long had the highest regard for the author's professional achievements and personal qualities. The book's orientation is neitheroverly sophisticated norwatered down, butrather ligor that creates an inherent iexibility for the structuring of a a mix o f intuition and derivatives course. The author begins with an introduction to forwards and futures and motivates the presentation with a discussion of their use in insurance and risk management. He looks in detail at forwards and futures on stocks, stock indices, currencies, interest rates, and swaps. His eeatment of options then follows logically from concepts developed in the eadier chapters. The heart of the text an extensive treatment showcases the author's of the binomial.option model and the Black-scholes equatio crystal-clear wliting and logicz development of concepts. Excellent chapters on financ ial engineering, security design, corporate applications, and real options follow and shed light on how the concepts can be applied to actual problems. The last third of the text orovides an advanced treatment of the most important concepts of delivatives discussed eadier. This part can be used by itself in an advanced derivatives course, or as a usef'ul reference in introductory courses. A rigorous development of the Black-scholes equation, exotic options, and interest rte models are . presente d us in g Brownian Motion and It's Lemma. Monte Carlo simulation methods are a l so discussed in detail. New chapters on volatility and credit risk provide a clea.r discussion of these fast-developing areas. D e rivatives concepts are now required for every advanced linarlce topic. Therefore, it is essential to introduce these concepts at an eady stage of MBA and undergraduate business o'r economics programs, and in a fashion that most students can understand. This text achieves tu s goa j jrj such an appealing, inviting way that students will actually enjoy theirjourney toward an understanding of derivatives. EotlAqoo

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I N TH E SECO N D ED ITI O N

PREFACE

general introduction j third of the book. The srstconrse is a u to the 'Iirst two-thirds and last options, swaps, and structured products), t futures, products (principally to derivative The second course is for those wanting a markets in which they trade, and applications. and the ability to perform their own analysis. deeper understandilg of the pricing models basic statistics and have seen calculus, The advanced course assumes that students know option-pricing framework as fully as and from that point develops the Black-scholes scienMBA-level course will produce rocket lo-week possible. N0 one expects that a students cheating of derivatives and it would be tijts, but mathematics is the language to pretend otherwise. order than it occurs in the book, Yu may Want to cover the material in a different severa.l possible paths allow Qexible use of the material. 1 indicate so I wrote chapters to possible to hop around. For example, 1 through the material below. In many cases it i.s siniulation chapters on lognormality and Monte Carlo wrote the book expecting that the might be used in a first derivatives course. 1 introduc ihe basic building The book has five parts plus appendixes. Part Chapters 2 and 3 and call and put optins. blockj of' derivatives: forward contracts hedging and invetfent strategies. examine these basic instruments and some common risk manageent tools ad discusses why Chapter 4 illutrates theuse of derivatives as chapters focus on undefstanding the firms might care about risk management. These pricing. contracts and strategies, but not on and swaps contracts. 1:! these Part 2 considers the pricing of forward, futures, pre-speified pzce, at a fture day. The contracts, you are obligated to buy an asset at a Chapter and how is it detrinined? Iain question is: What is the pre-specified price, commodities, assets, Chapter 6 discusses 5 examines forwards and futures on inancial forward contracts. Chapter 8 shows how rate and Chapter 7 looks at bond and interest from forward prices. swap prices can be deduced intuition about options prior to Part 3 studies option pricing. Chapter 9 develops 10 and 11 cover biomial option delving into the mechanics of option pricing. Chapters formula and option Greeks. Chapter 13 pricing and Chapter 12, the Black-scholes used by market-makefs when managing explains delta-hedging, which is the technique hedging relates to pricing. Clppter 14 looks the risk of an option position, and how compound including Asian options, barrier ojtins, at a few important exotic options, options, and exchange options. chapters are applied in Part 4. Chapter 15 The techniques and formulas in earlier from the which is the creation of new financial products covers fnancial engineering, compensation and equity pricing, derivatives building btcscks in earlier chapters. Debt 17 studies real optins-the 16. Chapter in covered Chapter options, and mergers are valuation and management of physical investapplication of derivatives models to the lnents.

hedging in depth. The material in this part Finally, Part 5 explores pricing and assumptions underlying the standard derivatives explains in more detail the structure and model and shows how the Black-scholes models. Clapter l 8 covers the Iognormal discusses Monte Carlo valuation, a powerful formula is an expected value. Chapter 19 20 explains what lt means to say that and commonly used pricing technique. Chapter

%

XZii

stock prices follow a diffusion process, and also covers lt's Lemma, which is a key result in the study of derivatives. (At this point you will discover that It's Lemma has already been developed intuitively in Chapter l3, using a simple numerical example.) Chapterzl delives the Black-scholes partialdifferential equation (PDE).Although the Black-scholes fonllltlais famous, the Black-scholes eqttatiol, discussed in this chapter, is the more profound result. Chapter 22 covers exotic options in more detail than Chapter 14, including digital barrier options and quantos. Chapter 23 discusses volatility estimation and stochastic volatility pricing models. Chapter 24 shows how and binomial analysis apply to bonds and interest rate delivatives. the Blaclscholes Chapter 25 covers value-at-risk, and Chapter 26 discusses the burgeoning market in credit products. WHAT

IS NEW IN THE SECOND

EDITION

rfhere are two new chapters in this edition, covering volatility and credit risk: * Chapter 23 covers empirical volatility models, such as GARCH and realized volatility',financial instruments that can be used to hedge volatility, such as variance swaps', and pricing models that incorporate jumps and stochastic volatility, such as the Hestop model. . Chapter 26 covers stnlctural models of bankruptcy risk (the Merton modell; tranched structtlres such as collateralized debt obligations', credit default swaps and credit indexes. throughout the book. Among the

There are numerous changes and new examples changes are the following: important more

. An expanded discussion of bond convexity * An expanded treatment of computing hedge ratios * An' expanded treatment of convertible and callable bonds * Discussion of the new option expensing expensing proposal

rules in

FAS 123R and the Bulow-shoven

. Discussion of a variable prepaid forward on Disney stock issue by Roy Disney . ln-depth discussion of a mandatorily convertible bond issued by Marshall & Ilsley, including pricing and structuring . The

use of sirulation

to price American options

. Additional discussion of implied volatility . Enhanced discussion of te link between discounted cash flow valuation and riskneutral valuation

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xxiv

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-

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THE MATERIAL

.4,

the material in this book. The matelial is There are potentially many ways to cover mathematical dificulty, which means that generally presented in order of increasing For example, tixedincome is related-material is sometimes split across distant chapters. exotic options in Chapters 14 and 22. Each of these covered in Chapters 7 and 24, and neighboring chapters. As an illustration of one way to use chapters is at the level of the chapters I skip in the courses 1 teach (withinthe the book, here is the material 1 cover constraints): some specisc topics due to time 15.42t5.5, 16, 1-6, 7.1, 8-10. 11.1-11.2, 12, 13.1-13.3, 14, * Ineoductory course: '

13, 18-22, 7, 8, 15.1-15.3, 23, 24, 25, 26. * Advanced course'. possible sets of chapters to use in courses that The table on page xxv outlines some sections of the book that provide background have different emphases. There are a few short-sales (Section 1.4), conreader should understand. These include on topics every (Sections 5.1 and 5.2), contracts prepyid fotward tinuous compounding (Appendix B), 7.1). (Section an'd zero-coupon bonds and implied forward rates

8. Swaps 9. Parity and Other Option Relationships l0. Binomial Option Pricing: l l 1. Binomial Option Pficing: 11 12. The Black-scholes Formula 13. Market-Making and Delta-l-ledgng 14. Exotic Options: l

A

NOTE

ON EXAMPLES

assist you this book display intermediate steps to Many of the numerical examples in it will also be possible for you to crein following the calculations. ln most cases basic assumptions. and compute the same answers starting from the ate a spreqdsheet generally rounded to three or four decimal However, numbers displayed in the text are signifcant digits. This creates a points, while spreadsheet calculations have many morewould obtain using spreadsheet, a those you dilemma: Should results in the book match equations? displayed obtain by computing tle or thoje you would will provide tlze results the numerical examples in the book rule, general As a spreadsheet. The displayed entering the equations directly in a will be you would obtain by logic of a calculatiqn, but a spreadsheet caleulations will help you follow the helpful in reproducing the linal rsult.

SUPPLEMENTS

accompanies materials for both instructors and smdents A robust package of ancillary the text.

15. Financial Engineering *

l6. Corporate Applications l7. Real Options 18. The Lognormal Distribution ..

l9. Monte Carlo Yaluation 20. Brownian Motion and It's Lemma 21 The Black-scholes Equation .

22. Exotic Options: FI 23. Volatility 24. Interest Rate Models 25. Value at Iisk 26. Credit lkisk

lnstructor's

Resomces

Forinstructors, an extensive set of online tools is available for download from the catalog P age for Derivatives Markets at www.aw-bc.com/finance.

i

AcxxowuEocuEx'rs

xxvi

% PREFACE Mark Cassano, University of CalAn online lnstructor's Solutions Manual by solutions to Ohio State University, contains complete gary, and Rdiger Fahlenbrach, problems. selected solutions spreadsheet to all end-of-chapter problems in the text and Indianapolis, features of University Will, The online Test Bank by Matthew W. and questions, short-answer five approximately ten to Vteenmultiple-choice qtlestions, for each chapter of the book. one longer essay question electronic formats, includingWindows The Test Bank is available in both print and Test Bank are and Microsoft Word files. The TestGen and r Macintosh TestGelt fles available online at hup://www.aw-bc-com/irc. Cao, Pennsylvania State UniOnline Powerpoint slides, developed by Charles Emm, Georgia State Uniand Ekaterina versity', Ufuk Ince, Unikersity of Washington; Copies of the slidej 4an the book. selected from art versity, provide lcture outlines and facilitate note taking duling class. be downsized and distributed to students to the computerized Test Bank filej (Testcontains The lnstructors Resource Disk the Test Bank liles (Word) and Powerpoint Gen), the Instructor Manual files (Word), 'liles.

Student Resources A printed Solutions ilanual by

Mark Cassano, University qf Calgary and Riidiger all the ven-numbered problems Fahlenbrach, Ohio State University, provides answers to in the textbook. Solutions, by Riidiger Fahlenbrach, New to this edition, Practice Problms and and worked-out solutions for each problems additional contains

Ohio State University, chapter of the textbook. functions in Excel are included on Spreadsheets with user-defined option pricing functions are written in VBA, with Excel the book. These a CD-ROM packaged with Visual Basic editor built into Excel. These the code accessible and modifable via the the book's Web site. spreadsheets and any updates are also posted on

ACKNOWLEDGMENTS book in 1994 by asking that the Kellogg Kellogg student Tejinder Singh catalyzed the derivatives course. Kathleen Hagerty and I Finance Department offer an advanced notes (developedwith Kathleen's initially co-taughtthat course and my part of the course third of this book. help and feedback) evolved into the last edition, I received invaluable assistance from kiidiger In preparing the second material with a cdtical of Fahlenbrach, Ohio State University, Fho read much the new other mistakes and offered valuable suggestions. Numerous eye, and who both caught the first edition. Colleagues in sttldents, colleagues. and readers provided comments on with their time include Torben Anthe Kellogg hnance department who were generous Deborah Lucas, Mitchell Petersen, Ernst dersen, Kathleen Hagerty, Ravi Jagannathan, Many Kellogg MBA and PIA.D.stuSchaumburg, Costis Skiadas, and David Stowell. Arne Staal, Caroline Sasseville, and Alex dents helped, but I want to especially thank

k.

xxvii

Wolf. Others who reviewed new material include David Bates, University of Iowa; Luca Benzoni, University of Minnesota; Milhail Chernov, Columbia University', and Darrell Dufie, Stanford University. Mark Schroder, Michigan State University, kindly provided code to calculate the non-central chi-squltred distribution, and Kellogg student Scott Freemon implemented this code in VBA. A special note of thanks goes to David Hait, president of OptionMetrics, for permission to include options data on the CD-ROM. ' I also received help and comments from George Allayanis, University of Virginia; Jeremy Bulow, Stanford University', Raul Guerrero, Dynamic Decisions', Darrell Karolyi, Compensation Strategies, lnc.; C. F. Lee, Rutgers University; David Nachman, University of Georgia', Anil Shivdasani, University of Nol'th Carolinai. and Nicholas Wonder, Western Washington University. I would like to particularly thank those who provided valuable feedback for the second edition, including Turan Bali, Baruch College, City University of New York; Philip Bond, Wharton School, University of Pennsylvania; Michael Brandt Duke University',Chades Cao, Pennsylvaniastate University',Bruce Grundy, Melbourne Business School, Australia; Shantaram Hegde, University of Connecticut', Frank Leiber, Bell Atlantic', Ehud Ronn, University of Texas, Austin; Nejat Seyhun, University of Michigan', John Stanseld, University of Missouri, Columbia', Christopher Stivers, University of Georgia', Joel Vandeni Dartmouth College; and Guofu Zhou, Washington University, St. Louis. I would be remiss not to aclcnowledge those who assisted with the srst edition, including Tom Arnold, Louisiana State University', David Bates, University of Iowa', Luca Benzoni, University of Minnesota', Mark Broadie, Columbia University', Mark A. Cassano, University of Calgay; George M. Constantinides, University of Chicago',Kent Daniel, Northwestern University', Jan Eberly, Northwestern University; Virginia France, University of Illinois', Steven Freund, Suffolk University', Rob Gertner, University of Chicago; Kathleen Hagerty, Northwestern University', David Haushalter, University of Oregon; James E. Hodder, University of Wisconsin-Madison; Ravi Jagannathan, Northwestern University', Avraham Kamara, University of Washington', Kenneth Kavajecz, Wharto:i School, University of Pennsylvania', Arvind Krishnamurthy, Northwestern University; Dennis Lasser, State University of New York at Binghamton', Cornelis A. Los, Kent State University', Deborah Lucas, Northwestern University', Alan Marcus, Boston College', Mitchell Petersen, Northwestern University', Todd Pulvino, Northwestern University', Ernst Schaumburg, Northwestern University', Eduardo Schwartz, University of California-l-os Angeles; David Shimko, Risk Capital Management Pqtners, lnc.; Anil Shivdasani, University of North Carolina-chapel Hill; Costis Sldadas, Northwestern University; Donald Smith, Boston University; David Stowell, Northwestern University', Alex Triantis, University of Maryland; and Zhenyu Wang, Yale University. The following served as software reviewers'. James Bennett, University of Massachusetts-Boston; Gordon H. Dash, University of Rhode Island', Adam Schwartz, University of Mississippi; and Robert E. Whaley, Duke University. Special thanks are due to George Constantinides, Jennie France, Kathleen Hagerty, Ken Kavajecz, Alan Marcus. Costis Skiadas, and Alex Triantis for their willingness to

'

AcltNowLEocu

xxviii %.P REFACE multiple times and for class-testing. Mark read and comment upon some of the material which I used both to compute the Broadie generously provided his pricing software, calculations. Heston model and to double-check my own Will, and Charles Cao for I thank Rdigey Fahlenbrach, Mark Cassano, Matt In addition, Riidiger book. for this their excellent work on the ancillary materials and KlishNovikov, Dnaitry Fahlenbrach, Paskalis Glabadanidis, Jeremy Graveline. Kaplin and Andy book for the checkers namurthy Subramanian served as accuracy jrovided programming assistance. Burghardt of Can-Futures, Andy Among practitioners who helped, 1 thank Galen Jacobson of the International Alex of Moor of E1 Paso Corporation, Brice Hill lntel, Tradelink, L.L.C. of Securities Exchange, and Blair Wellensiek intellecmal debts. From the many, Iwant With any bok, there are many long-term several classes from Robert Merton at tale good fortune to to single out two. I had the book is deeply in his debt, and M1T while 1 was a graduate student. Every dedvatives 1970s the are as essential today as they this one is no eycepton. His classic papers from with Dan Siegel, with worling amount learned an enormous were 30 years ago. I also 1991 death in at the age of 35 was Dan's options. whom I wrote several papers on real personally. profession, as well as to me a great loss to the made it clearfrom the outset The editolial and production team atAddison-Wesley lucky to have the project overI that their goal was to produce a high-quality book. was Donna Battista. Project Edity, Finance and tireless seen by Addison Wesley's talented myriad details and offered exof track expertly kept Manager Mary Clare McEwing Editor Mmjorie Singer Development board. sounding' cellent advice when l needed a significantly. Promanuscript improving the suggestions, Anderson offered innumerable manuscript into a physical marshalled forces to t'ul'n duction Supervisor, Nancy Fenton Services Publishing E1mStreet at excellent teams book. Among those forces were the edition, of the design first the compliments on and Techsetters. I received numerous second. Kudos are due to Gina Kolenda which has been carried tllrough ably into the and cover design. and Rebecca Light for their creativity in text tried hard to minimize errors, including have The Addison-Wesley team and 1 of course, 1 alone bear Nevertheless, above. noted the use of the accuracy checkers will be available at updates software and responsibility for remaining errors. Errata find do errors so we can if lnow you 1et Please us www.aw-bc.com/mcdonald. update the list. revision using Gnu Emacs and MITeX, 1 produced the original manuscript and authors. l am deeply patef'ul to the worldtools for extraordinarily powerful and robust this software. wide community that produces and supports Through both editions 1 thanks go to my family. My deepest and most heartfelt and tolerance. This book is have relied heavily on their understanding, love, support, children Claire, David, and Henly dedicated to my wife, Irene Freeman, and 111.M '

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xxix

Robert L. McDoltald is '?3W?llj Ndp?l?/lcr. Distillgltished Professor tTFf/lt-I?gcc atNorthJs/c,re?-/l University :5'Kellogg School ofManagemellt, lWpc?-dhe has rt-Illg/?/' since 1984. He is co-Editor of the Review of Financial Smdies altd llas !7c77Associate Editor of Analysis, Management #?c Journal of Finance, Journal of Financial and Quantitative Science, and tl/'/ze/--jt?l/?-??t'l/.. He tu a fh ill fct/nt//??fc..-t???? the U/?ve?-.W/yofNorth Carolina at Chapel Hill and a PIt.D. ?? Economicshnnl MlT

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and is pejorative. Suppose your family grows corn automatically think the term Yo provides insurance'. to mi11into cornmeal. The bet your friend's family buys corn your income. Your supplements this price; low sells for a earn $1 if your family's corn family the high cost of offsets buys is expensive; this friend earns $1 if the corn his unfavorable outcomes. The the bet hedges you both against corn. Viewed in this light, both of you. contcact has reduced risk for price of of contract simply to speculate on the Investors could also use this kind /74)/ the is It point: is key contract is not insurance. And that a llt7l it is corn. ln this case the ltsed, and JW/t?uses it, lt'Il detennines JW7cle?-olitself bltt Fitnj?it is ct?pl/?-t'7J everything. risk-redttcillg. Context is if you are new to the subject the impliAlthougl w've just defned a delivative, be' obvious dght away. You will come to a cations of the desnition will probably not through te book, smdying different deeper understanding of derivatives as we progress economics. products and their underlying lvbet''

Uses of Derivatives

derivatives? Here are some motives: What are reasons someone might use

and other users to reduce companies Derivatives are a tool for Risk manasement seller this in a simple wa#: The fnrmer-a risks. The corn example above illustrates low. contrac' t which makes a payment when'the plice of corn is of corn--enters into a h farmer who we therefore ty is hdging. This contract reduces the risk of loss for t. e forbiddingly complex, but many deriktives are It is common to think of derivatives'as Automobile is a derivative, for example. simple and familiar. Every form of insurance around a have an ccident. If you wrap fourcar insurance is a bet on whether you will valuable', if the car remains intact, it is not. tree, your insurance is in investment vehicles. As you will see later speculation Derivatives can serve as make bets that m'e highly leveraged (thatis, the book, derivatives can provide a way to of maling be large relative to the init'ifll cost the potential gain or loss on the bet can that the S&P bet example, if want to you specic view. For the bet) and tailored to a dedvatives today, can be from 1300 and 1400 one year ,

500 stock index will be between constructed to let you do that.

effect Sometimes derivatives provide a lower-cost way to transaction costs wish fund mut'ual of may a transaction. For example, the manager other a particular financial and paying brokers this entails paying fees to to sell stocks and buy bonds. Doing which we will discuss later. It is possible to spread, bid-ask the trading costs, such as economic effect as if stocks had acmally trade derivatives instead and achieve the same transaction the delikative might result in lower been sold and replaced by bonds. Using and buying bonds. costs than acttlally selling stocks regulatory restrictions, lt is sometimes possible to circumvent o' Regulatory arbitrage Derivatives are ften used, for examby trading derivatives. taxes, and accounting rules eliminate the risk of holding and of stock (receivecash ple, to achieve the economic sale

Reduced

Is A DERIVATIVE?

%

the stock) while still maintaining physical possession of the stock. This transaction may allow the owner to defer taxes on the sale of the stock, or retain voting lights, without the risk of holding the stock. Thse are common reasons for using derivatives. The general point is that derivatives provide an alternative to a simple sale or purchase, and thus increase the range of possibilities for an investor or manager seeldng to accomplish some goal.

Perspedtives on Derivatives How you think about derivatives depends on who you are. ln this book We will think about tluee distinct perspectives on derivatives: End-users are the coporations, investment managers, and perspective derivative enterinto who contracts forte reasons listed in the previous section'. investors speculate, reduce risk, costs, or avoid a nlle or regulation. End-users have a to manage riskreduction) and goal (forexample, care about how aderivative helps to meetthat goal.

The end-user

The market-maker

perspective

Market-makers are intermediaries, traders who will.

buy derivatives from customers who wish to sell, and sell derivatives to customers who wish to buy. ln order to make money, market-makers charge a spread: They ate like buy at a low price and sell at a high price. In tls rejpct markt-ralts Marketprice sell the retail price. wholesale and the low who higher at buy at grocers makers are also like grocers in th a t their inventory reqects customer demands rather than their own preferences: As lng aj shoppers buy paper towels, the grocer doesn't care whether they buy the decorative or super-absorbent style. Mter dealing with customers, market-makers are left with whatever position results from acommodating customer demands. Market-makers typically hedge this lisk and thus are deeply concerned about the mathematical details of pricing and hedging. The economic

observer

Finally, we can look at the use of derivatives, the activities of

the marlet-makers, the organization of the markets, the logic of the pricing models, and try to make sense of everything. This is the activity of the economic observer. Regulators must often don their economic obselwerhats when deciding whether and how to regulate a certain activity or market participant. These three perspectives are intertwined throughout the book, but as a general

point, in the erly chapters te book emphasizes the end-user perspective. In the late chapters, the book emphasizes the market-maker perspective. At all times, however, the economic observer is interested in mnking sense of everything.

Financial Engineering and Security Design the major idea-is that it is generally One of the major ideas in derivatives-perhaps possible to create a given payoffin multiple ways. The construction of a iven financial ngineerig. The fact that product from other products is sometimes called linancil

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TH E ROLE oF

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TO DERIVATIVES

need to hedge their implications. First, since market-makers this is possible has several 'I'he marketworks. market-mnking understanding how positions, this idea is central in that pays position ofsetting and ten creates an maker sells a contract to an end-user, posidon. hedged This creates a laimif it is necessary to pay the customer. how it can contract can be replicated often suggests given Second, the idea that a risk, initial the change dials effect, to tarft market-maker can, in be customized. The These changes pennit the cteation derivative. of paymentcharacteristics a premium, and given situation. of a product that is more appropziate for a about a given derivative by realizing Thiyd,itis often possible to improve inmition already understand. that it is equivalent to something we arbiage create a payoff, the regulatory Finally, because there are multiple ways to code, or in stdp. Dijtinctions existing in the tax discussed above can be dificult to is since a particular secttrity or derivative that regulations, may not be enforceable, the replaced by one that is tzeated differently but has regulated or taxed may be easily same economic prosle. that derivative products can generally be A theme running throughout the book is constnlcted from oter products.

MARKETS

sayin: that the Dow Jones lndustrialAverage has gone up We take for granted headlines arld interest rates have risen. But why 100 points, the dollar has fallen against the yen, index (such zise and fall of a pnrticular snancial do we care about these things? ls the and winners trak lndustrial Average) simply a way to keep score, to as the Dow Jones where watching sports; we watching the stock market like losers in the economy? ls of sound atld f'ury,but f'ull by told journalists, tale root for certain players and teams-a signifying nothing? often underappreciated, impact on Financial markets in fact have an enormous, consider tll role of fnancial markets we will everyday life. To help us understand the and both children 2.3 and Sarah Average have Average family, living in' Mytown. Joe income pays for employer in Mytown. work for the XYZ Co., the dominant lef4 is over goes cloting, and medical care. What their mortgage, trarlsportation, food, retirement. children's college tuition and their own toward savings earmarked for their of the markets and dezivatives play in the lives What role do global finarlcial 'rheir

Averages?

Financial Matkets and the Averages

affect their of the ways in which financiql markets The Averages are largely unaware lives. Here are a few: rieed for money to finance employer, XYZ Co., has an ongoing * The Average's dependnt on tll local bnnk forfunds because operations and investments. Itis not stocks and bondj in global rarkets. it can raise the money it needs by issuing '

MARKETS

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lisks. In addition to having property and * XYZ Co. insures itself against certain for its buildings, it derivatives markets to protect global casualty insurance uses against adverse interest commodity price changes. By and itself rate, currency, risks, able into bnnknlptcy, XYZ is less and less being likely to go to manage these unemployment. throw the Averages into likely to

* The Averages invest in mut'ual f'unds. As a result they pay lower transaction costs than if they tried to achieve comparable diversifcationby buying individual stocks. * Since both Averages work at XYZ, they run the risk that if XYZ does fall on hard times they will lose theirjobs. The mutual funds in which they invest own stocks in a broad array of companies, ensuring that the failure of any ope company will not wipe out their savings. * 'I'he Averages live in an area susceptible to tornadoes and insure their home. If theirinstlrance company were completely locak itcould notoffertornado insurance because one disaster would leave it unable to pay claims. By selling tornado risk in global markets, the inslzrance company can in effect pool Anytown tornado risk with Japan earthquake lisk and Flolida hurticane risk. This pooling makes instlrance available at lower rates. *

1.2 Tt4E ROLE OF FJNANCIAL

F1NANCIAL

Averages borrowed money f'rom Anytown bnnk to buy their house. The bnnk sold the mortgage to other investors, freeing itself from interest rate and default risk associated with the mortgage, leaving that to others. Because the risk of their mortgage is borne by those willing to pay the highest price for it, the Averages get te loWestpossible mortgage rate.

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In all of these examples, pnrlcular financial functions and rijks have been split up and parceled out to others. A bank that sells a molgage does not have to bear the risk of the mortgage. An insurance company does not bear a1l the Iisk pf a disaster. ltisk-sharing is one of the most important f'unctions of fnancial markets.

Risk-sharing Risk is an inevitable part of our lives and all economic activity. As we've seen in the example of theAverages, tinarlcialmarkets enable the fipgncial losses from at least some of these risks to be shared. Risk arises from natural events, such as eartquakes, Qoods, and hurricanes, and from unnatural events such as wars and polhical conQicts. Drought and pestilence destroy agriculttlre every yearin some part of the world. Some economies boom as oters falter. On a more personal scale, people are born, die, retire, find jobs, losejobs, marly divorce, and become ill. In the face of this risk, it seems namral to have arrangements where the lucly share with the unlucky. Risk-sharing occurs informally in families and communities. The insurance market makes formal risk-sharing possible. Buyers pay a premium to obtain various linds of insurance, such as homeowner's insurance. Total collected premiums are then available to help those whose houses bul'n down. The lucky, meanwhile, did not need inslzrance and have lost their premium. The market makes it possible for the lucky to help the unlucky.

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and interest ln the business world, changes in commodity prices, exchange rates, dollar becomes lf down. the burning house rates can be the financial equivalent of a It makes others and hurt. companies helped are are expensive relative to the yen, some risk, this exehange that the companies enabling so to sense for there to be a mechanism lucky can, in effect, help the unlucky lisk. Consider an insurance company that provides Even insurers need to share A large earthquake could generate clnims residents. earthquake insurance for California companies sufxcient to bankrupt a stand-alone insurance company. Thus, insurance claims. aainst larje ???c?'ker insuranc reinsurrs, buy, from to often use the reilsurance Iisks, risks enablinjinsurance bcome more t thereby Reinsurers pool diferentkinds of widely held. catastopliebbnds-bonds ln some cases, reinsurers fnmhersharerisks by issuing earthquake, that the issuer need not repay if there is a specifed event, such as a larg tisk earthquake willing accept Bondholders can to causing large insumnce claims. is if bond tlle there no interest payments on buy these bonds. in exchange for greater those exactly by risk erthquake be borne to allows earthjuake. An eazthquake bond investors who wish to bear it. Although there are mechanisms for sharing many kinds of risks, some have argued lisk-sharing is possible and desirable. Tlze ecolmrnist Robert that significantly more creation of entirely new markets for risk-sharing, the envisions 2003) Shiller tshiller, risks associated Fitfl house prices), incometrade including home equity insprance (to be need fully repaid ifwages decline in a particular that loans not linked loans (personal with occupation), and macro insurance (contracts payments linked tp ntional incomes). inciusive markets for While these markets do not yet exist. there is a trend toward more created an risk transfer. #or example, Goldman Sachs and Deutsche Bnnk httve recen with based claims buy possible payouts wlzich is to it market derivatives'' in teconomic economic statistics. The box on page 7 discusses this market. on You rnight be wondeling what this discussion has to do with the notions of diversiable and nondiversifable risk familiar from portfolio theory. ltisk is diversiliable risk if iy is unrelated to other risks. The risk that a lightning strilce will cause a factory idiosyncratic and hence diversisable. tf jny ikestors to burn down, for exnmple, lisk, this it has no signiticant effet on anyope. Rijk tat dpes not piece of share a small nondiversilia b le risk. % e ri sk of a stock spread vanish when across many investors is nondiversifable. example, is market crash, for Financial markets in theory setwe two purposes. Markets permit cliverslfable risk By deEnition, diversifiable risk vanishes when to be widely shared. This is ecient: markets permit nondiverstfiale risk, time, At shared. te same it is widely snancial which does not vanish when shared, to be held by those post willing to hold it. Thus, the ctmcdpf- and ??yl?-kcl' discttsed i'n this the l?7#twnc?7M/ ecoltomic idea ll?p#cl-fl/iWq p'??d/f. p-l/c-,/lt7?-ii/g mechallsms evelyone. book is 1/2:71the existelzce of .is

1.3 DERIVATIVES

IN PRACTICE

Derivatives use and the variety of derivatives have grown over the last 30 years.

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Growth in Derivatives Trading The introduction of delivatives in amarketoften coincides with an increase in priceriskin that market. Currencies were oficially penztted to float in 1971 when the gold standard was omcially abandoned. OPEc'S 1973 reductjon in tjye suppjy o ofI was followed by high and variable oil prices. U.s. interest rates became more volatile following inPation and recessions in the 1970s. The market for natural gas has been deregulated gradtlally since 1978 resulting in a volatile market in recent years. The deregulation of electricity began during the I990s. Figures 1.1, 1.2, and 1 show the changes for oil prices, exchange rates, and interest rates. The link between price variability and the development of derivatives markets is natural there is no need to risk .3

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A futures exchange is an orgaized and regulated marketplace for ading futures contracts, a lind of derivative. Figure 1.4 depicts f'utures contract volume for the three largest U.S. f'utures exchanges over the last 30 years. Clearly, the usage of futures contracts has grown signifcantly. Exchanges in other countries have generally experienced comparable or greger growth, ln 2002, Eurex, an electzonic exchange headquartered in Frankturt, Germany, traed 528 million contractsj the largest trading voltlme of any futures exchange in the world.

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different credit lisks. Hence bnnks could be expected to use interest rate derivatives, lisks in tose areas. Manufaccurrency derivatives, and credit dedvatives to manage turing firms that buy raw materials and sell in global markets might use commodity and cttrrency derivatives, but their incentives to manag: risk are less clear-cut because they are not regulated in the same ways as financial finns.

1.4 BUYING AND SHORT-SELLING FINAKCIAL

NASDAQ100 index

Namral gas

Eurodollars

Heating oil

Gasoline Gold

Wheat

Nikkei 225 Pork bellies Heating and cooling degree-days

Oats

Japanese yen

Electricity

Corn Soybeans

Copper

ASSETS

Throughout this book we will tnlk about buying and selling-and short-selling-assets such as stocks. These basic transactions are so important that it is Worthdescribing the details. First, it is important to understand the costs associated with buying and selling. Secopd, a very important idea used tlzropghout the bpok is that of short-sales. The concept of short-selling should be inmitive-a short-sale is just the opposite of a purchase-but for almost everyone it is hard to grasp at first. Even if you are fnmiliar with short sales, you should spend a few minutes reading this section.

Buying an Asst

ln recent years the U.S. Securities andExchange Comnzission(SEC), FinancialAccounting Standards Board (FASB), and the lnernational Accounting Standard Board (IASB) have increased the requirements for corporations to report on their use of derivatives. Nevezlheless, surprisingly little is known about how companies acmally use delivatives lisk. will be The basic sategies companies use are wemunderstood-and to manage perceived risk what of example, frction for described in this book-but it is not lcnown, do We companies the all frequently aggregate. in is hedged by a given company, or by hedging. hedging for rationale specific not or not lnow a company's We would expect the use of derivaives to vary by type of ftrm. For exnmple, financial firms, such as banks, are lzighly regulated and have capital requirements. They may have assets and liabilities in different currencies, with different maturities, and with

Suppose you want to buy 100 shares of XYZ stock. This seems simple: ((fthe stock price is $50, 100 shares will cost $50 x 100 $5000. However, this calculation ignores ansaction costs. First, there is a commission, which is a eansaction fee you pay your broker. A commission for te above order could be $15, or 0:3% of the purchase price. price'' is, surplisingly, imprecise. There are in fact two Second, the term prites, a price at which you can buy, and a price at which you can sell. The price at which you can buy is called te offer price or ask price, and the price at which youcan sell is called the bid price. Where do these terms come from? To buy stock, you can pick up the phone and call a broker. If the stock is not too obscure and your order is not too large, your purchase will probably be completed in a matter of seconds. Have you ever wondered where the stock comes from that you have just bought? lt is possible that at the exact same moment, another customer called the broker and put in an order to sell. More likely, howekr, a market-maker sold you the stock. Market-makers do what their nnme implies: They make markets. If you want to buy, they sell, and if you wany to sell, they buy. ln order to enrn a livin'g, market-makers sell for a high price and buy for a 1ow price. Ifyou deal with a rparket-maler, therefore, you buy for a high price and sell for a 1ow price. This difference between the price at which you can buy and te price at which you can sell is called the bid-ask spread-s In practice the bidrask spread on the stock you are buying may be $49.75to $50. Tllis means that you can buy for $50/shareand sell for $49.75/+%e. If you were to buy

lx'he table lists only a fraction of the contracts traded at these exchanges. For example, in June 2004, the Chicago Mercantile Exchange Web page listed futures contracts on over 70 different underlying assets ranging from butler to the weather in Amsterdam.

51fyou think a bid-ask spread is unreasonable, mskwhat a world without dealers would be like. Every buyer would have to lind a seller, and vice versa. The search would be costly and take ime. Dealers, because they maintain inventoly offer an immediate transaction, a service called imnediacy.

Table 1.1 provides examples of futtlres conacts traded aythese exchangesx Much commercii delivatives trading occufs in the over-the-counter market, where buyefs bnnks and dealrs rather than on an exchange. It is dicult ' gnd sellers transact with volume. However, in some markets, such as obtain statistics over-the-counter for to market is signiscantly larger than the over-the-counter the clear that currencies, it is market. exchange-traded

How Are Derivatives Used?

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INTRODUCTION

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BUYING

DERIVATIVES

commission twice, and you would pay the immediately and then sell. you would pay the bid-ask spread. difference between the Apparent rapid price :uctuations can occur because of the the bounce.'' bid is $49.75and the ask is If called bid-ask This is bid and ask prices. stock was last aded $50, a series of buy arid sell orders will cause the ptice at which the price has not changed, however, because to move between $49.75and $50. The changed. the bid and ask have not t'tnle''

commission Suppose XYZ is bid at $49.75and offered at $50, and the Example 1 100) + $15 $5015. lf you is $15. lf you buy 100 shares of the stock you pay ($50x Your round immediately sell them again, you receive ($49.75x 100) $15 $4960k receive from and what what you you pay tlip transactin cost--the difference between $51. k the bid and ask ptices-is $5015 $4960 a sle, not counting changes in

AND SHORT-SELLING

FINANCIAL

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There are at least tluee reasons to short-sell:

1. Speculation: A short-sale, considered by itself, makes money if the price of the stock goes down. The idea is to hrst sell higi and then buy low. (With a long position, the idea is to first buy low and then sell high.) 2. Financing: A short-sale is a way to borrow money, and it is frequently used as a form of hnancing. This is very common in th bond market, for example. 3. Hdging: You can undertake a short-sale to offset the risk of owning the stock or a delivative on the stock. This is frequently done by market-makers and traders.

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arld come f-rom. Incidentally, this discussion reveals where the terms like it sounds bid price backward. 'I'he terminology is that the You might at lirst think price at should be what you pay. lt is in fact what the market-nlaker pays; hence it is th what sell it is will forvhence market-maker what the offer is plice which you sell. The reflects the perspective of the market-maler. you have to pay. The terminology happens to your shares after you buy them? Generally they One last point: What carefully, ' are held by your broker. If you read the ffne print on your brokerage coneact shares to another investor. Why would your broker typically has the right to lend your topic, want to bon'ow your shares? The answer to that brings us to the next 'Ibid''

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anyone short-sales. associAlthough we have focused here on shares of stock, there are similar issues ated with buying any asset.

Shortlling exnmple, When we buy something, we are said to have a long position in that thing. For sell if we buy the stock of XYZ, we pay cash and receive the stock. Some time later, we that is lending, in the sense we pay money te stock and receive cash. This transaction receive may not be of The return we rate back the future. in receive and today money stock price lnown in advance (ifthe stock price goes up a lot, we get a high remrn; ifthe nonetheless. of loan but it is a ldnd goes down, we get a negative ret-urn), entails The opposite of a long position is a short position. A short-sale of X'YZ cash. time later, receiving Some we the borrowing shares of XYZ and then selling them, the lender. A short-sie can it it, and to ret'urn cash paying for stock, the back XYZ buy ft'om a be viewed, then, as just a way of borrowing money. When you borrow money advance. of in bank, you receive money today and repay it later, paying a rate interest set the This is also what happens with a short-sale, exeept that you don't necessarily know : ;

rate you pay to borrow.

These reasons are not mumally exclusive. For example, a market-maler might short-sale to simultaneously hedge and snance a position. use a Because short-sales cgn seem confusing, here is a detailed example that illusates how short-sales work. Example: short-sellingwine There are markets for many collectible items, such as :ne wines. Suppose there is a wine from a particular vintage and producer that you believe to be overpriced. Perhaps you expect a forthcoming review of the wine to be negative, or perhaps you believe that wines soon to be released will be of extraordinary quality, driving down prices of existing wines. Whatver the reason, you tlzink the price will fall. How could you speculate based on this beliep If you believe prices will rise, you would buy the wine on the market and plan to sell after the plice lises. However, if you believe prices will fall, you Would do the opposite: Sell today (at the high price) ad buy sometime later (at the 1owplice). How can you accomplish tlzis? In order to sell today, you must first obtain wine to sell. You can do this by borrowing a case from a collector. 'I'he collector, of course, will want a promise ihatyou will return the wine at some point', suppose you agree to returfl it one week later. Having reached agreemet, you borrow the wine and then sell it at the market plice. After one' week, you acquire a replacement case on the mtrket, then ret'ul'n it to the collector from whom you originally borrowed the wine. If te price has fallen, you will have bought the replacement wine for less than the price at which you sold the oliginal, so you make money. lf the price has risen, you have lost money. Either way, you havejust completed a short-sale of wine. The act of buying replacement wine and remnng it to the lender is said to be closing or covering the short position. Note th a t short-selling is a way to borrow money. Initially, you receive money selling the wine. A wek later you paid the money bak (youhad to buy a teplacefrom ment case to return to the lender). The rate ot interest you paid was 1c)wif the price of the replacement case was lom and high if the price of the replttcement case was high. This example is obviously simplifed. We have assumd several points:

* lt is easy to find a lender of wine. * lt is easy to buy, at a fair price, satisfactory wine to remrn to the lender: The wine you buy after one week is a pelect substimte for the wine you borrowed.

BUyl NG AN D SH O RT-SELLI NG

TO DERIVATIVES

k. INTRODUCTION

14

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The preceding examples were simple illustrations of the mechanics and economics of short-selling, and they demonstrate the ideas you will need to understand our discussions of derik.atives. It turns out, however, that some of the complexities we sldpped over are easy to understand and are important in practice. In this section we use the wine example to illustrate some of these practical issues.

credit rislt As the short-seller, you have an obligation to the lender to ret'urn the wine. The lender fears that you will renege on this obligation. This concern can be addressed with collateral: After you sell the wine, the lender can hold the money you received from the lender keeps the money selling the wine. You have an obligation to return the wine; ' . don't. that the in event you Holding on to the money will help the lender feel inofe scqre, but after thinking the matter over, the lender will likely want more from you thn just the current value of the wine. Suppose you borrow $5000woft of Wine. Whai happens, the lender will think, if the price of that particular wine lises to $6000one Week lat/ This is a $1000 losson your short-sale. In order to returu the wl, you Fill hve to pay $6000for wine youjustsold for $5000. Perhaps you cannot afford the xtra $1000and you will fail to rettlrnthe borrowed wine. The lender, thinking ahead, will be worried at the outset abou thispossibility and will ask you to prov tde lnole than the $5000th wie is worth, say extra $1000. Ius exea amount is called a haircvt, and serkes p proiect tlae lender against your failure to rettlrn the wie when the price lises.o In practice, short-sellers must have funds--called capital-ao be able to pay haircuts. I'he amount f capiti Places a limit on their ability tp short-sell. ''''

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Now consider a short-sale of stock. As with the previExample: short-selling stoclt selt iq receiving cash Fhp you short-sell stock you borrow the stock and ous eyopl, it the original today. At some f'uture tiate you buy the stoc k in the market and rettll'n to of te stock you shorttoday, equal to the market value owner. You have cash coming in market then-current its sell. ln the f'uture, you repay the bonowing by buying the asset at short-selling of a.loan. Thus, price and remrning the asset-this is like the repayment interest rate you pay is not that the except borrowing money, equivalent to a stock is stock plice. The rate of known in advance. Rather, it is detennined by the change in the security falls in price. In effect, if the interest is high ifthe security rises in price and 1ow which you borrow. With a shol-t-sale,you te rate of return on the security is the rate at rather than the buyer of a security. are like the issuer of a security 1.2 depicts the cash Suppose you want to Short-sell IBM stock for 90 days. Table shprt-setler must in t'urn dividends, the flows. Observe in particular that if the share pays wtne! This with alise did not make dividend payments to the share-lender. This issue and dividnd ordinary payment, dividend payment is taxed to the recipient, just like an it is tax-deductible to the short-seller. opposite of the cash flows Notice that the cash flows in Table 1.2 are exactly the l/2c opposite of bttying. literally from purchasing the stock. Thus, short-selling is

The Lease Rate of an Asset make payments to We have seen tat when you bonow an asset it may be necessary to stock are an exnmple of this. We will refer to the the lender. Dividends on of the asset. This concept will arise payment required by the lender as the lease rate discussions of frequently, and, as we will see, provides a unifying concept for oqr layer derivatives. under some circumstances The wine example did not have a lease payment. But Wine does not pay an explicit it might be necessary to make a payment to borrow wine. seeing bottles in the enjoys dividend but does pay an implicit dividend if the owner 'short-sold

15

ltisk and Scarcity in Shortelling

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scarcity As the short-seller, do you need to won'y about the short-sale proceeds? The lender is going to have $6000 of your money. Most of this, however, simply reqects your obligation, and we could ask a trustworthy third party, such as a bank, to hold the money so the lender cnnnot abscond with it. However, when you remrfl 4he wine, you fnrd?'dJl. This raises te question: What rate are going to want your money back, pltts you? the will the lender Over of interest pay course of the short-jale, the lender can en'rning. 6%. The lender could offer to pay yo 4% on the funds, invest your money, say, difference the 6% ealmed on the money and the 2% between keep thinking to qs a fee What happens the lender and paid if borrower negotiate? the 4% to you. point: The of is Here the interesting interest the lender pays on the cpllateral rate people how going depend want to borrow wine from the particular vintage is to on many

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f'Note that the lender is not concerned about your failure to perform when the price goes down because the lender has the money!

16

%.llq-rRoouc-rlohl

To

PROBLEMS

DERIVATIVES

practical matter, it may not be and prbducer, and how many are willing to lend it! As a wine, the lender will offer borrowed high demand for easy to 5nd a lender. lf there is willing for being to lend something that essentiily enrning a fee a low rate of interest, the lender might onelude the wine, is scarce. However, if no one else wants to borrow close to the market interesi of offer ' th an n othing and you a rate that a small fee is better rate.

different markets, the repo The rate paid on collateral is called different things in market. Whatever it is called, stock the in rate in bond markets and the short rebate another cost to your is of interest market rate the difference between this rate and the short-sie.

k

Jorion (1995) exnmines in detail one famous tderivatives disaster'': Orange County in Califonzia. Benzstein (1992) presents a histoly of the development of 5nancial markets, and Bernstein (1996)discusses the concept of risk measurement and how it evolved over the last 800 years. Miller (1.986)discusses origins of past Enancial innovation, while Merton (1999)and Shiller (2003)provide a stimulating look at posmarkets. Froot and O'Connell (1999)and Froot jiblelrl/rd developments in snancial (2001) examine the market for catastrophe reinsutance. D'Avolio (2001)explains the economics and practices associated with short-sales. Finally, Lewis (1989)is a classic, funny, insider's account of investment bnnking, offering a different (to say the least) perspective on the mechanics of global risk-sharing.

PROBLEMS the pric of plpthing Derivati.vesare financial ins%ments with apayoffdeterminedby lisk nd i ieduce speculation, management and else. They can be used as a tool for transaction costs or avoid tues and regulation. risk-sharing. One important f'unction of financial markets is to facilitate optimal d ith cpincid w an increase in The growth of derivatives markets over the last 50 years has shck, the ablidpment 1973 oil the zisks evident in variou markets. Events such as the created have a ew role of fixed exchange rates, and the deregulation of energy markets for dedvatives. ther cash) Ashort-sale entails borrowing asecurity. selling it, maki dikid:d (or coneepttlally the short-sale is payments io the security lender, and then retunng it. A of financing, used for speculation as a form opposite of a purchase. Short-sales can be details f short-selling in practic arl be undrstood of the hedge. Many o or as a way to short-seller and scarcity of shares ihgtcan be bkrowed. as a response to credit risk o f the lnderss' liig with Short-sellers typically leave the short-sale proceeds on deposit wit.h ,

1.1. Heating degree-day and cooling degree-day futtlres contracts make payments based on whether the temperature is abnormally hot or cold. Explain why the following businesses might be interested in such a contract: a. Soft-drink manufacmrers. b. Ski-resort operators. c. Electric utilities. d. Amusement park operators. 1.2. Suppose the businesses in the previous problem use f'umres contracts to hedge their temperamre-related risk. Who do you think might accept the opposite lisk? 1.3. ABC stock has a bid price of $40.95and an ask plice of $41.05.Assume there is a $20 brokerage commission. a. What amount will you pay to buy 100 shares? b. What amount will you receive for selling 100 shares? c. Suppose you buy 100 shares, then immediately sell 100 shares with the bid and ask prices being the same in both cases. What is your round-trip transaction cost?

collateral is called the short additional capital called a haircut. The rate paid on this the interest rate. rebate, and is less t11:,11

FURTHER

READING

this chapter. HowThe rest of this book provides an elaboration of themes discussed in 3, and4induce 2, discussion. Chapters the related directly to ever, certain chapters are derivatives, and show how forward and option conacts, which are the basic contracts in derivatives marketdetailhow in discusses 13 Chapter they are usedin risknanajement. derivatives can be combined makers manage their risk, and Chapter 15 explains how products. with instruments such as bonds to create customized risk-management sites for TheWeb thatlistthirconeacts. sites haveWeb The derivatives exchanges (Chicago Board of Trade), wwmcme. the exchanges in Figure 1.4 are www.cbot.com (New York Mercantile and Exchange), www.nymex.com com (Chicago Mercantile www.gs.com/econderivs. discussed at Economic derivatives are Exchange).

1.4.

Repeat the previous problem supposing that the brokerage fee is quoted as 0.3% 0 f the bid Or ask price. '

1.5. Suppose a security has a bid price of $1t0 and an ask price of $100.12.At what price can the market-maker purchase a security? At what price can a marketmaker sell a security? What is the spread in dollar terms when 100 shares are traded? '

1.6. Suppose you short-sell 300 shares f XYZ stock at $30.19 with a commission charge of 0.5%. Supposing you pay commission charges for purchasing the seculity to cover the short-sale, how much proht have you made if you close the short-sale at a plice of $29.87?

18

k. IN-rRoouc-rlolq

To

DERIVATIVES

1.7. Suppose you desire to short-sell 400 shares of JK.I stock, which has a bid price of $25.12and an ask price of $25.31.You cover the short position 180 days later when the bid price is $22.87and tle ask price is $23.06. a. Taldng into account only the bid and ask prices and interest), what proft did you earn?

(ignoringcommissions

b. Suppose that there is a 0.3% commission to engage in the short-sale (this is the commission to sell the stoek) and a 0.3% commission to close the short-sale (this is the commission to buy the stock bck). How do these commissions change the prct ilz the previous answer? paid nothing c. Suppose the 6-month interest rate is 3% and that you are much interest do you lose dpling the 6 proceeds. How short-sale the on months in which you have the short position? '

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1.8. When you open a brokerage account, you typically sign an agreement giving the b.rokerthe light to lend your shares without notifying or compesating you. Why do brokers want you to sign this agreement? 1.9. Suppose a stock pays a quarterly dividend of $3. You plan to hold a short position in the stock across the dividend ex-date. What is your obligation on that date? If of the you Jtre a taxable investor, what would you guess is the tax consequence deductible'?) dividnd the be would tax to particulm (In expect payment? you Suppose the company announces instead that the dividend is $5. Should you care that the dividend is different fromavhat you expected? 1.10. Short interest is a measure of the aggregate short positions on a stock. Check an service for the short interest on several stocks online brokerage or other snancial of your choice. Can you guess which stocks have high short imerest and wlzich have low? Is it theoretically possible for short interest to exceed 100% of shares outstanding? 1.11. Suppose that you go to a bank and borrow $100. You promise to repay the loan in 90 days for $102. Explain tls transaction using the terminology of short-sales. 1.12. Suppose your bnnk's loan officer tells you that if you tale out a mortgage (i.e., than you borrow money to buy a house) you will be permitted to borrow no more terminology of using ansaction the this Desclibe 80% of the value of the house. short-sales.

An

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chapter introduces the basic derivatives contracts: forward contrats, call options, and put options. T hese f'un damental contracts are widely used and serve as building blocks for more complicated dedvatives tht we discuss in later chapters. We explain here how the contiacts work and how to think about their risk. We also introduce an exemely important tool for analyzing derivatives positions-namely, payoff and proft diagrnms. The terminology and concepts introduced in this chapter are fundamental and W ill be used throughout this btjok. ,

2.1 FORWARD

CONTRACTS

Suppose you wish to buy a share of stock. Doing so entails at least three separate steps: (1) setting the price to be paid, (2)transfrring cash from the buyer to the seller, and (3) 'ansferring the share from the seller to the buyer. With an outright purchase of stock, :11three occur simultaneously. However, as a logical matter, a price could be set today and the ansfer of shares and cash would occur at a specifed date in the f'uture. This is in fact the definition of a forward contract: lt sets today the terms at which you buy or sell an asset or commodity at a specitic time in the future. A fomard contract does the following:

* Species the quantity and exact type of the gsjet or commodity the seller must deliver. * Speciss delivery logistics, such as time, date, and place. @ Specifies the price the buyer will pay at the time of delivery. * Obligates the seller to sell and the buyer to buy,subjet t the above spciscations. 'rhe

The time at which the contract settles is called the expiration date. asset or commodity on which the forward contract is based is called the underlying asset. Apart from commissions and bid-ask spread: (see Section 1.4), a forward contract requires no initial payment oc prerniumk The conact'ual forward price simply represents the price at which consenting adults agree today to transact in the f'uture, at which time the buyer pays the seller the forward price and the seller delivers the asset. Futures contracts are similar to fomard contracts in that they create an obligation to buy or sell at a predetermined price at a future date. We will discuss the instimtional and pricing differences between forwards and futures in Chapter 5k For the time being, think of them as interchangeable.

21

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for different values of the

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50

EXAM PLE: EQUITY-LI

AN D OPTIONS

TO FORWARDS

AN INTRODUCTION

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Economics of the CD Now we are in a posiion to understand te economics of this product. TIAiIIIC about what happens if te index is below 1300 at expiration. We pay $10,000and we receive $10,000 back, plus an option. we have forgone interest on $10,000in exchange for the possibility of receiving 70% of the gains on the S&P. Suppose that the effective rfhus,

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742

y Equity-lainlced CDs?

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Essentially, the buyer forgoes interest in exchange for a call option on the index. With this descliption we have reverse-engineered the CD, decomposing it in tenns of an option and a bond. The question of whether the CD is fairly priced turns on whether the $2742is a fair price for the index option implicii in the CD. Given information about the interest rate, the volatility of the index, and the dividend yield on the index, it is possible to plice the option to determine whether the CD is fairly priced. We perfonu that analysis for this example in Chpter 15.

la

.

51

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annual interest rate is 6%.,after 5 1/2 years, the buyer has lost interest of

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N KED CDS

With reverse-engineedng, we see that an investor could create the equivalent of an equity-linked CD by buying a zero-coupon bond >nd 0.7 call options. Wy, then, do products like this exist? Consider what must be done to replicate the payoff. If a retail investor were insure to an index westment using options, the investor would have to learn about options, decide what maturity, stlike price, and quantity to buy, and pay transaction costs. Exchange-trade d opt ions have at most 3 years to mamrity, so obtaining longerterm protection requires rolling over the pitipn at plp pnt. An equitplinked CD provides a prepackaged soluiton. It may provide a pattern of market exposure tht many investors could not otherwise obtain at such 1ow transaction

.

costs.

The idea that a prepackaged deal may be attractive should be falrtilir to you. bpilding blocks, as it wer-and they Supermarkets sell whole heads of letmce-salad also sell, at a premium price, leytuce already washed, torn into bite-sized piecs, and mixed as a salad. The transaction cost of salad preparation leads some consumers to prefer the prepackaged salads. What does the :nancial institmion get oui of this? Just as the supermarket earns prost on prepackaged salads, the issuing bank wants to earn a transaction fee on the CD. When it sells a CD, the issuing balzk bonows money tthezero-coupon bond portion of the CD) and receives the premium for wliting a call option. The cost of the CD to the bank is te cost of the zero-coupon bond plus the cost of the call option. Obviously the bank would not issue the equity-lirtked CD in the srstplace unless it was less expensive than Zternative ways to attract deposits, such as standard CDs. The equity-linked CD is risk'y because the bnnk has written a call, but th bank can manage this risk in several ways, one of which ij to purchase call options from a dealer to offset the risk of having written calls. Using data from the early 1990s, Baubonij et a1. (1993) estimated that issuers of equity-linked CDs earned about 3.5% of the value of the CD as a fee, with l about 1% as the transaction cost of hedging the wlitten call.1

l IA back-of-the-envelope calculation in Chapter 15 suggests the issuer fees for this product are in the neighborhood of 4% to 5%.

k

k

52

AN INTRODUCTION

FURTHER

AND OPTIONS

FORWARDS

To

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ard contract, wlzich has two mipor differences from the act'ual fonvard: 1. The forward contract has a zero premium, while the synthetic forward requires that

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2. With the forlard coneact we pay theforward jnce, while with the synthetic forward we pay the strike price. If you think about it, these two cosideratiops must be related. If we set the strike price low, we are obligated to buy the ipdex at dijcount relative to the fomard price. plie than the foioard pric ij a bne:t. In order to obtain this beneft Buying at a tower we have to pay the jositiv nt tjti Ijfmium, which stemj from th call being more

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so -50 -100

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-

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In words, the presentvalue of the bargain elementfrombuying theindex atthe strikeprice (the right-hand side of equation (3.1)) must be offset by the initial net option premium (the left-hand side of equation (3.1)1.Equation (3.1) is lnown as put-call parity, and one of the most important relations in options.

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Example 3.1 As an example of equation (3.1),consider buying the 6-month 10001)l@p ' S&R call for a premium of $93.809and selling the 6-month looo-strike put for ).1stnke )))) premium of $74.201. These transactions create a synthetic forward permitting us to C a '!'1 buy the index in 6 months for $1000. Because the actual forward price is $1020,this synthetic forward permits us to buy the index at a bargain of $20, the present value of !q.g . )ll which is $20/ 1.02 $19.61. The difference in option premiums must therefore be ). .'tq In fact, $93.809 $74.201 $19.61.This result is exactly what we would get )t.,$19.61. .'

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800

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950 1000 1050 1100 150 S&R Index Pdce ()

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1200

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expensive t.11a:1 the put. ln fact, in Figure 3.6, tlzeimplicit cost of te synthetic forwardthe price at which the profit on the ombined call-ppt position is zero-is $1020,which is the S&R forward price. Sirnilarly. it we set the sike price high, we are obligated to buy the index at a high price relative to the forward price. To offset the exta cost of acquiring the index using the high stdke options, it makes sense that we would receive payment initially. This would occur if the put that we sell is more expensive than the call we buy. Finally, if we set the strike price equal to the forward price, tlen to mimic the forward th: irlitial yremim ntust tualzero. I this ase, put nd call prmiums must

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A forward contract for which the premium is not zero is sometimes called an offmarket forward. This trminology arises since a true forward by desniitpn has a zero premium. Therefore, a forward contract with a nonzero premium must have a forward market (forward)price.'' Unless the stlikeplice equals te folavard price which is price, buying a call and selling a put creates an off-market forward. Iloffthe

We have seen earlier that buying the index and positions prost buying a put generates the same as buying a call. Similarly, selling a covered call calll selling generates the same profit as selling a put. Equation (buying the index and a (3.1) explains why tis happens. Consider buying the index and buying a put, as in Section 3.1.. Recall tat, in this and the index price equal to $1000. example, we have the forward price equal to $1020 equals of forward price the index price. Rewriting equation the Thus, the present value (3.1) gives Equivlente

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We can summarize this argumet by saying that the ndr cost t/-lllyf??' the f?7#c-'t: ttsing options ??2II'f eqttal (he n'f cost ofbttying #?dinde,x ldWlg afonvalzl contract. If at time 0 we en terinto a lonjforward positton expiring at iime F, Fe obligate ourselves to buying the index at the forward price, F(),w.'I'he present value of buying the index in the ftlture is just the present value f the forward price, PV(F0,r). q lfinstead we buy a call and sell a put today to guarantee tlle purchase price for the index in the f'utwe, the present valpe of the cost is the net option premium for buying the call and setng the put, CalltA-. FJ Puttff. F. plus the present value of the strike price, PV(#). trfhenotations Calltf, 7') and Puttff, r) denote the premiums of options wit.h stdke price K and wit t periods until expiration.) ,

-

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of diferent

PV(Fo,w) + Puttf,

F)

=

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=

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That is, buying the index and buying the put cost the same, and generate the same payoff, as buying the call arld buying a zero-coupon bond costing PV(#). (Recall from Section 2.1 that a bond does not affect protit.)

% Ilqsu RANCE,

70

AlqD OTH ER ARATEGI

COLLARS,

Similarly, in the case of PV(Fo,w)

writing

F)

=

PV(#)

That is, writing a covered call has the same proft as lending PV(#) Equation (3.1)provides a tool for constructing equivalent positions.

and selling a put.

In deriving equation (3.1), and in some earlier discussions, we relied the that if idea two differept investments generate the same payoff, they mgst have on the same cost. This commonsensical idea is one of 4he most important in the book. If eqpation (3.1)did not hold, there would be both low-cost and high-cost ways to acquire the index at time F. We could simultaneously buy te index at 1ow cost and sell the index athigh cost. This transaction has no risk (sincewe both buy and sell the index) and generates a positive cash flow (becauseof the difference in costs). Taking adkantage of such an opporlnit.y is called arbiage, and the idea tat prices should not pelmnit pricing.'' We implicitly illustrated this idea earlier in arbiage is called'tno-arbiage showing how owning the index and buying a put has the same profit as a call, etc. No-arbitrage pricing will be a major theme in Chapter 5 and beyond. a No arbitrage

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You might ask: Is there a lower-cost way to speculate tat the stock price will rise, still has the insurance implicit in the call? The answer is yes: You can lower the that cost of your strategy if you are willing to reduce your prot should the stock appreciate. You can do tlzis by selling a call at a higher strike price. 'T'he owner of this second call buys appreeiation above the higher strike price and pays you a premium. You achieve a lower cost by giving up some portion of prct. A position in which you buy a call :nd sell an othemise identical call with a higher stn'ke price is an example of a bull spread. Bull spreads can also be constructed using puts. Perhaps surprisingly, you can achieve the same result either by buying a low-strike call and selling a high-strike call, or by buying a low-stlike put and selling a high-strike put. Spreads constructedwith eithercalls orputs are sometimes calledverticalspreads. The terminology stems from the way option prices are typically presented, with strikes arrayed vertically (as in Table 3.4).

There are many well-lnown, commonly used strategies tat combine two or more options. ln this section we discuss some pf these strategies and explain the motivation for using them. The underlying theme in this section is that there are always trade-offs in designing a position: lt is always possible to lower the cost of a position by reducing its payoff. Thus there are many variations on each particular strategy. All the examples in this section will use the set of option plices in Table 3.4. We will assume the continuously compounded interest rate is 8%. ';'7* j'f)' ?' T' 'SX kjjj(r)'. j'rjf )' f' @' ;'(' )' j'g' 1* ;r' qltqqqr;);f y' yjjltatjjljijiiz'k'k. tjjzyt;jjrgr'-d r' y'. j': t;f q'lrf j'k)f (tf ytf tt'':'p' ' E' i ! : (q55:'7* j'y' '')'()' ('. jjj'yy (': (jf )' E i jEE ' E E 'E '' E'' E EE'E'' ' : '. (('E..(. E..E(.iE;.q.(.E(.q. ('!:'.lj.(! (! ; (q'(... .ti(i.ij'ir( ; ;1:.j(.'(' (..('(.(j.((?'iE'EE.(.( @ . (.!.((!'( .q 1* .(( Eyrj,j . ! .y i. E! E i .EE . E E.E. i. E E E -L,)q.;-;L-t3qq'.;.. i. !t.EIE.-.t-.I;ftii-.... E. .E .rt-' E .E E . . g .E .. .( ( y y . , r r . .. . q r . j y . . j g j ! ( j . j j y . g . j ;y. . ....y.y.y..jryr. yj gj. yj.;j j ;j..jy,;r)r-))j. y.. .j.kj.yj. .;Igtii;l j. yyy .i:k(..... . .))t!lkk IjEjjL...?.:)t-.j.y.-j . .j ..

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3.3 SPREADS AND COLLARS

'

COLLARS

An option spread is a position consisting of only calls or only puts, in which some options are purchased and some written. Spreads ar a common strategy. ln this section we defne some typical spread strategies and explain why you might use a spread. Suppose you believe a stock will pppreciate. Let's compare two Fays to speculate belief: entering into along forward contract orbuying a call option with the strike this on equal price to the forward price. The forward contract has a zero premium and the call has a positivepremium. A difference in payoffs explains the difference in premiums. J.f the stock price at expiration is greater than the forward plice, the forward contract and call have the same payof'f. If the stock price is less tan the forward price, however, the forward conact has a loss and the call is worth zero. Put-call parity tes us that the call is equivalent to the fonvard contract plus a put option. Thus, the call premium equals the forward the cost of the put, which is insurance against the stock price being less t11:,11

Puttff, F)

-

Alqo

Bull and Bear Spreads

a covered call, we have

Calltf,

-

SPREADS

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0.44

1.99

5.08

SAnother way to express the plinciple of no arbitrage is using profit diagrams. Given two profit diagrams, there is an arbtrage opportunity if one diagram is everywhere above the other.

-'.

(' y' t' L'

r Exam' pIe 3.2 To see how a bull spread arises, suppose we want to speculate on the 1i1 # stoc k p rice increasing. Consider buying a 4o-strike call with 3 months to expiration. lil

jt) From Table 3.4, the premium for this eal1 is $2.78. We can reduce the cost of the )' Position-and also the potential profit-by selling the zz-strike call. ).1 l'1 An easy way to construct the graph for this position is to emlate a spreadsheet: :;( l)( For each plice, compute the prot of each option position and add up the profits for the individual positions. lt is worth worling through one example in detail to see how this ',! ;( . . is done. . t') The initial net cost of the two options is $2.78 $.97 $1.81. With 3 months (!r'l ) . t) interest, the total cost at expiration is $1.81 x (1.0833)0.25 $1.85.Table 3.5 computes 'i) the cash flow a t expiration for both options and computes profit on the position by .,! .# )ij subtracting the future value of the net premium. lt Figure 3.7 graphs the position in Table 3.5. You should vel'ify that if you buy the )) %. f 4o-strike put and sell the 45-strike put, you obtain exactly the same graph. 'i: :

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INSURANCE,

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AN D OTH ER ARATEGI

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cash flow in the future. Hence, it is an option spread that is purely a means of bormwing lisk. The reasons for using a box or lending money: It is costly but has no stock price spread are discussed in the box on page 74.

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The opposite of a bull spread is a bear spread. Using the options from the gbove example, we could create a bear spread by selling tle 4o-strike call and buying the 45-strike call. The prost diagram would be exactly the opposit of Figure j.7.

Box Spreads A box spread is accomplished by using options to create a synthetic long fonvard at one price and a synthetic short fonvard at a different price. This sategy guarantees a

A ratio spread is constnlcted by buying ?n calls at one strile and selling 11 calls at a different stdke, with a11options having the same time to mat-urity and same underlying asset. Ratit? spreads can also be constructed with puts. You are asked to constnlct ratio spreads in pfoblm 3.15. Also, a ratio spread constructed by buying a low-stn'ke call and selling'two higher-strike calls is one of the positions depicted in the chapter summal'y in Figure 3.17. Since ratio spreads involve buying and selling unequal numbers of options, it is signicanc of this may not possible to construct ratio spreads with zero premium. be obvious to you now, but we will see in Chapter 4 tat by using ratio spreads we can construct paylater strategies: insurance that costs nothing if te insurance is not needed. The trade-off to this, as you might guess, is that the insurance is more costly if it is 'he

needed.

Collars A collar is the purchase of a put option and the sale of a call option with a higher strike plice, with both options having the same underlying asset andhaving the same expiratio date. lf the position is reversed (saleof a put and purchase of a calll, the collaris written. The collar width is the difference between the call and put strikes.

Q INSURANCE,

AND OTH ER WRATEGI

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cost i.e., the ftlttlre value of the stock price. In the glrday example above, this would require that you set te put stdke equal $40.808, to which gives a premium of $2.39k) The call premium at this strike is also $2.39!Ifyolt p'y to ??.If?'d against aII Iosses tp?) the stock, fncfllfll.,g interest, J'/?e??. a zero-cost collar 7itz-zero w/#/'. ' This is an implication of put-call patity, equation (3.1).lt trns out tfiat is also the theoyetical folward price. $40.808 lf we set the strike equal to the forward price, the call premium equals the put premium. ,

3.4 SPECULATING ON VOLATILITY The positions we have just considered are a11directional: A bull spread or a collar is a bet that te price of the underlying asset will increase. Options can also be used create pojitions rhatare nondirectional with respect to the underlying asset. With toa nondirectional position, the holder does not care whether the stock goes up or down, but on1# how much it moves. We now examine stradles, strangles, and buttertly spreads, whih are examples of nondirectional speculations.

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strangle The disadvantage of straddle is the high premium a cost. To reduce tlz premium, you can buy out-of-the-money optiond ratherthan at-the-inopey pptio. Such a position is called a strangle. For examjle, consider buying a 35-stlike put and 45stlike call, for a total premium of a $1.41.with a f'uttlrevalue of $1.44. Wese reduce your muimum loss if the options transactions expire with the stock near $40, but they also increase the stock-price move required for a pro:t.

Figure 3.11 shows the 4o-strike seaddle graphed against the 35-45 strangle. Tlzis comparijon illuseates a key pointp ln comparing Any two faidy priced option positions, t.here w ill always be aregion where each outperfonus the other. lndeed, this is necesiary to have a fairly priced position. J.nFigure 3.11, the sangle outpedbrms the saddle roughly when t stpck prie at expiration is between $36.57and $43.43. Obviously, there is a much broder range in which the staddle outperforms the sangle. How can vou decide which is j.jy e jiet. ter investment? The answer is that unless yo tt have particular view on the stock s performance, you cnnnot say that one position is preferable to the other. An option pricing model implicitly evaluates the likelihood that one strategy will outperform the other, and it computes option prices so that the two strtegis aie equivalently fair deals. An investor might have a preference for one sategy over the other due to subjective probabilities that differ f'rom the market's.

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The straddle writer can insure against large losses on the sladdle by buying options io protect against losses on both the upside and downside. Buying an out-of-the-money put provides insurance on the downside, protecting against losses on the at-the-mony written put. Buying an out-of-te-money call provides insurance on the upside, protecting against losses on the written at-te-money call. . Figure 3.13 displays the saddle written at a strike price of $40, iong Fith the of options to safeguard the position: A 35-strike put and a 45-strike cil. The net combining tese tlu'ee seategies is an insured written saddle, which is called abuiierfly spread, graphed in Figure 3.14. lt can be thought of as a wlitten stradtle for the timid (or for the pnldentl). Comparing the butterlly spread to the written straddle (Figure 3.14), we see tat the butterfly spread has a lower maximum pro:t (dueto the cost of insurance) if the stock at expiration is close to $40, and a higher proft if there is a lrge move in the stock price, in which case the instlrance becomes valuable. spread we gain imWe will see in Chapter 9 that by understanding the buttey portant insights into option prices. Also, the butterlly spread can be created in a variety of ways: solely with calls, solely with puts, or by using the stock and a combination spread in Figure 3.14 of calls andputs.s You are asked to verif'y this in Problem 3.18.

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20

25

30

35 40 45 XYZ Stock Price

50

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($)

lt might occur to you that an investor wishing to bet that volatility will be low could write a saddle and acquire insurance against extreme negative outcomes. That inttlition is correct and leads to our next sategy.

-20

20

25

30

35 40 45 XYZ Stock Price

50

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60

()

snchnically, a true butterlly spread is created solely with calls or solely with puts. A butterlly spread butterfly.'' created by selling a straddle and buying a strangle is called an t'iron

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yy-'. j'))t'ryjj'. Cr' :(' (' jj'. yyry)f (jyyy.f gjjjyjyjj,f t'r' (E' i:' ?' Thrll' 11q5r* iE' i'(1t:*1qEr1* ?)!;kry'.r( ,f j'. yy'r;yj. j'r j'j).ff y'.y'. sliitlr'k) jyjy' )!ifi' )))r:l)'tkC:r!!p(!!qrii7l )' (' 'tj)f 1* (:8)* yq' q' (lh'rggr''jyjk/))jf ikrjy'.jqltll i)' i':.(ii i'li: E r::;:q:7rlrq!:rp:!/rr7rqrF::7rrrq7rrErqqr!7srli'1. !' 1))8.9118p)(1: y'y' y.pyf $' 7.77(:1:* 1).*: ' . q't.! :q'1 iti ic: t't' y' r;ll'. yjl.ll'jy:rji )''q' )q' r'. rrjjy;;f r' r;)' tjt)..'r!iitl1q!. .. 'Ejt:.(. ! .. EE.i'.r.y.E.)1p.tti. (;@p! !E gky.)l ytjj: .y. r'yji..(g. y.(y)i (jjg (jq.. ;)t,:jLikjLjjflj.siqjlqttfjjlkLjj). . q. yjyg tT):-i..j.-!ii!11k'. .tyj.j. IIE:PI:L 411::;;k....-jj:yj . y;jy.j!..y.ygri 7-iIi-.......)y)...tt!iiI;!,,. ()j.:1f-:1f-;l)L;);-..kk.. -;L!11k;kII'. y.. . ..jjjjyjy,( (...Lq. tj )iIIE!SL. .1IIq!q;1iIl.-:$2: -'Lq..-..t.. !h.'#, ())(...jjyyy jjjgyyy,j jgjjr . . . . . -y' ,''''

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huying a 35-40 bull spread and a 40-45 bear

-

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($)

In general, consider the strike prices ffl, Kz, and Ks, where #1 Define so that #3 Kz -

= Ks

Examine Figure 3.15. It looks like a butterfly spread except that it is asymmezc: The ' . peak is closer to the high strlke than to the 1ow strike. This picttlre wgs created by buying two 35-stlike calls, selling ten 43-strike calls (witha premium of $1.525,using and buying eight 45-stHke calls. The position. is like a the jiuljtin iri Tble butterfly in that it earns a proft if the stock stays within a small range, and the loss is the same for high and low stoclt prices. However, the prolit diagrnm is now tilted to the right, rather tan being symmetric. Suppose you lnew that you wanted a position that looks like Figure 3.15. How would you lnow how malzy options to buy and sell to construct this position? In order to obtain this position, the s'trikes cleady have to be at 35)43, and 45. The total distance of the way from 35 to 45. In between 31 and 45 ls 10. Te num ber 43is 8t% (= 43-3/ jo ) fact, we can wlite 43 as '''

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or

Asymmetric Butterdy Spreads

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-

(0.2 x 35) + (0.8 x 45)

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Kz

=

ffl +

(1 k4Ks -

0.2, as in the above example. Forexample, if #1 = 35, Kz = 43, and Ks = 45, then for evel'y Kz call we write, we buy #1 In order to constrtlct an asymmetric butter, K5 calls. calls arfd 1 You should venf'y' tat if you buy two 35-st1ike puts, sell ten 43-srrike puts, and buy eight 45-strike puts, you duplicate the profit diagram in Figure 3.15. =

-

3.5 EXAMPLE:

ANOTHER

EQUITY-LINKED

NOTE

lion by issuing In Jgly 2004, Marshall & Ilsley Corp. (ickersymbol h41) raised $400kml* 2007.6 maturity Instead of maling matllring in August effecdvely bonds payment a irl cash, the bonds pay the holder in shares of Marshall & llsley's own stock. A bond that, under some circumstances, pays the holder in stock instead of cash is called a

6ln'orsimplicity we will refer here to the Marlhall & llsley security as a Chapter 15, it was actually a bond plus a fonvard contract.

''bond-''

As we will dscuss in

'

% INsuRANcE,

84

COLLARS,

CHAPTER

STRATEGIES

AND OTHER

%.

85

stock. Both graphs ignore dividends and other distributions. Based solely on compadng the mattlrity payoffs, investing in the bond is inferior to investing in the common stock. on the stock (an However, distributions on tlae bond (6.5% armually) are greater t.11:.11 analysis it is impossible valuation withoutperforming a annui dividend of about 2917);so Accounting indifferent. other, the or be to say Whetherinvestors should prefer one or bond to lead the upward and payoff lines shif't would the for the distributions would bond the outyerforming with stock prices, stock the at 1ow outperfonn 0.6699 shares of

Marshall & Ilsley note always settles in stock; hence it is called copvertible bond. bond. We will discuss this particular bond more in Chapter mandatorily convertible a interestinM but bond is 16the at this point because the payoff strucmre resemblej a rl'he

CO

SUMMARY

.

The bond pays an annual 6.5% coupon and at maturity makes paymets in shares, the numberof shares dependentupon the Erm's stockprice. 'I'he specifc terms pf the with mamrity payment are in Table 3.7. To interpret this payoff, note that when the Marshall & Ilsley stock price at maturity is between $37.32and $46.28,the payoff is a varying number of shares, selected so the bond is worth $25(e.g.,0.5402 x $46.28= $25). Figure 3.16 graphs the mamrity payoff of the bond as a function of the N.!Istock in price three years, against the payoff of owning 0.6699 shares of Marshall & llsley

at high pfices. The Marshall & llsley share price was $37.32on the day the bond was issued. The bond was designed so tat, at issue, it would sell for $25, which is the same price as 0.6699 shares. The bond was also designed so that bondholders would forgo 24% of the appreciation on the stock above $37.32. We have $37.32 x 1.24 $4,6.28,and 25/46.28 0.5402. Tlzis accounts forthe flatrangein thepayoffand the numberof shares exchanged above a share plice of $46.28. Finally, in order for the bond to underperform the stock above $37.32,yet to sell at the share price, it is necessary to compensate bondholders with additional payments. The 6.5% coupon accomplishes this. How would we pzice te bond? The p'aph in Figure 3.16 should remind you of graph is $y (1) owning 0.6699 shares a written collar. One way to construct the same with a strike price of $37.32,and calls selling 0.6699 dividends), (2) of stock (ignoring algebraic expression for the price strike of $46.28. (3) buying 0.5402 calls with a payoff is .

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1/1.2i high Note that the bond holder is implicitly selling a low strike call and buying divialso forgoes premium. bondholder option owed 'Fhe strike calls, and is therefore effectively wllich distribution, is the explain 6.5% dends on the stock. twf)factors the dividend plus the amortized option premium. By plicing these options and valuing the 6.5% distribution, we can arrive at a fair price for the bond. 'Fhis discussion leaves unanswered the question of why Marshall & Ilsley would issue such a bond. We have also not discussed a1l the details of the bond's strutttlre. We return to these issues in Chapter 15.

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Puts are insurance against aplice decline and calls are insurance agaibst a price increase. Combining a long or short position in the asset with an offsetting position in options (for example, a long position in tlae asset is coupled either with a purchased put or written call) leads to the various possible positions and their equivalents in Table 3.8.

k

86

INSURANCE,

AND OTH ER ARATEGI

COLLARS,

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PROBLEMS

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87

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Bond + C:.II

lnsured Ass:t

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Bond

Covered Written call

-zero-coupon

-zero-coupon

-

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(floor)

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-

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........................-..

(cap)

dr:z)j

Bull Spread

.

Calltr, F)

Puttf,

F)

=

PV(Fo,z

-

K)

Collar

Stock Price

.

Stock Price

Profit

Profit

Buying a call and selling a put wit the same strike price and time to eypiration creates an obligation to buy the asset at expiration by paying the strike price. This is a synthe zic forwarct. A synthetic forward must have the same cost in present val terms as a trtle forward. This observation leads to equatin (3.1): -

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-1::

.

Profit diagrams for positions discussd in the chaptvr: bull spread, collar, straddle, strangle, butterfly, and 2 : 1 ratio spread

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'''

(3.1)

Skaddle

Strarfgle

stockPrice

jtock Price juoyky

Profit

explains the difference in call and put premiThis relationsllip, called put-call rf7p#@, othemise options. lt is for identical one of the most important relationships in ums

Butterfly

Rat'io Spread

Stock Phce

stockPrice

derivatives.

There are numerous sategies that permit speculating on the direction of the stock the size of stock plice moves (volatility). Some of tese positions are summarized on or graphically in Figure 3.17. We also categorize in Table 3.9 various strategies according to whether they reqect bullish or beatish views on the stock plice direction or volatility.? Netscape PEPS were equivalent to a bond coupled with an option spread, illustrating that the tools in this chapter have applicability eyond speculative investing.

.'

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FURTHER

READING

In Chapter 4 we will see how flrms can use these stategies to manage zisk. We will further explore put-call parity in Chapter 9, in which we also will use bull, bear, and butterlly spreads to say more about what it means for an option to be fairly priced. Put-call parity was flrst demonsated in Stoll (1969).Merton (1973a)corrected the original analysis for the case of Amelican options, for which, because of early exercise, parity need not hold. Ronn and Ronn (1989)provide a detailed examinadon of price bounds and returns on box spreads. There are numetous practitioner books on option trading sategies. A classic practitioner reference is McMillan (1992).

7erable3.9 was suggested by David Shimko.

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-. . -. . . .

-

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-

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direction.

'

i Elill iil''.ijwj.jjk q'. .. . .... ..... .

.

.

.

Buy puts

No price view

Buy saddle

Price will increase

Buy calls

PROBLEMS

'j-/-hk/,-

- - .. . . . . ......... .

.

u

F -'.tti',-.

..

' ' ' dEl,:' Ii..ll'

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( ( jujrgqyy j j tjju r.;.. jry...E . (311:.r. ..# ... . .. . . . . . .. ...... ... Sell underlying Sell calls

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.

. . . . . . . . . .. . . . ...

-

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.

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.

.

.

.

3.1. Suppose that you buy the S&Rindex for $1000,buy a looo-strike put, and borrow $980.39. Pelbrm a payoffand prost calculation mirnicking Table 3.1. Graph the resulting payoff and prost diagrams for the combined position.

.

88

%.IlqsuRAxcE,

COLLARS,

AN D OTH ER ARATEGI

PROBLEMS

Es

.p/-t?pfc??l.5'

following 6-??7o?7r/7J5?a$?'J?-#>?-cc /c

assume

is $1020,t7?/#

6month

the cf/c/ve

r/7c.5'e#rc??)i'II???-./br

ttse

rate is 2%, the vS&k S&R options wr 6 ??7t???r-

nrcrEwr

3.12. Suppose you invest in the S&R index for $1000,buy a 950-stI4ke llW-strike call. Draw a prost diagram for zero-cost collar?

to expiration: Strike $950 1000 1020 1050 1107

Call

Put

$120.405

$51.777

93.809 84.470

74.201

71.802 51.873

101.214 137.167

a. loso-strike S&R straddle. b. Written 950-strike S&R straddle.

84.470

. ..

.

.

.

S&R straddle.

3.14. Suppose you buy a 950-strike S&R call, sell a looo-strile :-rike S&R put, and buy a looo-strike S&R put. a. Venfy' that there is no S&R price risk in this b. 'What is the irtitial ost of the position?

. '

mont

-

diagram by borrowing

'

s.

c. Consider buying n 950-stlike calls and selling ??7 1050-sike calls so that the premium of the position is zero. Considering your analysis in (a)and (b), what can you say about /1/??1? What exact ratio gives you a zero premium?

3.7. Vel'ify that you earn the same proft and payoff by (a) shorting the S&R index for $1000and (b)selling a loso-stn'ke S&R call, buying a 1050-stlike put, and borrowing $1029.41.

3.16. J.nthe previous problem we saw that a ratio spread can have zero irliti premium. Can a bull spread or bear spread have zero irlitial premium? A butterlly spread? Why Or Why not?

3.8. Suppose the premium on a 6-month S&R call is $109.20and the premium on a put wit the same strike price is $60.18.What is the jtrile price? .

'

3.9. Construct payoff and profit diagrams for the purchase of a 950-strike S&R call and sale of a looo-strike S&R call. Verify that you obtain exactly the samrm/r diagram for the purchase of a 950-strike S&R put and sale of a looo-strike S&R put. What is the difference in the payoff diagrams for the call and put spreads?

l

u

a. Buy 950-strike call, sell two 1050-strike calls. b. Buy two 950-stn'ke ealls, sell three loso-sfn'ke calls.

3.6. Verify that you enrn the same prolit and paypff by (a) buying the S&R index for $1000 and (b) buying a 950-strike S&R call, selling a 950-stlike S&R put, and lending $931.37.

3.10. Constnzct payoff and proft diagrams for the purchase of a 1050-stlike S&R call and sale of a 950-strike S&R call. Vel'ify that you obtain exactly the same proht diagram for the purchase of a loso-str1'ke S&R put aad sale of a 950-strike S&R put. What is the difference in the initial cost of these positions?

'ansaction.

3.15. Compute profit diagrams for the following ratio spreadj:

$1029.41and buying a 1050-strike put.

Why is there a difference?

S&R call, sell a 950-

c. What is the value of 4heposition after 6 months? d. Verify that the implicit interest rate in these cash :ows is 2% over 6

3.5. Suppose you s ort theS&R index for$1000 andbuy a loso-sfn'ke call. Construct diagrams forthis position. Verify thatyou ob,tain the same payoff payoffandproqt andprofit

.

c. Simultaneous purchase of a 1050-strike straddle and sale of a 950-strike

.

'

put, and sell a How position. close is this to a this

3.13. Draw profit diagrams for the following positions:

3.3. Suppose you buy the S&R index for $1000and buy a 950-strike put. Construct 'payoff and prost diagrals for this position. Verify that you obtain the same payoff nd prost diagram by inves ting $931 37 in zero-coupon bonds and buyig a 950-strike call. 3.4. Suppose you short the S&R index for $1000and buy a 950-strike call. Constnlct payoffand prostdiagrams for this position. Vel'ify tat you obtain the sam payoff and profit diagram by borrowing $931.37and buying a 950-strike put. .

89

3.11. Suppose you invest in the S&R index for $1000,buy a 950-sfrike put, and sell a loso-stn'ke call. Draw a profit diagram for this position. What is te net option premium? If you wanted to construct a zero-cost cpllar keeping the put stn'ke equal to $950,in what direction would you hay to change th call strike?

3.2. Suppose that you short the S&R index for $1000and sell a looo-strike put. Conseuct a table mimicking Table 3.1 which summarizes the payoffand profit of this position. Verify that your table matches Figure 3.5. For

%

3.17. Construct an asymmetric butterlly using the 950-, 1020-, and loso-stn'ke options. How many of each option do you hold? Draw a protit diagram for the position. .(

3.18. Verify tat th.e butterfly spread in Figure 3.14 can be duplicated by following eansactions (usethe option prices in Table 3.4): a. Buy 35 call, sell t'wo 40 calls, buy 45 call. b. Buy 35 put, sell two 40 puts, buy 45 put. c. Buy stock, buy 35 put, sell two 40 calls, buy 45 call.

i

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90

%

INSURANCE,

COLLARS,

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Golddiggerp agrej to t yt a e sold today for $420/oz. jckldyroductonin 1 year. we will assume in atl exiilllj that th toixkard coneact its nanially. fonvaid is the si with phyjical noted earlier, the payoff : .

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seller'' is the sum of te other two lines, @Hedged proht: The line labeled adding them vertically at every gold price. lt is flat at $40/oz.,as we wouldexpect fromrfable 4.2. Aquick way to add the lines togetheris to notice that the cold forward'' rrah has a seller'' graph has a positive slope of 1. and the

seller'' shows o Unhedged profit: Since costis $380/oz.,the line labeled lligher For exnmple, prices. and prot at zeto prost at $380,a loss at lower prices, ground, gold the Golddiggers has a long in at $4209profit is $40/oz. Since it has position in gold. ,* Prtt on the short fomrard position: The sh ort g old fomrd'' tte rpresents the profit from going short the gold forward contact at a fonvard pticeof $420/oz. :

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We profit from locking in the price if prices are lower than $420 and we lose if prices are higher.

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Hedging with a Fo-ard Contract t; olddiggis can lockin apriceforgoldin 1 yearby enteringinto ashortforFard contract, gtd for delivel'y in year. Suppose tht gpld t b delikred in ageeing tcy to sell its sell fofwfd al1 of and that

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PERSPECTIVE

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A possible objec:on to hedging with a forward contract is that if go'ld prices do rise, Golddiggers will still rceive only $420/oz',there is no prospect for greater profit. Gold instlrance-iae., a put option-provides a way to have higher profts at high gold prices while still being protected against low prices. Suppose that the market price for a 420strike put is $8.77/0z.2This put provides jloor on the price. Since the ppt premium is paid 1 year plior to the option payoff, we must tale into account interest cost when we compute prost in 1 year. 'I'he future value of the premium

z'rhisuses the Black-scholes formula for the c. 5.5%. J 4.879% and t 1 tyearl. =

=

=

put price with inputs S

=

420, K

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420, r

=

4.879%,

93

% INTRODUCTION

94

BASIC RISK MANAGEMENT:

RISK MANAGEMENT

To

is $8.77 x 1.05 $9.21. As with the forward contract, we assume financial settlement, although physical settlement would yield the same net income. Table 4.3 shows te result of buying tlzis put. H te price is less than $420, the tlie put. This is put exercised and Golddiggers sells gold for $420/oz.less the cost of Golddiggers sells gold at price is the $420, incom of If net tan greater gives $30.79. market price. the the put-performs better than shorting the forThe insurance strategy-bpying than $429.21. Otherwise the short forward gold of 1 is if price in the more ward year outperforms insurance. Figure 4.2 shows the unhedged position, profit from the put by itself, and the result of hedging with the put. :

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With tlze sale of a call, Golddiggers receives a prernium, which reduces losses, but the written call limits possible frots. One can debte whether tlzis really constimtes insttrance, but our goal is to see how the sale of a call affects the potential prost and loss

$30.79 $30.79 $60.79 $110.79

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PERSPECTIVE

willbe in differentregions; thatis, how likely is it thatthe goldprice will exceed$429.21? The plice of the put option implicitly contains information about te likelihood that the gold prie will exceed $420,and by how much. The probability distribution of the gold price is a key factor determining the pricing of the put. We will see in later chapters how the distzibution affects the put price arld how to use information about te probability ' distribution to help us assess lisk. Figure 4.3 compares the profit from the two protectiv sategies we have exnmined: Selling a fonvard contract and buying a put. As you w. uld expect, neither strategy is cleady preferable', rather, there are trade-offs, with each contract outpedbrrning the other for some range of prices. The fact that no hedging sategy always outperforms the other will be true of all fairly priced strategies. J.npractice, considerations such as transaction costs and market views are likely to govern the choice of a sategy.

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PRODUCER'S

What this analysis does not address is the probability tat the gold price in 1 year

=

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THE

for Golddiggers. Suppose that instead of buying a put, Golddiggers sells a 420-strike call and receives an $8.77premium. Golddiggers in this case would be said to have sold a cap.

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for Golddiggers hedged with a forward contract and hedged with a put option.

$30.79

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Comparison of payoffs

Unhedted seller purchased put Hedged seller

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450 350 400 Gold Price in 1 Year ($)

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96

INTRooucTlox

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unlwdgedseller written call sellerhedged with

@ Sell some of the gain. Both of these sategies reduce the asymmetry between gains and losses, nd hence lower the cost of insurance. The first strategy, lowring the strike price, permits some additional loss while the second, selling some of tlie gain, puts a cap on the potential

written call

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The manager's view of the market and willingness to absorb risk will undoubtedly in:uence te choice among tese alternatives. Managers optimistic about the plice of gold will opt for low-strike-plice puts, whereas pessimistic manager; will more likely choose high-strike puts. While corporations per se may not be risk-averse, managers public's may b. Also, some managers may perceive losses to be costly in terms of the of perception boss's them. of f11-11: perception the or the This problem of choosing the appropriate strike price is not unique to corporate risk management. Safe drivers and more careful homeowners often reduce premiums

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Figure 4.4 shows the payoff to this strategy. J.f we compute the acmal profit 1 year from today, we see tht if te golcl price in 1 year exceeds $420,Golddiggers will show Prots of

$420 + $9.21

-

$380

=

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'rhat is, Golddiggers sells gold for $420 (sincethe written call is exercised by the holder), receives the ftlttlre value of the premium, and has a cost of $380. If t.1+plice t gold is less than $420,Golddiggers will make Pgold+

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Reducing the strike price lowers the amount of insurance; therefore the put option will have a lower premium. Figure 4.5 compares prost diagrams for Golddiggers's hedging using put options with strikes of $400 (premium $2.21), $420(premium 4oo-stlike, low-premium option yields = $8.77),and $440 (premium $21.54). lowest profit if needed the highest proft if insurance is not ttllepriee is high) and theoptipn yields the Oo-strike, high-premium is low). price The insuratlce is needed (the neded. profit is insurance and needed, the if highest lowest protit if insurance is not

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* Reduce the insured amount by lowering the strik price of the put option.

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There are at least two ways to reduce the cost of insurance;

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ENT: TH E PRODUCER'S

BASIC RISK MANAGEM

MANAGEMENT

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On the downside, Golddiggers has exposure to gold but keeps the option premium. By writing the call, Golddiggers keeps the $8.77 call premium and 1 year latef makes $9.21 more than an unhedged gold seller. On the other hand, if the gold price exceeds $420,the call is exercised and te plice Golddiggers receives is thus capped at $420. Thus, for gold plices above $429.21,an unhedged sategy has a higher payoff than that of writing a 420-strike call. Also, for prices below $410.79,being f'ully hedged is preferable to having sold the call.

Adjusting the Amount of lnsurance Consider again Golddiggers's strategy of obtaining insurance against a price decline by purchasinga put option. A common objection to the purchase of insurance is that it is expensive.Insurance has a prernium because it eliminates the risk of a large loss, while allowinga prost if prices increase. The cost of insurance reqects this asymmetry.

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Hedged wit.h 420-strike put wjth zjlo-sjrrike put uedgedO:.II 440-strike put Hedged

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400 450 350 Gold Price in 1 Year

500

($)

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INTRODUCTION

BASIC RISK MANAGEM

TO RISK MANAGEMENT ;';' !' y' 'Fr'; (' 59* )' (C' F' ?' qf (f ))' ltf li :E t'i'')#' (' . it?tljlji ''

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call. The owner of that

$420.

ln either case, Golddiggers sells gold at $420. Thus, the collar'' is exactly like a fonvard conact. By buying the put and selling the call at the same strike price, Golddiggers has synthetically created a short position in a forward conact. Since a short forward and 420-420 collar have the same payoff, they must cost the same. This is 1W7ythe #?z??,I/??l- ol the 420-strike options t'lrc the Tllis exnmple is really just -anillustration of equation (3.1). ::420.-420

-rHE HEocE

Rvlo

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te call strike ptice below $440.78,in wllich case we would obtain a higher premitlm per call. To offset the higher premium, we could buy less than one call. The ade-off is that we cap the gold price on part of production at a lower level, but F maintain solpe participation at any plice above the stnke. lternatively we could raise the cap level (thesike price on the calll and sell would increase participation in gold price increases up to the than one call. more also effect of generating a riet short position in gold if prices rose the have level, but cap above the cap. rf'his

-twnn

synthetic i orwar d s a t p rices other 'than :420 We can easily extend this example to understand the relationship between option premiums at otlzer strike prices. In the previous example, Golddiggers created a synthetic forward sale at $420. You might think that you could benefit by creating a synthetic fonvard contract at a higher price such as $440. Other things being equal, you would rather sell at $440than $420. To accomplish this you buy the 440 put and sell the zl40 call. However, tere is a catch: 'Ihe zgo-strile put is in-the-money and the Mo-strile call is out-of-the-money. Since we would be buying the expensive option and selling the inexpensive option, we have to pay a premium. How much is it worth to Golddiggers to be able to'lock in a selling price of $440 instead of $42 ? Obviously,it is worjh $20 1 ye ar from toda# or $20+ (105) $19 05 in present value tenus. Since locking in a $410 price is f'ree, it should threfore be the case that we pay $19.05in net premium in order to lock in a $440price. In fct, looling at the plices of the Mo-strike put and call in Table 4.8, we hav premiums of $21.54for te put and $2.49for the call. This gives us ::2

,

Net premium

=

$21.54 $2.49 -

.

.

$19.05

=

Similarly, suppose Golddiggers explored the possibility of locking in a $400 price for gold in 1 year. Obviously, Golddiggers would require compensation to accept a lower price. In fact, they would heed to be paid $19.05-+e present value of $20-today. Again we compute the option premiums >nd we see that th 400-FtIike call sells for $21.26while the 400-stlike put sells for $2.21.Again we have Net premium

=

$2.21 $21.26 -

-$19.05

=

Golddiggers in this case receives the net premium for accepting a lower price.

Other Collar Strategies

Paylater Strategies A disadvantage to buying a put option is tat Golddiggers pays the premium even when the goldplice is high andinsurance was, afterthefact, unnecessary. One strategy to avoid this problem is a paylater sategy, where the premium is paid only when the insurance is needed. Wllile it is possible to constnlct exotic options in which the premium is paid only at expirain and only if the option is in ttze money, the sategy we discuss here is a ratio spread using ordinacy put options. The goal is to find a sategy where if the gold plice is high, tere is no net option prernium. lfthe gold plice is low, there is insurance, but the effecuve prernium is greater than with an ordinary inslzrance sategy. Ifthereis nopremiumwhen thegoldpriceis Mgh, wemusthaveno initialpremium. This means that we must sell at least one option. Consider the following sategy for Golddiggers: Sell a434.6-strikeput andbuy two4zo-strikeputs. Using ourassumptionsk the premium on the 434.6-strike put is $17.55,wllile the premium on the 420-strike put is $8.77. Thus, the net option premium from this strategy is $17.55z- (2 x $8.775)= 0. Figure 4.12 depicts the result of Golddiggers's hedging with a payter sategy. Whentheprice of goldis greaterthan$434.60, neitherputis exercised, and Golddiggers's profit is the same as if it were unhedged. When the price of gold is between $420 and $434.60, because of the written $434.60put, the firl'n loses $2 of profit fpr every $1 decline in the price of gold. Below $420 the purchased 420-.tri1e puts are exercised, and prost becomes constant. The nt result is an insurance policy that is not paid for unless it is needed. Also depicted in Figure 4.12 is the familiar rsult from a conventibnal insurance strategy of hedging by purchasing a single 420-sfrike pt. When the gold price is high, the pay 1a ter strategy wit.h a zero preinium outpebrms the sigte put. When the gold price is low, the paylater sategy does worse because it offets less insurance. Thus, the premium is paid later, if insurance is needed.

4.5 SELECTING

Collar-type strategies are quite :exible. We lave focused on the ease where the ftrm buys one put and sells one call. However, it is also possible to deal with fractional options. For exnmple, consider te 400.78-440.78 collar above. We could buy one put to obtain full downside protection, and we could vary the stn' price of the call by selling fractional calls at strike prices other than $440.78.For example, we could lower 'ke

THE HEDGE

RATIO

ln the Golddiggers and Auric exmuples, we pedbrmed all calculations in terms of one unit of gold, and made two important assumptions. First, we assumed perfectcorrelation between the plice of gold and the price of what each compahy wants to hedge. Second, we assumed that the companies lnew for certain the quantity of gold they would sell and buy. As a result of these assumptions, we effectively assumed that the hedge ratio is one,

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S ELECTI

MANAGEMENT

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350 400 450 Gold Price in 1 Year

Nlwo'w2 + (H

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where te hedge ratio is defined as te ratio of the fonvard position to the underlying E

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assumption-may be valid. We ftrst exnmine the effect of widget ' ln practice, neither price uncertainty on hedging, and discuss cross-hedging. We then exnmine qttantity I/??c'?-lc?lfy

NwPw

=

in te price of widgets

-

(4.1)

N.g Pgold

Ng Pgold+ StPgold

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gLH N g )20.2 g +

-

-

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where o'm is the standard deviation of the widget price, o-g is the standard deviaion of the goldprice, and p is the price correlation between widgets and gold. Note that pavo'g is the covariance between widget and gold prices. rfhe variance-minimizing hedge position, S*, is13 H.

aSSC (

Ng

=

-

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c'p Equation (4.3)has a saighforward interpretation. 'Fhe flrst tet'm hedges costs-i.e., we second enter into a long position on the nmount of gold we use in production, Ng. term hedges revenue, prov iding the quantity of gold forWards we use to hedge widgets. iS 9om regressing the Widget ljlice ii tli gold p1ice.14 The telql pGwxg the Coecient times the number of widgets, #w, ives us the mber of This regression coecient result ij that the hedge gold conacts to use in hedging widget price risk. tuantity, 'l'he

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In th Aul'i exampl we assumed that widget prices are sxed.However, since gold is used to produce widgets, widget prices might vary with gold prices. If widget and gold prices vat'y one-for-one, Auric's prots would be independent of the price of gold and Auric would haye tno need to hedge. 12 . . More realisttclly, the price of widgets ould chpnge wit.h the price of gold, but not one-for-one; othr factorq could affect widget plices as well. In this case, Auric might 5nd it helpf'ul to use gold derivaives to hedge the price of the widgts it sells as well as the price of the gold it buys. Using gold to hedge widgets would be an exnmple of crosshedging: the use of a derivative on one asset to hedge anotlzer asset. Cross-hedging adses in many different contexts.

lztrne-for-one''in tlzis context means tllat if the price of by$1 times the amount of gold used to make the widget.

115

The variance of te retllrn on the hedged position is given by

j.

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Suppose that we go long H gold futures contracts, each coveling 1 oz. of gold. I.f F is the gold fonvard plice, the retttrn on the hedged posidon is

j

j

o

where Pw is the widget price.

.

()

RATI

and gold. Conceptually, we can think of hedging widgets and gold separately, and then combining those separate hedges into one net hedget To generalize the Auric example, suppose that buying Ng ounces of gold enables produce Nw widgets. Profit per ounce of gold, without hedging, is to us

.

.... .. . ... .

The hedging problem for Aul'ic is to hedge the dterence

E: 'i'(:.E: EE /73);:E . .

H EDGE

N G TH E

gold rises by $1, the.plice of a widget

lises

l3rrhis can be derived by differentiating equation (4.2)with respect to H*. 14ne term pcrmjo'g measures the comovement of gold and widgetplices, which is typically measured using a linear regression. A common approach' is to regress prce changes on price changes: P..t

-

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=

+

ppg,l

-

#,g,,-1) +

lf/

'

where the subscript t denotes tlle value at time t. Other specilications, including ttle use of jercentage changes, or regressing levels on levels, are possible. The conect rejression to use dpendj (jn context. In general, regressions using clianges are more likely to give a correct hedge ceflicient since te goal of the hedge is (o haye changes in the price of the asset matched by changes in futures price. In Chapters 5 and 6, we will present examples of hedging stocks and jet fuel, and the appropriate regressions will be returns on returns (stocks)arld chang s on changes et fuell. Regressions of level on level are problematic in mafly contexa. Forexample, in the case of stoclcs, msset pricing models tell us that stock retttnts are related, but we would not expect a stable relauonship between twoprices. The appropriate regression is retul'ns on returns. A comprehensive discussion is Siegel and Siegel (1990, pp. 114-135).

116

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Ilq'rltooucTlox

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Rlsl

MANAGEMENT

sEt-Ec-rllqc

H*, is denominated in terms of units of gold. The net use of gold futtlres, H*, is the ditference in the number of coneacts needed to hedge gold inputs and widgt outputs. Note that if gold and widgets are uncorrelated-i.e., p = o-the hedge ratio is purchased), while fp > 0, the hedge of gold long fomard per ounce ounce one one (go ratio is less than one because the widget price itself provides an implicit long position in gold that is a partial hedge. When we hedge with H* f'uttlres, ctjgetj is obtained by substimting S* into

equation (4.2):

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Te uncertainty remaining in the hedged position is due to basis risk, which is lisk due to the hedging instzument (gold)and the hedged price (widgets)not moving as predicted. The variance o f pro fits ultimately depends upon te ability to hedge the price of widgets, which, since we are using gold to hedge, depends on the correlation, p, between widgets and gold. The larger that p is, the less is basis lisk. tls section we have shown that the ability to cross-hedge depends upon the 1.11 coaelation between the hedging instrument and the asset being hedged, and that we can determine the hedging amount as a regression coefcient. We will see in Section 5.4 that the same analysis obtains when we use stock index futtlres conacts to cross-hedge a stock portfolio.

QuantityUncertainty

Thequantity afirmprociuces andsells mayvary withtheplices of inputs orottputs. When t'obvious'' hedge ratio (for example, by hedging te expected this happens, usina the quantity of an iput) can increase rather tan decrease lisk. ln this sectin we xamine quantity uncertinty. Agricultural producers commonly face quantity uncertainty because crop size is affected by factors such as weater and disease. Moreover, we expect there to be a correlation between quantity and price, because good weather gives rise to bountiful harvests. What q'uantity of fonvard onacts should a corn producer enter into to minimize the variability of revenue? We will look at three examples of different relationships between price and quantity: The benchmark case where quantity is certain, an example where quandty and price are neyatively correlated, and an example where quantity and price are positively correlated.lD

t

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and quantity for an agricultural producer. Each row is equally Iikely. In scenario A, there is no quantity uncertainty. In scenario B, quantity is negatively c rrelated with prce, and irl scenario C, quantity is positively correlated with price.

t')@,

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addiion, for each ppssible price of corn there are two equally likely quantities, for a totat o f four possible price-quantity pairs. Table 4.9 illujtrates te three scenados. Note that in scenario B, average quantity is low when price is high (negativecorrelation), correlation). wherea i.nscenario C, average quantity is high whn price is high (positive

First considr ScenarioAwhere quantity iq certain: The producer always produces busels. t-etv and denotetheprice andquantityin 1 year. Revenueis SQ. Without 1m hedging, revenue will be either $3m(if the corn price is $3) or $2m (if the corn price is ..

$2). F

On the other hand, if the producer sells folavard lm bushels at the forward price 2.50 revenue is

=

Revenue

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=

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2.5m

=

-

:uatanted rvenue in this case. 'I'he calculation is illustrated explicitly in

We hke

.'

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ls-rhere are futures contracts intended to mitigate the problem of quantity uncertainty in an agriculttlral context. Corn yield futures, for example, traded at tlze Chicago Board of Trade, permit farmers to hedge variations in reoional production quantity, and provide alz aoericulmral eKample of a conact. We discuss quantos further in Chapter 22.

.jjy:;gjj::. jjy,,, .jjjgrj jjy

t t( t,! o.fluctl.oy t;y. ) j jjrj!y n..y j j ( j . ( ( . . y.. jj:jj,;(jjyrj(.. .. .y.( . jjrjjr.,j;jg;y. . gjt;j . .jjyjr. ..gjjjrjjy jjy2rj. .gjrj,;.kjIj,,. yjtytjjs :jgjjj.. :yjjj(. ;.j. ,jjgjj, jj;;. jjkgjytjjy j(yy(j:!rjjk4...j(ryyjj(gyrr,. ...jjtyyj. . gjjrgjjj. y.y . jjgyyyg:r. j;kgjjj. jjryjj,. jjyj!j. . jjjt,i .....-.....g jjgjj,.kjjyjr (jjy (jj, kjgjjjjs .kjrjj;; kjryjj)gjrj,, yyyyy.. .kjjyjj .gj;jj,..gjjji,.. yyjj.... . rjjkjj;;i (y)j. ..... -..-..................;..L.,k......L-.,...L.L.

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In all the examples we suppose that the col'n forward plice is $2.50*u and that there is a 50% probabili. ty tat in one year the corn price will be $2/b11or $3/bu. ln

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(Scenario A), revenuerof $2.5m is assured by sellinj forward 1 m bushels.

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118

Ilq'rstolluc'rlolq

CHAPTER

MANAGEMENT

Rlslt

To

ln general, if the producer enters into fonvard conacts Hedged revenue

RH4

=

(4.5),when there is uncertainty,

Using equation is

2 O'RCH)

2

y2

-1-

GSQ

=

:)

CS x

=

2 GS

+ (S x CS

0.25 x (1 + 0.6 + 1.5 + 0.8)

F)1

-

(4.5)

V gI.LI9SQ,S

(y.6)

SQ o's

g.

The standard deviation of total revenue, SQ, is o'sc, and the correlation of total revepue with price is psc,s. As in the preceding discussion of crossuhedging, the H tat minimizes the variance of hedged revenue will be H

.

=-

The variancecminimizing

calculation, we have psc,s have

pso,so'so Gs

H

This is the same as the second term in equation (4.3). The formula for the varianceminimizing hedge ratio in equation (4.7)is the negative of the coecient from a regression'of unhedged revenue on price. We can therefore determine the variance-minimizing B and C) in hedge ratios for the negative- and positive-conelation scenarios (scenarios Table 4.9 either by using equation (4.7)directly, or els we can nm a regression of revenue on p S ce. First, consider what happens in Scenario B if we hedge by shordng the expected quantity of production. As a benchmark, column 3 of Table 4.11 shows that unhedged revenue has vatiability of $0.654m.Fromrfable 4.9, expected production in the negative

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Column (7) of Table 4.11 shows that variability is reducd to $0.652mwhen hedging optimal hedge quantity is closer to no hedging t.1)a11 hedging. tis amount. to R111 In fact, we gain little by hedging optimally, but we increas: the standard deviation of revenue by 25% if we adopt the plausible but incorrect hedging strategy of shorting 975,040 bushels. Problem 4.21 asks you to venfy' that you obtain the same nswer by On ning a regression of revenue on price. Yo right guess by now that when correladon is positive (Scenario C), the optimal hedge quantity exceeds expected quantity. The fact that quantity goes up when price goes up inakes revenue tat much more varibl than when pric alciil: vaties, and a corrspondigly larger hedge position is reqired. Problem 4.23 askj #o to compute 2 millio bushelseven . j hedge in scenario C. The answer j s to jjmrt mmost th ptlm though production is never tat large.

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Ifwe short tltis quantity of corn, column 5 of Table 4.11 shows that there is sti.llvariability irl hedged revenue. Perhaps more surprising, the variability of total revenue acmally itcreases. The reason is that price decreases when quantity increases, so nature already provides a degree of hedging: The increase in qaan:ty pnrfially offsets the decrease in pric. Hedging by shorting the f'UIIexpected quantity leaves us overhedged, with a commensurate increase in variability.

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A producer sellig a rislty commodity, such as gold, has ll inherent long position in the commodity Assuming costs are :xed, the firm's profit increases when the price of the commodity increases. Such a f11-mcan hedge profit with a variety of sateocries, including slling forward, buying puts, and buying collars. A 51-r11tat faces price l'isk on inputs hs an inherent short position in te commodity, with prost that decreases when the price of the input increases. Hedocringstrateocriesfor such a 51-minclude buying forward, buying calls, and selling collars. Al1 of the sategies hwolving options can be customized by chanocing the option strike prices. Sateoeies such as a paylater can

l6Because Table 4.11 presents the complete population of outcomes, which are equally likely, it is appropriate to use the populauon estimate of the standard devition. In Excel, this is STDEVP as opposed to STDEV. ne calculation for o'sc is obtained as STDEVPO,1.8, 3, 1.6) 0.6538. =

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provide insurance with no initial premium, but on which the company has greater losses should the insurance be needed. Hedging can be optimal for a company when an extra dollar of income received in times of high prosts is worth less t1::,1:an exa dollar of income received in times of low prosts. Prosts for such a fil'm are concave, irl which case hedging can increase expected cash iow. Concave prosts can arise from taxes, bankrptcy costs, costly external nance, preservation of debt capacity, and managerial risk aversion. Such a 51-mcan increase expected cash flow by hedging. Nevertheless, firms may elect not to hedge for reasons including transaction costs of dealing irl derivatives, the requirement for expertise, the need to monitor and con'ol the hedging process, and complications iom tax and accounting considerations. FURTHER

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l-year continuously compounded 'Fhe l-year forward price of copper is $1/1b. option prices for interest rate is 6%. One-year copper are shown in the table below.l? 'rhe

Strike 0.9500 0.9750 1.0000 1.0250 1.0340 1.0500 In your answers, at a $1.00, $1.10,and $1.20.

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Call

Put

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$0.0178 0.0285 0.0376 0.0509 0.0563 0.0665

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consider copper prices in 1 year of

$0.80,$0.90,

4.1. lf XYZ does nothing to manage copper pri risk, what is its prot 1 year from now, per pound of copper? If on the other hand XYZ sells forward its expected copperproduction, what is its estimated prot 1 yearfrom now? Cnstnlct graphs illusating both unhedged ynd hedged profit.

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@ Telco installs telecommunications equipment and uses copper wire from Wirco as an input. For planning purposes, Telco assigns a sxedrevenue of $6.20for each unit of wire it uses.

READING

In this and earlier' chapters we have examined uses of forwards and options, tnking for g'rantedihe pricing of tlpse contracts. Two big unanswered questiops afei How are prices determlned? How does the market for them work? those ' In chapters 5 tlaiough 8, we will explore forwartt and ftures coniacts cticussing Capters 10 througti 13 we will Pricingas well as how market-makers function. 1.11 answer the same questions for optlons. Chapter 14 will discuss how exotic options can be used in risk-management sategies in place of the ordinary puis nd clls discussed in this ch>pter. Wlkarton and irms to assess their hedging. survey noafinanlal cmc regularlyBodnar A recent survey is summ arized in et al (1998) Bnrtram et al (2004)exanune hedgsng bhavior in an internadonal sample of over 7000 firms. Tufano (196, 1998), Petersen andrfhiagarajan (2000),andBrown ei a1. (2003)have smdied hedgingpractices in thegold-mlnlngindustly Otherpapers exnmlnlghedogincludeGczyeti. (1997), Mlayannis and Weston (2001),Maynnnis et a1. (2003), and Allaynnnis et a1. (2004). Guay and Kothari (2003) attempt to quantifyv derivatives usage using information in 51-mnnnual reports from 1997. Brown (2001)provides an interesting and dtailed by one (anonymous)'Iirm and Faulkender (2001) description of the hedging exnmines imerest rate hedging in the chemical industry. Gastineau et a1. (2001)discuss Statement of Financial Accouting Standards 133, wlzichcurrently governs accounting for derivatives. Finally, Reming (1997)relates some of the history of ttheficdtous) Auric Enterprises.

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4.2. Suppose the l-year copper forward price were $0.80instead of $1. If XYZ wer to srll folavard its expected copper production,whpt is its estimated profit on'e year from now? Should XYZ produce copper? What if ie forward copper plice is $0 45?

.

4.3. Compute estimated profit in 1 IIIIaI'it'XYZ buys a put opon with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case. 4.4. Compt: estimated prost in 1 IaTiI'XYZ sells acall option Fith a sfrike of $0.95? $1.00, or $1.05. Draw a graph of prost i each case.

.decisions

4.5. Compute estimated profit itl 1 yearif XYZ buys collars with the following s-rikes: a. $0.95for the put and $1.00for tlkecall. b. $0.975for the put and $1.025for the call. c. $1.05for te put and $1.05for the call. Draw a graph of protit in each case.

4.6. Compute estimated profit in 1 year if XYZ buys paylaterpf/l. premium may not be exactly zero):

PROBLEMS For the following problems consider the following three ftrms:

@A'I''Z mines copper, with xed costs of $0.50/1band variable cost of $0.40/lb. * Wirco produces wire. lt buys copper and manufactures wir. One pound of copper can be used to produce one unit of wire, which sells for the price of copper plus $5. Fixed cost per unit is $3 and noncopper variable cost is $1.50.

as follows tthenet

a. Sell one 1.025-strike put and buy t'wo 0.975-strike puts. j

g These

are opion prices from the Black formula assuming that the risk-free rate is 0.06, volatility is 0.l and time to ekpiration is one year. ,

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probability of a $600 loss each year. Firm B has a 50% probability of a $300 proft and a 50% probability of a $100 prot each year.

b. Sell two 1.034-strike puts and buy three l.oo-strike puts. Draw a graph of profit itl each case.

4.7. If Telco does nothing to manage copper plice tisk, what is its profh 1 year from now, per pound of copper tat it buys? If it hedges the price of wire by buying copper forward, what is its estimated prost 1 year from now? Construct graphs illustrating both unhedged and hedged prost. 4.8. Compute estimated profit in 1 year if Telco buys a call option with a strike of $0.95, $1.00,or $1.05.Draw a graph of profit in each case. 4.9. Compute esmatedprofit in 1 year if Telco sells aput option with a strike of $0.95, $1.00, or $1.05. Draw a graph of prost in each case. 4.10. Compute estimated prost in 1 year if Telco slls collars with the following sikes: a $0'.95for the put and $1.00for the call. b. $0.975for the put and $1.025for the call. c. $0.95for the put and $0.95for the call.

a. What is the expected pre-tax prost nekt year for ftrms A and B? b. What is the expected after-tax profit next year for fil'msA and B? 4.17. Suppose thattirms face a40% income tax rate on positive prosts and that netlosses reeive no credit. (Thus, if profits are positi' ve, after-tax income is (1 0.4) x proht, while if there is a loss, after-tu income is the amount lost.) Firms A and B have the snme cash flow distzibution as in the previous problem. Suppose the appropriate effective annual discount rate for both firms is 10%. -

a. What is the expected pre-tax profit for A and B? b. Whgt is the expected after-tax prot forA and B? c. What would Firms A and B pay today to receive next year's expected cash flow for sure, instead of the variable cash Qows described above? For the following problems use the BSCall option pricing function with a stock price of $420 (theforward plice), voladlity of 5.5%, continuously compounded interest rate of 4.879%, dividend yield of 4.879%, and time to expiration of 1 year. The problems require you to vary the stlike prices.

Draw a graph of profit in each case.

4.11, Compute estimated prost in 1 year if Telco buys paylater calls as follows (the net premium may not be exactly zero): '

one 0.975-strike dall and buy t'wo 1.034-st!ike calls.

a. Sell b. Sell t'wo l.oo-sfrike calls and buy three 1.034-sike

calls.

a. Suppose that Auric insures against a price increase by purchasing a +1.0strike call. Vel'ify by drawing aprost diagrnm that simultaneously selling 400-stIike put will generate a collar. What is the cost of yhis collar to a

'

Auric?

Draw a paph of profit in each case.

4.12. Suppose that Wirco does nothing to manage te risk of copper price changes. What is its profit 1 year from now, per pound of copper? Suppose thatWirco buys copper fomard at $1. What is its prost 1 yeay from now? 4.13. What happens to the vatiability of Wirco's prost if Wirco undertakes any strategy (buying ealls, selling puts, collars, etc.) to lock in the price of copper next year? You cn use your answer to the previous question to illusate your response. 4.14. Golddiggers has zero net income if it sells gold for a price of $380. However, by shorting a fonvard contract it is possible to guarantee a profit of $40/oz. Suppose a manager decides not to hedge and the gold price irl 1 year is $390/oz. Did the 511.11 em'n $10 in prost (relativeto accounting break-even) or lose $30 in prot (relative to the prost that could be obtained by hedging)? Would your answer be different if the manager did hedge and te gold price had been $450:7 4.15. Consider te exaple in Table 4.6. Suppose that losses are f'ully tu-deductible. What is the expected after-tu profit in this case?

4.16. Suppose that srmsface a 40% income tax rate on a11prohts. ln particular, receivefull credit. Firm A has a 50% probability of a $1000 proft and

4.18. Consider the exnmple of Auric.

losses a 50%

b. Find the strike prices for a zero-cost collar (buyhigh-sike low-strike put) for which the sikes differ by $30.

call, sell

4.19. Suppose that LA1N lnvestment Bnnk wishes to sell Auric a zero-cost collar of width 30 without explicitpremium (i.e.,there will be no cash payment fromAulic to LMN). Also suppose tat on every option the bid price is $0.25 below the lack-scholes price and the offer price is $0.25 above the Black-scholes price. LMN wishes to enrn their spread ($0.25per option) without any explicit charge to Auzic. What should the stlike plices on the collar be? Note Since the cpllar involves two options, LMN' is looking to make $0.50 on the deal. You ned. to 5nd strike prices tat differ by 30 such tat LNIN makes $0.50.)

4.20. Use the snme assumptions as in thepreceding problem, withoutthe bid-askspread.

Suppose that we want to construct a paylater strategy using a ratio spread. Instead

of buying aco-strike call, Auric will sell one Mo-sfrike call, and use theprerium to buy two higher-sfn'ke calls, such that the net option premium is zero.

a. What higher stn'ke for the purchased calls will generate a zero net option premium? b. Graph the proft for Auric resulting from this strategy. h

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4.21. Using the information in Table 4.11, verify that a regression of revenue on price gives a regression slope coecient of about 100,000. 4.22. Using the information in Table 4.9 about Scenario C: a. Compute ctotal revenue positive.

When

correlation between price and quantity is

b. What is the correlation between price and revenue? 4.23. Using the infonuation in Table 4.9 about Scenario C: a. Using youranswerto theprevious queson, useequatipn (4.7)to compute the variance-minimizing hedge ratio. b. Run aregression of revenue on price to compute the variance-mirlimizing hedge ratio. c. What is the variability of optimally hedged revenue? 424. Using the infonnation in Table 4.9 about Scenazio C: a. What is the expected quantity of production? b. Suppose you short the expected quantity of cor'n. What is tlze standard deviation of hedged revenue? 4.25. Suppose that plice and quantity are positively correlated as in this table: Price

Quantity

Revenpe

$2 $3

0.6m bu 0.934m bu

$1.2m $2.8m

There is a 50% chance of either price. The f'utures price is the effect of hedging if we do the following'.

$2.50. Demonstrate

a. Short the expected quantity. b. Short the minimumquantity. c. Short the maximum quantity. d. What is the hedge position that eliminates variability in revenue? Why?

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k. FINANCIAL

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From Table 5.1 it is clear that you pay interest when you defer payment. 'I'he interesting question is how deferrinj the physical receipt of the stock affects the ptiee; this d e fen'al occurs with both te folavard and prepaid fonvard conacts. What should cases'p you pay for the stock in those

CONTRACTS

ON STOCK

a foreign Aprepaid forward contract entails paying today to receive something-stocks, permits the prepaid of contracts forward The sale future. a currency, bonds-in the of period possession time. for retaining physical while a owner to sell an asset We will derive the prepaid forward price using three different methods: pricing pricing by present value, and pricing by arbiag. analogy, by

Pricing the Prepaid Forward by Analogy Suppose you buy a prepaid forward contract on XYZ. By delaying physical possession of the stock, you do not receive dividends and have no voting or conol rights. (We ignore here the valpe of voting and conol.)

the possibilility that the arrangements also differ with respect to credit risk. which arises f'rom his or her end of the deal. (And of course the 11511 of will side not transaction other the the person on wonied about yott f'ulfillingyour obligation.) person on the other side of the deal may be

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prepaid forward contract and an outzight purchase is tht wit.h the former, you price you pay is not necessarily the receive the stock at time F. We will see that the ttock price. Al1 arrangement in which you both pay for the stock and Forward contract: with te time F price specised at time 0. receive it at time F,

5.2 PREPAID FORWARD

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In the absence of dividends, whether you receive physical possession today or at time F is irrelevant: In eiter case you own the stock, and at time T it will be exactly as if you had owned the stock the whole time.4 Therefore, w//cn there c?'e no dividends, the #?-fcc ofthe prepaidfonvard ctmrmcl is r/7cstockprice today. Denoting the prepaid fonvard jrice for arl asset bought at time 0 and delivered at time F as FpPw, the prepaid forward price for delivety at time F is

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Pricing the Prepaid Forward by Discounted Present Value We can also derive the price of the prepaid fonvard using present value: We calculate the expected value of the stock at time F and yhendiscount that value at an appropriate stockprice at time F, 5'w,is uncertain. in computing te present rate of rempz. risk-adjusted price, eed appropdate of the stok value to use ap rpte. we If the expected stock price at tlme F based on information we liave at time 0 is EoSv4, then the prepaid forward ljrice is given by 'fhus

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Pricing the Prepaid Forward by Arbitrage Classical arbitrage describes a sitation in which we can generate a positive cash flow either today or in the future by simultaneously buying and selling related assets, with no net investment of funds and wit.h no risk. Arbiage, ill other words, is free money. A.n extremely important pricing principl, which we will use often, is tat the prfcd ofa thqt nt7 arbitrage is possible. be deriyative Hre is an exnmple of arbiiage. Suppose that the prepaid fomard price exceeds P J arbitraoeur w ill uy 1ow anci sell high by buying te stock price-i.e., ' Fo,w > %. 'qlfltf

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lsuppose that someone secretly removed shares of stockfrom your safe arld remrned them 1 year later. From a purely linalzcialpohlt of view you would never notice tlzestock to be missing.

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To understand the effects of dividends, we will compare prepaid dividends StockA pays no dividend, and othenvise identical stock B pays stocks: forwards on two days dividend from today, just before the expiration of te prepaid forwards. 364 a $5 We know that the prepaid forward price for stock A is the current stock price. What is the prepaid fonvard plice for stock B? Since the $5 dividend is paidjust before the delivery date for the stock 1 year from today, on the delivel'y date stock B will be priced $5 less than ock A. Thus, the price we pay today for stock B should be lower than that for stock A by the present value of Discrete

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the stock for Sz and selling the prepaid forward for &%.This ansaction mkes money arld it is also risk-free: Selling the prepaid forward reqires tat we deliver th stock at time T and buying the stock today nslzres that we have the stock to delivef. Thus, we the buyf of the prepaid earn Fopv s'atoday and at expiration we supply the stock to offset all and futilre fisk. Table 5.2 profts positiv earned today have forwar. We simation. this summadzes

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< &. Now suppose on the other hand that Xpv we can engage in arbiage ' eprning So shorting stock, Oe yar from and the by buying the prepaid forward stock folward arld via tp clse te short we us tat te prepaid now we acquire te stock reversed. simply table abo-ve the afe P osition. Te cash flows in Throughoutthebookwe will assume thatprices are atlevels thatpreelte rbitrage. This raises the question: Ifprices are such that arbitrage is not prohtable, who can alord to become an arbitrageur, watching out for arbiage opporttlnides? We can resolve rbitrage to earn livig, this paradox with the insight that in order for arbitrageurs ttan of degree equilibrium be there from must time; time to opportunities must occur disequilibzium.''s However, you would not expect arbiage to be obvious or easy to undrtAe. The transactions in Table 5.2 are the same as those of a market-maker who is hedging a position. A market-maker would sell a prepaid forward if a cujtomer wished to buy it. The market-maker then has an obligation to deliver the stock t a sxedprice and, in order to offset tis risk, can buy the stock. 'T'he market-maker thus 4ngages in the same ansactions as an arbitragetm except the purpose is risk management, not twptf arbitrage. Thus, the rl't//kmclDn described ?7 Table 5.2-sellilq theprepaidfonvard ??7t7r/l-?nckc?: buying l/kc stock--also describes the acfsn, ofa The no-arbitrage arguments we will make thus srve two funcions: 'Fhey tell us how to take gdvantage of rnispricings, and they desc/be the behavior f market-makers

In general, the price for a prepaid folavard contract will be te stock price less the present value of dividends to be paid over the life of the contract. Suppose there are multiple dividend payments made throughout the life of the forward contract: A stock n. A prepaid is expected to make dividend payments of Dti at times ti i = 1, receiving the receive time but without will entitle the stock F at fomard contract you to prepaid price is the dividends. 'Fhus, forward interim

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stiglitz(1980),in which this

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)' )y'. rj Example 5.1 Suppose XYZ stock costs $100 today and is expected to pay a $1.25 /1 quarterly dividend, w ith the srst oming 3 months from today and the lastjust prior to ytr tE) the delivery of the stock. Suppose the annual continuously compounded lisk-free rate is )'!qi 2.5%. prepaid therefore continuously compounded is A l-year quarterly 10%. The rate )q 'i) forward contract for the stock would cost tti 'j @ (qyj: k E i

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The calculation in this example implicitly assumes that the dividends are certnin. Over a short horizon tlzis might be reasonable. Over a long horizon we would expect dividend risk to be greater, and we would need to account for this in computing the present vale of dividends.

132

%.FINANCIAL

FORWARDS

AND FUTURES

FORWARD

.'y .'.' .'.'.

CONTRACTS

$

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6nere is significant seasonality in dividend payments, which can be important in practice. A large numberof U.S. firms pay qumerly dividends in Februmy May,August, and November. Gennan firms, by coneast, pay mmual dividends concentrated in May, June, and July.

CONTRACTS

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Tls formula shows thattheforward coneactis apurchase of the stock, with deferred P ayment. The hlterest adjustment compensates for tat deferral.

For the case of continuous dividends, equation (5.7),the annualized forward premium is simply the difference between the risk-free rate and the divideljd yield.

Discrete dividends: To obtain the forward price for a stock that pays discrete dividendj, we tale the f'uture value of equation (5.3). The forward price is the fumre value of the prepaid forward.

Creating a Synthetic Forward Contract

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Whereas for the prepaid fonvard we subtract the present value of dividends frorrt the current stock price, for the fonvardwe subact the fumre value of dividds from the future value of the stock price.

Continuous dividends: future value of equation

A market-maker or arbitragelzr must be able to olset the Iisk of a fonvard contract. It is possible to do this by creating a synthetic fonvard conact to offset a position in the act'ual forward onact. In this discussion we will assume tat dividends are continuous and paid at the rate and hence that equation (5.7)is the appropriate fonvard price. We can then create a , ' synthetic long forward contract by buying the stock and bolwwing to ftmd the position. To see how the synthe:c position works, recall tlzatthe payoff at expiraiion for a long fonvard position on the index is

When the stock pays continuous dividendsj we tale the

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We can annualize the forward premium and express it as a percentage, irl which case we Annualized

,synthtically

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F0,z

=

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It is important to distinguisk between the fonvard #?-cd and the p?z/nfa/l, for a fonvard coneact. Because te fonvard coneact has deferred payment, its nfrftzlp?-'?/7rll?i is zero; it is initially costless. The forward price, however, is the future value of the prepaid forward price. This difference between te fonvard price and premium is in conast with the prepaid forward, for wltich plice and the premium are the same: 'I'he prepaid forward price is the amount you pay today to acquire the asset in the future. Occasionally it is possible to observe the forward price but not the price of the underlying stock or index. For example, the ftltures coneact foi t S&P 50 index ades at times when tlw NYSE is not open, so it is possible to observe the f'utures price but not the stock price. The asset price imptied by the forward pricing fonuulas above is said to dene fair value for the underlying stock or index. (5.7)is used irl tlzis case to infer the value of the index. 'l'he forward preminm is te ratio of the forward price to the spot price, defined

F0 w'

-

In order to obtain this same payoff, we buy a tailed posiion in the stock, investing SOe-&T. This gives us one share at time F. We borrow this amount so that we are not arlything additional at time 0. At fime F we must repay Szer-&sT and we required to pay Table 5.3 demonsates that bon-owing to btty the stock replicates stock Sp. for sell the payoff expiration forwrd contract. te create a forward, we Just as we can use the stock apd borrowing to can also use the fonvard to create synthetic stocks and bonds. Tabl 5.4 demonseates that we can go long a folavard contract and lend te piesent vale ju gol'war tj p yoe to synthetically create the stock. The expiration payoff in this table assumes that equation (5.7) holds. Table 5.5 demonstrates that if we buy the stock and short the folward, we create cash flows like those of a risk-free bond. The rate of rettllnA on tltis synteic constnlction of which is summarized in Table 5.5-is called the implied bond-te repo rate. To summarize, we have shown that

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.

- .. E . .

Sze...sv

..

. ...... ..

E

.. . . . -. .. . -. -

..

-

-

-.

. - ..

-

..

- .

-

.

.. . - ..

.. .

.

- .y . . . ... . . .. .. . . . .. . .

.sv

-

.

..

.

. g. . . .

.

.

.y

.

..j..

.

,.. .

-Sv

-l-&c

e-JF ()

-s

.

.

+s 0 elr-lr

0

Sp

0

Szer-&tT

.

.

-

y . . .. .

j... .. . . g. y.. . jjjyyjjjj,. . . j. :

... .

. . . . ..j.yy kjjjjyrjjjj, (jjjjs jjjyjgjry rjjjrj,;i rjjjjygjjj,s y..r y . ..j...y..... y.y.

.

. . . .. y , ... .. .... . .. .. ........ . . . .. ........... . .

F0,z

.. . . .

-

. . .. . . .. . ... . . . . . . .

.

.

.. .. . -

.

. .

#()v

Similarly, suppose the market-maker wishes to hedge a long fonvard position. Then it is possible to reverse the positions in Table 5.6. The result is in Tale 5.7. in which you buy the underlying assetand shortthe offsettingforward

Aansaction

Synthetic Forwards in Market-Making and Arbitrage

gjg;i

a forward contract and short a synthetic

forward contract.

hort tailed position in ltock, receiving

We can rearrange this equation to derive other synthetic equivalents.

,j,j,jjjj,

E Eg ( ( E E '' . (E .( . 'j j.q . '.(.'.j...!.((gj(.j(.. .. ...g.j('...y.'.j.).!.j(.Ej.jj.j.g.....(j.'.(...E. j..j(k.j.... g.k.g.. j.jjr...j..jg.E(.......g..(..r j (gj j...g.. ... ..... g.... j...j E(. gg..j..j... g.....j... j g.. j.j.j.. .jg.... jj ... . y.yy.. j..y. . .. ... .y. j .. . .. y.. .. . .... . ... .. .. . . j. j... . . . ... j .. j.'yj... g l'' ' . E....... .. E... E. . Iiliq)ii. .... . j.(.. .g( . j q... . . .yq y-E g ..l('. . q .... q. . .. q... .. ;.. . .. . E.( . y. .... . .. ..E. ... .. g.. .y.. ... . q j..... . . . ! .. y . ..y yr ..iI:j2),k. . y y . y. r . . .y y y ;qE .y .. E g..Ey g .. y . r (y . y. . yy j y... -..j y . yy yy j y i.''.l..:' '';qI1dIqq'!!ii@:#kk'. ' ''' . 1iI;j:!j: . E ' . . . . . .. y y y j . . . ... y . )'.E y ' yy yy. ,....y....y.j... . .. . .. . gj.. j.j. . y. E .... ... .yy.y... y.j y...(.E... y. . yj... . yj. y.y....... y....j... yr . . E ..q . . y. .. . . ( . . .' .' .( '' . . . E.. ....E j..... y. y . .y. y.. E... q y y . ...E... ......E y..E.E...r........E.E.y.y. . y.......;. .... y....y. y.j .. y.. y...-j..y.Ejy......; y.....ryy...y... y.......j..... .y..........yy.. y. .E... ..... y.. .... ...(...... 2. .(.. . (:. y .. ) . . .... .. . .. ........ ..... ....... .... . . . . . ..... . . y .. .. .. . ... ........ . .

.E'k '.' ':

. .. .

F 0,T'

jtyyjjj jgjyjrjtyygj jr,.kjyyjs j;,.jkggrrj;gyjr jr,, jyjjj jj..;j .jy:jtry;jj yjji gjgy kyjjj jr,. jr,gyjjyjkyjyjj jr..;j jrrg;s ::

...jks..jr,.

)t-.-)-t..'.-kkkyk).... ... market-maker is Iong

Long forward

Stock

.

-.-.-..!..tp@

--.jt-..)--y-.-.;.)k;..

' :

sv

E E -.

' : : :

:

:...

:

Acash-and-canyhas nolisk: Youhave an obligation contractis called acash-and-carry. also the but deliver the asset. The market-maker offsets the short folavard asset own to position with a cash-and-cany An arbitrage that involves buying the underlying asset and selling it folward is called a cash-and-carry arbitrage. As you might guess, a reverse cash-and-carry entails short-selling the index and entering into a long forward position. If te forward coneact is priced according to equation (5.7),then prohts on a cashand-carry are zero. We motivated the cash-and-can'y in Table 5.6 as lisk management by a market-maker. However, an arbitrageur might also engage in a cash-and-cany If the forward price is too high relative to the stock price-i.e., if F().w > Sctr-llr-then an profit. arbitrageur or market-maker can use the strategy in Table 5.6 to make a lisk-free .

138

%.FINANCIAL

FORwARDS

FORWARD

AND FUTURES

A.n arbieageur would make the transactions in Table 5.7 if the fonvard were unW V SCCOFYOVVC to ZC St0Ck--i.C., if ROCSV-JIX > S T As a final point, you may be wondering about the role of borrowing and lending in Tables 5.6 and 5.7. When you explicitly account for bonowing, you account for the opportunity cost of investing funds. For example, if we ornitted borrowing from Table 5.6, we would invest %e-'T today and receive F0,w at time F. In order to know if there is an arbitrage opportunity, we would need to perfonn a present value calculation to compare the time 0 cash flow with the time F cash flow. By explicitly including borrowin g in the calculations, this time-value-of-money comparison is automatic.? Similady, by comparing the implied repo rate with our borrowing rate, we have a simple meastlre of whether there is an arbiage opportunity. For example, if we could borrow at 7%, then there is an arbtrage oppormnity if the implied repo rate exceeds 7% On the other hand, if our bonowing rate exceeds the implied repo rat, tere is no arbitrage oppormity. .

1

'No-Ybitrage

Bollnds with Transaction Costs

Tables 5.6 and 5.7 demonstrate that an arbitrageur can make a costless prot if Fo w + qoetr-llr This analysis ignores eansactioncosts. In pracce an arbieageur will face trading fees, bid-ask spreads, diferent interest rates for borrowing and lending, and the possibility that buying or selling in large quantities will cause prices to change. The effect of such costs will be that, rather than there beilzg'a single no-arbiage price, there will be a no-arbiage boundl a lowerprice F - and an upperprice F + such that arbitrage will not be proftable when the forward price is between thse bonds. Suppose that the stock and forward have bid and ask prices of Sb < S' and Fb < FJ, a ader faces a cost k of transacting in te stock or forward, and the interest rates forborrowing and lending are rb > r. ln this example we suppose that there are no transaction costs at time F, when the forward is either settled by delivery or cash-settled. We will lirst derive F+. An arbitragetlr believing the obselwed forward price, Fo,w, is too high, will undertake the ansactions in Table 5.6: Sell the forward and bnow to buy the stok. Fof simplicity we will assume the stock pays no dividends. 'I'he arbitrageur will pay tlle ansaction cost k to short the fomard and pay (5! + k) to the position is terefore acquire one share of stock. The required borrowing to snance payoff 2k. At time F, is 5'g + the

-(.%a +

lkje

rbT

+ F0,w

Repayment of borrowing

Arbiage

-

Sr +

Value of fonvard

Sv Value of stock

>

Fo

=

a (.% + 2k)c ay.

?ln general, arbitrageurs can bonmv and lend at different rates. Apro forma arbitrage calculation needs to account for te appropriate cost of capital for any particular transaction.

O N STOCK

$

139

Thus, the upper bound re:ects the fact that we pay a high price for the stock ttheask price), pay eapsactioncosts on both the stock and forward, and borrow at a high rate. We can derive F- analogously. Problem 5.14 asks you to vel'ify that the bound below wlzich arbitrage is feasible is F0,w < F -

=

zh

b

rlw

-

zkje

(5.11)

This expression assumes that short-selling the sto'ck does not entail costs other than bid-ask ansaction costs when the short position is initiated. Notice tat in equations (5.10)and (5.11),the costs 811 enter i.n such a way as to make the no-arbiage region as large as possible (for example, te 1ow lending rate enters F- and the high bonowing rate enters F+). This makes economic'sense: Trading costs cannot help an arbiageur make a prost. There are additional costs not reQected in equations (5.10)and (5.11). One is that signicant nmounts of tradlng can move prices, so that what appears to be an arbitrage enters a large ordr. Another challenge may vanish ifprices change when the lbiagetlr risk. tzades do not occur instantaneously, tlie arbiage cpn vaaish tf can be execuuon completed. before the trades are It is likely that the no-arbitrage region will be different for different arbiageurs at a point in time, and different across time for a given arbitragetm For example, consider the ading eansaction cost, k. A large westment bnnk sees stock order llow from a variety of sotlrces and may have wentory of either long or shol't positions irl stocks. 'I'he bnnk may be able to buy or sell shares at low cost by selwing as market-maker for a customer order. It may be inexpensive for a bnnk to short if it already owns the stocks, or it may be inexpensive to buy if the bnnk already has a short position. Borrowing and lending rates can also vary. For a transaction that is explicitly fnanced by boaowing, the relevant interest rates are the arbiageur ds margin al borrowing rate (if that is the sotlrce of f'unds to buy stocks) or lending rate (if stocks are to be shorted). However, at other tlmes, it may be possible to borrow at a lower rate or lend at a higher rate. For example, it pay be possible to sell T-bills being held for some other may effectiyely permit bonowing at a 1ow purpos as a source of short-term funds. Finally, in order to bon'ow money or secures arbitragelzrs must have available rate. Undertnking capital. one arbiage may prevent undertnking another. and I'he overall conclusion is not surprising: Arbiage may be diflicult, lisly, deviations omleieoreticipdcemaybeabiaged, butsmall Largedeviations costly. may or may not represent genuine arbitrage opportunities. E

.

'lhis

Quasi-Mbitrage

is prostable if this expression is positive, or F0,F

CONTRACTS

(5.10)

'l'he previous section focused on explicit arbiage. However, it can also be possible to undertake ilnplicit arbiage by substittlting a 1ow yield position for one with a higher return. We call this quasi-arbitrage. Consider, for example, a corporation tat can borrow at 8.5% and lend at 7.5%. Suppose there is a cash-and-carry ansaction with arl implied repo rate of 8%. There is no pure arbitrage opponunity for the corporation, but it would make sense to divert

140

%.Fl NANCIAL

Fo RWARD

AN D FUTURES

FORWARDS

lending from the 7.5% assets to the 8% cash-and-carry. tf we attempt explicit arbieage by borrowing at 8.5% in order to enrn 8% on the cash-and-cany the transactionbecomes unprostable. We can arbiage only to the extent that we are already lending; this is why it is quasi'tarbitrage.

Does the Forwrd

Price Predict the Future Price?

lt is eommon to think that the forward price re:ects an expetation of the asset's fumre price. However, from the formula for the forward price, equation (5.7),once we lnow the current asset price, risk-free rate, and dividend yield, the fomard price conveys no additional information about the expected f'uture stock price. Moreover, the fomard price systelhatically errs in predictig the future stock price. that has The reason is straightforward. When you buy a stock, yu invest lney interest-earning asset), been in invested othemise could have an oppormnity cost (it an and you are acquiring the risk of t-h stock. On average you expect to earn interest for the time value of money. You also expect an additional retllrn ms as compensation for the risk of the stock-this is the tisk premium. Aljebraially, the expected return on a stock is 'compensation

%=

a

r

-

?-

,

(.12)

Compensation for risk

Compensation for tme

CONTRACTS

$

ON STOCK

141

For example, suppose that a stock index has an expected ret'urn pf 15%, Fhile the l'isk-free rate is 5%. lf te current index price is 100, then on average we expect that te forward plice for dlivery in 1 yea.r will be only 105, index will be 115 in 1 year. of the fomard conact will on average enrn positive This holder however. means that a of bearing the risk of the index.S albeit profits, at the cost This bias does not imply tat a forward contract is a good investment. Rather, /7J. tells it us tat the risk p?'c/?3rfl?ion f/n asset cfln b created at zdm cost ttntf hence value. Though this seems surprising, it is a result from elementaly finnce that a zdm if we buy any asset and bon'ow the RIII nmount of its cost-a tzansaction that requires we earn the risk premium on the asset. Since a forward conact no westment-then has the risk of a fully leveraged investment in the asset, it earns the riskwemium. This proposition is true in general, not just for te exnmple of a folavard on a nondividendPaying stock. rfhe

An Interpretation of the Forward Pricing Formula 'l'he fomard pricing formula for a stock index, equation (5.7),depends on r J, the difference between the risk-gee rate and the dividend yield. This difference is called the cost of carrp Supposeyou buy aunit of the index that costs S andfund theposition by borrowing the risk-free rate. You will pay t'S on the borrowed nmount, but the dividend yield at will provide offsetting income of &S. You will have to pay the difference, (r 84S, on an ongoing basis. This difference is the net cost of carrying a long position in the asset', hence, it is called the 'Tost of carly'' Now suppose you were to short te index and invest the proceeds at the risk-ee would receive S for shorting the asset and enrn t'S on te invested proceeds, You rate. would have to pay S to the index lender. We will call 8 the Iease rate of the but you lease rate of an is what index; it you would have to pay to a lender of the asset. mmualized cash that the make bonower must to the lender. For a payment asset is te nondividend-pgying stock, the lease rate is zero while for a dividend-paying stock, the lease rate is the dividend. Here is an interpretation of the forward pricing formula: -

-

When you enter into a forward conact, there is no invstmnt', hee, you are not compensatd for the time value of mo-ney. However, the fo>ard contract retains the risk lisk. This ???dtwz' that #kdftp/4s/tzlW tontract of the stock, so you must be compensated for ??7lI.l eal'n the ?-'ky?'c/nfl/?/l. If the risk premium is positive, then on average you must expect a positive ret'ul'n from the forwrd contract. The only way this can hppen is if the forward Ijrice predicts too low a stock price. In other words thefonvard coiitract is t/.fl/yd//llf?'c stock price. a biased predictor We ean see this algebraically. Let or be the expected return on a nondividndpaying stock and 1et 1, be the effective annual interest rate. Consider a l-year forward forward price is conact. 'l'he

F() = Sfj(1 + r)

'l'he

Forward price

=

Spot price + Interest to carry the asset

The expected future spot price is

f

)

=

,V(1 +

J)

where Efj denotes as of time 0.3'Thus, the differepce between the forward price and the expected f'uture spot price is -

F0

=

Asset lease rate

Cost of carry OCSI

ttexpectation

.E'0(5'1)

-

S0(1 +

ce)

-

S0(1 + r)

=

Sza

-

r)

1- is the risk p?'c??7!I?n The expression a on the asset i.e., the amount by which the lisk-free asset. This equation veries that thefolnvard asset is expected to outperform the tftr l-lj'i: Tr (7JJ t, t. l-isk (f t)/ /J l-i ()t? is 47 ittlt? b4'l/7 tr :7???t?If?) t tlttr yp?-tr??7ittl71 t7?$ tIltr ttl

The fomard contract, unllke the stock, requires no investment and mkes no payouts and therefore has a zero cost of cany One way to interpret the forward pricing fonnula is that, to the extent the fonvard contract saves ou' having to pay the cost of carry, we is what equation (5.13)says. are willing to pay a higher price. rfhis

-

Accounting fordividends in this example would not change the magnitude of the biks since dividends would lower the expected f'umreprice of the index and te forward price by equal amounts.

142

%.FINANCIAL

FORWARDS

Fu'ruREs

AND FUTURES

%.

cox-rRAc'rs

143

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5.4 FUTURES

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Fptures conacts are essentially exchange-eaded fomard contracts. A; with commitmentto buy orsell an underlying assetatsomefumre representa futures contracts they are standardized and have specised exchange-aded, f'utures date. Because are Futures and locations, dates, procedures. may be traded either eleconically or delivel'y orders shouting is alled open sellers to one another ttllis and with buyers pits, in trading clearinghouse The role clearinghouse. of the associated Each exchange has outcry). an is to match the buys and sells that take place during te day, and tp kep track of the obligations and payments required of the members of the clearinghouse, who are called clearing vcp//pua. After matching trades, the clearinghouse typially ecomes the counterparty for each clearing member. Mthough forwards and f'utures are similar in many respects, there gre diferences.

settled * Whereas fclrward coneacts are settled at expiration, futures contracts are calledmarking-torparket. whopis whatto who of owes daily. The determinaon Frequent marling-to-market and settlement of a futures conact can lead to pricing differences between the futures and an otherwise identical fotavard. pjile to offset * As a result of daily setllement, f'utures coneacts are liquid-it ij poidon. ample, For enteling into by opposite date the afnobligation on a given cgncel conact, ftltures 500 you cap your if you are long the September S&P obligation to buy by entering into alz offsetting obligation t selt the jejtember S&P 500 contract. tfyou use $e snme broker to buy ad to dell, yur obliation is cancelled.g of'lic iall y * Over-the-counter forward conacts can be customized to suit the buyer or seller, wherea futures contracts are standardized. For exnmple, available futures conacts may permit delivery of 250 uaits of a particular index in March or June. A fonvard contract could specify Apzil delively of 300 units of te indey. lisk is different with the futures * Because of dily settlemeny, the nature of credit structured ln so as to minl'mize te effects of fact, fmures cpnacts are contract. risk. credit There are typically daily price limits in futtlres markets (andon sme stock ex* changes as well). A price limit is a move in the f'utures price that triggers a temporary halt in eading. Fr exnmple, there is an initial 5% limit on down moves in the S&P 500 fumres coneact. An offer to sell exceeding this limit can trigger effect. I.f that is a temporary trading halt, after wlaich time a 10% price limit is in omplicated, rules be The and 20% 15% limits. subsequent there cn exceeded, are exist. such rules that importnt but it is to be aware E

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specific example.

The S&P 500 Futures Contract The S&P 500 futures contract has the S&P 500 stock index as the underlying asset. Futures onindividual stocks have recently begun trading in theunited States. See the box above. Figgre 2.1 shows a newspaper quotation for the S&P 500 index futures contract along with other stock index futures conacts, and Figure 5.1 shows its specifcations. The notional value, or size, of the contract is the dollar value of the assets underlying one contract. In this case it is by desnition $250 x 1300 = $325,000.10 The S&P 500 is an example of a cash-settled conact: Instead of settlin by act'ual delivery of the underlying stocks, the contract calls for a cash payment that equals the proft or loss as (/' the contract were settled by delively of the underlying asset. On the expiration day, the S&P 500 fumres contract is marked-to-market against the actual cash index. This final settlement against the cash index guarantees that the futures price equals the index value at contract expiration.

loBecause the S&P 500 index is a fabricated number a value-weighted average of individual stock prices the S&P 500 index is treated as a pure number rather than a price and the contract is dehned at maturity to have a size of $250x S&P 500 index.

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AND FUTURES

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2000 units of tlle index wlkich we could now sell for only $2000 x 1099. Thus, we lose Suppose tlzat over the flrst week, the futures price (1099 1100) x $2000 drops 72.01 points to 1027.99, a decline of about 6.5%. On a mark-to-market basis, we have lojt

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Specificatons for the S&P 500 index futures

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lt is easy to see why the S&P 500 is cash-settled. A physical settlement process would call for delivery of 500 shares' (or some large subset thereog in the precise percentage they make up the S&P 500 index. This basket of stocks would be expensive to buy and sell. Cash settlement is an inexpensive alterna:ve.

Margins and Marlcing to Market Let's explore the logistics of holding a fumres position. Suppose the f'uttlres price is 1100 and you wish to acquire a $2.2million position irl the S&P 500 index. The notional value of one contract is $250 x 1100 = $275,000;this represents the amount you are agreeing to pay at expiration perfutures conact. To go long $2.2million of the index, you would million = 8 long futures contracts. The notional value enter into $2.2 m1'11ion/$0.275 of 8 contracts is 8 x $250 x 1100 Q $2 000 x 1100 = $2.2 million. A broker executes your buy order. For every buyer there is a seller, wltich means that one or more westors must be found who simultaneously agre to sell fomard the total number of open positions (buy/sell pairs) same nupber of units of the index. is called the open interest of the conact. Botll buyers and sellers are required to post a performance bond with the broker to j j ensure that they can cover a specified loss on the position. This deposit, which can enrn interest, is called margin and is intended to protect the counterparty against your failure to meet yotlr obligations. The margin is a pelformance bond, not a prmlum. Hence, futures contracts are costless (not counting, of course, commissions qnd the bid-ask 'rhe

spread).

To understand tlze role of margin, suppose that there is 10% margin and weelly

sttlement (in practice, settlement is daily). 'Fhe margin on futures conyacts with a notional value of $2.2 million is $220,000. tf the S&P 500 f'utures price drops by 1, to 1099, we lose $2000 on otr futures position. The reason is that 8 long conacts obligate us to pay $2000 x 1100 to buy

.

-$2000.

=

-

$2000 x

-$144,020

=

v'

We have a choice of either paying this loss directly', or allowing it to be taken out of te margin alance. It doesn't matter which we do since we can recover the unused margin balance plus interest at any time by selling our position. lf the loss is subeacted from the margin balance, we have enlmed one week's interest and have lost $144,020.Thus, if the continuously compounded interest rate is 6%, otlr margin balance after one week is

$220 000c0.06x1/52 $144 ()%

$76 ;g3.99

=

-

,

,

,

Because we have a 10% margin, a 6.5% decline in the fumres price results in a 65% decline in margin. Were we to close out our posiion by ntering into 8 short index futures contracts, we would receive the remaining margin balance of $76,233.99. The decline in the marginbalance means tlle brokeras signcantly less protection shoull we default. For this reason, pnrficipants are required to maintain the margin at a minimum level, called the mnintenance margin. Tltis is often set at 70% to 80% of te initid margin level. In this example, where the margin balance declines 65%, we would have yopost addiional margin. The broker would malte amargin call, requesting additional margin. If we failed to post additionl mrgin, tlze broker would close the P osition by selling 1000units of the index, and reitlrn to us the remaining margin. In practice, marldng-to-markt and seftling up are performed at least daily. Since margin you post is the broker's protecuon againsi your default, @.major determlnant of margin levels is the volaiility of the undrlying aqset. 'I'he mlnlmum margin on ihe S&P 500 conact has generally been less ttia!l t e 10% F assume in this example. In Aurust 2004. for example. the minimum marcin on tlle X;P 500 futtlres . of the conact. contractwas about 6% of the noonal value 'fo illus eate the egect of priodic setttemet Tale 5.8 reports hypotheticalfutures plice moves and acks the margin posiuon over a period of 10 weeks, assuming weeldy marklng-tp-market and a continuously compounded rlsk-free rate of 6%. As te party agreeing to buy at a dxedprice, we make money when ihe ptice goes up and lose when rfhe opposite would occur for the seller. the price goes down. The lo-weekproft on the position is obtained by subtracting f'romthe final margin theposion balancetllefuturevalue of the originalmargin hwestment. Week-loprofto 5.8 therefore Table is in .

*.''''

.

.K.

.

$44990.57

-

'

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..

.

,

l l'l'he exchange's clearinghouse deterntines a minimum marocrin,but individual brokers can and do demand higher marocrins from individual customers. The reason is that the broker is liable to the clearing corporation for customer failure to pay.

-72.01

$220,000c0.06x10/52 =-

.

.

sjep562.6:

What if the position had been a fonvard rather than a futures position, but with prices the same? In that case, after 10 weeks our profit would have been

(1011.65

-

1100) x

$2000

-$176,700

=

i

k. FINANCfAL

146

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wee jrom jong gosjtkon 8 5&P contracts. The last column does not include additional margin payment. The final row represents expiration of the contract.

C) Vtkiitif i

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FORWARDS

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35.35 -24.45

1090.32

41.54

1106.94

16.62

1024.74 1007.30

96,102.01 166,912.96 118,205.66

9.89

201,422.13 234,894.67 243,245.86

4.04 -1.

7.44

36,248.72

4.35

44,990.57

-

....

147

Conversely, suppose that the interest rate declined as the futures price rose. Then margin balance on a long position grew, the proceeds would be invested at a lower the as Similarly, as the balance declined and required additional hnancing, tls nancing rate. would occur at a higher rate. Here a long futtlres contract would pn average perfonu worse than a long forward contract. Th is comp alison of the folavard and futures payoffs suggests that when the interest is rate pojitively correlated with the futures price, t futures price will exceed the price on an oth.erwise identical folavard contract: The investor who is long futtlresbuys at a higher price to offset the advantage of marking-to-market. Similady, when the interest rate is ngatively correlated with the forward price, the futures price will be less than an otherwise identical fonvard price: The investor who is long f'utures buys at a lower price to offset the disadvantage of marling-to-market. As an empirical matter, forward and futures plices are very similaniz The theortical difference arises frm uncertainty aboutthe intereston mark-to-inarketproceeds. For short-lived conacts, the effect is generlly small. However, for long-lived contracts, the difference can be significant, especially for long-lived interest rate f'umres, for which there is sure to be a c rrelation between the interest rate and the plie of the underlying asset. For the rest of this chapter we will ignore the difference between fomards and

Arbitrage in Practice: S&P 500 Index Arbitrage

Why do the lmres and forward prosts differ? The reason is that with th futures proceeds. Given the prices in Table contrct, interest is eained on the mark-to-market prices on average re below the forwards because 5.8, e loss ij larger for futures t.11% With they ftlnd losses as a fonvrd, by ntrast, losses occur. iriitiz prie and w hv to settlement tugnifies t.11: daily the interest on are not fnded until expiration. Enrning been cosistent gains on there Had contzact. gain or loss compared t!) tat on a fomard exceeded have would the forward prost. prot t e Iiosition in this exanple, te fumres and ultimate futures ontract payoff forward to a lfndix 5.B demonseates that the of number conacts f'utures adjusting the exampleby so s to undo can be equated in this te rriagnlf'y'ing effect of imerest. .

k.

futures.

71,046.69

-86.24

cox-raAc'rs

'

.

.

Comparing Futures and Forward Prices A.nimplication of Ajpendix 5.B is that if the interest rate were not random, t en forward randomly? andfutums prices wouldbe the same. However, whatif theinterestrate varies when the unexpectedly increases interest the rate Suppose, for example, tat on average margin the correlated. balarce positively the t'wo are f'uttlres plice increases; i.e., higher. the interest rate price) f'utures was increased just as would grow (due to an margin balance would shrink as te interest rate was lower. . On average in this case, a long ftlttlres position would outpelorm a long forward coneact.

The S&P500 f'uturescontractprovides a contextfor illustrating practical issues that arise when we try to apply the theoretical pricing formplas to determine the fair price of a futures contract. 1.rlorder to compute the theoretical fonvard price using equatiop (5.7), we need to determine tllree things: (1)the value of the cash index (&), (2) the value of dividends to be paid on the lndex over the life of the contract (J), and (3) the interest We can use readily available information to see whether the observed futtlres price

is close' to that given by equation (5.7). On August 30, 2004, the closing S&P 500 index vale was 1099.15 and th December futures price was 1099.30. The annualized dividend yield on the index was approximately 1.75%, which we will assume is expected to be constant over titne. The December contract expirs on December 17, hence there 0.2986) until expiration. were 109 days (F What interest rate is appropriate? Two interest rates that we can easily observe are which the yield on U.S. Treasury bills and the London Interbnnk Offer Rate ILIBORI, institmions. Ninety-day LDOR can be inferred is a borrowing rate for large snancial from Eurodollar f'utzlres,wlzich we will discuss in Section 5.7. Te yield to maturity on a Treasury bill matuling in December was 1.56%, while implied 90-day LDOR was '

=

rfhen

'rhe

l2See French ( 1983) for a comparison of fonvard and futures prices on a variety of underlying assets.

% FI NANCIAL

148

Fu-ruREs

AN D FUTURES

FORwARDS

1.86%from September to December. Using the T-bill rate,l3 the teoretical is '

Sze The theoreticl

(r-J)T

=

156-0.0175) 1099.15c (0.0

x 109/365

=

futtlres price

Sze

jtjpp j.5,(0.0186-0.0175) x

=

.

109/365

=

1098.53

QuantoIndex

jtlpg 5 j

. . ..

.

As illuslated by quations (5.10) and * There are transaction costs of arbiage. (5.11), tansaction costs create no-arbitrage regions, ratherthan no-arbitrageprices.

In practice, a representative bid-ask spread on the index f'utures contact rnight be 20 to 30 basis points (abasis point on the S&P futtlres conact is 0.01) and 0.25% to 0.5% on the stocks in the index when traded in signilicant quptities.

*

Different interest rates will ieflect diferences

in issuer credit

'nsk.

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stocks, there is also execution risk-the possibility that prices move during the timebetween the brder being placed and the stoc k betng acmally prchased.

example uses yields as if they were quoted as continuously compounded, which tlley are not. l3-l-his However, forshortperiods and low interestrates, there is almostno dfferencebetween effective annt!al and continuously compounded rates. For example, if the effective annual rate is 2%, te continuously 0.0198, or 1.98%. ln practice, interest rates are quote us n compounded equivalent.is ln(l E vriety of arane conventions for annualizing the rate.

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Listing for the Nikkei 225 futures contract frtlrl til(, bjllII itrt?: t lournal, July21 2004.

With margin

rf'he

'

!

.

requirements, daily setlling up, and clearinghouse guarantes, ihe credit risk of LIOR. ln a futures contract is not the snme as that of either'rfreasury bills or pricesk pract ice, LIBOR is frequently used to compute fonvard

notie entiresoo-stock * Because of transaction costs, an arbiageurwill usuallybuy it.l4 f'umres contract and the offsetting position index, but instead a subset of in stocks may thus not move exactly together. When buying a large number of

.

'

.

' :

Contracts

the Chicago Mercantile Exchangeds Nikkei 225 f'utures contract-see a newspaper quottion in Figure 5.2 and the details summarized in Figtlre 5.3-is a stock index coneact like the S&P 500 conact. However, there is one vel'y important te currency difference: Setllement of the contractis in adifferentcurrency (dollars)t.14:.:1 of denomination for te index (yen).15 To see why this is important, consider a dollar-based investor wishing to invest in the Nilkei 225 cash index. Tis investor must undertak two transactibj: changing dollars to yen and using yen to by the index. When te posidon is dld, the investor reverses these transactions, selling the index and converting yen bak to dollars. There lisk in this transaction: the risk of the index, denoninated in yen, and are two sources of the risk that the yen/dollar exchange rate will change. From Figure 5.3, the Nikkei 225

Future dividends on the s&P 300 stocks are uncefiip. ' Fof plicing a 3-month . . . ftltures contract, one could use equation (5.6),wit.h actal recent ash dividends on underlying stocks for Dti as proxies for fortcbminj dividends.Ttir is still the lisk change over the next 3 ronths. iik is for Th will that dividends jrater a .. . .. longer-dated futures contracts. :

LIFETIME(pEl1 0?Eh1 HIGH L0W SFITLEIH6 HICH t0W ljT

Nikli 225 Stock Average (rzlE)-$5x index 55 177: g. 9719, 5:5 sejt l:3lg. z:37. ,1 7?g. !13 E!t vnI2,597;v()l lnn2 6637tlpen lnt 31,39).,-1.M jnom Ijj suu-g; u ztuw; tjas: suxay,

3:295

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Nikkei 225 index futures contract.

,112..:1r'

lll),.E,L. ,(:j!l. ::,11; dt::)p d::::. C!i!;: ::;!:. 'Ik-::. 'Ip-::. ,!:) (IL!;::. iL. ::;2!:. e (:hd;:. n d::ilE e ql:-.lL. :!i,.-. n :liir;' (irt''t;l' e iL. .

.1r

Where traded

Size Months Tradimg ends

Settlement

Chicago Mercantile Exchange

$5 x Nikkei 225 lndex

Mar, Jun, Sep, Dec . Business day prior to determination of settlement price Cash-settled, based upon opening Osaka quotation of the Nikkei 225 index on the second Friday of expiration month

.02)

=

lzlAnother way to trade te cmsh index is with the use of Standard and Poor's Depository Recepts These are unt investment trusts that are backed by a portfolio intended to mimic the S&P 500. lnvestors can convert units of 50,000 SPDR shares into the actual stock and can convert stock into SPDRS. This keeps SPDRS close to the S&P 500 index, but in practice SPDRS may be mispriced relative to the cash S&P 500just as futures are.

149

srstglance

At

.

There are Jwt? theoretical ptices, depending upon which interest rate we use; te act'ual December futtlres plice, 1099.30, is between these two prices. Does this mean that there is an arbitrage opportunity? There are a number of considerations in interepreting these differences in prices:

.

%

Arbitragetlrs will need to take into account these considerations. Ultimately, the only way to know if arbiage is prostable is to assess specisc prices, ading costs, and bolwwing and lending rates.

priee using LIBOR instead of the T-bill rate is (r-J)T

CONTRACTS

(SPDRS).

lserhere is also ayen-denominated Nikkei 225 futtzres contract that trades at the Osakaexchange. Since it is purely yen-denominated, this contract is priced according to equation (5.7).

.

i..

150

k

Fl NANCIAL

FORwARDS

UsEs

AN D FUTURES

op

IhloEx

%.

Fg-ruqEs

151

.',.'''

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4111k,.,),.,.* ;'(' ;'E' $' y' (' qjq!ttyf ;'''r'?' E ' : ; I (IE ji

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)'qt:tyo k' (' y' y'y)o r;l'f ,'))yt))'. j'y'-j-. yy'yy j-' y' )' q )!lr .ty)' j-j;', ltqr.r)l t'tyy'yjyyyo ;. r tt-jji-'-'.'''IiiEt-.-p'-.k-..p..... (. j qt)itr tyy.yy-.y-y, . t.;j!jjjj;j4;r. .::::11 . Iii.LL' 4:::j1: ttyg. dt::: .j lyjyy dr:::,, Ir..,l :lr:. .1rE d:::), .t:7)q:iiiii:(''. iiilIr..l t!!;: .y-. iiIr..,p Ip-::. 1#..,. ....y1( r.-yy.-j....y.y...jy..yj.yj.y. IEEEE..I--IFIr..,r..'r iiIg..lk .Igi;lr..,l e 't::::t':rd::::'.rlq!(2I. !(yIl..,. e tE;.1t:. e 1( !E2I o Ir..-k-/-k-/-' s 'L..ll a a 'r.. a t!!;: S T100 and Fzy $110.

rj,rqro j )k' :

futures contract is denominated in dollars rather than yen. Consequently, the contract insulates investors from currency risk, permitting them to speculate solely on whether the index lises or falls. This lind of contract is called a qttanto. Quantocontracts allow investors in one country to invest in a different countl'y without exchange rate risk.

Nikkei contract provides an interesting variqtion on the The dllar-denominated of construction a futtlres conact. Because of the quanto feature, the pricing fonuulas we have developed do not work for the Nikkei 225 contract. We will discuss quantos and the necessary modilication to plice a quanto f'uttlres coneact in Chapter 22.

5.5 USES OF INDEX FUTURES An index f'utures contract is econorfiically lile borrowing to buy the indey. Why use an index f'utures conact if you can synthesize one? One answer is thai index f'uttlres can actually ading a basket of the permit trading the index at a lower transaction cost t.11a11 stocks that make up the index. lfyou are taking a temporaly position in the index, either for investing or hedging, the transaction cost saving could be signiscant. In this section weprovide t'woexamples of the use of index f'utures: asset allocation and cross-hedging a related portfolio.

Asset Allocation Asset allocation strategies involve switching investments among asset classes, such as stocks, money market instruments, and bonds. Trading the individual securities, such as the stocks in an index, can be expensive. Ourearlierdiscussion of arbitrage demonstfated tatwecan usefomards to create synthetic stocks andbonds. Thepractical implicationis that a portfolio manager can invest in a stock index without holding stocks, commodities without holding physical commodities, and so on.

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General asset allocation We can use forwards and futures to perform even more sophisticatd asset allocation. Suppose we wish to invest ourportfolio inrfreasuly bonds (long-tenu Treasury obligations) instead of stocks. We can accomplish this reallocation with two folward positions: Shorting the fonvard S&P 500 index nd going long te fonvardrltbond. The firstansaction converts ourpor/olio from an indexikestmentto a T-bill investment. The jecond transaction converts the portfolio f'roma'Tlbill investment to a T-bond investmet. This use of futtlres to convert a position from one asset category (stocks) to another (bonds)is called jt futures overlay. Futures overlays can have benehts beyond reducing transction costs. Suppose. invejtment invest managetnet compahy has portfolio managefs Wh sessfully an

they believe t be mispttced. The magers are jttdgd on therelative to the S&P 500 stock index and consistently outperform th indx by 2% per is 2%). Now suppose that new year (in the language of portfolio theoly their clients of the ompany like the performance record, bt want to invest in bonds rather stocks. The invejtpentmanagement company could ;re its stockmanagers andhire t.11a11 bond managers, but )ts..existinl investment manasers are the reason for the compalw's . success. The company can use a futures overlay to continue to invest in stocks, but to provlde a bond ret'urn ipstead of a stock rettll'rl to investors. By investing ip stocks, shorting index futves, and going long bond ftllres, the managers continue to invest in t stocks, but the client receives a bond rettlrn plus 2% rather t.1:a1: a stock return p us 2%. This use of f'uttlres to tfansfonn an outpedbrming portfolio on one asset class into an outperforming portfolio on a different asset class is called alpha-porting.

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swltthing from stocks to T-bills As an example of asset allocation, suppose that we have an investment in the S&P 500 index and we wish to tempornrily invest in Tbills instead of te index. Instead of selling all 500 stocks and investing in T-bills, we can simply keep our stock portfolio and take a short forward position in the S&P 500 index. This converts our cash investment in the index into a cash-and-cany creating a synthetic T-bill. When we wish to revert to investing in stocks, we simply offset the fonvard position. To illustrate this, suppose that the current index price, %, is $100,and the effective l-year lisk-free rate is 10%. The forward price is terefore $110. Supppse tllk4 in 1 yar, th index price could be eithr $80 or $130. 1.fwe sell the index and invest in T-bills, we will have $110in 1 year. Table 5.9 shows that if, instead of selling, we keep the stock and short the forward contract, we enrn a 10% feturn no matter what happens to the vzue of the stock. ln this example 10% is the rate of return implied by the forward premium. tf there is no arbitrage, this return will be equal to the risk-free rate.

Cross-hedging with lndex Futttres Index futures are often used to hedge portfolios that are not exactly the index. As discussed in Section 4.5, this is called cross-hedging. Suppoje that we have a portfplio that is not cross-hedsins with perfeu correlation the S&P 500, and we wish to shift the portfolio into T-bills. Can we use the S&P 500 fumres contract to do this? The answer depends on the correlation of the portfolio with

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te S&P 500. To the extent the two are not perfectly coaelated, there will be residual risk. Suppose that we own $100million of stocks with a beta relative to the S&P 500 of 1.4. Assume for te moment that the two indexes are perfectly correlated. Pedkct correlation means that there is a perfectly predictable relationship between the two indexes, not necessarily that they move one-for-one. Using the Capital Asset Pricing Model ICAPMI,the return on our portfolio, rp, is related to its beta, pv, by r +

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of the hedgedposition. Theresult, as you would Table 5.10 shows teperformance is that the hedged position earns the risk-free rate, 6%. expect,

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The hedge quantity is denominated in ten'ns of a quantity of fmuies contracts. The second equality follows because Covto, ?'sapl/o'zap the slope oecient when we regress j is te portfoli returri on the S&P 500 rettlrn', t.e., it is the portfolio beta wit mspectto the S&P 500 index. (5.14)is plso the formula we used in copcluding that, with perfect correlation, we should short 509.09 contracts. Notice that the hedge ratio in equation (5.14)depends on the ptio of the market. of alue the portfolio, 1p, to the notional value of the SP 100contract, N. Thus, as the v portfolio chanes value relative to the S&P500 index, it is necessary to change the hedge ratio. This rebalancing is necessary when we calculate hedge ratios using i relationship based on rettlrfls, which at percentage changes. When we add H* futures to the portfolio, the variance of the hedged poitfolio,

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Assume also that the S&P 500 is 1100 with a 0 dividend yield and te effective annual 1166. risk-free rate is 6%. Hence the futures price is 1100 x 1.06 1.fwe wish to alloate from the index into Treasury bills using futures, we need to short some quantity of the S&P 500. There are two steps to calculating the short futures

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Cross-hedging with imperfect correlation The preceding example assumes that the portfolio and the S&P 500 index are perfectly correlated. In practice, con-elations between two portfolios can be substantially less than ne. Using the S&P 500 to hedge such a portfolio would introduce basis risk, creating a hedge with residual lisk.16 Denote te return and invested dollars on the portfolio as rp and Ip. Assume that short S futures contracts, each with a notional. value N. The futures position we earns the risk premium, rstt:p r. Thus, the return on the hedged position is

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iit the s&P 500 futures. We obtain te variance-minimizing position in te s&P 500 by lli regression (fr om June 1999 to June 2004) of the daily till using equation (5.14). A s-year tt( NASDAQ return on the S&P 500 return gives i):r.( /2 0.7188 1l) 1.xAso 0001 + 1 4784 x (?-sap ?-) (g(pto g iE. 'I -

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An asset manager who picks stocks is often Rlsk management for stock-pickers maling a bet abot the relative, but not the absolute, performance of stock. For on a riskexample, XYZ might be expected to outpedbrm a broad range of stocks . . . ln value will decline XYZ adjusted basis. lf the economy suffers a recession, however, help used islate in this case to even if it outperforms other stocks. Index f'umres can be the relative performance of XYZ. Suppose the return of XYZ i given by the CAPM: .

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5.6

CONTRACTS

Currency fumres and forwards are widely used to hedge against changes in exchange rates. The pricing of currency contracts is a straightfomard application of the principles we have already discussed. Newspaper listings for exchange-traded currency contracts are shown in Figum 5.4. Many corporations use currency f'umres and forwards for short-term hedging. An importer of consumer electronics, for example, may have an obligation to pay the manufacmrer 7150 million 90 days in the f'uttlre. The dollar revenues from selling these '

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products are likely kown in the short run, so the importer bears pure exchange risk due to te payable being fixed in yen. By buying 7150 rnillion forward 90 days, tlle importer locks in a dollar price to pay for te yen, which will then be delivered to the manufacturer.

Currency Prepaid Forward Suppose tat 1 yer ior today you

want to have Y1. A prepaid forward allows you to pay dollars today t acquire Y1 in 1 year. What is the prepaid forward price? Suppose th yep-denfninated interest rate is ry and the exchange rate today ($/Y) is ab. We can work backward. tf we want Y1 in 1 year, we must have e-ry in yen today. To obtain tat many yen today, we must exhange xze-ry dollars into yen. Thus, the prepaid forward price for a yen is '

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underlying on stock. By deferring delivery of the fc same as thatfor a #?'c#t7i#/tp?-u?J?W j. . lmmediately, msset,youlose income. In ihecase of currency, if you received the currency you could buy a bond denominated in that currency and enrn interest. The prepaid .

l7You can, of course, also compute the correlation coecient

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The tenn axsz in this context represents the expected abnormal return on XYZ. If we aecording to equation use the S&P 500 as a proxy for the market, then we can select H will position be that on average, we earn ax'xz + ?'. (5.14). The result for the hedged The lisk of the position will be given by equation (5.15). Since the correlation of n individual stock and the index will not be close to 1, there will be considerable remaining lisk. risk. However, the portfolio will not have market

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can pay today for the yen, and borrow in dollars to do so. To have 1 yp in 1 yar, we need to invest

forward price reQects the loss of interest from deferring delivery, just as the prepaid forward price for stock re:ects the loss of dividend income. This is why equation (5.17) is the same as that for a stock paying a continuous dividend, equation (5.4).

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in dollars, and we obtain this amount by borrowing. 'Fhe required dollar repayment is '

Suppose that the yen-denominated interest rate is 2% and that the ' .).) current exchange rate is 0.009 dollars per yen. Then in order to have 1 yen in 1 year, we L; would invest today q!)Example

which is the forward exchange rate.

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5.6 lq Exaeple iq of buying 1

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Suppose that the yen-denominated interst rate is 2% and the dollarExample 5.5 denominated rate is 6%. The current exchange rate is 0.009 dollars per yen. The l-year fomard rate is =

.

The example shows that boqowing in one currency and lending in another creates If w ffset this bofrowig and lending the same cash flow as a foiward conact. acmat ansactipn lj cllel covered the resulting with fonvald contract, ition an pos

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FUTURES

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Businesses and individuals face uneertainty about f'uture interest rates. A manager may plan to bon'ow money 3 months h'om today but doesn't know today what the interest rate will be at tat time. There are forward and fumres contracts tlaat permit hedging interest rate Iisk by allowing the manager to lock in now a borrowing rate for 3 months in the f'umre. The principles underlying interest rate contracts are exactly those we have been discussing, but interest rates seem more complicated because there are so many of them, are also dependin: upon whetheryou investfor 1 day, 1 month, 1 year, or30 years. 49 implied forward interest rates between any two points in the fumre. Because of this complexi, Chapter 7 is devoted to interest rates. However, the Eurodotlf contract is Eurodollar strip (theset of futtlres prices so important that we discuss it brie:y ere. provides time) basic interestrate informadon that matulities pointin different at with one is commonly uged to plice otherfumres contracts and to price swaps. Figure 5.5 shows a newspaperlisting forthe Eurodollarfutures contract and the companion l-monthlwmolt

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CHAPTER

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1.lzthe ase of a ctlrrency fol-ward, the dividend yield forgone by holding the forward contact instead of the underlying asset, is the interest rate you could enrn by investing ?-y, where ?-y is the in foreign-currency denominated assets. Thus, for currencies, interest foreign rate. Afonkard conact is equivalent to a leveraged position in an asset-borrowing to possible wit.h it is combining the forward the By to create contract oter assets asset. buy synthetic stocks and bonds. These equivalents are summarized in Table 5.12. Since a fomard contact is rislty but requires no investment, it enrns te risk premium. The forward price is therefore a biased predictor of the future spot price of the asset, with the bias equi to tlle lisk premium. The fact that it is possible to create a synthetic forward has two important implications. First, if the fonvard contract is mispriced, arbitrageurs can take offsetting positions in te forward conact and the synthetic forward contract-in efect buying low and selling high-and make a l'isk-f'ree proft. Second, dealers who make markets in the forward or in the underlying asset can hedge the tisk of theirposition with a syntheic offsetting posion. Wit.h transacion costs there is a no-arbiage region rather t.11%a

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20It might occur to you that the Eurodollar contract pays us at the time we borrow, but we do not pay interestuntil the loan matures, 91 days hence. Since we have time to earn interest on the change in l contract per $1 million borrowing. We thevalue of the contract, the hedge ratio should be less t.111111 discussthis complication in Chapter 7.

161

reQects the costs and benets delivery are deferred. The pricing of fonvard conacts delivery. 'I'he seller receives and payment later, so the plice is of this deferred payment owed the seller, and the buyer receives possession later, so the Mgher to re:ect interest prelaid forward conact dividends received by buyer. A reflect the not price ij lower to effects. The hence, it today; price of a prepaid separates these t'wo requires payment forward is

Three-month Eurodollar coneacts have maturities out to 10 years, which means that it is possible to use the contract to lock in a 3-month rate as far as 10 years in the future. The September 2007 futures price in Figure 5.5, for exmnple, is 95.1 1. Apositiop in tlzis coneact can be used to lock in an annualized rate of 4.89% from September 2007 to December 2007. The Eurodollar contract can be used to hedge interest rate risk. For a borrower, for example, a short position in the contract is a hedge since it pays when the interest rate rises and requires payment when the interest rate falls. To see this, suppose that 7 months from today we plan to borrow $1 million for 90 days, and that our borrowing qate is the same as LYOR. The Eurodollar futures price for 7 months from today is 94; this implies a 90-day rate of (100 94) x 90/360 x 1/100 = 1.5%. Now suppose that 7 monts hence, 3-month LIBOR is 8%, which implies a Eurodollar futtlres price of 92. The implied 90-day rate is 2%. Our ex'a borrowing expense over 90 days on $1 million will therefore be (0.02 0.015) x $1m = $5,000. Tlzis extra borrowingexpense is offset by gains on the short Eurodollar contract. The Eurodollar fumres plice has gone down, giving us a gain of $25 per basis point, or $25 x 100 x (94 92) = $5,000. The short position in the futtlres contract compensates cost.20 In the snme way, a long position can be usd us for the increase in our borrowing to lock in a lending rate. The Eurodollar f'utures price is a construct, not the price of an asset. ln this sepse Eurodollar f'utures are different fl'om the futures contracts we have already discussed. Although Eurodollar LDOR is closely related to a numier of other interest rates, there is no one specific identisable asset that underlies the Eurodollar futures coneact. dollars, and comparable rates are quoted LIBOR is quoted in currencies other t.1::.r1 in different locations. In addition to LDOR, there are PDOR (Paris), TDOR (Tolyo), and Euribor ttheEuropean Banking Federation). Finally, you might be wondering why we are discussing LIBOR rather than rates on Treasury bills. Business and bank borrowing rates move more in tandem witlztaliltl)R than with the government ' s bon'ow ing rate. Thus, these borrowers use the futures coneact to hedge. LDOR is also a better measure of the cost of funds ff a markt-maker, so L1BOR.is typially used to price folavard contracts. We will f/ther discuss Eurodollar f'utures in Chapter 7.

The purchase of a stock or other asset entails agreeing to a price, making payment, and taking delivery of the asset. A folavard contract sxesthe price today, but payment and

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% FINANCIAL

FORwARDS

PRo BLEMS

AND FUTURES

contracmal differences between forwards and futures can lead to pricing differences, but in most cases forward prices and futures plices are vel'y close. In addition to hedging, forward and f'utures contracts can be used to synthetically switch a portfolio invested in stocks into bonds. A portfolio invested in Asset A can remain invested in Asset A but enrn the returns associated with Asset B, as long as there are fonvard or fumres contracts on A and B. This is called a f'umres overlay. The Eurodollar fumres contract, based on LDOR (London Interbarlk Offer Rate) is widely used for hedging interest rate risk. Because the Eurodollar f'utures conact does not represent the price of an asset (at settlement it is 100 L1BORI,it cnnnot be xpricedusing the formulas in this chapter.

%.

163

5.2. A $50 stock pays a $1 dividend every 3 months. with the Erst dividend coming 3 months from today. The ontinuously compounded risk-free rate is 6%. a. What is the price of a prepaid forward contract that expires 1 year from today, immediately after the fourth-quarter dividend? b. What is the price of a forward coneact that expires at the same time? 5 3 A $50 stock pays an 8% continuous dividnd. risk-free rate is 6%. *

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-

a. What is the price of a prepaid forward coneact that expires 1 year from today? .

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b. What is the price of a fonvard contract that eypirs at the same time?

READING

Chapter 6 continues our exploration of forward markets by eonsidering commodity forwards, which are diferent from linancial forwards in important ways. Chapter 7 then exnmines interest rate forwards. Whereas forward contracts provide a price for delively at one point in time, swaps, discussed in Chapter 8, provide a price for a series of deliveries over time. Swaps are a natural generatization of forward contracts.

'I'he pricing principles discussed in this chapter will also play important roles when option plicing in Chapters 10, 11, and 12 and financial engineerine in Chap-. discuss we 15. ter . To get a sense of the range of traded coneacts, look at the fumres page of the Ftz/f and also explore te-Web sites of f'umres exchanges: te ClzicagoBoard Street Trade of (wwmcbot-com), the Chicago Mercantile Exchange (wwmcme.com),the York Mercantile Exchange (wwwwnymex.com),and tlle London International New Financial Futures Exchange (wwwoliffe.com),among others. These sites typically provide current prices, along wit information about the contracts: What the underlying asset is, how the conacts are settled, and so forth. The site for One Chicago (www. onechicago-com) provides infonnation about single stock f'uturesin the United States. It is well accepted that forward prices are determined by the models and considerations i tlzis chapter. Siegel an (j Siegel (1990)is a standard reference book on fumres. Early papers that examined fumres pricing include Modest and Sundaresan (1983),Cornell and French (1983),which emphasized tmx effects in futtlres pricing (seeAppendix 5.A), and French (1983),which compares forwards and f'utures when both exist on the same underlying asset. Brennan and Schwartz (1990)explore optnal arbitrage when there are transaction costs and Reinganum (1986) explores the arbitrage pssibilities inherent in time t'ravel. There is a more technical academic literattlfe focusing on the difference between fonvard and futtlres contracts, including Blak (1976a),Cox et al. (198 1). lichard and Sundaresan (1981),and Janrw and Oldlield (1981). W

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PROBLEMS 5.1. Construct Table 5.1 from the perspective of a seller, providing a descriptive name for each of the transactions.

5.4. Suppose the stock price is $35 and the continuously compounded interest rate is 5%. a. What is the Fmonth forward price, assuming dividends are zero? b. If the 6-month forward price is $35.50, what is the armualized forward premium? c. Ifthe forward price is $35.50,what is the annualized yield?

continuous

dividend

5.5. Suppose you are a market-maker in S&R index forward contracts. The S&R index spot price is 1100, the risk-free rate is 5%, and the dividend yield on the index is 0. a. Wh at is th no-arbitrage folavard price for delively in 9 rponths? b. Suppose a customer wishes to enter a short index f'utures position. J.f you take the opposite position, demonstrate how you would hedge your resulting long position using the index and borrowing or lending.

;

c. Suppose a customer wishes to enter a long index f'uttlres position. If you tale theopposite position, demonstrate how you would hedge your resulting short position using the index and borrowing r lending. 5.6. Repeat the previous problem, assuming tat te dividend yield is 1.5%. 5.7. The S&R index spot price is 1100, the risk-free rate is 5%, and the dividend yield pn the index is 0. a. Suppose you observe a 6-month forward price of 1135. What arbitrage would you undertake? b. Suppose you observe a 6-month forward price of 1115. What arbitrage would you undertake? 5.8. The S&R index spot price is 1100, the risk-free rate is 5%, and the continuous dividend yield on the index is 2%.

164

%.FINANCIAI-

FORWARDS

P Ro

AND FUTURES

no-arbitrage

B LEMS

k:

165

bound is given

a. Suppose you observe a 6-month forward price of 1120. What arbitrage would you undertake?

5.14. Vel'ify that when there are transaction costs, the lower by equation (5.11).

b. Suppose you observe a 6-mont.hforward price of 1110. What arbitrage would you undertake?

5.15. Suppose the S&R index is 800, and that the dividend yield is 0. You are an arbitrageur with a continuously compounded brrowing rate of 5.5% and a continuously compounded lending rate of 5%.

5.9. Suppose that 10 years from now it becomes possible for money managers to could travel engage in time travel. ln particular, suppose that a money manager 12.5%. Treasul'y the bill 1981, l-year rate Janual'y when was to a. If time travel were costless, whatriskless arbitrage strategycould amoney managerundertake by avelingback and forth between Janual'y 1981 and January 19822

b. lfmany money managrs undertook this strategy, what would you expect to happen to interest rates in 1981? c. Since interest rates wcrc 12.5% in January 1981, what can you coclude about wheter costless time travel will ever be possible? 5.10. The S&R index spotprice is 1100 and the continuously compounded risk-free is 5%. You observe a g-month forward price of 1129.257.

rate

a. What dividend yield is implied by this forward price? b. Suppose you believe te dividend yield over the next 9 months will be only 0.5%. What arbitrage would you undertake? youbelieveihe dividend yield will be 3% overthenextg c. Suppose would you undertake? arbitrage What

months.

5.11. Suppose the S&P 500 index f'utures price is currently 1200. You wish to purchase four futures contracts on margin. a. What is the notional value of your position? b. Assuming a 10% initial margin, what is the value of the initial margin? 5.12. Suppose the S&P 500 index is cuaently 950 and tlle initial margin is 10%. You wish to enter into 10 S&P 500 futures contracts. a. What is the notional value of your position? What is the margin? b. Suppose you enrn a continuously compounded rate of 6% on your margin balancea your position is markd to market weekly, and the maintenance margin is 80% of te initial margin. What is tlae greatest S&P 500 index f'uturesprice 1 week fwm today at wlzich will you receive a margin call? 5.13. Verify tat going long a forward conact and lending the present forward price creates a payoff of one share of stock when a. The stock pays no dividends. b. The stock pays discrete dividends. c. The stock pays continuous dividends.

value of

the

a. Supposing that there are no transaetion fees, show that a cash-gnd-carry arbitrage is not proftable if the forward plice is less tan 845.23. and that a reverse cash-and-carry arbitrage is not protable if the forward price is greater than 841.02. b. Now suppose that there is a $1 transaction fee, paid at time 0, for going either long or short the forward contract. Show that the upper and lower no-arbitrage bounds now become 846.29 and 839.97. c. Now suppose that in addition to the fee for the fonvard contract, there is also a $2.40fee for buying or selling the index. Suppose the contract is settled by delivery of the index, so that this fee is paid only at time 0.

What are the new upper and lower no-arbitrage bounds? d. Make the same assumptions as in the previous part, except assume that the coneact is cash-settled. This means that it is necessary to pay the stock index transaction fee (butnot the fonvard fee) at both times 0 and 1. What are the new no-arbitrage bounds?

e. Now suppose that transactions in the index have afee of 0.3% of the value of the index (thisis for both purchases and sales). Transactions in the forward conact still have a fixed fee of $1 per unit of th inex at time 0. Suppose the contract is cash-settled so that when you d a cash-andcan'y or reverse cash-and-carry you pay the index ansaction fe both at time 1 and time 0. What are the new upper and lower no-arbitrage boundj? Compare your answer to that in the previous pal't. Hint: To handle the time 1 transaction fee, you may want to consider tailing the stock position.) 5.16. Suppose the S&P500 currently has a level of 875. The continuously compounded return on a l-year T-bill is 4.75%. You wish to hedge an $800,000portfolio that has a beta of 1.1 and a correlation of 1.0 with the S&P 500. a. What is the l-year futures price for the S&P 500 assuming no dividends? b. How many S&P 500 f'utures contracts should you short to hedge your portfolio? What ret'ul'n do you expect on the hedged prtfolio? 5.17. Suppose you are selecting a fumres contract with which to hedge a portfolio. You have a choice of six contracts, each of which has the same variability, but with 0, 0.25, and 0.85. Rankthe f'umres contracts correlations of highest basis risk, from to lowest basis risk. with respect to -0.95,

-0.75,

-0.50,

166

k. FINANCIAL

FORWARDS

5.18. Suppose the current exchange rate between Germany and Japan is 0.026/Y. 'I'he euro-denominated annual continuously compounded risk-free rate is 4% and the lisk-free rate is 1%. What yen-denorninated annual continuously compounded yerl/euro price? forward arld euro/yen 6-month te are 5.19. Suppose the spot $/7 exchangerate is 0.008, the l-yearcontinuopsly compounded dollar-denominated rate is 5% and the l-yeai continuously compounded yendenominated rate is 1%. Suppose te l-year fonvard exchange rate is 0.0084. Explain precisely the eansactionsyou could use (beingcareful about currency of denomination) to make money wit.h zero initial investment and no risk. How much do you make per yen? Repeat for a fonvard exchange rate of 0.0083.

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APPENDIX

5.B;

APPENDIX

AN D FUTU RES

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Appendix available online at www.aw-bc.com/mcdonald.

APPENDIX

FORWARDS

5.B: EQUATING

AND FUTURES

Because the futures price exceeds te prepaid forward price, marling-to-market has the effect of magnifying gains and losses. For example, the futtlres price on a nondividendpaying stock is Fa,w = Sterl'. J.f the stock price increases by $1 at time 0, the gain on crW. Thus, in order to use futures to precisely hedge a the futures contract at time F is F) it is necessm'y to hold fewer fumres time settled being at position (withthe hedge volatility induced by the future the offsetting ffectively extra forward contracts, than 5.13, fewer than eight contraets, to Table long in the example In we can go value factor. marldng-to-market. of effect for male up the Table 5.13 shows the effect of this adjustment to the f'uttlres position and how it is adjusted over time. lnitially, we go long

8 x e -0.06x9/52 y qjess =

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Sometimes it makes sense for a commodity to be stored, at least temporarily. Storage is also called carry, and a commodity that is stored is said to be in a carfy market. One reason for storage is seasonal variation itl either supply or demand, which causes a mismatch between the time at which a commodity is produced and the time at which it is consumed. With some agricultural products, for example, supply is seasonal (there is a harvest season) ut demand is constant over the year. ln this case, storage perrnits consumption to occur throughout te year. With nattlral gas, by contrast, there is high demand in the winter and low demand in the summer, but relatively constant production over the year. This pattern of use and production suggests that tere will be times when natural gas is stored.

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The implied no-arbitrage

/.

181

In some markets, consistent and reliable quotes for the spot price are not available, or are not comparable to forward prices. ln such cases, the near-term forward price can be used as a proxy for the spot price, S. upward-sloping fonvard curve--occurs when the By desnition, contango-an risk-free downward-sloping forward rate. Backwardation-a lease rate is less than the culwe-occurs when the lease rate exceeds the risk-free rate.

qi r' EL:r?STr:rq?TI)Y 1* jjtf y''yy. yyjf j'l'y' ))-*-* jryf y' ;y' yyf j.'yj)'. (5* ;'jjf (' (t' j'. p' rjjjjyjjjjj,jjjj.d j'jyf )' r:: (. r (;((iq! ttlf 't'jjltyd y' . ;(!(.q( tjt'ytjtj. .gjjjy' . . tjjjyytjjjyjyjj(.'. I g..j.).j.j(j ;.. j j ttr )yy. . ( ryl'. tlqj y k j y y . r y yj j: . . .j casla-and-carry ''ltrx, arbitrage witla a commodity for whcb the Iease . . . . k'ti-:LL...--j,.,.....-,.... t -//1L.' :s Szer-<T restricton is jjtE:

k

Storage Costs and Forward Prices Storage is not always feasible (for exnmple, f'resh strawberries are perishable) and when technicily feasible, storage is almost always costly. When storage is feasible, how do storage costs affect forward pricing? Put yourself in the position of a commodity merchant who owns one nit of the commodity and ask whether you would b Willing to store this unit until time T. You face the choice of selling it today, receivin %, or selling it at time F. lf you elect to sell at time F, you can sell forward (toguarantee the price you will receive), and you will receive Fo,w. This is a cash-ud-cany The cash-and-carry logic with storage costs suggests thatyyll vill store (7?11, ifthe present vc/l/c ofselling at l??7d F is at least as great a l/7t# ofselling today. Denote te f'uttlrevalue of storage costs for one unit of the commodity from time 0 yo F as (0, F). lndifference between selling today and at time F requires

;

so

=

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gov

Revenue from selling today

This relationship

-

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(0, r)j

frm selling at time F

in turn implies that if storage is to occur, the forward price is at least F 0,r ZL s0 erT +

(6.13)

(0 r) ,

the special case where storage costs are paid continuously and are proportional to te 1.11 value of the commodity, storage cost is like a continuous negative dividend of and we can write the forward price as ,

y 0,T'

=

s0 e(r+)r

(6 j4l '

%.CouMoolTy

182

FORwARDS

AND FUTURES

CARRY MARKETS

0), equations (6.13) and (6.14) reduce to our When there are no storage costs ( fnmiliar fomard pricing formula from Chapter 5. When there are storage costs, the folavard plice is higher. Why? The selling price cost of storage (interest) must compensate the commodity merchantforboth the snancial the forward curve can rise faster and the physical cost of storage. With storage costs, dividend negative in that, instead than the interest rate. We can view storage costs as a hold the receiving cash flow for holding the asset, you have to pay to asset. of

-et'ur'n

Example 6.1 supposethattheNovemberprice of cornis $2.50/bushel.the efective 1.'j! l'1mont hl y imerestrateis 1%, and storage costs perbushel are $0.t5/zponth.Assuming that lij lj) is stored Npvepber to Fbwary, the February forward price must compensate

itlf

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col'n

om

qjl )jt owners for interest and storage. The E

jCi 'l

/;i' Ejl! lE

$0 05 + ($0.05x 1.01) + ($0.05x 1.012)

:

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:

sturevalue

:

=

.

= '''

k t) Thus, the Februaly forward price will be

;j-@j1 jtEt l ji)

2.50. x.

of storage costs is

($0.05/.01)x

$0.1515

(1.01)a + 0.1515

=

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-

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-

=

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l Problem 6.9 asks you to verify tat this is a no-arbitrage price. Keep in mind that just because a commodity can be stored should (or will) be stored. Pencils were not storedbecause storage necessary: Aconstant new supply of pencils was available to meet equation (6.13)desclibes the fonvard price vhen storage lccl/r'. cmmodity is stored are peculiar to each commodity.

183

corn, you can sell the excess. However, if you hold too little and run out of corn, you must stop producing, idling workers and machines. Your physical inventory of corn in this case has value-it provides insurance that you an keep producing in case there is a disrnption in the supply of corn. In this simation, corn holdings provide an extra nonmonetal'y return that is sometimes referred to as the convenienceyield-s You will be willing to store corn with alower if you did not enrn the convenince yield. What are the implications t.11:.r1 rate of of the convenience yield for the forward plice? Suppose tat someone approached you to borrow a commodity from which you derived a convenience yield. You would think as follows: I lend the commodity, 1 bealing interest cost, saving storage cost, and losing the value 1 deri've from having a physical inventoly I was willing to bear the interest cost already; thus, 1 will pay a commodity borrower storage cost less the ctwvdnfcncc yield.'' Suppose the continuously compounded convenience yield is c, proportional to the value of the commodity. The commodity lender saves c by notphysically stoling the commodity; hence, tile commodity borrower pays compensating the lender c fgr convenience yield less storage cost. Using an argument identical to that in Table 6.8, we conclude that the forward price must be no less than

=

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F0,F

k

Storage Costs and the Lease Rate

Fo w

The discussion of commodities to this point has ignored business reasons for holding commodities. For example, suppose you are a food manufacturer for whom corn is an essential input. You will hold an inventory of corn. If you end up holding too much

s0d

(r+-c)F

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=

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-

sa

This expression implies tat te folavard price must be below no cash-and-carry arbitrage.

Etvirtual

The Convenience Yield

=

,

This is the restlicon imposed by a reverse cash-and-cany in which the arbitageur borrows the commodity and goes long the fomard. Now consider what happens if you pelform a cash-and-cany buying te commodity and selling it folavard. If you are an average investor, you will not enrn the convenience yield (itis enrned only by those with a business reason to hold the commodity). Youcould try to lend the commodity, reasoning that the borrower could be a commercial user to whom you would pay storage cost less te convenience yield. But those who earn the convenience yield lilely already hold the optimal amount of the commodity. There ??l7ybe ?lt? wayforyou to /tz,77 the convenience yield w/3Nlpc/ft/?-/?lfng a caslvand-carly. Those Fho do not earn the convenience yield will not own the commodity. rfhus,-/br an tzvcmgp investor, the cash-and-carry has the cash flow/

does not mean that it was not econornically pencil demand. Thus, Whether and when a

Supjose that there is a c?)rry mrket for a commodity, so that iys forward price is given by equation (6.13).What is the leas rate in this case? Again put yourselfin the shoes of the commodity lender. Ifyou lepd the commodity, you are saved f'rom having to pay storage cost. Thus, the lease rate sould equi the negative of the storage cost. In other words, the lender will pay the borrower! In effect, storage'' for the commodity lender, who the commodity borrower is providing point the the The lender maling a payment to back commodity in future. receives at a dividend. negative borrower the generats a

1: Sze (r-5)r

-

e(r+)F

Szer-bksT '

if there is to be

?'

s'rhe term convenience yield is desned differently by different authors. Convenience yield generally means a return to physical ownership of the commodity. ln practice it is sometimes used to mean te lease rate. ln this book, the lese rate of a commodity can be infen'ed from the fonvard price using equation (6.11). 61n this expresson, we assume we tail the holding of te commodity by buying cF units at time 0, and selling off units of the commodity over time to pay storage costs.

.

k

184

% COMMODITY

FORwARDS

ANo

FUTURES

GOLD !;' ;';' TEjjjjj::j;;k;'E '')4/*2r'1:* #' k,f )' j'(' ;':'',' (('E: ;..' : ! jE )y--,;jjjjj'. ('jjjjy:jyyti q' j')' f' jf jkf ' ' ''(((''''('g('' !'. (' )' yry'r')?rll'lf jjjlf )': j' k jj(g g :j(i.j!(yjkkjjjjkl .:.- i! jg..(.jrq .ji!r .(.(.. ' E.. . . .i. - . p'E . i..- -. - ! E.E. E- .,.!....-..........-.-...............i .

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The conveniene yield produces a no-arbieage region rather tan a no-arbiage price. The observed lease rate will depend upon both storage costs and convenience. Also, as in Section 5.3, bid-ask spreads and trading costs will further expand te no-arbitrage region in equation (6.15). As anoter illustration of convenience yield, consider again the pencil example xof Section 6.4. 1.nreality, everyone stores a few pencils in order to be sure to have one available. You can think of this benestfrom storage as the convenience yield of a pencil. However, because the supply of pencils is perfectly elastic, the ptice of pencils is Exed at $0.20. Convenience yield in this case does not affect the forward price, but it does explain the decision to store pencils. The difficu'lty with the convenience yield in practice is tat convenienc is hard to obselwe. The concept of the convenience yield serves two purposes. First, it explains example, why a commercial user might store a commodity ' P atterns in storage-for when the average investor will not. Second, it provides an additionlll parameter to better explain the forward culwe. You might object that we can invoke the convenience yield to explain alt)' forward curve, and therefore the conceptof the convenience yield is vacuous. While convenience yield can be tautological, it is a meaningful economic concept and it would be just as arbitrary to assume that there is never convenience. Moreover, the ' upper bound in equation (6.15)depends on storage costs but not tlle convenience yild. Thus, the eonvenience yield only explains anomiously low forward pricej, and only when there is storage. We will now examine particular commodities to illusate the concepts from the previous sections.

6.7 GOLD FUTURES Gold is durable, relatively inexpensive to store (comparedto its value), widely held, and actively produced through gold mining. Because of transportation costs and purity concerns, gold often trades in certificate form, as a claim to physici gold at a specisc location. There m'e exchange-traded gold futures, specications for which are in Figure

6.1.7

X gold futtlres conact. Figure 6.3 Figure 6.2 is a newspaper listing for the fonvard curvef'umres contracts-the al1 available gold graphs the f'umres prices for (Newspaper listings 2001 2004. for most to for the first Wednesday in June, from expiration Figtlre of available dates, 6.3 the do f'ull set so futures contracts not show

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0 7Gold is usually denominated in troy ounces (480grains), which are approximately 9.7% heavier than themore familiar avoirdupois ounce (437.5grains). Twelve troy ounces make 1 troy pound, which weighsapproximately 0.37 kg.

Fu-ruREs

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20 30 40 Months to Maturity

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Soltrce: Futures data from Datastream-

% COMMODITY

186

Gouo

AND FUTURES

FORwARDS

is constructed using more expiration dates than are in Figure 6.2.) What is interesting about the gold forward curve is how relatively uninteresting it is, wit the fonvard price steadily increasing with time to mattlrity. From our previous discussion, the forward price implies a lease rate. Shortlsales and loans of gold are common in the gold market, and gold bon-owers in fact have to pay the lease rate. On the lending side, large gold holders (includingsome central bnnks) put gold on deposit with brokers, in order that it may be loaned to short-sellers. 'I'he gold lenders earn the lease rate. The lease rate forgold, silver, and other commodities is computed in practice using reporting services. Table 6.9 shows - equation (6.12)and is reported roudnely by fnancial the 6-month and l-year lease rates for the four gold forward culwes depicted in Figure 6.3, computed using equation (6.12);

Ie 6 Z Here are the details of computing the 6-month lease tate for June 2001. Gold futures plices are in Table 6.9. The June and Septembr Eurbdor ftlttlfes IZCCS On this date Were 96.09 and 96.13. Thus, LIBOR h'om Vne s-mont.h September t and 0.988+, from 91/90 96.09)/400 x September (100 was to 0.978%. The June to December interest December was (100 94.56)/400 x 91/90 1.9.763%, or 1.0197362 annualizd. rate was terefore (1.00988)x (1.00978) 1 Using equation (6.12),the annualized &month lease rate is therefore EXamp

*

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=

-

=

-

6-month lease rate

$.0197632 =

(jy.5)

(269/265.7)

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=

Fu-ruREs

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187

Gold Investments ((fyou wish to hold gold as part of an investment portfolio, you can do so by holding physical gold or syntetic gold-i.e., holding T-bills apd going long gold futures. Which should you do? lf you hold physical gold without lending it, and if the leas rate is positive, you forgo the lease rate. You also bear storage costs. With synthetic gold, lisk. on the other hand, you have a counterparty who may fail to pay so there is credit lgnorin.j credit risk, however, synthetic gold is generally te preferable way to obtain gold plice exposure. Table 6.9 shows that the 6-month annualized gold lease rate is 1.46% in June 2001. Thus, by holding physical gold instead of synthetic gold, an investor would lose this 1.46% ret-tllql.s Inlune 2003 and 2004, however, the leaserate was about tf storage costs m'e about 0.10%, an investor wouldbe indifferentbetween holdingphysical and synthetic gold. 'I'he f'uttlres market on those dates was compensating investors for storing physical gold. yield from gold. Some nonfinancial holders of gold will obtain a convenience Consider an eleceonics manufacttlrer who uses gold in producing components. Suppose that running out of gold would halt production. lt would be natural in this case to hold a buffer stock of gold in order to avoid a stock-out of gold, i.e., running out of gold. For this manufacturer, there is a retllrn to holding gold-namely, a lower probability of stocking out and halting production. Stocldng out would have a real financial cost, and lease rate-to avoid that cost. the manufacmrer is willing to pay a price-te -0.10%.

Evaluation of Gold Production Suppose we have an operating gold mine and we wish to compute the present value

of future producon. As discussed in Section 6.2, the present value of the commodity rate--of received in the f'umre is simply the present value--computed at te lisk-free the forward price. We.can use the forward curve for gold to compute te value of an --;-' -' -y' 937:1147'744747'/744* ,j'. ..' i'E.q.:' -' -'. jj:::y:,,f --' C' :')' L' k' ....' Etj'lf dl:::::,,f dl:::::,hf 'y''C;' j'k' k'))))yy'.' ).(;'' ?' (E E EE ' ((E -

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S EASONALITY:

AND FUTURES

FORWARDS

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As discussed in Section 6.6, storage is an ecopornic decision in which tere is a . trade-offbetwee selling today and selling tomorrow. Ifwe can sell cprn today for $2/17t1 and in 2 months for $2.254m,t storage decision entails comparing the price Wecan get Wecan get in 2 months. ln addition to interest, toda# wit te ' prsilt ' value of ihe plice ''' . . .. . anysis. incllde' need to toragecosts in our we An equtllbrium with olpe current selling and spme stopge tquires thgt col'n tjj pricesb expted to rise t e interest rate plus storage costs, Wllichimpjj es tjyat tjj ere upward trend in the prlce between harvets. While corn is beipg stred, the an willbe . . . fonvrd jrice shoutd behave s in equatipn (6.14),rising at interest plus stopge costs. if supply ad demand Onc th harvest begins, storage is no loger necsary; harvest price will be the fiom remain constant year to year, the same vl'y year. The Iising after the harvest. will that level harvest, only begin again plice fatl to at to corn The market conditions we have described are graphed in Figure 6.5, which depicts a hypothetical forward ctu've as seen from time 0. Between harvests, the fprward price '

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how the possibility of mine closings due to 1ow plices affects valuation.) Note that in equation (6.16),by computing the present value of the forward price, we compute the prepaid forward price. 4*p Example 6.3 t'''j Suppose we hay.e a rnining project that will produce 1 ounce of gold )) jj every year for 6 years. The cost of this project is $1,100today, the marginal cost per compoundd interest rate ounce at the time of extraction is $100, and the continuously y't '1 .hitis 6%. #, We observe the gold forward prices in the second column of Table 6.10, with ')g implied prepaid forward prices in the third column. Using equation (6.16),we can use j . t,! these prices to perform the necessary present value calculations.

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THE MARKET

6.8 SEASONALITY: FORWARD

CORN

Corn in the United States is halwested primadly in the fall, from September through November. The United States is a leading corn producer, generally exporting rather than importing corn. Figure 6.4 shows a newspaper listing for corn futures. Given seasonality in production, whpt should the forward curve for corn look like? Corn is produced at one time of the yem but consumed throughout the year. ln order to be consumed when it is not being produced, corn must be stored. Thus, to understand the forward curve for eorn we need to recall our dijcussion of storage and can'y markets.

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k. COMMODITY

190

AND FUTURES

FORWARDS

NATURAL

of corn rises to reward storage, and it falls at each harvest. Let's see how this graph was constructed. The corn price is $2.50initially, the continuously compounded interest rate is 6%, and storage cost is l.s%/month. The folavard price after n months (where?7 < 12) is ' .

(0.005 +0.015)

x?l

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Over.

Farmers will plant in anticipation of receiving the harvest price, which means that it is the harvest price that reqects the cost of producing corn. The price duling the rest of the year equals the hazwest price plus storage. In general we would expect those storing replenish inventory from corn to plan to deplete inventory as hanst approaches and to the new harvest. This is p.simplihed version of reality. Perhaps most importank th: tpply of corn varies from year to year. When there is a large crop, producers will expect corn to be 'stored not just over the current yeaq but into the next year as Well. If there is a large lise continupusly until year 2. To harvest, therefore, we might see the forward curve look of let's possible at real rn pries. behavior understand the corn, bettet Table 6.11 shows the June forward curves for col'n over a lo-yer peliod. Some clear patterns are evident. Fkst, notice that from December to March to May (columns 3-5), the futures price lises every year. We would xpect there to be storage of corn

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during this period, with the futures price compensating for storage. A low current ptice suggests a large supply. Thus, when the near-luly price is low, we might also expect storage across the coming halwest. Particularly in th years with the lowest July prices (1999-.-2002),there is a pronounced rise in price from July to December. When the price is unusually high (1996and 2004), there is a drop in price from July to December. Behavior is mixed in the other years. q We can al>o examine the July-December price relationship in the following yetr. In 6 of the 10 years, the distant-December plice (column 8) is below the distant July plice (column6). The exceptions occur in years These patterns are generally consistent with relatively low current prices (1998-2001). with storage of corn between harvests, and storage across harvests only occasionally. Fi n ally compare prices for the near-luly contract (the lipt column) with those for the distant-December contract (the last column). Near-term prices are quite variable, ranging from 216.75 to 435.00 cents per bushel. In December of the following year, however, prices range only from 239 to 286.50. In fact, in 7 of the 10 years, the price is between 251 and 268. The lower variability of distant plices is not surprising: lt is dificult to forecast a hmwest more than a year into the fumre. Thus, the forward prke is re:ecting the market's expectation of a normal harvest 1 year hence. the forward tfwe assume thatstorage costs ar approximately so.o3/month/bushel, price in Table 6.11 never violates the no-arbitrage condition

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GAS

Natural gas is another market in which seasonality and storage costs are important. The namral gas f'utures contract, introdued in 1990, has become one of the most heavily traded f'utures contracts in the United States. The asset underlying one contract is 1 months worth of gas, delivered at a specifc location (differentgas contracts call for delivery at different locations). Figure 6.6 shows a newspaper listing for natural gas f'umres, and Figure 6.7 details the speciscations for the Henry Hub contract. Nattlral gas has several interesting chapcteristics. First, gas is costly to ansport internationally, so prices and fonvard culwes vary regionally. Second',once a given well has begun production, gas is costly to store. Third, demand for gas in the United States is highly seasonal, with peak demand arising from heating in winter months. Thus, there is a relatively steady stream of production with variable demand, which leads to large and predictable price swings. Whereas corn has seasonal production and relatively constant demand, gas has rlatively constant supply and seasonal demand.

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gltispossible tohave lowcurrentstorage and alargeexpectedharvest, which wouldcause the December price to be lower than the July price, or high current storage and a poor expected harvest, which would cause the July price to be below the December price.

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Underlying

Soltrce: Futures data from Datastream.

,

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Where aded Size Monts Trading ends

'

72 consecutive months Third-to-last business day of month prior to mattlrity month As uniformly as possible over the delivery

Delivel'y

month

Figure 6.8 displays 3-year (2001)and 6-year (2002-2004)strips of gas futures prices for the srstWednesday in June fz'om 1997 to 2000. Seasonality is evident, with high winter prices and low surnmer prices. The 2003 and 2004 strip shows seasonal cycles combined with a downward eend in prices, suggesting that the market considered prices l that year as anomalously high. For tlze other years, te average pzice for each comitlg year is about the same. Gas storage is ostly and demand for gas is highest in the winter. The steady rise of the forward curve during the fall months suggests that storage occurs just before the heaviest demand. Table 6.12 shows prices for October through December. The monthly increase in gas prices over these months ranges from $0.13to $0.23. Assuming that the interest rate is about 0.15% per month and that you use equation (6.13),storage cost in would satisfy November 2004, ,

6.947

=

6.759c0.0015+

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Source.. Futures data from Datasteam.

implying an estimted jtorage cst of $0.178 in November 2004. You will find differentimptedmarginal storagecosts ineach year, butthis is to be expectedif marginal stpredr storage costs kary with the tuantity Because of the expense iri nspol-ting gas internationally, te seasonal behavior of the forward curv can kary in differet parts of the world. In tropical areas where gas is usedforcooking and elecicity generatiop, the forward curve is relatively flatbecause demand is reladvely llat. In te Southern hemisphere, where seasons are reversed f'rom the Northrfl hemisphre, the forward curve will pak in June and July rather than =

December and January.

%.COMMODITY

194

FORwARDS

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Recent developments irl energy markets could alter the behavior of the natural gas forward culwe in the United States. Power producers have made greater use of gas-fred peak-load electricity plants. These plants have increased summer demand for natural gat and may permanently alter seasonality.

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6.10 OIL Both oil and nattlral gas produce energy and are exeacted from wells, but the different physical characteristics and uses of oi1 lead to a very different forward curve than that for gas. Oi1 is easier to transport than gas. Transportation of oi1 takes time, but oil has of a global market. 011 is also easier to store than gag. Thus, seasonals in the price the X light oil contract unimportant. Specications for crude oil are relatively are shown in Figure 6.9. Figure 6.10 shows a newspaper listing for oi1 f'utures. The X fprwcd cul've n four datts is plotted in Figure 6.11.

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size

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Trading ends Delivel'y

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Sottrce: Futures data from Datastream.

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On the four dates in the figure, near-term oil prices range from $25 to $40, while 7-yeaT frward pfice in each case is btween $22 and $30. The lonprun forward the price is less volatile than the short-run forward ptice, which makes economic sense. ln the short rtm, an increase in demand will cause a price increase since supply is xed. A supply shock (suchas production restrictions by the Organization of PetroleumExporting will cause the price to increase. In the long run, however, both supply Countries EOPEQ) dmand time have to adjust to price changes with the result that price movements and attepuated. The forward curve suggests that market participants in June 2004 did not are price remain the at $40/barrel. to expect

6.11 COMMODITY

SPREADS

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Some commodities are inputs in the creation of other commodities, Which gives rise to commodity spreads. Soybeans, for example, can be crushed to produce soybean meal and soybean oil (anda small amount of waste). A eader with a position in soybeans and an opposite position in equivalent quantities of soybean meal and soybean oil has a the crush.'' crush spread and is said to be Similady, cnlde oi1 is resned to make petroleum products, in particular heating oil and gasoline. The refining process ntails distillation, which separates crude oi1 into different components, including gasoline, kerosene, and heating oil. The split of oil into these different components can be complemented by a process known as :trading

Etcrackinf''

%.COMMODITY

196

FORWARDS

H ED G l N G

AND FUTURES

oil hence, the difference in price between crude oil and equivalent amounts of heating spread. called the crack and gasoline is mixes of outputs. Oi1can be processed in different ways, producing different number of Spread terminology identifes te number Of gallons of oil as input, and the gallons of gasoline and heating oil as outputs. Traders will speak of lt; E5 :E5 1 and :Q-1-1'' crack spreads. The 5-3-2 spread, for example, reflects the prost from tnking of gasoline and 2 gallons of heating oil. 5 gallons of oil as input, and producing 3 gallons oil could use a futtlres crack spread heating and gasoline produeing Apeoleum rener prices. This sategy would entail going long and oil of output to lock in both the cost and heating oil ftltures. Of gasoline of quantities appropriate the noilf'utures and short possible is and it to produce other outputs, prouction other inputs to coursethere are hedge. perfect spread is .not crack the a such as jet fuel, so 'l''he

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STRATEG 1ES

$

197

(see again Figure 6.7). In

some cases, One commodity may be used to hedge another. As an example of this W discuss the use of cnlde oil to hedgejet fuel. Finally, weather derivatives provide another example of an instnlment that can be used to cross-hedge. We discuss degree-day index

coneacisas an example

of such derivatives.

Basis ltisk Exchange-traded commodity f'tlttlres coneacts call for delivery of the underlying commpdity at specisc locations and specisc dates. The acmal commodity to be bought or sold may reside at a different location and the desired delivery date may lot match that of the futtlres conact. Additionally, the grade of the delivrable under the fumres conact may not match the grade that is being delivered. This general problem of the t'utures or forward contract not representing exactly what is being hedged is c alted basis ? s k Basis risk is a generic problem wit commodities because of storage and ansportation costs and quality differences. Basis risk can also arise with financial fumres, as for example when a company hedses its oWn boaowing cost with the Eurodollar contract. Section 5.5 demonsated how an individual stock could be hedged with an index futtlres contract. We sw that if we regressed the individpal stock return on the index provided a hedge ratio that mini u ze tj tjje rettlrn, the resulting regression coecient variance of the hedged position. In the same way, suppose we wish to hedge oil delivered on the East Coast with X oil contract, which calls for delivery of oil in Cushing, Ollahoma. the variance-minimizing hedge ratio would be the regression coeflicient obtained by regressing the East Coast price on the Cushing price. Problems with this regression are that the relationsllip may not be stable over time or may be estimated imprecisely. 'i

Suppose we consider buying oil in July and selling gasoline and heatq Eyample 6.4 k.i/j ing oil in August. On June 2, 2004, the July futtlres plice for oil was $39.961-e1, jt( 42 gallons per barrel). The August ftltures prices for unff ti 01- $0 9514/gallon (there are and $1.0171/ga1lon.The 3-2-1 leaded gasoline and heating oil were $1.2427/ga1lon l) :l ')(E) buying 3 gallons of oil and . crack spread tells us the gross margin we can lock in by EE ;(' these prices, the spread is heating of Using oil. 1 and producing 2 gallons of gasoline . tj! tl.y) j i jj = !t'' (2 x $1.2427)+ $1.0171 (3 x $0.9514) $0.6482 i(()j t . . yjy calculation we made no interest adjustment for j or $0.6482/3 = $0.2161/gal1on.In this 11 the different expiration months of the futures contract. % .

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X. Two of these options pay There are crack spread options trading on of based on te difference between the price of heating oi1 and crude oil, and thq plice gasgline and heating oil, both in a 1:1 ratio.

6.12 HEDGING

STRATEGIES

contmodity In this section we discuss some of the complications thatean arise when using 3.3 we discussed f'utures and forwards to hedge commodity price exposure. In Section of quantity uncertainty, where, for example, a one such complication: the problem farmer growing corn does not know the ultimate yield at the time of planting. Other issues can arise. Since commodities are heterogeneous and often costly to transport hedge a risk with a commodity conact that is imperfectly and store, it is common lise lisk to basis rf.k: The price of the being hedged. This gives correlated with the differently tan the price of the commodity underlying the f'utures conact may move cornmodity you are hedging. For example, because of transportation cost and time, the price of namral gas in California may differ from that in Louisiana, which is the .to

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Another example of basis lisk occurs when hedgers decide to hedge distant obligations with near-tenn futtlres. For example, an oil producer might have an obligation to delixer 100,000 barrels per month at a xed price for a year. The natural way to hedge this obligation would be to buy 100,000 barrels per month, locking in the price and supply on a month-by-month basis. This is called a strip hedge. We engage in a stlip hedge when we hedge a strenm of obligations by offsetting each individual obligation with a f'uttlres contract matching the maturity and quantity of the obligation. For the oi1 producer obligated to deliver every month at a fixqd price, th hedge would entail buying the appropriate quantity each month, in effect tnking a long position in the strip. Arl alternative to a strip hedge is a stack hedge. With a stack Vdge,we enter into f'utures contracts with a single maturity, with the number of conacts selected so that changes in the pre-wnt valtte of the future obligations are offset by changes in the value ttstack'' of futures contracts. In the context of the oil producer with a monthly of tlzis delivery obligation, a stack hedge would entail going long 1.2 million barrels using the near-term contract. (Actually, we would want to tail the position and short less than 1.2 million barrels, but we will ignore this.) When the near-term contract mattlres, we

200

k. COMMODITY

PROBLEMS

A&D FUTURES

FORWARDS

There are many other examples of weather risk: ski resorts are hafmed by warm winters, soft drink manufacturers are harmed by a cold spring, summer, or fall, and makers of lawn splinklers are harmed by wet summers. In al1 of these cases, fin'ns could that make payments based upon hedge their risk using weather derivatives--contracts cross-hedge specific l-isk. their characteristics of weather-to realized weather-related delivatives based measurements. The payoffs for weather are on An example of a weather contract is the degree-day index futtlres contract traded at the Chicago Mercan tile Exchange. A heating degree-day is the maximm of zero and the difference between the average daily temperamre and 65 degrees Fnhrenheit. A cooling d.egree-day is the maximum of the difference between the average daily temperature and 65 degrees Fn hrenhe it and zero. Sixty-five degrees is a moderate temperature. Athigher temperatures, air conditioners may be tlsed, whil at lower temperamres, heating may be used. A monthly degree-day index is constructed by adding the daily degree-dys over the month. The futures contract then settles based on the cumulative heating r cooling degree-days (the'twoare separate contracts) over the course of a month. 'I'he size of the contract is $100 times the degree-day index. As of September 2004, degree-day index contracts were available for over 20 cities in the United States, Europe, and Japan. There are also puts and calls on these ftlmres. With city-specisc degree-day index contracts, it is possible to create and hedge payoffs based on average temperatures, or using options, based on rgpges of average temperamres. If Minneapolis is unusually cold but the rest of the contry is nonnal Iflalce a large payment that will the cooling degree-day contract for Minneapolis will consumption of Notice that in this increased the energy. the holder for compensate would provide example) nattlral not price contract (for a sucient hedge, scenario a gas much effect would alone Minneapolis have unusual cold not in on national energy since ,

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At a general level, commodity fonvard prices cm be described by the same formula as Enancial forward prices: (r-J)F F(),w Sze =

(6.21)

For fnancial assets, is the dividend yield. For commodities, is the commodity lease mrc-the return that makes an investor willing to buy and then lend a commodity. Thus, for the commodity owner who lends the commodity, it is like a dividend. From the commodity borrower's perspective, it is the cost of borrowing the commodity. As with snancial fonvards, commodity forward prices are biased predictors of the f'uttlre spot price when the commodity return contains a risk premium. While the dividend yield for a financial asset can typically be observed directly, lease rate for a commodity can typically be estimated only !?),obselwing #?c./?3j)(7?'# the fonvard curve provides important information aboqt the commodity. The price. Commodities are complex because every commodity market differs in the details. Forward eurves for different commodities reiect different properties of storability, stor-

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201

age costs, production, and demand. Electricity, gold, corn, natural gas, and oil all have distinct forward curves, re:ecting the different characteristics of their physical markets. These idiosyncracies will be reflected in the commodity lease rate. When there are seasonalities in either the demand or supply of a commodity, the commodity will be stored (assumin g this is physically feasible) and the forward curve for the commodity Willre:ect storage costs. Some holders of a commodity receive benests from physical onvenietce yield. The convenience ownership. This benest is called the commodity's ownership different different creates for investors, and may or may not returns to yield price. The yield forward convenience re:ected in the be can lead to no-arbieage regions plice. costly no-arbitrage It also be than to short-sell commodities with a rather can a yield. convenience signifcant ,

FURTHER

READING

We will see in later chapters that the concept of a lease rate-which is a generalization of a dividend yield-helps to unify the pricing of swaps (Chapter 8), options (Chapter 10), and commoditplinked notes (Chapter 15). One particularly interesting application of the lease rate arises in the discussion of real options in Chapter 17. We will see there that if an extractable commodity (suchas oil or gold) has a zero lease rate, it will never be extracted. Thus, the lease rate is linked in an important way with production

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A useful resource for learning more about cornmodities is the Chicago Board of X and the CBOT) Trade (1998). The Web sites of the valious exchanges (e.g., particular commodities and trading and with information about useful also resources, are hedging strategies. Siegel and Siegel (1990)provide a detailed discussion of many commodity f41t'ures. There are numerous papers on commodities. Bodie and Rosansky (1980)and Gorton and Rouwenhorst (2004)examine the risk apd return of ommodities as an investmept. Brennan (1991),Pindyck (l993b), and Pindyck (1994)examine the behavior of commodity prices. Schwartz (1997)compares the performance of different models of commodity price behavior. Jarrow and Oldfield (1981)discuss the effect of storage costs on plicing, and Routledge et a1. (2000)present a theoretical model of commodity

forward curves.

Finally, Metallgesellschaft engendered a spirited debate. Paperj wlitten about that episode include Culp and Miller (1995), Edwards and Canter (1995j, and Mllo and Parsons (1995).

PROBLEMS 6.1. The spot price of a widget is $70.00per unit. Forward prices for 3, 6, 9, and 12 months are $70.70,$71.41,$72.13, and $72.86. Assuming a 5% continuously compounded annual risk-free rate, what are the annualized lease rates for each maturity? Is this an example of contango or backwardation?

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202

%.COMMODITY

FORwARDS

P Ro

AND FUTURES

prices for 3, 6, 9, and 12 6.2. 'T'hecurrent price of oil is $32.00per barrel. Forward Assuming a 2% continuously months are $31.37, $30.75, $30.14, and $29.54. annualized rate for each malease the coltapoundedannual risk-free rate, what is backwardation? turity? Is this an example of contango or tisk-free rate of 3% annually, at what lease 6.3. Given a continuously compounded commodity price? tRecall the pencil rate will forward prices equal the current wotlld there be contango or example in Section 6.4.) lf the lease rate were 3.5%,

backwardation? continuously compounded lease 6.4. Suppose that pencils cost $0.20 today and the compounded interest rate is 10%. The continuously rate for pencils is 5%. The costlessly. stored be uncertain.and pencils can pencil price in 1 year is what payment do you have to make 1 a. lf you short-sell a pencil for year, Enancial investor to store to the pencil lender? Would it make sense for a equilibrium? pencils in b. Show that te equilibrium forward plice is $0.2103. prices are ruled out by arbitrage in the c. Explain what ranges of forward and cannot four cases where pencils can and cannotbe short-sold and can be loaned. price is 310.686, and 6.5. Suppose the gold spot price is $300/oz.,the l-yea/fonvard 5%. is compounded-risk-free rate the continuously

a. What is the lease rate? gold is not loaned? b. What is the ret'ul'n on a cash-and-carry in which cash-and-earry in which gold is loaned, earning c. What is the return on a the lease rate? continubusly compounded interest rate is For the next three problems, assume that the (payableat the end of the quarter). quarterly 6% and the storage cost of widgets is $0.03 widgets: Here is the forward price ulwe for 2004 Dec 3.000 6.6.

2005

Mar 3.075

Jun

3.152

Sep 2.750

Dec 2.822

Mar 2.894

2006

Jun

2.968

explanations for the shape of this forward curve? a. What are some possible cash-and-carl'y entered b. What annualized rate of ret'ul'n do you earn on a into in December 2004 and closed in March 20052 Is your answer sensible? of return do you earn on a eash-and-carry ntered c. What annualized rate 20052 Is your answer into in December 2004 and closed in September sensible?

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a. Suppose that you want to borrow a widget beginning in December 2004 and ending in March 2005. What payment will be required to make the transaction fair to both parties? b. Suppose that you want to borrow a widget beginning in December 2004 and ending in September 2005. What payment will be required to make the transaction fair to both parties? 6.8.

a. Suppose the March 2005 fonvard price were $3.10. Describe two different transactions you could use to undertake arbitrage.

b. Suppose the September 2005 forward price fell to $2.70and subsequent forward prices fell in such a way that there is no arbitrage from September 2005 and going forward. Is there an arbitrage you could undertake using forward contracts from June 2005 and earlier? Why or why not? 6.9. ConsiderExample 6.1. Suppose theFebruary forwardprice had been $2.80.What would the arbitrage be? Suppose it had been $2.65. What would the arbitrage be? In each case, specify the transactions and resulting cash flows in both November and February. What are you assuming about the convenience yield? 6.10. Using Table 6.10, what is your best guess about the current price of gold per ounce? 6.11. Suppose you lnow nothing about widgets. You are going to approach a widget merchant to borrow one in order to short-sell it. (That is, you will take physical. possession of the widget, sell it, and return a widget at time F.) Before you ring the doorbell, you want to make a judgmentabout what you think is a reasonable lease rate for the widget. Think about the following possible scenatlos. a. Suppose that widgets do not deteriorate over time, are costless to store, and are always produced, although production quantity can be varied. Demand is constant over time. Knowing nothing else, what lease rate might you face? b. Suppose everytlzing is the same as in (a) except that demand for widgets varies seasonally. c. Suppose everything is the same as in (a) except that demand for widgets varies seasonally and the rate of production cannot be adjusted. Consider how seasonality and the horizon of your short-sale interact with the lease rate.

d. Suppose everything is the same as in (a) except that demand is constant over time and production is seasonal. Consider how production seasonality and the horizon of your short-sale interact with the lease rate. e. Suppose thatwidgets cannotbe stored. How does tlzis affectyour answers to the previous questions?

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coupons and are issued at aprice close to their matulity value (i.e.,they are issued at parl. Notes have 10 or fewer years to maturity and bonds have more than 10 years to maturity. The distinctions between bills, notes, and bonds are not important for our purposes; we will refer to all three as bonds. You will also see Treasury inflation protected securities ITIPSIin Figure 7.1. These are bonds for which payments are adjusted for in:ation.

Finally, the most recently issued government bonds ar called on-the-run; other bonds re called off-the-run. These ten's are used ftequently in talking about government bonds since on-the-run bonds generally have lower yields and greater trading volume than ofthe-run bonds. Appendix 7.A discusses some of the conventions used in bond price and yield quotations. In addition to government bond information, there is also a listing for STIUPS. Trading of Registered lnterest and Principal of Securitiej-is A S'rllps-separate a claim to a single interest payment or the principal portion of a govelmment bond. These claims trade separately from the bond. STRIPS are zero-coupon bonds since they make onl# a single payment at mattltity. 'CSTRIPS'' should not be confused with the forward strip, which is the set o? forward prices available at a point in time. We need a way to represent bond prices and interest rates. Interest rate notation is, unfortunately and inevitably, cumbersome, because for any rate we must keep track of three dates: the date on which the rate is quoted, and the period of time (thishas beginning and ending dates) over which the rate prevails. We will 1et ?'? (?I r2) represent the interest rate from time rl to time tz, prevailing on date t. If the interest rate is current-i.e., if t rl-and if there is no lisk of confusion, we will drop the subscript. ,

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We begin by showing that the zero-coupon bond yield and zero-coupon bond price, columns(1)and (1) in Table 7.1, provide the same infonnation. A zero-coupon bond is a bond that makes only a single payment at its mamrity date. Our notation for zerocoupon bond plices will mimic that for interest rates. The price of a bond quoted at time

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maturing at tz, is Ptfj(fl 1:.)(). As with interest lo; with the bond to be purchased at rl and = /1 rates, we will drop the subscript when tfj plice P(0, 1) = 0.943396 means that you would of bond I'he l-year zero-coupon 1 You could also pay P (0, 2) = 0.881659 pay $0.943396today to receive $1 in year. = to receive $1 in 3 years. today to receive $1 in 2 years and P (0, 3) 0.816298 ?'crl//wl on a zero-coupon bond is simply The yield to maturity (orintenzal rate of bond, we end the percentage increase in dollars eacned from the bond. For tlae l-year interest quoting 1 = 0.06 more dollm's per $1 invested. If we are up with 1/0.943396 rates as effective annual rates, this is a 6% yield. 1 = 0.134225 For the zero-coupon z-yearbond. we end up with 1/0.881659 interest effective raye of could call this a z-year more dollars per $1 invested. We lf basis. we want this 13.4225%, but it is conventiontl to quote rates on an annual could assume annual yield to be comparable to tlae 6% yield on the l-year bond, we 2) = 0.065. r(0, that + r(0, 2))2 = 1.134225, wlch implies compounding and ,

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?3(0,

r2)(lY

rj

)1JI

=

/1)

P,

.P(0, r2)

Corresponding to l-year and z-yearinterest rates, rc(0, 1) and ?3(0, 2), we have prices of l-year and z-yea.r zero-coupon b on d s, P()(0 1) apd /$(0, 2). Just as the interest rates imply a forward l-year interest rate, the bond prices imply a l-yearfonvard zero-coupon bond price. The i.mpliedforward zero-coupon bond price must be consistent with the implied forward interest rate. Rewriting equation (7.3),we have ,

lmpliedForward Rates (1) or We now see how column (3) in Table 7.1 can be computed from either (2). The l-year and z-yearzero-coupon yields are the rates you can enrn from year 0 to is also an implicit rate that can be earned from year 1 and from year 0 to yem' 2. There with the other two rates. This rate is called the year 1 t year 2 that must be eonsistent implied forward rate. . 2. We Suppose we could today guarantee a rate we could earn from year 1 to year for 2 0(0, invested and $1 1)) years earns know that $1 invested for 1 year earns (1+ column

/$(/.1t?.) ,

=

1 (1+ r0lrl

,

/.2)1 , 2

-,,

(1+ ?-0(0, J'1))f' (j. /()((),uljj/a

=

./3(0,

/.a)

=

.j.

The implied forward zero-coupon bond price from zero-coupon bond prices maturing at tg and /1 .

/1

po

,

j.j

)

to h is simply the ratio of the

210

k

INTEREST

Boxo

AND FUTURES

RATE FORwARDS

k

.gs

tt

C

jtt )

=

0.943396 + 0.881659 + 0.816298 6.95485%

E.

j

=

%:

.1

,

=

211

j'yyl Example 7.2 Using the infonnation in Table 7.1, the coupon on a 3-year coupon . (y Etty tj.j bond thatsells at par is jy: j j j 6ggg y('.

Example 7.1 Using information in Table 7.1, we want to compute the implied forward interest rate from year 2 to year 3 and the implied folavard price for a l-yer zero-coupon bond purchased in year 2. The implied forward interest rate, rp(2, 3), can be computed as (1 + 0.07)3 (1+ ?-o(0 3)13 ()84s.74)5 ?-0(0, 2)) a 1 + ?-0(2, 3) (1 + () ()65)2 (1+ =

BAslcs

-

.

or equivalently as ,-0(2,

1+

3)

.Jb(0, 2) =

/$(0, 3)

0.881659 =

0.816298

=

Equation (7.5) computes the bond price by discounting each bond payment at the appropriate for a cash flow with that particular matulity. For exapple, in equation rate (7.5), the coupon occuring at time ti is discounted using the zero-coupon bond price Pt (?,.ti). An alternative way to write the bond price is using the yield to mattllity to discount a11payments. Suppose the bond makes p?l payments per year. Denoting the per-period yield to matufity as y,,,, we have

1.0800705

The implied forward l-year zero-coupgn bond price is 1 .P0(0. 3) 0.925865 ?-0(2, 3) = 1+ P0(0, 2)

k

=

11

Bt (r, F, c,

i= l

Coupon Bonds Given the pdces of zero-coupon bonds--column (1) in Table 7.1-we can price coupon 7.1-the coupon bonds. We can also compute the par coupon--column (4) in Table need to know rate at which a bond will be priced at par. To describe a coupon bond, we of the bond payments, and ed dte the date at wh'ich the bond is being priced, the start principal. Som ptactical of and the amount of the payments, and number amount the discussed complexities associated with coupon bonds, not essential for our purposes, are

in Appendix 7.A.

We will let Bt (h h, c, ?l) denote the time t price of a bond that is issued at h mattllit.y payment, and makes l everlly mamres at tz, pays a coupon of c per dollar of will spaced payments over the life of the bon, beginning at time 11 + h tjj/l. We different 1. is than $ we can maturity payment assume the maturity payment is $1. If the that amount. al1 npltiply by payments just time t Since the plice of abond is the present value of its payments, at issuance the price of a bond maturing at F must satisfy ,

,

-

lt

Bt t, F, c,

??)

/))

cP t,

=

+ Pt (/, F)

With i being the index in the summation. Using where ti t + i CT tj/n, solve for the coupon as (7.5)i we can #, (r, F) Bt t, T, c, n) C n )-!f=lPt t, ) =

-

-

1, so the coupon on a par bond is given by 1 Pf t, T) j7.'l l P J t tt ) -

C=

=

,

.

+

1

(1 +

#,?,)'l

to compute the quoted annualized yield to maturity, y, as y = m x y,?,. bonds, for example, make two coupon payments per year, so the annualized Goverment twtce maturity the semiannual yield to mamrity. is yield to ' ' ' . . . Th diffetence between cjation (7.5)and equation (7.7) that in equation (7.5), at the ppropriate fte for a ash flow occning at each copon payment is discontd tht timek I kaiio (7.7), oe rat is ttsed to dicimnt a1l cash flowj. By definition, kuatin (7.7)can be misleading, thetwo expresssions give the sam price. Ilowkf, invstor the maturity, yield the is not ret'urn since to an earns by buying nd holding y,?,, prvtdes insight into how th cash flowj from a ond. Mfeover, eqtio (7.7) no a bonds. ith be replicate d w zero-coupon bondcan

It is common

ts

'

Zeros from Coupons We have strted with ero-coupon bond prices and dedued the prices of coupon bonds. In practice, the simation is often the reverse: We observe prices of coupon bonds and coupon bond must infer prices of zero-coupon bonds. This procedure in which bond prices of is called bootstrapping. prices are deduced from a set coupon Suppose we observe the par coupons i Table 7.1. We can then infer the first zero-coupon bond ptice f'rom the srstcoupon bond as follows:

equation

1 (1 +

=

=

c

; (1+ j'lt')

'zefo

i= 1

A par bond has Bt

??)

=

0.06).P(0, 1)

This implies that P(0, 1) = 1/1.06 = 0.943396. with a coupon rate of 6.48423% gives us

1 Since we lnow 17(0, 1)

0.0648423P(0,

=

=

Using the second par coupon bond

1) + 1.0648423#(0,

0.943396, we can solve for #(0, 2):

2)

212

k. INTEREST

RATE FORWARDS

#(0, 2)

AN D FUTURES

=

1

BOND

0.0648423 x 0.943396

-

1.0648423 0.881659 =

Finally, knowing P(0, 1) and P(0, 2), we can solve for #(0, 3) using the 3-year par coupon bond with a coupon of 6.95485%:

1

=

(0.0695485x 17(0, 1)) + (0.0695485x #(0, 2)) + (1.0695485x /7(0, 3))

#(0, 3)

=

1

-

(0.0695485x 0.943396)

-

(0.0695485x 0.881659)

1.0695485

= 0.816298 There is nothing about the procedure that requires the bonds to trade at par. In fact, we do not even need the bonds to all have different maturities. ltbr exanple, if we had a l-year bond and tw0 different 3-yellr bonds, we could still solve for the tlu'ee zero-coupon bond prices by solving simultaneous equations.

lnterpreting the Coupon Rate A coupon rate-for example the 6.95485% coupon on the 3-year bod-detennines the cash flows the bondholdaer receives. Howevr, excpt in special cas, it does not , jy earps by hol yjy coaejpond to the rate of retfn that-an ihvestr citlt notjyja jjqyjtj. iinplied ertai; i.e., the i'pr/aidrates for rates interst Suppose ar a momnt tt Imagine and 2. that we buy actually in l that will Table 7.1 the in rates years occur are thef all reinvesting maturity, it bond and hold to the 3-year coupons as are paid. What stoj calulations, let's through the and discuss going eal'n? Before of do we rate rettlrn the intuition. We are going to invest an amount t time 0 and to reinvest al1 coupons by buying more bonds, and we will not withdraw any cash until time 3. f/? eject, Ij?e are ctpn-lrlfcrf?z,ga 3-)'ear zdm-ctplfp/n bolld. Thus, we should ellrn the same return as zero: 7%. This buy-and-hold return is different than the yield to maturity on a The coupon payment is set to make a par bond fairly priced, but it is not 6.95485%. of the acmally return we enr'n on th,bond exept in the special case when the interest rate is constant over time. Consider frst what would happen if interest rates were certain, we bought the 3-year bond with a $100plincipal and a coupon of 6.95485%, and we held it for 1 year. The price at the end of the yepr would be 106.95485 6.95485 Xl = V 1.0700237 (1.0700237)(1+ 0.0800705) ,

%

213

We earn 6%, since that is the l-year interest rate. Problem 7.13 asks you to compute yor z-yearreturn on this investment. By year 3, we have received three coupons, two of which have been reinvested at the implied forward rate. 'I'he total value of reinvested bond holdings at year 3 is

6.95485 x ((1.0700237)(1.0800705) + (1.0800705)+ 1) + 10

=

122.5043

The 3-year yield on the bond is thus

.

wlzichgives us

BAslcs

l/3

122 5043

.j

m

100

gogg

As we expected, this is equal to the 7% yield on the 3-year zero and dlfferent from the coupon rate. This discussion assumed that interest rates are certain. Supppse that Fe buy and hold the bond, reinvesting the coupons, and that interest rates ar not certain. Can we still expect to earn a ret'urn of 7%? The answer is yes if we use interest rate forward contracts to guarantee the rate at which we can reinvest coupon proceeds. Othemise, the answer in general is no. The belief that the implied forward interest rte equals the expeted f'uture spot interest rate is a version of the expectations hypothesis. We saw in Chapters 5 and 6 that forward prices are biased predictors of f'uttlre spot prices when the underlying assethas a risk premium; the same is true for forward interest rates. When you own a coupon bond, th rate at which you will be able to reinvest coupons is uncertain. If the resulting risk. carries a risk premium, then the expected rettlrn to holding the bond will not equal the 7% remrn calculated by assuming interest rates are certaink The expectations hypothesis will generally not hold, and you should not expect implied fomard interest rates to be unbiased predictors of f'uttlreinterest rates. In practice you can guarantee the 7% ret'urn by using forward rate agreements to lock in the interest rate for each of the reinvested coupons. We discuss forward rate agreements in Section 7.2.

'3-year

= 99.04515

Continuously Compounded Yields Any interest rate can be quoted as either an effective annual rate or a continuously compounded rate. (Or in a variety of other ways, spch as a semiannually compounded rate, which is common with bonds. SeeAppendix 7.A.) Column (5)in Table7.1 presents the continuously compounded equivalents of the rates in the yield'' column. ln general, if we have a zero-coupon bond paying $1 at mamrity, we cn write its price in terms of an annualized continpously compounded yield, I'CCCQ, t), as3 'zero

P,

1)

=

e

--rccfn tj; '

The l-period return is thus l-peliod

ret'unl

=

6.95485 + 99.04515 l00

= 0.06

=

1

3In future chapters we will denote continuously compounded interest rates simply as r, without the cc superscript.

%

INTEREST

RATE FORWARDS

FORwARD

AN D FUTU RES

We can comput

(0, t)

1 =

-

l

ln g1(P

(0, /')1

the continuously compounded 3-year zero yield, for example,

1

1n(1/0.816298)

=

as

0.0676586

Alternatively, we can obtain the same answer using the 3-yea.r zero yield of 7%:

ln(l + 0.07)

=

0.0676586

z4??yof the zero yields or implied forward yields in Table 7.l can be computed as effective annual or continuously compounded. The choice hinges on convention and ease of alculation. RATE AGREEMENTS, EURODOLLARS, AND HEDGING

7.2 FORWARD

.5%

211 days 1.5% rquarterly = 2% rquarterly =

Borrow

$100m

+100m

-l01.5m

-102.0m

'

' Depending upon th# interst rate, tbere is a variation of $0.5min the borrowing

cost. How can we hedge this uncertainty?

rate agreement

(FRA) is an over-the-counter

%

215

Firstconsiderwhathappens if theFlkAis settledin Septemln arrears ber, on day 211, the loan repaymentdate. In thatcase, thepaymentto theborrowershould

be

l-IrRAl

(rquarrly -

x notional principal

Thus, if the bonmving rate is 1.5%, the payment under the F'RA should be

(0.015

$100m

0.018) x

-

-$300,000

=

Since the rate is lower than the FRA rate, the borrower pays the FRA counterparty. Similarly, if the borrowing rate turns out to be 2.0%, the paymertt under the FRA should be -

$100m $200,000

0.018) x

=

Settling the FRA in arrears is simple and seems like the obvious way for the contract to work. However, settlement can also occur at the time of borrowing. lf the FRA is settled in June, at the tim FRA seulement at the time of borrowlng the money is borrowed, payments will be less than when settled in arrears because the borrower has time to enrn interest on the FlkAsettlement. In practice, therefore, the FRA settlement is tailed by the reference rate prevailing on the settlement (bprrowing)date. (Tailing in this context means that we reduce the payment to reqect the interest ealmed between June and Septemben) Thus, the payment for a borrower is uarterly (?q1 + rquarterly -

Notional principal x

?-frRA)

If rquarterly = 1.5+, the payment in June is

-$300,000 l + 0.015 =

Forward Rate Agreements A forward

AND HEDGING

FRA seulement

(0.02

We now consider the problem of a borrower who wishes to hedge against increases in the cost of borrowing. We consider a fir'm expecting to borrow $100m for 91 days, beginning 120 days from today, in June. This is the borrowing date. The loan will be repaid in September on the loan repayment date. In the examples we will suppose that or 2%, and yhat the te effective quarterly interest rate at that time can be either l is implied June 91-day forward rate (therate frbm June to September) 1.8%. Here is the risk faced by the borrower, assuming no hedging;

120 days

EURODOLLARS,

rate. The actual borrowing is conducted by the borrower independently of the FlkA. We witl suppose that the reference rate used in the FRA is the same as the acmal borrowing cost of the borrower.

Thus, if we observe the price, we can solve for the yield as rcc

RATE AGREEM ENTS,

-$295,566.50

-$300,000.

contract that guarantees a

borrowing or lending rate on a given notional principl amount. FRAS can be settled either at the initiation or mamrity of the borrowing or lending trapqaction. If settled at mattlrity, we will say the FRA is settled in arrears. ln the example above, the FRA could be settled on day 120, the point at which the borrowing rate becomes known and the borrowing takes place, or settled in arrears on day 21 1, when the loan is repaid. F''ltAs are a fomard contract based on the interest rate, and as such do not entail the

actual lending of money. Rather, the borrower who enters an F.RA is paid if a reference rate is above the FRA rate, and the borrower pays if the referene rate is below the IRRA

In order to fnake this payment, By denition, the future value of this is the borrower can bonow an extra $295,566.50,which results in an extra $300,000loan payment in September. If on te other hand rquanerly = 2.0% the payment is ,

$200,000 $196,078.43 1 + 0.02 = The borrower can invest this amount, which gives $200,000in September, an amount that offsets the extra borrowing cost. If the forward rate agreement covers a borrowing period other than 91 days, we simply use the appropriate rate instead of the 91-day rate in the above calculations.

%.INTEREST

216

Synthetic

RATE FORwARDS

RATE AGREEM ENTS,

FORWARD

AND FUTURES

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FltAS

.'.'. t'tj'. (yjjy'' i t.ty

.'1*. .

.

-

.

.

Suppose that today is day 0 and we plan to lend money 120 daks hence. By using a forwltrd rate agreement, we can guarantee the lending rate we will receive on day 120. In particular, we will be able to invest $1 on day 120 and be guaranteed a 91-day rettlnl of 1.8%. We can synthetically create the same effect as with an FlkAby trading zero-coupon

bonds. ln order to accomplish this we need to guarantee cash Cows of $0 on day 0, on day 120, and +$1.018 on day 211.4

-$1

First, let's get a general sense of the transaction. To match the FlkAcash flows, we cash going out on day 120 and coming in on day 211. To accomplish this, on day 0 want will need to borrow with a 120-day maturity (togenerate a cash outflow on day 120) we and lend Fit.h a 211 day mattlrity (to generate a cash inCow on day 211). Moreover, we Want the day 0 value of the bonowing and leqding to be equal so that there is no initial cash flow. This, description tells us what we need to do. In general, suppose that today is day 0, and that at time t we want to lend $1 for the period s, enrning the implied folavard rate ?:(/, t + J) over the interval from t to ?'otr, will denote the lonannttalized t+ t + s. To simplify the notation in this section, percent return from time t to time s. Recall lirst that

-

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:

:

:

'.

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Total

#(0, t +

1. Buy 1 +

rn(/, t + s4 zero-coupon bonds maturing at time t + s.

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The resulting cash flows are illustrated in Table 7.2, which shows that transactions made on day 0 synthetically create a loan commencing on day t and paying the implied forward rate, ?'c(?, t + s), on day t + s. This example can be modihed slightly to syntetically create the cash flows ft'om a fonvard rate agreement that settles on the borrowing date, day /. To make this modification, we sell at time t the bond mattlring at time t + s. The result is presented in Table 7.3. Note that if we reinvested the 17RAproceeds at the market rate prevailing on day t, /3(1, t+ we would receive ?';(/, t + n t, t + on day t. .)

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flows of 1 + rzt, t + at t + s. This synthetlcally creates the cash flows from entering into a forward rate agreement on day 0 to Iend at day t.

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1* ;'t'q j S i) ! li'j 1.018 x /7(0, 211) $0.97561 ).qj yj( )ii:')r In order to have zero cash flow initially and a cash outdow on day 120, we borrow tt 0.97561, with a l7o-day ppturity. This entails borrowipg pn 120-day bond, since ! ;li.) ))'.r () 9-/561 (E11.. lji P )) (0, 120) ird

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Example tq@

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Consider the exnmple above and suppose that #(0, 211) 0.95836 0.97561, which implies a 120-day interest rate of 2.5%. In order to receive $1.018on day 211, we buy 1.018 211-day zero-coupon bonds. The cost of this =

=

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4The example in the previous section considered locking in a borrowing rate, but n this section we lock in a lending rate; the transactions can be reversed for borrowing.

:;:::::::::::::

ljj. j)! The result on day 120 is that we pay $ 1 to close the short pos jtjon on the 120-day bond, ;.j 1) since we bought that many 211-day bonds. and on day 211 we receive $1 .018

%

218

%

INTEREST

AND FUTURES

RATE FORwARDS

To summarize, we have shown that an FRA is

just like the stock and currency

forwards we have considered, both with respect to pricing and synthesizing. tf at time 0 we want to lock in a borrowing rate from time t to time t + s, we can create a rate forward synthetically by buying the underlying asset (the bond maturing at t + s) and borrowing (shorting)the bond maturing at day t. ln general, we have the following conclusions concerning a rate forward covering the period rI to /2: @The forward rate we can obtain is the implied fomard rate P, (J0. /*1)/ Ptn(r0,r2) 1

i.e.,

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by borrowing to

buy the prepaid forward, i.e., by 1. Buying 1 +

rto (lI

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.'

FORWARD

tz4 of the zero-coupon bond maturing on day tz, and

2. Shorting 1 zero-coupon bond maturing on day

/1

.

we iszczo,

.

RATE AGREEM ENTS,

EURODOLLARS,

AN D HEDGI NG

$

219

have

-

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I

I 1 ! t

*.'''

Number of Eurodollar contracts

=

-

A

100 l + 0.0 18

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t?

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=

98.2318 x (92.8

94) x $2500

-

-$294,695

=

8% (rquanerly2.0%), our total gain on the contracts will be =

.

(92.8

-

(100

-

?'t.lBoRlq

x 100 x

$25

Thus, the payoff on te Eurodollar contract compensates us for differences between the implied rate (1.8%)and actual LIBOR at expiration. To illustrate hedging with this contract we again consider two possible 3-month borrowing rates in June:. 1.5% or 2%. If the interest rate is 1.5%, borrowing ost on $l00m will be $1.5m,payable in September. If the interest rate is 2%, borrowing cost will be $2m. Suppose that we were to short 100 Eurodollarfutures contracts. Ignoling markingto-market prior to June, if the 3-month rate in June is 1.5%, the Eurodollar futures price will be 94. The payment is

((92.8

-

94)

x 100 x

$25) x 100

-$.

=

300,000

We multiply by 100 twice: Once to account for 100 contracts, and the second time to convert the change in tlRe futures price to basis points. Similarly, if the borrowing rate

s'rhiscalculation treats the Eurodollar contract as if it were with daily settlement, discussed in Appendix S.B. associated

98.2318 x (92.8

-

92) x $2500

=

$196,464

Notice that the amounts are different than with the FRA: The reason is that the FRA payment is automatically tailed using the 3-month rate prevailing in June, whereas with the Eurodollar contract we tailed using 1.8%, the LIBO:R rate implied by the initial futures price. We can now invest these proceeds at the prevailing interest rate. Here are the results on day 211, when borrowing must be repaid. lf LIBOR = 6% (lkuarterly = 1.5%), . we save $300,000in bolwwing cost, and the proceeds from the Eurodollar . contract are -$294,695 x (1.015)

-$299,

=

8% (rquarterly 2.0%), we owe an lf LIBOR vested proceeds from the Eurodollar contract are =

=

$196,464 x (1.02)

=

extra

l15

$200,000in interest and the in-

$200,393

Table 7.4 summarizes the result from this hedging position. The borrowing cost is close

to 1.8+.

Convexity bias and tailing ln Table 7.4 the net borrowing cost appears to be little a less than l You might guess that this is due to rounding error. It is not. Let's examine the numbers more closely. .8%.

a forward contract, ignoring the issues 6We

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=

This is like the payment on an FRApaid in arrears, except that the futures eontract settles in June, but our interest expense is not paid until September. Thus we have 3 months to arn or pay interest on our Eurodollar gain or loss before we actually have to make the interest payment. Recall that when the FRA settles on the borrowine date, the oavrent is the oreselt vallle of the change in borrowing cost. The FRA is thus tailed automatically as part of the agreement. With the Eurodollar contract, by contrast, we need to tail the position explicitly. We do this by shorting fewer than 100 contracts, using the ihplied 3-month Eurodollar rate of 1.8% as our discount factor. Thtls, we enter into (5

lf LIBOR

q..t

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Eurodollar futures contracts are similar to FRAS in that they can be used to guarantee FRAS and Etlrodollar contracts, a borrowing rate. There are subtle differences between understand. that however, are important to Let's consider again the example in which we wish to guarantee a borrowing rate for a $100mloan from June to Seplember. Suppose the June Eurodollar ftltures price = j go over 3 luonths. As we saw in is 92.8. lmplied 3-month LDOR is 100-92.8 4 single short Eurodollar the payoff contract at expiration will be5 Chapter 5, on a

:

'

Now consider the gain on the Eurodollar futures position. lf LIBOR = 6% (rquarterly 1.5%), our total gain on the short contracts when we initiate borrowing on day l20 will

Eurodollar Futures

..

assume here that it is possible to short fractional contracts in order to make the example exact.

J

k. INTEREST

220

FORWARD

AN D FUTURES

RATE FORWARDS

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Gain plus interest

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Net

= 1.5%), we pay $1.5m in borrowing cost, pnd we 6%, (rqumerly the Eurodollar contract, for a net borrowing expense of $1.799115m. lose $299,115on ttprofit'' from the Eurodollar hedge, relative to the use of an FRA, of $1.8m Tlzisis a

If LIBOR

=

-

$1.799115m $884. ((fLDOR 8% (rqumerly 2.0%), we pay $2.0min borrowing cost, but male Eurodollar contmct, for a net bonowing expense of $1.799607m.We $200,393 on the FItA, of- $1.8m $1.799607m $393. relative proEt, to an make a =

=

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=

It appefkrs that we systematically come out ahead by hedging with Eurodollar futures instead of an I7RA. You are probably thinking that something is wrong. IJy the Eurodollar As it mrnsout, what we have just shown is that the ?-t-Ild implied p/-dl/t-I#l, ratefor l/7c san'e ft/t7?s.To FRA (??w/d#./-()?-1?t7?-#.) ctwlpwcfcalnot eqltal the the interest perspective: When rate tul'ns out to be high, consider this, the borrower see proceeds payoff the positive and Eurodollar short has can be reinvested conact the a interest realized the rate t'urns out When the high date rate. the payment loan at until payoff negative and has contract Eurodollar short low, the be we to can fund this loss a until the logn payment date by borrowing at a low rate. Thus the settlement structtlre of the Eurodollar contract works systenatically in favor of the bon-ower. By tvtnling the argument around, we can verify that it systematically works against a lender. The reason this happens with Eurodollars and not FlkAs is that we have to make decision before we lnow the 3-month rate prevailing pn day 120. When we tailing the fixed by tail a amount (1.8%in the above example), the acmal variations in the realized of the borrower and against the lender. The 17RA avoids this by work in favor rate tailing-paying the present value of the change in boaowing cpsts-using automatically date. borrowing the actual interest rate on the the rate In order for the futtlres plice to be fair to bot.h the borrowr and implicit in the Eurodollar futures price must be higher than a comparable FRA rate. This difference between the FRA rate and the Eurodollar rate called convexity bias. For the most part in subsequent discussions we will ignore convexity bias and treat the

tender,

.is

EU RO DO LLARS, AN D HEDGI NG

ENTSZ

$

221

is that in many FRAS as if they are interchangeable. The reason Eurodollar eontract and in a proht of results bias convexity example, small. ln the above cases the effect is For short-tenn contracts, the of several hundred dollars out of a borrowing cost $1.8m. efft can be important.? effect can be small, but for longer-term coneacts te before the final contract settlement. We ln practice, convexity bias also matters markng-to-market a fumres contract can lead to a fumres price saw in Section 5.4 that fumres contract is marked to market that is different from the forward price. When a with the ftitures price, there is a systematic and interest rates are negatively correlated This leads to a f'uturesprice that is greater advantage io being short the f'umres eontract. with the Eurodollar contract in this what exactly happens the forward price. Tltis is t.14a11 receives a payment that can be invested example. When interest rates rise, the bonower interest rates fall, the borrower males a paymentthat at the higher interest rate. When benest. Marldnginterest rate. This works to the boaower's can be f'unded at the lower implied by the why rate the another reason therefore to-market prior to settlement is comparable l7IA. Eurodollar coneact will exceed that on art otherwise

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RATE AG REEM

but The Eurodollarfutures contractis based on LDOR, contract example, the Treasury-bill futures there are other 3-month interest rates. For bill. A borrower could use either the Treasury is based on the pdce of te 3-month hedge their bon-owing rate. Eurodollar contract or the Treasury-bill f'utures contract to Which conact is preferable? potential to default. Thus, LIBOR Banks that offer LJBOR time deposits have the increase in the interest rate that is includes a defaultpremium. (The default premblm an will default.) Pzivate companies compensates the lender for the possibility the borzower will also include adefaultpremium. thatborrow can also default, so theirborrowing rates

I-IBOR versus

T-bills

3-month

then it would be possible to perfectly tail the 7If future interest rates were known for certain in advance, is due to interest on the difference between about the error rates, position. However, with uncertant'y the error is that Given we tail by the forward rate, the realized rate, F, and the fonvard rate, rforward. measured by F ( 7 rorward ) 1 + rfonvard -

The expected error is E

F ( 'F

-

1

rcorwardl

*1-1-'?-rf)rqsrrkrtl

( 1 + rorwaru) a J

s

2.)

(g

-

E (groovaru)

2

(1 + rforward)z

the standard

(7.2 example can be 2% or 1.5%, so is the vafiance of the interest rate. Rates in our where 0.00252 0.0000062.5. approximate'ly 25 basis points or 0.0025 and the variance is thus deviation is Convexity bias is thus 0.00000625 $603.09 $ l00m x l 18a example was ($884+ $393)/2, or $638.5. The actual average convexity en'or in the =

=

.0

k. INTEREST

222

RATE FORWARDS

DURATIOX

AND FUTURES

The U.S. government, by contrast, is considered unlikely to default, so it can borrow at a lower rate than Enns. In addition, in the United States and other countries, government bonds are more liquid than corporate bonds, and this results bonds.S in higher plices-a liqttidity prc/nflf?n-for government The borrower will want to use the futures contract that has a price that moves in tandem with its own borrowingrate. lt makes sense that aplivate borrower's interestrate will more closely track LDOR than the Treasurpbill rate. ln fact, the spread between corporate borrowing rates and Treasuries moves around a great deal. The problem with hedging borrowing costs based on movements in the T-bill rate is that a private firm's bolwwing costs could increase even as the T-bill rate goes down; this can occur during times of financial distress, when investors bid up the plices of Treasury securities ttllight to quality''l. Thus, LYOR is commonly used relative tp other assets (a so-called private high-quality, interest rate. benchmark, markets in as a Figure 7.3 shows historical 3-month LDOR along with te differnce between LIBOR and th 3-month T-bill yield, illustrating this variability.g It is obvious that the spread varies considerably over time: Although the spread has been as low as a few basis points, twice in the 1990s it exceeded 100 basis points. In September of 1982, when Continental Bnnk failed, the spread exceeded 400 basis points. A private Lmolt-based 'E '( i kjj;'!111::111:)* !1!Ik-k-jII;' y' i't' p' 1* t' jjy'j (' jjk'. jt'tyd (yy'' ( ' E' ql. y!E!!!j,;;f syg-'yf tjjjy,:,,.f 11!1/*. tttd t;yf r' lqc'if j j'.!((j'. ' gq j(.... j:ji j'jjtgtytyj j;;. 41111:::1;.17. .', . .j'.j. j .jjjy ty ! . j. r t!. . . IljlEjz:. .(..q.. .. jjy;.yjjj;y.. yy yg.. .yyt. jjygj-.g.(y.j...yq.,y..gjq y. )!i.r.r: ..2 . . . t-jyty;yyjy, .1lIi2:!!q-.. , . . . . ..

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8In the United States, another reason for government bonds to have higher prices than corporate bonds is that government bond interest is exempt from state tmxation. gThe TED spread tt'T-Bills over Eurodollars'') is obtained by gong long T-bill futures and short the Eurodollar futures contract'

coxvExl'ry

k

borrower who had hedged its borrowing rate by shorting T-bill futures in August of 1982 would by September have lost money on the T-bill contract as Treasul'y rates declined, while the act'ual cost of borrowing ILIBORIwould have remained close to unchanged. This example illustrates the value of using a hedging cpntract that reflects the acmal cost of bonowing. The Eurodollar futures contract is far more popular than the T-bill futures contract. Trading volume and open interest on the two contracts were about equal in the early 1980s. H.owever, in August 2004, open interest on the Eurodollar contract exceeded 6 million contracts, while the T-bill contract had zero open interest. This is consistent with LDOR being a better measure of private sector interest rates tan the T-bill yield.

lnterest Rate Strips and Stacks Suppose a borrower plans to borrow $100mby rolling over 3-month debt for a period of 2 years, beginning in 6 monts. Thus, the borrowing will take place in month 6, month 9, month 12, etc. The borrower in this simation faces eight unknown quarterly borrowing rates. We saw in Section 6.12 that an oilhedgercould hedge each commitment individually (a strip) rcould hedge the entire commltmentusing one nakldated contract (a stack). The same alternatives are available wit interest rates. One way to hedge is to enter into separate $100mFRAS for each f'uttlre3-month period. Thus, we would enter into one F'RA for months 6-9, another for months 9-12, etc. This strip hedge should provide a perfect hedge for f'uttlre borrowing costs.

'

Depending on market conditions, using a strip is not always feasible. Forexample, forward prices may not be available with distant mattlres, or liquidity may be poor at distant maturities. Rather than individually hedging the boaowing cost of each quarter, ttstack'' of short-tenn Flus or an alternative in the context of this example is to use a Eurodollar contracts to hedge the present value of future borrowing costs. In the above example, we will be borrowing $100 million per quarter for eight quarters. To effect a stack, we would enter into fprward agreementt on the 3-month rate, maturing in 6 months, for slightly less ihan $800 million (We enter into less than$800 million' of forward rate agreements due to tailing for quarters 2 through 8.) As with the oil example in Section 6.12, the obvious problem with a stacking stratborrowing costs in distant quaders may not move pedctly is egy basis risk: Quarterly with borrowing costs in near quarters. Once we reach the frst quarter of borrowing, a11of the forward agreements mature and we need to renew our hedge. We now face seven quarters with unlnown borrowing costs and therefore we enter into fomard agreements for slightly less than $700million. is exactly The constantrenewal of the hedgingposition necessaly to effect astackandroll like that in the oil example. .

J7

a;q?)

,.

AxD

7.3 DURATION

AND CONVEXITY

An important charactelistic of abond is the sensitivity of its price to interestrate changes, which wemeasureusing duration. Duration tells us approximately how much the bond's price will change for a given change in the bond's yield. Duration is thus a summal'y

i

224

% INTEREST

DURATION

AND FUTURES

RATE FORWARDS

of bonds with different coupons, measure of the risk of a bond, permitting a compmison times to mamrity, and discounts or premiums relative to principal. In this section we also discuss convexit'y, which is another measure related to bond price risk.

tl

#(y)

C/?n (1 + -

=

f

.:/,?7)

izzz1

+

,,

''

price =

-

Upit chgpge in yild

jxzj

=

-

(7/?n

i

(1 +

Unit change in yield 11

1

1

t)'/p?

i -

B(vj 1

-1-

yjln

j.j

??l

l . #(y)

(l

-1.

y/??l)f

+-

Alf'

11 ???

(1

...h

.y//31)/

We obtain another measure of bond price risk-Macaulay duration-by multiThis puts both bond price and yield changes in expression with a clem- interpretation:

i= l

??l

(1 +

c/???.j

i

Macaulay duration

.'.://,1()?1+1

m

-

1 + y/lll

(1 + y/???) '

+

?.1

Arf'

-

z?2 (1 + y/???)''

Example 7.4 Consider the 3-year zero-coupon bond in Table 7.1 with a yield to maturity of 7%. 'Fhe bond price per $100of matulity value is $100/ 1.073 $81.82979. =

$100/1.07013= $81.60691, Atayield of 7.01%, one bsis pointhigher, tebondpriceis per $100of maqlrity value. a change of As an alternative way to derive the price change, we can compute equation (7.9) with C = 0, M = $100,11 = 3, and ??? = 1 to obtain -.-$0.02288

$100

3x 1.073

1 + yjnl Chanee in bond orice x Unit change in yield S(y) -

=

-

(7.9)

q

Equation (7.9)tells us the dollar change in the bond plice for a change of 1.0 in y. It is natural to scale this either to refect a change per percentage point (in which case equation (7.9) by 10,0001. we divide equation (7.9) by 100) or per basis point (divide Equation (7.9) divided by 10,000 is also known as the price value of a basis point (PVBP). To interpret PVBP for a bond, we need to know the par value of the bond.

x

x

-

M

71 =

y/F?l)f+l ?'

1

1 - 1.07

=

=

.y/??1.11

-

?n

Change in bond price

Modifed duration

plying equation (7.10)by 1 + percentage terms and gives us an

The chapge in the bond price for a unit change in the yield, y, islo

hangeiti bond

=

1 #()?)

-

11

i

(7/!??l + ( 1 y/???)f

j..j

??)

/$lr

??

-1-

??2

(1

...1-

yjnln

To interpret this expression, note that (C/??1)/(1 + yjlnli is the present value of the is th bond payment, which occurs in f/??7 years. The quantity C/?n/((1 + 'jlnlBl'lj therefore the fraction of the bond value that is due to the fth payment. Macaulay duration is a bveighted twc/wgc of #7e time (nl/???/Jd?' ofperiods) ulltil #l' boltd ptky/?l'?lll occlfr, with the weights being the percentage of the bond price accounted for by each payment. This interpretiion of Macaulay duration as a time-to-paymnt measure explains why For a zero-coupon these meajures of bond price sensitivity are called bond, eq ation (7.11)implies that Macaulay duration equals time to maturity. Macaulay duration illustrates why maturity alone is not a satisfacto/ risk measure for a coupon bond. A coupon bond makes a series of payments, each with a different maturity. Macaulay duration summarizes bond price risk as a weighted average of these EEduration.''lz

-$228.87

=

a change of 1 basis point, we divide by 10,000 to obtain almost equal to the actual bond plice change. This -$228.87/10,000 k. illustrates the importance of scaling equation (7.9)appropriately.

In order for this to rflect -$0.02289,

=

lo-fhisis obtained

225

-l.

M

(1 + y/,n.)

%

When comparing bonds with different prices and par values, it is helpful to have of price sensitivity expressed ypr/-dollar of 1747/7:'/ btain this by price. k measure a This gives us a dividing equation (7.9)by the bond price, #(y) and multiplying by change in the bond price measure known as modified duratlon, which is lepercelltage for a unit change in the yield:

Dmation Suppose a bond makes ??l coupon payments per year for F years in the amount C(m, and pays M at maturity. Let ),/7?7be the per-peliod yield to mattlrit.y (by convention, y is the armualized yield to mamrity) and 11 = m x T the number of periods until matulity. The price of the bond, S(y), is given by

AN,D CoNvExlTy

by computing fe derivative of te borldprice with respect to the yield, dBlsl/dy.

l lerhis measure of duration is named after Frederick Macaulay, who wrote a classic history of interest rates (Macaulay, 1938). l2The Excel duration functions are Dltration for Macaulay duration and MDttratit for modified duration.

k. INTERES'I-

226

Du RATI o lq

AND FUTURES

RATE FORwARDS

Co

AN D

NvExl'rv

k

227

.0725)3

Returning Ekample 7.5 q'r :. j:gj; )(. . 7% bpnd is ytifor ihe )'.) 'j

to Example 7.4, using equation

' E

1.07

-$228.87

x

-

1)E,

1

(7.11),Macaulay duration

k

-3.000

=

$81.62979

Example 7.6 Consider the 3-year coupon bond in Table 7.1. For a par bond, the yield to mamrit'y is the coupon, 6.95485% in this case. For each payment we have x

(;.4;.54:5

payment 1

1.0695485

%pavment 2 -

=

%payment 3 Thus, with

11

=

3 and

??l

=

(1 x 0.065026) +

=

0.0695485 a

=

0.065026

=

0.060798

(1.0695485) 1.0695485 (1.0695485)a

=

The act'ual new bond price is 0.02% of the bond price.

$100/(1

k.

Although duration is an important concept and is frequently used in practice, it has a concept'ual problem. We emphasized in the previous section that a coupon bond is a collection of zero-coupon bonds, and therefore each ash flow has its own discount rate. Yetboth duration formulas are computed assuming that all cash flows are discounted by a single artificial number, the yield to mattlrity. In Chapter 24 we will examine alternative approaches to masuring bond price lisk.

Dttration Matching Suppose we own a bond with time to matulity 11 price 5.1 and Macaulay duration DI We are considering a short position in a bond with mattlrity tz, price Bz, and Macaulay duration Dz. We can ask the question: How much of the second bopd should we shortsell in order that the resulting portfolio-long the bond wit.h duration Dj and short the bond with duration Dz-is insensitive to interest rate changes? ,

0.874176

1, Macaulay duration is

(2 x 0.060798) + (3 x 0.874176)

=

2.80915

The interpretation of the duration of 2.8 1 is that thebond responds to interestrate changes 2.81 yeazs to maturity. Modihed duration is as if it were a pure discount bond wit.h k. 2.80915/1.0695485 = 2.626482.

Equation

(7.12)giyes

.

us a formula for the change in price of each bond. Let N B l + NB1

and, using equation (7.12),the change in price due to an interest rate change of

(#l(.71+ 6) fl (.:1)1+ = -Dl Bt (.:1)6/(1+ yl ) -

Since duration tells us the sensitivity of the bond price to a change in the interest given change in it rate, can be used to compute the approximate bond pzice change for a changes from y the yield the bond and is bond prie B the on (y) rates. Supppse interest tfle d duration, D, for mpd yield. The formula the small chpng is in where E E, a to y + can b written 1 (S()' + 6) :::::::::: ..i!E((;1p

,

denote the quantity of the second bond. The value of the portfolio is

-

N (S2(y2 +

6)

#D2.P2()'2)c/(1

Eq is

,:212)1

-

+

.y2)

where Dl and D2 are Macaulay durations. If we want the net change to be zero, we choose N to set the right-hand side equal to zero. This gives N=-

.l()/)1

-

.........

$81.060. The prediction enor is about

=

D1 S1 (.,1)/(1 + )?l) D2#2(y2)/(1 + J?2)

.............

..

s(y)

s

Letting Macaulay duration be denoted by Dslac, we have Duac = D(1 + y). We can therefore rewrite this equation to obtain the new bond price in tenns of the o1d bond price and either duration measure:

W'hen a portfolio is duration-matched in this fashion, th net investment in the portfolio will typically not be zero. 'Fhat is, either the value of the short bond is less tn the value of the long bond, in which case additional Enancing is required, or vice versa, in which case there is cash to invest. This residual can be financed or invested in vel'y shortterm bonds, with duration approximately zero, in order to leave the portfolio duration

matched.

lt Example 7.7 Considerthe 3-yearzero-couponbond with aprice of $81.63persloo )!))maturity value. The yield is 7% and the bond s Macaulay duraton is 3.0. f the yield jyt tl were to increase to 7.25%, the predicted price would be ,

y...t

i

/(7.25%)

=

$81.63 (3/1.07)x $81.63 x 0.0025 -

=

$81.058

Suppose we own a 7-year, 6% coupon bond with a yield of 7%, and r1 Example 7.8 )EJ lind duration-matched short position in a lo-year, 8% coupon bond yielding the it want to )).. '.''ti7 5%. Assuming annual coupon payments, the Macaulay duration and price of the two

228

% INTEREST

RATE FORWARDS

AND FUTURES

DURATION

bonds is 5.882 years and $94.611, and 7.297 years and $103.432,respectively. Thus, if we own one of the 7-yea.r bonds, we must hold 5.882 x 94.61 1/(1.07) - 7.297 x 103.432/(1.075)

equation (7.14)with C

0, ??7

=

Convexity

=

1, and M

3 x 4 x

=

-0.7408

=

Using equaton

the lo-year bond. The short position in the lo-year bond is not epough to pay for the 7-yea.r bond; hence, investment in the portfolio is 1 x 94.611 0.74 jy x jts zsr 17.99. If the yield on both bonds increases 25 basis points, the price change of the portfolio is units qf

-

.

-1.289 + (-0.7408)

x

=

Convexity The hedg

ip example 7.8 is not perfect because duration changes as the interest rate to which duration changes as the bond's yield

Convexity

11

1

B ()')

g=l

C/m

i i + 1) ln p,

(1+

.'.:/??7)

j.j.:t

+

/7

(n + 1) ???a

(j

.j.

M yjjtjtt-vg

By +

E:)

=

#(y)

(D x B (y) x

+ 0.5 x Convexity x B)')

x c2

7.@

Consider again example 7.7. We want to predict the new price of a 3-year zero-coupon bond when the interest rate changes from 7% to 7.25+. Using

13At the original yields, we computed a hedge ratio of 0.7408. Problem 7.19 asls you to compute tlle hedge ratio that would have exactly hedged the portfolio had both interest rates increased 25 basis points and decreased 25 bmsis ponts. The two hedge ratios are different, which means that one hedge rato would not have worked perfectly. l4-rhis is obtained by taking the second derivative of the bond price with respect to the yield to maturity, dlBs,jjdl'l, and normalizinR the result bv dividing by the bond price. l5If you recall calculus, you may recognize equation

price. See Appendix 13.%

(7.12)as

a Taylor series expansion of the bond

-

j'jy'yjj;-', '))' 77(* 1* ;'))r' t'(' (' q' 5,* )!..tl:')i (' (q!' ltf E' ?' jyyy', y' j'g.t' jjyd ;'rlf ((' j)' jjjjr:::;;;ky'-' 1* i'lqlft.qiplllqslkllijqrlq)l qjf i IE itryf (;4* yjd j.tt!r))jyy'. g' r?''t. ...'.' )' (' ttf y' j)' r:' t'j't' FE iE 7:7r.rr77)42:77( ' tjyt.yjyjjtdsjtjtyj'jy jjjjyll'tttf )' yjf jyf E l:lliti yjyy'jjj;i:jjj::d 1*.)* y' 1$$* '(tj'' '. .'. '!( E (:' ' E; E ktE ijjjt,,,jjjr'. E ;'('( ''(.!EE''?( t' Er; '. .. !.. ' i ' ... .E.' i. EE (qE..' q' .. (.E.. . (.ti .;:t( ;(.i i('i(:1((:.qi '):ijqq l E E! . .i E.; i ; #!.'- . i'-i lr:ii i!jjj):. (q , . .i E... ...Ei iE. pE :@. .j)! j jtjr ! . @ )..,j.. .r.yyjgtjy.ry(.j.)). E . jq . j( (: C1!i':t)E ; t ytjy;..y. yyy t ..'@t--It)-...y.yq.jy ljjjjj!j, .. j. ..yjjjjq:js jjjjjs j.yy IE .yygyy.q . ):j;;iiqf')., y y yy i.. . (E y. j..j..j.yyg jjy. jjL...,...,...j...-,..,.,.t= .. .yjjyjyy . .. . .' jyy .... .-. (' .

.''

rr-trtr'jjy;f : .

E

where D is modifed duration. Here is an example of computing a bond price at a new yield using both duration and convexity.

Example

=

-jj,'.

We can use convexity in addition to duration to obtain a more accurate prediction of the new bond price. When we include convexity, the price prediction formula, equation 15 (7.12), becomes 6)

(7.15),the price at a yield of 7.25% is

:';'F' -' y'j'ykytjj,'. lji'j,yyr r;--jjyjr')E '!)f I'E :'i':f'

(7.14)

-

10.48 12

=

Figure 7.4 illustrates duration and convexity by comparing three bond positions tathave identicalprices atayieldof 10%. Duration is the slope of thbond price graph at a given yield, and convexity is the curvamre of the graph. The 10% lo-yearbond has the lowest duration and is the shallowest bond price curve. The othertwo bonds have almost

changes.l'Coltvexitymeasures the extent C anges. The formpla for convexity is14 h =

81.63

The predicted price of $81.060is the same as te acttlal price at a yield of 7.25%, to an accuracy of three decimal points. ln example 7.7, the predicted price was slightly lower ($81.058) than the act'ual new price. The difference without a convexity correction occurs because the bond's sensitivity to the interest rate changes as the interest rate changes.l6 Convexity corrects for this effect. %.

Q

-0.004

-1.735

1

x

.07(3+2)

$81.63 (3/1.07)x $81.63 x 0.0025 + 0.5 x 10.48 12 x $81.63 x 0.00252 = $81.060

(7.25%)

.

$100,convexity of the bond is

=

100 l

%.

AN D colqvExl-ry

.

-

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Comparison of the valueof three bond positionsas a function of the yield to maturity: 2.718 1O-year zero-coupon bonds, one lo-year bond . paying a 10% annual coupon, and one 25-year bond paying a The 70%coupon. (D) and duration (c) of each convexity yield of 10% at a bond in the Iegend.

are

.

.

y

y

.

jrjj,skyjj j:gy)j jlyg)j ,(r2: j:yg)j qj.rjjs j:r:jj gjrr;js (yjj,

,-,,,,,,,,,

1.s 1.4 1

'

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a

-

egj 1.2 ;

1.1

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.

.......jfj;;,.

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=

=

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-w

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*'->.

'j

;:

j:gyjj j;jgj ijyjr.tj:y gjrg;j lgygjts gjj;;j (o n (s c no.91) 1o% coupon lo-yek bond CD 6.14, C 52.79) 10% Coupon 25-year bond CD 9.08, c 139.58) =

1 -39 ll-tltl (19-()9519-19

pts rznds cf 1Cl :15-(5515-B 1,333758 l',..ts l5-M 35,374 Dt t -17,W1 Est vl 723,779;vell'cil 3446117npenInt 14368,7524 Qf 5 Yr. Tfeaeury Notes ((BD-$1M1;pts 32* lr 11$41 llthl 1:%.13 :1.1+1.,q -1M 112-15 IA.Z.9 l,lszJM 5ept +3,$$1. Est ynl 408,924;vnIhlon1l,223; cpQR lnt 1,7:5.116, 2 Yr. Treasury Notes ((B.f'b$21%; ;ts 311s ef 11 7 l.tls4l (+1.87 137,212 s'pt 95-235 195-21 (5-162 195-17 vcIhlrt MJM; Q)eR IRt 127,21% Est vcI29,3224 -21.9

-1,03.

note is a commonly used benchmark interest rate, hence

.

(Futures plice x conversion factor) + acclued interest

-2

.-4

interest rate on the lo-year Treasu!'y l-/-f'he thelo-year note futures are important.

This description masks a complication that may already have occurred to you. The delivery procdure permits the short to deliver any note maturing in 6.5 to 10 years. Hene, the deliverd te can be one of many outstanding not:, Witha rane of oupons Which bond does the futures price represent? andmturities. . atl bonds Of that ttltl be delivered, there will generlly be on: that is tlie most advantageous for the short to deliver. This bond is called the cheapest to deliver. A description of the delivery procedure will demonstrate the importance of the cheapestto-deliver bond. In fulflling the note futures contract, the short delivers the bond in exchange for payment. The payment to te short-the invoice prc for the delivered bond-is the f'utures price times the conversion factor. The conversion factor is the price of te bond if it were priced to yield 6%. Thus, the shol't delivering a bond is paidlB

Example 7.10 Consider two bonds maling semiannual coupon p'ayments. BondA 7% bd Vtll exactly 8 years to maturity, a price of 103.71, and a yield of is opon bond would have a price of 106.28 if its yield were 6%. Thus its conversion 6.4%. factor 1.0628. Bond B has 7 years to mattlrity and a 5% coupon. Its current price and yield are 92.73 and 6.3%. lt would have a conversion factor of 0.9435, since that is its plice at a 6% yield. % .

'

.

'

.

.

.

.

'l'his

ts

l8A PPendix 7.A contains a definition of accrued interest.

$

REpuncl-lAsE

AN D FUTURES

I NTEREST RATE FORWARDS

Now suppose that the fumres contract is close to expiration, the observed futures price is 97.583, and the only two deliverable bonds are Bonds A and B. The short can decide which bond to deliver by comparing the market value of the bond to its invoice price if delivered. For Bond A we have

lnvoice price

-

1.0628)

market price

=

(97.583x

market price

=

(97.583x 0.9435)

103.71

-

0.00

=

For Bond B we have Invoice price

-

-0.66

92.73

-

=

These calculations are summarized in Table 7.5. Based on the yields for te two bonds, the short breaks even delivering the 8-year 7% bond and would lose moneydelivering the 'F-year 5% coupon bond (theinvoice price is less than the market price). In this example, the 8-year 7% bond is thus the cheapest to deliver. ln general there will be a single cheapest-to-deliverbond. You mightbe wondering why both bonds are not equitlly cheap to deliver. The reason is that the conversion factor is set by a mechanical procedure (the price at which the bond yields 6%), taling no coincidence, two account of the current rlative market prices of bonds. Except by deliver. cheap be equally will to not bonds Also, al1 but one of the bonds must have a negative dlivery value. If tFo bonds ontf no-arbiage had a positive delivery value, then arbitrage would be jossible. The confguration in general has one bond worth zero to deliver (Bond A ip example 7.10) and the rest lose money if deliverd; To avoid arbitiaje, the fumres prie is Plice of cheapest to deliver Fumres price + Conversion factor for cheapest to deliver =

,'-1I:i2t:' )' E' (' ,;' '77*; k' yrjj;;f j,y,,.f jj,,.,rjjf 'j't' Eklkqd ,'y' y'. if )' .jjjyy' .,,jg;,' ,,,,,,,,,* .jjjyy.' g,,,,.yjjf ;,y,.,' q' .y'. .y-t' .t'.y ,.' (,,1:* -1iiE:jr'' (' i';'(51* q:''''r(' j'j-;,t'! tif jjr:::,,f jtf E' ' ' i q ' @ ' r' rtf jjj,,,:j,f jjjf y' yyyf '( ('( i(q. '(' ( ' '7q:77r)C7:' i:@idI.'. (jjifd -1I::iiiI!.' -1IiiiidI:)' Eiilrd tk::iiIlt'. tilli::E::ll.'d )'. .g:yj,,,,,' .jy'-y'. .j' ( E'' TE 'q'(! .i!!i,ii:ipE;;;'(' )j'': yyqqti'l yy'' y'. .( (E. '(. !'' ( ' !. t'( (.(.' !' ' r '' ' i IE i rj. ;' ;li )' lii:d lq';j ')yj' y' L-j2L,,LLL-'. .-.LLj;itqq?:. ' j..y:.. :.;.yy;' ' . ' q.q. . ..y ; 'jj..(.g( ..y (.(iy.t.. jjjjj;;y q .r'.'. r.r'. . ;;. l). k'tlylljt'i y.:jqjjrr#.. y yj .qy.yj. y.r..yyj'j y j .;j.y j y .. . ty y j j. j y . . . j jjjj)k. y . y y . jjyy. yj )jy j j y. y. ) .,. .y. y.y;-,-tyyyjy,yj,y(,. jyjj.y t-. j jj-jj))... jjijhg ..j,q;jtL:!'.!' (@ i . .. .j.y.j.....-.jj...(' . )' . ... . .. --r' 5* -;jyy,;' 'it:?!,' F' :'(' j,:yyrrjjf jj:ry:r,f j,:y:gy,,f jj::yyyyf jjygjg,,f (' -.i,kb3'. -'''j,::gy:jjf lrf j,yjyg,,f jjyyyg,,f jjyjjg,,f tj'''''q'q ' :: ' !@ EIIZE !: E:C.i E ': E . . !. ).)Fyi q;Eq).j ..'.,,' ,;jjjjj;jjj'. .,,,' ,,,jg;y'

.'

.' .'. E '-

.

.

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-

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1E. -E ..

.

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,

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-

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.

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.

-

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-j(., ..j,tyyjy.

).(L.

E

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... . . .. . ..

.

..

y .-)..-)j........... -.

j -

,

.

.y . y

.- -

.rjk::y kjjyg,j, j,,,.. jj::ggr,,,

jj::rrr,,;!jj;;y

,!jk,,;,,

-,,,,.y,,. ,,,,

.

jj:yyr:,,, xjjjyyy-jj-,,-jj,jj,,

,,::yy:,,,

jjjjjjj ),jjkkjj$)jkkjk;.. . (k;gj:. ..

,

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,ijjj)j;.jj:g!g.j;gjkgy.

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103.71

Price at 6% (conversionfactor) Invoice price (futtlresx

106.28

conversion factor) lnvoice market

103.71

-

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233

This discussion glosses over subtleties involving transaction costs (whetheryou already own a bond may affect your delivel'y proft calculation) and uncertainty before the delivel'y period about which bond will be cheapest to deliver. Also the T-note is deliverable at any time during the expiration month, btg trading ceases with 7 business days remaining. Consequently, if there are any remaining open contracts during the last wek of the month, the short has the option to deliver any bond at a price that might be a week out of date. This provides a delivery option fof the short that is also priced into the contract.. There are other complications, but sufice it to say that the T-bond and T-note contracts are complex. The T-bond and T-note futures contracts have been extremely successful. The contracts illustrate some importantdesign considerations for afumres contract. Consider srst how the contract is settled. If the contract designated a particular T-bond as the underlying asset, that T-bond could be in short supply, and in fact it might be possible for someone to corner the available supply. (A market corner occurs when someone buys most or all of the deliverable asset or commoditjc) A short would then be upable to obtain the bond to deliver. ln addition, the deliverable T-bond would change om year to year and the contract would become more complicated, since traders would have to price the f'utures differently to re:ect different underlying bonds for different maturity

dates. An alternative scheme could have had the contract cash-settle

against a T-bond

index, much like the S&P 500. This arrangement, however, introduces basis risk, as the T-bond futures contract might then track the index but fail to track any particular bond. In the end, settlement procedures for the T-bond and T-note contracts permitted a range of bonds and notes to be delivered. Since a high-coupon bond is worth more than an otherwise identical low-coupon bond, there had to be a conversion factor, in order that the short is paid more for delivering te high-coupon bond. 'T'he idea that there is a cheapest to deliver is not exclusive to Treasury bonds. The same issue arises with commodities, where a f'uttlres contract may permit delivery of commodities at diferent locations or of different qualities.

jj,yyyrjjj

The futures price is 97.583. The shor't would break even deliveringthe 8-yer 7% bond, and los moey deliveringthe 7-year 5% bond. Both bonds make semiannual coupon payments.

'y.

Et

k.

jj:r:y:j,,

,rjj::.

jj::ry:jjy

jjijjj

$,::)y,,

...jj.j;

.

-'.''Market price -

'' EE ' : 'E E

E

-

.,j;j;y E y..-).....yi gjj(jjgy.. .:.i.:.l E...i i..'.kjjjyyjjj. (r,.(..j: E.'t..)....t....t kk-j. .,1tjjjjjjj;.. ' :ji.,y'' kt(jrjj4j;,. '. . ...E..y. ..' '..'. '...'i....''. . .,1;..1j. .'. . ..'..'.... '.:.. )((E':jk;ii,.,k:g .jjjkkj,..... '.E- .r..:.. jrgjj.' ..: .. )tt...'t.'.'E '....t.'.(y.g. '.'r. (r.Ey kjryjj.. .'r .t(..E..E .......... .-.r... .t''r!!iiii@r'''. E' . ! E . ... (E . . .'...t j o (,.r;, ! i ;E E(E (( i qiyrty, i E ltt q!. ;7, yjtyyjy (E ( ( )) )( r.,j.tji -1k5111,). -''ti!ik!i''-. y )( l ; ; ( (E( yjr, ; ) ' i Lq E ; ) y (E g. j.. y,,t . -. . . . ;,[email protected]' ! : . . -. - ' '. 'r!IIEqr7'' .. - -.1.tE1.k2.. :11I::.:k1iE '. .- . . .. - . .. .. . . -.-'. .. . .. -11:E@),,' . ). r pllEgt!l,p. r. . . . . . . .(.E (i..E . . . . . ; . . . lllkqt,,,;..... .t(y.. . ..r- . .' -. r...(.-- .... .-.. . .r...;-.... ......:.... ..(.... p.....-.. ... -. .. -. .-. (. -.:. . ... ... y- - , ....-.g.. (....... ..... ... .....--. ... ;. . ...'

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AGREEM Elq'rs

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....

92.73 94.35

...,

.(.

,

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. - . ... .

7.5 REPURCHASE

AGREEMENTS

An extremely important kind of forward contract is a repurchase agreement, or repo.19 A repo entails selling a security with an agreement to buy it back at q tixed plice. It is sale coupled with a long folward contract. Like effectively a reverse cash-and-carry-a it is equivalent cash-and-carry, to borrowing. The particular twist with a reverse any repo is that the underlying security is held as collateral by the counterparty, who has bought the seculity and agreed to sell it at a xed price. Thus, a repo is collateralized bon-owing. Repos are common in bond markets, but in principle a repurchase agreement can be used for any asset.

92.09 -0.66

l9For a detailed eeatment of repurchase agreements, see Steiner (1997).

'

,1'* ''' .''' ''''''

j'

% INTEREST

234

RATE FORwARDS

AlqD FUTURES

CHAPTER

!q Example 7.1 1 Suppose you enter into a l-week repurchase agreement for a 9))i)tk. and .. . The cunent price of the T-bill is $956,938, you agree to t month $1m Treasury bill. C repurchase it ip 1 week for $958,042. You have borrowed mony at a t-Week rate of ')q tl. 958.042/956.938 1 0.1 15*. receivine cash todav and oromisine to repay .cash plus (j. )'t interest in a week. The security provides collateral for the loan. k. -

=

.;1..

11..'...'*

''

:i

16.,,...3'.

.

The party who initiates the repo owns the asset when the trasaction is corpleted and is therefore the snancial owner of the security. During the repo, however, the counA longer-tenu repuychae areement tel-party Owns the bond. Most repos are Overnight. is

Called

a term repo. ' . .' . . . The eounterparty is said te have entered into a reverse repurchase agre:ment, or contraci. This is a loan of cash for th'duration reverse repo, and is short the fol-ward of the agreement with a security held as collateral. lt can also be desribd a a cashan

.

0

.

-cal'ly

If the borrower does not repay the loan, the lndef keeps th secutitf. Thus, the counterparty's view of the lisk of the transaction differs according to the quality of the collatei-al.Collateral with a more variable plice and a less litjuid market is lower quality from the perspective of the lender. Because collateral quality varies, evel'y security can have its own market-determined repo rate. government securities, and can Repurchase agreements are most common collateral collateral--called security a spcial as be negotiated to require a specic collateralsecurities of of vadety with any government as a rcpltrc/?tueagreenent--o ?'eptl?'c/?/.d agreement. Generi collateral repos have greater called a general collateral :exibility and hence lower ansaction costs. The repo rate on special collateral repos will generally be below that on general collateral repos. Suppose that there is demand for a specisc bond' as a speculative investment. The owner of such a bond can engage in a repurchase agreement, nd high low repo rate means that the demand for the bond will drive the repo rate down. cash received the for the bond that exceeds the original bondholder can ea.rn interest on . bond. of specialness te repo rate, thereby profting from the In addition to a repo rate that reQects collateral quality, dealers can also char/e a haircut, wlaich is the amount by which the value of the collateral exceeds the arpount of th e l 0an. The haircut reflects the credit lisk of the borrower. A 2% haircgt would mean that the bon-ower receives only 98% of the market value of the security, providing an additional margin of protection for the counterparty. Repurchase agreements are frequently used by dealers to Enance inventory. In the ordinary course of business a dealer buys and sells securitis. 'Fhe puycflase of a jecurity requires funds. A dealer can buy a bond from a customer apd thep repo it ovruight. The money raised with the repo provides the cash needed tl pay te sller.. 'I'he dealer then has a cost of carrying the bond equal to the repo rate.20 The counterparty on this .for

20The repurchase agreement in this exgmple provides financing. The dealer still is the ultimate owner of the bond and thus has plice risk that could be hedged Wt.h futures con%cts.

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eansaction is an investor with cash to invest short-term, such as a corporation. The investor buys the bond, promising to sell it back. This is lending. The snme techltiques can be used to finance speculative positions. Hedgefunds, for example, use repurchase agreements. A hedge f'und speculating on the price difference between two Treasury bonds can fnance the transaction with repos. An example of this is discussed in the Long-rrerm Capital Management box on page 236. How do we engage in a transaction lilie this-long bond A and short bond B-in practice? The answeris thatweundertake thefollowingtwo transactions simultaneously: The longposition: Buy bond A and repo it. Use the cash raised in the repo to pay for the bond (recallthat dealers nance wentory in this fashion). When it is time to reverse the repo, sell the bond and use the cash raised from the sale to buy the bond back and close the repo position tthink of the sale and close of the repo as happening snultaneously). Note that a low repo rate for this bond works to the arbitrageur's advantage, since it means that the reptlrchase price of the bond is low. The arbitrageur also benefts from a price increase on te bond. Borrow bond B by entering into a reverse repurchase agreeThe shortposition: ment. We receive the bond (collateralfor the loan) via the reverse repo, sell it, and use the proceeds to pay the counteparty. At the termination of the agreement, buy the bond back in the open market and ret'ul'n it, being paid the repo rate. Since we receive interest in this transaction, a high repo rate works to our advantage as does a price decrease on the bond.

Since the investor is betting that there will be a reduction in the plice difference between the two bonds, it is necessazy to enter into both legs of the eansaction. The arbitrageur would like a low repo rate on the purchased bond and a high repo rate on the sold bond, as well as a price increase of the purchased bond relative to the short-sold bond. ln practice, haircuts on bot.h bond positions are a ansaction cost. Haircuts yre a capital requirement imposed by the counterparty, which means that an arbitrageur must hkve capital to undertale an otherwise self-hnancing arbitrage transaction. Differences in repo rates on the assets can be alz additional transaction cost. Even if the plice gap between the two bonds does not close, the arbitrage can be prohibitively costly if the difference inrepo rates on the two bonds is suciently great. Cornell and Shapiro (1989) document that in one well-lnown episode of on-the-rurl/off-the-run arbitrage (seethe box on page 236), the repo rate on an on-the-run (short-sold) bond went to zero, making arbitrage costly even though the price gap remained when the on-the-rtm bond became off-the-run.

The price of a zero-coupon bond wit.h F years to maturity tells us the value today of $1 to be received at time F. The set of these bond prices for different maturities is the zero-coupon yield curve and is the basic input for present value calculations. There are

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Basic interest rate concepts, are ftlndamental in Nnance and ar used throughout this book. Some of the fonpulas in thts chapter will appear again as swap rate calculations in Chapter 8. Chapter 15 shows how to prie bonds that mak payments denominated in foreigncurrencies orcommodities, andhow topricebonds containing options. Whilethe bond price calculations in this chapter are useful in practice, concepts such duration as have conceptual problems. ln Chapter 24, we will Fee how to build coherent, internally a consistent model of interest rates and bond prices. Useful references for bond and money market calculations are Stigum (1990)and Stigum and Robinson (1996).Sundaresan (2002)and Tuckman (1995)are sxed-income int topics in this chapter i more depth. Convexity bias is studied by texts tat Burghardt and Hoskins (1995)and Gupta and Subrahran#am (2000). Grinblatt and Longs'taff (2000)discuss the market for STRIPS and smdy the pricing relationships between Treasury bonds and STRIPS. The repo market is discussed in Fleming and Garbade (2002,2003, 2004). .go

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Forwardrate agreements (FRAs) permitborrowers and lenders to hedge theinterest by locking in the implied fomard rate. lf the interest rate changes, FRAS require a rate payment re:ecting the change in the value of the interest rate as of the loan's laturity day. Eurodollar contracts are an alternative to FlkAs as ahedging mechanism. However, Eurodollar contracts make payment on the initiation date for the loan rather than the maturity date, so there is a timing mismatch between the Eurodollar payment and the interest payment date. This gives rise to convexity bias, which causes the rate implied by the Eurodollar contract to be greater than that for an otherwise.equivalent FlkA.rfreasury bill contracts are yet another possible hedging vehicle, but suffer from basis risk since

PROBLEMS 7.1. Suppose you observe the following zero-coupon bond prices per $1 of maturity payment: 0.96154 ll-yearl, 0.91573 (2-year), 0.87630 0.82270 tzyearl, t3-yearl, 0.77611 ( ' For each matulity year compute the zero-coupon bond yields (effective armual and continuously compounded), the par coupo rate, and the l-year implied forward rate. -year).

7.2. Using the information in the pmvious problem, nd the price of coupon a s-year bond that has a par payment of $1,000.00and annual coupon payments of $60.00.

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The swap counterparty is a dealer, who hedges the oi1price risk resulting from the swap. The dealer can hedge in several ways. First, imagine that an oil seller would like to lock in a fixed selling price of oil. ln this case, the dealer locates the oil buyer and seller and serves as a go-between for the swap, receiving payments from one party and passing them on to the other. In practice the sxedprice paid by the buyer exceeds th tixd price received by the seller. This price difference is a bid-ask spread and is the dalef's fee. Figure 8.3 illustrates how this transaction would work with financial settlement. 'he o il seller receives the spot price for oi1 and receives the swap price less the spot price, on net receivipg the swap price. The oil buyer pays the spot price arpd yecives the spot price less the swap price. The situation where the dealer matches the buyer and book'' transaction. The dealer seller is called a back-to-back transaction or bears the credit risk of both parties but is not exposed to price risk. A more interesting situation occurs when the dealer serves as counterparty and hedges the trapsaction using forward markets. Let's see how this would work. After entering the swap with the oil buyer, the dealer has the obligation to pay the price and receive the swap price. lf the spot price rises, the dealer can lose money. spot oil. The dealer has a short position in 1- and z-year The nattlral hedge for the dealer is to enter into long forward or fumres contracts to offset this shol't exposure. Table 8.1 illustrates how this strategy works. As we discussed earlier, there is an implicit loan in the swap and this is apparent in Table 8.1. The net

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The swap price, $20.483,is close to the average of the two oil forward prices, $20.50. However, it is not exactly the same. Why? Suppose that the swap price were $20.50.The oil buyer would then be commiitlng $t.50less than the to pay $0.50more than the forward price the srstyear and would pay /'/?t? relative Thus, price the second r/lc/t/rlszt-lpz bttyer J$)()If/# have forward to cttrve, year. the There implicit lending in the swap. iltterest-h-ee Ioan ct/lfpl/eprclr/y. is to p77t'l#c an Now consider the actual swap price of $20.4831m1e1.Relative to the forWard curve prices of $20 in 1 year and $21 in 2 years, we are overpaying by $0.483 in the first year and we are undepaying by $0.517 in the second year. Therefore, the swap is equivalent to being long the two forward contracts, coupled with an agreement to lend $0.483 to the swap counterparty in 1 year, and receive $0.517in 2 years. This loan has the effect of equalizing the net cash flow on the two dates. 1 7%. Where does 7% come The interest rate on this loan is 0.517/0.483 from? We assumed that 6% is the l-year zero yield and 6.5% is the z-yearyield. Given these interest rates, 7% is the l-year implied forward yield from year l to year 2. (See Table 7.1.) By entering into the swap, we are lending the counterparty money for 1 year beginning in 1 year. ((fthe deal is priced fairly, the interest rate on this loan should be the implied forward interest rate.

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The Market Value of a Swap 'When the buyer hrst enters the swap, its market value is zero, meaning that either party could enter or exit the swap without having to pay anything to the other part.y (apartfrom commissions and bid-ask spreads). From the oil buyef perspective, the swap' consists of two forward contracts plus an agreement to lend money at the implied fol-ward rate of 7%. The forward contracts and forward rate agreement have zero value, so the swap does as well. Ohce the swap is struck, however, its market value will generally no longer be for two reasons. First, the forward prices for oil and interest rates will change over zero, time. New swaps would no longer have a fixed price of $20.483;hencq, one party will tmvind the swap. owe money to the other should one party wish to exit or Second, even if oil and interest rate forward prices do not change, the value of the ltlttil theslist payment is ?pltz#c. Once the first swap swap will remain zero only payment is made, the buyer has overpaid by $0.483relative to the forward curve, and hence, in order to exit the swap, the counterparty would have to pay the oi1buyer $0.483. Thus, even if prices do not change, the market value of swaps can change over time due to the implicit borrowing and lending. A buyer wishing to exit the swap could negotiate terms with the original counterparty to eliminte the swap obligation. An alternative is to leave the original swap in place and enter into an offsetting swap with whoever offers the bejt pri. The priginal swap called for the oil buyer to pay the fixed price and receive the floating tze; the offsetling swap has the buyer receive the fixed price and pay floating. The original. obligation would be cancelled except to the extent that the xed prices are different. However, the difference is known, so oil price lisk is eliminated. (There is stitl credit risk whe the original swap counterparty and the counterparty to the offsetitng swap are different. This could be a reason for the buyer to prefer offsetting the swap with the oliginal counterparty.) To see how a swap can change in value, suppose that immediately after the buyer lises by $2 in years 1 and 2. Thus, the year-l enters the swap, the forward curve for oi1 forward. price becomes $22 and the year-z forward plice becomes $23. 'I'he oliginal swap will no longer have a zero market value. Assuming interest rates are unchanged, the new swap plice is $22.483.(Problem 8.1 asks you to verify this.) The buyer could unwind the swap at this point by agreeing to sell oil at $22.483,while the original swap still calls for buying oil. at $20.483.Thus, the net swap payments in each year are 's

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swap original

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E

254

% SwAps

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RATE swAps

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lf interest rates had changed, the new swap price. uting P

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The examples we have analyzed in this section illustrate the fundamental characteristics of swaps and their cash flows. In the rest of the chapter, we will compute more realistic swap prices for interest rates, currencies, and commodities and see some of the ways in which we can modify the terms of a swap.

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Cgmpanies use interest rate swaps to modify their interest rate exposure. ln this section we will begin with a simple example of an interest rate swap, similar to the preceding oil swap exapple. We will then present general pricing formulas and discuss Ways in wlzichthe basic swap structtlre can be altered.

However, the interest payment on the loan is due at the end of the yem The interest floating interest payment would therefore occur at the rate determination date for beginning of the period. As with an FRA we can tlzink of the swap payment being made at the end of te period (wheninterest is due). With the finaneiatly settled oil swap, only net swap payments-in this case the difference between LDOR and 6.9548t:,-.-:/e actually made between XYZ and the counterparty. lf on party defaults, tey owe to the other party at most the present value of net swap payments they are obligated to make at current market prices. This means that a swap generally has less credit lisk than a bond: Whereas principal is at risk with bond, only net jwap,payments are at lisk in a swap. The swap in this example is a constnlct, making payments as (f there were an exchange of payments between a sxed-rate and soating-rate bond. ln prtice, a fund might bonds fixed-rate and wish own Eoating-rate manager to have exposure while continuing to own the bonds. A swap in which a fund manager receives a qoating iate in exchange for the payments on bonds the fund continuej to hold is called arl asset swap. 'te

7kSirnple Interest Itate S4vap that every Suppose that XYZ Corp. has $200mof Coating-rate debt at LtBoR-meaning would preferto that Lolkvbut xed-rate debt with year's current have pays could effect mattlrity. There several XYZ change. this 3 years to are ways First XYZ could change their interest rate expospre by retiring th floatipg-rate debt and issuing Exed-rate debt in its place. However, an actual purchase ad sale of debt has transaction costs. Second, they could enter into a strip of forward rate agreements (FRAs) in order to guarantee the borrowing rate for the remaining life of the debt. Since the FRA for each year will lpically carry a different interest rate, the company will lock in a different rate each year and, hence, the company's borrowing cost will vary over time, even though it will be fixed in advance. A third alternative is to obtain interest rate exposure equivalent to that of fixed debt by entering into a swap. XYZ is already paying a Qoating interest rate. They rate therefore want to enter a swap in which they receive a Qoqting rate and pay yhefixed rate, which we will supposr is 6.9548%. This swap is illustrated in Figure 8.4. Notice the similarity to the oi1 swap. ln a year when the fixed 6.9548% swap rate exceeds l-year LIBOR, XYZ pays 6.9548% LIBOR to the swap counterparty. Conversely, when the 6.9548% swap rate is less than LIBOR, the swap counterparty pays LIBOR 6.9548% to XYZ. On net, XYZ pays 6.9548%. Algebraically, the net interest payment made by XYZ is

yearxYz

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=

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The notional

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-

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-6.9548%

=

Swap payment

$200m: It is the amount on which the interest

payments-and, hence, the net swap payment-is swap term or swap tenor.

based. The life of the swap is the

There are timing conventions with a swap similar to those for a forward rate agreement. At the beginning of a year, the borrowing rate for that year is known.

pricinjan

ie

swapcounterparty

To understnd the pricing of the swap, we will examine it from the perspective of both the counterparty and the firm. Wefirst consider the perspective of the counterparly who we assume is a market-maker. The market-maker is a counterparty to the swap in order to earn fees, not to take on interest rate lisk. Therefore, the market-maker will hedge the transaction. The marketmaker received th fixed rate from the company and pays the ioating rat; the danger for the market-m ke i is that the Qoating rate will rise. The risk in this eansactioncan be hedged by entering into forward rate agreements. We express the time 0 implied forward rate between time ti ad tj as rfjti tj4 nd ihe realized l-yer tate as 15 The currenl l-year rate, 6%, is known. Wit.h the swap rate denoted R, Table 8.2 depicts the lisk-free (but time-varying) cash flows faced by the hedged market-malcer. How is R determined? Obviously a market-maker receiving the fixed rate would like to set a high swap rate, but the swap market is competitive. We expect R to be bid ,

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% swAps

256

INTEREST

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R 8.0071% 6% R 7.0024% 0 + + 1.0652 1.07-6 1.06 This formula gives us an R of 6.9548%, which from Table 7.1 is the same as the par 3-yeaT bond! In fact, our swap-rate calculation is a roundabout way coupon rate on a bond yield. On reqection, this resuti shuld be no sul-prise. Once the compute to a par borrower has entered into the swap, the net effect is exactly like borrowing at a Exed rate. Thus the fixed swap rate should be the rate on a coupon bond. Notice that the unhedged net cash flows in Table 8.2 (the swap payment'' column) can be replicated by borrowing at a Qoating rate and lending at a hxed rate. ln is d#?l/p//d?l? other words, all interest rate to lvt?/'rt/ld?l.at ajloatilg rate to /plf)? a l-'rcl-/w/d bolld. R

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that the borrower could also have used forward rate agreements, locldng in an escalating interest rate over time: 6% the year, 7.0024% the second, and 8.0071% the third. srst By using interest rate forwards the bonower would have eliminated uncertainty about future borrowing rates and created an uneven but certain stream of interest payments over time. The swap provides a way to both guarantee the borrowing rate and lock in a onstant rate in a single transaction.

Computing the Swap Rate

8.0071%

-

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ttnet

The borrower's

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just the opposite of the market-maker's.

borrower continues to pay the Qoating rate on its :oating-rate debt, and and pays fxed in the swap. Table 8.3 details the cash flowsk

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*'e.. .-......'.... '...i x''%.....,.,.L. '.):).......'E..E. ... ..iE't....... . ....

6.9548%

-6.9548%

6.9548%

-6.9548%

6.9548%

-6.9548%

.

.. (.....

.

.

,

,

31

P (0, ti )(A

r()tp-l

-

,

i= l

ti )j

0

=

where there are ?? paypents on dates /1 lz, tv. The cash 'llows R 0(1/-1 tij can also be obtained by buying a fixed-rate bond paying R and borrowing at the Qoatingrate. Equation (8.1) can be rewritten as ,

R

=

.

.

.

=

,

,

)-)'j!.j p (0 tt )r (/)-j tiv) ,

17-I,?

,

#(0, t)

The expression 'F71.1 /7(0, tilrt-L ti) is the present value of interest payments implied by the syip of fonvard rates. The expression .P(0, ti) is just the present value of a $1 annuity when interest rates vary over time. S)'=j Thus, the swap rate annuitizes the interest payments on the floatingvratebond. We can rewrite equation (8.2)to make it easier to interpret: ,

11 t'(' jyyy;:jgf j';'jy)f '' i hf j'k,j-'r' k;jyy,rf (,jj;;' ;r' j,y,,.:,jf qE #;'. (r' Tq'ly'l'f .jjyr)' )' k' ,'t'iikiiIiir7i:'' :!jE;;' .jjyyy' 'q' (t' q' ( ')! q'll:!' '' i. (g'E.ik j!gjllgj;f .!jy::' .!jj::' yf )' ( jij:E(((.(';'.('' .. .'(qEi';'.:;'((.. 'r)('q.E'q: (',: '.)( (. i';.(((. ( (E.(E(.Eq''.' '..'' ('('.'E''(iqj.7?'jyyq r' '(Ejjjjjj,' jjf j;:yyy;,,f ijIi(:-'. IE (j.7(! )' t')'((';..* yky'. ljtjtf y' j'j-'-' ' : '..i' ij!!q'j.i ))1 'q'..i.('(i (r(@' , E - - E- E -.; ;)j;' .E kL.qk E :- . . Er:....E :. EE i $. ..E '. . -; E :. .(;; . y ! j ) jy--. i r. ( y . . . q . . y q j q yj.', . . q ( : y ( 71.1j ;-7tir . g . ( y ( y . . :.yyy jg . - j(-y..y .yyy..jy (j-,j-k. . . . . .q ;. . . .. rr .lkjj;y yt,yg.k(. (.r-..- . ; j.,.--(.(i. .. r . ...r. . . . ..

.

,

R

'=''*',,. '?q' ,,j-,,,-jj,,,,'

General

We now examine more carefully the general calculations for determining the swap rate. We will use the interest rate and bond pfice notation introduced in Chapter 7. Suppose there are ?? swap settlements, occurring on dates ti, i /?. The implied forward 1, interest rate from date ti-k to date ti, known at date 0, is rztil ti). (We will treat /-()(4-1 ti) as l'qt having been annualized; i.e., it is the return earned from ti-L to ti. The price of a zero-coupon bond mamring on date ti is #(0, ti). The market-maker can hedge the ioating-rate payments using forward rate agrements. The requiremeny that the hedged swap have zero net present value is

uwt?p

.'!'.........#''>'-.. -' -t'( ,,j,,;,,kk,,,,' ;'. -' ;''''7j'F' j,:yyr:j,f ,j:yyyr,' jj::rr;,,f :'1** j,:ry::,,f '-f

257

:

.

.

.

'

' ' .

'

:

.

'

r=

P (0, ti ) .P(0, tj3 >-.ly-l tt

i= l

r ti

-

j

,

ti )

Since the terms in square brackets sum to one, this form of equation (8.2)emphasizes that the lixed swap rate is a weighted average of the impliedfomardrates, wherezero-coupon bond prices are used to determine the weights. There is another, equivalent way to express the swap rate. Recall from Chapter 7, equation (7.4),that te implied folavard rate between times and tz, ?'()trl r2), is given rl by the ratio of zero-coupon bond prices, i.e., ,

?-()trl?2) ,

=

#(0,

/.1)/#(0,

?2)

-

1

.

%. SWAPS

INTEREST

.'

%. 259

RATE swAps

-1::::211)* (,:,,f -,:y,,' :7;7!7)* 'k' 2* F )' )' ''-,,,ll:f (' ')-).'.?)' q' ;'qyf 1* ,'1)* #' E')' Irr,'.-.-----'---,-.f ll:..''l:...!lkf 'I-..,!l.f t-' p' (' j'j't'q'.$1k::.' .tIk::.' t')'y'ygjjjl,,.d ii)f iizf r' r'''qrf IIL.,.,.X liq::::,'f IIEEE!E,X (:lr,----.f 11:::::.,* r'. .y' p' 'j' yyt'. j'g' ...y..yy...g.y.yy...y..y.:yy....g.yg...y........j....jg.g.yyy' )' y' .yt' ''75* p'y .k-;-'; Ei(i:(( ;' ( ( (q E '' ( E t'l'y .'q'(!'7*q.' (iqj:'(i( );.;@.;:!i,i ''67-,r(.!..' '.E E(:''7t': '7')77(7,'E qrl .i q: ' ('.'!((';; . . ( (!( . ;i. (. . .. -.. . ! l:. !:.;.k . ;. ;'..iiEEi...it. . .y;.y'y, (.y'j., 't'.)l;y(?.r))t(t77IIE.7..-. -j)). .y q qj :j. (. y-;yy(-' - ; '. ' ; j . .y ry y y t y y))y y y. yyyyy)-. ryj,..jjy.yy-. )' y.yy lr jy.yj )yj rysl iliiiiii 1.*:..11. r. y.k,,,)r(i.i;yj,,,,. ,I,,,,,1r,,,,., .y:-,,.j;;;-..;-..,...,-;;);;;,-L-)-.. ll....:ll y..y..j j;,,,.,,j 11r ;;!i55.L. IIEE!!!.IIF::EI:,. .1It::. . . .. .. Il....,ll y..:1Iq-.a l.r.'. Il:.,. . .... s . . y., '')' -j' ;' 'j''l.:i!,,-f -ls!!,,-f -1:::::11:* -.:i!!,-f -1:::::.1:* 'l:::::,,f -.' ;'11* .,,,)jr..,.' -'

Therefore equation ?1

(8.1)can -

.''.

be rewritten

E i -i .

11

P(0, ti4LR

rt-k

,

ti )1

.P(0, t)

=

R

-

P (0, /f-I P (0,t)

i= 1

i= l

)

+ 1

-

=

x-x?,

---.--k;iiiki..

- . -.

..

.

..............

'

.

..

.

.......

.

-

.

.

.............

. .

' '' ' ' E : EE E .E : ' : . ' : E ' E ( . . .E.. : E ... .. : . . . -- - - -. .. . . -. . -..

........

......

' '

E

'

: ..

.

=

l

This is the valuation equation for a bond priced at par wit.h a coupon rate of R. The conclusion is that the rate is l cottpon rate t??l a pJ?- coupon !747,?4@. This' result is intuitive since a firm that swaps frortl floating-rate to exposure sxed-rate bond. ends up wit the economic equivalent of a sxett-rate 'wcw

The Swap Cttrve As discussed in Chapter 5, the Eurodollar f'umres coneact provides a set of 3-month forward LYOR rates extending out t0 years. It is possibl: to use thij set of forward interest rates to compute equation (8.2)or (8.3).As discussed in Chapter7j zero-coupon bond prices can be constructed from implied fonvard rates. The set of swap rgtes at different maturities implied by LIBOR is called the swap clf?qzd. There is an over-the-counter market in interest rate swaps, wltich is widely quoted. The swap curve should be consistent with the interest rate curve implied by the Eurodollar fumres conact, which is used to hedge swaps.l . Here is how we construct the swap cul've using the set of Eurodollr prices.z Column 2 of Table 8.4 lists 2 years of Eurodollar f'uttlres prices from June 2004. The next column shows the implied 91-day interest rate, beooinning in the month in column 1. For example, using equation (5.19), the June price of 98.5558 implies a June to

l'T'heEurodollarcontractis a futures contract, while a swap is asetof fonvard rate agreements. Because of convexity bias, discussed in Chapter7, the swap curve constructed frop Eurodollaffutures contracts following the procedure described in this section will be somewhat greater t.11%th obsen'ed swap Bu'rghardt and Hoskins (1995)and Gupta and Subramayam (2000). curve. This is discussed by zcollin-Dufresne and Solnik (2001)point out that the credit risk implicit in the LIBOR rate underlying the Eurodollar fumres contract is different than the credit risk of an interest rate swap. EEBOR is computed as an average 3-month borrowing rate for international banks wit.h good credit. Banks tat experience credit problems are dropped from the sample. Thus, by construction, the pool of banks represented in the Eurodollar contract never experience a credit downgrade- A firm with a swap, by contrast, could be downgraded.

'' ' ' : Ei -

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.

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E. . . . . .

.

-

. -.-.-.)..);

-.::2:2::2:,,.

l,;''' 11. 11: 11: a ll;...l.-...llk ll;.'.s prices with maturity dates given in the first column. Prices are from June2, 2004.

. .

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Jun-04 Sep-04 Dec-04

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E ( .. ' E ' ( ( (( . . . ( ..' (..'.(' . . E . ' . .-- ..-..E E )E (( 'E jisj,' EE 1* ..'r)'t E.'. E '. . ' ' ..,q..yr. .-' '.( .... . . ( '.y... .. ... . . .( .'' ' j1!:' iIigilk). ' . (.q)t.'..'.g E'. ..' j'!.E' ' ' q( t. ..;L. j.. ' .'E'...' .11q'qIr.q1r.)I,' k;(Ijt..pI!()It,' (.E .) '..qi l.( j..Eq.... .... ... ...' ..'L dk)jIi. '. rt.. ltgq!)l ..t....-. ...'..;.' E t . . ...l.'.).E. t . E.. 1) y..'t.. jr)tyE; ..L .) jijj!!;'41t(Ii. 'tjIt;iq-' k:)jIk;.. ' litEE E (IEq E(y (E'. E . E .q. E . )q .)....y( ...(.ty.. E Eyy E g g .--. ( q i l!. .( r;k'.'. ( E ) ji E ,yj q . .' r yy w i E. . '' ' .. . . ' . . . . . E E . . . . q ( q i . . .y. y Ej y E. E j E E( . E g jty y E y j t E y ! . y . j . q ( E . . . . . .. . y '. ;r..g.. y. jj;;jg!:l . ... y . .. j . .. (. y y.. ..y(.... y y. .r. . gjjjjgjyi,.p.! y; ( . .. ...y.q . .y...q.(... y y. . y.) (.E q r...... r. y E E yy .. yy E y y j . rr q .. . y y jjjjjy--. . y . .. . . . . y . .. . . ... . j .... . ... ...yy . y y.. . ... y. . jy. . .. .. . y .y .... .y. .$jjjy2!ry rrjijjj;;, . j. rjjjkjr,r .. .. .y ... ( .g.....) j. yy.y...j.yyj.yy. jy..y..j.yyy.y.yy.j y.y.yyjj..... yy..y.. y... yy.. jyy..j. jy.y...j yyyy. ry.y.y..g...q..y....j..y.)j.(. . yjkty,,,j .. . ..j. . yg;y,k,. . (. yy..r.jj.. j,, . . . . .. ...y..y . .j . y. .gjj,gjj y. .. y.y.j( .yg..k.. jjgjjjj jj..jjg!jjyyg;jjy gjjj.rjjyjjjs . . . qjj;tyjj/pj.:...jj..y.......(. y;yj,y. jk;;y,u .(k,jjjy .. yy.. .jj.....:y.. .y.qj. yjj j jjgj)....j.j.. y.gjjjgjjy..j.. . yjjjkyjy .. jjryjjj.. kjy,..,yjj.jjyjjjy. gjjykt. .. .yjjtjkkr;kj. ..jy.. jjjj,r jjjs t jjjjjjgri . .$j44jjj yjjjj(jjj;jy.....jg tjykyjjyjjyjgsjs.jsy gjjg,,jjryjjr jj;;;r.. ...j,j gy....j.. jjjjjjj jjjs jjjgjj.jjj ;jj.gjj. jjjjyj jy..jjk .tjjjy,tjjy yj.;j;s jjjyjj.. )jj......j.j... . sjr..... jjjjjjjj j.jsjy... jj..j .y y....g.jjjjjjjjj (.... gjjy,jy .tjjjjyyyj . gy ..g..j. jj . ...E .. . y..g i ..: g. .. .. ..... . . . . q. . . j4j......:.g( .. ( . r . ..... (. ..jjtgjy. .y....(j;j.. .L..)jjj,j. . . . . . jj,L g . . y. .. ..j.. . . iiEiL' . 1.22 .... .....y .jy.y. .....jjjg,j.:.. y g jj.... j... .jjjq.. . .. . ... ..j.......y ygjayyg yssayaysyyy.ysy . .yyjyjjj. ..y ..yjjy. yjjgj. . y syaysyysay.. .y. . j jf).. .g y. 1817 jjy.jyjyj.. j.yjg,j.j jjj,..yjgjgy.. (Iil?'.jg ,i. .jjkjy(. yjj j.... jtjj j. j t...... gjjjj d ',# ...'..... ...t, kjjy.kjyjyjjyjy yjyjjyyyjj. d:) . jjj...y.:. .. jj(jgj. ... tlir' :iL . jjggjjj . :i .. ..g '. .)....-......'..t j .. . .. .. t... ... .... ...... 1.......,,..... i, .. .. .... ... )....., ... . . . . . . .. .. ... . . . . . . . . .. . . . .t. .

i

E

.

12

P (0,ti ) + P (0,??,)

' '' ' E ' E .E ' :: . : '

i - E : ' : .. '.

..

//) 2-anl 170(0,

You may recognize this as the formula for the coupon on a par coupon bond, equation (7-.6),from Chapter 7. This in t'ul'n can be rewritten as R

.

! ::

i .

:

E.Ei.. .......

Setting this equation equal to zero and solving for R gives us l #0(0, r,,) R

tyy'. )y'.,::;,jjijj$,'. t)).)( ikk)-..r l.ti t1ii1i).jttrtj'.'( j..j t@. '

...jyjyyy..

. .

-.

-

tjjyjj.

....

...

...

..

. .

..

...

.

.

.

,.

.

. .

1.461 1%

0.9964

1.7359%

0.0063

0.9914

2.0000% 2.2495%

Ma1=05 Jun-05 Sep-05

97.495 97.025 96.600 96.235

0.0075 0.0086

0.9851 0.9778

0.0095

0.9695

Dec-05 Mar-06

2.6997%

95.910 95.650

0.0103 0.01 10

0.9603 0.9505

2.8995%

2.4836%

3.0808%

Source: Eurodollars futures prices from Datastream.

Septembr quarterly interest

rate of

(100

-

98.555)

91 1

4 o0 --59

=

0.0037%

Column 4 reports the corresponding implied zro-coupon bond price. In the second row, the prie is the cost in June of $1 paid in September. The third row is the June cost of $1 paid in Decembervand so forth. The furth row is 1

1

1

0.9851 1.0037 x 1.0050 x 1.0063 which is the June cost of $1 paid in March. The December swal rate, expressed as a quarterly rate. is the 'Iixe d q uarterly intrest rate from Vne through March, with swap payments in June, September, and Decerber (the months in wlzich the quarterly rate prevailing over the next 3 months is lnown). This is computed using equation (8.3): =

0.9851 = 0;54)% + 0.9914 + 0.9851 Multiplying this by 4 to annualize the rate gives the 2.00% in the swap rate column of Table 8.4. 1

-

0.9964

In Figure 8.5 we graph the entire swap curve againstquarterly forwardrates implied by the Eurodollar curve. The swap spread is the difference between swap rates and

%.swAps

260 ':' y' yyf #' ;'tyj'j 9'f771*. 7* i'..;$,! ;;' ;;)' f' y' $' q' .j' 'tf trf yyf jk'. .y' ;'r'!'i.' j,'. pf tqf @' i'(' )' '' (E(E(E E llE ;E ' ' !E: !E E ; (';(il'fji

INTEREST

RATE SWAPS

%

261

-'.'-yy'.-.'-)).-j' -' )t' :''':':qrf -'. -y' -' -:''rii::::;;k,).. j'(' Ejf ;'r'r' :'''1* C'

r)'. ...'' r' j'y'jf f/ZIE.J?X li(i( (''(' !(E ( ('(i'(. (i ((( i ('i((E'''. q' ; 7.(!( q'; fl' ''iillf p' qtf 1lIiFi1Illt)'. i1iit2-2iIIr'. tj'tyf r''t)f (' (r' y' 1iII)'. i'q#' )y' y'y t'. t')@' jy-';yy)-yyyyy'.' ' ' tjyg'j . . . ; i.. (. .' .. .. ( .. . . . j(j'(jtktk ( 44. q j qy.iq. , ( j. (j(i.!g . q . q. j. g j .y ))jk)))-. j.y)yj y..kli1iiiilrL. .y(jkj.;(i! jj, ;j; yyj((yg. j yjy. ,.y j.jjyyj. ..i..jjy. II.iF'III. y.iiiiiEEi. '.)tr)s)(. yyj .IEt.r.ljtj. yiq .j.y. jjy....,g.jjyyyyy. 9-! ! ty rE ( ,kk: pk'r '. . .,: .!. : p. i;i;..j.!..(y . . . .. . ............ . .. .. . . .. .

r' it'.' ''. .''''' .

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-

.

E)!E1

-

E E

...

. .

: .

.

-

-.

'

.

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.

' E : EE. : E : E E''' EE ' lii''(E .E ' E ' . . :. EE i E.: E i ' E :.. ' . .. :

-

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:

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.

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E

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--

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.

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-. . .. . ... . . . . -:.,: . .-..:

-

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j.

E '' E : . :: . : --

:

receives floating. This investor is paying a high rate in the early years of the swap, and, hence, is lending money. About halfway through the life of the swap, te Eurodollar forward rate exceeds the swap rate and the loan balance declines, falling to zero by te end of tlle swap. The sxedrate recipient has a positive loan balance over the life of the swap because the Eurodollar f'umres rate is below the swap initially-so the fixed-rate rcipient is receiving payments-and crosses the swap price once. The credit risk in this swap is therefore borne, at least initially, by the Eked-rate payer, who is lending to the 'Iixed-rat recipient. The implicit loan balance in the swap lsk illustrated in Figure 8.6.

.. E. q - y E-. -

-

-.-. - ..

-

.. .

-

Forwrd (3-mntli

interest rate cul-ve mplied by the

Eurodollar strip, swap

rates, and constant maturity Treasury yields for june 2, 2004.

Deferred Swaps We can construct a swap that begins at some date in the future, but for which the swap rate is agreed upon today. This type of swap is called a deferred swap. To demonstrate this type of swap, we can use the information in Table 7.1 to compute the value of a z-period swap that begins 1 year from today. The reasoning is exactly as before: The swap rate will be set by the market-m er so that the present value of the :xed and

Maturity Soltrce: Datastream.

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The Swap's lmplicit Loan Balance An interest rate swap behaves much like the oil swap in Section 8.1. At inception, the swap has zero valu to both parties. If itlterest rate! change, the prsent value of the 'Iixedpayments and, hene, the swp rate will change. The market kalue f the swap is the difference in the present vale o? payrents between the old dhkp rate and the new swap tat. For example, conjider th 3-year swap in Table 8.3 (seepage 256). If interest rates lise after the swap is entered into, the value of the existing 6.9548% swap will fall for the party receiving the sxedpayment. Even in the absence of interejt rate changes, however, the swap in Table 8.3 changes value over time. Onc the first swap paynzent is npade, 4he swap acquires negative value for the market-paker (relativeto the use of forwaos) becauje in the second year the market-maker will make a net cash payment. Similarly, the swap will have positive value for the borrower (againrelative to the use of forwards) after the 'lirst much'' (relative payment is made. ln oyder to smooth payments, the borrower pays curve) the forward in the first and receives refund in the second to year a year. The l-telltillji ?-tf cl g/'t? $)t7/7 ij trt)l t JJ':/ I?ll /t? t!?l rt??-/? (7,7 t lll /4/ p7t??? l6; 2 ? -1$?:7 Ifll l :7 il tlt)?l ttr t? tltli ti (7, 7:7 l tl-cl r 421

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Suppose that nondividend-paying stock A has a price of $20, and r Example 9.6 )# of $25. Because neither stoc k p a y s dividends, t( nondividend-paying stock B has a price equal lf A is the underlying assei and B is the prices. their 1i their prepaid forward prices ti) k tl strike asset, then put-call parity implies that '

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and Puts?

The preceding discussion suggests that labeling an option as a call or put is always a al1 the time in matter of convention. lt is an important convention because we use it being calls interpret general Nevertheless, puts, and in options. as about can we talking analogy. using why by an vice versa. We can see When you go to the grocery stofe to obtain banaas, you typically say that you involves handing cash t the grocer and are buyiltg bananas. The act'ual transaction receiving a banana. This is an exchpnge of one asset (cash)for another (a banana). We could also describe the transaction by saying that we r sllink ctu/? (in exchange for bananas). The point is that art exchapge occurs, and we can describe it either as buying the thing we receive, or selling the thing we surrender. Any transaction is an exchange of one thing for another. Whether we say we are buying or selling is a matter of convention. Tlzis insight may not impress your grocer, but it is important for optins since it suggests that the labeling we commonly use to distinguish calls and puts is a matter of convention. To see how a call could be considered a put, consider a call option on a stock. This worth is the right to exchange a given number of dollars, the strike price K, for stock > K, we S, if the stock is worth more than the dollars. For example, suppose that if S S K. We can view this as either of t'wo ansactions:

earn

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exercise when S * Buying one share of stock by paying K. In this case w This is a call option on stock.

Options to Exchange Stock Executive stock options ar sometimes constructed so that the strike price of the option option is the price of an index, rathf than a lixed cash amount. The idea is to have an competitfs, fgther yhat thq outperforms pays e the wh company that pays offonly of this, krfijl hyjthtil As al1 stock have plices a gone up. off simply because . off compenation that given is optlons of Microsoft, pay chairman suppose Bill Gates, options and if if only exercise these will He outperforms Google. only if Microsoft i.e., of share Googl, the exceeds price of Mierosoft, skslrr, the share price s'hlsp'T > From Gates's perspective, this is a call option, with the payoff ksooc,

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This is a call from Schmidt's perspective. Here is the interesting twist: Schmidt's Google call looks to Gates like a Microsoft put! And Gates's Microsoft call looks to Schmidt like a Google put. Either option can be viewed. as a put or call; it is simply a matter of perspective. The distilctioll /pdrks/dcn t'l??# d-vwap/d depends I1p-l what I#?c label #?' underlying asset a pltt and a call 11 thl's what Jvc label as the strike asset.

at Are Calls

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Buy prepaid

OPTIONS

Now consider the compensation option for Eric Schmidt, CEO of Google. He will receive a compensation option that pays off only if Google outperforms Microsoft, i.e.,

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of stock. Again we exercise when @ Selling K dollars in exchange for one share worth less than the stock. This is a put sell when dollars the we S > A'-i.e., are sike the of stock asset. share as option on dollars, with a Under either intepretation, if S < K we do not exercise the option. If the dollars

sell them for the stock. are worth more tha the stock, we would not exchange Now consider a put option on a stock. The put option confers the right to < K; we earn K S. We dollars. Suppose S one share of stock for a given number of can view this in either of two ways: @ Selling one share of stock at the price K. -

290

%.PARITV

AXD OTHER

Op7'loN

RELATIONSHIPS

GEN ERALIZED

@ Buying K dollars by paying one share of stock. This is a call right to give up stock to obtain dollars.

where we have the

PARITY AN D EXCHANGE

$

OPTIONS

291

Though they will be exercised under the same circumstances, the dollar-denorni-

nated euro call and the euro-denominated dollar put differ in two respects:

If S > K we do not exercise under either interpretation. If the dollars are worth than th stock, we would not pay the stock to obtain the dollars. less

* The scale of the two options is different. The dollat-denominated euro call is based and the euro-denominated on one euro (whichhas a current dollar value of $0.90) dollar put is based on one dollar. * The currency of denomination is different.

Currency Options

W can equate the scale of the two options by holding more of the smler option or less of the larger option: We can either scale up the dollar-denominated eur cltlls, i of them, or we can scale down the euro-denomiated holding 92 dollqr puts, holding .0 0.92 of them. To see the equivalence of the euro call and the dollar put, consider the following two transactions:

The idea that calls can be relabeled as puts is notjust academic; it is used frequently by ansaction involves the exchange of one kind of currency currency aders. A currency for another. In this context, it is obvious to market participants that referring to a particular currency as having been.bought or sold is a matter of convention. Labeling a particular option a call or a put depends upon which currency you regard as your home

1 Buy I l-year dollar-denonzinated euro call options with a strike of $0.92. lf we exercise, we will give up $1 for 'Ctl1ju The cost is 1 x $0.0337 $0.0366. 9.92 . 2. Buy one l-year euro-denominated put option on dollars with a strike of 61.0870. 't'he cost of ihis in dollars is 0.90$/6 x 60.04t7 $0.0366. When the option expires, convert the proceeds back from euros tp dollarj.

Ctlrfency.

In the ollowing example we will show that a dollar-denominated call option on which gives you the right to pay dollars to receive euros, is equivlent to a euros, . euro-denominated put option on dollars, which gives the ri g ht to sell a dpltar for euros. Obviously, the strike prices and option quantities must be chosen appro)ri j tely yor tjjere to be an equivalence. We will say that an option is if the strike prie ;nd Ifmium option is denominated in dollafs. An if the strike priceand are premium are in euros. Suppose the current exchange rate is -Y0 = 0.90$/E, and consider ihe following two options: 2

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Table 9.3 com/ares the payoffs of these two option positions. At exercise, each 1 Position results in surrendering $1 for 6 0.92 if a:l > 0.92. Thus, the two psitions must the else there arbitrage opporttmity. is an cost same, or We can summarize this result algebraically. price of a dollar-denominated foreign currency all with stlike K, whn the current exchange rate is ab, is Cslab, K, F). Te price of a foreign-currency-denominated dollar put with strike ia when the

E'dollar-denominated''

lfeuro-denominated''

'the

1. A l-year dollar-denqntinated call option on euros with a strike price of $0.92 and premium of $0.0337.ln 1 yelm the owner of the option has the right to buy 61 for $0.92. The payoff on this option, in dollars, is therefore

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OpTloN

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calls in one currency are the same as pts in the other-is commonplace among currency traders. While this observation is interesting in and of itself, its generalization to all options provides a fresh perspective for thinking about what calls and puts actuatly are.

This insight-that

TO STYLE,

OPTIONS MATURITY,

WITH RESPECT AND STRIKE

We now exnmine how option prices change when thre are changes in option charactelistcs, such as exercise style (Amelican or Ettrupean), the strik jiice, ynd time to iand without exp i ra tiplj Reparkably, we can say a great deal without a pricing rodl making any assumptions about the disibution of the underlying asset.ao Thus, vhatever the particular option model or stock price distribution used for valuing a given option, we can still expect option prices to behave in certain ways. Here is an example of the lind of questions we'will address in this section. Spppose have three call options, with strikes of $40,$45, and $50. Hpw dp the premiums on you these options differ? Common sense suggests that, with a call option on any underlying asset, the premium will go down as you raise the stlike plice; it is less valuable to be able to buy at a higher plice/ Moreover, the decline in the premium cannot be greater tan $5. (The zight to buy for a $5 cheaper price cannot be worth more than $5.) Following tltis logic, the premium will drop as we increase te strike from $40 to $45, and drop again when we increase the strike further from $45 to $50. Here is a more subtle question: ln which case will the premium drop more? lt t'urns out that the decline in the premium from $40 to $45 mltst be greater than the decline from $45 t $50, or else there is an arbitfage opportunity. In tis section we will explore the following issues for stock options (someof the properties may be different for options on other underlying assets): .

* How prices of otherwise identical American and European options compare.

3The so-called of rational option prcing,'' on which this section is based, wtts lirst presented in 1973 by Robei't Merton in an astonishing paper (Merton (l973b)). Ths material is also superbly exposited in Cox and Rubinsten (1985). 'theory

4If

RESPECT

TO STYLE, MATU RITY, AN D WRI KE

* How option prices change as the time to expiration

* How option prices change as the strike price

Cstab, K, F)

9.3 COMPARING

M PARI N G OPTI ONS WITH

you are being fastidious, you will say the option premium ctllot acrct7-p as the strike goes up. Saying that the option premium will decrease as the stlike increases does not account forthe possibility that a1l the premiums are zero, and hence the premium will not go down, but will remain unchanged, as the stzike price increases.

$

293

changes.

changes.

A word of warning before we begin this discussion'. f.f you exnmine option price listings in the newspaper, you can often find option prices that seemingly give rise to atbitrage oppormnities. There are several reasons for this. One is that some reported option price quots are stale, meaning that the comparison is among option prices recorded at diffent times of the day. Moreover, an apparent arbitrage oppo>nity only becomes genuine when bid-ask spreads (seeTable 9.1), commissions, costs of short-selling, and market impact are taken into account. Caveat ar itl'ag eur!

Ettropean Versus American Options Since an Amelican option can be exercised t any time, valuable as an otherwise identical European option. (By that the two options have the spme underlying asset, strike piion pprplipte to a Eurojtan Any eyeycise jtotegy ption cannot be lls an Americ ofitioi Th Ameria

it must always be at least as othelavise identici'' w mean price, and time to expiration.) a alwys be duplicated with valuable. Thus ave

CAmer(S,K, F)

kCEurt5',

fC, F)

F)

: #Eurts,

K, F)

fkmertq,K,

We will see that ther >r times when the right to eady-xercise Anrican and uropean opttons have the same vdue.

(9.8a) (9.8b) is wprthless, and, hence,

Maximum and Minimum Option Prices hw expensive or inexpensive an option can be. Here

It is often usef'ul to understandjust are some basic limits. Calls

The price of a European call option

e Cannot be negative, because the call need not be exercised. * Cannot exceed the jtock price, because the best that can happen with a call is that you end up owning the stock. value. * Must be at least as great as the price implied by parity with a zero put Combiningthese statements, togetherwith theresult

aboutAmerican

optons never

being worth less than European options, gives us S k CAmer(S,K, F) k

where present

CEurt5',

values are taken

K, F) k maxgo, PV0,w(F0,w)

over the life of the option.

=

PVo,w(#))

294

% PARITY

AN D OTH ER OPTION

RELATIONSHI

Co

PS

undiscounted strike price, since that is the most it * Cannot be worth more than the drops to zero, te put pays K at some point). stock plice the worth (if can ever be implied by parity wit.h a zero eall value. * Must be t least as great as the price Also, an American put is worth at least as much as a European put. This gives us K

K, F) k maxlo. PMIK)

pEurts, t fkmertq, K, F) k:

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ssome options. such as compensation options, cannotbe sold- ln practice itisommon to seeexecutives exercise options prior to expiration and then sell the stock. The dscussion in this section demonstrates that such exercise would be irrational if the option could be sold, or if the stock could be sold short.

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OPTIOI.I

That is, if interest on the strike price (whichinduces us to delay exercise) exceeds the present value of dividends (whichinduces us to exercise), then we will for certain never early-exercise at that time. If inequality (9.12)is violated, this does not tell us that we kvillexercise, only that we cannot rule it out. lf dividends are suciently great, however, early exercise can be optimal. To take for $100, an exeme example, consider a go-strike American call on a stock selling $90 to acquire the which is about to pay a dividend of $99.99. If we exercise-paying $100 stock-we have a net position wol'th $10. If we delay past the ex-dividend date, the option is worthless. 1.fdividends do make early exercise rational, it will be optimal to exercise at the last moment before the ex-dividend date. By exercising earlier than that, we pay the strike price prematurely and thus at a minimum lose interest on the strike price.

for puts When the underlying stock pays no dividnd, a call will not rnight be. To see that early exercise for a put can make eady-exercised, but put a e economic sense, suppose a company is bankrupt and the stock price falls to zero. Then until expiration will be worth P%,pK). lf we could a put that would not be exercised early-exercise. we would receive K. If the interest rate is positive, K > PV(#). Therefore, early exercise would be optimal in order to receive the strike price earlier. We can also use a parity argument to understand this. The put will never be exercised as long as P > K S. Supposing that the stock pays no dividends, parit.y for the put is Early exercise

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vzueless (asin the aboveexample of abanknlptcompany), parity cannot rule out eady exercise. This does not mean that we vill early-exercise', it simply means that we callnot rule it out. We can summarize tllis discussion of early exercise. When we exercise an option, necessat'y we receive something (the stock with a call, the strike plice with a put). condition for early exercise is that we prefer to receive this something sooner rather than later. For calls, dividends on the stock are a reason to want to receive the stock earlier. For puts, interest on the strike is a reason to want to receive the sike price erlier. Thus, dividends and interest play similar roles in the two analyses of early exercise. In fact, if we view interest as the dividend on cash, then dividends (broadlydesned) become the sole reason to early-exercise an option. ln Similarly, dividends on the stlike asset become a reason npt to eady-exercise. of strike and dividend the in the case puts, asset, on the case of calls, interest is the dividends on the stock are the dividend on the strike asset.

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when make some general statements about The point of this section has been to conditions it might occur. Early exercise is a under what underlying early exercise will not occur, or the strile price, dividends on the of trade-off involving time value money on outwhen liguring to exercise the position. ln genefal, 10 asset, andthe value of insurance on in Chapters further exercise will discuss early requires an option pricing model. We and 11.

Time t Expiration

expiration? lf the options are change as we increase time to expiration. 1.f How does an option price decline with an increase in time to American, the option price can never increase time to the price can go either up or down as we the options are European, expiration. expirationis atleastas valuable AnAmerican callwithmoretimeto American options expiration. An American call with 2 years identical call with less time to as an otherwise option with 1 year to example, can Etlways be t'urned into an American to expization, for the after 1 year. Therefore, z-yearcall is at least expiration by voluntarily exercising it l-year call. as valuable as the longer-lived Amedcan put is always worth at least as The same is true for puts: A much as an otherwise equivalent European put. nondividend-paying stock will be at least as A European call on a European options call with a shorter time to expiration. This oecrs' valuable as an otherwise identical otherwise identical call has the same price as an because. with no dividends, a European options may be less longer-lived European American call. With dividends, however, it optimal make that options. Economic forces valuable than shorter-lived European short-lived than a worth option mor European early can make a to exercise an option long-lived European option. weeks stock that will pay a liquidating dividend 2 To see this for calls, imagine a since it is value will have call with l week to expiration will from today6 A Euppean have expiration call with 3 weeks to exercisable prior to the dikidend. A Europeanexpiration. This is example of a longeran value at the stock will have no no value since that if the options were Note option. shorter-lived lived option being less valua le than a diyidepd. exercise the 3-week option prior to the Amezican, we wouldsimply shorter-lived Euyopean valuable than Longer-lived European puts can also be less the present worth will be is banknapt company. The put expiration. Logerputs. A good example of tltis a value calculated until time to value of the strike price, with present American, they options were lf shorter-lived puts. the lived puts will be wort.h less than strike price. the worth and hence would be would al1 be exercised immediately

shareholders. A tirm is worthless Iirm pays its entire value to 6A liquidating dividend occurs when a after paying a Iiquidating dividend.

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298

RELATIONSHI

OPTION

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present value of the strike price therefore decreases with time to mamrity. Suppose, Ker however, that we keep the present value of the strike constant by setting Kt When the strik grows at the interest rate, the premiums on European calls and puts on maturity.? We will demonstrate this a nondividend-paying stock increase with time to calls. identical for is demonstration the for puts; To keep the notation simple, let Pt) denote the time 0 price of a European put Ker We want to show that #(F) > Pt) if mattlring at time r, with strike price Kt will demonstrate > 1. show an arbitrage if #(F) S P (r). -F To this, we if P (F) :; Pt/l-buy the put expensive-i.e., longer-lived not is put more lf the expiration. At tim: with the t the sell and expiration t put to years F with years to shorter-lived the and is ignore value > Kt its St expire. If will can we zero written put option from thij point on. If & < Kt the put holder will exercise the short-lived option and our payoff is St Kt Suppose that we keep the stock we receive and bonow to second option expires at time nance the strike price, holding this position until the F Here is the important step: Notice that the time-r value of this time-r payoff is erT-tj s r K? sT Kp. Table 9.5 summarizes the resulting payoffs. By buying te long-lived put and selling the short-lived put, we are guaranteed not to lose money at time F. Therefore, if P (r) : P (F) there is an arbitrage opportunity. A practical application of this result is discussed in the box on page 299.

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We discussed at the beginning of this section some statements we can make abnut how option prices vary with the strike price. Here is a more formal statement of these propositions. Suppose we have three strike prices, #1 < Kz < Ks, with corresponding call option prices CLKL ), C(ff2), and C(A%) and put option prices /'(/('1), PKqj, and Plj. Here are the propositions we discuss in this section:

1. A call with a low strike price is at least as valuable as an otherwise identical call with a higher strike plice:

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C(fC1) R: C(fQ)

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1. If equation (9.13)were not true, buy the low-strike call and sell the high-sfrike call (this is a call bull spread). If equation (9.14)were not true, buy the high-strike put and sell the low-strike put (aput bear spread). 2. lf equation (9.15)were not true, sell the low-strike call and buy the lgh-strile call (a call bear spread). Ifequation (9.16)were not trtle, buy the low-stn'ke put and sell the high-sike put (a put bull spread). 3. If either of equations (9.17)or (9.18)were not true, there is an asymmetric buttey spread wit positive profits at a11prices. We will illustrate these propositions with numerical examples.

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call, which is a bear spread. arbitrage by buying the 55-strike call and selling the so-stlike and never have to pay more than $5 in the future. This Note that we receive $6 initially % is an arbitrage, whatever the interest rate.

There is a different proposition, strike price c-lvdA/y. 11 and Ks,' between is equation Kz Since (9.17). way to wlite the convexity inequality, is that strikes, of other the two we can write it as a weighted average

Now consider the

fhird

Kz

=

(1

#l +

lf'a

-

where

Kz

With this expression for

Example 9.7 Suppose we observe the call premiums in PanelA of Table 9.6. These values violate the second property for calls, since the difference in strikes is 5 and the difference in the premiums is 6. lf we observed these values, we could engage in

The difference in

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= Ks

SIn fact, if the options are European. the second statement can be strengthened: option premiums must be less than tlle preselt lz(Jlt? of the difference in strikes.

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change in the option premium ($6) exceeds the change in the strike price ($5).Panel B shows how a bear spread can be used to arbitrage these prices. By Iending the bear spread proceeds, we have a zero cash flow at time 0; the c.ash outflow at time F is always

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(9.20)

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Here is an example illustrating convexity.

'l@Example 9.s (( l..

.

If

r,

-

50, Kz 59

=

-

59, and Ks

-

6s,

z

0.4 x 50 + 0.6 x 65

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65-59 65-50

-

() 4. hence, .

,

k

302

PARITY AND OTHER

OPTION

RESPECT

wITH

To

STYLE, MATU RITY, AN D WRI KE

Q

303

The formula for may look imposing, but there is an easy way to hgure out what is in any situation. ln this example, we had the prices 50, 59, and 65. lt is possible to express 59 as a weighted average of 50 and 65. The total distance between 50 and 65 is 0.6 of the total distance. Thus, 15, and the distance from 50 to 59 is 9. which is 9/15 59 write as can we 59 (1 0.6) x 50 + 0.6 x 65

'iy Call prices must then satisfy l tj C(59) :G 0.4 x C(50) + 0.6 x C(65) t)' .y.yytj observe the call premiums in Table 9.7. The change in the option premium l'it' jyysupp ose we 0.567, and the change from 59 )) per dollar of strike price change from 50 to 59 is 5.1/9 ')t 65 is 3.9/6 0.65. Thus, prices violate the proposition that the premium decreases to E .. :

.

Op-rlolqs

CoM PARING

RELATIONSHIPS

.

=

=

=

=

ll l'ilat a decreasing rate as the strike price increases. ) ) To arbitrage this mispricing, we engage in an . )!'r1

-

This is the interpretation of in expression (9.20). Here is an example of convexity with puts.

asymmetric butterfly spread: Buy jy, j) calls, buy six 65-strike calls, and sell ten 59-stfike calls. By engaging in t) fqur so-strike tt $) a butterfly spread, Panel B shows that a prot of at least $3 is earned. Q. ''i

)q Example 9.9 )) and Ks 70. tr) t'l since t't E =

55, 50, Kz See the prices in Panel A of Table 9.8. We have #1 violated 0.75 and 55 0.75 x 50 + (1 0.75) x 70. Convexity is =

=

=

=

-

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.

#(55)

=

(1 0.75)

8 > 0.75 x 4 +

-

x 16

=

7

.

i) To arbitxage

this mispricing, we engage in an asymmetric butterfly spread: Buy three puts. T'he result is in Panel 5o-s,,a e pu ts buy one vo-sikeput, and sell four ss-strike tj of Table B 9.8. k (j.'1 (.:E '

.

'itt )j

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50

Stlike

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50)

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Sell four 55-strike puts Buy one 7o-strike put

gNote that we get exactly the same arbitrage with any number of calls as long as the ratio at the various and bought 0.6 65 strikesremains the same. We could also have bought 0.4 50 calls, sold one 59 call,

calls.

32

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Sp4

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-

55)

4(Sr

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55)

0

k: PARITY

304

AND OTHER

In this case, we always make at least 4. Figure 9.2 illustrates the necessary shape pdce. of culwes for both calls and puts relating the option premium to the stlike

Put-call pality is one of the most important relatioris in option pricing. Pmity is the observation that buying a European call and selling a European put with the same strike price and time to expiration is equivalent to maling a leveraged investment in the underlying asset, less the value of cash payments to the underlying asset over the life of the option. Different versions of parity for different underlying assets appea. in Table 9.9. ln very case the value on the left-hand side of the parity equation is the plice of the underlying asset less its cash llows over the life of the option. The parity relationship can be algebraically rean-anged so that options and the underlying asset create a synthetic bond, options and a bond create a synthetic stock, and one lind of opti3n together with the stock and bond synthetically create the other kind of option. 'l'he idea of an option can be generalized to permit an asset other than cash to be strike asset. This insight blurs the distinction between a put and a call. The idea the and calls are different ways of looking at the same contract is commonplace in that puts

Exercise and Moneyness If it is optimal to exercise an option, it is also optimal to exercise an othelavise identical this option that is more in-the-moneykConsider what would have to happen in order for not to be true. Suppose a call option on a dividend-paying stock has a strike price of $50, and the stock price is $70. Also suppose that it is optimal to exercise the option. This means that the option must sell for $70 $50 = $20. Now what can we say about the premium of a 4o-strike option? We lntw from the the discussion above that the change in the premium is no more than tlae change in that oppormnity. This arbitrage means is there else sfrike price, or an -

C

(40)S $20 + ($50 $40) $30

currency markets. Option prices must obey certain restrictions when we vary the strike price, time to maturity, or option exercise style. American options are at least as valuable as European options. American calls and puts become more expensive as time to expiration increases, but European options need not. European options on a nondividend-paying stock do become more expensive with increasing time to maturity if the strike price grows at the interest rate. Dividends are the reason to exercise an American call early, while interesy is the reason to exercise an American put early. A call option on a nondividend-paying stock will always have a price greater than its value if exercised', hence, it should never be exercised early. There are a number of pricing relationships related to ehanging strike plices. In particular, as the strike price increases, calls become less expensive with thir price

=

-

it. Since the zm-strike call is worth $30 if exercised, it must be optimal to exercise for also tltis puts. is true the Following same logic,

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Illustration of convexity and other strike price properties for calls arid puts. For calls the premium is decreasing in the strikez' with a slope Iess than one in absolute value. For puts the premium is increasing in the strike, with a slope lessthan one. For both, the graph is convex, i.e., shaped Iikethe cross section of a bowl.

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Calls -+-puts

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50

%. 305

SUMMARY

CHAPTER

RELATIONSHIPS

OPTION

45 40

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foreign currency, rf; and the current bond price, %.

25

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Futures contract Stock, no-dividend

.

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0

20

40

60 Strike Price

80

100

Stock, continuous Currency Bond

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vtl = CK

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PVa,w(Div) e-T%

dividend

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306

k. PARITY

AND OTHER

OPTION

deereasing at a decreasing rate. The absolute value of the change in the call price is less than the change in the strike price. As the sttike plice decreases, puts become less expensive with their price decreasing at a decreasing rate. The change in the ppt price is less than the change in the strike price. FURTHER

PROBLEMS

RELATIONSHIPS

READING

cltapters. Two of the ideas in this chapter will prove particularly important in later calls we also understand tatif we ideais put-call parity, which tells us The srstkey option since the understand pricing easier to understand puts. Tllis equivalence makes it applicable dlrectly option of to the are pricing techniques and intuition about one kind another-also for exchange options--options asset one to other. The idea of exchange will show up again in later chapters. We will see how to plice such options in Chapter 14. A secpnd key idea that will prove important iq th determination pf fgtprs influecing early exercise. As a practical matter, it is more work tp price p.lzAmerican than a European option, so it is usef'ul to lcnow when this exa work is not necessaty Less obviously, the determinants of eady exercise will play a key role in Chapter 17, where we discuss real qptions. We Fill see that certain kinds of inveqtmentprojects are analogous to options, and the investment decision is like exercising an option. Thus, the early-exercise decision can have important consequences b.eyond the realm of hnancial options. Much of the material in this chpter ean be traced to Merton (1973b),which contains an exhaustive eeatmentof optipn prqperties that must hold if there is to be no arbithis mgterial. trage. Cox and Rubinstein (1985)also provides an excellent eatlent of .

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307

c. What difference between the call and put prices would eliminate arbitrage? d. What difference between the call and put prices eliminates arbitrage for strike prices of $780,$800,$820,and $840? 9.4. Suppose the exchange rate is 0.95 $/6, the euro-denominated continuously compounded interest rate is 4%, the dollar-denominated continuously compounded interest rate is 6%, and the price of a l-year 0.93-strike European call on the euro is $0.057l What is the price of a 0.93-strike European put? .

9.5. The premium of a loo-strike yen-denominated put on the euro is 78.763. The current exchange rate is 95 Y/C. What is the strike of the corresponding eurodenominated yen call, and what is its premium? 9.6. The plice of a 6-month dollar-denominated call option on the euro with a $0.90 strike is $0.0404.The price of an otherwise equivalent put option is $0.0141.The annual continuously compounded dollar intrest rate is 5%. a. What is the 6-month dollar-euro forward price? b. If the euro-denominated annual continuously compounded interest rate is 3.5%, what is the spot exchange rate? 9.7. Suppose the dollar-denominated interest rate is 5%, the yen-denominated interest rate is 1% (both rates are continuously compounded), the spot exchange rate is 0.009 $/Y, and the price of a dollar-denominated European call to buy one yen with 1 year to expiration and a stlike price of $0.009is $0.0006. a. What is the dollar-denominated European yen put price such that there is no arbitrage opportunity? b. Suppose that a dollar-denominated Eropean yen put with a strike of $0.009 has a premium of $0.0004.Demonstrate the arbitrage. .

PROBLEMS 9.1. A stock eurrently sells for $32.00. A 6-month call option with a strike of $35.00 has a premium of $2.27.Assuming a 4% continuously compounded risk-ee rate and a 6% continuoujdividend yield, whatis t e priee of the associatedput option? 9.2. A stock currently sells f6r $32.00.A Gmonth call option wit.h a strike of $30.00 has a premium of $4.29,and a 6-month put wit.h the same sttike has a premium of $2.64. Assume a4% continuously compounded risk-free rate. Whatis the present value of dividends payable over the next 6 months? 9.3. Suppose the S&R index is 800, the continuously compounded risk-free rate is 5%, and the dividend yield is 0%. A l-year 815-strike European call costj $75 and 8l5-stIike European put costs $45. Consider the sategy of buying the a l-year call, and buying the 815-seike put. the selling 815-stdke stock, position held until the expiration of the a. What is the rate of rettlrn on this options?

b. What is the arbitrage implied by your answer to (a)?

c. Now suppose that you are in Tokyo, trading options that are denominated in yen rather than dollars. lf the price of a dollt-denominated at-themoney yen call in the United States is $0.0006,what is the plice of a yen-denominated at-the-money dollar call-an option giving the right to buy one dolllm denominated in yen-in Tokyo? What is the relationship of tlzis answer to youranswerto (a)? Whatis the price of the at-the-money dollar put? 9.8. Suppose

call and put priees are given

by

Strike Call premium Put premium

50

9

55 10

7

6

What no-arbitrage property is violated? What spread position would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage.

;

308

% PARITY

OPTION

AND OTHER

PROBLEMS

RELATIONSHIPS

50 16 7

55 10 14

What no-arbitrage property is violated? What spread position would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage. 9.10. Suppose call and put prices are given by Strike Call premium Put premium

50 18

7

55 14 10.75

100 9

4

21

Demonstrate an arbitrage.

14.45

105

5 24.80

Find the convexity violations. What spread would you use to effeet arbitrage? Demonstrate that the spread position is an arbitra/e. '

.

.

9.15. Thepriceof anondividend-paying stockis $100andthecontinuously compounded sk-freerate is 5%. A l-year European call option with a strike price of $100 x l of $11.924.A 112 year Etlropean call option e0.05x $105.127has a premium ,0.05x1.5 of price strike $107.,788has a premi.um of $11.50. $100 x witha =

9.50

9.11. Suppose call nd put prices are given by 80 22

c. Now suppose that Apple is expected to pay a dividend. Which of the above answers will change? Why?

=

60

Find the convexity violations. What spread would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage.

Strike Call premium Put premium

309

b. supposeyou have a put option that permits you to give up one share of Apple, receiving one share of AOL. In what circumstane might you th is pu t? Would there be a loss from not eay-exercising early-exercise Apple had a zero stock plice? if

9.9. Suppose call and put prices are given by Strike Call premium Put premium

k

'

. . '

9.12. ln each case identify the arbitrage and demonstrate how you would make money by creating a table showing your payoff. stock with the same time to a. Consider two European options on the same the and 95-strike call costs $4. call The $10 costs go-strike expiration. b. Now suppose these options have 2 years to expiration and the continuously compounded interest rate is 10%. The go-stlike call costs $10 and the 95-st1ikecall costs $5.25. Show again that there is gn arbitrage opportunity. Hil'tt--It is important in this cs that the optins are European.) call sells fors l5, a loo-strike call sells c. Suppose that a go-strike Europeap sells call for $6. Show how you could use an los-strike Of and f $10 a asymmetric butterlly to proht from this arbitrage opportunity. 9.13. Suppose the interest rate is 0 and the stock of XYZ haj l positive dividend yield. ls there any circumstance in which you would early-e:rcise an Amerian XYZ call? Is there any circumstance in which you would early-exercise an American ,

XYZ put? Explain.

9.14. ln the following, suppose that neither stoek pays a dividend. permits you to receive one share of a. Suppose you have a call option that Apple by giving up one share of AOL. In what.circumstance early-exercise this call?

might you

9.16. Suppose that to buy either a call or a put option you pay the quoted ask plice, denoted Ca K, T) and Pa (A', F), and to sell an option you receive thebid, CbK, F) and PbK, F). Similady, the ask andbid prices forthe stock are Sa and %.Finatly, suppose you can bon'ow at the rate ru and lend at the rte L. The stock jys no dividnd. Find the bounds between which you cannot proEtably perform a parity arbitrage. 9.17. J.nthis problem we consider whether parity is violated by any of the option prices in Table 9.1. Suppose that you buy at the ask and sell at the bid, and that yotlr continuously compounded lending rate is 2% and your borrowing rate is 4%. lgnore transaction costs on the stock, for which the price is $106.79. Assume that (IBM is expected to pay a $0.14 dividend in early May. For each sfrike and expiration, what is the cost if you a. Buy the call, sell the puk short the stock, and lend the present value of the strike price? b. Sell the call, buy the put, buy the stock, and bon'ow the present the strike ptice?

9.18.

.value

of

onsider the April 95, 100, and 105 call option prices in Table 9.1.

a. Does convexity hold if you buy a butter:y spread, buying at the ask plice and selling at the bid? b. Does convexity hold if you and selling at the bid?

sell a butterlly spread, buying at the ask price

c. Does convexity hold if you are a market-maler either buying or selling a butter:y, paying the bid and receiving the ask? d. Suppose the ask plice for the loo-strike call had been $9.20 instead of $9.00. ls there a eonvexity violation? For whom?

.

PROOFS

310

%

PARITY AN D O'rH ER

Ola-rlorq

REl-A-rlohlsiqlfzs

APPENDIX 9.A: PARITY BOUNDS OPTIONS FOR AMERICAN

P R: C + PV(#)

S

-

would be worth K S. For However, suppose that the put were exercised early. Then it buying the call qnd selling the example. if we thitk of synthetically creating the stock by eket the stock price in the expiration, put, there is a chance that we will pay K before the present value replace plummets and the put is early-exercised. Consequently, if we Of the strike with the undiscounted strike, we have a valid upper bound for the value of argument) that the put. lt will be true tand you can vel'if'y with a no-arbitrage -

-

S

When there are no dividends, we have C + K S as an upper bound on the put, and European parity as a lower bound (sincean American put is always worth at least as the much as a European put). The parity relationship can be written as a restriction on put price or on the call price S C + K S t P 2 C + PV(#) P + S PV(fC) R: C R: P + S K Thus, when there are no dividends, European palit.y can be violated to the extent of interes't on the sike price: Sinc this will be small for options that are not long-lived, European parit.ycan remain a good approximation for American options. Dividends add the complication that the call as well as the put may be exercised early. There exists the possibility of a large parity violation because of the following tlwhipsaw'' scenario: The eall is exercised eady to capture a large dividend payment, the stock price drops, and the put is then exercised early to capture interest on the strike price. 'Ihe possibility that this can happen leads to a wider no-arbitrage band. With dividends, the parity rlationship beeomes (Cox and Rubinstein, 1985, p. 152) -

-

-

-

-

S S k: P k: C + PV(#) C + K + PV(D) S PV(D) P + S PMLK) k: C A: P + -

-

-

-

=

=

options. 'I'he exact pariyy relationship discussed in Chapter 9 only holds for European especially when parity, put-call However, Amelidan options oftn come close to obeying options have shol-t times to expiration. With a nondividend-paying stock, the call will not be exercised early, but the put receipt of the strike might be. The effect of early exercise for the put is to aceelerate the times to maturity, parity will price. Since interest on the strike price is small for short nondividend-paying stocks. American options on come close to holding for short-lived American options. The put can We now 1et P and C refer to prices of Ameriean have and the European valuable than we put, be more

P S C+ K

dividends. The lower bound bound as in the European case with no and it may be optimal to optimal to exercise the call to avoid dividends, be not may early-exercise the put. $100 and S $100, and the K Consider the worst case for the call. Suppose D $100). We will not exercise the stock is bout to pay a liquidating dividend (i.e., exercised after the dividend is paid, will be call, since doing so gives us nothing. The put relationship then states The worthless. So P = $100. once the stock is 100 0 100 100 + l00 C R: P + S D K

-

K

The upper bound for the call is the same as in European parity, except without dividends. The intuition for the upper bound on the call option (the left-hand side) is that we can avoid the loss of dividends by early-exercising the dall; hence, it is the same

=

-

-

-

=

in this case. And indeed, the call will be worthless

PROOFS OF APPENDIX 9.B: ALGEBRAIC RELATIONS STRIKE-PRICE wwmaw-bc.com/mcdonald. Appendix available online at

-

=

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which early and showed that the method provides a tractable way to price options for exercise may be optimal. The binomial model is often referred to as the Rubinstein pricing model.'' We begin with a simple example. Consider a European call option on the stock does not pay dividends and of XYZ, with a $40 strike and 1 year to expiration. XYZ Iisk-free interest rate is 8%. We its current price is $41. The continuously compounded wish to determine the option price. could expire Since the stock's ret'urn over the next year is uncertain, the option price is stock either in-the-money or out-of-the-money, depending upon whether the valuation for the option should take into account more or less than $40. Intuitively, the each in assign values and possibilities case. lf the option expires out-of-the-money, both its value is zero. lf the option expires in-the-money, its value will depend upon hw far charcterize the unceftainty in-the-money it is. To price the opticm, then, we need to about the stock plice at expiration. Figure 10.1 represents the evolution of the stock price: Today the price is $41.and is in 1 year the price can be either $60 or $30. This depiction of possible stock prices the option. price called a binomial tree. For the moment we take the tree as given and ttcoyukoss-

Later we will learn how to construct such a tree.

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Note that Portfltos A and B have the same payoff: Zero if the stock price goes down, in which case the option is out-of-the-money, and $20 if the stock price goes up. Therefore, both portfolios should have the same cost. Since Portfolio B costs $8.871, then given our assumptions, the price ofone t?/#(??2 mltst be $8.871. Portfolio B is a synthetic call, mimicking the payoff to a call by buying shares and borrowinj. The idea that positions tat have the same payoff should have the same cost is called the Iaw of one price. This example uses the 1aw of one price io detennine the option price. We will see shortly that there is an arbitrage opport-unity if the laF of one Price is violated. The call option in the example is replicated by holding 2/3 shqres, which implies tha t one option has the rijk of 2/15 shares. The value 2/3 is the delta (A) of tlte option: the option payoff. Delta ' is a key concept, and we that replicates the number pf shares '''' '''' ''' ' . . ''' . ... . . . .. . . . will say much more about it later. Finally, we can say sofnething about the expected return on the option. Suppose XYZ has a positiye risk premium (i.e.,the expectd ret'urlz on XYZ is greater than the rist-free rate). jince we creie the synthetic call by borrowing to buy the stock, the call is equivalent to a leveraged position in the stock, and therefore the call will have an expected ret'ut'n greater thap that on the stock. The option elasticity, which we will discuss in Chapter.lz, measures the amount of leverage implicit in the option.

'T'he Binomial Solution l Of course, it is not possible to buy fractonal shares of stock. As an exercise, you can redo this call options (Portfolio A) to examplee multiplying al1 quantities by 3. You would then compare three = (Portfolio B). buying two shares and borrowing $ I 8.462 x 3 $55.387

In the preceding example, how did we lnow that buying 2/3 of a share of stock and replicate a call option?

borrowing $18.462would

316

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BINOM

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d

-

Note that whep there are dividends, the fonuula adjusts the number of shares in the replicating portfolio, A, to offset the dividend income. Given the expressions for A and B, we can derive a simple formula for the value of the option. The cost of ereating the option is the net cash required to buy the shares the cost of the option is LS + B. Using equations (10.1)and (10.2), and bnds. ,

rfhus,

have . we

LS + B the stock if it goes up, In this tree 11 is interpreted as one plus the rate of capital gain on dividends, the total there down. capital if it are loss of (1f goes and # ij one plus the rate dividend.) return the capital gin or loss, plus the the option whe the stock goes up or down, of value represent the and Cd Let Cu the respeetively. The tree for the stock implies a corresponding tree for the value of 0ption:

ts

-

=

e

-rh

c

e

(r-)

,,

fl

The assme 81stoc tt f rice movements opportunities. In particular, we require that 11 >

e

d

-

#

-

,

If

cd

+

jj

e(r-)/l

-

( jtl 3) .

1/ r-

d

and d shoutd not giv rise to arbitrage q .

(r-J)/l

>

( jtj y;

d

.

To se why this condition pust hold, suppose 8 = 0. lf the condition wre viplated, we 1/), erh or we would borrow to buy th tock (if k: would shozt the stock to hld bonds (if *ay we wpuldearn an arbitrage prost. Therfore the assi'ed process d A:.erh). 10.21 asks #ou to vel'if'y equilibrium. Prblei could not be consistet Vthany possible also hold when &> 0. ihat th coditipr lst A is the number of shares in th replicating prtfolio, it can also Not iat bcse in th ' '( ttjck prie, If tlie stock interpreted a the jenjitivity of the option to a chane . . . .' . . ' . erice by A. This inteiretation changes by $1, ihenthe option plice, LS + B, hafi/ P will be quite importt later. 'tiihr

,

E.

erl' The problem is to solve lf the'length of a period is h, the interest factor per period is the option for A and B such that our portfolio of 2 shares and B in lending duplicates payoff. The value of the replicating portfolio at time h, with stock price% is .

.

.

,

Z S/1 V Z B

At the pliees %

=

ds and

SIt

=

portfolio will satisfyz a., a successf'ul replicating

(A x ds x e h) + (2 x US x e h) +

( x

erh)

=

(s

erhj

=

.p

x

cs c ,f

Example 10.1 Here is the solution for A, B, and the option price using the stock 1.4634, # $60/$41 price tree depicted in Figure 10.1. There we have 11 of $40 and strike price had call addition, the a opon $30/$41 0.7317, and 8 0. In and Cd 0. Using /? 1. Thus C,f $60 $40 $20, 1 year to expiration-hence, equations (10.1)and (10.2),we have $20 0 :/3 h $41 x (1.4634 0.7317) =

=

=

=

=

=

-

'

j

-

=

.

-

proportional 2The term eSh arises in the following equations because the owner of the stock receives a reinvested shares. in that is dividend we assume

=

=

=

1.4634 x $0 0.7317 x $20 1.4634 0.7317 -

s

=

,-0.08

-

-$18.462

=

,

% BI NOMIAL

318 '

A Oh1E-PERI o D BI N o M IAL

I

PRICING:

OPTION

.

). Hence, the option price is given by gj

j.

,.)

j

%

.

hs + B

t).

=

$41 $18.462 $8.871

2/3 x

=

-

it is

to

't/h?e./b?-

A

Note that t/-l/care illterested t???fyill #?' optiol #/#cc, lnow only the option price, we alld B; that is just an intermediate step. If we want to can use equation (10.3)directly: :0.08 1.4634 :0.08 0.7317 :-0.08 $0 x 1.4634 $2n x 1.4634 0.7317 + ts + B 0.7317 ?2t)/. llecessaty

k.

319

Tlle option is underpriced Now suppose that the market price of the option is $8.25. We wish to buy the underpriced option. Of course, if we are unhedged and the stock price falls at expiration, we lose our investment. We can hedge by selling a synthetic option. We accomplish this by reversing the position for a synthetic purcased call: We sho rt 2/3 shares and invest $18.462of the proceeds in Treasul'y bills. The cost of this is

-$8.25

2/3 x

Option premium

$41

$18.462

-

$0.621

=

lnvest in T-bills

proceeds short-sale

At expiration weahave

-

-

=

TREE

=

-

= $8.871 interested Throughoutthis chapterwe will continue to report A and B, since we are replicating portfolio. not only in the price but also in the

Purchased call 2/3 short-sold shares Sell T-bill Tbtl payoff

Arbitraging a Mispriced Option

price? Because we have What if the observed option price differs from the theoretical usin the stock, it is pojsible to take advantage of the a way to replicate the option to buy low and sell high. mispricing and fulfill the dream of every trader-nalely, pdce option is anythig thr than the The following examples illustrate that ifthe theoretieal price, arbitrage is possible. Suppose that the marketprice forthe option is $9.00,instead lisk thpt the stock plice at but tis leaves us with the option, of $8.871. We can sell the the''stock. expiration will be $60 and we will be required io deliver ( at the same time we sell the option ynthet zisk buying bf c a we can address this to create the synthetic optin by buying 2/3 actualoption. wehave already seen how sell th emal option and buy the simultaneously shares and borzowing $18:462.If we synthetic, the initial cash flow is

The option

is overprked

2/3 x

$9 00

$41 + $18.462 $0.129 =

Borrowing

costor shares mispriced. We earn $0.129,the amount by which the option is expifation. We have Now we verify that there is no risk at Receive option premum

Stock Price in 1 Year (&) Wzitten call

$30

:

2/3 Purchased shares Repay loan of $18.462 Total payoff

$0 $20 j

=.$

0

$0

By hedging the written option, we eliminate risk.

..-$20

-$2

$60

$40 j $0

Stock Price in 1 Year (&) $30 $60 -$20'

$0

'

$20

-$40

$20 $0

$20 $0

We have earned the amount by which the option was mispriced and hedged the

1

risk associated with buying the option.

A Graphical Interpretation of the Binomial Formula The bipprnial solution for A and B, eqpations (10.1)and (10.2),is obtained by solving two equations in two unknowns. Letting Clt and % be the option and stock valge after one binomial period, and supposing J = 0, the equations for the portfolio describe a line with the formula Clt '

:=i

X

. .

Sh

'

rlt e B '

'

.

.

.

. E

.

; !' .

'

'

Thiq is graphd Msline AE D in Figure 10.2, which shows the optiop payoffas a fupction of the stock price at expiration. We choose A and B to yie td a p ortfolio that pays Cd whe % = ds and C,, when = % uv. Hence, by conitructiop this line rlws through points E and D. We can control its height by the slope of a payoff diagmp by yarying th number of shares, L, varying the number of bonds, B It is apparent that a line that runs through both E and dS). Also, the point A is the value of the D must have slope A = cu Cdjlus when % = 0, which is the time- value of the bond position, cr B. Hence, portfolio /' er B is tlw y-axis iptercept of the line. You can see by looking at Figure 10.2 that ally line replicating a call will have a positive slope (A > 0) and a negative intercept CB < 0). As an exercise, you can verify graphically that a portfolio replicating a put would have negative slope (A < 0) and positive intercept LB > 0). 'and

.

-

-

A ONE-PERIOD

%

320

OPTION

BI NOMIAL

(' '(' 4(* jjjjg:qqy'. t'. t'(kj(g;jjjjgj'..t' tj . t)!r'.. @li;.;.;t'.

j'jkr': jjd jj': p' l'!ti y' (' f' '.' tlf .r' ?' )(' EE E E' iEE qE E ' EE' E ! iEi ! ;;( yt(jjj'. q' )?' L;L,I?'L'''I;)L;;';;'L' jjjjt,,kjjjjf !!1jrj::;:;;r.' jlf ftj' k i:t'5E t'l)r (' .y' y(' y' tl' q)(;(i);'$r' @' i:)'' ii'('E)( ' ( ' r@' jtd rl;f t'1* j'y' E( 7E ' qIF;IE j)('. y' jjd yjd yjyjy'.' yjd jtg.yd '7'lt')'7j!)7* r' )i' (' ('. (! ;p' :i E 'i'((!E' i.E E.iE.(..9( . .. ; (E .E'. q'.('. i'( .(' . .( ' ' j):)!: EE g(' l !( i ( E E ! 'j; j yj g(y ';ry j y Eq' l:yyi ).1(! .yE.. li 'E:'i(...(' E'EE EE.Ei ! Ejjtj(:(i''i;ij ri.' Ei(j E (:;j:42 y)tg j ': j)jji j .j).. j j yy(ty. j y jr j . . yj . j j j . j . . . j . . r,yyl j yygj y. jyys : jjjjjj!jrj y;y . jjjrq:tjjr:y .jr;. . .y yyjy ....................q.E..tE..... (- ,k-,-y.k(y .. . g j .yy jjjjrs l2rq(. j(g.jjyj..jyyyy . , ... j. jy.gy.g . . )ir)).-

rgt-kjj-k-------it-kjk-yyiytitt'yf !' y' E(' ;'y' j'jyf y,' td j'1* ';'Eil)f jy)d ;'t'':'''t' :qll ' ' Ei E

.'.'

'' .

:

.

'g((

:

' iE

E

E' ' '' . :

:

.. .

i

E

:

:'

.

.

.

..

'

: 'i

'

:

: : .j,jjjjjg. .

'

' ' '' : . ..: .

'

'

y

.

. : :

.

.

r

.

.

:

.

.

.

------.

tmdiscounted What happens if we use p'e to compute the expected Doing this, we obtain p*)dS = er-&bh.s = Ftgl-lt /? *11S + (1

'( '

'

.

'. 'i Ej!jjjjggjjjjk.( :. E: :

.....).-..-..--:

,.........:..;...-............

.......

'

'

:

.

.

..

.

-

! :. . : i: ..

: 'y

y

j.

'g : :

g

:. .

r

y.

, .

,

-

.

,

... .

The payoff to an expring cail option s the dark heavy line. The Payoffto the option at the points d. and us ar Cdand (at point D). The portolio consisting of A shares and B bonds has intercept erDB and slope h, and by construction goes through both points E an d D The slope of the Iine is calculated as Kttaun between points f and D, which gives the formula for A.

c,,

=

stock price?

ltS -K

o

----------------=----

1 1 I I I

k

=

.

1

.:

(Q

=

0

16

K

M

x tltrlrjept t

.

' ,

:'=

Run

its d -

.

ltise

=

%

Cd

-

.>,

(stock

lce naer tme petiod)

t

in Chapter 5, equation 5.7.) Thus, wc calt ct//l7./pl/ld about p* is that it is the probability for which the expected tree. In f. ct, one way to think stock price is the forward price. stock price. We will call p* the risk-neutral probability of an increase in the risk-neutrgl pricing will more discuss and we Equation (10.6)will prove very important in Chapter 11.

Constructing a Binomial Tree t<ee.3 Recall that the goal of the tree We nOw explain thr construction f the binomial stock the price in an economically reasonable about is to charactelize future uncertainty way. uncertainty about the future As a starting point, we can ask: What if there were no period must equal the forward plice stock ncertainty, the next stock price? Without is the price for forward the formula pdce. Reeall from Chapter 5 that F;

going up or down. J.n fact, So far we have not specifed the probabilities of the stock calculations. Since the strategy ptice probabitities were not sed anywhere ilt the option whichever way the stock moves, of holding A shares and B bonds replicates the option irrelevant for pricing the stok is the probability of an up or down mbvement in the

option.

Although probabilities are not needed forpricing the option, there is a probabilistic er-sjh (10.3). Notice tat irl equation (10.3)the terms in terjretation of(Ifequation from follotvs positive and both ctr-xl/f #)/(If are d) sum to 1 tthis #) and prbabilities. Let inequality10.4). Thus, we can intemret tlese terms as -

-

-

er-&tlt

#

,j

=

d

-

. .

j/

..

(10.5)

d

Equation (10.3)cat then be written as

c=

(10 7)

the expeeted stock price equals the

Risk-Neutral Pricing

-

$

when w use p* as the probability of an up move, for the fomard priee of the stock fotwardprice, er-ths. wederived this expressionthefonval-dprice ll-l,g the lpfl/t/?nft'I/

slope

'

TREE

-

Opdoxl

rayoa

.

BINOMfAL

l

PRICING;

-r

e

(,

'.c

,f

+

(1

-

pslca)

(10.6)

Itis peculiar, though, This expressionhas the appearance of adiscountd expectedvalpe. risk of the option is at the though risk-free rate, discounting even at the because we are position in the stock leveraged option is call stock the of a risk (a least as great as the probability that since B < 0). In addition, there is no reason to think that p* is the true ' the stock will go up; in general it is not.

,+?,

=

s'fetr-ll/' P

( l 0.8)

this, under certainty, Thus, without uneertainty we musthave St-v = Ft.twl,. To interpret plice must rise stock the risk-free Thus, the rate. the rat of return on the stock must be at the risk-free rate less the dividepf-tyttld,-r-c-wt.what we man by Now we incomorate unciiainty but we Erst need to desne is the annltalized about return stock the unertainty of uncertainty. A natural measpre ?zp/?w, which stock comptnded '/tz?/#t'I?'# dviation we wil'l denote ofthe ct???/'#?4Itx/,Wy the that return will be stock how deviation sure we are measures by t7'. T. e standar of a rettln chance will greater have with larger a o' a close to te expected return. Stocks far from the expected retutn. and down We incorporate uncertainty ipto the binomial tree by modeling the up difference from the with forward the the Tlice? plice to relative moves of the stock 11.3 that Section will in We see related to i'Vvi-kidfd forwardprlce being length h is of deviation period standard is over a the if the annual standard deviation c, the o'.I. In other words, the standard deviation of the stock return is proportional to aqltare root of time. We now mod.el the stock price evolution as

zeviation.

,+,,,+0.47f

ltz z-,tsie-c.'ii dz F, =

=

n-depth discussion. 3Thl-sdiscussion is intended as a quick overview. Section 11.3 contains a more

.

322

%.BI N o M IAL

OpTl

o h1 PRIcI

Two o R M o RE B l N O M lAL P ER IO DS

I

Nc:

yjtyf I!IEEEEE,X .')(y';' llliq:ill,:.. t . rr. . .. lrfp .

.'....' '.t . --;t!i#'. ..--' :' -'-' -.t'

Using equation (10.8),we can rewrite this as 11 #= =

-

c(r-d)/l+(rx/#

-.

(10.10)

e(r-J)/J-o.x/'

.

binmial tree in Wq began this section by assuming that the stock plice followed the selected and to make the of prices stock $60 were $30 Figure 10.1. The up and down is the same everyting where example example easy to follow. Now we present an and down the moves. up except that we use equation (10.10)to construct 1, so that Suppose volatility is 30%. Since the period is 1 year, we hake equation ( Using and 0. 1' 0.08, (10.10), o'x$41, 0.30. We also have Sj =

=

=

we get = =

..

:. :.

E

: :

-

. :

1+().3x4

$zjjelgs-ox 1-0.3x4T $41, (oms-ojx

,$.j9

= =

954.

.

$ap,jlts

(10.11)

.

10.1, the option price will be different

Because thebinomial tree is'different than iFigure

= $59.954/$41 = Using the stoek prices give in equation (10.11),we have lt = $40, we have Ctt = $59.954 1.4623 and d = $32.903/$41 = 0.8025. With K $40 = $19.954,and Cd = 0. Using eqtlations (10.1)and (10.2),we obtain

.

. . .- .. .. .

i

.

.. :

E

.

:

-'

. .

''

.

::' :.' ...

. .

E

:

-

. ..

. ,L

:

' .- . -.

:.: . . ...

.

-

,

. .

, ... . .

-

.

.

: ! .

:.

:

-

'

-

.

-

-

!

.

..

-.

.

:.

..

'-71111.

'

.

-. . . . .

.

.- . .

.

-

.

.

.

-

Binomial tree fr pricing a European call option; assumes .S' $41 #= $40.00, o' 0.30, r 0.08, F= 1 years, J 0.00, and h 1 000. At each . node the stock price, opton price, A, and B are given. Option prices in bold ltwlfc signifythat exercise is optimal at that node.

659.954

.00,

=

sp-ps4

.00

$41.00 $7.839

=

=

=

=

Example

k(S dz

.

.

=

Ross-lkubinstein tree.

=

.

:

'''

.

=

' ''

i

.

volatility if This is the formula we will use to construct binomial trees. Note that we set volatility, = = with zero equal to zero (i.e.,c = 0), we will have 11St dSt Ft,t-gt Thus, volatility does not bill. Zero with Treasury rise time, still a will just as over the price that prices are klovn fn advance. mean that prices areflzt- it means torward tree.'' In We will refer to a tree constructed using equation (10.10)as a Coxthe Section 11.3 we will discuss alternative ways to costruct a tree, including

Another Oneeriod

%. 323

')'liE1q@' il'tl (' ?' ;f 'fEjjjjy(jy'yjjjy'. j'qf ;'ftljt'tjyjjf tff yt' E' .111:::11,.*----.* :.'(' jtjf )j' )' jlf pjpi.f ql'l.li '#' q' lsrllil-'klylllki:iljij. !)' Tpf rkf qE!' s' (('.)i E i E(()))15 (1;1r714:':r55551((;*6): #' ' j . jj!tr r' )t'. q' (' k' t'4111k,::111'*. j'yjf )' yjkqrf ;'tyf (:E(('Eiq'.!';;' E k ..' jjtj'(yjjjj.(. ''. rg'..jf j.jjj)' jjg'yj)f ,'q' q5''@)@ ''E.i'i'E'. .TE!EE.E ' E(-( (... Ei E.(E'E( . E..:( (E'E( '.; !((( .(T!' liqqi t':f''E')1* ;:k#))-'' i'r p)'ljl.'. j'ijfkjr'. ;'1111.*. y' p!' (jj)'i:. t@ ..;. E:'!.. g((j. ,;'!EiE!!1.r-'---------;:.'' )' )yyy(t;yyjy'. y'. yijtf tj'tjy' .r. r.. ... . t(. ..E.(q..y.EE(.E( . k(Ekk.'..iir . . . '''E . ttt. . ;:. (i ..E . ;i (E; TEjtE fjjjy (!jtrt. !: jyEq t.E! .tjE . (.. ( E.. t.q . . y-; j . r ,;..j j; )( (ytj . ' jyj.:;jyj . . ' i .@1I1EqFi).y., . i -E !1111:::::;1--. E . -E. l.hi . i ;:.)(.. .E;.. :i.; .. ; .ipi ....t(-: . -r.!! ..: 8k)i ' .'-'

F

s

0.738

x-yzz..,jts

$a,'z.9()a sO.0O0

* In order to price an option, we need to lnow the stock price, the strike price, the siandard deviationof retunas on the stock (in order to compute lt and #), the

dividendyield,

and the

lisk-fTee

rate.

* Using the risk-free rate, dividend yield, and (r, we can approximate the futtzre distrlbutionf ihe stock by creating a binomial tree usig equation (10.10). it is possible to plice the option using equation . Once we have the binomial (10.3). 'I'he solution alsp provides the recipe for synthetically creating the option: Buy A shates of stock ( quation 10.1) and borrow B (qptttion10.2). 'ee,

* The formula for ihe option price, equation (l.3), can be written so that it has the appearance of a discounted expected value.

-

$19.954 0 -

. 2.

=

$41 x

=

(1.4623 0.8025)

0.7376

-

1.4623 x $0 0.8025 x $19.954 1.4623 0.8025

There are still many issues we have to deal with. The simple binomial eee seems simple to provide an accurate option price. Unanswered questions include how to too handle more than one binpmigl period, how to price ptg options, how to price Amelican hand, wecan now t'ul'nto those questions. options, etc. Withthebsicbinmialfonnulain

-

B

=

e -0.os

-$22.405

=

-

Hence, the option prke is given by L.S + B

=

0.7376 x $41

OR MORE

BINOMIXL

ERIODS

We now see how to extend the binornial tree to more than one period. -

$22.405 $7.839 =

This example is summarized in Figure 10.3.

Summary .

10.2 TwO

We have covered a great deal of ground in this section, so we pause for a moment to review the main points:

A Twoeriod

Ettropean Call

We begin flrst by adding a single period to the tree in Figure 10.3*,the result is displayed in 'Figtlre 10.4. We can use that tree to price a z-yearoption with a $40 strike when the current stock price is $41, assuming a1l inputs are the same as before. Since we are increasing the time to matulity for a call option on a nondividendpaying stock, then based on the discussion in Section 9.3 we expect the option premium

:

Q BINOMIAL

324

Two o R

l

PRICING)

OPTION

F' :' (iF'iqlf qElrf lrttf q' (' g,' 6* jjf j'g'. ' E i'' E( ' i'E tf 'q' ')r1*,'7!7( yyjf j'y' 14* d!rli;kf k'. )' J' t'1* ti' q' F' ' ltr *47:7(':)* (77q* (' 1* g' jyt.yjjf E' ' ' EEE;E' E FlE; E ('.;'('E.E.;.E'yE.(.( ' gjy'. r' .y.jyjjrjjjyjy,'..' !!g(;.,' t')')' y' 1* )' j)td j'?'11*. llllt---illrf :;E!y).g '' . . E('''' r'! E E y' jsyiyr'' 11:E1;;* (' .-' '1iI:1:1i,k' ((Ej tjf r' j'j)'. t)jf L'-'jik'nk,Ljl-(1111* '(. . ';;.lli . 'pq (JE yjj(jj'. y' ')t'7* '(('. E'. ((.( . .(f;'.' E..(.jtE. .jj.E.i:g;.'. ...(@ ' ijii!ii!tf . g y' r;' ( E. (y y ;k''j E'q. (. E ; i g. EE. i( y y yy jy.jy. yj Ey.y'. E EjE ( .. y.jj .jjjq)(gy.y ;fi : EE l 1 . ( . i. jEE!'E @ ( . k':1t1l-. g j ' . (' yyj, j ( '' y (jy (Ejj .IIII:j:;I;IE. j jjyj. : EE E(.1 . . ..jjgEjy.y.jjjjj.gj;g.jj.j.j..E.j.Ey j.jj llryE. E. .q( ....E' !'. E-(. . l'. 11II::pliIi( E; p ' . .;. ((ri..; .. ) . ; g. r !!.. ... . . . ('l. .. . ( . t ... . ... .. .. .. ..... ... . . . .... . .... ..... r y .. . . . .

,'' :'-2,kj4;2* :;:' -k-f'-'.j' ;':'j,rf r' -' -,r' -,y' -).y)' :7* 7* (.':E C' ;':' :'i'7)* ''('

.'.'

ililq!qf ' :

:

.

.

.

2:

'

TE

: .

'

.

:. . lE :'

-

.

.

:.

' .

' :

.

:

'

:

' :

: :

: .

. . . ..

.

'

(E ' :

E

:

.

'

:

:

:

.

.

:

E': ' ..

.

:

'

'

'' E : '.

....

. .

.

i

L,.

g

-

.

:

,

.

y

:2;:;,-. 11:r4 say-ggp

:

.

-

.

. ..

. . .

$:!11,;.

.

Binomial tree for prcing

a European call option; = $41 assumes 5% # = $40.00, tr = 0.30, .00,

-

.

-

-.

-

.

. -.

,,.

-

. . .

(r:)I

--:!ij;:-

j,r

=

,$23.029

B

0.08, T = 2.00 r 0.00, and years, J h = 1.OOO.At each node the stock price, and B option price. are given. Option prices in bold ztwllc signify that exercise is optimalat that node.

/.

1.000

s

-$36.925

=

=

sud

$41.:0: (s.) 510.737

=

B

.%

A

=

=

usu,

11If

=

$48.114 sttd Sdtt) $8.114

=

4/2

A

0.374

=

=

To see how to eonstruct the tree suppose that we move up in ,

$59.154. lf we reach this price, t en we can move further up or down

(10.9).We get

s

sttd

Year 2, Stock 'rfc ma.xlo, S K) =

=

=

$41

=

$8t.669 x

$41

$48.114

=

$26.405

=

*

''''

.

=

=

..

are at expiration, the option value is

$48.114 Again we are at expiration, so the option value is

$26.405Since the option is out $59.954At this node we use

of the money, the value is 0.

equytipn (10.3) to compute Year 1, Stock Price the' option value. (Note that once we are ar this node, the stock price, IIS, is stock price, dS, is $48-114.) $87.669, and the

$59 954,0.08-0.3 $48 114

'tup''

tldown''

.

.

=

x

$87.669 Since we

$47.669.

Year 2, Stock 'zce $8.114. Year 2, Stock Prce

$59.954c0.08+0.3$87.669

The subscript tttt means that the stock has gone up twice in a row and the subscript @.pdthen down. means that the stock has gone up once Similarly if the price in one year i s Sa = $32 93 we have

Sdu

e(0.08-0.3)

''''

=

=.

=

How do we price the option when we have two binomial the call option . periods? The key insight is tht we work back-wardthrough the binorial iee. In order up and down to use equation (10.3),we need to know the option prices resulting il moves in the subsequent period. At the outset, the only period where we lnow the option plice is at expiration. Inowing the price at expiration, we can determine the price in period 1. Having determlned that price, we can work back to period 0. Figure 10.4 exhibits the option plice at each node as well a's the details of the replicatinj portfoli at each nde. Remember, however, when we use equatin (10.3), it is not necessary to cofpute A and B in order to derive the option price.s Here are details of the solution:

-

=

c2x(0.08-0.3)

.

.

-

1111

=

x

$41

.

tis pfice of $10.737, compared to increase. J.nthis exanple the two-period tree will give a 10.j. t $7.839iri Figure

year 1, to Su according to equatiri

$41

=

e(0.08+0.3)

Prking

-$9.111

B

tree

x

x

=

=

$26.405 fku) 50.000

=

$41

d x $41

x

If

x

c2X(0.08+0.3)

-$19.337

yaz-qt)a sdj $3.187

construuins th.

325

.

II2

=

5'w/

=

0.734

$

We lso could have used equation (10.10)directly to compute the year-z stock c0.08+0.3 = 1.462 and d = /0.08-0.3 = () 803. we have Prices. kecallthat 11

$59.954 (.6,1) =

BI N O M IAL PERI O DS

followed by an up move, we would have a nonrecombining tree/ A recombining tree has fewer nodes, which means less computation is required to compute an option price. We will see examples of nonrcombining trees in Sections 11.5 and 24.4.

:

..

,

.

. . ......

:'

.

. g

.

:

'

.

,.

:

:

: .

MORE

l(d e

-0.08

$47 669 x .

e0.08

j () 803 + $8.114x 1.462 1.462 0.803 .462

-

-

'

-

-

e0.08

0.803

=

$23.029

,

$32.903c0.08+0.3 $y8 jj4. =

.

and Sdd = $32 903c0.08-0.3 .

=

$26 405 .

generates the same stock Note that an up move followed by a down move (u%,#) recombining price as a down move followed by an up move sdttj.This is called a than different price a down move tree. lf an up move followed by a down move led to a

4In cases where the tree recombines, the representation of stock prce movements is also (and,some argue, more properly) called a Iattice. The term tree would then be reserved for nonreombining stock movemenB.

5As an exercise, you can verify the A and B at each node.

326

% Bl hlo

M IAL

O p'rl olq

Year1, Stock Prce value:

PRI

Two o R M O RE BI N O M IAL P ER I O DS

cl Nc: I

$32.903Again we use

=

(10.3)to compute the option

equation

t't1*. k' ;'tyyyyjjj;f ' ttyd tjd r)','. ttyyyyf t'. rr'. ii'.:iill:f tty)', t!r,r1@iiiq'. .

.'' .'..' . --r'.. -'.. -..t' -'(E'IX tqf

'

::

,0.98

-

=

.

-

-

=

-

-

=

-

-

'

'

'

'

:.

jr

;'

E' ' . ..

'

-.

- E

.. .

'

'

'.

yyj yl: E' E

. -

-

:

:'

-

'

:

E :''

'

E

:

.

E'

.

: '

:

-;

.

yyy j. ty . (g y jyy .. yg tiiid!i!t . . E:

.

- .. -.

- -,,. , ---2..

.

'-tlilt.r

..

. -

EE-

-

-

-

.

-);-.

.

.

. - -. . .. .

-.-

-

.

-

-.

. --

.. . -

. .. .k Binomial tree for priclng a Europepn call option; assumes .S' $41 # $40.00, tr 0.30, r= 0.08, T= 1 years, J 0.00, and h 0.333. At each node the stock price, option prie, A, and B are given. Option prices in bold ltolk signifythat exercise is Op timal at that node. . ... .

. .

.

.-

. ...

- .. .

. .. .

.

---..-.

.--.

.

-

.

.

..

::;;;'-. LL;;;''.

$34.678 $61.149 $22.202

.00,

::2:

=

-

A

=

B

.00

=

* The option price is greater for th

than for z-yea1'

e l-year option, as we would

t

expect.

* We priced te

option by worling

backward through the tree, starting at the end and

working back to the first pedod.

A

B

'

=

=

=

'

y

=

$52.814 $12.814

0.922

-$33.264

A

B

=

$43.246 :5.700

0.706

A

-$21.885

B

=

=

0.829

-630.139

=

$35.411 $2.535

A

B

=

537.351 $0.000

0.450

-$13.405

=

$30.585 $0.000

A

-.

The generalizaton to many binomial periods is saightforwardk We can represent only spreadsheet or computr programcan a small number of binomial periods here, but a handle a vel'y large nurtlber of binomial nodes. A.n obvious objection to the binornial calculations thus far is that the stock can option only have two or three different values at expiration. lt seems unlikely tat the time problem is to to divide the price calculation will be accurate. The solution to this expiration into more periods, generating a more realistic tree. To illustrate how to do this. at tle same time illustrating a tree with more than two periods, we will re-exnmine the l-year European call option in Figure 10.3, which has a $40 sfn'ke and initial stock price of $41. Let there be three binomial periods. Since it is will assume that other of a period ij /7 1. c We a l-year call, this means that the length 0.3. inputs stay the same, so ?' 0.08 and tz Figure 10.5 depicts the stock price and option price ee for this option. The option price is $7.074, as opposed to $7.839in Figure 10.3. The difference occurs because

1.000

=

s4)..(j(m $7.074

nodes. ln particular, at a given point * The option's A and B are different at different into f'urther j the money. in time, A increases to 1 we go eady exercise was pot permittd. However, * We priced a European option, so made no ditf erence. At evet'y np de p rior would have permitting eprly exercise S K; hence we woutd not have to expiration, (he option price is peater t.11a.11 exercised even if the option hal been American. option it is straightforward to value an option @ Once we understand the two-peliod The importantplinciple is to workbackward periods. binomial two using more t.111111 through the tree.

Many Binomial Periods

= -538.94

=

$50.071 $12.889

=

Notice the following:

327

,--8211-.* .-.' .'

-1:,!1;* -' --' F' jy'yyyf L.,jf )' j'jy'y 7* 1* 7)* il'. jfd j;d (.(.j)' (' yyf ''!' q' )(' jqf (ftf :pii!!;;f :,ii;;' j'#)' ';'fiif (' ;'t'@' t'())* k(; ttf E E' E )' )j(' q' ,.,:'' jj'. jjyf g' ! ' '' !E'' E' E 7* ;'.j'k' I.II::jII.i'. qqlikqljt'. 'IIIk::iIIi'-' !''I:i::;1ki'j'(j'. ;;' !( ( 'E'TE i.(''iiCij'(''' .'E'EE'i'('.E''E'('!.(E''.('; CJSTI7rI'C/IITIIE'')'';E;7JTI'IT7T'F)'')' E r,f -'1I!I:4;-. y'. i'tj j.4' y' tyf (p''!( (i.'g.'.(i;''' ' .i'' '(E''E.(E'(E' '(' i'E(E.i q ;(..'E ': tiqlpgpllpiilqlqg .y.' ' . ' '. #i''' j5 (( E.. ;'((;;...( gjq.kk, 2E...j;.j..;., .jj.( k. r' qi ' E1 (..( ' i ! ! (Ei j;lj yqqj.yg (y;. ()): .jgyyjEtt.,. jyy .-t y . jyy)tt(.' y --)i.. . . yy . y; ( )-.j -11111.... i. yj. -.t, . ,t((--.. - ...y-.yyyy - j-..j. r-r --. ;- q - .:.) ;).;@ ! (:..-( (.. - . ... ttk.-. . .. . .. . . .

1.462 e0.08 0.803 + $3.187 $0 x 1.462 e $8 114 x 0.803 1.462 0.803 $41 Again using equation (10.3): Year0, Stock f'rcc ,0.08 1.462 c0.08 0.803 $10.737 + $3.187 x -0.08 $23.029x 0.803 1.462 1.462 0.803 -0'0S

$

B

=

=

0.000

$0.000

.

$26.416 $0.000

the nurperical approximation is different; it is quite common to see large changes in a binonzialprice when the number of periods, ?l, is changed, particularly when n is small. Sice the lgth of the binomial period is shorter, tt and d are smaller than before 1). Just t be clear about (1.2212 and 0.8637as ppposed to 1.462 and 0.803 with h second-peliod the nodes computed: how procedure, here is the are =

s tt ' Sd

.j

=

yyjstsx

jyaoao.p

2277

()0sx 1/3-0.3../-/3'

$41,

-

=

y5(jwj .

$35.411

The remaining nodes are computed similarly. Th: optiop ptice is computed by working backward. The risk-neutral probability of the stock price going up in a period is 1/3 e().08x

1.2212

-

-

() 863.7 0.8637

=

0.4568

AMERICAN

%.Bllq o M lAL O p-rl o lq

328

PRI

cl lq c: I

The option price at the node where S

e-0

'

08xl/3

=

(($12.814

x 0.4568) +

-

$5

=

%:

329

node where the stock To illuseate the calculations, consider the option price at the computed is node as ptice is $35.411.The option price at that e0.08x1/3 j (pjp, 8637 0.08x1/3 d () = $5 046 08x 1/3 363 x $8 0 + $1.4()1 x e1.2212 0.8637 0.8637

$43.246,for example, is then given by

($0x (1 0.4568)j)

Op-rlolqs

,74)o

-

.

-

'

'

.

'

.

1.2212

Option prices at the remaining nodes are priced similarly.

10.3 PUT OPTIONS Thus far we have priced only call options. The binomial method easily accommodates option prices using put options also, as well as other derivatives. We compute put option call plices; the only the same stock price tree and in lmost the same way as of computing the Instead expiration: difference with a European put option occurs at price as mtxto, S K4, we use maxto, K S4. Figure 10.6 shows the binornial tree for a European put option with 1 year to expiration and a.strike of $40 when the stock price is $41. This is the same stock pzice tree as in Figure 10.5.

-

-

previously had to consider. Figure 10.6 does raise one issue that we have not the price is stock $30.585, option pdce is $8.363. lf Notice that at the node where the make would sense to exercise at that node. The option this option were American, it = expiration, but it would be worth $40 $30.585 when held until is worth option should be more $9.415 if exercised at that node. Thus, in this case the American will option. We now see how to use valuable than the otherwise equivient European options. the binomial approach to value American -

*$8.363

-

-

OPTIONS

10.4 AMERICAN

optimal, the binomial Since it is easy to check at each node wheter early exercise is of the option if it is left metod is wemsuited to valuin Ameriean options. The value it for another period, equation (i.e.,unexercised) is given by the value of holdingby maxto, S K4 if it is a call (10.3). The value of the opion if it is exercised is given and max(0, K S4 i.fit is a put. Thus, for an American put, the value of the option at a node is given by I'alive''

l'.E ;'(y' g' (' ,'(' 4* @:q' j'q(')' )' ''''' 1/4(1* 1* ')'. )-' j')'(p'-ptk-f 4* y' r' ((' (11611* )' ).)-)'. y' ''' qrqd t:d E E ti ! ' ! ' !q! ' .(i()(. '

.'-

.-.' .''. --.' '(;lr'q7!qrq'' -.!15!1.* -' -':'-'-' II;'F!!'?X -' -'-'.' ,',' ;':';'

pi-' -'i1:I:-:;i1r'' .'!.iIr::1;:;;'. kL'. q' .j;j' liliti!!:f lllli:ril!!f j'j'; .kjjjgg;jjjj''jlf 1* y' jE'' (' ('''' !

' '

:

'i i j'(;

('q q'(' j((' ';'q'('Tq'7' (

'

'

EE' ' ' ' ' ' ' : E' ' ' qE( Jj.E E EEE ' ' ' ' ' ''EE'''E'' E : ' E: : : E (i E(. E. E1) ! ;.:. :(. E:EE. . (.Ei (E. E!!!E. ' .( (.EE.(: (g E. :(E. . ... . .. . !.;: t);;)i(J))'i'qiEEl.)E(i . : -. . - y t.E))!j1(:')()./1E' (. ... E.. yEq (: q. yy...;. y y: y.. yyy qy..E- yq yy j- y-y 2 Ejllryjt.:. . . y y -. y- y yy. y .. .. y g . .y y . y y . ;1.()E)E :-' y -(@ . -r 'lE'..;.q(E . - -i-y.E . ; -. -..-- -..--. -. - . . -.. . . . . : . . . .. . . . . . :'

(

(rl '(:.. . !.' j..( . . .. .. . . . y ; y .j jyy j ( y y.jyjy jy. yy(y-yjj)jjq)j. jtyyy.y.jyjj.yj.yjjy..y..j..j)yjjyyyj.jyj.yyyjjs . l:. .!11!. y.. y .. ..ytj y jyj.i . !; yy . 7 :;i .t-. )... .k. . .i.; t,F-.; tr. .'. . . .. '

'

.

.

-.

''

'

'

:

'

'

.

((

'

'.

. ,;gjjjj..

..

-

:

.

.

.

.

.

. -

-

:

:

-

.

-

.

-:!ijq;:

.'

.

.........

.

....;.

. .. . .

...........

..

......

...

. ................

.

-

... . . -.

.. ..

....

Binomial tree for pricing a, European put option; asjumes .S' 541 # = $40.00 , c = 0.30 ! r= 0.08, T= 1.O0 0.0b, and Years, = At each 0.333. h node the stock price, option price, A, and B Option are given.boid llwllc in Prices signifythat xercise is optimal at thai node.

::;;;;-. ::;;;-'' -.,:E11;. .-q:.,-.

y(j.(;(s $61.149 50.000

.00,

=

A

Ps,

K, t4

=

maxLK

0'000

= .$0.(jx

.n

-

-

u,

e-rh

r, t + ltlp. g#(,/5',

where, as in equation (10.5), $so.oy1 $0.741

=

$41

.

A

B

=

$52.814

-0.078

A

=

B

=

,$0.000

.$4.659

s4a.z46 $1.401

cc()

$2.999

,

*

-0.294

=

$15.039

A

= =

$8.809 .$37.351

$35.411 $5.046

$2.649

-0..550

B.

= =

$24.517

$30.585 $8.363

A

-1.000

cs

B;=

(r --J)/, tt

---

--

K, t + /,)(1

-

p*)j)

(10.12)

tj

tf

of the put option valued Figtlre 10.7 presents the binomial tree for the American version where the stock price node the at in Figum 10.6. The only difference in the eees occurs value. Amelican optio at that point is worth $9.415,its early-exercise is $30.585. unexercised is the option if We have just seen in the previous section that the value of 'rhe

-0.171

B

#=

e

+ pds,

t38947 526.416 $13..584

$8.363. through the eek When The greater value of the option at that node ripples back is $35.411,the value is price stock the where ode omputed at the the option price is the reason is that the price is greater at the greater in Figure 10.7 than in Figure 10.6., subsequent node Sdd due to early exercise. the European The initial option pdce is $3.293,greater than the value of $2.999for node. Sdd exerise early the at entirely is to value due option. This increase in ekample. At each In general the valuation of American options proceeds as in tlzis when exercised, we node we check for early exercisek H the value of the option is greater of option unexercised. assign that value to the node. Otherwise, we assign the value the We work backward through the tree as usual.

'

%.Bl N o M lAL

330

O PTl o N

PRI Cl NG:

op-rloxs olq OTHER

l

' E: E ' ' E E' 5*'' E' ' E :' ' E E' ' )' (' ;'t' j':'' :'q' @' ,'y' ''i' 7)/* jif ' : ''IE'''q7. qrqq' EE: : 'I(E'E'Eq( EE k' q'. ,'t)f ,-.t'.j' yy'' )' ;'p' (' F:77'7q4. l'qljf @' :' E E E'(.( ;':! i(qq' (' :,ji!,k' jyy' y'.'j?' t';f )' kj'')i p' ll'rjElqq':pljll ' CE ll.: 'p (Ei i(ii ( 'E. . .(:(E.'E'E.('i.' (j.'-i(EEi.E.E!;i!i.E(EE!' FE'' . '( . . .i ..(i .. :. .:('': i:1: . . . - .-.- .- .E . --. ..- -. .... -. .. . E.-.. i. -..E.E. !':. . .. -. E :.rEE. i ;E . q q . (..,j.. j k. . . . - r! ty r . , q . ! ;E . y y ( y .....y q-y j . . -. ( y-. yy y . E . -. )(' . E!E ; El'i1::::;;,.. -. . .. ;. ...... .- -. -. ; - -. --..;;.. r. jyjt--.E.E! . .)(E r. . .. !..E . . .. . . ..... ! ... .

.'.-!.(11-.* ,'')' 77* ''--;k' ;;,r' F' ;''Ekjjjrjrjjjj'i -' :''t'E'

djrE!;kf ,;' )' )('. ))yy'. .,).!y' j'. )' jyj'gj t'rr!p!l;'f iff r' )'. tyjf y' yy't tj'rj. ';'(' C-'. '-' @-' 11111* r:;;;;,.f t''1I1k:--jI1p' li'tf )'

.'' ..' tpf t.'lf t't.'1-!

'

'

.

'

.

'

'

'

'

:

'.' .

.

.

.

'.

'.

'

-

-

.

..

:

:

. .

..

.

.

. .

-

..

.. .

.

.. .

.

. . .

. .

.

.

')y .

.

.y

-

.-

. . . . . . . ..... .

. . . . . .. .. . . . .

.

-

-

.

...

.

.

-

'

.

-

-

-

.

.

.

-.

.

-

.

,

.

-

.

..

.

-:,114;-

.

.

-

,-.

-

$61.149 $0.000

=

0.000

=

.00,

=

.n $().cc() =

.$40.00,

=

=

::;;;--.

s().(o()

.

cr0.05xl/3 :.45.7x $87.74-7+ (1

.00

5 sc 0,/1

=

A

=

B

=

331

0.457) x

-

=

$32.779j $56.942 =

56.942, we exercise the option at that node.

>

352.814

-0.078

.

.

Since 57.101

.

50.741

%.

(10.2), and the option price by equation (10.3).The risk-neutral probability is given by equation (10.5).6 Figure 10.8 displays a binomial tree for an American call option on a stock index. Note tlptbecause of dividends, early exercise is optimal at the node where the stockprice 0.457.,hence. when S $157.101, is $157.101.Given these parameters, we have p' unexercised is of the option the value

:

'. ( (.yjy-j .7.. 1111F44!1. r .q r llllkipllti. .. jjLiii,btq... .. yj. j' j,;j.r.j(y.ry.jjjy.j'. kk#it1I). iiiikdijk. (...; ..l(. .. ;.(ti r-. . . . . . . .. . .. . .. Binomial tree for pricing an American put option; assumes 41 .K 0.30, o. 1 T= 0.08, r years, J = 0.00, and h = 0 333 At each node the stock price, option price, A, and B are given. Option prices in bold ltwlic signify that exercise is - () Ptimal at that node. .

AssE'rs

0.0O0

$4.659

;'r' 1111:* :!iiF;k' (j'. pll'y)j ttf k' yyf .;' j)'yy!. y)y'-)'jijjkj,-jjjr' y'' lllli:pttf r. r'. tlf t'1* kgjjjjjjjji-jf r' '. (. jy)y)yj@ . (.iill:;llit. (..... r. t-... ( yjjjj ky Iliiij!!:. 7 #1 t.(j..(... ...g;r......-yy!r.k( . .; . 1(. ... .... .

tf '!

:':.' ;;' :,' ;.' -,yr' -j:'.':' -.' j';';:.' -' ;';' ''#' C' Cif ' )' f' p.yf 71* :''(SY 1' E )::: :.(E ''EpE : : ;

.'.

'r' t.@' r.yyf ,')' lrr' ;'(7* r)f /'i't't'-)'i! (74E itf j'j))).#j;yt-(y'kjtr();)-. ')y!' kp-!,' ;'@' (' p' )!7*:1* (7117r1q324)*1(!(t* :1i5r;' y' )' ' l('' i!'((E'E (:i1). i(((E' t'. (E(:'.li'.i ;'';qE'E'(;'E'.q.EEEE(EiFE(EE:.q''qE'.'.iE'E'' .(bb-'E''( lE'!'7':C'S'F'T7'III''!C'I?'CCII')'!TC ' qj .' .l' . E IE E. - - - E ; E --. : --- i ; jjjjjr:jjjk; - ---- -i 5. . - .y E- r. ' '- . t::. ;r-!t'1::::;;;kE. rrr.. :.)----)---:,-,y -; g.(.r -g r; j@-. . E :r.. qE, .:k;jjjjg. ( (,:;; ;!. . . y i ..... i yy j r))i . ..-. j. : j)).-. j. .-i-..;j-., Ei:. -; t..,. i!. q .. E ... ..!. . .. . . .. .

-' ..' .-.' .--:j11--* ......-)' ''

'

$4.a.:,46

5 41 (0tj .

s A

-0.332

=

B

$16-891

=

$8.809 ,37.351 $2.649

.

$5.603 -0.633

A

.2

=

1

:

'

E':' '

''

:' ..

'

!: .

. ..

.

:' :

- -

i

'qqq'lq ' '

' ''

' :E' : .

:

-

.

-

628.018

:

-.

.

-y

.

-

:

..

. . .

-

-

.......... . ..

.

:. .

......

.

. . . . . .. . .. . . . . .

E.

.

.

: .

.

.

:. .. . ...

-

-

.

.

.

.

.

Binomial tree for pricing an American call option on a stock index; assumes . $1 10.00 # $100.00, c 0.30, r 0.05, 'F 1 years, t 0.035, and h 0.333. At each node the stock price, option price, A, and B are given. Option pricej in bold ltwllc sighifythat xerise is optimal at that node. =

=

'

.

.

'

-.

-

.

'

:

-0.171

.$35.411

:

.

-

=

=

B

'

1.401

$3.293

'

'

-

$157.101 $.57.101

A

,

=

::;;;),' ::;;;;-. q!l).. ::;;;.-.

$87.747

B

=

=

0.988

-698.347

=

.00

=

530.585 $9.415 A

B

=

ON OTHER

$1a1.4s8 $33.520

=

-1.000

A

=

=

$38.947 $26.416 $13..584

10.5 OPTIONS

=

ASSETS

tpderlying 'l'h model developed thus far an be modifed easily to price pptions op examples of this seyion we present assetp other than nondiyidend-paying stpcks. In curppcies, and futures cpptracts. how to do so. We yamine optiops on stock indexes, In every case the general procedure is the same: We compute the option plice using equaion (10.6). Th diffrelce for different underlying assets will be the construction of the bipomial tree apd tlw risk-neutral probability. The valuation of arl option on a stock thatpays discrete dividends is mor involved and is covered in Chapter 11.

B

= -146.185

$132.779

0.911

$32.779

=

s1 1() ()co

s).11.1()6

.

$18.593

A

B zz

= -:$57

$14.726

0.691 4O8

A

B

.

=

0.833

-$77.871

=

$qz j).g() $6.616

$,a,,4)4 $0.000

.

A

B

=

0.447

-534.984

= ;

$78.576 50.000 A 0.000 B $0.000 =

=

$66.411 $0.000

Option qn a Stock lndex Suppose a stock index pays ontinuous dividends at the rate &. This type of option has in fact already been covered by our derivation in Section 10.1. The up and down index replicating portfolio by equations (10.1) and moves are given by equation (10.10),the

Inmitively, dividends can be taken into account either by (1) appropriately loqkeringthe nodes on the lisk-neutral probabilities unchanged, or (2) by reducing the risk-neutral jrobability tree and leaving and leaving the ee unchanged. The forward tree adopt the srstapproach.

332

%.Bl rqoM

IAL

Op-rlox

PRIcI NG:

op-rloxs otq OTHER

I (l':)i j).1@1 q' f' ..(jyyt'yjjyjy'' (' .j' jyyyf jty'kj. (1* 'J'' 1)* ;'lf 11* p' j'J' t'. r' y' E lrli':r'jf (jf ')f 1111k---1110*.* t'y'tf (' y' /')'!r?qprr7;qrsf 1!* (Cq.q i (' ('(('; f ' '('lrrlr'lq:l'rr?!ll!r'l q (qlik );f.it: @ yyf yyj'yyjj ?' j'( #' ;'t' i' ( q@ q( . ;. (' ! (' jyy'.' t'rrf y' jj': j'#' g.)'y' qtjt'i)ki (241L..* )))' lll'icf .j' 'lf ?p)i'. .q'qr r'Jlplqiiillrq-.. tq'..' . it).. tt.((t;'. q q I r (qi . ;' y ( .. ; .. ;. . j y. y I yyjj. y ry . rjy yj. ry.l1ii:itiIiil. .. yyyj .jg ti. ' (11:1 jyy.yygjjy..j.j.yjjy jj.; j.yytj. jyyj...yyg . j yjg g ( . 1IiiiE!!i-. . E ' i E ),(l1IIiFq5F!. (k.d!1I::::;;ki... 'Ei ..:! '. !tj,; :jtjkk--. . )jEj).. i.tl!'. . t(. . .. . ::..;E'.r.!i(!.;i;.jr .. . . yrjyyj'jjjtyf 'r' ---' -.' E' L' ?' j',':'(' ''(' '4'j,yjjjj', 11:::::,.* j'@)' 'C' rjjfzj'fE. :'(js'jgsf (#' tyf )' ;';'j'jjj;gggjjj;'r' tgf ''. y' T)IT;EE)E;E '7 E E E '' ' E : EE (iE'. '':..E E''' :i jE E E trrTE !E i;. .t (''i'i EE '!: EE: . E :(Ei '.!'' i . i .' i' EE EE . ' y-y.' . j E(-lyjky yyyg jq y q.g. g..q: E : yE.q. !, .gq.r. . y.(. : j;. i. ; !q y . . y j j E ; E

.'.'' .'.

'.'-' .,.'

Options on Cmrencies xzer-rfjh spot plie. Ab, the forward price is F() /, With a cqrrency with .. (. the oreign interest rate. Thus, we construct the binornial tree using =

,

Ifx

where r.f is

=

-

.

(r-ry)/l If

-

-

-

(E: E qE :

'

'

.

. -

.

-.

.

..

. .

'

..

'

'

. .. . . .

;

.

-.. . ....... .. .. .

-;.(

-.

.. ..

,.

'

.

-

: . y

'. .

i. -. E ..

-

-.

..

.

.

-

-.

E

. .

. -. -E - .

Binomial tree for prcing an American put opton on a curreny; assumes

-:ji5;;,

.

.Q5Iq, . # $1 0, c 0.10, r 0.055, T= 0.5 years, & 0.031, and h 0.1 67. At each node the stock price, OPtiOri Price 2. and B are given. Option prices in bold ltollc signifythat exercise is optimal at that node.

'

#

$1.148 A

s

=

=

=

.j

(j ().j )

#

B

=

$1.107 $0.000

q

A

=

$0.867

$1.(sa $o.oaz

B

-0.915

= =

$1.009

$1.012

$1: 02O yatuo

$0.088

.

-0.995

Z

=

X

=

1 O90

A

B

jtl.ngj so zas -0.995

=

=

$1.090 $0.940 $0.160

. (.

..

' .

. .

.

..

.

Solving gives?

Options on Futures Contracts

c

---

We now considt- options on futures conacts. We assume the forward price is te same based on the forward price, we simply add as the f'tlturej pric. Since we build the tree Thus, the nodes are constructed as up and down movements around the current price. lI

= ezr

d

=

-a4' e

Note that this solution for u and d is exactly what we would get for an option on a stock index if J, the dividend yield. were equal to te risk-free rate. In constructing the replcating portfolio, recall that in each peliod afutures contract requ'ed to enter a f'utures pays he change in the futtzresplice, and there is no investment and the lending, #, contract. rheproblem is to find the number f fumres contracts, that replicates the option. We have ,

-

-

F4 + er /, x B F) + er x B

=

Cd

=

C tt

fI

AH

k

A x dF A )M: (IfF

.s().()(o

$0.525

-0.774

=

O.0O0 0.000

-0.459

A

$().oss B

=

y1.(s8 $0.021 =

sjmsj

,

Notice that if we think of ?'y as the dividend yield on the foreign currency, these two equations look exactly like those for an index option. In fact the solution is te same as for an option on an index: Set the dividend yield equal to the foreign risk-free rate and the current value of the index equal to the spot exchange.rate. Figure 10.9 prices a dollar-denominated American put option on the euro. The cuaent exchange rate is assumed td be $1.Q5lq and the strike is $1.10/C. The eurbdenorninated interest rate is 3.1%, and the dollar-denominated rate is 5.5%. Because volatility is low and te option is a-the-money, early exefcie i t/timal at three nodes prior to expiration.

'

(:!1t.. ::;1!,:.

$0.000

=

,

333

!- . gq ! y

:. :

.

.1

A x dxerfh + erh x B = Cd A x uxerlh + erh x B = ClJ The risk-neutral probabilityof an up move in this case is given by d

.

=

There is one subtlety in creating the replicating portfolio: Investing in a income obligation denominated in means investing in a money-market fund or sxed in Chapter 5.) Talinj into account previously that cuaency. (We encountered this idea obligation, the two equations are interest on the foreign-currency-denominated

=

'

..

.

=

*

'

''

:

.

. .

....' . .. '

. .

. -

..11111...

.$1

ttcurrency''

p

.

.

xe r-rytl-b. xer-rfth-vs

=

dx

3

'

:

.

%.

AssE'rs

F(II

s

-r

= e

c

1 ,,

-

11 -

-

d

yf

#) u

d d oc If

-

-

1 d

. W'hile A tells us how many f'utures conacts to hold to hedge the option, the value of the option in this case is simply B. The reason is that the f'umres contract requires no

Rerhe

interpretation of A here is the pumber of futures contracts in the replicating portfolio. Another interpretation of A is the price sensitivity of the option when the pfice otthe underlying asset changes. These two interpretations usually coincide, but not in the case of options on futures. The reason is that thefumrespliceattime/ reflects apricedenominated infumredollars. Theeffectontheoption priceof a l f'umrs pricechange today is given by e-r ; .To see this, consideran opton that is one bnomial period contillkled)

334

k

BI Nou

IAL OPTION

op-rlolqsolq OTHER

1

PRICING:

.'p . .

.y'k .y'; )' t't' .,11:t::;;;1..*.* 4.::111;* :'tjyj;f @' )' ;',rt' t-,qi'iljq',i ''irf !' j:yf Flrqi.lif F' 67* ' ii ' E iC( j)'i )' 'y' k' ),r$' ql;f )4(* i'qr'.)f i'tl t23'1;* ' )t)i)t. 11* p!' E i ' '' i ; @ E ll()lSi ' 'irl:li f!E i@! jjr,;))y'. jjf gj'ykf rlf .'('#'(qf (q:1ti h' qE(79. llrpl7lllpp7sqiipli;E I' (r(E'(:7.'i;(!' fi: )' .1iI:::Il,s'-' ytf ,:55E!1;* 1!IIE:!q' ))' )t' f'jj;j;;-. (?' .(C!' t'(')'.'l't'. 1111r* )'j)'(.)ypjjjk,kkjjj;'. l'q (' y),tf jlll#itill:)f r' tjf )jj.'-t-. t','?p.,pf rjity.jf .ri. '. . ..... . . . . ' '. ' . .. ' t.. ))t)(. Fi (i,i (..y. ..!jy i (.. .r'ktl g . r ( . t-.j.j : t-; yyrrj. ).t !j(j; yy . j 'y)).-------()).-j j y Ilil!EE!k. t y y . t))y. i. . :--1111!:. t, . r!. iy.; . )-.;.t. '.pl';y)... ) ....E y. . . ... .;.....

AssE'rs

%

.-,,-......--))141,)));4r*...-' -,' -r' .''

,'-)))j,)'. --' --(' -' .''. 'fi'.li' :'y' j'tf ky.;)'-.,r'--i(2' ;)' )' ''

ll?l'ti7')).. .'.'.

bond. We can again price the investment, so the only investment is that made in the oP tion using equation (10.3). probability of an up move is given by The lisk-neutral

p*

l

=

d #

-

If

-

.

(10.14)

-

E' ' : .E ! :

-

-

. .

:

E

E

.

.

.--- -.. ..'. -- E .. .

.

..

Options o Cmmodities

' . ..

it also possible to have Many options exist on commodity futures conacts. However, is and bonowing the lending market for is a options On the physical commodity. lf tere saightforward. is option priclng such an commodity, then, in teory, is concepmally similar Recall from Chapter 6 that the lease rf7/d for a commodity commodity, the lease rate. lf you buy the yleld. you pay If you bon'ow to a dividend the perspective of from Thus, lease the receive rate. the cornmodity and lend it, you stock index, with the commodity like is the option, a someone sythetically creating the lease rate equal to the dividend yield. 10.8 (imagin Because this is concepmally the same as the pricing exercise Figure 30%), we volatility of and of 3.5%1 lease a rate ine a commodity with a price of $110,a do not present a pricing example. the physical commodity ean ln practice, pricing and hedging an option based on could use it to market-maker exists, contract a be problematic. If an appropriate futures commodities often have physical in transactions hedge a commodity option. Otherwise, commodity Short-selling may not assets. a fnancial greater transaction costs than for dicult. then is Market-maldng discussed 6. in Chapter be possible, for reasons

E

-

. -

'

..

' ' E;

' : : :

E

:

E(E' E

'

E

'

f : .!

.

.

..

E' ' iE' : :.Ei::: .: . ! : . .

':

E: . :

-

. -

.

-jj

-E

-

!--

.

E'

E -. .

--

-

E

rrrlllltg--lllliilpp,-, tree fr . pritihj Binomial

- qE(E)))-. ,r,.rsi ....L...Ety.-.........:....-.,.................,......-......j.E-...

.

..

. .

.

........

.

-- -. .

.

.

.

--- -

-

. ....

-

..

.

.

-. -.

.

an American call optton on a futures contract; $300.0 0 assumes # $300.00, tr 0.1 0, r 0.05, F= 1 years, 8 0.05, and h 0.333. At each node the stock prie, option price, A, and B are given. Option prices in bold ltwllc 5i9 nify that exercise is optimal at that node. -

option on a gold futures contract. Figure 10.10 shows a tree for pricing an American call for early exercis is Early exercise is optimal when the plice is $336.720.The inmition holder option pays nothing, is that when an option on a f'uttlrescontract is exercised, the differnce of nyered into a f'utures contract, and receives mark-to-market proceeds the bility to the exercise is between the strike price and the futures price. The motive for mark-toumarket pruceeds. earn interest on the .

iqilsE

=

-

-

--.

:,:111;. ;,!i5;;. ::!11,,::;;;'-' ::E11!,-

. -

...

$.5.6.7.53 $336.720 $36.720

A

,

=

B

=

=

1.000

$36.720

=

.00.

$317.830 521.843

=

=

A

=

B

=

actl.tlto

$aco.(oo $12.488

A

B

=

=

$317.830 $17.830

0.768

$21.843

=

0.513

A

612.488

B

$8.51s =

0.514

$8.515

=

yp,8a.17c A

B

sz8a.17o

$4.066

=

=

$0.000

0.260

$4.066

$267.284 $0.000

A

B

=

=

0.000

50.000

$252.290 $0.000

Options on Bonds Finally, we will brie:y discuss options on bonds. We devote a separate chapter later to discussing fxed-income d rtva t.ives, but it is useful to understand at this point some of the issues in pricing options on bonds. As a srstapproximation we could just say that bonds are like stocks that pay a discrete dividend (a coupon), and price bond options using the binomial model. However, bonds differ f'rom te assets we have been discussing in two important respects.

from expiration and for which

lf

F

>

#F A

K- Then

>

lIF

=

K (#F F(lt #) -

-

-

K) =

1. The volatility of a bond decreases over time as the bond approaches maturity. The prices of 30-day Treasury bills, for exnmple, are much less volatile t.11%the prices of 30-yer Treasury bonds. The reason is that a given chmzge in the interest rate, other things eq'ual, changes the price of a shorter-lived bond by less.

1

-

But bve also have B

=

e-rl'

F

(,I

.z?-r/, (s

-

-

K)

l lt

-

-

d + dF d

-

K)

If JJ

-

-

l

d

gcl

the futures price changes by $1, the option price From the second expression, you can see that if e-rh changes by

2. We have been assuming in all our calculations tat interest rates are the same for a1l maturities, do not change over time and are not random. While these assumptions may be good enough for pricing options on stoks, tey are logically inconsistent for pricing options on bonds: If interest rates do not change unexpectedly, neither do bond prices.

k

336

%.BI NoM

lAL OPTION

CHAPTER

l

PRICING:

' t'(E1lIi2E2E:lI,k''''r!!diI:@''''. ;'I1iE@!!' j'(' j)f E' i1.( ((' !T(' i.(EE('f;'r :'iiil'. tii'fq:iI'-'. :111.#:.112*. )' Eil1(EE!1iEii'il'f' j)f ;;' q' T' (t)' '1* 797J))* (ifi'';lEjf ()(i ):EE'(E.Er ( i'')'.((7*'' '77r777E577* E ; jE i'itiq,'. t'y'qf ptf @' (1* $' i @ (@ jqi lk q.'(.E..' . ('.i'EE' !(7'.@ .1q 'E.. j . (. . . . y .:. . . .y.. .y..y.y.,,.. . . .tjsj yyy;ty yty.. . .y yy y..y. y.gy...g L'.r!E7/'t;t).) .jyy... ).iE.....yE. . ) lliitit.) . ')..E.j.jj,qjjjj) jytE.,;j.Eyy)jqj.yEj,..jqjyyyyy.,yq..rjygjy,.yEjyy,y,y:t.y.tyy...yq.,y:.yyy,(,yEy..y..tyj. j @tI-)t'E.i . .LLL..? . qjj). ;j, . . g ......'.)'..--..........-.-.............;-k:'..--.....!t7i1lr)-----j(-.,-----).. qqj tyyy . . jj tjyy j:.jjjyj:y;jjr.jjji jj.j yji j.. -t.3L;1-kq;'6..L;.-.r..

.'

-' ''!' r' :';'q,n'jjj. ,'t'.j' -q'. -'-'iii!f,-. -'. -.11.1,,:*. -'1iii!:-'. :;''f ;''717''7;77r:7)757)* Cqitf j'6* .11:11111:9*. 5* :tf yjf ;'4* (' rprfj. ' Ek;q

111:*. ::2!912,7.11,..*. Elll)d -''ii:iiiII!.t'EIIlEEF;'t::.'. 'iiIt::'. j .'.'rlj''rplljlls'q)f ).'i)!.. ;)()jJ.)7 ,.......Iit:2:.....-.'.' iiidi2-.... .. !-)7tL(-..)'! ...:..'...-..-.......--....B... ...

'.' 'sii-r.: .;'. ,''' ;,'

much. bond, these issues On the other hand, if we have a 3-year option to buy a s-year bond payments are discrete, so coupon might be quite important. Another issue is that approximation. the assumption of a continuous dividend is art different nugh that ln general, the concepmal amd practical issues with bonds are 24. Chapter bonds wgrrnt a separate treayment. We will remrn to bonds in

Summary discussed in this section. Here is the general proeedure covering the other assets the price of the underlying asset using * Constnct the binomial tree for Fk.t+l, e tt Ft.t-vlte+(rJ# If St or S: (10.15) F',t+ll --.(ws4- Or # e dSt Ft,t-he St price formulas, the Since different underlying assets will have different forward underlying assets. 'ee will be different for different option is unexercised, can then be computed * The option price at each node, if the as follows: Ft,t-h 1St d ff # tt (10.16) ..yo.x/k7

=

=

-c.v'#

=

=

-

--

r-&lll

=

d

-

11 d -

and, as before, 2*

c

=

e-rh

(/5/

sc

11

+

(1

-

pslcdj

(10.17)

the current node. For where Ctt and Cd are the up and down nodes relative to the value if American option, at each node take the peater of this value and

an exercised.

risk-neutral Pricing options with different underlying assets requires adjusting the Mechanically, underlying asset. of the probability for the borrowing cost or lease rate stock index with an this means that we'can use the formula for pricing an option on a summalizes 10.1 the substitutions. Table yield. appropliate substimtion for the dividend .

.

:

:

: .:

.

-

..

.

:

.

:

.' . .

.

E:' . .. . :

. .

... .

:

. .

' r

:

.

y..

:.

:

..

. .. . ; .ti)y

,

!

:

'' E : .: E .y jj

. .. y

;

..,y:

.

'

: .

,

....

: E .

:

:,. ;.jyy ,yt)

.-)ytik--.

:

y. . y . ,. .. :

..

.

:

: y: r

.

j

;

.

:

.... y .

yjjjj:y;jj

-.j::'.;.).

,.

.

.

.

((

:

(

.k!!r!!!i..2l..

:

((

E

'

(( ( ( -' '

-'

.-

. .

-..

-

-

.

- .. .

.

-.

.

.. .

.

i

- .. ...

.

:

i.!ii;:-

-.

.

.

.

j:y;jjr.)j j:r;j;..j kjgj yjy yjygjyi jjyj,.j jgjj j:y;j jjj j.. jjy .jo jjy;j j..jjjjj..j jjjj jk jytjy jr.jjty jjr.)j jjjjjjty;jj:y)j tjy

j

stock index.

k (t E.ii:t''' ;.tiii1:-:11Ili:. 'i1It::. ..111(E:..t11..ii1i::''.Illlii!li;' i '. . .' .. .. . . . . . . . .. . . . Domestic Stock index

( :) g ( ; : E- ,.- .. ..- . . . .

;'.'...'

'' :. :.. . .

'

:

using the simplebinomial ln some cases, itmay be reasonable to plice bond options option on a 29-year bond. The model in this chapter. For example, consider a 6-month the volatility underlying asset in this case is a 29.5-ye% bond. As a practical mattey, sftptl. because Al$o, difference betwqen a 29.5- and a 29-year bond is likely to be very inierest short-term sensitive the to it is short-lived, this option will not be particulady will maqer interest not rate and the 6-mont.h rate, so the correlation of the bond pzice

%.

SUMMARV

-.

.

.

-

..

.

-.

( ( ( ((k :-' ( j j i. . .. .. ''1It::. ' q...'i.i'...p. r . .. ...'.'-

( E.g E.j ...E'''iL.. E

.

.

.

..

.

...

l'isk-free

-. . . ..

..

.

g:E

.

-.

.

..

-. .

.

-r

-.

.. .

.

-

-

. . ..

.

-

umres contract Commodity Coupon bond

Domestic

tisk-free

-

.... . .

rate

Domestic risk-free rate Domestic risk-free rate

Currency

:

g

-.

. .

rate

Domestic risk-free rate

.

-. .

( L ( j j j jljl( (( ; ( !,IIE'iIIL.. i1.Ei!,,-. iiii.. -'I::i!iII... ' k''ti!!i-. !lll '. ..I:tE'IIC.. il i'. . '. .. ' . .. .. . . .. .. Dividend yield '''iii;.''iii.-

.' .

-.

... . .......... -

.

.

. ..

.

g:

j

-... -. .

.

-

.

.

Foreign

.

.

-

..- . .

.

lisk-h'ee

-'

-. . . - . -

.

.

.

.

-

' ' '' . .. .

-.

.

' - ' '

.. . .

rate

Domestic risk-free rate

Commodity lease rate Yield on bond '

In order to plice options, we must make an assumption about the probability distribution binomial distribution provides a particularly simple stock of the underlying sset. point distribution: At in time, the stock price can go from S up to If S or down price any If and d are given by equation (10.10). the dS, where factors movement to binomial stock price Given movements, the option can be replicated by holding A shares of stock and B bonds. The option price is the cost of this rplicating portfolio, A,C + B. For a call option, A > 0 and B < 0, so the option is replicated by borrowing to buy shares. For a jut, A < 0 and B > 0. If the option price does npt equal this theoretical plice, arbiage is possible. 'I'he replicating portfolio is dynmic, changing as the stock price moves up or down. Thus it is unlike the replicating portolip for a fonvard contract, which is fixed. The binomii option pricing fonnulahas an intemretation as a discounted expected value, with the risk-neutral probability (equation10.5) used to compute the expected payoff to the opiion and the risk-free rate used to discount the expected payoff. Tlzis is lcnopn as risk-neulal pricing. Thebinomial modelcanbe usedto priceAmericap andEtlropean calls andputs on a variety of underlying assets, including stocks, indexes, f'utures, currencies, commodities, and bonds. rfhe

FURTHER

REAUING

This chapter has focused on the mechanics of binomial option pricing. Some of the undrlyingconcepts will be discussed in more detail in Chapter 11. Thre W Will have more to say aboutrisk-neutralpricing, the Enkbetween thebinomiltree and the assumed stock price distribution, how to estimate volatility, and how to llice options when the stock pays a discrete dividend. The binomial model provides a foundation for much of what we will do in later chapters. We will see in Chapter 12, for example, that the binomial option pricing

'

'

338

k. BINOMIAL

OPTION

PRICING:

PROBLEMS

l

%.

339

0. Let tt 30%, r 1.3, 8%, F 1, and ts 10.8. Let S $100, K $95, tr option. /3 the binomial for American At 0.8, and 2. Construct tree put d an premium, and node provide A, B. each te

small. formula gives results equivalent to the Black-scholes formula when 11becomes the understand also pricing, you Consequently, if you thoroughly understand binomial generalize binomial will trees to see how to Black-scholes formula. ln Chapter 22, we tmcertainty. handle two sources of and Bartter ln addition to the original papers by Cox et a1. (1979)ad Rendleman binomial the exposition of excellent (1979), Cox and Rubinstein (1985)provides an model.

=

=

=

=

=

=

=

=

=

10.9. Suppose s'g $100,K 1. and r =

=

$50,r

=

7.696%

(cntinuously compounded),

=

0,

=

1.05. What is the 1, we have II' 1.2 and # a. Suppose that for h binonaial option price for a call option that lives one period? ls there any problem with having d > 1? =

PROBLEMS

=

=

0.6. Before comptlting the option l and d b. Supjose now that tt pri, what is your guess aboui how it will change from your previous answer? Does it change? How do you account for the result? lnterpret your answer using put-call parity. 1.4 and # 0.4. How do you tIIiII:Z the call option price c. Now let 11 will change from (a)? Hciw does it hange? How do you gccount for thiSb Use Ijt-ill parity to explain your aswer. .4

=

a11rates are In these problems, 11 refers to tlze number of binomial periods. Assume otherwise. xplicitly states continuods ly compoun ded unless the #roblem

10.1. Let and 11 u

$100, K

=

$105,1,

=

8%, F

=

0.5, alzd J

=

=

0. Let

u

=

1.3, d

0.8,

=

.1.

=

$100, K

=

$95, r

=

8%. F

=

0.5, and

=

=

t

r

0. Let If

=

1.3, d

0.8,

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

=

0. Let 11 1.3, 1, and $95, o' 30%, r 8%, F 10.6. Let S $100, K 2. Construct the binomial tree fpr a European put ption. At d 0.8, and /) each node provide the premium, A, and B. of 10.7. Repeat the option price calculation in the previous question for stock prices else lixed. What happens $80, $90, $110,$120, and $130,keeping everything increases? price stock the A to the inital put as =

=

=

10.10. Let S $95, ?$100, K 3. and n = 0 F 1 year,

=

=

=

=

=

=

=

'' .'

=

8%

(continuouslycompounded),

o'

=

30%,

=

=,

7

1.

=

call is.$16.196. a. Velify tlat the price of a European arbiage? b. Suppose you observe a call price of $17. What is the price of $15.50.What is the arbitrage? c. Suppose you observe a call 10.3. Let S $100, K $95,r 8%, F 0.5, and $ 0. Let tt 1.3, d 0.8, and ?1 1. is a. Venfy' that the price of a European put $7.471. b. Suppose you obselwe a put price of $8. What is the arbitrage? of $6. What is the arbitxage? c. Suppose you obsetwe a put price 1.3, 1, and 8 0. Let If 30% r 8%, T 10.4. Let S $100, K $95, tr node each option. At call for binomial tree a 2. Construct the d 0.8, and l provide the premium, A, and B. of 10.5. Repeat the option price calculation in the previous question for stock plices happeps else evelything What keeping and sxed. $130, $80, $90,$110,$120, to the initial option A as the stock price increases? =

=

=

and B for a European call? a. What are the premium, A, b. What are the premium, L, and B for a European put?

10.2. Let S and ?;

=

a. Vel'ify that the binomial option price for an American call option is $18.283. Verify that there is never eady exercise', hence, a European call Wouldhave the irie plice.

b. Show that the binomial option price for a European put option is $5.979. Venf'y' that put-call patity is satised.

c. Verif'y that the price ' of an American put is $6.678. (

E

10.11. Repeat the previous problem assuming tllat the stock pays a continuous dividend compounded). Calculate theprices of theAmelican of 8% peryear (continuously and European puts and clls. Which opons are eady-exercised?

10.12. Let S .

F

'

=

=

$40, K

=

0.5 year, and

$40, ?12

=

=

8% (continuouslycompounded),

o-

=

30%,

=

0,

2.

a. Consttuct the binomial tree for the stock. What are b. Show tat the call price is $4.110. c. Compute the prices of Amrican

tt

and #?

apd Europea!l puts.

10.13. Use the snme data as in the previous problem, only suppose that the $5 instead of $4.110.

catl price is

a. At time 0, assume you write the option and fonu te replicating portfolio to offset the written option. What is the replicating portfolio and what are the net cash flows from selling the overpriced call and buying the synthetic equivalent?

;

340

% BI rqo M IAL O p'rl olq

APPEN D IX

P RI cl lq G:

b. What are the cash flows in the next binomiz period (3 months later) if the call at that time is fairly pliced and you liquidate the position? What would you do if the option continues to be overpliced the next period? c. What would you do if the option is underpriced the next period? 3%, If 10.14. Suppose that the exchange rate is $0.92/6. Let ?'s 4%, and rq and K $0.85. d 0.9, F 0.75, n 3, =

=

=

=

=

=

1.2,

=

$1.00.

dollar interest 10.16. Suppose that the exchange rate is 1 dollar for 120 yen. is 1% compounded) and rate (continuously te yen rate is 5% (cpntinuously compounded). Consider an at-the-money American dollar call tat is yendenominated (i.e.,the call permits you to buy 1 dollar for 120 yen). The option has 1 year to expiration and the exchange rate volatility is 10%. Let n 3. =

a. What is the price of a European call? An American call? b. What is the pzice of a European put? An American put?

=

prices for American options instead of

10.20. Repeat

the previous problem ciculating European. What happens?

AND OPTION

10.A: TAXES

PRICES

It is possible to solve for a binornial price when there are taxes. Suppose that each form of income is taxed at a different rate: interest at the rate rf capital gains on a stock at the rate Tg, capital gains on options at the rate To, and dividends at the rate Td. We assume that taxes on a11forms of income are paid on an accrual basis, and that there is no limit on the ability to deduct losses or to offset losses on one form of income against gains on another form of income. We then choose Lt and Bt by requiling that the afer-tx ret'ul'n on the stock/bond portfolio equ the after-tx rettzl-rlon the option in both the up and down states. 'Fhus we require that

g&-mTgvt-v

10.17. An option has a gold f'utures conact as the underlying asset. The current 1year gold ftltures price is $300/oz.,the strike plice is $290,the risk-free rate is 1. What 6%, volatility is 10%, and time to expiration is 1 year. Suppose ?1 replicating the for the gold? What portfolio is price call option of is the on a entails call always option VtReplicating option? Evaluate the statement: call a asset.'' underlying buy.the borrowing to =

5%, J

=

5%, F

=

1,

The solutions for

=

=

=

a. What is the price of a European call option with a stlike of $95? b. What is the price of a European put option with a strike of $95?

L

and B are then

1

-

=

vo

E1+

+

?-/,(1

z.)1 Bt

-

.

(10.18)

(.%+/,) /,(x%)1 14,+/,

To

-

) 4 , sib) / j (5.-1 -

S*l 1 -L r # S-1 1/ 1 (S ) A # $ I (S ) $ To 1 d u -

B

=

1 1 Ti JL ?-/, 1 ro -

-1--

/

-1

-

Tg

us'p

-

1

-

-

T

-

Tg

-

o + J(1

-

Tdl

-

This gives an option price of

$t

=

=

AJ

z $t+h(.%+/,)

A

=

$1000?Why do a. What are the prices of European calls and puts for K equal? plices be the find to you $10002 b. What are the prices of American calls and puts for K c. What are the time-o replicating portfolios for the European call and put? 5%, 8 3. Let 3%, and T 30%, ?' 10.19. For a stock index, S $100, c ?7 3.

z'd)1

-

-

=

30%, r

+ &&(1

,%) -

-

c. How do you account for the pattern of early exercise across the two options?

=

=

,

'rhe

=

=

=

=

APPENDIX

a. What is the price of a g-month European put? b. What is the price of a g-month American put?

=

341

.

10.15. Use the same inputs as in the previous problem, except tlwt K

=

$

10.21. Suppose tat tt < er-&th showtlaat there is an arbitrage opportunity. Now er-nsll showagain tlaat there is an arbitrage opportunity. supposethat d >

a. What is the price of a g-month European call? b. What is the plice of a g-month American call?

c

TAXES AN D OPTI ON PRI CES

$100, o' 30%, ?- 3%, and & 5%. (You $95, K c. Now 1et S have exchanged values for the stock price and strike price and for the interest rate and dividend yield.) Value both options again. What do you notice?

=

10.18. Suppose the S&P 500 futures plice is 1000, and /1 3.

1O.A:

=

1 1 Ti 1 + rlt 1 ro -

('%+/,)+ Lp*$t+lt +

(1

# jstwi, (S,+/,)1 +

-

-

(10.19)

-

where 1+ +

#=

?-/,

1 1

-

Ti -

-

Tg If

-

8 d

1

-

Td -

1

-

To

# (10.20)

In practice, dealers are marked-to-market for tax purposes and face the same tax rate on a11fonns of income. ln this case taxes drop out of all the option-pricing expressions. When dealers are the effective price-setters in a market, taxes should not affect prices.

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and tr 0, any in-the-money option should be aexercised In the special case when r = when the stock price is twice the exercise xercise 0.5r, tlten we l'mmediaiely. If J =

=

price.

Thedecisionto exerciseis morecomplicatedwhenvolatility is positive. lnthis case the implieit insurance has value that varies with time to expiration. Figure 11.1 displays $100, the price above which early exercise is optimal for a s-yearoption with K 500 usirig computed binomil volatilies, different 5%, and J 5%, for thzee ?' stock price, stepsk Recall from Chapter 9 that if it is optimal to exercise a eall at a given shows 11.1 the lowest prices. thus stock Figure all then it is optimal to exercise at higher which is price, this lowest oscillation in stock price at which exercise is optimal. The approximate that binomial and down movements evident in the figure, is due to the up the behavior of the stock', with an ifnite number of binomial steps the early-exercise schedule would be smooth and continuously decreasing. Compacing the three lines, we obsel've a signilicant volatility effect. A s-yearoption with a volatility of 50% shold only be exercised ifthebtockplice exceeds about $360. Ifvolatilit is 10%, theboundary =

=

=

' :

.

.-.-

-

''' -

'' .

-

.

. .r.

!. . . . ;: . : . . .. -..:.111-,211. :

:.' .

. -

!

-

'

.

-'

FE-.' .

i

-

-

. . ,.-1j11).

'

.

. .

-

: ..:

'

:.

.

-

.r...,..;

;.

'

. -

-

. ..

.

-

. -. : -)jj,....

-

: : E-. .. -

.

.

.

-

,-j..

.

-. .

-

i

.

-.

E'

. . . .

-

.

'

: -' ' -'. .

.

100 90

''

.

o-lo

=

.

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'

...,. . ..

au .

.

.

.

.

.

#-. q

.

.

.....z.

..

C'

.

.

,/.

y'

.

.

,

,)

'

.

.

=

=

J

=

..

5 ogo.

z .

.

'$ ,q&,,

-

vo '

;

'.

.

.

80

...

.

.

boundaries for of 10 o/o, volatllltles . 300/o, an d 50% f'or a 5-year Amerlcan p t O tlO n t In a II Cases P /( $ 100, r 50/0,an d

'

' '

.

:

.

.-;::;;(;;ty.

.:-.y

.

' .

''

'

:'E .' .' . :

E'

-

;

. ' .' . Early-exercise .

.


E Fi g ure 11 4 is . example

A multi-period

'ee

.

'

.

jyjs

x j /3

1.

.(;

*

g6gg

=

0.5247

.

To comput te p t ttae nod Whre the ftk ljrice ij optifl yricete aext period ai 26.9%. This gives

expected

e-0.269xl/? x $34.678+ (1 0.5247) (0.524.-/ -

x

$6t.149,W

discount the

$12.814) $22.202 =

$43.246,the discount rate is 49.5%, and the option price is ,-0.495x1/3 x $12.814+ (1 0.5247) x $0j $5.700

W'hen the stock prie is

lsee,for example,

$

:

.

:g y . yy - -- - - - . ..

.. -

PRl C1 N G

'

j( g

:

-

-

,0.15

y

y

fhr

=

'

. .

. : :

,

---).

'

E

.

:

:

:

=

iS

#

.

''

.

.00

ihe

e

.

''

EE E ' '

'

:

=

tisk-neual probabilides: At Now we face the proble> with using real as opposed to discount te opon whai rate do we discount this expected payoff? It is not conect to equivalent to a leveraged is the opdon stock, because expected the ret'urn the a, at on the stock. investment in te stock and, hence, is riskir t1):.11 Denote the appropliate per-period diseount rate for the option as y. To compute y, weighted average of the we calz use the fact that the requiredremrn on any poAolio is te porlblio.l In Chapter 10, w saw that an option is etpivalent ret'tlr'nson the assets in the exfected remrn to holding portfolio consisting of A shres f stock and B bods.

lz/:

'

.

:

-

=

=

-

#. Using p, the actual

'

.

.

.

. .. . .' Binomial treefor pricing a European call option; assumes :=F 41 # $40.00, o' 0.30, r 0.08, F= 1 0.00, and years, h 1.OOO.This is the same as Figure 10.3.

.

=

If

MCFYYiO On tYS

.'. .'..' .''

E'!E : E' E ' E ; . EE .. . . E 'y' ggy . (j;Eq yj. gyy y. y .y . y y --.'

: '

L.L.. :.

.

ah

RISK-NEUTRAL

:'#' tj'yyj. y' 'j'q' (' j7* 5)* ff ;q(;t;'. .j' ')' i'q!' ?' r' 1* if :774r:y' yq''!qlrrr'qqlq:qqll (/k t'j'('(1* qt' p;f (.;((('@ (j(;.j. ; ..(qq (tq.'')rqiqpp))i y(.'j' (::1 k( ' (' ( ij q'qE((ill i E.('( (( !'.'i( rq:' ?' yjjj'( ' i'!t:' ifi@( $* l'r' plilt-:-.dl!f L-?iLL-'. j'y' ;';'(kl))'. )' j'. jjj'. jjf (' .i5. i'p' i'. 1111(*. . t'r'7IiiI.(' .'.( '.. ' . . . .. . jj. . ..j t()j)jj).' )-',' t'. jyj' j.j . jy..y'.j.j.j. .y(.'j..yy'. .li. .' tqil.E...( yyy..(..jyy. j. 'jy tyyg' L;L . .). j r..ygjgj yjjjg' ;: . jjjj.jj j .. j j j g j g . g j . .. . . j ! . t . 1IIiiii.. (E .jj .. j j : . . . illlk:tliii. i '. !!!!iqqit-. t .. ' (.:. .....-.'. : p- r.. )-... .;...t......)-.., (.. .''yijj'y '.' '''r:' '' -.-.'ki.. --.-----' -' ;''F' ' i 'I.II::::Cii..X --' ---j'-' -' y' $),* y',,;jyjjjy j','gjjjgjjjjy. lff .4* (' @E 'E : E :E E ; :E E E : . . : . . . . . . . . (E !'E).t.ijy.EL;E'CiI. iEq)@-f/i)@

Solving for p gives us

D l NG

y.5:2,,(j,

-

=

%.BlxoMlAL

350

TH E

11

Pnlcllqc:

OpTlox

)' ;'!' (rr.'!ri ;j'* j'jjf y' q' 1* $' srl' '11:* @' )' 1.1* l't: E' qr!f )5;* q' y' j('. )'. q' iff t.;Ff j(' )' tyf t'tif i'1* 77!717'r'''q!' qq!rf 1(* r'jjjjtgjgjjjj'i tf ttlf (' jgjjjjy;d y' t:jd j'jg;'.yyyyy. ;yj'. y' r' jf yyd j'q j:' jjgfyjyjjyjjjjgtjjy'yi'y j'. )'. (jj' ryjjj;;d yj'tr (jj'r(yjy'. yj'. j'?:'jjd y'. E !E .(' (' y' ' ( !';)i2 ' ( !.E!'(.'((..'('' i.!'i' 'i prl !E(: . ; .' i' E!i.!!.i'(.pi(:: ( :; t'y'. . . '.;' . . . '. .. (.( ( y;(. yyy (j(' p:i;kl. ;yq:jty jy;y. jjytjjrrjjjjjjyy . . j .j j.f) .!-sl.; . . ; : : j j( ( ;j,jjjjj, j yjjy;j.jjjy gyjk jjjj.yy jyl j j.yyy j.yjgy y . . y . jj jy. j. . g (jy-jyjyi tti . yj.y jjjjjj;jj;j,. yjy . .. )i.-'t'E .jyygjy.yygg. E!).--?.t).F-.-......I...-..:...d.-..-'-''.'.'.'.'i:.. ;y .gjj.y.y . tj; .y ygjj.gjj.j.j..y.y . . .. ;',.y' jjrgjjjjd :'('ilfllhLflli'Elprlilrl''lilrl;E)f ky'. ,,,4444,,* ..yjj' ,g;;;;j,'

.' '

E jE : .

.

EE tE

E:' ' :' '' .

;

E: . .

:

E'E 7 iE E E!E E .E. i!

:: E: EEE' E

:

:

'' : .. .

. ...

,,;jjjjj.

.

.

.

y

.$jjy,.

'..)!,-.

...-.;:2ii))...

....

.

. y.

y

yj

yjy y yy y y kik---t y .g y g gyy . . . -

...

...................

Fi'

:

E' EE: iE. E.'i.. .y .. . .

:

.

'

' 5;

j( ,,..-.-.'.L-..i-....,..,..-.'.L.L-..----11.LX.i

. .. , g....E..g jjjjjgjgj,E E E E EE . -

'

. .

,

,

:. . :

-

.y

:

y

,-,

-

.

.-.

$34.678

.

Binomial tree for pricing an American call opton; assumes .S $41 # $40.00, c 0.30, r 0.0% T= 1 0.00, and years, h 0.3334 The continuously compounded true expected return on te stock, a, is 1594. At each node the stock 'price option price, and continuouslj compoun d e d true discount rate for the OPtion v, are given.

y

x/x

=

661.149

$22.202 v 0.269

.00,

=

11.3 THE BINOMIAL

TREE

M lAL TREE AN D LOGN ORMALITY

Q

AND LOGNORMALITY

The usefulness of the binomial pricing model hinges on the binomial tree providing a reasonable representation of the stock price distrlbution. In this sectio we discuss the motivation for and plausibility of the binomial tee. We will define a lognonnal distribution and see that the binomial tree approximates this distltiution.

=

=

=

gg;;;j,,

:.

Bl NO

.00

$52.814 $.12.814

sso.()71 $12.889

=

=

y= 0.323

=

y

.

o.a.s7

y

0.495

=

$37.351 30.000

,$35.411

62.535

,

NlA

=

$43.246 $,5x()

$41.000 $7.074 =

y

v

=

O.4qs

y

wx/x

The Random Walk Model It is often said that stock plices follow a random walk. In this section we will explain what a random wall is. In the next section we will apply the random walk model to stock prices. To understand a random wall, imagine tat we flip a coin repeatedly. Let the random variable J' denote the outcome of the qip. If the coin lands displaying a head, F = 1. If te coin lands displayig a tail, F = If the probability of a head is 50*, the coin flips, with is After th flip denoted Yi, the cumulative total, the fair. n we say Zn is -1.

.

$30 s8s $0.000 .

')'

=

(11.8)

X/X

,

$26.416 50.000 y NIA =

;''

lisk-neual These are both the same option plices as in Figure 10.5, where we used Plicing. though the e. To compute th price at the node We coptinue by . wt Z. ng back . . . . option jrice 4henext period xpected where the stock prie is $50.071,we dlsctwnt the at 32.3%. Thus, -0.323X1/3

e

+ (1 g() sg4'jy $:2,p,.:2,g:). .

-

0.5247) x $5.7004F:r $12.889

Again, this is the same price at this node as in Figure 10.5. The act'ual discount rate for the option changes as we move down the ee at a point i.ntime and also over time. The required return on the option is less when the stock when it is $43.246(49.5%).The discount rate increases price is $61.149(26.9%)t.11:.r1 equivalent to a leveraged position in as the stock price decreases because the opton is option moves out of the money. the the stock, and the degree of lvrage increases as possible obtain opton prices using standard to These exapples illustrate tat it is iehnlqus. denerk owekef, ihereij noraqn to dp so. Moredisconted-cash-qw risk-neuttat pricinj Worl mns that it is not necessa to estimate over, the fact that option. Sinee expected remrns are a, the expected ret'urn on the stock, when pricingan option pricing a great deal.easier. hard to estimate preciqely, this makes Appndix 11.B goes into more detail about risk-neu'al p/cing.

It t'urnsout that the more times we :ip, on average, the farther we witl move from where we start. We can understand inmitively why with more iips the average distance fl'oll the stnrting point increses. Think about the first flip and imagine you get a head. You move to +1, and as far as the remaining :ips are concerned, this is yt?lfr ncw starting point. Afier te second flip, you will either be at 0 or +2. If you are at zero, ti is as 1.f you started over,' however if you are at +2 you are starting at +2. Continuing in tllts way your average distance from the starting point increases with the number otdips.z Mother way to represent the process followed by Z,, is in terms of the change in Z?,

-

Z?,-l

=

F,,

explicitlyas

We an rewlite this more

Heads:

Zn

Tails :

Z,,

-

-

Z,;-I

=

Z?;- I

=

+1 -

(1 1.9) (11.10)

1

zzkfter?? llips, the average squared distance from the starting point will be ll. Conditional on the rst flip being a head, your average squared distance is 0.5 x 0 + 0.5 x 22 2. Ifyour lirst flip had been a tail, your average squared distance after two moves would also be 2. nus, the unconditional average squared distance is 2 after 2 Qips. lf Dtlt represents your squared distance from the starting point, then =

Dn2 = 0 5 x (D ?:.

l

+ 1)2 + 0 5 x (D lt .

-1

-

1)2

=

2)2

a-)

+ j

Since D()2 0, this implies that Dcl n. 'Fhis idea that with a random walk you drift increasingly farther from the starting point is an important concept later in the book. =

=

%.Bl N o M IAL

352

O p'rl olq

PRI

cl N G: 11

TH E Bl NOM IAL TREE AN D LOGNORMALITY

With heads, the change in Z is 1, and with tails, the change in Z is 1. This random walk ls illusated in Figure 11.5. 'I'he idea that asset prices should follow a rgndom walk was articulated in Samuelson (1965) In ecient markets, an asset price should reflect all available infrmation. by dehnition, new infonnation is a surprise. In response to new information the price is equally likely to move up or down, as with the coin flip. The price fter a period of time is the initial price plus the cumulative up and down movements due to informational .

3. The stock on average should have a positive return. The random walk model taken literally does not permit this. It ttlrns out that the binomial model is a variat of the random walk model that solves a11 these problems at once. The binomial model assumes that continuously rehtrns are a m??#t7?nwalk. Thus, before proceeding, we 'Iirst review some ct/n7pt7lf?7#c# compounded rettlrns. of continuously properties

surprises.

.of

Modeling Stock Prices as a Random Walk rf'he idea that stock prices move up or down randomly makes sense', howe'ver, the description of a random wnlk in the previous section is not a satisfactory description of stockprice movements. Suppose we take the random walk model in Figure 11.5 literally. Assume the beginning stock jrice is $100,and the stock price will move up or down $1 each time we flip the oin. There are at least three problms with this mbdel:

Continuously Compounded Returns Here is a summary of the important properties of

?-/,/.1-/,j.nSt+kIS:) =

((f we know the conThe exponential function computes prices from returns exponentiating both sides of obtain St-vltby compounded tinuously reprn, we can This gives equation (11.11).

fl' !)' 7* 'f'' i)tjd :4* 4q':.,1:.2:1t.* :jI:r,:I(.' 1:17,-7777:!77*'. yyd t'(t' 1' ;'@' i'iq)d f' lyqr'jld .-,-L-'. .!1t:y' )' !4111::::1;17*. j'('qy'pt!rrd t'. ..j):)-,-jr:'y' EI' .11:2y:11.* rjlryr:ltr,:lk.d jj)'. t'(t' :j' yyd yj'. ;'''' -': j'jI--'E 'j' -' --' pidlir'trlq:irE (ililqlrpEliEl.hl'') '71* )' ''F' ,'''CCX :'q' ,1::2(442::* ,1:::::,,* ')' :'' 1* ;2* $!:li'(! -'E k' :y' y' 'j'. Eky-,,g'. ,,,' ....' .-1;,;,,.*

.'

' i

'

'

-- .

-

'

'

..

--

-

. . -. -

:. .

-

' :

E-

.:. .

jppprqqi)i !!!!(' ...

.

-. . :-2,..--

...

......................

.......

.

-.

-7; '

:' -' ' : .

-y : . -

-g

:

.

.

.

.

.,.

,

!

r

.

-.---.

.

.

.-

-- -

t)y. . t.

-

-g

-q -

.

Illustraton of a random walk, where the counter, Z, increases by one when a fair coin flip comes up heads, and decreases by one with

tails.

'

-

- . .

i .

.

.

-

''

.

:.;

.

-.. -..

;

'' : .

:

i

.

.

-.

.

.

.

41::22((r::.

.-;;jjij!:'.

()1). (;:Eig,. S

21(:.. ql:Esr,:

.11,.2,1:. .1j,.(,l:. ;;!;I(,

St e rl'J+

x%.14,=

.

.

compounded returns.

Let St and St-o be The logarithxnic function computes returns from prices stock prices at times t and t + /7. The continuously compounded ret'urfl between t and t + /7, l't,twl, is then

2. The magnitude of the move ($1) should depend upon how quiclcly the coin Qips occur and the level of the stock price. If we flip oins nce a second, $1 moves li E ' :IEi 77:77777:77')777:. !'' LE E Eq'' i iql ti!E.';i. i';(''(.'.(EE !'.l'FiE:r.E' E' ' EE i( E /1 pjl 711)!r1i.EEEE q! jj)l l '''f-k (j:1)9: . 'E . ; (Ei'E.i ;! '!(E ';EEE.!'Ei' EE 'q; . . ig E E. !. E ltjjE (i E.@ ; i ! E-E ;. . E E ;-,jjjj.! t...E i!.; .E.Ej.);j (: :E..yi.p. i.j jjLILL. E! . E. .( ( ; y . . g j)). . y;q; IE j( jg .. jj.;(qj;j )g)(.. .j .;,-(!-j;;.. yy y;..yj.yEg;j.! -.E i;,. lq yjy. (qy yj l'lEj . ;j. ijEq.. .))Eh1p)....(; : .. .. . llllq:piliil ; ., . .(. !(.:;E. y( ()))y)k)).. ....y. j q y-i ... ... ..... r. . . ... .. .r.. , .r..

continuously

(See also Appendix B at the end of this book.)

1. If by chance we get enough cumulative down movements, the stock price will become negative. Because stockholders have limited liability (theycan walk away from a bnnknlpt srm),a stock price will never be negative.

.

353

are excessive; in real life, a $100 stock will not typically have 60 $1 up or down movements in 1 minute. Also, if a $1 move is appropriate for a $100 stock, it likely isp't appropriate for a $5 stock.

-

ijqd td t(;';' (:'I1I1k2L:IIir'' r' j;'. )' j'yyd '.ytyy'yy'tr;iyyyq' t'.:': (' (;' i'trd lllt!i!jkd y' -.1.-J111r*);,lkiiiiq)'.' rjjtd tjj')jy .t)' td 1il':

Q

(11 1g )

yy

Continuously compounded returns are additive compounded retul'ns over a number of periods-for

.

Suppose we have continupusly

example, h,twlt, r?+/,,r+2/,, etc. The continuously compounded ret'urfl ovet long period is the sum of continuously compounded returns over the shorter periods, i.e.,

o

11

-5

.13)

( 11

.

?-l

l-t--

==

l lltgt-ih

i= l

-10

A ontinuously Continuously compounded returns can be less than compounded rettlrn that is a large negative number still gives a positive stock price. The reason is that er is positive for any r. Thus, if the log of the stock price follows a random wnlk, the stock price cannot become negative.

-100%

-15 -20

Here are some examples

-25 -3O

-nh

0

50

100

150 200 250 300 350 Number of Coin Flipsyn

400

450

500

illustrating these statements.

Supposethe stockprice onfourconsecutive

Example 11.1

days is $100,$103,$97,

and$98. The daily continuously compounded retul'ns are ln(103/100)

=

0.02956; ln(97/103)

-0.06002;

=

111(98/97)

=

0.01026

THE BINOMAL

k. Bl NoM

354

lAL

11

op-rlorqPRlcllqc:

ret'urn from day 1 to day 4 is ln(98/100) continuously compounded returns: daily of ))jThis is also the sum the = tt + ?'2,3 + r3,4 = 0.02956 + (-0.06002)+ 0.01026 compounded

t) The continuously

)E)

-0.0202

k..

r1,z

TREE AND

3d5

$

LOGNORMALITY

with the square root of time. This is why The standard deviation therefore scales pricing model. appears in the binomial

-0.0202.

=

o'.ll i

k The Binomial Model

Suppose that the stockpdce today is $100and tat 1 yearfwm today qj)) Example 1 However, the 100)/100 't,(it is $10. 'Fhe percentage rettzlaa is (10 compounded continuously a (1:continuously compounded return is 1(10/100) .jj.retunsof k. .2

-1

=

=

-

understand the biqmial model, whih is We are now in a position to better

-90%.

-0.9

SL

.

yjt

s

cauu xzy

g

(p-jjjysg

ujy

-2.30,

=

),'

-230%.

Taking logs, we obtain 1n(&.g,/& )

=

(?' -

jh

(T''UV

:1:

(11.16)

.

compounded ret'urn from t to t + rt,t-vlt, the Since Lnst-vlt/'t) is the continuously continuouslycompoundedremm. the binomial modelis simply aparticularway to model certain of is which g(?' ()J,), and the other of which is That returp has two parts, one pdce moves (+cVI1. uncel-tainand generates the up and down stock tvee problems in the random wnlk solves the equation (11.16) Let's see how discussed eadier: down the binomial tree stock price cannot become negative. Even if we move 1. compounded retul'n continuously negative, resulting large, many times in a row, the will give us a positiv Iric. smaller, therefore up and down 12 2. As stock price moves occur iore frequently, gts volatility is the sarr n rtter how nl cqjtnlti, moves get smaller. By thpeicentage ,

Example 11.3 Suppose tlaatthe stock price today is $100 and that over 1 year the Using equation (11.12),the end-of-year continuously compounded ret'urn is = percentageret'urrt is price will be small but positive: ,I = 100=5'00 $0.6738. -500%.

'rhe

0.6738/100 1 -

%.

-99.326%.

=

'l'he

The Standard Deviation of Returns equauon Suppose the continuously compounded return over month i is rmonthly,f From nnnual the return is (11.13), we can sum eontinuously compounded returns. .

'rhus;

- 12

t-nRon jjj y, L

p.annua j z. i==1

The variance of the

nnnual

return is terefore rm'ontluy'f

(11.14)

f= 1

Now suppose that remrns are uncorrelgted over time; that is, the realizaion of the return expected retut'ns in subsequent periods. With this in one peliod does not affect each assumption, tlle variance of a sum is the sum of the(7.2variances. Also suppose tat nnnual variance, denote lf let ten te we month has the same vaziance of remrns. from equation (11.14)we have 'the

2

=

(7.2

12 x

mOn

gjjy

Taldngthe squarerootof both sides andrearranging, deviation in terms of the annual standard deviation: tr Jmonfllly =

thrare. sincretisftlowafndomwalk, marlybinomiplperiods the stock pric is $t00 r $5. whether the same Plicechange ts probability of an up move, so we 3. There is a (r J) term, and we cart choose the the stock price is positive. -

12

Vartrannual) = Var

'

-

we can express the monthly standard

./--12

If we split the year into /7 periods of length 12(so that 12= over the jeriod of length h, (r2,, is

1/77),the standard deviation

expected change in ean guarantee that the thought of #s a random FnlkqFigtqe 11.6 Tclillumate that th binprnial 7ee cm be when the continuously compounded rt'urn folloWs illuspts the stock pri that results particular path through a 500-step binomial ee, with is a ranom walk. The hgure one of coin flips as in Fi'gtlre 11.5. the par' ticular path generated by the same sequence

Lognormality and the Bpmial

lognormal distribution, which is commonly used to The binomial tree approximates a model stock prices. lognormal distribution is te probaFirst, what is the lognormal distribution? The assumption that continttously ct/l/lpt/lln#c# rcrffm, bility distribution that arises from 4he tree, we are nonnally ditributed. When we averse the binomial ::lu on the stock t7?z l of h 8 X-hIn the o' ) components ' implicitly adding up binomial random remrn variables random of binomial the 0), sum (x) or, the same thing, h limit (as ?1 compounded rettlrns in a binomial tree are continuously is normally distributed. Thus, -

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neutral probability of reaching each nal period node. There is only one path-sequence tlze top or bottom node (IfIIIf or ddd), but there are of up and down moves-reaching intermediate reaching node. For example, the ftrst node below the top each three paths huld) can be reached by the sequences IIIf#, ltdlt, or dutt. Thus, there are more pats that reach the intermediate nodes than the extreme nodes. We can take the probabilities and outcomes from the binomial tree and plot them against a lognormal distribution with the same parameters. Figure 11.8 compares a three-period binomiz approximation with a lognormal distribution assuming that the initial stock price is $100, volatility is 30%, the expected ret'urn on the stock is 10%, and the time horizon is 1 year. Because we need different scales for the discrete and continuous distributions, lognormal probabilities are graphed on the left vertical axis and binornial probabilities on te light vertical axis. Suppose that a binomial tree has 11 periods and the risk-neutral probability of an up move is p*. To reach the top node, we must go up n times in a row, which occurs with a probability of (p*)'l The price at the top node is Sttn There is only one path through the tree by which we can reach the top node. To reach the flrst node below the top node, we must go up 12 1 times and down once, for a probability of (p*)'i-1 x (1 p*). The price at that node is Since the single down move can occur in any of the this there periods, can happen. The probability of reaching the t.h node are n ways /7 p*)f The price at this node is Sun-id The number of is the below top (p*)'?- x (1 reach node this is ways to

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We can construct the implied probability distribution in the binomial tree by plotting the stockprice at each final period node, Sun-id against the probability of reaching that node. The probability of reaching any given node is the probability of one path reaching that node times the number of paths reaching that node: Probability of reaching fth node

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Figure 11.9 compares the probability distribution for a 25-period binomial tree with the orresponding lognormal distribution. The two distributions appear ctose; as a practical matter, a 25-peliod approximation works fairly well for an option expiring ip a few

months.

Figures l 1.8 and 11.9 show you what tlle lognormal distribution fpr th stockprice looks like. The stock price is positive, and'the distribution is skewed to the light; that is, there is a chance that exeemely high stock prices will occur.

Mternative Binomial Trees :

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Thre are other ways besides equation (11.16)to constnlct a binomlal tree that gpproximates a lognormal distribution. An acceptable tre must match the standard deviation 'pf the continuously compounded returfl on the asset and must generate a.n appropriate distribution as the length of the binomial period, /?, goes to 0. Different methods of constacting the binomial lzree will result in different 11 and # stock movements. No matter how we construct the tree, however, we use equation (10.5)to determine the lisk-neutral ' probability and equation (10.6)to determlne the option value.

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Is the Binomll Model Realistic? Any option plicing model relies on an assumption about the behavior of stock prices. As *e seen in this section, the binomial model is a form of the random wnlk stck prices. The lognormal random wnlk mdel in this adapted model, to lodeling othek stck plice nmng things, that volatility is constant, that section assumesj A1I of thse independent time. tht and do returns are over not occur, movements the data. violated in assumytions appear to be We will discuss the behavior of volatility in Chapters 18 and 23. However, there is evidence that volatility changes over time (seeBollerslev et aI., 1994). It also appears binomial model has the property that on occasion stocks move by a large amount. period length, h, becomes smaller. smaller the that stock price movements become as terefore a feature of the data inconOccasional large price movements-ljumps''-are sistet wtth the biomial model. We will also discuss such moves in Chapters 19 and 21. Finally, there is some evidence that stock rettlrns are correlated across time, with positive .have

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J.n pratice we need to figure out what parameters to use in the binornial model. 'I'he directly. most important decision is the value we assign to t7', which we cnnnot objen?e t One possibity is to measure o' by computing the standard deviation o condnuously compounded historical returns. Volatility computed from historical stock retxlrns is historical volatility. Table 11.1 lists 13 weeks of Wednesday closing prices for te S&P 500 composite index and for M, along with the standard deviation of the continuously compounded returns, computed using the StDev fucdon in Excel.4 Over the l3-week period in the table, the weelly standard deviadon was 0.0309 and 0.0365 for the S&P 500 index and IBM, respecdvely. These are weekly standard dviations since they are computed from weeldy returns; they therefo/e measure the variability in welly returns. We obtain nnnualized standaid devi/ons by tuliplying the weeldy standard deviations by W-i giv hlg mmual standai d deviauons of 22.32% for the S&P 500 index and 26.32% for IBM. ' We can now use these armualized standard deviations to construct binomial eees with the binomial period, h, set to whateker is appropliate. Don't be misled by the fact that the standard deviauops were estimated with weelly data. Once we annualize the esthnated standard deviations by multiplying by VV Fe can then muluply again by .Wto adapt the annual standard deviation to any size binomial step. 'I'he procedure outlined above is a reasonable way to esdpate volatility when conand identicalty distribtd as in thelogatinuously compoundedreml'ns areindependt arenotindepepdent-as randomwalkmodel in S.ectipn11.3. However, ifretllrns lithmic with som commodiies, forexample-volatility estimation beomes mom omplicated. If a high price of oil today leads to decreased demand and increased qupply, we would expect prices in the futum to come down. In this case, the volatility over F years will be t7'uf, reQecting the tendency of prices to revert from exeme yalues. Extra less than care is required with volaety i.f te random walk model is not a plapsible economic model of the asset's price behavior. ,

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E ; El iljjj;j' E ' '. ' E :jj(gyjjj,' ;' .. . '. . .E' ( ' q (i . ; (ijjjjjj;;i ' ( E (E('. ( (('y)i''.' .l.@ . . . .. :.' !.(ljjjljjyy:,jjl .y......(.(('.....(' y..y....'(.'... !.'.).' ('E...' .( !E.E.'. !.('.'l.' k' ('(''y'.E... i1 kjjyi,jrg ..(jj y.y.y..(... . y(..y(y.(..((. E.. ..y...(( ..y....... . ...j.j.. . (Ij:yrj))i ('. .... .(..(''.E'. . (('y(.!.(.i' . . . .. .. y. y..... yjj.''.'.j. j '. .' ..' (' . ' ' . i . . . . j ' .. yjjj.s . . . . (.y. ... (j ( q j) q jjjjjyr,r . j . .'C..''i..'.' (....y.. (.'.(.y.'..''.'(.......y..y..(y. ...y.. ... .q... yjkjj,. . gjj;yy,-,. . . . yy . . . jyjjjjgy:jj,y. . jy.; . tjjjj, rjjjjyrjjjj,y .. jfffjqqjifq)j. .. jg.(j.yy..i jyjjy,. . .... .. EEi''..l' yjyy.( j. .j.....j. j...ry j.yy...j........()jjj.. . g..yy.. )...... @l..('E.'..('(..)....E..E..(;'...q.(.....(.(......... ... .... (. . . . ... . . .

.

VOLATILITY

: --1 :

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11.4 ESTIMATING

D lVl

JE (: i( 7E )'(yjjiljj:iil'd jy'jy': y)'.j'( ;'C'. t'tl.t?qf .( ;. .:!th,---.' 11:::::,,* ljf yytr-t'jy y' j't' jqf r!'l'9'' ' '(.ig kjyljjjjjyi.jjjr.d )'')y'. ))' j.l!))'jyyy);yjjjjy.y,. tyjjjjgjjjj,;d tjjjggrrr'. )' yj'. 1* )'j)jq)).ykjjj)jy))' tttf (jgjj;d ..' t)'. )#iiiiiiikii#'---' iiiiiiiiiik:.' ''jyjf y...jy--'-t: 'qjjf yj'!;f yj' .r!:,,,-,.' )gji:!ithjj;;t)ji(jy'. ttl'. tljijljyyjjljtjj,i;d yyyyyyjj't-'y.'yjjttf ' : iiiiiiiE tt)y )yg.(;k#jjjj). tjyyji , ;yy.' ,;jjjjj( ij-. yjy..rj..yj.yy.y, yj r ! gy(r y j . . . y j y . .. ..1:::: yy . jl',,.,.l . Cl'ir.. jjkjj ; j il'...!ll j-jjjjjj. I,.''' -l::E::,,i,r''' 'lE!ii,,r .tIt::( 111 'I'..',lr..'l.. ll''..!lid'Eii!,'11,-:::, iik s a ii Il:.,,.ll ... )q .!ty t-. . y;j.g.jjq:ry:,,, s jrjj...y..j.j,jj )(l.'li. '-; . '.ii!i:LL1'. .. .. . .. . . . .. index and from 3/5/03 IBM, to for the S&P 500 5/28/03.

.jyjkj((.;'. ..' .y-'. j,:!jjjjj:;'. -' -jjjjy,,' -' ,,jk,;j,-pkk-;,,,' -j'.

correlations at the short to medium tenn and negative correlation at long horizons (see Cnmpbell et al., 1997, ch. 2). 'I'he random wnlk model is @useful stnrling point for thinking about stock price behavior, and it is widely used because of its elegant simplicity. However, it is not sacrosarlct.

PAYI N G DISCRETE

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03/19/03 03/26/03 04/02/03

874.02 869.95 880.90

04/09/03 04/16/03 04/23/03 04/30/03

865.99 879.91 919.02

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81.55 81.46 82.88 85.75 84.90

.

0.0868 -0.0055

-0.0011

-0.0343

0.0516

0.0340 -0.0100

0.0207 0.0230 -0.0288

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0.0160 0.0365

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0.2632

0.0318

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86.68

88.70 86.18

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std. deviation std. deviation

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05/28/03

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11.5 STOCKS PAYING DISCRETE

DIVIDENDj

Al t ough it may be reasonable to assume that a stotk index ljys dividndj contiuously, stocks pay dividends in discrete lumps, quarterly or annually. In addition, individt!al How ovef shrt horizons it is freqntly possible to predict the amount of te dikidd. should we plice an option when the stock will pay a known dllr dividpd dudng the life of the option? The procedure we have alrady developed for creating > binomial 'ee can accommodate this case. However, we will also discuss a preferable alternative due to Schroder (1988).

(i

4We use weelly rather (11m1daily data because compudng daily statistics is complicated by weekends and holidays. ln theory the starldard deviatlon overthe 3 days fromFriday to Monday shouldbe orter t111111 over the l day from Monday to Tuesday. Using weekly data avoids this kind of omplicauon. Further, using Wednesdays avois most holidays.

Modeling Discrete Dividends When no dividend will be paid between time t and t + //, we create the binomial tree as in Chapter 10. Suppose that a dividend will be paid between times t and t + and that

'

362

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lAL OpTl

o?q Pnlcl

STOCKS

11

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its ftlttlre value at time t + h is D. 'I'he time t forward price for delivery at t +

IL

is then

E'EE' t(' j.grf $' t');' )' lfltE'lit!.pqrlf jjjsf Ejj' ))41:r* fp' ''(' (' E E E Cf (;')(t! t';''' kjtyjjj'ry,f .';'5)* :prEf qf 711(f?* jf tltqi ilrql;i EE E jjt'.' tjjjyyf (;' q' ?' ()(* 1* ((' ilf i'(?' ; iq .'E E.E E !;7i ';')#';':i i'!' i'' Eli i!(E.( Ek,; EE('E : .7:7* ;ftlSE qpqf yyf r' .j' j'jyf . rr;c)E'l(l'' E.r yjyf .jjjjy', )((rt' .' (' y' jlt'qqqsf (jjj;'. t't'y'j'jtf .. .. . @E;. .(.E.. qii . E. E.i.qi ; 'El yE;E j.ii.. jj.(.il . 151 . . . y; .y2 ; j. ( .gjy yyj.;.y Ejyjjj.. r!11:i:;k'-.L--..'-.. ; . . (t:'),,!qjd:kjy!)iqyjrtj#@ jj, )j jjg .. @...-..,:#?t)k.-.)'...rt)..'..:.)?....-;.)!ri)..).E;. .(jj )tytjg ;))j.))L;;.. . .jjyyjgj..q-jy.yy, .t . .'.r..'i''i#l.!'ilil... :'.11t2:2Il.......-....lIiiplt.... jyjjlyj;y tjjtj:-..).......)-..j. y' j'(' 7* :';'?k'' S' Elf

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=

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Since the stock price at time t +/7 will be ex-dividend, we create the up md down moves based on the ex-dividnd stock price: 5'lt t Sd t

=

=

el'l

LuterhD) C-GW -

sterl'

tjjogtg

Dj

-

How does option replication work when a dividend is imminent? When a dividend is paid, we have to account for the fact that the stock earns the dividend. Thus, we have + 0) A + erhB =

c,,

sul

sd l +

erhe

:

.

0) h +

cd

=

jy,r

jtry;j

j:yjjg jryg,j

Dividend:

Ctt

A=

tyjji

ryjjr-

5

0

$67.417 *55.203 y,jr,,oyo -

$47.679

$50.071

sa9.c41

=

The solution is -

ryjj,

0

=

=

363

:.

-. .

The ttee depicts a discrete $5 dividend paid between the first and second binmial periods. There are eight discrete terminal nodes (six of them distind) . rather than four. Assumes .S'= $41, (r 0.3, r 0.08, t 1 year, and h 0.333. =

Q

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DlVl D EN DS

:'

.'

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PAYI N G DISCRETE

$33.720

$41.000

Cd

$45.553

s;t sf -

sdc f s''lcd t 5ili

$37.300

-

B

l-lt

=.a

e ---

11

=

vtd

LDe

---

?-

/1

$35.411 532.216

-

Because the dividend is known, we decrease the bond positionby the present value of the certain dividend. (When the dividend is proportional to the stock price, as with a stock expression for the option index, we reduce the stock position, equation (10.1).) price is given by equation (10.17).

$32.216 $26.380

rfhe

$22.784

Problems with the Discrete Dividend Tree 'Fhe practical problem with this procedure is that the tree does not completely recombine after a discrete dividend. ln all previous cases we have examined, we reached the snme price after a given number of up and down movemepts, regardless of the order of the movements.

Figute 11.10, in which a,dividend wit a period-z value of $5 is paid between periods 1 and 2, demonstrates that with a discrete diyidend, the order of up ap tl down povements afects the price. In the third binomial period there re six rather than four possible stock prices. To see how tlle tree is constructed, period-l prices are l/3+0.3x4'f7'z

$41c0.08x

=

$5o (pl

$41e osxl/3-0.3xs/l7J

=

$a5 4ll

.

().08x1/35) x e0.3xx?'f?-J $55 2()a ($50.071c I/3 5) x e-tl.3x./I7!' 0.0sx $a9 04.1 ($50.071c -

=

=

'ee

,

.

.

The peliod-z prices from the $50.071node are -

ud and du nodes do not recombine. There are six distinct prices in the final ljefiod as each set of ex-dividend pricej generates a disunct tree (tltreepricej arise frl iv top from the bottom two prices i pefiod 2). twoprices in period 2 and three prices arike Each liscre te dividend auses the to bit-urcate. There is also a concpt'ual problem wit.h equation (11.20).Since the amount of the dividend is fixed, the stock price could in principle become negative if ther have been lprge downward mokes in the siock prior to the dividend. . This exmnple demonstrates that handling fixed dividends requirs care. Wenow constructing a tree using eqgation tttrn to a method tat is comptationally easier t.1)a14 (11.20) and that iill not generate negative stock prices.

.

.

Repeating this procedure for the node S = $35.411gives prices of $3t.300 and $26.380. You can see tat there are now four prices instead of three after two binomial steps: The

.

.'

A Binomia l Ree Uslg the Prepaid Forward Schroder (1988)presents an elegnt method of constructing a tree for a dividend-paying stock that solves both problems encountered with the method in Figure 11.10. The key insight for this method is that if we lcnow for certain that a stock will pay a sxed dividend, then we can view the stock price as being the sum of two components: the

% Bl NoM

364

Op-rlo?q

IAL

CHAPA'ER

11

PR1cl Nc:

dividend, which is like a zero-coupon bond with zero volatility, and the present value of the ex-dividend value of the stock-in other words, te prepaid fonvard price. Since te dividend is ltpwn? all vqlatility is atibuted to tlle prepaid folward component of te stock price. g where F is Suppose we know that a stock will pay a dividend D at time To < the prepaid stock price base of option. Then expiration movements date the o the we The one-period fonvard pdce for the prepaid fomard pricek Fff'w St De-rlb-tt Fherh. this gives us up and down movements of before, As is Ft,t-v fonvard

have a greater volatility. We use the approximate cF

e

r-yo'W'

d

=

correction

S FP

=

0.3392

dividend affects Figure 11.11. Note

=

erh-o'.ll

FS

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365

.

De-rTn-ts However, the actual stock price at each node is given by st = + Figure 11.11 shows the construction of the binomial tree for this case. Bot the observed stock price and the stock price less the present value of dividends tte prepatd forward price) ftre included irl the sgure.Note tat te volatility is 0.3392 rather t11:.11 0.3 as in Figure 10.$. The reason for this difference is that the random wnlk is assumed to apply to the prepaid forward price. If the acmal stock price is observed to have a volatility of j0%, then the prepaid forward price, which is less than the stock price, must

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k.

1.2492. Look at the node where the stock price is $61.584. Tlzis is a flrst that If cum-dividend price, just before the dividend is paid. The nodes in the last period are constzucted based on the ex-dividend price, for example

=

=

x

You pay be wondering exactly how te

-

tt

ty s

$41 = 0.3 x $36.26

,

=

=

SUMMARY

($61.584

-

$5) x 1.2492

=

$70.686

As a final point, we obtain risk-neutral probabilities for the tree in the snme way as the absence of dividends. It is important to realize that we cons%ct the binomial tree in probability of Leprepaidfolward for wllichpays no dividends. Thus, the lisk-neutfal equation prepaid price is given by the moke forward in (10.5)jjust as in the case n p paying stock. nondividend of a

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: i ' i( E EiE EEE '' E IEEE qE 'i'E'EEi ! qr. EE .Ct! f:( !' i' 'i.'.'(.':..'.'E'E!E'q'q!(' ' @'! ((Ej.ji(E '(i(.'( li '''. ( 1* 'r;'i E . y-. (. g ;.j.()j.).L.;Lj;;LL);L;LqLL;) jyyjyy. y y. .. ; , ...y . ( .j)y.yy yy,. j yj. yrj . ;.yy y. j .y. yjj gjyyj; q.;ljEE y,.k jj;.j((..y.k,jj(; ;.y . yjyyyjy jy .. yy . jj jj. jg . g y .y:j,;jyjj, . .yy yyy. j . y j;t: . Iliit:tli:ll . - . . -- j.- .- .-j- -.jyE-:j.jj-)( /:/: ..q LkbbiqLkLk. . . . ' i;.,.! ... r.!. .. . E' .

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.. . .. .. . . . Biomial tree for pricing an American call option on a stoc k paying a distte dividend of $5 in 8 months; assumes

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5. $41 #= $40.00, tr 0 r3392 r

. -

-. .

-

.

.-.

.

FP

70.686

=

$.50.686 FP

$61.584 =

56.584

$21..584

.00,

.$50.164

=

=

,

FP

1 years, = 0.00,' and h 0:333. At each nde .the stock price, prepaid forward price, and opton price are 9iven. ption prices in bold ltwllc signfy that exercise is optimal at that node.

$47.777

45.296

=

pp

$11.308

0.08,

=

.4:.7y,:

,$'

y yyy .

.00

T

.

=

'

FP

$41.000 =

FP

36.260

$43.246 =

38.246 4ly

E

sa

$5.770

-

FP

$35.485 =

$32.293

Ff'

30.616

$1.501

32.293

:z

$0.000

$30.851

F j,

=

:,s

.

a,s)

stlc(o

FP

$21.827 21.827 50.000 =

Both call and put options may be rationally eyercised prior to expirauon. The, eadyexeicise decisionweighs threeconsiderations: dividends onthe underlying asset, lnterest on the sfrikeprice, andtheinsurancevalue of keeping tlzeoption alive. Calls will beearlyexercised in order to capture dividends on the underlying stock; interekt and insurance weigh agninst early ek ercise. Puts will be early-exercised in order to captttfe interest on the strike price; dividends and insurance weigh agninst eady exercise. For both calls and puts, the early-exercise criterion becomes less stringent as the option has less time to matllrity. -isk-neual optton valuation is consistent with valuation using more tradition plicing it is not lecessazy to estimate discounted cash flow methods. With l'isk-neueal tlzeexpected refllrn on the qtock in order to price an option. With traditional discounted cash flow methods, the coiect discount rate for the option varies along the binomial pricing. with liskrneulal tree; thus, valuation is considerably more complicated t.11a11 'l'he binomial model, which approximates the lognormal distri u tin is a random wnllcmodel adapted to modeling stock prices. The model assumes tat the continuously compounded refllrn on the stock follows a random wnlk. The volatility needed for the binomial model can be estimated by computing the standard deviation of continuously compounded returns qnd almualizing the result. 'lil e btnomial model can be adapted to price options on a stock that pays discrete (i divi en s. bi scre te dividends can lead to a nonrecombining binomial tree Ifwe assume that the prepaid forward price follows a binomial process instead of the stock price, te tree becomes recombining. ,

t

.

366

k. BI NoM

FURTHER

IAL

Op-rlorq

PRlcllqc:

k

PRo BLEMS

11

11.4. Consider a one-period binomial model wit.h 12 1, where S $100,130%, and 0. Compute American put option prices for K o' $110, $120,and $130. =

READING

model, which we discuss The binomial model can be used to delive the Blaekcholes lisk-neutral pricing will become evident in of in Chapter 12. The practical importance valuation hinges upon risk-neutral pricing. Chapter 19, when we see that Monte Carlo and show how the option price may 11.4 ln that chapter we will also reexamine Figure

0.08,

=

$100,

=

a. At which stdkets) does early exercise occur? b. Use put-call parity to explain why early exercise does not occur at the other strikes. .

be computed as an expected value using only stock prices in the linal period.

The issue of how te stock price is distributed will also arise f'requently in later chapters. Chapter 18 discusses lognormality in more detail and presentq evidence that ha pter 20 will examine in more stock prices are not exactly lognormally distributed. what happens when gets particular in detkil the question of how the stock plice moves. vel'y small in the binomial model. We will ret'ut'n to the detenninants of eady exercise in Chapter 17, when we discuss real options. 'T'he literature on risk-neutral plicing is fairly technical. Cox and Ross (1976)was pricing and Harrison and Krepj (179) stdied the the ftrst paper to use lisk-neutral good treatments of this topic are Huang and Litzeberger economic underpinnings. Two inspired Appendix 11.B-and Buter and Rennie (1996). (1988, ch. 8l-their treatment and Campbell et al. (1997) Cochrane (2001)summarize evidence on the distribution of stock prices. The original Samuelson work on asset prices following a random walk (Snmuelson, 1965) remains a classic, modern empirical evidence notwithstanding. Broadie and Detemple (1996)discuss the computation of Amrican option plices, and also discuss alternative binomial approaches and their relative numerical eciency.

=

=

=

367

c. Use put-call pality to explain why early exercise is sure to occur for all strikes greater than tat in your answer to (a). 11.5. Repeat Problem 11.4, only set J 0.08. What is the lowest stlike prie at which early exercise will occur? What condition related to put-call parity is satisfed at this strike price? =

11.6. Repeat Problem 11.4, only set ?' 0 and J 0.08. What is the lowest strike price (if there is one) at which eady exercise will occur? If early exercise never occurs, explain why not. =

=

For te following problems, note that the Binomcall and Bilompttt functions are array functions tat rettzrn the option delta (A) as well as the price. If you know A, you can comppte B as C SL. -

11.7. Let S $100, K $100, o' 30%, 1, 0.08, t stock has Suppose te an expected ret'ul'n of 15%. =

=

=

=

=

1, and J

=

0. Let l

=

10.

y

a. What is the expected ret'urn on a Europemz call option? A European put option? b. What happens to the expected ret'urn if you increase the volatility to 50%:/

PROBLEMS Many (butnot a11)of these questions can be answered with the help of the Binomcall and Binomlhtt functions available on the spreadsheets accompanying this book.

$'1t0 1, where S 11.1. Consider a one-period binomial model with h prices call option ?orK G' 30% and J 0.08. Compute American $80, $90, and $100. =

=

=

,

=

.

,

?' =

=

0

,

$70 ,

0.08. What is the greatest strike 11.2. Repeat Problem 11.1, only assume that 1' occur? will exercise What con dition relaye t pt-call parity price at which early price? is satisEed at this strike =

=

0.08 and

t5

=.

0. Will

=

=

=

=

all c. Use put-call parity to explain why early exercis: is sure to occur for lower strikes tan that in your answer to (a).

?'

=

=

a. At which strikets) does early exercise occur? b. Use put-call parity to explain why early exercise does not occur at the higher strikes.

11.3. Repeat Problem l 1.1, only assume that ever occur? Why?

11.8. Let S $100,o' 30%, ?' 0.08, t 1, and & 0. Suppose the true expectd /1 stock is the 15%. 10. Compute Eufopean call prics, A, and B Set return on sikes of and ff $70,$80,$90,$100,$110,$120, $130. For each jtrike, cmpute ihe exyected ret'urn on the option. What effect does the sike hav on the option s

early exercise

y

expected

return?

11.9. Repeat the previous problem, except that for each sttike price, compute the expected retfn' on the option for times to expiraion of 3 months, 6 months, 1 year, and 2 years. What effect does time to mamrit.y have on the optiqn's expected feuarn? '''

.

. .

. .

.

.

.

11.10. Let $100,o' 30%, r 0.08, t 1, and & 0. Suppose the true expected ret'urn on te stock i 15%. Set ?1 10. Compute Europea put plices, A, and B for stlikes of $70,$80,$90,$100,$110,$120,and $130. For each stlike, compute the expected ret'urn on te option. What effect does the strike have on the option's epected return? .%

=

=i

=

=

=

=

11.11. Repeat the previous problem, except that for each strike price, compute the expected remrn on the option for times to expiration of 3 months, 6 months, 1 year,

368

k

Bl lqoM lAL OpTloN

11

PRIcI Nc:

APPEN DIX

and 2 years. What effect does time to maturity have on the option's return?

expected

0. Using equation (11.17) 1, and 11.12. Let S $100, c' 0.30, 1. 0.08, t tenminal node and Sudn- to compute to compute the probability of reaching a lisk-neutral distribution of year-l stock prices as in the price at that node, plot te ?1 10. 3 and n Figures 11.8 and 11.9 for

11.13.Repeat S3

the previous problem for n $80? 5'1 > $1202

50. What is the risk-neual

=

ure 11.4.

0.08, and & a. Using S $100,r prices? l-year forward and

expected stock plice using the binomial ee in Fig-

tisk-neual

=

0, what are the 4-month, 8-month,

=

lisk-neual

expected sttwk b. Velify your answers in (a)by computing the equation second, in the and binomial Use period. third price flrst, (11.17) to determine the probability of reaching each node. plice using the

11.15. Compute the l-year fomard lem 11.13.

so-stepbinomial

tree in Prob-

=

$

PRICI NG WORK?

369

What do you observe about the pattern of historical volatility over time? Does historical volatility move in tandem for different assets? stock 1. 0.08, apd t 0.30, r 11.20. Suppose that S $45, g $50, K will pay a $4 dividend in exactly 3 months. Compqte the price of European and A me lican call opions using a four-step binomial tree. 'l'he

=

=

=

=

=

11.A: PRICING OPTIONS WITH TRUE PROBABILITIES In tlus appendix we demonseate algebraically that computing the option price in a way using a as the expected rettlrn on the stock gives the correct option price. consistent dennition of )z, equation (11.6),we can rewrite equation (11.7)as the using

1

(A5' + #)

=

=

=

=

d

(,W

d

e-o'./f

=

rlt

-

d

C,, +

tt

rh

=

e e Cd + d

(z

-

e

r(t

Cu

-

Cd4

!1 11 d ctzA5' + e s,s rj d Since LS + B is the call price, we need only show that the expression in large parentheses is equal to one. From the desnitions of A and B we have -

erh d C,,+ jj JI tt d -

We can rewrite

j

(hs

erh

-

-

d

-

-

Cd=e rIt (A5' + B4

(11.4)as

=

11

e

-

1, compounded), t 8% (continuously 11.16. Suppose $95, r $100, K using the Explicitly 30%, arld & construct an B-periodbinomial tree 5%. cr Cox-lkoss-Rubinstein expressions for 11 and #: =

RISK-NEUTRAL

probability that

'

'

DOES

APPENDIX

11.14.We saw in Section 10.1 that the undiscounted equalsthe fonvard price. We will verify tis =

WHY

=

=


.' t .. .' . . . . . ' t.. E )' 1ikg!:;. 1 1::22:1,.' ' . . g y. ' ' j1Iiqti1).. ( E. . . E . . (' . j .. l.j E . E.E. (( '' 41::(2:1,.. ' kllzEl . j' . . . . '..' E E (' iy .. Ili:r'rzll).y 111:((111 'E' E E' . ..EE' '' 'E i' ' iE ' ' q E . . E lilrj))k E jjjll):),k . ' ; . 'E.E....;Ey.....yqi.i.y(..i..(g..E..(.((..E..'.E.y.(.E. . g E . (114. '.(jjlj;:jpp (;1(:7-.: E q.'. .... i('. y. . .. . '. ( y... ( q .Ey .. ... . (E .. (.('.y( j(.(k. g .. q.. .. . ('. . yg... :::;44!2-. jjjkjjjjl y .. . .. . ' :tii!j. !ll!sl;::. .. .!'. ..... '. .. E . . . . .. . .IjyrjE . . . jjjjgjrj,: y . . . E E ' . . y . j jjjjjjrj.' . . . . ... . ...... .' ... E' ' . . .. .. . . E (. . .. ... .jjyy,' . ( E jj2jjrrr,' l'' ' ' . . Ej:jjjjjy,' . .. (. . . E E . E '. E ... ' .( . ( .E.' ( . jj:yyjjjj, qygjy(E..q.E.. .E..kE.yjyy.yyy... .q.j!y...i.y.i...yyyy..'.j.j y. .. .E jj:yg:jj;.. ' ' .'... . E.E....... .. . . :,jjg:,' jjj,. .. . g 'kjjjgj!r;' .'.. .. q . .g..q (.y ' ' jjggjjky.jjjt.,. . . .. j jjjjr)kj.. ' j)jkjy,.' ' ' (. .. ... .. .y y y . . . . y y jjjjkjjjjjj;, .. . . . . yg y . . y . . . y gqgjq kjzytyrj,jyyjj,;pj.. j yy E.y.y j .y.y. y.. ..... y .j E. . . . . ( ( i jE E kj ( jjyyijyql q lc... . E y y,. y.y.iIl57''EyEi.E-)'.).) y . E Fichyt ( l . . j . . .(. ( j r j y (( y yp(:1:1121!.:.....* ( k g g juus.tu.k M/4::11..* j: ( y11::11.* u dt1t1i. u 'y'..'7I::1l:1k.' ii:rllL. u y1IEr1l.' iiliiJ.Eq(:'1EEIi.)I::1il.. 4(1211. m ;.IL.'1EE1l).' il...' ..'.').C' (itlht//tlll'llii/h''(iE:2i(2@)l-. u '1E!li ' (2t!);t'.ii:i::. *1E112*.*:) ikt!:l..ii:i7q1) -.-.'L.'L': ' ii122i-. 'ILJ11.'' =** ' $1E11.E1::11. '' iiEt1L..'.'-.'.):-(t dktlr: 'iitilg ' ' ...'.'.L.... ' E1tE' ' ..' 1I (' **e** dlz::: .1:.. il:llp (114:1!1(1 4::) .11:::1$.. lt. ). ' 44:11. ' llftlL 11411'. jE-..ry.. 1IEll. !1(;1' (111111t.. (IEr'qlg73lgqlL'ltjil. (ILi;i2.'' *(11. 1E111; .q.(E .,tEjjyjjjy.:).;j(yjjyE......Ejjj. E E..jtjgj;gjjr;jjyj(.. gjjg;(,.E.. E..Ej E gjgyj..E . ..jjjjj:..(....y.((y(....--. . ...-.t .y.y(.y ... . . i.j),j;.. jjygjj.. ,jyj(j.( ( ).; jtjjik. . . gr gjgjr. .t;....;;..gjyjjj.(.... ..kjjjjjj.. ...j....( . .-.: jkjjj,:... yy.g j.j. . .tjjrjjjy( gjy(jjjjjj... ptitifft-iijEr'. r .jjyjk,.. .....-j.(jjjjkj(jjjj. y.... gj(gjy(j(y. . .. j.. . yl.y..yl,yyyy ,y . . . ... .t118 t strted. j yty y y, y y g ggj, .'i,tiyy,y ':.)ji'.#(E#.

111::(((22:!:. 't

.

.'d-.

'

jtrjjt, .. .k:!:

..,,..

. ,

-

.71...

.

dl:::,,

41:22jjj).

.

.

-

.

:

t,''

.ji

'

:

.. . .

-

. .

.

j . ... 'i::;,- .i::),

.

E

,(:t:,-

.

. . .

..

...

'.'

.

: .

.

:

' ''

' . . .

'

'

.

-

.

.

:

..

.

..

...

'

.

:

-

.

....

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.

.

'

: . .

..

. . . .

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.

.

:

... :

-

ett ttt-i''tsjll,

Let S $41, K $40, o' 0.3, r 8%, F 0.25 (3 months), r Example 12.1 ' call price, we obtain l the Black-scholes and J 0. Compung f'j jl lnt!l + (0.08 0 + () aa) x 0.25 td oxo.:?.sx N e$41 ',@ x 0.3 0.25 tt t ().32 1l) /..!. + (0.08 0 ln(4n) :2 ) x 0.25 li $3.399 $40 x e x x t) 0.3 0.25 ki j, =

=

=

=

)!1

=

=

-

':?

-

i' ..

-a.()jx0.25

-

-

=

-

6.969 6.966

lThecilpHceherecubp-omputedusingieBlack-scholes

BSCaIIIS',X', c, r, t, J)

=

formulacall spreadsheetformula,

BSCall(41, 40, 0.3, 0.08, 0.25, 0)

=

$3.399

BSCZI:

378

% TH E Bl-Acu-scHol-Es

FonM

$

APPLYI N G TH E FO RM U LA TO OTH ER ASSETS

uLA

There is one input which does not appearin the Black-scholes formula, namely the expeeted ret'urn on the stock. You might guess that stocks with a high beta would have a higher expeeted returfl; hence, options on these stocks would have a higher probability of settlement in-the-money. The ltigher expected return would seem to imply a higher option price. HoWever, as we saw in Section 11.2, a high stockbeta implies a high option beta, so the discount rate for the expected payoff to such an option is correspondingly of the key insights from the Black-scholes analysis-is that net result-one greater. larger intlevant: is The average payoff to options on high beta stocks is exactly beta 'l'he

offset by the larger discount rate.

379

'When Is the Black-scholes Formula Valid? Derivations of the Black-scholes formula make a npmber of assumptions that can be sorted into two groups: assumptions about how the stock price is distributed, and assumptions about the economic environment. For the version of the formula we have presented, assumptions about the distribution of the stock price include the follow-

ing: Continuously compounded returns on te stock are normally distzibuted and independent over t'ime. (As discussed in Chapter 11, we assume there are no in the stock price.) Ejumps''

* The volatility of continuously compounded remrns is ltnown and constant.

Put Options

@Fumre dividends >re known, either as a dollar amount or as a tixeddividend yield.

The Black-scholes formula for a European put option is

ps zt7

'

o. r '

1

J)

r,

=

Ke-rTx-dz)

se-sTN-dt)

-

(12.3)

where #1 and dz are given by equations (12.2a)and (12.2b). Since the Black-scholes call and put prices, equations (12.1) and (12.3),are for European options, put-call parity must hold: P (S, .J(', c',

l',

F

)

,

CS K

=

.

(7'

,

,

1*,

T

)+

,

Ke-c

X

Se-&T

-

(124) .

This version of the formula follows from equations (12.1)and (12.3),together with the (This equation says that the probability of a fact that for any x, 1 Nx) = N-x). normitl being above x, 1 Nx), equals the distribution standard from the random draw #(-x).) below of being probability a draw -

-

--:,

Example 12.2 Using the same inputs as in Example 12.1, the put price is $1.607. We can compute the put price in two ways. First, computing it using equation (12.3), we obtninz

$4:4:-0.08x0.25x $4j.-ox0.25x

f!z+ (() ()8 ln(4())

()

-

.

-

0.32

-

:t

)0 :2.5

Assumptions about the econornic environment include these: 'l'he

@

risk-ft'ee rate is known and constant.

* There are no transaction costs or taxes. It is possible to short-sell costlessly and to borrow at the

lisk-free

rate.

Many of these assumptions can easily be relaxed. For example, with a small change in the formula, we can permlt' t.h vo l a tility and interest rate to tal'y over time in a known way. In Appendix 10.A we discussed why, even though t ere are tues, tax rates do not appear in the binomial formula; the same argument applies to the Black-scholes

fonnula.

about te stock price As a practical matter, te flrst set of assumptions-those disMbution-are the most crucial. Most academic and practioner research oh option pricing concenates on relaxing these assumptions. They will also be our foctts when we discuss empirical evidence. You should keep in mind that almost any valuation procedure, including ordinary discounted cash flom is based on assumptions that appear song; the interesting question is how well the procedure works in practice.

.

0.3 0.25

f!1) 1n(4() + -

(0.08 0 + -

0.32)0 2

.

:5

12.2 APPLYING THt FORMULA

:;:z$j.6(c

0.3 0.25

Computing the price using put-call parity, equation

#(41, 40, 0.3, 0.08, 0.25, 0)

=

(12.4),we have

3.339 + 40c -0.08x0.25

-

41

k

= $1.607

2The put price here can be computed using the Black-scholes

BSPut(5', K, (r, r, t, d)

=

put spreadsheet fonnula, BSPut:

BSPut(4l, 40, 0.3, 0.08, 0.25, 0)

=

$1.607

TO OTHER

ASSETS

The Black-scholes formula is often thought of as a formulafor pricing European options on stocks. Specifcally,-quations (12.1)and (12.3)provide the priee of a call and put option, respectively, on a stock paying continuous dividends. ln practice, we also want to be able to price European options on stocks paying discrete dividends, optipns on We have already seen in Chapter 10, Table 10.1, f'utwes, and options on c=encies. adapted that the binomial model can be to different underlying assets by adjusting the adjustments work in tlle Black-scholes formula. dividend yield. The snme Black-scholes the We can rewrite db in formula, equation (12.2a).as 3nSe-&T(Ke-rT)

dj

=

cJF

+ 1(r2r 2

'

%.THE

380

APPLYI N G TH E FORM U LA TO OTH ER ASSETS

FORMULA

BLACK-SCHOLES

When #1 is rewritten in this way, it is apparent that the dividend yield enters the fonuula Se-&T, and the interest rate enters the formula only only to discount the stock price, as Ke-rl-. Notice also that volatility enters only as c'2F. to discount the strike price, as Se-&T The prepaid fonvard prices for the stock and strike asset are Foppuj Ke-r-. Then we can write the Black-scholes formula, equaion (12.1), andFoPvK) entirely in terms of prepaid forward prices and trVF:3 =

$

381

Compared to the $3.399plice computed in Example 12.1, the dividend reduces price by about $1.64,or over half the amount of the dividend. Note that this option the of a European option. An American option might be exercisedjust prior to price is te dividend. and hence would have a greater plice. the

=

P

c'(F() w(s)

FP ,

0,z

K) o- F) .

FP

(5')A'(t/1) c,w lng/'p sljFp 0,F

=

,

:/1

-

0,F

=

dl

ga

=

-

o.

J-r

FP

o,w

KlNdz)

(#)) +

We can price an option on a currency by replacing the dividend yield with the foreign interest rate. If the spot exchange rate is x (expressed as domestic currency per unit of foreign currency), and the foreign cuaency interest rate is ?-y, the prepaid forward price

1c2r 2

cJ-r

for the cun-ency is

This version of the formula is interesting because the dividend yield and the interest rate do not appear explicitly; they are implicitly inoporated into te prepaid forwgrd pdces. To price options on underlying assets other t.11%stocks, we can use equation (12.5) in conjunction with the forward price formulas f'rom Chapters 5 and 6. For all of the examples in this chapter, we will have a sfrike plice denominated in cash, so that FP

0.F

(X')

=

Options on Cttrencies

sz'0,

#

C@ ,

,

c'

-

,

?.

F

-

0.25 (3 8%, and r $40, tr 0.3, r Suppose S $41, K r Example 12.3 't:!1months). The stock will other makes month, but 1 payouts in over dividend no ))t pay a $3 tt .the dividend is vdue of l) i the life of the option (hence,8 0). The present .l( =

=

=

=

yt'g tt

=

$3/-

()gg x j / j :t '

=

jnxjKj

$2.98

lj?settingthe stockprice in the Black-scholes formula equal to t)tBlack-scholes ca.ll price is (1 . $1.763.

$41 $2.98 -

=

$38.02,the

$

P (x, K, o- ?' F r.f ) ,

,

(r G

d1

2

(126)

)

.

:? yo' IF ry + 1

-

WF

o.V-f

-

,

=

Cx K c r, F, r.) + ,

,

Ke-rT

-

xc-rzF

,

$0.92/6, K $0.9, o' Suppose the spot exchange rate is x )1 Example 12.4 ).) ' ' euro-denominated rate), F 1, and dollar 3.2% interest 6% 0.10, (the r ltlze rr t!q price of a dollar-denominated ekuo call is $0.0606,and the price of a tt. imerest rate). t- donar-denominad etuo put is $0.0172. % =

'1

=

=

=

=

=

'r'he

Options on Futttres The prepaid forward pri.e for a futures conlact is just the present value of the f'utures . .' . . . .. cokttat option f'utur:j European Thus, by . sing the futures price We l4ce pric on a a P q r as the stock price and setting the dividend yield equal t te tk-free rat. The resulting formula is also known as the Black formula: .

C(F about the relative time-r values 3We can also 1etV(T') o' T represent total volatility-uncertainty of the underlying and stlike assets--over the life of the option. The option price can then be written solely in tenns of F1vS), FOPpLK), and 7(F). This gives us a minimalistversion of the Black-scholes fonnula: To plice an option you need to lnow the prepaid fonvard plices of the underlying asset and the strike asset, and the relative volatility of tlte two.

+

Ke-r-Nd

-

'

where PV0,w(Div) is the present value of dividends payable over the life of the option. Thus, using equation (12.5),we can price a European option with discrete dividends by subacting the present value of dividends from the stock price, and entering the result into the formula in place of the stock price. The use of the prepaid forward plice here should remind you of the approach to plicing options on dividend-paying stocks in Section 11.5.

Pvmiv)

=

)

1

This formula for the price of a European call on currencies is called the GnrmanKohlhagen model, after Gnrman and Kolllhagen (1983). The price of a European currency put is obtained using patity:

,%

'.i

d3

xe-rf-Nd

=

W en a stoc k m ales discrete dividend payments, the prepaid'forward plice is Fvlj = PVo,w(Div)

=

=

./

dg

Options on Stocks with Discrete Dividends

=

?

.70d

=

)

-

,

Ke-rT

(&

.-ryw

w@) Using equation (12.5),the Black-schole formula becomes

K c r F

?-)

=

#1

=

ln(F/#)

=

d,

Fe-rTNdkj

=

-

+ 1o'2F

clF d,

-

c,/f

Ke-rl-Ndzj

2

(12.7)

%.THE

382 &'

op-rlolqGREEKS %

FORMULA

BLAcx-scHoLEs

Theta 04 measures the change in the option price when there is a decrease in the time to maturity of 1 day.

The put price is obtained using the parity relationship for options on futures; Ke-rT Fe-rl#(F K tr ?' F ?') = C(F, K, (r, r, F, r) + -

3

'

?

'

lkho (p) measures the change in the option price when there is an increase in the interest rate of 1 percentage point (100basis points).

jk'i 1)*

tu and Suppose the l-yearfutures pricefornatural gas is $2.10 q Example 12.5 1, 0.055, T 0.25, r $2.10, K $2.10, tr the volatility is 0.25. We have F ltl and 0.055 tthedividend yield is set to equal the interest rate). The Black-scholes call Q 0 plice and put price are both $0.197721.

Psi (T) measures the change in the option price when there is an increase in the continlpus dividend yield of 1 percentage point (100basis points).

l)Ei jE!

=

=

=

=

=

=

12.3 OPTION

383

GREEKS

Option Greeks are fonuulas that express the change in the option price when an input inputs/ One importpnt use of to the fonnula changes, tnking aj :xd all the other lisk exmuple, bnnk with market-mnking For a Greek measures is to assess exposure. porifolio of options wold Want to understand its exposure to stock price changes, a interest rates, volatility, etc. A portfolio manager wants to know what happens to the vlue of a jrtfol'i of stock index options if there is a change in the level of the stock index. An options investor would like to lnow how interest rate changes and volatility changes affect prot and loss. Keep in mind that the Greek measures by assumption change only one input at a time. In real life, we would expect interestrates and stock prices, for exaple, to change 012% one input together. The Greeks answer the questfon, what happens when one t7?7t changes? The actual fonnulas for the Greeks appear inAppendix IZ.B. Greek measures can be computed for options on any kind of underlying asset, but we will ftus here on stock Options.

Etvolatility''

and Ausef'ul m emonic device forremembering some of these is that tttime.'' often and Also used letter, do is frst to denote the share the snme as tlrho.'' and is first letter in the interest rate We will discuss each Greek measure in t'urn, assuming for simplicity that we are tnlking about the Greekfor aptlrchased option. The Greekfor a wlitten option is opposite in sign to tat for the same purchased option. E'vega''

Gr''

Gtheta''

We have already encountered delta in Chapter 10, where we defined it as the number of shares in the portfolio that replicates the option. For a call option, delta is positive: As the stockprice increases, the call price increases. Deltais also the sensitivity of the option price to a change in the stockprice: 1f:.14option is replicated wit.h 50 shares, the option should exhibit the pric sensitivity of approximately 50 shares. You can tink of delta as the share-equivalent of the option. Figure 12.1 represents the behavior of delta for three options with different times illustrates that an in-the-money option will be more sensitive to to expiration. 'I'he sgure the syock price t.11m1 an out-of-te-money option. tf an option is deep in-the-money (i.e., t.he stoclprice is Mghrelative to te strike price), itis likely to be exercised and hence the option should behave much like a leveraged position in a f'ull share. Delta approaches 1 in this case apd the share-equivalent of the option is 1. If the option is out-of-the mopey, itis unlikely to be exercised and the option has alow pric, beh:ving like aposition wit case delta is approximately 0 and the share-equivzent is 0. A.n very few shares. In at-the-money opuon may or may not be exerised and, hene, behaves like a position with betwen 0 and 1 share. This behavior of delta can be Feen in Figure 12.1. Note that as time to expiration increases, delta is less at ltigh stock prices and greater at low stock prices. This behavior.of delta reflects the fact tat, for ihe depicted options that have greater time to expiration, the lilelihood is gregter that arl out-of-te money option will evenmally become in-the-money, and the likelihood is greater that an in-the-money opion will eventually become out-of-the-money. We can use the inyepretation of delta as a shgre-equivalent to intrpret the BlackScholes price. The fonnula both pries the option nd also tells us what position in the stock and borrowing is equivalent to the option. The formula for the call delta is Delta

'

'tis

De6nl'tion of the Greeks measured are a matter of convention. Thus, when we The units in which changes will also provide the assumed unit of change. desne a Greek measure, we 'are

Delta (A) measures the option price change when the stock price increases by

Gamma (r) measures the change in delta when the stock price incress

$1.

by $1.

Vega measures the chanye in the ootion orice when there is an increas in vplatility pint.D of one percentage ..1$.

.

.

.

.

.

'

. .

..L

'

.

.

.

.

.

.

'

.

.

.

..

A the Greeks are mathematical delivatives of the option price formula with respect to the 4speciqcally. ts inpu and is not a Greek letter. are also sometimes used to mean the same thing sktvega'' .

ttlappa''

1lS Vega.

xlambda''

If we hold e-tNdqj

e

=

-spyy

1

;

shares and borrow Ke-rTNdz)

dollars, the cost of tlzis portfolio

is

S e -&TNd

l

)

-

Ke-rTNd

2)

%.TH E BLACK-SCHOLES

384

'..' ''. .'' (i '

.'

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309% Msumes tr 0. r 8%, and

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1.0 (j.9

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a mqrtths + 1 year . 3 years -s

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3 months -+. 1 year . a y ears

0.1 () 20

25

30

35

40 Stock Prie

45

50

55

-0.9 -1.0

20

25

30

35

60

($) :';' (y'(yyE j'q)' --.' -': -' -'. -p' (' --'t)' yyd -!?'. f' ;'')'(.1)111* 'iq-ki;-;' (' 75* 1* jjqjljjd std j'j't' $' y' k-'!i'I::::;:;iE. #' I'E .:'q' !' :p' r;f :!' '(r' jjk('. jrjj.jljl!.jjd ljd ;!' j'(' tyyjjtjd !9*......:il:r:ll:::l..f :.l:r:).lr::l..f i!El:.f pyf q' (rrf (l'f)f (' iE'-rr!!rIqIi@)!(F':.@t'i.' lirs'k!)pi ?' irFlrf F':. Tf f' jjd ytjy'tjjjj y' ).titt))'. )' ;'C'..'kiik''

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40 45 Stock Price (s)

50

55

60

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Gnmma-the change in delta as the stock price changes-is always positive purchased call or put. As te stock price increases, delta increases. This behavior for a both be Figures 12.1 and 12.2. For a call, delta approaches 1 as the stock seen in can For increasest price a put, delta approaches 0 as the stock price increases. Because of parity, put-call gnmma is the same for a European call and put with the same strike price time expiration. to and Figure 12.3 graphs call gnmmas for options with three different expirations. Deep in-the-money options have a delta of about 1, and, hence, a gamma of about zero. (lf delta is 1, it cannot change much as te stock price changes.) Similarly deep out-ofthe-money options have a delta of about 0 and, hence, a gamma of about 0. The large gamma for the 3-month optio in Figure 12.3 corresponds to the sieep increase in delta for the same option in Figure 12.1.

i

:

-

I'his is tl)e Black-scholes pzice. Thus, the pieces of the formula tell us what psition in the stock and borrowing syntheticdly recreates the call. Figure 12.1 shois that delta changes with the stock pdcek so as the stock price moves, the replicating portfolio changes and must be adjusted dynmically. We also saw this in Chkter 10. Delta for a put option is negative, so a stock price increase reduces the put price. relaiionship Thls afl be seell in Fiure 12.2. Since the put delta is jut te call delta e-&T minus (fromput-call parityl, Figure 12.2 behaves similarly to Figfe 12.1.

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20

25

30

35

40 Stock Price

45

($)

50

55

60

% THE

386

BLACK-SCHOLES

op'rlolq GREEKS %

FORMULA

Options generally-but not always-become less valuable as time to expiration decreases. Figure 12.5 depicts the call price for out-of-the-money, at-the-money, and inthe-money options as a function of the time to expiration. For the at-the-money (strike = $40) option, time decay is most rapid at expiration. For the others, time decay is steady. Figure 12.6 graphs theta explicitly for three different times to expiration, more

Theta

A delivative for which gamma is always positive is said to be convex. If gnmma is positive, then delta is always increasing, and a graph of the plice f'uncdon will have curvattlre hk'e that of the cross section of a bowl. An increase'in volatility raises the price of a call or put option. Vega measures sensitivity of the option price to volatility. Figure 12.4 shows that vega tends to be the options, and greater for options wit moderate ta wit short at-the-money for g'reater expiration.6 of put-call parity, vega, lilce gnmma, is the same for calls Because times to price and time to expiration. with strike the artd puts same When calculating vega, it is important to be clear about units: How large is the assumed change in volatility? lt is common to expreqs vega as the change in option price for a one percentage #t?f?ll (0.01)change in volatilit.v Figure 12.4 follows this

Vega

shoWing that time decay is greatest for the at-the-money short-term option. Time decay can be positive for European option! in some special cases. Deep-inthe-money kall optios on an asset with a high dividend yield and deep-in-te-money puts are two examples. ln both cases we would want to exercise the options early if possible. Since we cnnot, the option effectively becomes a T-bill, appreciating as it gets close to expiration. This efect is evident in Figure 12.7, in which the in-the-money tso-strikel put becorizes more valuable, other things equal, as expiration approaches. Figure 12.8 on page 390 graphs the put theta explicitly, illustrating the positive theta. When interpreting theta we need to know how long is the assumed change in time. Figures 12.6 and 12.8 are computed assuming a l-day change in time to expiration. It is also common in practice to compute theta over longer peliods, such as 10 days.

convention.

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.

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.

387

Rho is positive for an ordinary stock call option. Exercising a call entails paying

Rho

the fixed strike price tti receive te stock; a higher intrest rate reduces the present value of the strike. Similrly, for a put, rho is negative, ince the put entitles the owner to

3 montlu 1 year -+.a years

-F-

O 25 '

0.20

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=

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12

0.05

10 0

20

25

30

35

45 40 Stock Price ($)

50

55

60

8 6 '

4

-1- Strike + Strike ..A- Strike

2 6Be aware that neitlzer result is true for very long-lived options. Wit.h a 20-year opton, for example, of prices vega is greatest for out-of-the-money calls and lower than that for a 3-year call for the range

in the hgure. ?vega is the derivative of the option price with respectto o'. lt is expressed as the result of a percentage point change in volatility by dividing the derivative by 100.

0

0

0.5

1.0 1.5 Time to Expiration

2.0

(years)

2.5

=

$30

=

$4O

=

550 3.0

$

388

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,(jy'. -' .',,yy'(j;'j),-. ........' ''' '

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:

. ,.

-'rq)p(q.-.

-

.

'

.112:.

:

il:i,t.-EiIi.

0

-..

..... ....

'.. -.

' E 'E ' EE ' E i':' EEE: : : E E

' ' : : ' '. '

E..--:.

.

.

. :

. . . ..

.

..

.... . .

' i

'

-

E

E ;E E: E .

'

'

:. :

. E EE!E! E: E . :E . E.E. E : E. . .: y E .-. .. ..- .- 5 . .... .. .. -

. .. .

..

-

. :

..

.. .. ... . .

-

.

...

..

. .

.

'

. .

' : .

'

E

.

.

:

E E. - E y.

-.::

E

.

.. . -

''' E E: .

: ' E'

E.i. j:

-. .

-

:. ...

.

-

- .

-

.

.

E

:

i qq!. y. .

-

--- .. - E

-

.

jj:yyyyyy:jj.

=

-0.004

Strike

=

strjxe

+

=

wstrike

j)

..

. '+-

10

=

30 4:

so

8

=

=0.

=

=

.

jj:yj .jjyyyyjjk, yjty.yjjr, .yj.,kjj, jjjyy:),,tjjjs

diferent times to expiration. Assumes $40, tr 3094, . r 8% and

-().c(s

389

' E' . ' . E ' i E ;(E . : . . :

-

. yy . y yy. . E .... ..y

.

...

k.

GREEKS

opTlox

FORMU LA

BLACK-SCHOLES

THE

.

7

-0.006

6

-0.008

5

-0.01

4

-0.012

3

-0.014

-+. 3 months + 1 year .. 3 years

-0.016 -0.018

20

25

1 30

35

45 40 Stock Pdce ($)

50

55

0

60

receive cash, and the present value of this is lower with a higher interest rate. Figtlre 12.9 shows that as the time to expiration increases and as a call option becomes more in-the-money, rho is greater. interest Figure 12.9 assumes a one percentage point (100basis point) change in te

0

0.5

optionposiuons is easy to evaluate. stock, where th quantity underlying

expiratipn. Figure 12.10 shows that the absolute value of psi increases with time to matttrity, but time with short effect to little a An increase in the dividend yield has that Figure mattlrity. Note with time to dividends lost by not owning the stock increase 12.10 is a mirror image of Figure 12.9.

Greek Measures for Portfolios l/7e individual polfolio The Grcck measure of a porlfolio is l/?d stml of /d Greeks of lisk of complicated This relationship is important because it mans that the ct/?llpa?ld?yr..

2.0

2.5

3.0

(years)

11

opuons with a single we have

)f ,

tt

ati Af i= 1

The same relaon

Psi

up, is positive.

For a portfolio containing of each option is given by

Aportfolio=

rate.

Psi is negative for an ordinary syock call option. Acall entitles the holder to receive exercising the opion. stock, but withoutreceiving the dividends paid on the stockpriorto the received is lower, greater the dividend yield. Thus, the present value of the stock to be in exchange forcash. stockin f'uture te Owning aput entitles an obligation to deliver the dividend yield goes the when down dlivered goes The present value of the stock to be psi for a put yield is Hence, greater. so the put is more valuable When the dividend

1.0 1.5 Time to Fxpiration

q Example 12.6

j'lt

*.

.

holds

eue for te other Greeks as well.

Table 12.2 onpage 392 lists Grekmeasttres for a40-45bu1l spread. zm-stnke call less those for the 45-stnke cll. '

'

.

.

.

t Greeks for the spread are Greeks for the ')) . E(

.

.

.

.

.

%.

Option Elasticity An option is an alternative to investing in the stock. Delta tells us te dolla.r risk of the option relative to te stock: If the stock price changes by $1, by how much does the option price change? The option elasticity, by comparison, tells us the risk of the option relative to the stock in percentage terms: lf the stock price changes by 1%, what is the percentage change in the value of the option?

%.THE

390

op-rlolqGREEKS %.

FORMULA

BLACK-SCHOLES

''.' ' '' ...'

l!)'q?lf (rjjf (' j((q' ;')' f' JFY @' !' 1* (isif i'('( j:' (k!)@i (' (r' pir!lE lijllrii'Ii' E( ( ' !'' E(:! ( ?' tqf ;l(: (' ( (' (( E!''(fkl)( $::. (' JE( 'pi 1I!1:pFq'. ;)-)))t(''111L1'-i1Il'. ))' lili:'iiittf @q' t';'. .y q))'' . (';'' . !.. ..'i j . (: i jy.' .y ' . .y . y.... ... @ ' y ijt. ..)(.;.tr@;)/ry)!kt.j)/.Fii':)

:''' ----' .-it-'. -' IFX -'. t'!'!il:::::;'E;'''T' 69* '5* iyjEE ' ' -l:::::.,f 'E' fr''f 7* jjf f' -.'E kf E E :I' l);E i 'l.'jjE.

391

'.'''

:'

y' y.yjyy'. (' :'j'' k' jjf 'q s' Eiilr'ii.Ep' 87*:.71 .'(. tjrjrf y'. yyyjy'jtyjy,jytyj'y.yjjjj.. yt'. qn (' q' p''' !' lrf )'))#' :?'q77:!77!: ! )'1 i i'' ! ! qrpr!: i F' i j)y'. jjtf ltkttf .yry:' )' r' r-r!:f tllf TE ' prq jjjjjf rf t')'j'7* j'1* )'yy(y'.r'' ykyyf tjjyy,r,j.f y' '!qf jl' t'y'-' )' y'y .r.' 'r' .q':' (l.'. i(ti:. ,i' ! !(-...r q.qii'; (:k i 'j ..','... r' tt'. ;.. ,jij(jkjjjjj. jg ( y .. jjjjj:tjjjj, r .yjj,j yj),...yrj 'yjj;..:j (yjy j,.j;j . t.!j:j:jj:;r;jk. ) ). jy ' E!Ijjjqr; .. (: . j. . jj. y j . t)q . . jjjij!jjj j . y jg. . :.L-,....,.;. . ..,...........LL.L y yj. yy.yy p'ltq.. . y.tj. . ... ....... Theta for put options --' g'-:jjjjj;;k,,. --'' '''rtf ?'r' :'yjgyjjy'. Eyf y'ljq-yyfjt'l'yjjjf y'. f:)' l''tE :'C' F' y' q' ;'(' tFf ?' jE i ': -',,.yjjyy,,' ' -:;;;;jjjjj'!.':''

.' E'.

E: :

- i'; ..;tE i. (E; ' :

'

'

'

-

.

-

-

-

..

.

-

.

i,,

.

.......,.......-...-..

............

,, ...

,....

...

'

.

. .

.

- . . .. .

:'E'' E EE .: E EEE!;'

'

E '' EE !E- E' E 'E- :'. E'E'EE E' -' E ' 3(E'EE E:(E E@i i ri . -.' E. .. ' . !E .. E . ; . ' ' . -. . - . - - -.EE.! -!: -. . -.E ... E-Ei,;-jjjjj. j .q . j-jjlj.. y. y y.. .yqt:E)..--.-. . . ---t............-....

i

.

@

(

'

. .

-

. ..

-

.

-, .

-

-

.

a

jjyyyg,y

.

.. . .

().01

with different expirations Mtdifferent stock prices. Assumes 309% # 40, (r r 8%, and & = 0.

.

() 008

-+-

w.

0.006

=

=

.

--

- E . .. -.. - - -EE .y -E E qy .. -y y ky. jy. .g.- y.y;( y - yyyj. . , . ... .. .. . -.

.

:

-

.

.

:. .

. y. . -.

.

..

. .. . - . . ..

..

a months 1 year 3 years

.

-

'

.

'

:

,

:

'

-

-

-k.'

' ''

.

,

..r..

..i !'E -E.

'

'

.' .

..

. :

.. . -) k -LL.' :. ilE jj- (q. - E i E.. .... lpjr-i..-;------(..jj-

. '.. . '

''

:' ' ' ' : : ' : ' ' E:E ' E' E :: ;' (Ei.ii ' f(i: '' '''' .E . .: E : . E Ei EE E E .. EE . .. . . .. . . . .j.E.ttj(.(j.j..Eg.rj(q.j.j'.j.y(.,.;:..jE().yj!.t.r.;y'(;y,yt.'jt.yyy(',.()j ...'. ..E.;i.)''i''!(.j(.E.jy.kE:E;Eji!'t;E.i:(jyy!,,r.EF.Er.),';:j.'.y(.y'#.itk.Ett).E,t)yyj(@.yj.yk...

liliiiii. r'.iiI!: y. ''.! lr ..14444. -

-@(. -..-(--. - . . -. . . - - -. . .. .

' . . . . . . .. . Rho for call options with different maturities at difrent stock prices. Msumes # $40, o' 3096, r = 8%, and 8 0. .

.

'u

E

' E(E 'E E. ..

--- .

. ..

. -

..

.. .

.

. .-

.

.

-.-.

. .

.

--

.

-

-

--

.

.

.

.

.+-

0.8 0.7

=

.'

...

.. .

.

a months

1 year 3 vears

=

0.6

=

=

..+-

.

..:

)lEq:i:-.lllg.:l,..

0.004 0.002

0.5

0

0.4

-0.002

0.3

-0.004

' -0.006

0.2

-0.008

0.1

-0.01

O

20

.'

.

c ltange

35

i n op tion price

changein stock

=

50

40 45 Stock Piice ($)

If the stock price changes by

Dollar risk of the option Pjj js ' . .

30

25

)' t')'.

6,

the

change

in the option

. . .. . price x option

tlelta

.

.

,r

.

.

.

.

.

.

.

The option elasticity, dnoted tjj'jjj

.

11 $6.961. Deltayis 0.6911. lf we own optlon; io buy 1000 shares of stock, th delta of the . . . ..j') ... .. . . . . . is l position . t 1000 x A 691.1 shares of stock ). . of 691 shares. l.f '' Thus the option position at this stock price has a ).;t t)t :

. .

=

-

-;

I'share-equivalent''

,,

jrjlthe

stock pzice changes by

)) l),i

$0.50,we expect an option price ch>nge of a 1000 x A x

$0.50

=

$345.55

k

50

.55

60

(5)

Q

C by f), is the ratio of these two:

k % change in option pnce jyygyggggy ............ % change in stock price

.l

r;:;:;::::yc .%

S

:;;;;;;;;;

SL

C

The elasticity tells us the percentage change in the option for a 1% change in the stock. lt is effectively a mepsure of the leverag implicit in the option. For a call, f2 k: 1. We saF in Chapter 10 that a call option is replicated by a levered investment in the stck. A leverd position i asset is always rislier than the underlying ajst> r. Also, th implicit levefge in the fition becomes greater as the option is mor out-o'f-the-money. Thus, f2 decfeases as the strike price decreases. For a put f2 S 0. This ocurs because the replicating position for a put option involves shorting the stock. .

'

and gamma. This accurate measure of the option price c hange is obtained by using both delta 'delta-gamma approximaton'' is discussed in Chapter 13.

8A more

45

rislt of the option The optlon elasticity computes the percentage change option plice relative in the to the percentage change in the stock plice. Th percentage in plice stock change the is simply ejS. I'he percentage change in the option price is dollar hange the option price, cA, divided by the option plice, C'. the in

=

=

.

.

40 Stock Price

Percentage

=

E

3

6A

=

j)

30

25

@

ttd Example 12.7 thatthe stockprice is S $41,the strike pdce is K $40, suppose r lisk-free 1, the time to expiration is F 1* 71) 0.08, rate is 0.30, the C)tVOlatility is o' option price the chapter, is in the d the dividend yield is & 0. As we saw ellier li .j an =

20

60

55

gMathematically,

this follows since S

-

Se-'t N(#l )

>

C(J).

k

392

oraTlox

FORMULA

THE BLAClt-SCHOLES

%.

GREEKS

393

('E

.'''.';'

1, and ) Example 12.8 Suppose S 0.30, r 0.08, F $40, ty $41, K bjtt8 0. The option price is $6.961and A 0.6911. Hence, the all eljticity ij

.'-.-:t;;;1!!::'-' -'..' .*1* :''''

;':i!ii'-' -' -'E(-''' ;')E-''!!1::::;kiE. 'j'y'E .111:::11..--* y' -'E E.'Ef 1* '.F'i !' ;'S' :'fqf f!' )' :!i!;;f rf q' 7* J' j)' r' jld y' (' (7* p' fL,'.Lf'. jj':ti Jj-j).lg(' t'!' j')''j' y' (' lryd )''i'E )' tk.' ))' Err':.(: 7741* )t!-rl' !'' rrrf '?' j)f ',-'' t'' t'ryf i'jqlE!ElE:srrtrlsf rq!prf ;':('ptf 5!(?' ( (E (E E lq !!q('g.(!l' i'Eqq !E(I(;'y((( itg g' !lr' ' ' ''E 'EE(; !' r'('.()('.). . .((.(;g..( (kiE....' ((T' q! ()17)! ;EE'EE.E'E.CE .. . . . ! .. . .. .i ! ll)..E i . EE Ey (q .y ;E; !iE.EEE IE ;, !. )!(@. E.EE.E.EE.i; EE.;E.EE'.E(. .. q.k )p ;-!!E:, gi ;. i! r! '):ktjt:.jy)y. . ;.y(.j .. 5:-. y; ((j. (y y: ;.q . .. -' j j)!'iiE. ; . y p @ E' t':'--:'qi ;ti.. .. ..p - ii...;.. l :- ' ;; ;.; ' . . .. . . . . . -'. ----:;-.ik(t?'. --' '

(21(::,-* jjf j't)t.tjk))y()yt'..' yrf li-.llf )' :2::-*-))r*. ))---('-' )-tk(i'k7'7I!It'. 1111k,-k1111*. ()' y' r'rqrq!qd r)' y)jr-' ('yj. 1* (')r t'!!$1iqiqq,.' r r r!.jyj.. l ' '. (jy j(j.y. .tj:. )y;.k. .y. j j.ty .t'-t' r;ty .rk. . y . 1IIE:1iilt:.. . . 11115E1!:.. . ' '-).'. ).; i '. ... '. ri pi.1111!.. .i. ))--. . .-.. )).. .

ilf .' '

'

''

:

.

E.

-.

:

.

:

''

-

.

. .

-. -

.

-

-..

-

-

'-

.

-

.

.. . ..

E. -

-

'

.

:

--

:

:

' .

- -

'

.

.

-

' '

=

...,.......z

..

;:,

' :

' ' :... : : : .

: .

.. :

:

: ---'-

.

. . . ...

:.

:

- .

.

''

'.

-.

. . ; -. : . .

- -. . . .

'i ' '

yE tjE E

:

.

-

.

:

- -

:.

-.

.

tsi.

. -. .-

-

.

.. .

- -

. - .

.

Psi for call options with differet maturities at different stock prices. Msumes # = $40, = c 309% r 8%, and = 0

.

=

. .

--...

n 4

=

.

=

.''''

f2 't'he

-0.6

$41 x 0.69 1l $6.961

=

=

$2.886and 2 of $41 x f) $2.886

.

4.071

=

-0.3089.,

put has a price of

t). ti) #t jil j

z

=

=

.()

)i' (t 'j,. ;. tji

() n

=

=

'

'

'

hence, the elasticity

is

.

-0.3089

%.

-4.389

=

=

Figure 12.11 shows thebehavior of elasticity for a call, varyingboth the stockprice andtime to expiration. The 3-monthout-of-the-money calls haveelasticities exceeding 8.

-0.8

For longer time-to-expiration of the option.

-1 -1.2

The volatility

options, elasticity is much less sensitive to the moneyness

The volatility

of an option

qf an

volatility of te stock:

-1.4

-+. 3 ponths .+- 1 year u- 3 years

-1.6

.

Joption

j'((y'. y' j)'(.j'i-i', .j' yyd (' jjjLj)jj,j'. jyyyjj'. jj;jjjjj'. t')'

E'yE ()' yjd (' ;(j' j)d t';' y' ;')(' yjd j'. g' :'j')' yyjgyj,d ggjyy;d f' ?' jjj'. y' )' p' j'(' i'@' q' ljd .yjjg;gjjj.' yy'g (.k(@14*;.1415((((4!* ll'ir'qd 'itd j'( t'()ky' ir;d )' yy'yj'. jjydyyjyjjy'j jgj.d jgjjjjjgyd (jj.' jyjy', (rt'tlli j'. y.jjd j'yjjj'.

Jstock

X

lQl

(12.9)

y' kygygjjjjjyytk,yd

-1.8

35

30

25

20

50

45 40 Stock Price ($)

55

60

.'.

=

option is the elasticity times the

.

' !' i'E )(i :E ( E ((' !'' !E(' ' iE(7*i i 7Ei ' (''('!' i' ('' :' ('qq'qr'q!?:?l'p'l'?'!!lq:rqqq;:'q):'' '(' (q' f.' ' )) ti ;@.: . jyr.j('j j(.q'(..;...(.(. .r'. (..g.. .k.y.(jjyyj.r'(.jj(jj.y(..j.(.j (.y.y..g.jy.jy....' t:.y '(..j,.tyg y j j j yjy y . j (yr gyg... g..g. g -tr.-tEi). .jj.g..jgjj.g.yj...j.ygjj.. j ..tyjjyyy yggy..yy .j !tq.?.-.r.......--p.i..-)4.-..-....'.'.'.'.!:.b. tjttj ...IitipI:.. . . . ..E--i.(.... yy y j ...j.yyjjjj.gyy jg jjjjjjjjjjjj iit5it.. . j;g. jyyjyjj .. y..y.yjyy . ....yj :E 1:(:E

(E

,

:..

.jjjjj:gy;E i j (

:

i

E

.......

:

'

.

E ' EE EE'EE''E:E' ' ': E.'.' E:5* EE! 7' .EE E qE.. E: E. E E . . .. . . y. . ,: .. y. y : yy .. . gj,;j(jjj. yjyy y y y. y. y --..-)yl:---.--..2g y . g .g . .. '

:

.

-

:

:

..

. :

-

-

.

:

'

:

.

-g

, ,.j

,

....

y :g y y y y... y .j y yy y g . g. yg. . . g.g .

,

.

.-....-,.

.

:'C' f' 1* ;':' t;d t'E ----' -.' -1::::::::;.* y' '#i''-' 57()):-:7':::1::77';:5:'7r:4r775Fq7'7:F* ;'i'7* r' k' k;yyjjjd '5* 1* ;'i't?d Etd ld lqdiltErjjlljyjtjjjjjjt'q)pq)d prd qd p' jjyty;d Ilr...d !' I,:...r.kd jjy,..!jjd )' jy'. --.Lkkkk---' :.'' iiiid 1* qd ttd j'. t')'--')$r);.r'k'jjd jjy.,,yjjd jj:yyr:j,'. yyjjjg;,d jtjtjj'. r' ;'('jri'kiiijjjjrj't. j'(' )' ..'

))';1 ' '' ' ' ' yy'jjijkjjjjjj'td y(y'. t'tlyyjd yj'. 1*. f'' LLjiibtibbiiiL.'-'-.. tjj)'jy'jyjjyj'. s--d r' t'. t'(')'l'y iti-..ii77tif'-' td (yyjjy.r.d yyza';d t)jk'-

.'.*1 .'. .'' .. 5

'

.

''l)E '

q (i ((t.t: ((.i. !. E r! (' i @ t ti ltlll. . . . .1. . jj,.... ( 'E'ltprEp; yyi!.i !p(rlli ; ; yi .) i j!: .y '.(ljyj. .;yji.yj. r (y ; .y.j ;j!.yjy -LLjjL. .. r . Ei!!i;kI,:''. . ji yjj r tg y . .jjjyy, l j . 7t'iiIli. . yy j y y y . . iize. jjy j j.jj. j . . e (:)---::: !. yy 'I..,,il. ''Eiii,,''Ess,'11 11. ll:''' Il'...,'l lli::::,,Il--.::.r:i!i;.iiii!i!i:.. .(jjyj. a 'Ir...,l:..'!.t.E!r'l '. ..1111..1.11... llq::::),. iii1i22.ea ) t.. e i :... ..-)r ill ..!....;.. t-. :k .. ..);)-.------. .k t. . . 1:4 . ' r . .. .. ... . . . . ... ..... . . . . where .S' $40, tr 0.3, r 0.08, and T 91 days, wth a purchased 4o-strike call and a written 4s-strike is the difference call. The column titled between column 1 and column 2. ;tlpE

-

!

- -. . .

'

.

: -

' ---)-

.

'

.

'

.

.

-

-.

-.

i EE . E EEIE E . -

-

::E E

-

-

.

.

.. ..

.....,....-....

-.---

-

:

..

...

-.

- - - ... . ..

.

.

-

.

:

i

-- . . g -

!

..

-

. .. .

.

-' -.-.

-

.

!

E.;E E . .

..' !..: EE.' . E ;(E E 5' : E.E' t.E..i E-...E... . ...!.'; . . i .E . : E y -

'

:

iECE

E

.

)tE

-

i

-

...y . qy y--. - ... . -

.!1--'

-.

.- - - .-----. .

E. Eiy

.

.

.

-. .

-.

.

jj:jjgyjjj

-1:::2:11.

.11.::.

'':::::.,.

jj::yyyyyyyy

.

.

.

=

=

=

=

-

.

.

Elastity for a call

--..' ''fr'r.E -' '-'-' f'Ftq;r:trd jjyjjgj!d .j' -' ''-'--. -,,,' ..'

. .

.

option for different stock prices and times to expiration. Assumes /( $40, t:r 0.3, =

f

=

'

jqE ' EE'E pi E EE

:'

,

jjjy

.

,

.g g

g

.gjggyg;jj,;j,,

jygggggs

.

3 months -v. 1 y ear -F-

35

.a

years

30

=

0 06

3nd

d

=

0.

'

25

''combined'?

(lly(7 y

qy

Ej :

yygj j

E

((

yrur

jjjyjyjjjqx yE

ls..g:

jyty ) tptwu y

1

)i

2.7804

Delta

0.5824

Gnmma

O0652 0.0780

Rho

0.0511

0.0257

u

j

yy

15 10

0.3009 0.0088

0.0106 -0.0040

-0.0134

-0.0173

,

1:8094

0.053 0.0674

.

Vega

(jj

1 0.9710 0.2815

-

Plice

Theta

jy jtjjj tjuijyjj y y) jk

(4yjjj wp. g tlru) jq xq jiyyy

20

0.0255

E

5 :

0

20

25

30

35 40 Stock Price

45

($)

50

55

60

k. THE

394

BLAcx-scHouEs

FORMULA

PROFIT

whre If)I is the absolute value of f2. Since elastiity is a measure of leverage, this calculation is analogous to the computation of the standard deviation of levered equity valu to quity. Based on Figure by multiplying the unlekered beta by the ratio of f11.111 option could easily have a at-the-money volatility, stock with 30% 12.11, for a an a of 120% volatility or more.

of an option Since elasticity measures the percentage sensitivity relative stock, it tells the option us how the risk premium of the option compares the to of lhat of the stock. In Section 11.2, we computed the discount rate for an option. We to wereimplicitly using option elasticity to do this. At a point in time, the option is equivalent to a position in the stock and in bonds; hence,the return on the option is a weighted average of the rell'n on te stock and tlze lisk-freerate. Let a denote the expected rate of rettlrn on the stock, y the expected return the option, and r the lisk-free rate. We have

The risk premium

on

'

y

LS =

ct5'l

)

a +

AS

CSl

The elasticty

=

Qct +

(1

risk premium

and

395

se does not change

.

=

'

/1

rz)fLi

(12.13)

i= 1

'I'he elasticity of the portfolio is the percentage chapge in the portfolio divided by the P ercentage change in the stock, or Qportolio

=

E'J-:us .!.

,, =

?,

S

nCi

'Af

J'l.ja.j j c j

i= l

c?

,,

T'WQ'

=

Using equation (12.10),the lisk premium of the portfolio, y elasticity times the risk premium on the stock, a r:

or

$

The elasticity of a portfolio is the

of a portfolio

oh d.f )(27,-,1

Q)?*

-

MATURITY

veighted twcmge of the elasticities of the portfolio components. This is in contrast to the Greeks expressed in dollar tenns (delta,gamma, etc.), for which te portfolio Greek ' is the .If?!; of the component Greeks. 'fo understand this, suppose there is a portfolio of lt calls with the same underlying stock, where the th call has value C and delta Li, and where oh is the fraction of the portfolio invested in the th call. The portfolio value is then S)' l ah Q For a $1 change in the stock price, the change in the portfolio value is

Since LS/CCS) is elasticity, this can also be wlitten y

BEFORE

equivalent to a levered position in the stock, and that leverage per jo the Sharpe ratio.

).

-

DIAGRAMS

(12.14)

j.j -

r, is

just the portfolio

-

y

'

-

1--=

(a

r) x f2

-

(12.10)

premium on the stock tnes f2. Thus, the risk premium on the option equals the Using our earlier facts about elasticity, we conclude that ifthe stock has a positive risk Premium then a call always has an expected rettzrn at least as great as the stock and that, other things equal, the expected ret'urn on an option goes down as the stock price goes up. ln terms of the capital asset pricing model. we would say that tlle option beta goes down as the option becomes more in-the-money. For puts, we conclude that the put alwys has an expected ret'urn lss than that of the stock.

y

-

r

=

Qportfoliotl

-

r)

(12.15)

lisk

,

The

sharpe ratio

premium to

of an option volatility:

The Sharpe ratio for any asset is the ratio of the risk Sharpe ratio

Using equations

(12.9) nd (12.10),the

J

..-'

)*

?. o (a ?.) a (12.12) f)c o' Thus, the Sharpe ratio for a call equals the Sharpe ratio for the underlying stock. This equivalence of the Sharpe ratios is obvious once we realize that. the opdon is always -

-

=

DIAGRAMS

BEFORE

MATURITY

In order to evaluate investment strategies using options, we would lile to be able to answer questions such as: If the stock price in 1 week is $5 greater than it is today, what will be the change in the price of a call option? What is the prost diagram for an' optiori position in which the options have different times to expiration? Our previous discussion of option strategies in Chapter 3 examined only expiration values. Now we will examine the behavior of option prices prior to expiration. To do tllis we need to use an option pricing formula.

(12.11)

=

Sharpe ratio for a call is

Shatpe ratio for call

12.4 PROFIT

=

loerhereis one subtlety: While the Sharpe ratio for the stock and option is the same at every point in time, it is not necessnn'ly the sam'e when measured using realized remrns. For example, suppose you perform the experiment of buying a call and holding it for a year, and then evaluate the after-the-fact risk premium and standard deviation using historical returns. A standard way to do this Nkould be to compute the average lisk premium on the option and the average volatility and then divide them to create the Sharpe ratio. You would find that the call will have a lower Sharpe ratio than the stock. This is purely a result of dividing one estilnated statistic by another.

k. TH E BLACK-SCHOLES

396

PROFIT

FORMULA hf F' p.jjjj(,kjjjj('i'(' q' pr';r qff f' t)' @' (ll:y::ltr.:l..f r' (ll:rr:llr:,..f q' !1!7lqi(' El' )' )'1l';$li' ')j(' ;'y' ))#,;-..-.).;'. j'k' 7::71( T' J'T'))EITT i (!' i.!(('(: ( !fi (!;( i ( !'' (!:(:7$q* (ip)f r' (j)!' t'll' y' ,411:::::2..*. (ii.' '(.(jl )i.'. Ilii!j!!-f .-' (' l'y 11111*. j'jjf ;'y')'(-' )y' ,jj:Ljjjj', ('; jytt'-iiggyjjjrt.yf j:y,;gjjjj';i y'. )' 'f1*77* yjyjyt'j. t'.r' (rlf .(i'(): i . ' ' (': 'Ii.'(.' l ..' . i )(.k' ('' ' '. '. (()( ' rrt('. .y k(y jjjk::jjjrj,: y, .:. p..jy (yy(..y( yy(yy...yrjy.jy(.j(.yy y. .y .(i'i j.y.jjjjrjj:ryg. ( (. y .! .( y)')' j yy ;. j E .y . . . .j. . jk;-;. t''l' ' ).rr .. jLjL. . ( 7* ''q' :'tr'rf ?' -' ---.'.-' ;'':' j'y -'. '!' (' ),' ?' -..r' (y'E -' !'' ''5* (6f' : : i;iE E fiE (' )' ptf ()' y' 41::22:2::,,* (( ; . ----' ---,)')t--'.. ..' .'' -'

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what the Consider the purehase of a call option. Just as with expiring options, we can price. stock Table particular in time point >ndfor a value of the option is at a particular four prices five at stock for option different call of 12.3 shows the Black-scholes value a expiration, time given to price stock for the varying a different times to expiration. By keeping everything else the same, we are able io graph the value of the call. Fieure 12.12 olots Black-scholes call ootion prices for stock prices ranging from $20 to $60, including the values in Table 12.3. Notiee that the value of tlle option prior the option at expiration. to exp irat ion is a smoothed version of the value of 12.12 does not show us how the value of Figure depicted in The payoff diagram order original to dt that, we can subtract the cost of cost. In the option compares to its l interest.l the option, plus In order to determine proftability, we need to answer two questions that were initial cost of the option position, and unnecessal'y for the jayoffdiagram: What is the proht, the we take the valu of the position and what is.the holding peliod? To compute interest. including subtract the cost of the position,

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12.3 costs $6.285at a stockprice of $40. q)Example 12.9 The l-year option in Table will have fallen to ll 1f after 1 day the stock price is still $40, the value of the option 0.08/365 '))( 'i.E'. = -$O 012 $6 274 and the l-day holding period prost is $6.274 $6.285 x e iiE'j' .'rhis loss reflects the theta of the option. If the stock price were to increase to $42, the ! li option premium would increase to $7.655,and the l-day holding period prolit would be

$

,

Payoff diagram for a call

=

MATURITY

-y-,

option for diferent stock prices and times to expiration. Msumes # $40, r 8%, & 0, and c 30%. =

BEFORE

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40

45

Stock Price

($)

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--

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t') $6.285 x .$ $7.655

c0.08/365=

-

$1.369.

FCX

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the holding period prot at a price of $40 would be $4.t55 $2 a86 svenif the stock price had risen to $42, the holding -pj l;iperiod ret'urn would still be negative Thege' prbst calculations are illuseated a jrj (11 in Fir 12.13t % After 6 months, ); 0sxa.5 'ili lij $6.285 x e ?-

-

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discussed in Chapter 2, this is like assuming the option is financed by bonowing.

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The option premium g'raphs in Figures 12.12 and' 12.13 can help us understand the behavior of delta and gamma discussed in Section 12.3: In a1l cases the slope of the call option graph is positive. This corresponds to a positive delta. In addition, the slope becomes greater as the stock price increases. Delta increasing with the stock plice corresponds to a positive gamma. 'Ie fact tat gnmma is always positive implies that the graphs will be curved like the cross section of a bowl, i.ek, te option price is convex. A positive gamma implies convex curvature. A negative gamma implies the opposite ! (concave) curvamre. ,

Calendar

y

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We saw in Chapter 3 that there are a numberof option spreads thatpermit you to speculate on the volatility of a stock, including straddle, strangle, and butterlly spreads. These

'

Q Tl4E

398

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Assumes we sell p 91-da# Mo-trike call with premium of $2.78, and buy a 365-day 4o-strike call with premium of $6.28. Msumes $40, 30%, 8%, and r o' & 0. .

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DIAGRAMS

-4 20

25

30

35

45 40 Stock Price (5)

50

55

60

-+-

20

25

30

.

spreads all contain options with the same time to expiration and different strikes. To speculate on volatility you could also enter into a calendar spread, in which the options dates. you buy and sell have different expiration X'YZ'S stock price will be unchanged ver the speculate that Suppose yo want to seaddle written alternative mortths. An 3 or a written butterlly spread is simply to a next will remain unchanged and you will stock price the that hbpe call in'the sell or put, a to option does move into ihemoney, you that the ptential if is premium. The the cost earn can have a large loss. To protct against astockpriceincrease when selling acall, you can simultaneously buy a call option with the snme strike and greater time to expiration. This purchased calendar spreadexploits thefactthatthe written near-to-expiration option exhibits greater time decay than te purchased far-to-expiration option, and therefote is profitable if the zm-strike call with 91 days stock price does not mvek For ekample, suppose you sell a price of $40, stock expiration. zm-sfrike At with eall to 1 expiration and a buy year a to call. l-year the The profit call and for $6.28 91-day premiums for te the are $2.78 diagram for this position for holding peliods of 1 day, 45 dgys, apd 91 days is displayel in Figure 12.14. You can see tat you enrn maximum pr flt okt 91 daf if the stock price does not change.

i10

45

Stock Price

($)

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55

'

''

We can understqnd the behavior of prost for tlzis position by considering the theta of the two options. Figure 12.6 shows tat theta is more negativ for te 91ray call if the stck price does nt change (-0.0173) than for the l-year call (-0.0104). over the course of 1 dayi the position will make money since the written option loses the purchased option. Over 91 days, the written 91-day option will lose more viue t-1::.r1 . its full value (its price declines from $2.78to 0), while the l-year option will lose only about $1 (itsprice declines from $6.28to $5.28)if the stock price does not change. The difference in the rates of time decay generates proft of approximately $1.78. The prot diag'ram also illustrates tat at a stock price of $40,delta for the position is initially positive. Over 1 day,the pnximum prpst pcurs if the stpck price rises by a reflects tl,e fct thatthe delta ofie tten l-ak catl if 0.525 and amount. ' This ' . o? the purchased .. ...l-year . call 0.66t5, for a nei ppsttive deltg of Withthe that .'. ' .. .. .. of the graph . . the purchased below 40 reqetd l-year holding period, the 91-day portlon stock unprfitable 40 increasingly price falls. Above wh h te optiop, ic becoms as the gain on the purchgsed l-year option is offset by the loss on the expiring 91-day call. Since it is expiring, the delta of the 91-day call is = 1 for stock plices above 40, which results in the graph ttlrning back down to a negative slope above 40. As the stock price continues to increase, however, the delta of the purchased l-year call increases toward 1, so the slope of the net position approaches zero. rfhus,

small

E

ts

.

.

'

' . .

:

. .

:

.

.

t.tt90.

,

;

$

400

THE

Bl-Aclt-sclqoLEs

IMPLIED

FORMIJLA

12.5 IMPLIED VOLATILITY

t(.

and Figure 12.4 shows that option prices, particulady for Volatlity is unobservable, near-the-money options, can be quite sensitive to volatility. Thus, choosing a volatility to use in pricing an option is difficult but also quite important.lz O ne approac h to obtaining a volatility is to use the history of retllrns to compute historical volatility (seeSection 11.4). A problem with historical volatility is that histl'y is not a reliable guide to the future: Markets have quiet and turbulent periods and predictable events such as Federal Reserve Board Open Market Committe meetings can create periods of greater than normal uncertainty. There are sophisticated statistical models designed to improve upon simple volatility estimates, but no matter what you do, you cannot count on history to provide you with a reliable estimate ofhltltre volality. In many cases we can observe option prices for an asset. We can then invert the question: Instead of asking what volatility we should use to price an option, we can compute an option's implied volatilit'y, which is the volality that would explain the observed option price. Assuming that we observe the stock price S, strike price K, intrest rate r, dividend yield J, and time to expiration F, the implied call volatility is the that solves

jitjBy tlial and error (or by using a tool such as Excel's Goalseek), t,t ja t, o' 28.7% gives us a call price of $8.07.

=

CS, #,

,

r, F,

)

Table 12.4 lists ask prices of calls and puts on the S&P 500 index, along with using the Black-scholes formula. These S&P options are Europan style, so the Black-scholes model is a/propriate. Notice that, although the implied volatilities in the table are not all equal, they are all in a range between l3% and 16%. We could describe the general level of S&P option prices by saying that the options are trading at about a 15% volatility level. There are typically numerous ptions on a given asset', implied volatility can be used to succinctly describe'the general level of option prices for a given underlying asset. There is one clear pattern in the implied volatilities in Table 12.4. For a given expiration month, volatilitis decline as the strike price increases. This occurs with both the calls and puts. Fof example, the volatility for the November calls declines from 16.3% to 12.84% as the strike rises from 1100 to 1150. The decline is evident but smaller for the later expiration months. This systematic change in implied volatility across strike

implied volatilities computed

:'-' 7j'('* j'';'1* T'qf C' :IIr:.:Il,'' 1iI:.--i'' gjtf ()-' p' k.))j'. k' t'. '1/*(' E( (!jl .! i1! kliid (iiiqd t'tt;f t,f 4112Ej,,)* f' )' (' i':'q'' !' ,)' ,;'

5' 'i q !E E(' ('''(E'i :' ' '!Cll'7pq !)?1T'q(7':F!'l' E.-y...r.---...--t(jj.jyt:.;..yy!yy.'')tjy.y.yy...y...g ((r !..''' (( !!' (.'i i (''( ( !j t. )5 ' ( ri r' )t;' :11I:r' i1IiL11-''.' iiliiEi@l.'' r'. (@:-7Iiiiiii:': '..'. (1II..1iii.' (111).* tqjllElE:'l.-d ;:!(ijiI;' )' .:'iiiIi'. yrf )'. ('t'iq'tt't'i'. )'rj-.t-(jjjjjj;.... '..t . ..!.-yy ))).' kllijillltd tl;i!!:d '.' iiiii@!2ik'. ttr.ttfplittyqlllit.tl7l:tt.' ('. i'yjytrty.lrtjj yy.jj)j.)j.y;yy..q.y,q.y.y-ryy;yy.y..-y.(yyq ' ' i . d:::;p tl::d::::,kpij;l:E 4(:2),. d:::,, lC. jjjj;g,r ij,l,1E: ,!!;.I: Ir..1!14; k j::gjjj ,:::,.11 ii lr..lk .t;tl!r i;Iiiq:1r:.i(,lE!!t 1E::,. d:El)p cl. jyry,j '.r'.-,'LikiiijL-. s Ig::;,. .'-'-.'''-' .'' ). ... r. ).-. -.--. . . . ; -.-. :. j;jrs.,r..,j 'li:tt'i . . Option prices (ask)from www.cboe.com; . .. .. 1 $1 127.44, r 2%. '.111r.*. .-'

.''.ik') . . --' -' --..-' 't'

(C

.

E' :

-

-'.

''

.

.

Volatility

'

-

-' .

.

.

: :'

'

'

:

:

.

-. .

.

-

.

.

-

.

.

.t!iik

.

. .

--.-.-

. .

-

rfhe

t) y ield is zero. We can compute the option price as i!

$8.07 BSCa1l(50, 45, c, 0.08, 0.5, 0). =

I2of the five other inputs to the Black-scholes formula, we also cannot observe dividends. Over short horizons, howeveq dividends are typically stable, so past dividends can be gsed to forecast f'uttlre dividends.

.

- - -. .. .

.. . . . . ..

-//''

...

=

.

-//''

4::2). 1!11,k (:;4!!. ::;!:. d::E),

1:::2),

..1(

,,.

- -- -

--.:14.

,.

assumes

.859%

.

=

=

iII(!r,).' ' ' '. .1112,. :iikj!llg. $1Ij!,' ' (jillEEF?!i,. .11;..:11r..1111. '' ,I1j::!!. kkl:tillt.t .' ' (jII).tziirj-,. .. '. jiii.y!llt:.r;ztg ..'.. '...'.. ()114. ..('. ... .. ''. tlli!ir: . ;. '. ). ....-.-.tt ly'.'.. rjilj--. '.'..(y ..'.yy.i ... ... ..;... i......( .. . .. . .;' . . . ..iy .....-. )..) y.. .... . . . .(.:-...y...;.( .). )). .(...t . ..yt . ..-..;. .. r--.z.. .;..-.;,.... ;.-. .-.. ti.. ... r ....t t ...t .. ..y.. . tq.....41:11k. . .... .. .) . j .. )...... ..y :,.. . t. ' ....t. . :iiri1l.. ., ..t..-.........t L?'b-.-L' :11::112. iIEEll.. 11!5:112... ..)'.. 41::,,.. ;........ ltlii;. . 1(L) ..:iEjli(. ....L.. IIEi!(I;..:!iij2jtIjjj!;,. tIi2t.... .-.. 111.. .jii.....11l2(:!j/'.t. .)-... 111.. 11L.. tii.. . .)..-.4(L. ..1:). t!tli;-. t!tii;*''-*.*,';)4.. .-r......). )...t.. ) ... d1:Ei,-. )...(.......... .. .-.. ...111..... ...... tii:(it:''. . ..... ( t ....t-..... .. . .... . ... . . .. . . . . . . . 0.1630 6.80 0.1575 1100 11/20/2j04 34.80 '

E E. .

.

;

-

tlil:qqqqlz:!E

' -.'. . E' E ' E. ' E. . . ' -. E E. -... ... . . . . EE. . .-. E E. - . . - . .E- .. . . .. . - . . ... . . . .E . . . . -Egk. . . . EE . . . . ... . .. -..-. E . ... . . ...:...)E ..

'

-

-

. E

.

.

--.......

..,:. . . ..r..t. (.-...y-.y.. ...t ;E. -...-..iE .-t........Ei..

.)*)' lq Example 12.10 Suppose we observe a4s-sfrike 6-month European call option with of $8.07. stock price is $50, the interest rate is 8%, and tlle dividend premiuin a jj

'

,jk,,,

-

.

..

'(EqlE' E

':E' :E ' : : : ' .. .

-

- - -.

.

-.

' ' ' .

' (E 7E E! ' '' E E EE Ei ?' E : E . it',--rkjjy..-.t(..gkkjjjjt--.. L(L(-(-);L;;j)L.-L

.-

.....y

,

'

:E ' ir

' ' (E E.

..r

'

-

Computing an implied volatility requires that we (1)obsel've a marketprice for an option and (2) have an option pricing model with wlzich to infer volatility. Equation (12.16) so it is necessary to use an iterative cannot be solved directly for the implied volatility, prpcedure to solve the equation. Any pricing model can be used to calculate an implied volatility, but Black-scholes implied volatilities are f'requently used as benchmarks.

IE);

'

:. .

..-

'

-

Computing

o. w.

(12.16)

By desnition, if we use implied volatility to price an option, we obtain the market price of the option. Thus, we cannot use implied volatility to gssess whether an option price is correct, but implied volatility does tell us the market's assessment of volatility.

Implied

401

we ;nd that setting

=

'

Market option price

%.

VOLATILITY

..

......

..-..

.....-..

...

.

..

-.;. .

.,11:7,12....

-..

-..

...-

-

.(1p

..

.

1100 1125 1150 1100 1125 1150

E.. E. .. -..

-

. .

((IjI(q)iI,. :11i.;211. '(1ji::';. .I1Ii:E '

' . ;. . . . ..

'''':.........-...

..)..E..

j...

-

.--gr..

..

.

..

'....:......

..'...

....--

.-

.. .

.

.

1125 1150

-..

......

'....

. .

.llIr.'II,..:lII' (iij. ' .iiI..-ii..'

. ?''-''.j.. .k-g.-..:., j..-..E . ...t..-..... . . . .... . . .. -,11,//-... ... ...E .i.. -............(. ....

.:..y...-..j...--.

E

...

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iE .

. ..-.yt .....-..) -y...........

.:

-..)

.:.-.kk.,. .

E

-.

.1.:7'...-,. -..(-:.-

..)

..

..-t-..

..

'!dd,./'.

.E(.....(

..

. ..

11/20/2004

5.80

0.1434 0.1284

12/18/2004

41.70

0.1559

13.80

0.1389 0.1539

12/18/2004

24.50

22.50

0.1436

12/18/2004

13.00

0.1396 0.1336

35.50

0.1351

1/22/2005

49.10

0.1567

20.40

1/22/2005

33.00

0.1518 0.1427

1/22/2005

20.00

0.1463 0.1363

29.40 41.50

0.1337

11/20/2004

17.10

14.70

0.1447

29.20

l3An implied volatility function is available with the spreadsheets accompanyin.g this book.

-

E. '

. . . -. .

i.

E

.......

..:112(:!9//...

E. .-t

402

k

THE

BLACK-SCHOLES

PERPETUAL

FORMULA

kesv.

14

is cled volatilit'y S prices oecurs generally for different underlying assets and the resulting line can take price, strike When you graph implied volatilit'y against the smirks.'' Explaining these and 'Trowns,'' different shapes, often described as is a challenge for option for them account patterns and modifying pricing models to in Chapter 23. f'urther issue pricing theoly We will discuss this important ppt-call pality in mind. keep helpful to 'When examining implied volatilities, it is strike t'??/Jtime to evkpiratiolt IWr/lthe w??ld If options are European, then plfr. and calls prices of European puts and because is p?7lt./have l/?c same Ibw/fW volatilty. This true opporttmity. Thus, arbitrage there is relationship or else an cal1s must satisfy the parity rather but cqll, to differences in skew is not related to whether an option is a put or a volatilities and re not exactly call Although put the strike price and time to expiration. not be profitable would arbitrage enough that parity equal in Table 12.4, they are close options in-the-money for example, the after transaction costs are taken into acount. For dtfference for that more pccount can in Table 12.4, bid-ask spreas are as wide as $2, a alone. option than percentage point o implied vo 1a tility on one tsmiless''

a

12.6 PERPETUAL

OPTI ONS

AM ERICAN

$

403

OPTIONS

AMERICAN

formula prices options that are oply exercised at expiration. ln this The Blackcholes based on Merton (1973b), for the prices of calls and puts fonnulas, section we present will call such options perpetual options. expire. We are also ltnown as that never options. expirationless American options are harder toprice than European options because it is difcult to characterize the optimal exercise strategy. Using the binomial model, we saw in Section 11.1 tiat for a finitely lived call option on a dividend-paying stock, the stock price at which it is optimal to exercise the option declines as the option approaches expiration. It is this changing optimal exercise price that makes it hard to derive a vqluation formula. With perpetual Amelican options it is possible to dedve a valuation formula beinlinity. Since time to cause such an option always has the same time to expiraion; option problem will look exercise the the is expiration constant, same today, tomorrow, exercise the option is constant. price optimgl which it is Thus, the forever. at to and The optimal exercise strategy entails picking the right exercise barrier and exercising time the stock price reaches that bnrrier. the option the 'rhey

srst

Using lmplied Volatility

need to price an if lmplied volatility is important for a number of reasons. First, you price, you can use implied volatility option for which you canllot obsel've a market with the prices of aded options. Market-makers, for to generate a price consistent of similar options. Second, as we example, will price options consistently with prices used volatility often is as a quick way to describe the level implied previously discussed, prices are sometimes quoted in tenns of option prices on agiven underlying asset. Option volatility skew provides a measure of of volatility, rathef than as a dollar price. Third, volatility is unbservable, a standard how well option pricing models work. Because whether, for a given is approach to testing the validity of the Black-scholes model to see The existence of prices. al1 strike underlying asset, implied volatilities are the same at assumptions tre not a perfect and volatility skew suggests that the Black-scholes model models in Chapter 23. description of the world. We will discuss alternative pricing that informarecognize volatility To better understand the importance of implied markets. You can think of options tion about futtlre volatility is uniquely provided by elsewhere. available volatility that is not option markets as providing information about based on an index of implied In fact, there is now an exchange-laded futtlres contracts about stock priees volatility for the S&P 500. Just as stock markets provide information volatility, and, in about information and permit trading stocks, option markets provide perspective, we should expect effect, permit the tading of volatility. Viewed from this learn from information about volatility to be 0ne of the most important things we can depth in Chapter 23. option prices. We will discuss volatility in more

Barrier Present Values As a prelude to valuing a perpetual option, consider computing the present value of $1 payable when the stock plice reaches a level, S. We call H the barrier level and call th ttbarlier present yalue.'' lt value today of $1 paid when te stock plice reaches H the depending this, simple formula which differs there for is upon whether H is a t'urns out value, S. stock below the current above or tf S is above S (i.e., S has to rise to reach S), the value today of $1 received when bnrrier present value-is S reaches S-the Value of '

$1 received when S first reaches H from below

S =

;

(12.17)

S

where /? j

1 =

-

?'

8

-

-

0-2

2

+

?'

1 2

8

-

-

-

c2

lf H is below S (i.e., S has to fall to reach S), the

2 value of

2? .

+

w

o'z

$1 received

when

S reaches

H is Itz

Value of

$1 received when S first reaches H from above

=

-

S

where strikes. Thus, 14The Black-scholes formula is defived assuming that volatility is constant across volatility. implied in changes track model to it is intemally inconsistent. to use the Black-scholes benchmark. Nevertheless, it is a widely used

ltl

-

hz

=

-

1 2

1-

-

-

-

c2

r

1 2

-

-

&2

-

2

2? .

+

=

c2

(12.18)

404

%.THE

BLAcK-scHoLEs

FURTHER

FORMULA

Suppose we have a perpemal American call with strike K. If we decide to exercise the K. From option whenever S reaches the banier S, then at exercise we receive H equation (12.17),te value of receiving H K when S reaches S is -

-

K4

-

=

K

-

Price of perpetual put

-

H

ln order to finish computing the value of the call we need to specify S, the price at which the ca.ll should be exercised. We simply need to pick a value for H that makes the value of the call as great as possible. If we make H too small, then we prematurely throw away option value (i.e., protection against a subsequent price decline). If we make H too large, the we forgo dividends for to long while waiting to exercise. lt is possible to show that the exercise level S* that muimizes the vlue of the call isl5 H* 1, we have S*

=

11l

K

=

hl

1

-

K. Maling this substittltion, the value of the perpetual

>

Price of perpetual call II'J = 0, then H* Paying stock.

405

h2 /?2 1 which implies that the price of the perpetual put is H*

lI

(S

>

%.

Again selecting the exercise level S* to maximize the value of the put, we get

Perpemal Calls

. Since 111 call is

READING

=

/7I

K

121

l

-

/r;l

1 S 11l K -

(12.19)

-

x; i.e., itis neveroptimal to exercise acall option on anondividend-

Perpetual Puts For a perpet'ual put, using equation given by

(12.18),the value if we exercise when S

CK

S)

-

=

Ita

S -

S

where h

z

=

-

1 9

?-

?-

-

-

1 2

-

-

-

(7'2

ls-rhisis accomplished by differentiating the expression tozero, and solving for H.

t:r

2

-

2

e?

-?

+

(y

-

2

wth respect to H, setting the derivative equal

H is

=

K

.

112

1 S

-

-

1

-

hz

2

K

:t 11

(12.20)

Under certain assumptions, the Black-scholes formula p/ovides an exact formulaapproximated by tlle binomial formula-for pricing European options. The inputs to the Black-scholes formula are the same as for tlze binomial formula: the stock price, strike price, volatility interest rate, time to expiration, and dividend yild. As with the binomi formula, the Black-scholes formula accommodates different uderlying assets by chanjig the dividend yield (seeTable 10.1 fr a summary). Option Greeks measure the change in the option price (or other option characteristic) for a change in an option input. Delta, gamma, vega, theta, rho, and psi are W idely used in practice to assess the risk of an option position. The option elasticity is th percentage change in the option's price for a 1% change in the stock price. Th volatility and beta of an option are the volatility and beta of the stock times the option elasticity. Thus, an option and the underlying stock have the same Sharp ratio. Of te inputs to the Black-scholes formula, volatility is hardest to estimate. ln inferthe market's stimate practiceitis common to usethefo=ulainbackwrdfasMonto of volatility from the option price. This implied volqtility is computed by fpding the volatility for which the fonuula matches observed market prices for options. In theory, all options of a given matgrity should have the same implid volatility. In practice, they' do not, a phenomenon known as volatility skew. Mthough there is no simple formula for valuing a fnitely lived American option, there are simple fprmulas in the special case of perpemal puts and calls.

FURTHER

READING

Chapter 13 will explore in more detail the market-maker's perspective on options, including how a market-maler uses delta to hedge option positions and the circumstances under which market-makers earn prcts or make losses. Chapter 14 extends the discussion in this chapter to include exotic options. In Chapters 15, 16, and 17, we will use option pricing to explore applications option pricing, including the creation of strucmred products, issues in compensation of capital options, structure, tax management with options, and real options.

406

% THE

of the Chapters 18 through 21 delve more into the mathematical underpinrngs again discussed in Black-scholes model. The barrier present value calculations will be detail in Chapter 23. much in volatility more will discuss We 22. Chapter The classic early papers on option pricing areBlackand Scholes (1973)andMerton detils of how the binomial model converges to the Black-scholes model (1973b). derived inMerton (1973b).The IiIIIC arein Cox et al. (1979).Theperpemalputformulais and Siegel (1986). between theperpet'uf catl and putfonuulas is discussedby McDonald

J

=

0, and r

=

$120,c

=

30%,

?'

=

0.08, and

t5

=

0.

price for 1 year to mamrity and for a a. Compute the Black-scholes call maturity. What happens to the option price times to variety of very long x'? as F O.OOI. Repeat (a). Now what happens to the option plice? What b. Set accounts for the difference? -->

$120, K

=

=

$100.c

=

30%, 1-

=

0, and

=

0.08.

call price for 1 year to matulity and for a a. Compute the Black-scholes variety of very long times to maturity. What happens to the price as

F

-->

(x)?

O.OOI. Repeat b. Set r difference? =

(a). Now what

happens? What accounts for the

12.5. The exehange rate is Y95/E, the yen-denominated interest rate is 1.5%, the volatility is 10%. euro-denominated interest rate is 3.5%, and the exchange rate yen-denominated euro put with 6 months a. What ij the price of a go-strike to expiration? call with 6 months b. Whatis theprice of a l/go-stlike euro-denominatedyen

to expiration? and your answer to (b), c. What is the link between your answer to (a) convrted to yen?

=

30%, r

=

0.08, &

0.03, and F

=

=

=

.,

,

0.75.

=

=

(

,?

=

$100 x /.

=

g.

12.8. Make the same assumptions as in the previous problem. a. What is the g-month forward price for the stock? b. Compute the price of a 95-stlike g-month call option on a flatures ccmtract. c. What is the relationship between your answef to (b) and the price you computed in the previous question? Why? 30%, 1, 0.08, F 0.5, and the stock is to pay a single 12.9. Assume K $40,c dividend of $2 tomorrow, with no dividends thereafter. . #' option? IIPPOSe price of call What the European Consider S is a. $50 a an otherwise identical American call. 'What is its price? =

=

=

b. Repeat,

=

12.4. Let S

$95, c

=

=

=

$100, K

$100, K

=

a. Compute the Black-scholes price of a call. b. Compute the Black-scholes price of a call for wlch S ,-0.08x0.75 .-0.03x0.75 ,G () r K Lj.y5 $9,5x , that for How does your answer compare to (a)?

approxi12.2. Using the Binomcall and Binolnput functions, compute the binomial rpations for th options in Examples 12.1 and 12.2. Be sure to compute prices 8, 9, 10, l 1, apd 12. What do you observe about the behavior of the for n binomial approximation? =

40%,

=

0.06.

=

c. What is the price of a l-year los-stke option, wflere the underlying asset is a futtlres contract maturing at the same time as the option?

PROBLEMS

a

$100, o'

=

v

407

a. What is the price of a los-strike call option with 1 year to expiration? b. What is the l-year forward plice for te stock?

12.7.' Suppose S BSCaII, Bsput, Callpel'ln answering many of these problems you can use the f'tmctions forthe accomjuyingfunctions Greeks (seethe spreadpetual, and Putpelvetttal and the sheets on the CD-ROM aecompanying this book). 12.1. Use spreadsheet to vel'ify the option prices in Examples 12.1 and 12.2.

stock. Suppose

12.6. Suppose XYZ is a nondividend-paying

'l'he

12.3. Let S

%.

PRo BLEMS

FORMULA

BLACK-SCHOLES

=

=

.

t y suppose S

=

$60.

c. Under what circumstance would you pot exercise the option today? 12.10.

decay is greatest fpr an option close to expiration.'' Use the spreadsheel f'unctions to evaluate this statement. Consider both the dollar change in the option value and the percentage change in the option value, and examine both in-the-money and out-of-the-money options. EErfime

12.11. ln the absence of an explicit formula, we ean estimate the change in the option price due to a chage in an input-such as o'-by computing the following for a small value of c: BSCaIICS,K, o' + 6, ?', /, ) BSCaIIIS, K. ty e, r, t, J Vega 26 -

-

'

-

'

'

=

.

a. What is the logic behind this calculation? Why does 6 need to be small? b. Compare the results of this calculation with results obtainedfromfsfc//kkgtz. 0.03, and F 0.08, & 30%, ?' 12.12. Suppose S $95, o' $100, K 0.75. Using the technique in the previous problem compute the Greek measure =

=

=

=

=

=

408

% THE

BLACK-SCHOLES

c. Suppose ?- 0.07 with al1 other inputs the same. What happens to the price and exercise barrier? Why? . =

$40. What are delta, gamma, vega, theta, and rho? $45. What are delta, gamma, vega, theta, and rho? different? ((f so, why? c. Are any of your answers to (a)and (b)

a. Suppose S b. Suppose S

50% with all other inputs the same. What happens to the d. Suppose o' exercise barlier? Why? and price =

=

=

=

=

=

12.19. Consider a perpetual put option with S 0.03. and

$45. What are delta, gnmma, vega, theta, and rho? different? lf so, why? c. Are any of your answers to (a)and (b)

= 0.07 with all other inputs the same. What happens to the plice and exercise banier? Why?

=

as it does.

0. Using l-year-to-expiration European 30%, $ 8%, c 12.17. Assume 1' sell two Bo-stn'ke puts, buy one 95oPtions, construct a position where you strike put, buy one los-stHke call, and sell two lzo-strike calls. For a range of stock prices from $60 to $140, compute delta, vega, theta, and rho of this position. As best you can, explain inmitively the signs of the Greeks. =

=

12.20. Let S

=

$100, K

$90, (7'

=

=

$50, K

=

$60. r

=

0.06, tz

=

0.40,

=

30%, 1-

8%,

=

=

5%, and

a. What is the Black-scholes call price? b. No* price aputwhere S $90, K $100,(7' 1. and F =

=

=

r

=

30%, r

1.

=

5%,

=

8%,

=

c. What is the link between your answers to (a) and (b)? Why? 12.21. Repeat the previous problem, but this time for perpet'ual options. What do you notice about the prices? What do you notice about the exercise barriers? APPENDIX 12.A: THE STANDARD NORMAL DISTRIBUTION T he standard

p7t/?7n/l

probability densityhtnction is given by

4@)

=

12.18. Consider a perpetual call option witll S 0.03. and =

50% with all other inputs the same. What happens to the d. Suppose c price and exercise bnrrier? Why? =

30%, 0. In doing the following calculations, use a 8%, o' 12.16. Assume r stock price range of $60-$140,stock priee increments of $5, and two diferent times to expiration: 1 year and 1 day. Consider purchasing a loo-strile straddle, i.e., buying one loo-strike put and one loo-strike call. call and put separately, for the of a. Compute delta, vega, theta, and rho the expiration. and times to different stock prices rho and of the purchased straddle (dothis by theta, b. Compute deltg, vega, options). As bst you can, explain individual of the adding the Greeks straddle Greeks. of the intuitively the signs and rho of the straddle with 1 year to expiration theta, c. Graph delta vega, stock of the price. In each case explain why the graph looks as a f'unction =

0.40,

=

c. Suppose r .

=

d. Are any of your answers different in this problem from those ip Problem 12.142 If so, why? =

0.06, cr

=

=

=

b. Suppose S

$60, 1,

b. Suppose & 0.04 with a1l other inputs the same. What happens to the price and exercise barlier? Why?

=

=

=

a. What is the price of the option and at what stock price should it be exercised?

12.15. Consider a bull spread where you buy a 4o-strike put and sell a 45-strike put. 0.08, J 0, and F 0.5. ' Suppose o' 0.30, r and rho? a. Suppos S $40. What are delta, gamma, vega, theta, =

$50, K

=

=

=

=

409

=

.

12.14. Consider a bull spread where you buy a zm-strile call and sell a 45-strike call. 0.5. 0.30, 1, 0.08, $ 0. and r Suppose o'

$

0.04 with a1l other inpts the same. What happens to the b. Suppose exercise bnrrier? Why? and price

=

=

=

=

DISTRI BUTI ON

a. What is the price of the option and at what stock price should it be exercised?

corresponding to a change in the dividend yield. What is the predicted effect of dividend yield? a change of 1 percentage point in the 12.13. Consider a bull spread where you buy a 4o-strike call and sell a 45-strike call. 0.5. Draw a graph with Suppose S $40, (7' 0.30, 1. 0.08, J 0, and F the depicting proft on the bull spread after ranging $60 from $20 to stock prices months. and months, 6 1 day, 3 =

N O RMAL

TH E STAN DARD

12.A:

APPEN D IX

FORMULA

l

HB

lx

e-

Jra

(12.21)

'

distl-ibtltion d/zc/cpl, evaluated at a point x, fOr eXThe cumulative stalldard llorlnal nmple, tells us the probability that a number randomly drawn from the standard normal distlibution will fall below x, or -$.-

N @)

-!.-

$ (x)#-'r

jg

>

-

-x

-x

2x

-Y e -J.v2#

(jr,!p,:) .

410

%.TH E BLACK-SCHOLES

APPEN DlX

FORMULA

Excel computes the cumulative distribution using the built-in f'unction NORMSDIST. Note that #/(x1) = /la'j ).

Call theta

OCCS, f('. o' r, F Pt

-((T-l) N(# I ) = &Se

In this section we present formulas for the Greeks for an option on a stock paying Greek measures in the binomial model are discussed in continuous dividends.l6 Appendix I3.B.

Delta measuz'es the change in the option price for a $1 change in the stock price: DCCS, K, c', ?', F t, ) = Call dblta = e -

p

Put delta

,

).P(5', #, c,

=

-j(,z.-?)s(#j;

?-,

F

1,

-

8)

put '

ye-rv-tj

ytfygjg

2

r

t

-

)

t,

-

C a 11theta +

=

J5'd-tlT'-''

-

Vega measures the change in the option price when volatility also use the terms Ianlbda or kappa to refer to this measure:

;

Call vega

(r) Call gamma Put gamma

=

pzctu

'

t, J)

-

'

0SI

p2#(5', K, =

t7', r,

F

t,

a

=

)

=

F

-

Rho

Call gamma

Theta measures the change in the option priee with respect to calendar time tixed time to expiration (F):

N (-a-)

With some effort, the second can be verilied algebraically:

Se

-Jrx?

(d l )

=

Ke-rvx'otzj

p.P4u$',K,

r, F

F

tr, r,

-

=

-

-

t,

()

t,

-sv-tjytyj

=

Se

=

Call vega

'

; p

.j

chang

(p) -

Call rho

=

jjj:)i jgkjlr.jjy)t jgjj. jyy)j:::::::2

(1),holding

l61f you wish to derive any of these formulas for yourself, or if you lind that different authors use formulas that appear different, here are two useful things to know- The lirst is a result of the normal distribution being symmetric around 0: -

t7',

Rho is the partial derivative of the option price with rspect

(p)

1

DCLS, K, =

t

The second equation follows from put-call parity.

N (a-)=

in the

Some writers

changes.

It is common to report vega as the change in the option p6ceiael'pel'celtagepoillt in the volatility. This requires dividing the vega fonuula above by 100.

e---ttN'dL) us'ty

-

change

Vega

Put vega K o-, r, T

411

-

,

pz'(,s,K, c., ?-, T t prKe-''T-''

=

$

-(5(w-,)x(-#j

-d

=

Gamma measures the change in delta when the stock price changes:

Theta

theta

GREEKS

If time to expiration is measured in years, theta will be the amlltalized option value. To obtain a per-day theta, divide by 365.

Delta (A)

Gamma

2) # (y.j

).Ke-rT-l)

-

GREEKS

FOR OPTION

12.B: FORMULAS

APPENDIX

FOR OPTION

/, &)

-

,

=

FORMULAS

12.B:

pC S, K, c'

,

?-,

F

t, J)

-

pr

0 P S, f('. c'

,

r, F

. ...............................

0r

t, $j

-

.

to the interest rate: -r(w-,)

=

(F

-

yyjy)

t) Ke

-vv-tj

::::::::

.......

jgk (jlr'' .......

.S ()j

jrk

jsjr

x (.tyz)

These expressions for rho assume a change in 1, of 1.0. We are typicatly interested in evaluating the effect of a change of 0.01 (100basis points) or 0.0001 (1 basis point). To report rho as a change per percentage point in the interest rate, divide this measure by 100. To interpret it as a change per basis point, divide by 10,000.

Psi

(#)

Psi is ihe partial derivative of the option price with respect to the continuous dividend

yield:

'j)d ,'Ejl' jj:yry:jjd ,j:yyr::' ijd y' :'1* E' 11:::(j1(.* Ejd k' ;'f' E;' j''''#' yjjd 11118* i'yr'.' 'jk(-,' ,')' j'qgjjltry'..rjyt'.,qd E;' ;-' J' ?' ljdepldjjjld.ttltll.lEjttd y' @E.' F' ;',' ld (,' :'',' q$' Cttd (' 5* ''').' jii' j-,,' ')' h' k' E-' i'Ep' qfililiijjjjjlii'j' .E' F' ;',jj.jj)jjjjjyyjjjyjjyyj(;.' qqt.gld jyjqtqjEjyjy'.!ytyrrj)rtd #' q.?(jjhy-:,t@jy!!L' i'j'lli'!:d :jjky' .4* ;'(' 11::(::)j* 112::::11* j'1* q' 41::2::1,* 41::22:1* kd y' jjjj!jjjjjjjjj!q'd (' t';'))' jy'y' i'Ejyi.!.d '.' $' )' E' ?' ,y--.:j' f' C' ''kkd ;,' tti:d j'k' q' itE..d jrd j..$5:*:21y.,--,j,*-,?..-,2,.*;:j--4(4,,.4,1,,,1,,,,,,..* k);;d ,'' (t' Ejd itd :''j;' g' ;)(y' !' ()' yrd 'jtd jl.kttjisllttl'sd )y' ;;;t);,' ;';'?' 'jd ljd jyd j'.jd y' jtrjrjg').d )#(-'.:' )jk..' 4:!)* r' .j' j;d ?' k14,1E1!.* ''i:.' ;kr:.' jkd kplti'd ,jjjd y' 7* py')kkd frd i'';' k' j?,3Lit:L-LL$j;L,q;L.',,3b-',LL' 4* ;)'-' jid ;.jq)' C' yyryyd j(' jyd j?d p;)d j'yjd lytyj'j' .j' (' j(' 1* g' )jj' jytjigr-'jlt'.d )' jygd 1;* .jjyr' t'?'p' rd /''jr' ;'.;''(' k' l'td 'y' )'q' )-*-* j';)')'('ld $' )' t'jjjllr::,id ()lrq:51(.' (iit.d rl!ll.llt,;rll.d (qI12qjI::''' y(' j!' jyd gyd jpd (',j' jjd (j'.' j'p'gjd )))' j.jyt' yyd q' r)' .y' f' !!.' jt)!yjpd )y'y@qq.' td ;f' (* ly'p;d $' y!.y' (' p' pt'lll!?d q.)kjzqtd :i#l('k)#)t-.@Ttr'!r' (((' .,' l'd );' r.p)k)'-j((' ;p'(k..' @' t')kp;!-' it'ljd tE..r.;$It' ('(' p.''-?j-'-'' .-' )))!@j(r:-' qqjd trd i'yrd )?' j.)d pgppd .'' jtd .))' ;';' t,j.'-kj'fq.'jqLi,.,' (k.;' tktd 4* i)' ?' ktd lylj:--d j.yd yjd ;'-jt' y)' j:r.:r..d jlyyiij,j'.)d ');'')'@';E,,,.kj;.'.jjj)' ' yd 11)* i.ld jd 1* ljd )' j'(ijtd gydyy'kjt'jtd (jd .y' ;yjgt'yrtj(y..' qrd j;d piljyd Tjt)jjd jj;.yq;)jy'.' S!yd j-' j)'t))tjjp,' g' C'-' .:' ytd ysjsgikt.jd 1* )jjtj;',' ))' .y' y' (' lgd )' f' 1* J' /'i'(qlIIEEqElk:-..' (' p' y' 'y' )' ygd yyd r' k' qtd y.')' ;'!'r' p))r'..' y)d ):r*:)* .;;' y' t);tj).).(jygj' ljd kiltd ):k)d 4)* (j' 2* 1* .r' )d t.;td ty'-)d )(' tritgdyrjy.ykjdyjydtjrrysjjyyryyjjr--lk-jjjjd (' ttd tkd ttyd q' yyj)d $' pjd y-':'-' 4* .(jryj)jjj(j,.' qj'jyyd tryd ykjjttjjjjjyly:.rrjpd (;jjjyj(();'.' rits.lpylqqqiytjpppy.yytsqljjqidirypplpptqkryt't)j,!d 4/2.4* )' ((' 4()* kttd yqjd t'yjjk.pd rid .jjjy,rjj.' t'i''J'ltd rjd td r(.:jli'i' tjd t.pd t't' t.(j)(y' lllfl'-ip''d t;()* r:.yy..' t'?d yjjy--;d ;)f'j');jL(;:j',.' t)d rqd t'tpd 'jirk'lpd ttd rrd r'jjd r' .,I'IIIIIII'-,.'''''II)IIIIF'''X (jjgl;:lpd (::j),;,,* 41::(j11L.* .(' tjd jjd r' rylltji.)rtd

ryjd ''''(q!II((q'''' 111:::E(q(2::1,1* (E1IIEiF;ll:-..' ((IIEiiEi(:,' r' rid rjtd rd .((i!!!1'd1.,-''!:!IIi((.' ((111(((((((::)11.* ,:E1E11!11,* rqlllll...d

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412

%.THE

Bl-Acl-scldol-Es

Call psi

=

FORM uLA

OCLS, ff, o-,

?-,

F

t,

-

()

-j(w-,)s(#j;

=

-(F

-

tj'e

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.'''.' .'y' .'.'j' .'kt'. ''--.'-

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E ..-E

=

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:

:

.

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.

.

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.

.i.

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0. We will let F denote the expiration ?' 0.08 (continuouslycompounded), and time of the option and t te present, so time to expiration is F t. Let F t ct:c 91/365. The price, delta, gnmma, and theta for this call are listed in Table 13.1. Because te market-maker has written the optipn, the sign of the Greek measures for th position is opposite those of a purchased option. In particular, the wlitten option is like shorting shares of stock (negativedelta) and the option gains in value over time (positive theta). Because deltais negative, the risk for te market-maker who has written a call is that the stock price will rise. Suppose that the market-maker does not hedge the written option and the stock price rises to $40.75. We can measure the profit of the market-maker by marlking-tomarket the position. Marking-to-market answers the question: If we liquidated the position today, what would be the gain or loss? In the case of an option price increase, the market-maler would need to buy the option back at a higher price than that at which it was sold, and therefore would lose money. At a stock price of $40.75,the call price would increase to $3.2352,so the market-maker profit on a per-share basis would be $2.7804 $3.2352 Figure 13.1 graphs the overnight proit of the unhedged written call option as a function of the stock price, against the profit of the option at expiration. In computing overnightprost, we are varying the stockpriceholding fixed al1other inputs to the BlackScholes formula except for time to expiration, which decreases by 1 day. lt is apparent from the g'raph tat the lisk for the market-maler is a lise in the stock plice. Although it is not obvious from the graph, if the stock price does not change, the market-maker will profit because of time decay: It would be possible to liquidate the option positio by buying options at a lower price tlle next day than the price at which they were sold originally.

able to proft based on the markup. The store mantains an inventory, and the owner is Market-inakers immediately. television buy and wnlk who want in a to satisfy customers supply immediacy, permitting customers to trade whenever they wish. Proprietary trading, which is conceptually distinct from market-mnking, is tradtypically expect ing to express an inkestment strategy. Customers and proplietary traders market the whether depending up or down. In prostable goes be upon positions to their bid and selling at the ask. 'I'he position the buying profit mayket-makers at by contrast, of a market-maker is the result of whatever order flow arrives from customers. A difference between appliance sellers and financial market-makers is that an appliance store must possess a physical television in order to sell one. Alinancial marketgenerating inventory as makeq by contrast, can supply an asset by short-selling, thereby needed. ln some cases market-makers may trade as customers, but then the market-makr market-maker. is pay ing the bid-ajk spread and therefore not serking as a

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Without hedging, an active rparket-maker will hgve an arbitrary position geperated by tisk. An advepe price fulfllins customer orders. An arb 1trary po rtfolio has uncontrolled the market-maker. Consequently. market-makers movehas the potential ioofbnnkrupt tlaeir positions. zisk hedge aitempt to te . Market-makrs can control tisk by delta-hedging. As in Chapter 10, the marketmaker computes the option delta and takes an offsetting position in shares. We say that such a position is delta-hedged. ln general a delta-hedgd position is not a zerovalue position: The cost of the shares required to hedge is not the sarile as the cost of capitf to the options. Because of the cost difference, the market-maker must invest .

position. is that such a hedged position should earn ihe risk-free derivatives key in A idea it, and you have no risk rate: You have money tied up so you should earn a ret'ul'n on used this argument explicitly in our discussipn so you should earn the risk-free rate. We in binomial pricing in Chapter 10. The implicitly and 5, Chaptr in pricing of fonvard notion that a hedged position earns the risk-free rate is a linchpin of almost alldelivave thei.r pricing models. lt was the ftmdamental idea exploited by Black and Scholes in derivation of the option pricing model. With the help of a simple numerical example, we can understand not only the intuition of the Black-scholes model, but the mathematics as well. Delta-hedging is key lisk of an option position. If we to pricing because it is the technique for offsetting the provides us with delta-hedging think of option producers as selling options at cost, then is Delta-hedging replicated. option is when it is an understanding of what the cost of the plicing. option understanding thus both a technique widely used in practice, and a key to

mnai tain a delta-hedged

Option Risk in the Absence of Hedging buy a 9l-day call option, the market-maker ftlls this order by If a customer wishes = $40, K = $40, (y = 0.30, selling a call option. To be specifc, suppose that S 'to

!' :.jj* .j' fijtry'l-ylgijt. 'jk' irf illl,llilr.llllr:r';d ;'t')'ptf t''(';'@' j/f i'7)* y)yyj'. 6j'i;! lf )' (jI1p:r--' 'r''iii,;;''Iii,;;!';' Irf 1* ()* !' ;'f'TT7T'rF''I7'r?''''TF'qTF7TTT:'F'FF7SJTFTY 1i)l' f7)q1q9/)(t*''1717q: jf )q:11.* 57* (7* ' ' ' '!'j!Sr ')''i'( i. .'('' ' i'...(i!'i((i(i'; ;' .(.! EE((''('(. (E'''(!'.'.E''(( '(' y' )'(y))t(.. (' '.';l iE(E'. .F;z'(q; t'J'qitpf kliti!id y'. yjl';yrjyy .j' )' . ty't)trq..)jk)y;y. yjq)yy'. rtf E . . . .. t'?'jfkjljjf t'.; i i r..( .r (.. E..( :;i.'kkt;i;y;t,)t?,@kt).yy;kti?E.,i!..,'i,.Er).t,(.,q;,r,(ity) ! )q$! !j) q C1'i j.TEi..j; ytyjjj; ( (jjjyy...;j; y tj Price and Greek information for call option with tj(yjy) ) . .qi'..)? .jj yg.....j-. i.f'lx*il''il. l(.....-.; a = $40, # $40, tr 0.30, r 0.08 (continuously . 91/365, and J 0. compounded), F:'y'jgf ;'y' F' ''C;' f

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the stock price moves $ 1. This is why delta did not accurately predict the change in the option price: The delta itself was changing as the stock price changed. The ultimate change in the option price is a result of the average delta during the stock price change, not just the delta at the initial stock price. As you might guess, we can use gamma in addition to delta to better approximate the effect on the value of the option of a change in the stock price. We will discuss this adjustment later.

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Consider the 4o-stn'ke call option described aboke, written on 100 shares of stock. The market-mker sells the option and receives $278.04.Since A 0.5824, the market-makr also buys 58.24 shats. (We will pftit fractional share purchases in thij example.) The net investment is Day ()

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At an 8% interestrate, the market-maker has an overnight financing charge of $2051.56x

Delta and Gamma as Measures of Exposttre Since delta tells us the price sensitivity of the option, it also measures the market-mAer's which sggests that a exposure. I'he delta of the cll at a stock price of $40 is 0.5824, by approximately option of the value increse should the $1 incrase in the stock price terefore the option plice would increase Jirice $0.5824. A $0.75increase in the tock option's value is the act'ual increase = in However, the 0.5824 $0.4368. by $0.75 x

$0.4548, greater by $0.0180. This discrepapcy occurs because delta varies wit.h the stock price: As the stpck price increases and the option moves more into the money, delta also increases. At a stock price of $40.75,delta is 0.6301. Thus, the delta at $40 will ullderstate the act'ual change in the value of the option due to a price increase. Similarly, delta will overstate the decline in the value of the option due to a stock price decrease. tf the stock price had fallen $0.75 to $39.25, the option price would have declined to $2.3622,wlaich would result in a gain of $0.4182to the market-maker. x 0.5824 Using delta, we would have predicted a plice decline of option delta decreases the because decline. actual This than the occurs which is greater 0.5326: price this is The delta at the stock new plice declines. as Gamma measurs the change in deltawhen te stockprice changs. In the example above, the gamma of 0.0652 means that delta will change by approximately 0.0652 if

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Without at Erst worrying about rebalancing the portfolio' : Marlting-to-marltet to maintain delta-neueality, we can ask whether the market-maker made money or lost money ovemight. Suppose the new stockplice is $40.50.The new call option price with 1 day less to expiration and at the new stock price is $3.0621.Overnight mark-to-market proft is a gain of $0.50.computed as follows: Day

,1

58.24 x ($40.50 $40)

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the portolio new delta is 0.6142. Since deltahas increased, = additional shares. This transaction requires an 58.24 3.18 buy 61.42 must we investment of $40.50x 3.18 = $128.79.Since the readjustment in the number of shares 'l''he

Day 1: Rebalanclng

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profits entails buying at the current market price. it does not affect the mark-to-market for that day. .

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61.42 x ($39.25 $40.50)

Gain on 61.42 shares

-

call option

Gain on written

$306.21 $232.82 1) x $2181.30

-$76.78

=

$73.39

-

-(c0..08/365

Interest

-

.

=$0

48

= -$3.87

!

.

Overnight proht

$128.79to buy more shares and $0.45to pay interest expnse. The

$129.74

$128.79 $0.45

-

$0.50

=

-

Since the bnnk is willing to lend us the value of our securities, we are free to pocket the $0.50 tat is lef't over. 'fhis exnmple demonstrates that the mark-to-market prolit equals the net cash flow genepted by always bon-owinjto fully fundtheposition. Another way to see the equality of mark-to-marketprost and net cash flow is by examining the sources and pses of funds, and the extent to wllich it is necessary to injct additional cash into the position in order to maintain the delta-neutral hedge. We can calculate the net cash flow frpm the portfolio aS

Interpreting the Prolh Calculation ConcepAt the end of day 1, we show a$0.50 protitfrom the mark-to-marketcalculqtipn. p' ortfolio whieh the the measuring extent loss proht to of the think as or tually, we can show a When hedge. delta-neutral maintain we order infusions a in to requires cash portfolio. of the cash take this out positive proft, as in case, we can To see that mark-to-market proft measures the net cash ipfusions required to maintain the delta-neutral position, suppose that a lender is willing t a1l times to lend buy 58.24 shares of stock, wlzich us the value of securities in the portfolio. lnitially, we option premium, so the net cash the by offset is $278.04 costs $2329.60,but this amount portfolio of value the also is (stockless the option), our net This we require is $2051.56. amount.z this so we can bon'ow As time passes, there are three sources of cash flow into and out of the portfolio: 1. Borrowing: Our borrowing capacity equals the market value of securities in the portfolio; hence, borrowingcajacity changes as thenetvalueof thepositionchanges. On day 0, the net value of our securities was $2051.56. On day 1, te share price rose and we bought additional shares; the market value of the position was 61.42 x $40.50 $306.21 $2181.30.Thus our borrowing capacity increased by capacity, but there is $129.74. The change in the option value changes boaowing options. number of changing the no cash flow since we m.e not 2. Purchase or sale of shares: We buy or sell shares as necessary to maintain deltaneutrality. In the above example, we increased shares in our polrtfolio from 58.24 to 61.42. The price at the time was $40.50,so we spent 3.18 x $40.50 $128.79. =

=

3. Interest:

419

extra amount the bank will lend us-is $129.74. change in our bonowing capacity-the 'l''he difference between what the bank will lend us on the one hand and the cost of addidonal shares plus interest on the other is

Net cash flow

-

%

NG

We pay interest on the borrowed amount. On day 1 we owed

$0.45.

2In practice the market-maker would be able to borrow only part of the funds required to buy securities, with market-maker capital mald ng up the difference. .

Change in borrowing capacity additional - cash used to purchase interest

=

shares

Let Af denote the option delta on day i, Si the stock price, Ci the option price, and

Mjq the markefvalue of the portfolio. Borrowing capacity on day i is MY. hence, the change in borrowing capacity is MVf

MVf-l

-

=F:

2; Si

-

Ci

Si-j

(Af-I

-

The cost of purchasing additional shares is Si(Af depends on the previous ay's borrowing, rMVf-1

-

MV/-I

-

-

Si(A?

-

.

-

-

Ci ;

Ci-k )

-

-

-

-

-

), and interest owed on day i Thus, on day i we have

Af-l

-

A/ Si

Af-I

-

) rMVf-l LLCSL Ci (Af-l Si-q Ci-k ) Si(Af = (Cf Ci-j Si-j Af-l Si ) rMVf-l ) =

Net cash flow = MVf

=

-

-

Af-I)

-

rMVf-1

-

The last expression is the overnight gain on shares, less the overnight gain on the option, less interest-,this result is identical to the proft calculation we pedbrmed above. ln the numerical example, we have

11-71 -

MVa

-

5'l(AI

-

A0)

-

?'MV:

$2181.30 $2051.56 $128.79 $0.45 $0.50 = =

-

-

-

This value is equal to the ovenzight proft we calculated between day 0 and day 1. Thus, we can interpret the daily mark-to-market protit or loss as the amount of cash tat we can pocket (if there is a proft) or that we mst pay (if there is a loss) in order to fund required purchases of new shares and to continue borrowing exactly the ainount of our securities. When we have a positive proft, as on day 1, we can take money out of the portfolio, and when we have a negative prost, as on day 2, we must put money into the portfolio. A hedged portfolio that never requires additional cash investments to remain hedgedis self-financing. One of the questions we will answer is under what conditions delta-hedged portfolio is self-fnancing.

420

%.MARKET-MAKING

DELTA-HEDGI

AND DELTA-HEDGING

Nc

k

Figure 13.2 graphs overnight market-maker prost on day 1 as a function of the

Delta-Hedging for Several Days

stock price on day 1. At a stock price of $40.50,for example, the profit is $0.50,just as g'aph is generated by recomputing the lirst day's prost per share for a in the table. variet.yof stock prices between $37 and $43. The gpph veres what is evident in the delta-hedging market-maker who has written a call wants small stock price table: and can suffer a substanti loss with a big move. ln fact, should the stock price moves move to $37.50,for exaple, the market-maker would lose $20. If the market-maker had purchased a call and shorted delta shares, the prost calculation would be reversed. The market-maker would lose money for small stock prtce moves and make money with large moves. The profit diagram for such a position would be a mirror image of Figure 13.2.

We ean continue the example by letting the market-maker rebalance the portfolio each lay. Table 13.2 summarizes delta, the net investment, and proft each day for 5 days. The proft line in the table is daily prost, not cumulative profit. What determines the pattern of gain and loss in the table? There are three effects, attributable to gamma, theta, and the carrying cost of the position. 1. Gamma: For the largest moves in the stock, the market-maker losej money. For small moves in the stock price, the market-maker makes money. The loss for large moves results from gamma: If the stock price changes, the positi becomes unhedged. In this case, since the market-maker is short the option, a larj move lises, the delta of the call incregses and the call generates a loss. As the stock price loses money faster than the stock makes money. As th stck pfie falls, the delta of the eall dereases and the call makes money more slowly than the fixed stock position loses money. In efect, the market-maker becomes unhedged net long as losses on the stock price falls and unhedged net short as the stock price rises. days 2 and 4 are attributable to gamma. For a11of the entries in Table 13.2, the per share. garnma of the written call is about

'l'he

'l'he

rfhe

-0.06

2. Theta: Ifaday passes withno change in the stockprice, the optionbecomes heaper. This time decay works to the beneht of the market-maker Fho could unFind te position more cheaply. Time decay is especially evident in the proft on day 5, but is also responsible for the proht op days 1 and 3. 3. Interest Cost: tn order to hedge, the market-maker must purade carrying cost is a component of the overall cost.

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Overnight profit as a function of the stock price for a delta-hedged has writlen a call.

market-maker who

k. MARKET-MAKING

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Option delta lnvestment

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Call ($)

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Stock ($)

lnterest

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jjy,.,yjy,,,yjj jjg,,, jjy,,, jj,,..yjj .:jjj;; jjr,., jjr.,,)jy,,,yjj .gygj,;j,, delta-hedges, assuming the stock price moves up or down 1 o' ..

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.

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1,725.95

1,894.27 -0.38

-0.41

DELTA-HEDGI

NG

$

423

Using Gnmma to Better Approximate the Change in the Option Price

:

-'

-

.

TH E MATH EMATI CS O F

0.51

0.46

0.42

0.38

0.02

0.01

0.01

0.00

Delta alone is n inaccurate predictor of the change in the option price because delta changes with the stock price. When delta is very sensitive to the stock price (gammais large), the inaccuracy will be relativety great. When delta is not sensitive to the stock price (gammais small), the inaccuracy will be relatively small. Since gamma measures the chqge in delta, w can use gamma to develop a better approximation for the change in the option price. If the stock price were $40.75instead of $40, the option price would be $3.2352 instead of $2.7804. For the purpose of computing the change in the option price, we want to know the average rate of price increase between $40 and $40.75,which we can approximate by averaging the deltas at $40 and $40.75: AAverage

ln the previous example, the stock price changes by varying amounts and our daily proht varies substantially. However, in Figure 13.2, there is an up move and a down move for the stock such that the market-maker exactly breaks even in our proft calculations. If the stock always moved by this amount, the portfolio would be self-financing: No cash in:ows are required to maintain delta-neutrality. lt turns out that the portfolio is self-snancing if the stock moves by one standard deviation. In the binomial option pricing model in Chapter 10, we assumed that the stock Serh-''ll where o'X-h is the standard deviation per move d up to or down to interval of the rate of remrn on the stock. Suppose we assume the stock moves up or down according to this binomial model. Table 13.3, which is otherwise the same as the previous example, shows the results of the stock moving up, down three times, and ten each day. I. the up. You can see that the market-maker comes close to brealcing even approximately portfolio is stock moves according to the binomial model, therefore, the self-fnancing.

=

$40.75 y computing

C($40) + 0.75 x Aywerage

(13.1)

When we average the deltas at$40 and $40.75,wehave to compute deltas at two different stock prices. A different approach is to approximate the average delta by using only the delta evaluated at $40together with gamma. Since gamma measures the change in delta, we can approximate the delta at $40.75by adding 0.75 x F' to Ag0: A40.75=

+ 0.75 x 1-'

40

Using this relationship, the average delta is A4() + (A4() + 0.75 x AAverage =

2

r)

j

= A40 +

Serh-bTvl

,

2

We could then approximate the option price at

C($40.75)

A Self-Financing Portfolio: The Stock Moves One o'

A40 + A40.75

=

x 0.75 x

(13.1),we can then approximate the

Using'equation

C($40.75)

=

C($40) + 0.75 x

= C($40) + 0.75 x

r

call price as

AAvecage

1 r. x 0.75 x F'

A4o +

= C($40) + 0.75 x A4: +

1

j

x 0.75

2

x In

(13.2)

The use of delta and gammato approximate the new option price is called adelta-gamma

13.4 THE MATHEMATICS

OF DELTA-HEDGING

Clearly, delta, gamma, alad theta :11 play a role in determining the prolit on a deltahedged position. ln this section we exnmine these relationships more closely in order to better understand the numerical example above. What we do hely is a kind of financial forensics: Once we larn how the stock price changed, we seek to discover why we earned the profit we did. .

approximation/

3You may recognize that we have already encountered the idea of a delta-gamma approximation in Chapter 7, when we used duration (delta)and convexity (gamma)to approximate the price change of a bond.

k. MARKET-MAKI

424

lq G AN D

D ELTA-HEDG

TH E MATH EMATI CS O F

l NG

?' )t@' )' 7* )j;lfEkample 13.1 lf the stock price liss from $40 to $40.75, the option price injyj )) creases from $2.7804to $3.2352.Using a delta approximation alone, we would estimate. .)(jC($40.75) ) as .1(.i .p.tt . C($40.75) l) C($40) + 0.75 x 0.5824 $3.2172 l.!j '. lj) j i))I delta-gamma approximation, we obtain Using a j J'jE

=

=

.

L (&) + 8

(SJ+)

= L$)

2

=

=

-

'

$

425

+

1

(13.4)

-E:1-'(u%)

2

The olkion price at te new stock plice is the initial option price, C&), plus the average delta times the change in the stock price, or

CS+l,4

.

1 )(t C($40.75) C(740) + 0.75 x 0.5824 + $3.2355 x 0.752 x 0.0652 li 2 tti li fhe delta-gamma approximation is signifcantly closer to the trtle option price at $40.75 t,, than is the dlta approximation. y Similarly, for a stock price decline to $39.25,the true option price is $2.3622.The t! i(' delta approximation gives

NG

simply the average of LS:) and A(u%+/,), or (usingequation 13.3)

E

'

DELTA-HEDGI

Using equation

Cst

=

)+

6

A(&) + A(&+?,)

2

(13.4),we can rewrite this to express A(&+/,) in terms of r(&): Cstwllt

gi.

=

C'tj

+ E7A(q%)+

l a j.6 P(u%)

(13.5)

..'

.

q. . j: E .

:

:.

k..)

.

E E .

f'' C($40) C($39.25) j 'l tiy ) The delta-gamma approximation gives ,y).

=

). iy j'!

' iltii 'j

.

c($39.25) C($40) =

-

0.75 x 0.5824

-

0.75 x 0.5824 +

=

$2.3436

prediction.

1

0.752 x 0.0652 :5- x

=

$2.3619

-

. approximation is more accurate. )i Again, the delta-gamma

Delta-Gamma Approximations We now repeat the previous arguments using algebra. For a move in the stock price. we lnow that the rate at which delta chlnges is given by gamma. Thus, if over a time interval of length /? the stock price change is 'Gsmall''

e

=

-,+/,

The gamma correction is independent of the direction of the change in the stock price 62, which is always positive. When the stock price goes because gamma is multiplied by change predicts little call price, and we have to add something alne the in too a up, delta the stock price prediction. When the goes down, delta alone predicts too much to correct and the option plice, again have to add something to correct the of a decrease in we

&

-

The new predicted call plice is not perfect because gamma chgnges as the stock price changes. We could add a tel'm correcting for the change irt gamma, and a term correcting forthe change in the change in gamma, and so forth, but the gamma correction alone dramatically improves the accuracy of the approximation. You might recognize equation (13.5)as a second order (becauseit uses the frst and second derivatives, delta' for the change in the call price. Taylor series and gnmma) Taylor series (?pp?'tW???t#ft??? Appendix discussed in I3.A. approximations are Figure 13.3 shows the result of approximating the option plice using the delta and delta-gnmma approximations. The delta approximation is a straight line tangent to the option ptice culwe and is always below the option plice culwe. Because of this, the delta approximation understates the option price, whether the stock price lises or falls. The delta-gamma approximation uses the squared stock plice change, which generates a curve more closely approximating the option price curve.

then gamma is the change in delta per dollar of stock price change, or r(&)

=

Lvt-hl

-

A('%)

Rewriting this expression, delta will change by approximately the magnitude of the price change, 6, times gamma, cr: A(u%+/,) = A(.%) + 6P(u%)

lf the rate at which delta changes is constant (meaningthat gamma is constant), this calculation is exact. How does equation (13.3)help us compute the option price change? 1-fthe stock price changes by EF, we can compute the option price change if we know the average delta over the range 'St+;, to St lf r is approximately constant, the average delta is .

Theta: Accounting for Time 'Fhe preceding calculations measured the option lisk that arises from plice changes alone. Of course, as te price changes, time is passing and the option is approaching maturity. The option's theta 0) measures the option's change in price due to time passing, holding the stock price lixed. For a period o length /?, the change in the option price will be 011. For example, consider the 91-day option in Section 11 3 ad consider the effect of a day passing, with no change in the stock price. Since the variables in the option pricing formula are expressed as annual values, a one-unit change in F t Since h is 1/365, the implied daily option price change is implies a 0 of .

-

-6.33251.

-6.33251/365

-0.01735.

=

$

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The calculations in Tables 13.2 and 13.3 use actual option price changes over 1 day, not approximations. However, by using equation (13.6) to approximate the change in the option price we can better understand the profit results in those tables. Theviue of the market-maker's investmeptvlong deltashares and shortacall-i.s the value of delta shares of stock less the value of the call, or

(5)

Lt St

t)

(13.6)

The experiment re:ected in this equation is t periods until expiration and a stock price of St, we want to express the new option price as a function of the old plice and the delta, gnmma, and theta evaluated at the original price and time to expiration. This formula is quite accurate for relatively smil changes in time and in the stock price. -

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Now recall equation Substimting equation Ltst-l'

Cstjj

-

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changein value of stock

-

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= Example 15.2 Table 13.4 shows the results from using equation (13.6)to predict the option price change when prices move by $0.75 (6 = +0.75). ln this exnmple, Note that in each case the formula L = 0.5824, In = 0.062, and ho = predicted the plice the change, is close to the actual price. but slightly overstates %

CSt)

-

Suppose tat over the time interval h, the stock price changes from St to St+. The change in the v/ue of the portfolio is te change in the value of the stock and option POS it.iOIIS, less interest expense:

this: Starting with an option with F

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1 2 F'r + 0th + 2

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-

-

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)q

(13.7)

'l'his expression gives us the prot of the market-maker when the stock price changes by E over an intelwal of length h. lt is quite important and variants of it will appear later in the book. On the right-hand side of equation (13.7)we see the effects of grlmma, theta, and interest:

k. MARKET-MAKING

428

AND DELTA-HEDGI

NG

TH E BLACK-SCH -1czrJ.

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Since the gamma of the 1t Gamma: The effect of gamma is measured by 2 writing will market-maker lose positive, the call the by call is money in proportion The larger the stock the of the stock price change. move, te greater is the to square loss. 2. Theta: The effect of theta is measured by the option writer benefits from theta.

Since theta for the call is negative,

-0th.

If

ty

length 11is

is measured annually, then a one-standard-deviation Therefore a squared one-standard-deviation CstX-h.

2

E7

Market-maker profit

(Al St Cstjj. The option writer has 3. lnterest cost: lnterest is measured by investment because shares Lt are more expensive than one option. Hence, a net interest is a net cost. -

Since 0 is negative, time decay beneits the market-maker, whereas interst and gamma work against the market-maker. Let's examine how 6, the change in the stock plice, enters equation (13.7). Note first that because we have delta-hedged, E itself does not affect market-maker prot. 62. Consequently, However, prolit does depend on the sqttared change in the stockplice, as w saw in Table 13.2, it is the magnimde-not the direction of the stock price movethat determines prost. Table 13.5 calculates equation (13.7)for various moves in te stock pzce. Because equation (13.7) depends only on the squared stock price move, the calculation is the same for moves up and down. If the stock price moves $0.6281,equation (13.7)is exactly zero. We have already seen in Table 13.3 that the market-maker approximately breaks even for a one-standarddeviation move in the stock. Here we at the same result. Let's explore this idea further. '

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.. .

.

'.

.

-.

.

:: E' .

:

.

yyy . .,. y. ;''.... ''i ' i..' r ..pjyri .y 1iiIii!k.. yt... qri . ..j ... )(k))).. '.'. .r. yjjjjjri, . '! (' .. . E..:.) . ..... .... . . . . ... .

..

j.

. --

.

-.

-

. .

. .. .. . . . . . . . . . . . . . .

.

.

-

. . ..

..

.

-

.

...

-.

-.

--

.

-y

j'. ) j -

..

.

. . - -. .

-

. . .. . .

-

$

429

=

ty

2

2

h /1

we can rewrite equation

1 2

-c -

=

2

move over a period of move is

(13.8)

(13.7)as

St2 L + 0t'+ r (A?,%

-

Cstl

/7

(13.9)

This expression gives us market-maker pront when the stock moves one standard deviacs-h = $0.6281.From Table tion. As an example, let 11 1/365 and o, 0.3. Then equation stock plice exactly zero! (Prblem 13.13 asks 13.5, with this (13.9)is move accident equation that is verify this.) lt is not an (13.9) zero for thij price move. you to Section 13.5. explain tis result in We In Table 13.5, the loss from a $ 1 move is substantially larger in absolpte valpe than gain from no move. However, small moves are more probable tn bt: moves. 1.f the we think of returns as being approximately nonnally distributed, then siok pricemoves greter than one standard deviation occur about one-third of the time. The market-maker thus expects to make small profits about two-thirds of the time and larger losses about one-third of the time. On average, tllp market-maker will break even. =

=

13.5 THE BLACK-SCHOLES

ANALYSIS

We have discussed how a market-maker can measure and manage the risk of a portfoli6 containing options. What is the link to pricing an option?

The Black-scholes Argument

;'.'i(' ',jjyyy' ..' .,,,' ,,,jj' .,,' -jjjjjjjj;j'; --.----' -'' -'-'.' --' y'j'.jtIjj;jj;yE j':'r'L' )Fy'. j,yyg,,f j,::yyrjjf jj::rygyf j,yyjgr,f j,::yyrj;f jjyjjy,,f -' j,:ryy:j;f jj:j,,f jj:yyr:,jf jj::yyf

.. . .

62,

Substituting this expression for

-1'11

'

O LES ANALYSIS

'

. .

-

..

-

.

. -

jjg.,jjyy,

jj:y:y:ys jjyyyg,,s

.!jy::.jj:::,j,,,,,jy jjj ,,,,

tjggjjgjj,jjy

jj:y:g:,,, k;yjyj,g

dffqrent-sized stock price moves on the profit of a delta-hedged market-maker.

1.283 1.080 0.470 0.6281 0.7500 1.0000 1.5000

y;;;;;,,, jy;jj.rjj::y

0.000 -0.546

#' a stock moves one standard deviation, then a delta-hedged position will exactty break even, taling into account the cost of funding the position. This dndingis not a coincidence', it reflects the arguments Black and Scholes used to derive te option pricing; formula. Imagin :e for example, that the stock always moves exactly one standard deviation evel'y 'minute; A market-maker hedging every minute will be hedged over an hour or over any period of time. Blackand Scholes argued that the money invested in this hedged position should earn the risk-free rate since the resulting income stmam is risk-free. Equation (13.9) gives us an expression for market-maker proft when the stock moves one standard deviation. Setting this expression to zero gives 1 2 z 1C(u%)) I 0 - 2 St F'; + 0 + LLt't -o'

-

=

-1.970

-6.036

'lThere are 365 x 24 x 60 525, 600 minutes in a year. Thus, if the stock's annual standard deviation is30%, the per-minute standard deviation s 0.3/ 525,600 0.04%, or $0.016 for a $40 stock. =

=

430

% MARKET-MAKING

TH E BLACK-SCHOLES

AND DELTA-HEDGING

f

+ P

=

(13.10)

l'cst)

to characterize the lldzWt??- of :7,7 option. This is l/2c equation Black J??# Scholes II-EW is the The Greeks F', A, and 0 are partial derivatives of the option price. Equation (13.10) equation Black-scholes partial differential equation, orjust Black-scholes well-lnown . . .. ll). We Fill see (s opppse to the Black-scbesfonnula for the price of a Eurpean in-later chatrs that this relationship among the Greeks is d fundamental in valuing risky cash flows qs is c-rr when valuing risk-free cash flows. that the underlyquatin (13.10)embodies numerous atsumptions, among thin uzviuend, O))ti0Z ZC the interest iZg ZSSG UOCS JROt itsclf does hot liy a PZJ! t dividcfld: devitio standar over a small rate d vltility are constant, and the stock moves one holdj for calls, pu, Ariierican time itefkl. Wit these assumptions, equation (13.10) will consider in hapter options, Europah options, and most of the exotic variats we j wi 14. With simple modifcations, an equation like (13.10) 1 also hold fof ftionson ('jivtdnd-paylng' stocks, currencies, fumres, bonds, etc. The link between delta-hedging and pricing is one of the most important ideas in financ. -

.''''

.'x

'

.

.

.

Equation (13.10)holds forAmerican options as well as for European options, but it does early-exercise the option. Consider a deep-innot hold at times when it is optimal to the-money American put option and suppose the option should be exercised early. Since = r = 0, and 0 = 0. early exercise is optimal, the option price is K S',hence, A. In this case, equation (13.10)becomes -1,

-

(?' x (-

-

431

In practice. because of ansaction costs, it is expepsive for a market-maler to trade shares for every change in an option delta. Instea, a delta-hedger will wait for the positio to become somewhat unhedged before trading to reestablish delta-neueality. In the binomial model in Chapter l0, and in the preceding discussion, we assumed that market-makers mintain their hedged position and that stock prices move exactly one standard deviation. In real life the stock price will rarely move exactly one standard deviation over yhe course of a day. What does the market-maker lose by hedging less

equently?

Boyle and Emanuel (1980)considered a market-maker who delya-hedges at set intervals, rather than evel'y time the stock price changes. Let xi denote the number of standard deviations the stock price moves-we can think of xi as being drawn randomly from the standard normal distribution. Also 1et Rvt. denote the peliod-/ ret'urn to a deltahedged market-maker who. as in our eadier example, has wlitten a cll. Bo#le and Emanuel show that this ret'urn can be written as5

1 2 2 1)/7 S tr In xi2 (13. 11) 2 where r is the option's gamma and, is the time interval between hedge readjustments. 1 is 2', hence, From Boyle and Emanuel, the variance of x/ Rki

=

-

-

-

MaR,il

Delta-l-ledoinog of American Options

1) x 5'/1 +

$

What ls the Advantage to Frequent Re-l-ledgm'g?

lf we divide by 11and rearrange terms, we get 1 2 a St rf + t'St -t7' 2

ANALYSIS

1 x 0 + 0 2

=

1,

x CK

-

&)

Note that -rSt appears on oth sides of the equation. Thus, we can rewrite the equation

2

2

(5'c. r//)

2

(13.12)

=

-

1 (S ac a r/365)2 2

: The dally remrfl of the market-maker who hedges hourly is the sum of the hourly returns. Assuming for the sale of simplicity that S and r do not change much, that variance is

as

var Since this equation is false, something is wrong. We began by assuming that the put early-exercised. From the discussion of early was so far in-the-money that it should be received on the strike exceeds the loss interest that exercise in Chapter 10, this means should be exercised but you own it, option the if Thus, of the implicit call option. lose interest on the sfn'ke you are not then exercise it, you delta-hedge it, and do not option and delta-hedged, and the owner the written have receiving. Similarly, if you of rK. arbitrage earning prct does not exercise, then you are where early exercise is not optimal. region valid in only Thus, equation (13.10)is a is irrational and there behavior then exercised, exercised is but not lf an option should bq risk-free and thus no reason the should position rate delta-hedged earn is no reason why a hold. should that equation (13.10)

1 2

We assume-as in the binomial model-that the stockreturn is uncorrelated across time, so that xi is uncorrelated across time. N ow l e t's compare hedging once a day against hedging hourly (supposetrading occurs around the clock). The daily variance of the return arned by the market-maker who hedges once a day is given by Var(.RI/365,l)

=

-

24

24

J''l i= 1

R;,,i

=

1

J'''l2 g.ao. ar'/(24x 365)j -

2

izzzz 1

1 = 24 x Var(Alp65, I ) -

Thus, by hedging hourly instead of dily reduced by a factor of 24.

the market-maker's

serhisexpression can be derived by assuming that the stock price variance /,, and subtracting equation (13.9)from (13.7). vc

move,

6,

total return variance is

is nonnally distributed with

% MARKET-MAKING

432

Arqo

TH E

DELTA-HEDGING

Whateverthe hedgingintelwal, abouttwo-thirds Hereis teinmitionforthisresult. of the price moves will be less than a single standard deviation, whereas one-third will be greater. Frequent re-hedging does not avoid these large or small moves,since they can occur over any intervi. However, frequent hedging does permit better averaging of the effects of these moves. Whether you hedge once a day or once an hour, the typici stock price move you encounter will likely not be close to one standard deviation. However, if you hedge every hour, over the course of a day you will have 24 moves and 24 opportunities to re-hedge. The average move over this period is likelier to be close to gins from small moves and losses f'rom large moves witl one standard deviation. tnd to average over the course of a day. ln effect, the more f'requent hedger benests f'romdiversiEcation over time.6 'l'he

''

j'

q Example 15.3. Using Boyle and Emanuel's formulas to study the market-maker (4.1 ' tty problem in Section 13.3, the standard deviation of profit is abotg $0.075for a market) mafter who hedges hourly. Since hedging errors are independent from hour to hour, the j.. daily standard deviation for hourly hedger would be $0.075 VV $0.37. If the x an g) 'lrjt.j mr ket-maker were to hedge only daily as in otlr example, the daily standard deviation %. k' would be about $1.82. '

'

f.

..

E

E

=

E

.

As you would expect, the mean return on a delta-hedged position is zero, even if

the hedge is not frequently readjusted.

Delta-lqedglg

in Practice

The Black-scholes analysis outlined here is the linchpin of modern optiop pricing theory alld practice. Market-makers use equation (13.10)to price options, subject to qualiscations mentioned above. . We have seen, howeker, that delta-hedging does not eliminate lisk. One problem, wllich we emphasized above, is that a delta-hedged portfolio wi th negat ive amma can sustain losses due to large moves in the price of the underlying asset. Consequently, a delta-hedging market-maker needs to wort'y about gamma. Anoter problem, discussed in the box on page 434, is that firms can unexpectedly change their dividend payments. There are at least four ways a market-maker can try to reduce the risk of exeme price moves. Note that some of these seategies require the market-maker to acquire :

6-fhis resembles the problem faced by an insurance company. lf the company insures one large asset, the standard deviation of the loss is /1 greater than if it insures ?; small assets, with the same total insured value in each case. Similarly, we can view the return over each hedging interval as being an independent draw from a p'robabilit'ydistribution.

BLACK-SCH

O LE5 ANALYSIS

$

433

specific option positions, which means that the market-maker may have to pay the bidask spread. Since the bid-ask spread is revenue for the market-maker, paying the spread is undesirable. Firstjust as market-makers can adopt a delta-nutral position, they can also adopt position. This position cnnnot be achieved with the stock alone, since gnmma-neutral a the gamma of the stock is zero. Thus, to be gnmma-neutral the market-maker must buy or sell options so as to offsetthe gamma of the existingposition.? We provide an example of gnmma-neutrality below. Second, in a related strategy, market-makers can use static option replication, a strategy in which options are used to hedge options. In our delta-hedging example, the market-maker might not be able to buy an exactly ofsetting call option to hedge the written catl, but by selectively setting the bid and ask prices for related options, might be able to acquire an opion position requiring only infrequent rebalancing. To take a simple example, if the market-maker were able to buy a put with the same strike price and maturity as the written call (e.g.,by setting the bid price to attract any sellf of that op:on), then by buying 100 shares to offset the risk of the position, tlze inarket-maker would have used put-call parity to create a hedge that is both gamma- arld deltauneual for the life of the options. Third, a market-maker can buy out-of-the-money options as instlranck In our example of delta-neual hedging of a written option, the market-malcer could buy a deep-out-of-the-money put and a deep-out-of-the-money call. The two options would be relatively inexpensive and would protect against large moves in the stock. One problem with this solution is tat, since option positions in the aggregate sum to zerd, the market-maling community as a whole can buy protective options only if investors in the aggregate are witling to sell them. hwestors, however, are usually thought to be insurance buyers rather than inslzrance sellers. Fourth, a market-maker can create a financial product by selling tlle hedging error, by Carr and Madan (1998). For exnmple, to hedge a negativ-gnmma, discussed as delta-neutral position, the market-maker would make a payment to a counterparty if the stockmakes a small move in either direction, and receive payment from the counterpm't)/ if the tock makes a large move in either direction. This is effectively a variance swap, which we will discuss in more detail in Chapter 23. The point is that the market-maker can potentially solve the delta-hedging problem by creating a new product.

Gamma-Neutrali'ty Let's explore gnmma-hedging in more depth. Suppose we wish to both delta-hedge and gamma-hedge the written option described in Table 13.1. We cnnnot do this using just

RMarket-makers sometimes buy and hold over-the-counter options issued by a firm. For example, in the late 1990s some finns sold put options on their own stock. These options were reportedly held by dealers and delta-hedged. One possible motivation for dealers to undertake such a transaction would be to acquire positive gnmma. See McDonald (2004)for a discussion.

%

434

MARKET-MAKI

D ELTA-HEDG

NG AN D

TH E BLACK-SCHO

IN G

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options forevery zm-strike 3-month Thus, we need to buy 1.2408 of the 4s-strikezmonth position are in the last column of from resulting the option we have sold. The Greeks 17.49 shares of stock to be both buy need 1749, we to Table 13.6. Since delta is delta- and gamma-hedged. Figure 13.4 compares this delta- and gamma-hedged position to the delta-hedged delta-hedgedposition position, discussed earlier, in which the same eall was written. The position gamma-hedged The losses. always cause has the problem that large moves stockprice increases. the if male and money can loses less if there is a large move down, positive vega. Why Moreover, as Table 13.6 shows, the gamma-hedged position has a ltot gamma-hedge? would anyone

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44

46

48

50

436

%.MARKET-MAKING

MARKET-MAKI

AND DELTA-HEDGING

ganuna-hrdging requires the use of There are two reasons. First, as noted already, the required option position obtain additional options. The market-maker will have to example, a11profits enrned this In spread. bid-ask from anothermarket-maker, paying the who sells the 45-stIike market-maker zjo-strike the call will go to pay from writing the with many options bought call used in gamma-hedging. In a large portfolio, however, the net exposure would not gamma-hedging and sold, naturally offsetting one another, require many option transactions. and calls, then in The second reason is that if end-users on average buy puts gamma,'' end-users to use marketthe aggregate they have positve gamma (the negative aggregatemustthenhave in market-makers the makingparlance). By definition. gamma-hedge, could not a11 market-maker gamma. Thus, while in plinciple any one will not insurance, they buy lf want investors to market-rakers could be gamma-neutral. theaggregatebegnmma-neutral. cannotin market-makers begamma-neutral, and, hence, in ln addition to gamma, other risks, such as vega and rho, can be hedged the same fashion as delta. 'tbuy

13.6 MARKET-MAKING

AS INSURANCE

write options can sustlin The preceding discussion suggests that market-makers who that option marketlarge losses, even if they delta-hedge. This conclusion suggests with making (and delivatives market-maling more generally) has more in common insurance than you might at first think.

lnsurance individuals pay an lnsurance companies attempt to pool diversisable risks. All insured who suffer lotses, those used compensate then prerniums to are insurance premium. The spreads the pain rather thereby Insurance premium. while those without losses lose the losses. of inevitable the bear brunt individuals to than forcing a few unlucky ln the classic model Of insurance company, risks are independkt. Suppose an

an households each Prfivides ZSUFRIRCC fOr a lafge nllmber Of identical in ftre 'Fhe of which has an independent 1% chance in any year of losing $100,000 a = fa''' the is 000. This 000 1% $1, x $100, expected loss for each house is this from collects amount the insurance company that if insurance premium, in the sense annual insurance daims. In general each household, it will on average be able' to pay of doing business, less insurance will be priced to cover the expected loss, plus costs premium. interest earned on tle claims However, evn with diversication, there is a chance that actual insurance which insurance the in household, insured case exceed $1000per in a particular year will promises unless it has access to additional f'unds. company will not be able to fulfill its lisklisk-management insurance company-has product-the a Thus, the seller of a its obligations to management problem of its own-namely to be sure that it can meet conscience management; for of notjust matter obligations is such a customers. Meetng

irlstlrlmce

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NG AS INSU RANCE

$

437

if there is a signifcant chance that the insurance company will be bankrupted by claims, there will be no customers in the srstplace! Insurance companies

have two plimal'y ways to ensure that they meet claims:

1. Capital: Insurance companies hold capital, i.e., buffer fund in case there is an unusually large number of claims. Because of diversifcation, for any given bankruptcy probability insurers can use a smaller reserve fund per insured house as 4henumber of insurd houses groFs. Capital in the form of reserves has traditionglly been an important buffer for insurance companies against unexpectedly large claims. 2. Reinsurance: There is always the possibility of a loss that can exceed any flxed amount of capital. Am insurance company can in ttll'n buy insurarice against large losses, in order to be able to make large payouts if necessaly lnsurance for insurance companies is called reinsurance. lnsurance companies buy insurane from reinsurance firms to over the vent that claims exceed a certai amount. Reinsurance is a put option'. The reinsurance claim gives the insurance company the right to sell to the reinsurers claims that have lost money. Reinsurance does not change the aggrejate need for capital, but it does permit fimher diversihcation.

Market-Makers Now consider again the role of market-makers. Suppose that investors, fearful of a market crash, wish to buy out-of-the-money puts. Out-of-the-money put writers are martiet moves. lt is precisely when large market moves selling insurance against large delta-hedging Qccur that breaks down. Just lile an insurance company, a market-maker requires capital as a cus hion ag ainst losses. Since capital has a cost, markt-mkers may also raise the cost of writien options that require a disproportionately large commitment of capital per dollar of premium. buying out-of-the-money plk options Reinsurance foramarkey-makerwouldentail lisk market-maker, but ultimately if the nancial industry is a: another to to move some be capital in the of losses. writer of insurance, event tere must net of capital and the analogy insurance becomes more obvious The importance to derivatives markets. For example, consider think about weather delivawhen we new sld-resort institutions have hedged Financial operators against warm winters, softtives. (II-iIIk manufacttzrers againstcold andlawn sprinklermanufpcturers against wet summers, Ultimately, the bank must find a counterparty willing to absorb the risk. tf summers. weather insurance, on a global basis they are like tradithe risks think about in you in the United States can be diversified with weather Weather insurlmce. contracts tional capital committed to insurance absorbs the ultimately the global Asia, and conacts in defined are thus the npttlral party to capital markets broadly risk. Global reinsurance lisks. these absorb Some risk, however, is not globally diversisable. Consider writing puts on the S&P 500. lf the U.S. stock market suffers a large decline, other markets around the world are likely to follow. Ultimately, it is capital that safeguards the financial industly

438

%.MARKET-MAKING

PROB LEMS

AND DELTA-HEDGING

Delta-hedging plays a key role, but in the end there is always risk that must be absorbed by capital.

Market-makers buy and sell to satisfy customer demand. A derivatives market-maker resulting position. By desnition, the can use the underlying stock to delta-hedge the depend does position not on the directiol; in which the stock cettlrn on a delta-hedged lnagnitltde the stock prtce move. A position of the Price moves, but it does depend on small stock price moves and loses mak. for es money with zero delta and negative gamma is positive, it makes position of the the gamma money for large stock price moves. l.f small Either loses moves. way, the lelta-hedged money for money for large moves and deviation. standard stock position breaks even if the moves one Using a delta-gamma-theta approximation to charactelize te change in the value . of the delta-hedged portfolio, we can demonseate that there are three factors that explain the profitability of the portfolio. First, gamma measures the tendenc? of the portfolio to become unhedged as the stock price moves. Second, theta measures the gain or loss market-maler will have on the portfolio due to the passage of time alone. Third, te portfolio. interest income or expense on the If we assume that the stock price moves one standard deviation and impose the sattsfies a parcondition that the market-maker earns zero profit, then a fai.r option price relationship, equation theta. and relationship This (13.10), ticular among delta, gamma, is the foundation of the Black-scholes option plicing aniysis ad applies to derivatives in general, notjust to calls and puts. Ultimately, market-maldng is risky and requires capital. If customrs on average buy puts and cftlls, and if we think of options as insurance, then market-makers are in the capital, since if arl exeeme event same business as insuranc companies. This requires will fail. occurs, delta-hedging

FURTHER

439

call with 91 days to expiration. What is delta? lf the 13.1. Suppose you sell a 45-stIike option is on 100 shares, what investment is required for a delta-hedged portfolio? What is your overnight prost if the stock toronow is $39? What if the stock price is $40.502

13.2. Suppose you sell a m-strike put with 91 days to expiration. What is delta? Ifthe option is on 100 shares, what investment is required for a delta-hedged portfolio? What is your overnight prost if the stock price tomorrow is $392 What if it is $40.50? 13.3. Suppose you buy a 40-45 bull spread wit.h 91 days to expiration. 1.fyou deltahedge this position, what investment is required? What is your bvernight prost if the stock toinon'ow is $392 What if the stock is $40.50? 13.4. Suppose you enter into a put ratio spread where you buy a 45-strike put and sell two zm-stlike puts. Ifyou delta-hedge this position, what investment is required? What is your overnight profit if the stock tomorrow is

$40.50?

$39? What if the stock is

13.5. Reproduce the analysis in Table 13.2, assuming that instead of selling sell a 4o-strike put.

a call you

13.6. Reproduce the analysis in Table 13.3, assuming that instead of slling a call you sell a 4o-strike put. 13.7. Consider a zm-strike 180-day call with S $40. Compute a delta-gamma-theta approximation for the value of the call after 1, 5, and 25 days. For each day, consider stock prices of $36 to $44.00 in $0.25 increments and compare the acmal option premium at each stock price with th predicted premium. Where are the two the same? =

13.8. Repeat the previous problem for a 4o-strile lo-day

put.

13.9. Consider a 4o-stlike call with 91 days days to expiration. Graph the results from the following calculations. .

a. Compute the actual price wit.h 90 days to eypiration at $1 intervals from $30 to $50.

READING

The main example in this chapter assumed that the Black-scholes formula provided the correct option price and illusated the behavior of the fonnula, viewed from the perspective of a delta-hedging market-maker. ln Chapters 20 ad 21 we will start by building a model of how stock plices behave, artd see how the Black-scholes fonnula is derived. As in this chater, we willconclude thatequa:on (13.10)iskey to understanding option pricing.

b. Compute the estimated plice with 90 days to expiration using a delta approxlmation. c. Compute the estimated price with 90 days to expiration using a deltagamma applpximation. d. Compute the estimated price with 90 days to expiration using a deltagamma-theta approximation. 13.10. Consider a 4o-strike call with 365 days to expiration. Graph te results from the following calculations.

PROBLEMS ln the following problems assume, unless otherwise stated, tat S 1' 8%, and J 0. =

%.

=

=

$40, o'

=

30%,

a. Compute the actual price with 360 days to expiration at $1 intervals from $30 to $50.

440

%.MARKET-MARING

AND DELTA-HEDGI

b. Compute the estimated price with 360 days to expiration using a delta approximation. expiration using a deltac. Compute the estimated price with 360 days to ganuna approximation. d. Compute the estimated price with 360 days to expiration using a deltagamma-theta approximation.

13.11. Repeat Problem 13.9 for a 91-day 4o-strike put. 13.12. Repeat Problem 13.10 for a 365-day

=

APPENDIX 13.A: TAYLOR APPROXIMATIONS

0.5 and K

=

GREEKS

$

I N TH E BINOM lAL MODEL

441

SERIES

We have seen that the change in the option price can be expressed in terms of delta, gamma, and theta. The resulting expression is really just a partiular approximation to the option price, called a Taylor selies approximation. Let Gx, y) be a function of two valiables. Taylor's tleorem says that the value of the fttnction at the point Gx + 6.v, y + e'y) may be approximated using derivatives of the f'unction, as follows:

zm-strike put.

13.13. Using the parameters in Table 13.1, verify that equation 13.14. Consider a put for which F that equation (13.9)is zero.

13.B:

APPENDIX

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(13.9)is zero.

$45. Compute the Greeks and verify

13'.15.You own one 45-strike call with 180 days to expiration. Compute and graph the l-dayholding period prot if you delta- and gamma-hedge this position using a 4o-strikecall with l80 days to expiration. 13.16. You have sold one 45-strike put with 180 days to expiration. Compute and graph the l-day holdingpeziod prctif you delta- and gamma-hedge this position using the stock and a 4o-strAe call with 180 days to ekpiration. 13.17. You have written a 35.-40.-45 butterlly spread with 91 days to expiration. Compute and graph the l-day holding period prolit if you delta- and gamma-hedgethis position using the stoct and a zm-strike catl with 180 days to expiration. 13.18. Suppose you enter into a put ratio spread where you buy a 45-strike put and sell two 4o-st14ke puts, both with 91 days to expiration. Compute and graph the l-day holding period proit if you delta- and gamma-hedge this position using the stock and a 4o-strike call with 180 days to expiration. 13.19. You have purchased a 4o-strike call with 91 days to expiration. You wish to delta-hedge, but you are also concerned about changes in volatility; thus, you want to vega-hedge your position as well. and vegqa. Compute and graph the l-day holdingperiod pmht if you deltazm-sirike call with 180 days to and stock the using position this a hedge expiration. b. Compute and graph the l-day holding period proft if you delta-, gamma-, and vega-hedge this position using the stock, a zm-stlike call with 180 days to expiration, and a 4s-st1ike put with 365 days to expiration. 13.20. Repeat the previous problem, except that instead of hedging volatility risk, you wish to hedge interest rate risk, i.e., to rho-hedge. In addition to delta-, gamma-, and rho-hedging, can you delta-gamma-rho-vega hedge?

(13.14) The approximation may b extended indenitely, tives. The nt.h term in the expansion is 11

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means take the fth derivative with respr to and the (?) lth derivative with fespect to ;?. Therfaylor slies is useful whe the approximation is reasonably accurate with ew terms. equation Forourpurposes, itis enoughtbnotethatthedeltargnmmaapprofmation, (13.6), looks like a Taylor selies >pproximation that stops with the second derivative. You may wonder, however, *hy there is no second derivative with respect to time. You my also wonder why the approximation stops wit the second delivative. These questions will arise again and be answered in Chapter 20.

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APPENDIX 13.B: GREEKS BINOMIAL MODEL

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The Black-scholes Greeks are obttined by differentiating the option price. However, the general binomial option plice clculation is not afofmulabut an algorithm. How can option Greeks be computed using the binomial ee? We can use some of the relations between delta, gamma, and theta discussed in this chapter to compute the binomial

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oj standard coneacts like these, you obtain a nonstandard or exo (jj,. yjjyys options can prvide precise tailoring of risk exposures, and they permit investment strateg jes (jjauujt or costly to renlize with standard options and securities. In tllis chapter ld nds of exotic options, including Asian, bnrrier, compound, gap, discuss basic some we options. In Chapter 22 we will consider other exotic options. and exchange

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Using the delta-gamma-theta approximatipn, equation price at time lh and node ltds ps

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and at This is an approximatign since we wish to lnow gamma at time 0, not at time h, approximation the binomial steps, the price %. However, even with a kmall number of reasonably well. works With theta we are interested in the pure effeet on the option price of changing time. We can calculate this using delta and gamma. Deine =

-

y. yyg .E. EE.i k.j:..g.yyyj...jy(.g...y..;..gy. .y.jy ...;E.)'. . ..!.E.E;E... .y.y. . ... (g. .jkqjk.gj.j.jy.j jj.q(..... !.ir(.g..(Eyj.jj E.y;yj...qEj.j.jy.g j;i. .;E..iy . ;..!.j.E. !.yE. E.jyq..qyg.! y..y.jr;.yyy.y.yj'. E .(! .. q ( . l'.!.. i.. .;.Eq.yEq.(qE.. ..rEjq.....yg jt:j.jyg;.y..j.q..jgjg.g.q(j..ijj.irg...j;..y.qr:;g..y y. ...fl'.t'.y . . .. :( .

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we have

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at the initial node is computed using the formula

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!; i...;.yj ( (.;('(.((;r.;. . tjr. .jj'. i.y jrj...t-kt'. (k g;r. ));-;-;)j;('. ,LLL)3;;t). ;.;. ..y;ry(.jg jLbbL'.L;(LL:;. ).)jfL)L.-j;Lj;))Lf)));. k. .)'. y'jjjLLLL)jL,, j.jjjjjk)).. j!r. . . i(t)'. ;j;)-. p'-jyr'. j'.trf.. jy .. jr ('jy yj;'p( jyy; .. $jyy(:--)p).-j.t(); y. y(y. i p)t'. lk (;y. pprj.q,jjtr ( ;'..k rj. .y;tj.(y(y4 (j.Tt('t;. kt)'. )t;)LL);jjLLLjj.jjq;L3L;;L'. !(yg j.(-y.yly r(ri(jjj'g .j.r'.y.y(. .. li t..yjjy;()y. j .y'. .yt.)-. t.-'l.; ttjj(-tj)j'i . j...y'. y.j y.1:. ktj'. .gig..t'. ..k .yy.. . .jjjy.. . i '. ))yjt4)j));..'.g.jjrrg'..);.)),)'!j)))jjiykrj)))'. .)))y;;y))'. ..ryr-t;i ! j.yi.r jy)'.... r.yi )j));,'LL)k;'.)j))'. )t'-(j)'. )g'. y-r)('. jj'jj-jjjtj.'.. f-ji -. (1:. E. y( . j tjy . ttj2'rjjjy. . tj. )tr);j)'. .r-'. .4y;.. : r. '.:.! (. '-yy.y';j(.. LLt;. lijy. '.k.jk i ..(;it-'. y't(q..);' p'j('qrj!.(i r .C'll...! )-:yjjj)); ',).. l r('t.)l.l ttp?)).-. !'g !l1ti?):y(k). ; .qr! .I!.;.y. iIlii',l(p'!l )'k( y.q.q..;'g (;; ty'. ft-j.pl'l);h))')'y(t-) i-i(liE..'.; (.yg! q i-.; : y;y'. ( yr(i';! k.j)). (lt! jti 1y qq! ; h'-j pp I.ihtl : y:)'. )-(yp('k('(; q..r'!-kq.. (rzg'iq .q$r(2());)-. ; .( .t'tpr 1tii L''-.i . . . r(kk'. (. lyi !jg.g;r((!.. j;. j!r)'. E lg '; '.jjj'.)'jj;-r.: !t-kj k:ti r. .. . j.@ (( yj.)I ...,)';r.. qi.-.r'q'j.)'' :j.j' j.y'It(,jy pj'(k j,y:-. l j; ..;,,! . i!.llj.jqqjlL,L, tt);)k'. t!:ri.t))'. ljj)-. k( .jj :. ,(!iy'i'. ptt-rp'. 4yt'. (t..i t;(jlti (i.tq j;( p$)yjr). yi g(!j:'y .yyy. .r;kjjp)p)'. tl.tl. t,-.);i)'. jyyijr'y .jl.k .y ; ..jt'. yl)'. 2$t)*. y'i t't.ytdqr-jy . jjt;'. :@;. yl'. ti igp'-j kE t'! y 1*. @ j!y ; .jy..y'i . : g'. ..t..j)'. ji-; jy1*j),jy krry,pj-tpt/rdjtj-jttp ! (-p.y ), . j ! ((jk. tt'I qj;.;..(k. jtr'ti . j.gi ...)rj; . j t.. @ , ;p'. ty t'j . y trt)'. ''.'i''.'.'..'.''l';jijr))k;;ti.pzpr'ltjjttr t))-. yt'p y '. y : y'. : j k j. ! )tt')'t.. j . . ; k ; ( jjty. ( ; lr j-'jj')j, r y)()'. . y'y'.. r . ; tjyjjj,'-;;'jj-jy.. ry!py jj t'. t'. t)))'; tr'. yjy.y yy;( .t r r;j jyyjg'. ( , yyj t k. y ( . j y j. r j, q . )'. r tji kjy );:'--:'yj y.yj. jr j tq.., q ,j yr k j j . . y ryy ))'yytj'((q((jIjq((y,y , . )r('. .. yjjy .. . j. y y jy(('. . y E y y r y r y j . .. )9*. y y '. j tj'. ('j)j . r;.;y, j)t't k'. j yyyy-'. jyy jrtt;.. yj-'-)'. y..;! (.. . .:y j . j.j. ..j('. y. .. ..y..;.t-.@. . .. . ..;'tyyy'. ; .yy.!yy;.rj ygj yj yj .. !. y.(.yy t. (yjjgjy)y.--.jryyjy-,-... y. j'-j))'. r . jjj;)()LL-. . l.. y jj'. ., j'. y l . . . . p. ry ! .. . t,-j;'kj(j. .. )f;. . j ( . : ((.... t)'. ;;1*. k. .jyj'. j jy.;j)'. tjj'. i';k t'; 13.. l .. tjjj,'. ,. jyrt)y-tjj', .qi .t! . q. .'L3Lb. . .; y'..; lr y'. g . j'. )'. r; t. . .:. . )'.; .: . . . , )'. . . yyyjyj'y. y.yyyyyjjjjjyy'. jjkjjjy'. ;.)'. t r. .:f'.t'j j'.. . t:'.. rj .q:(@)t ..LLLLLLL-LL. )'-.... .. jt'. . ...(3kqbjL. :f(;)yt)pj;r.)l)ry)) . yjjjjy'. . ... .. tky'. .. Lj')j';;'j-'L'. . LL;', ... I;j,)r,.;,)yyyk-f.jr, ..tkltiiqk . kf .. )'.. t', yij-j)/)j)'. '.-LLL.--.-LLjLL'.'''..'.'.LL''-'LLL.'--;3.'kLk . ))l)pl g'.. .LL(.,j-L.b.L(..L..L,.jL-L,.j.;L-.LL):-..(.-.,jLLLL .y'. q('yjyt-,t'. '.. )')yj(ij' r-jjlljxhsialjj-k--rl .ykitj..j..) . ..(j()r.-! j)littqttk j:ltk-kk),k):k-::-j)k-').---;t-tjqj!y)----.)pjp,.. r'. . .ri j ..-,;,,,.-j,i-.LLL--q, .. i.j-r..gji.(.!,2-.). '' .'.' .. ..ti-... .i .@q'. -),)...l.:@(:T-''9 p) qjjy. ..)'. Fjktr)L)4k.-...... .!. '-ljyjgkllti ))t-jj )'. '1l )'p#tkt))-;)).)))k;;)))!.)())jy.,..t.--))...-.?.).. t )j!---?i-ik;.t r--.;.t;., !- ())q), )!l)i.'..-.,..t..))--.h)r t-..#yt-;))yt).' .;qLLj)LLLL)LLLLjLj-.;Lj-jL))-.3tjL(.b ...L..;jLL))Lj. )y...LL-.;,L.;)..L (t,L-)L(,,LL--,; t(,--tytt)),j:):)... .. t! .t)y)...: .

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.'.'-.' .' . . -

r' )' :'(.' (' ''i' dl:s!,,-f !' y' tErll'.. E.' jtlpf jtf (:144!(/)* (jf jf ''qqf ;'q' )' :'. k:' kf ' ' :' ; E : ' ' j! )' 1* j)f (' ''.' q'r!!f 1.)' ('E((E EE E t;f (::;!r:' i!'l:t::17;'.-' 'ri,rlkf r' '1i1t2-:21I1'. '; EE 'q' 'iE...'..:.,'.',........').,..'.''.,(.y.y.).,qy.,j.,....,.'..,,,,.yr,:.........r.(yg...y.j.).,yj.:..'.,,(-.j.;,;yjr..,,.pyy..tyjj, !IiI:qp'.' )'. i ;( !tlEjjg ' r'('7* t'lllttitif ' '(E'Eq'E.'E 'EE.('((.' (.((.i :'(i.EE'' q(E.'. E ( .'(.''(;'i'''E. q('q'q''''('!' .' j. q'.!y. ''(

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!

444

% Exo-rlc

OPTIONS:

AslAN

l

opiions. 'I'he challenge is to invent new options It is not hard to invent new kinds of did in the preceding example) n that that are potentially attractive to buyers (whichwe without too much difculty. In Chapters 10 and 13 we saw can be p riced and hedged That analysis 1edus to see how how a market-maker can delta-hedge an option position. of synthetically manufactudng the option. the priee of an option is equivalent to tl cost there is a certain relationship among the In particular, an option is fairly priced when Greeks of the option. and delta-hedged in the same Options with exotic feamres can generally be priced options.' As a consequence, exotic derivative products aze quite comway as ordinary well understood. Iiz - in practice and the technology for pricing and hedging them is mon is an anachronism. fact, since many such options are in common use, the term We will coniinue to use it, however. mathematical details of particular The goal in this chapter is not to master the understanding of the trad-offs in dsign and products, but rater to gain an intuitive of the formulas appearin the chapter apedixt Priqing. Consequently, most ordinary ojtions in minor Since exotic options ar oen constructed by tweaking exotic useful as benchmarks for exotics. To understand ways, ordinary options are options you should ask questions like these: exotic compare to that of a standard option? * How does the payof'f of the approximated by some portf.olio of other options? * Can the exotic option be exppnsive relative to standard options? Understanding . Is the exotic option cheap or understanding its pricing and use. the economics of the option is a critical step in the use of the exotic option? * What is the rationale for option be hedged? An option may be desirable to a o How easily can the exotic unless the risk arising from market-maliing can be customer, but it will not be sold controlled. ,

op-rloxs k

445

There are many practical applications in which we average prices. In addition to the 5rm cares about the average exchange rate (aswith XYZ), averaging is where cases also used when a single price at a point in time migt be subject to manipulatin or price swings induced by thin markets. Bonds convertible into stock, for example, often base te terms of conversion on the average stock plice over a 20-day period at the nd of the ond's life. Settlement based on the average is called an Asian tail, since the averaging occurs pnly at the termination of the contract. As we will see, Asian options are worth less at issuance than otherwise equivalent ordinary options. The reason is that the averaged price of the underlying asset is less volatile than the asset price itselt and an optio on a lower volatitity asset is worth less. '

'Iexotic''

'

14.2

ASIAN

OPTIONS

price over some period of An Asian option has a payoff that is based on the average which means that the option, path-dependent time. An Asian option is an example of a stock arrived at its the which by the path value of the option at expiration depends upon XYZ'S problem. hedging linal price.z Such an option has the potential to solve

diflicult to hedge even thaugh l However, as we will see in Chapter 22, there are options that are quite they are easy to price. pricing model. ln the binomial model 2Youcan think of path dependence in the context of a bnomial stock price moves paths--occurring irl a and down of ltdlt series up and dtttt m.e a of Chapter 10. stock price. Thus, both yeld the sarne payoff for a fnal the which lead order same different butHowev'er, to path-dependent option, these two pats w u1d yield different nal with a European option. intermediate stock pfices were different. optionpayoffs because the

XYZ'S

Hedging Problem

Let's think more about XYZ'S currency hedging problem. Suppose that XYZ has a monthty euro inqow of C100m, reiecting revenue from selling products in Europe. Its costs, however, are primarily sxedin dollars. Let xi dnote the dollar ptice of a euro in month i. At the end of te year, the converted amount in dollars is 12

C100m x

xie

r( 12-f

)/ l 2

i= 1

We have numerous strategies available for hedging the end-of-year cash flow. Here are a few obvious ones:

* Strip of forward contracts: Sell euro forward contracts maturing each month over the year. The premium of this strategy is zero. * Euro swap: Swap euros for dollars. We saw i.n Chapter 8 tat, except for the timing of cash flows, a swap produces te same result as hedging with the strip of forwards. A swap also has a zero premium.

i

@ Strip of puts: Buy 12 put options on C100m, each maturing atthe end of a different month. The cost is the 12 option premiums. As we saw in Chapter 2, the difference between the fonvard and option strategies the ability is to profit from a euro appreciation, but we pay a premium for the possibility enrning prost. You can probably think of other strteis that of as well. The idea of an Asian option stems from expression (14.1): What we really care about is the future value of the sunl of the converted cash Qows. This in t'urn depends on the sum of the month-end exchange rates. l.f for simplicity we ignore interest, what we are trying to hedg is 12

12

-q i= 1

=

12 x

Ef-1x 12

(14.2)

The expression in parentheses is the month-end arithmetic average exchange rate, which motivates the idea of an option on the average.

446

% Exo-rlc

AslAlq

Op-rlorqs: I

Options on the Average As a logical matter there am ight basic kinds of Asian options, depending upon Fhether ariihmetic the otion ij a put or a call, whether the average is computed as a geometric or underlying ' average, and whetherthe average assetpdce is psed i place of the pric rf the. . alternatives. asset or the stlike price. Here are details about some of these lt is most common in practice to defne the average as arithmetic Suppose we record the stoek price every /1 periods from time 0 cvcmkc. an is delined as to F; there are then N = F/ periods. The arithmetic average

The definition

of the average

A(F)

=

-

N

E

..

$55 + $72 + $61 + $85 )( sg ,5() ti4 4 j;E' )itT he g eometric average is illtlj. . $55 x $72 x $6l x $85)0*25 $6,7.315 'k, g:

=

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averagein Chapier

The payoffat mattlrit.y average is used as the asset price or the strllte of the underlying asset plice the either stock price as computed using the average can be option is called an price, the used is asset the When the pric. strike the as average or as option is called the strlk' plice, the used is the qs e option. price When average average the geomeeic based options of variants the four optiolt. strike Here on are average an

average.

Geometric averag price call

=

Geometric average plice put

=

Geometric average strike call

=

Geometric average strike put

=

maxlo, G(F)

mlkxgo,K maxlo, Sr

-

f-j

G(F)j

-

-

maxlo, G(F)

G(F)) -

5'w)

(14.6) (14.7) (14.8)

strike'' refer to whether the average is used price'' and The tenus in place of the asset plice or the strike price. In each case the average could also be computed as an arithmetic average, giving us our eight basic linds of Asian options. The following example illustrates the difference between an arithmetic and geometric average. itaverage

taverage

SBecause the sum of lognormal variables is not lognormally distributed, there are no smple pricing formulas for options based on the arithmetic average.

simple) (relatively

options

formulas forpricingEropean

19.

Table 14.1 shpws values of geometric average plice calls and puts. If the number of avrages, N, is one, then the average is the Enal stock price. Irl that case the average price call is an ordinary call. Intuitively, averaging reduces the volatility of G(F) relative to the volatility of the price stock at expiration, Sp, and thus we should expect the value of an average price

There are easy pricing fonnulas for options based on the geometric average (see the chapter appendix).

Whether the

has

Comparing Asian Options

(14.4)

'v

k

=

based on the geometdc average. We further discuss options based on the arithmetic

i= 1

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(

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incpnvenient.3 It While arithmetic averages are typically used, they are mathematically gonietric tkv/rtzqj?c the in ue practice, to is omputationally easier, but less commori dehned as stockprice, which is G(F)

.

EE

(14.3)

Silt

447

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OPTIONS:

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the ayerage. This is option to decrease with the number of stock prices used to compute price ojtion as evident in Table 14.1, which shows the decline in value of the average the frequency of averaging increases. of Table 14.1 also shows that, in contrast to average price calls, the price an average of stock prices The average strike call increses with the number of averaging periods. time F, Sp. lf price at between times 0 and F is positively correlated with the stock the makes averaging average G(T) is high, Sp is likely to be high as well. More frequent correlation between Sv and G(F). To the reduces valuable because it option strike more if the average is computed only using the final see this pattern, consider what happens stock price. The value of the call is

maxp,

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t.t'.)')' tf'')')) -'.-'.i:iiL'-'' t..,).).t).r,y.iy.qrjp,r,#,..('...... . . t !. 1Ii1i11-2l. ! ).t...t.f))..tl''r .1IE::.:!;; lliiii!!. :i;,r.' . !f!i;k Il'... ''r''''lr''':li Il..'' ll....,,. .:If::' lli(k! !i::r! ;i! s l,..'':'l a .1It:E. 11:::::.,. a l'''''iiq . a.,1k::: es e l...'' a.,Ik::. klE!s!,,r',....:ll .11t::. !.;-... .. '.'. '.'L-6''';''tib--.. ' ...!'.. '.' ii @))-' . . . . . ..... . . . . . . . .. for XYZ.The price in the second row is the sum of premiums for put.sexpiring after 1 month, 2 months, and so forth, out to 12 months. The first, third, and fourth row premiums are calculated assuming 1 year to by 12. Msumes the maturity, and then multiplied current exchange rate is $0.9/6, option strikes are 0.9, rs 6%, rq 3%, and dollar/euro volatility ij 10%. ..' '' '''f -' --.' .---' -..'

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profits are reduced if the euro euros and its costs are fxed in dollars, received for of number dollars a euro is lower. We could depreciates-that is, if the exchange rate received. construct an Asian put option that puts a Eoor, K, on the average would be option The per euro payoff of this

If XYZ receives

max 0, K

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1 12

12

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zero the average strik option increases.

An Asian Solution for X'YZ

E'

E

'.

= S., and the value of the option is If only one stock price observation is used, G(T) is reduced and the value of con-elation averaging the for sure. With more frequent

off when When would an average strike option make sense? Such an option pays and the option of the the life price the asset over average there is a difference between where simation insurance could for in used option a be asset price at expiration. Such an accumulated entire sold the and time then of period we accumulated an asset over a position at one price.

:

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Gepmetric average plice put Arithmetic average price put

..'...

rliiitifi

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0.2753 0.2178

0.1796 0.1764

the lz-month average. The Asian put is cheaper since there will be situations in which some of the individual puts are valuable (for example, if the exchange rate takes a big swing in one month that is reversed subsequently), but the Asian put does not pay off.

The geometric option hedges less well than the arithmetic option since the quantity being hedged (equation14.1) is an arithmetic, not a geometric, average. Finally, be aware that this example ignores several subtleties. The option strikes, for example, might be made to val'y with the forwqrd curve for the exchange rate. The effect of interest in equation (14.1)could also be taken into account.

(14.9)

/=1

of $0.90per euro, we For example, if we wanted to guarantee an average exchange rate than ='z: less that, we would be paid wouldset k $0.9. lf the average exchange rate was repatriate e1.2bover the course thedifference between $0.9and the average. Since we 61.2b. of a yefmFe would buy contracts covering Do yourecqgnize the kind of option described by equation (14.9)? The average arithmeiic, the average is used in place of the asset price, and it is a put. Hence, it is is A,5-ft'Ill pttt. an al-ithltletic t'Ivc/wtc p-ice premiums for There are other hedging strategies XYZ could use. Table 14.2 lists expensive oplion. the year-end most is exping at several altenaatives. The single put could be quite exchange yegr-end rate the As discussed earlier, it has basis risk because the risk: basis less signfcantly strategies have different from the average. Two other of strip Asian puts The arithmetic put. and the strip of European puts expiring monthly whereas the Asian option protects month-by-month. exchange 1ow against rates protects

14.3 BARRIER

OPTIONS

A barrier option is an option with a payoff depending upon whether, over the life of the option, the price of the underlying asset reaches a specifed level, called the balwier. Barrier puts and calls either come into existence or go out of existence the first time the asset price reaches the barrier. If tey are in existence at expiration,' thy are equivalent to ordinary puts and calls. It can be tricky to desne what it means for the stock price to reach a barfier. See the box on page 450 for a discussion. Since barrier ppts and calls never pay more than standard puts and calls, they are expensive tan standard puts and calls. Bnrrier options are another example of more no option. path-dependent a Banier options are widely used in practice. One appeal of barrier options may be their lower premiums, although the lower premium of course reQects a lower average payoff at expiration.

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Figure 14.1 illustrates how a barrier option works. The stock price starts at around $ 100, ends at $80, and hits the barrier of $75 abouthalfway throughthe year. lf the option and pay $15($95- $80) at were a 95-strike down-and-in put, the option would lnock in would be worthless at expiration. expiration. lf the optiop were a down-and-out put, it would knock-in at $75 but still be worthless at lf the option were a down-and-in call, it price. strike expiration because the stock price is below the

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Standard call

Since these option prenziums cannot be negative, tlzis equation demonstrates directly that barrier options have lower premiums than standard options.

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Chapters 9 and 11.

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Suppose astockwith aprice of $100willpay a $5 dividendin 91 days p Example 14.2 (rI 0.249). An option with a strike price of $90 will expire in 152 days (F 0.416). y(j 0.08. The vglue of a European call on the stock is Assume o. 0.3 and r

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Figure 14.3 shows a gap call option with payoff S 90 when S >. 100. The gap in payoff the occurs when the option payoffjumps from $0 to $10 as a result of the stock changing from $99.99to $100.01. plice S when S < 100. This option Figure 14.4 depicts a gap put that pays 90 require, option be strucmred that for some stock prices, a to demonsates can a gap Figure 14.4 wit holder expiration. You should option compare from at the payout Figure 4.12-the gap put looks vel'y much lile a paylater strategy.6 Note that the owner of ihe put in Figure 14.4 is required to exercise the option when S < 100.7 The pricing formula for a gap call, which pays S- A-1when S > Kz, is obtained by modifcation of the Black-scholes formula. Let #1 be the strike price ttheprice simple a

value of an American call is computed as the present value of equation (14.14),with (0.416-0.2..1.9))ucza price for the compound option equal to 5 90(1 e exercise the i.;.g tt.. 3.805, and time to maturity 0.249 for the compound option and 0.416 for te underlying oPtion. The price of the compound option is

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Compound options provide yet another variation on possible currency-hedging strategies. lnstead of buying a 6-mont put option on the euro, we could buy a call option on wait arld a put option. In effect, this compound option is giving us the opportunity to see what happens. whether to buy the put option. Here is uPP ose that after 3 months we will decide 3-month one way to structure such a transaction. We could figure out what premium a BlackThe still at 0.9. put with a stlike of $0.9 would have, if the exchange rate were of strike $0.9would Scholes formula tells us that a 3-month at-the-money option with a have a premium of $0.0146.(This value compares with the premium of $0.0188for the 6-month option from Table 14.3.) Now we can use the compound pricing formula to plice a call on a put, setting the stlike to equal $0.0146.The price of this compound call is $0.0093. So by paying less than two-thirds the premium of te 6-month at-the-money optin, tve can buy an option that permits us to pay $0.0146for a 3-month option. By seleting this strike, we have constructed the option so that we will exercise it if the exchange rate is below 0.9. If the exchange l-ate goes up, we will not exercise the option and save the premium. lf te exchange rate goes down, we will acquire an in-the-money option for the plice of an at-the-money option. Mariy other structures are possible.

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more fully discuss' gap and related options in Chapter 22.

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Suppose an exchage call maturing r periods from today provides the right to obtain l unit of risky ajset 1 i ekchange for l unit of risky asset 2. (We could tlzink of this as, for example, the right to obtain the Nikkei index by giving up the S&P 500.) Let St be the price of risky asset 1 and Kt the price of lisky 2 at time r, with dividend asset yields 8s and &K and volatilities o's and c K Let p denote the correlation between the k contlpuously compounded remrns on the twoassets. he payo g to tus oyttjon is K;) mllxto, The formula for the price of an exchange option (seeMargrabe, 1978) is .

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Note that for a givn strike, #1 we ean always lind a tligger, Kz, to make the option premium zero. Thus, gap options permit us to accomplish something similar to the paylater sttategy discjsed in Section 4.4. ,

the option holder pays at expiration to acquire the stock) and Kz the paymet trigger (theprice at which payment on the option is triggered). The fonupla is ihen KLe-rrxdzj (14.15) Se -&1'Nd l ) #2, o-, n F, t) CS, nse-'l-jltqe-rl-) + 1o-2r 2 #

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Consider an option to receive (IBM shares by giving up Microsoft 'l' shares. Wecan view this as an IBM call wtth Microjft as the stlike aset. On Kvember lt.q ')ll $27.39.Thus, oneshareof VM 15, 2004, thepriceof IBMwassgs.gzandMicrosoftwas i :j;yj 3.5020 shares of Microsoft. For IBM and ltt had the same dollar value as 95.92/27.39

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Problem 14.19 asks you to think about the circumstances under which XYZ might

hedge currency risk using exchange options.

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,

.

SUMMARY

.

.

j'

-

461

Suppose the option permits exchanging equal values of Microsoft for IBM, based on the November 15 prices. We could then exchange 3.5020 shares of Microsoft for l share of price of a l-year exchange call would be mM.

.

.

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0.20302 + 0.22272

k

= 0. 1694

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=

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SUMMARV

22.27% for Microsoft, with a return s in ce January 2003 had ben 20.30% for IBM and .

An exotc option is created by altering the contract'ual yerms of a standard option. Exotic options permit hedging solutions tailored to specilic problems and specvlation tailored to particular views. Examples of exotic options include the following'.

@Asialt options asset over the tlle underlying average strike

have payoffs that life of the option. asset (an average optiol. Averages

are based on the average price of the underlying The average price can be used in place of either price t7rrft/nl or in place of the strike price (an can be arithmetic or geometlic.

have payoffs that depend upn whether the price of the underlying * Barrier (//7/*J4777, reached has asset a bnrrier over the life of the option. These options can come into when te existence knock-in option or go out of existence kllock-ollt option barrier is reached.

* Compound options are options on options: Put or call options with put or call options as te underlying asset. * Gap optiolls are options where the option payoffjumps at the pricewhere the option comes into the money. * Exchange options are options tat have risky assets as both the underlying asset and the strike asset. It is helpful in analyzing exotic options to compare them to standard options: ln

what ways does an exotic option resemble a standard option? How will its plice compare

;

462

k

Exo-rlc

OpTloxs:

PRo BLEMS

I rnight someone use

to that of an ordinary option? When standard option?

%

463

a. What are the possible geometric and arithmetic averages after 1 year? b. What is the price of an Asian arithmetic average plice call?

the exotic option instead of a

c. What is the price of an Asian geometric average price call? FURTHER

READING

14.6. Let S

ln Chapter 16 we will see some pore appliations of exotic options. ln Chapter 21 we will discuss the underlying logic of pricing exotic options and in Chaptr 22 we will discuss additional exotic options. General bboks covering exotic options include Brifs and Bellala (1998), Haug -(1998), Wihuott (1998), and Zhang (1998). Rubinstein (1991b)discusses exchange options, Rubinstein (1991a)discusses compound options, and Rubinstein lpd Reiner (1991a) discuss bacrier options. PROBLEMS To answer many of these questions you can use the exotic option functions in the spreadsheet accompanying this book.

14.1. Obtain monthly stockplices for 5 years for tllree stocks. Compute the arithmetic and geometric average month-end plice for each stock. Which is greater?

14.2. Suppose you observe the prices (5,4, 5, 6, 5l. What are the arithmetic and geometricaverages? Now you-observe (3,4, 5, 6, 7). What are the two averages?

What happens to the ditference between the two measures of the average as the standarddeviation of the observations increases?

1. 14.3. Suppose that S 0.30, J 0, and F 0.08, o' $100, r $100, K Constnlct a standard two-period binornial stock price tz'ee using the method in Chapter 10. =

=

=

=

=

=

a. Consider stock price averages computed by averaging the 6-month and l-year prices. Whatare the possible alithmetic and geomeic averages after 1 year? b. Construct a binomial tree for the average. How many nodes does it have after 1 year? llint'. While the moves I4# and #If give the same year-l price, they do not give the same avergge in year 1.) c. What is the price of an Asian arithmetic average price call? d. What is the price of an Asian geometric average price call? 14.4. Using the information in the previous problem, compute th pries of a. An Asian arithmetic average strike call. b. An Asian geometric average strike call. binomial tree. Assume 14.5. Repeat Problem 14.3, except construct a lrcd-periol averaging the prices Asian options based that every 4 months. are on

$40, K

=

$45, c

=

0.30,

=

1.

=

0.08, F

=

1, and 8

=

0.

a. What is the price of a standard call? b. What is the plice of a knock-in call with a barrier of $44. Why? c. What is the price of a lnock-out call with a barrier of $44? Why? 14.7. Let S $40, K 4 5 100). =

9

$45, o'

=

0.30,

=

?'

=

0.08,

=

0, and F

(0.25,0.5, 1, 2, 3,

=

'

'

a. Compute the prices of knock-out calls with a barrier of $38. b. Compute the ratio of the knock-ut call prices to the plices of standard calls. Explain the pattern you see. puts assuming a barrier of $:I4.

14.8. Repeat te previous problem for up-and-out

14.9. Let S $45, o' 0.30, ?' 0.08, and 8 0. Compute the value $40, K with lnock-out calls of a bnrrier of $60 and times to expiration of 1 month, and 2 monts, so on, up to 1 year. As you increase time to expiration, what the price of the knock-out call? What happens to the price of the happens to call relative knock-out to the plice of an otherwise identical standard call? =

=

=

=

=

14.10. Examine the prices of up-and-outputs with strikes of $0.9and $1.0in Table 14.3. up-and-outs appear to have the same With barriers of $1 and $1.05,the premium as the prdinary put. However, with a strike of 1.0 and ihe samebnrriers, the up-and-outs have lower premiums than the ordinary put. Explain why. What would happen to this pattel'n if we increased the time to expiration? .go-strile

14.11. Suppose S

=

$40, K

=

$40, c

=

0.30, r

=

0.08, and J

=

0.

;

a. What is the price of a standard European call with 2 years to expiration? b. Suppose you have a compound call giving you the right to pay $2 1 year from today to buy the option in part (a). For what stock prices in 1 year will you exercise this option? c. What is the price of tllis compound call? d. What is the price of a compound option giving you the right to option in part (a) in 1 year for $2?

sell the

14.12. Male ttle same assumptions as in the previous problem. a. What is the price of a standard European put with 2 years to expiration? b. Suppose yo have a compound call giving you the right to pay $2 1 yea.r f'rom today to buy the option in (a). For what stock prices in 1 year will you exercise this option? c. What is the price of this compound

call?

464

k. Exo-rlc

OpTloxs:

l

PROBLEMS

d. What is the price of a compound option giving you the right to sell te option in part (a) in 1 year for $2? 14.13. Consider the hedging example using gap options, in particular the assumptions and prices in Table 14.4. a. Implement the gap plicing formula. Reproduce the numbers in Table 14.4.

$0.8 and Kz $1. 1.fvolatility were b. Consider the option with #1 of this option be? What do you thirlk will would the what price zero, happen to this premium if the volatility increases? Vel'ify your answer using your pricing model and explain why it happens. =

=

14.20. A chooser option (also known as an asryou-like-it option) becomes a put or call at the discretion of the owner. For example, consider a chooser on the S&R ' lndexfor which both the call, with value Cst K, T t4, and ihe put, with value l), have a stlike price of K. The index Pvt K, T pays no dividends. At the choice date, ,1 the payoff of the chooser is -

,

-

,

,

maxlct.%l K, T '

'to

,

,

,

,

=

=

=

=

=

=

=

=

=

=

=

=

=

=

30%, r 14.21. Suppose that S $100, cr contract which, 6 months from today, will at-themloney call option. (This is called a r, o', and are certain not to change in the =

a. Val'y &frpm 0 to 0.1. What happens to the price of the call? b. Vary 8c from 0 to 0.1. What happens to the price of the call?

=

=

=

=

=

=

=

=

a. What is the price of an exchange call with S as the

underlying

asset and

Q as te stlike price?

b. Now suppose c o

=

0.40. What is the price of the exchange call?

ca Explain your answers to

(a)and (b).

K, T

-

Jl))

=

=

= 8%, and 0. Today you buy a give you one 3-month to expiration forward start option.) Assume thjtt next 6 months.

c. What would you pay today for the forward start option in this example? d. How would your answer change if the option were to have a strike price that was 105% of the stock price?

-0.5

0.30, ?' 1, and 8 0. Also let Q 14.18. Let S 0.08, F $40, $40, o' 0.30, 8c 0, and p u:cq 1. Consider an exchange call with S as the price o'c of the underlying asset and Q as the price of the strike asset.

,

a. Six months from today, what will be the value of the option if the stock price is $1002$50? $200? (Use the Black-scholes formula to compute the answen) ln each case, what fraction of the stock price does the option cost? b. What investment today would guarantee that you had the money in 6 months to buy an at-the-money option?

=

to 0.5. What happens to te price of the call? c. Vary p from d. Explain your answers by drawing analogies to the effects of changing inputs in the Black-scholes call pricing formula.

P't,

.

=

=

/1),

b. Suppose that the chooser must be exercised at /1 and that the underlying options expire at r. Show that the chooser is equivalent to a call option ,-dtT'-Jl) with strike price K and maturity F plus put options with strike Ke-r-bl--tbb expiration and rice rl P

0.08, r 1, and $ 0. Also let Q 14.17. Let S 0.30, r $60, $40, o' this problem will and In 0.5. 0, 0.50, compute prices of we p o'c c exchange calls with S as the price of the underlying asset and Q as the plice of the stlike asset. =

-

a. lf the chooser option and the underlying optims expire simultaneously, what ordinary option position is this equivalent to?

,

0.08, F 1, nd & 0. Also let Q 14.16. Let S 0.30, r $60, $40, o' What is the price of a standard 4o-strike and 0.5. 0.04, 0.50, p o' c c call with S as the underlying asset? What is the plice of an exchange option with S as the underlying asset and 0.667 x Q as the strike price?

465

14.19. XYZ wants to hedge againstdepreciations of the euro and is also cpncenaed about the price of oil, which is a signiicant component of XYZ'S costs. However, there is a positive correlation between the euro and the price of oil: The euro appreciates when the price of oil rises. Explain how an exchange option based on oil and the euro might be used to hedge in this case.

14.14. Pfoblem 12.11 showed how compute approximate Greek measures for an technique Use this option. to compute delta for the gap option in Figure 14.3, ranging stock prices for from $90 to $110and for imes to expiration of 1 week, How months, and 1 3 easy do you think it would be to hedge a gap call? year. 14.15. Consider the gap put in Figure 14.4. Using tlze technique in Problem 12.11, compute Vega for this option at stock prices of $90 $95 $99 $101 $105 and $110, and for times to expiration of 1 week, 3 months, and 1 year. Explain the values you compute.

%

14.22. You wish to insure a portfolio for 1 year. Suppose that S $100, ty 30%, considering simple You strategies. The f/zlll?-t7/lcc 8%, and 0. two are ?price strike option maturity entails buying with l-year at put a one a strategy that is 95% of the stock price. The rolling pl-lf?wncc strategy entails buying one l-month put option each month, with the strike in ech case being 95% of the then-curren t stock plice. '=

=

=

=

E

.

'

a. 'What is the cost of the simple insurance strategy? b. 'What is the cost of the rolling insurance strategy? Hint.. See the previous problem.) c. lntuitively, what accounts for the cost difference?

466

%.ExoTlc

OlnTloxs:

14.A:

APPENDIX

I

PRICING

FORMULAS

F0R

EXOTIC

OPTIONS

$

467

where the correlation between 1n(&) and G(F) is given by

APPENDIX 14.A: PRICING FORMULAS FOR EXOTIC OPTIONS

1 6N + 1)

#=

discussed in this ln this appendix we present formulas for some of the options

chapter.

Asian Options Based on the Geometric Average (an average price option) or

The average can be used in place of either the asset price the strike price (an average strike option).

rate is 1- and the stock has a dividend Suppose the lisk-free price options computtthe average using N equally spaced prices from 0 yield and volatility o'. We geometric average plice option time to F, with the first observation at r/#. AEuropean formula for Black-scholes a call by setting the dividend can then be valued using the volaiility equal to yield and Averase

(14.18)

IN + 1 * Use the current stock price as the stlike price. * I'he dividend yield remains the same.

Compound Options Letting p denote the correlation coefcient between normally distribtged zl and z2, we denote the cumulativ bivariate standard normal distributiim as '

Probtzl

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bonds, and a put, and so forth. ln Chapter 10 we saw that an option could also be synthetically created from a position in the stock and borrowing or lending. If prices of act'ual claims differ from their synthetic equivalents, arbitrage is possible.

Financial engineering is an application of the Modigliani-Miller idea. We can combine claims such as stocks, bonds, fomards, and options and assemble them to create new claims. The price for this new security is the sum of the pieces combined to create it. When we create a new instrument in this fashion, as in the ModiglianiMiller analysis, value is neither created nor destroyed. Thus, financial engineering has no value in a pure Modigliani-Miller world. However, in real life, the new instrument may have different tax, regulatoly or accounting characteristics, or may provide a way for the issuer or buyer to obtain a particular payoff at lower transaction costs than the alternatives. Financial engineeling thus provides a way to create instrtlments that meet specifc needs of investors and issuers. As a starting point, you can ask the following questions when you confront new nancial instrments:

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t?-V0)J

=

P is the current price of a r-periodzero-coupon bond. There are two important, equivalent interpretations of Pt. First, Pt is a discount factor, since it is the price today for $1 delivered at time r. Second, Pt is the prepaid forward prke for $1 delivered at time t. These re different ways of saying the same

thing:.

Zero-coupon bond price

=

Discount factor for

$1 c.:= Prepaid forward price for $1

Financial valuation entails discounting, which is why zero-coupon bonds are a basic

building block. The notion that prepaid forward prices are discount factors will play an impottant role in this chapter.

* What is the payoff of the instrument?

Coupon Bonds

e Is it possible to synthetically create the same payoffs using some combination of assets, bonds, and options?

Once we have a set of zero-coupon bonds, we can analyze other fixed payment instruments, such as ordinary coupon bonds. Consider a bond that pays the coupon c, 11 times over the life of the bond, makes the matulity payment M, and matures at time F. We will denote the price of this bond as /(0, F, c, ?l, M). The time between coupon payments is F/?;, and the fth coupon payment occurs at time ti i x F/n. We can value this bond by discounting its payments at the interest rate appropriate each for payment. This bond has the price

* Who might issue or buy such an instrument? * What problem does the instrument solve?

=

AND DESIGNING NOTES STRUCTURED

15.2 PRICING

11

We begin by examining sucttlred notes. An ordinary note (or bond) has interejt and maturity payments that are fixed at the time of issue. A structured note has interest or mattlrity payments that are notExed in dollars but are contingentin some way. Sucttlred notes can make payments based on stock prices, interest rates, commodiyies, or currencis, and they can hav optios embedded in tem. The equity-linked CD discussed in Chapter 2 is an example of a structured note, as it has a maturity payment based upon the performance of the S&P 500 index. ln tMs sectipn we discuss structlzred notes without options. In the next section we will introduce notes with options.

Zero-coupon

Bonds

The most basic tinancialinstrument is a zero-coupon bond. As in Chapter 7, 1etrstz, rj) represent the annual continuously compounded interest rate prevailing at time s S l0, for a loan from time tz to time 11 Similarly, the price of a zero-coupon bond purchased at time 1(j, mattlring at time r1 and quoted at time s is Pstf), rl ). Thus, we have .

.:(0 F c

?7,

M)

ce-rtit'i

=

lt

c#?; + MPv

= i'= l

This valuation equation shows us how to price the bond and also how to replicate the bond using zero-coupon bonds. Suppose we buy c zero-coupon bonds mattiring in 1 year, c maturing in 2 years, and so on, and c + M zero-coupon bonds maturlng in F years. This set of zero-coupon bonds will pay c in 1 yar, c in 2 yeqrs, and c + M in F years. We can say tat the coupon bond is engineered from a set of zero-coupon bonds with the same maturities as the cash flows from the bond. In practice, bonds are usually issued at par, meaning that the bond sells today for its maturity value, M. We can structure te bond to make tllis happen by setting the M coupon so that the price of the bond is M. Using equation (15.1),1(0, F, c, ?), M) if the coupon is set so that ,k

'

=

,

Ps (/0 t l ) ,

=

e -rx

(Jn.Jl

)(?1-J())

When there is no lisk. of misunderstanding, we will assume that the interest rate is quoted at time tv = 0, and the bond is also purchased then. We will denote the rate

+ Me-rl-lr

i= l

c

=

M

(1

-

,,

Pr

)

(2/-1 #J,

(15.2)

We have seen this formula before. lt appetred in Chapter 7 and it was also the formula for the swap rate, equation (8.3)in Chapter 8.

k. Fl NANCIAL

476

ENGI Iq EERI NG AND SECURITY

DESIGN

PRI Cl N G AN D DESI GN 1NG STRUCTU RED NOTES 11:(:)1*

t' rate is therefore 1.5%). From h' ,t'

Equity-lainlced Bonds ln this section we discuss pricing of various types of equity-linked bonds. Specihcally, le of XYZ we consider a bond that, instead of paying M in cash at maturity, pays uncertnl'n mattll'ityvalue. stock at mattlrity. With this change in terms, the bond has an for such Moreover, this change raises questions. What does it mean a bond to sell at paid in cash or in shares of XYZ? par? tf there are coupon payments, should they be stock For regulatory and tax purposes, is this instrument a or a bond? -/?tz?z

l .i.j iis IIE;; :( .)t

.

:::4:.

-

. 7,) -.t): -

477

(15.3),the s-yearprepaid forward price for XYZ

equation

' i(

tlt

$

$100

1:.

E

$1.20e-0.0l5x? $,7942 =

-

i jE)

.

f= l

l.).t 1'' Thus a zero-coupon equity-linked bond promising to pay one share of XYZ in 5 years $,# )'q would have a price of $79.42. % '''

.

We now add cash coupon payments to the bond. Represent coupons of c each and ashare atmaturity as #(0, F, c, ll, Sp). The valuation equation for such a note-the analog of equation (15.1)-is

equitplinld bond Zero-coupon Suppose an equity-linked bond pays the bondThere are no interim payments. What is a fair stock time F. share of holder at one price for this bond? this valuation problem is the same as Although the language is now that of valuing a prepaid forward contract, which we analyzed in Clpter 5. In both cases the investor pays today to receive a share of stock at time F. In te context of this chapter, we could also call this instrument a zero-coltpz ctylf/l/-llp/# bond. ecall from Chapter 5 that the prepaid forward price is the present value of the forward price, Fapw= #wFo,w. This relationship implies that for a nondividend-paying stock, -rT (r-1)F = St since J = 0. The prepaid fozward price is the stok price. F()P :.:= e &e w

cash coupon payments theplice of abondpayingn

'different,

/1

B (0, F, c,

l

Sv)

,

P

(j.j

Pz + F(),w

c

=

.

4)

=l The value today of the mattllity payment, which is one share of stock, is the prepaid forwardprice, Fopv. If the stock pays dividends, then in order for the bond to sell at par the current plice of the stock it must make coupon payments. In particular. the ntp/c ntttst /JJy coltpons vith a prd-w/' value eqttal to the prc-d/lr value ofdividends t/l/dr the ftfd ofthe p7tvc.To see this, use equation (15.3)to rewlite equation (15.4):

'

)1 Example 15.1 Suppose that XYZ stock has a pricy of $100and pays no dividends, )!! and that the annual continuously compounded interest rate is 6%. In //7c absence of () ) dividellds, #?' plepaidfonvalvl p? ce ctzl tals //2c stock price. Thus, we would pay $ 100 iit ,). %. t to receive the stock in 5 years.

/1

'i' E

BCQ,T, c,

E

-i

?? ,

Svj

11

#n + Sz

c

=

Pt

xl)f;

i= l

= l

'

-

,

ln words: The price of the bond, B, will eqpal the stock price, %, as long s the present value of the bond's couons (the ftrst tenn on the right-hand side) equals the present value of the stock dividends (the third term on the right-hand side). ln general, if we wish to price an equitplinked note at par, from equation (15.4), price B will equal the stock price, %, if the coupon, c, is set so that bond the '

This example shows that if we issue a bond promising to pay one share of a nondividend-paying stock at maturity, and tle bond pays no coupon, then the bond will sell for the current stock price. In general, a bond is at par if the bond price equals the maturity payment of the bond. The bond in Example 15.1 is at par since the bond pays one share of stock at maturity and the price of the note equals the price of one share of stock today. Suppose the stock makes discrete dividend payments of Dt; Then we saw in Chapter 5 that the prepaid fomard price is

;

c

&

=

-

FP () v

(15.5)

'

,,

)Zf-,Pz

That is, the coupon must amortize te difference between the stock price and the prepaid forward plice.

.

lt

F'0,

,%

T

=

-

J'''llbi o,f

(15.3)

f= l

q Example (j

:' ..

Consider XYZ stock as in Example 15.2. If the note promised to note would sell for coupon equal to the stock dividend-the

15.3

'E

If the stock pays dividends and the bond makes no coupon payments, the bond will sell at less than par. Example Suppose the price of XYZ stock is 15.2 continuously compoundedinterestrate and the annual $1.20,

j'

$100, the quarterly dividend is is 6% (thequarterly interest

.

quarterly-a ltyjpay $1.20

t.).$100.

.

..

k

.

Notice that equation (15.5)is the same as the equation for a par coupon on a cash bond, equation (15.2). lnstead of 1 Pv in the numerator, we have s'tl Ff%. The former is the difference between the price of $1 and the prepaid forward price for $1 -

-

478

k

FINANCIAL

ENcl NEERING

PRI Cl N G AN D

DEslcx

AND sEcunl'ry

.

12

z:

c

=

F'aP;f +

zy i= l

=

ypv

-

,

=

1

'''

ynp

0,Jf

.

(15.6)

1

-

d

-

',

d-f

Sf=le

X -,Jf

(15-8)

,

Bonds

Now we repeat the analysis of the previous section, except that instead of paying a share the note pays one unit of a commodity. We ask the snme of stock at maturity, we questions about how to structure this note. We will see that the commodity lease rate 'suppose

rate is positive.

=

=

Fo%, we have

FX 0,F

-

;

E3i l P z:z

Comparing the equations for c and c* makes it apparent that the appropriate discount factor for a coupon is determined from the lease rate on the underlying asset. In the case of a bond denominated in cas the lease rate is the interest rate, while in the case of a bondcompletely denominated in shares, the lease rate is the dividend yield.

Commoditpl-inked

Since by desnition of the prepaid forward price, Pv F0,w &

becomes

1

.

Suppose we have a commodity with a current price of Sv and a forward cash interest price of Fo,z., and we have a commodity-linked note paying a cash coupon. For the note to sell at par, we need to set the coupon so that

C

jgjgpj e-sq i:zzz1

=

=

'

-T

We can compare this expression for c* with thatfor the coupon on an ordinary cash bond. ln the special case of a donstant interestrate and assuming a $1 par value, equation (15.2) C

4:) $455,-0.0625x3 lyyy :,m8 szx ))t ow y'i ). l.1This amount is less than the spot price of $400because . the lease =

becomes C=

Suppose the spot price of gold is $400/oz.,the 3-year forward price p Example 15.4 and the 3-year continuously compounded interest rate is 6.25%. Then a is $455/0z.9 (tt ) 17 t zero-coupon no te paying l ounce of gold in 3 years would sell for 'lt)

E ' .

When we pay coupons as shares rather than cash, the coupons have variable value. Thus, it is appropriate to use the prepaid forward for the stock as a discount factor rather tha.n the prepaid fonvard for cash. dividend yield, $, this equation In the specialcase of aconstantexpectedcontinuous +

479

commodity-linked Zero-coupon bonds Suppose we have a note that pays one unit of a commodity in the future, with no interim cash flows. What is the price of the note? Once again, the answer is, by definition, the present value of the folavard price, or the prepaid' forward price. As we saw in Chapter 6, the difference between the spot price and the prepaid fonvardplice is summarized by the lease rate. Thus, the discount from the spot price on a zero-coupon note reqects the lease rate.

-''

y uuf

$

t:(

P

z-nw

The number of f'ractional shares that must be paid each year for the note to be initially priced at par, i.e., for J/0 = %, is

cs

l N G STRUCTU RED HOTES

replaces the dividend yield. A commodity-linked note will pay a coupon if the lease rate is positive, and the present value of coupon payments on the note must equal the present value of the lease payments on the commodity.

delivered at time F. The latter is the difference between the price of one share and the prepaid forward plice for one share delivered at time F. ln praetice, dividends may change unexpectedly over the life of the note. The note issuer must decide: Should the dividend on the note change to match the dividend paid by the stock, or should the dividend on the note be sxedat the outset using equation (15.5)? The price should be the same in either case, but a different ptrty bears dividend risk. An alternative to paying interest in cash is to pay interest in fractioni Interest in-kind the coupon could be the value of 2% of a share at the time of For example, shares. payment, rather than a fxed $2. To price such a bond, we represent the number of fractionl shares received at each coupon payment as c*. The value at time 0 of a share Thus, the formula for the value of the note at time tfj, 4/0is received at time t is F

D ESI GN

exactl#as with a dividend-paying stock. The coupon serves to amortize the lease rate. Thus, the lse mrc plays /'/7c role of a lfW#e??tf yield ?l pricing a ct?/?/?nt/#i/)/-ff/l/cc# rfhe note. present value calculation treats the lease rate exactly as if it were a dividend yield', what matters is that there is a difference between the prepaid fonvard price and the current spot price.z Suppose the spot price of gold is $400/oz.,the 3-year fonvard price @ Example 15.5 lli) rate is tr is $455/0z.,the l-year continuously compounded interest rate is 5.5%, te z-year

2As we saw in Chapter 6, a lease rate can be negative if tere are storage costs. In this case, the holder of a commoditplinked note benefits by not having to pay storage costs associated with the physical commodity and will therefore pay a price above mattlrity value (in the case of a zero-coupon note) or else the note must carry a negative dividend, meaflng that the holder must make coupon payments to the issuer.

k. FINANCIAL

480

ENGI N EERI NG AN D SEcuRl'ry

PRI Cl N G AN D

DEslcrq

ltt 6%, and the 3-year rate is 6.25%. The annual coupon is then determined as j() jj $400 $3-77.208 $8.561 c )(q ,-0.0625x3 ,-0.055 tl + e-0.06x2 + 'h.) l11 The annual oupon on a 3-year gold-linked note is therefore about 2% of the spot jtq ri price. . . k ii.l'

-

D E5I GN

I N G STRU CTU RED NOTES

$

481

What if a bond pays one unit of the commodity per year, forever? We lnow that if it pays St in perpettlity it is worth %. Thus, if it pays $ it is wot'th

=

=

'E

.

f).)

A 2% yield in this example might seem like cheap financing, but this is illusory and stems from denominating the note in terms of gold. When the yield on gold (thelease rate) is less than the yield on cash (the interest rate), the yield on a gold-denominated note is lss than the yield on a dollar-denominated note. This effect is reversed in cases where the interest rate in a particular currency is below the lease rate of gtld. l Japan during the late 1990s, the yen-denominated interest rate was close to zer, d the coupon rate on a gold note would have been greater than the interest rate on a yen-denominated

note. ln-tind As wit stocks, we can pay fractional units of the commodity as a interest payment. The present value of the payment at time t is computed using periodic prepaid price, Fnp,. Thus the value of a commodity-lin ked note at par is forward te exactly the snme as for an equity-linked note paying interest in-lind: Interest

This is the commodity equivalent of a perpetuity. The cnclusion of this section is simple: Commodity-linked notes are fonnally like equity-linked notes, with the lease rate tnking the place of the dividend yield.

Currency-tainlced Bonds What happens if we change the currency of denomination of the bond? As you can probably guess by now, the foreign interest rate, being the lease rate on the foreign currency, takes the place of the dividend yield on the stock. Suppose that we want to compare issuing a par-coupon bond denominated entirely in dollars and a par-coupon bond denominated entirely in another currency. We will BF denote zero-coupon bond prices denominated in te foreign cuaency, ?'s(l) the 1et foreign interest rate, and PtF the price of a zero-coupon bond denominated in the foreign Ctlneency.

As you would expect, a bond completely denominated in a foreign currency

qa-

c

=

F() ,f + F(),z

(15.6).

'

-,>

.->

-r

I

=

-

PF T pp = i' 1 t -

''

Perpetuities A perpetuity is an innitely lived coupon bond. We can use equations (15.7) and (15.8)to consider two perpemities: one thatmakes annual payments in dollars and another that pays in units of a commodity. Suppose we want the dolltr perpettlity to have a price of M and the commodity perpemity to have a price of %. Using standard j, (x) in equation perpettlity calculations, if we let F (15.8) (thls also means that (x)), the coupon rate on the dpllar bond is ?) 1 Mer 1) 'M c M =

gy jy?J

C=

p

p

i= l

The fonuula for c* is givn by equation

1

F

lt .

will

have a coupon given by the formula

=

'-e-r

where is the effective annual interest rate. Similarly, for a perpetuity paying a unit of a commodity, equation (15.7)becomes

In other words, foreign interest rates are used to compute the coupon. What happens when the principal, M, is in the domestic currency and the interest payments are in the foreign currency? Once again wejust solve for the coupon payment tat makes the bond sell at par. There are two ways to do this. First, we can discount the foreign currency coupon payments using the foreign interest rate, and then translate their value into dollars using the cuaent exchange ratet of foreign currency). The value of te fth coupon is Pfc, ab (dnominatedas and the value of the bond is

s/unit

-p

?1

B (0, F, c F

,

?7.,

M4

=

xvcF

PF t;

.

Alternatively, we can translate the f'uturecoupon payment into dollars using the forward currency rate, Fp,?, and then discount back at the dollar-denominated interest rate, Pt. The value of the bond in this case is 11

BQ, T,

c

F ,

?l,

M4

=

cF

F flvt # p + MPp

f

=

where is te effective annual lease rate. Thus, in order for a commodity perpettlity to be worth one unit of the commodity, it must pay the lease rate in units of the commodity. (For example, if the leae rate is 2%, the bond pays 0.02 units. of the commodity per yealc)

+ MPv

i= 1

l

The two calculations give the same result since the currency forward rate, from equation (5.18) is given by

482

%.Fl NANCIAL

ENGIN EERI Nc AN D SEcu

BoN Ds WITH

Rl-ry DEslclq

Tllefotz3al-dpt-icefol-fol-eigll exchallge is set so tllat it plc/cd' no ttf/rtr/lct, vhether wc ?Ll l T7r?-l tlltr (llI?-?-l? 2t2)7Llll ff tI1tr7l (lisc t?tfll f, t7?* tlis (24)1ll1 t f7?l f tll tr?l ?L)ll 17t,?-l tIlt! (lI I1*1*3,1 ?(2)/. The coupon on a par bond with foreign interest and dollar principal is given by F C

=

p.f yg

1

Pp

-

s

i'=xl 0,h p

(j5

.

j(p

K

Ji

=

OPTIONS

Options in Coupon Bonds One common kind of equity-linked note has a stnlcture where, at matuhty, te holder can receive some fraction of the retul'n on the stock but does not suffer tt loss of principal if the stock declines. We obtain this seucture by embedding call options in the note. Let y denote the extent to wVc h the note participates in the appreciation of the underlying stock; we will call y the price participation of the note. ln general, the value J/0 pf a note with sxedmaturit.y payment M, coupon c, maturity F, strike price K, and price participation y can be written lt

J'')Pt

+ yBSCall(&,

483

.

These conditions the equation

imply that

&

JZ() =

Sj

=

M

K, o', r, F, 84

(15.11)

i= 1

Eqution

(15.11)assums that the principal payment is cash. lt could just as well be shares. Equation (15.11)also assumes that the note has a single embedded call option. Given equation (15.11),we could arbitrarily select M, F, c, K, and y and then value the note, but it is common to structure notes in particular ways. To take one example, suppose that the initial design goals are as follows: 1. The note's initial price should equal the price of a shar, i.e., 70 2. The note should guarantee a ret'ul'n of at least zero, i.e., M

=

=

A', and thus the price of the note satisses

=

3In addition to convertible bonds offered by firms, there are bonds offered under many names for different kinds of equity-linked notes-for example, DECS (Debt Exchangeable for Cornmon Stock), PEPS (Premium Equity Participating Shares), and PERCS (Preferred Equity Redeemable for Common Stock), a1l of which are effectively bonds plus some options position.

yBSCaIIIUV, %, c',

r, F, $)

(15.12)

J=zl

Given these constraints, equation (15.12)implies a relationship between the coupon, and price participation, y. Given a coupon, c, we can solve for y. and vice versa.

optionsin Equitpt-inked

c,

Notes.

With an equity-linked note, the maturity value is shares rather than a fixed number of dollars. The plice of a note at par paying qne unit of a share at expiration is '1

. =

P

c i= 1

Pz + Fa + y Bscallt5' w

,

K c' r ,

,

,

r

J)

,

(15.13)

Compare equations (15.12)and (15.13). lnstead of paying 'Sf)dollars at expiration, the equity-lirlkednote pays one share. lf the share pays no dividends, then (assuming y : 0) 0. To the th share extent the equitplinked note can sell at par only if c pays y dividnds, it is necessary for the note to ofer either coupons or options. =

=

Valuing and Sttuctuting an Equity-l-inked

CD

We have already described in Section 2.6 an example of an equitplinked CD, but we did not analyze the pricing. The CD we discussed has a 5.5-year mgturity and a retunz linked to the S&P 500 index. CD Suppose the S&P index at issue is Sft and is 55.5 at maturity. pays n6 coupons (c = 0), and it gives the investor 0.7 at-the-money calls ()z = 0.7 and Aftr 5.5 years the CD pays K = Pricins

'lnhe

the CD

'

,%).

Sj + 0.7 x max Using equation

-

Sj x P5.5 + 0.7 x BSCal1(u%, So, t)-,

where P5 5 +

=

(15.14)

(5'5.5 %, 0)

(15.11),the value of this payoff at time 0 is

%.

J/.

Pz + Szpv +

c

=

vz

4/0 MPp + c

$

lt

We now considerthe pricing of bonds with embedded options. Such bonds are common.3 Any option or combinatioh of options can be added to a bond. The option prernium (if a purchased option is added to the bond) is amortized and subtracted from the coupon. J.fthe option is written, the amortized premium is added to te coupon.

=

OPTIONS

3. The note should pay some fraction of stock appreciation above the initial price, i.e.,

l

The currency formula is the same as that for equities and commodities. tf we tlzink of the foreign interest rate as a dividend yield on the foreign currency, equation (15.10)is the same as our previous coupon expressions.

15.3 BONDS WITH EMBEDDED

EM BEDDED

?-,

5.5,

t)

(15.15)

d-rY5'5 *

To perform this valuation, we need to make assumptions about the interestrate, te volatility, and the dividend yield on the S&P 500 index. Suppose the 5.5-year interest rate is 6%, the s-yearindex volatility is 30%, the S&P index is 1300, and the dividend yield is 1.5%. We have two pieces to value. The zero-coupon bond paying $1300 is WO rl

$1300c -0.06x5.5

=

sjsy6: .

'

484

k

FI NANCIAL

ENGINEERI

NG AN D SECU RITY DESIGN

BoN Ds WITH

lf we

The 0.7 call options have a value of

0.7 x BSCall($1300, $1300,0.3, 0.06, 5.5, 0.015)

$309.01

=

=

$56.39less than the $1300initial investment. This difference suggests that the sellers enrn a 4.3% commission (56.39/1300)for selling the CD. This analysis makes it clear why the CD does not provide 100% of market appreciation. At 100%, the value of the CD would exceed $1300,and the bank would lose money by offering it. The bank is a retailer, offeling.the CD to the public in order to make a proft from it. The bank's position is that it has borrowed $934.60and written 0.7 call optins. You of eqation (15.15)as the vholesale cost of the CD-it is the theretical cost can tI-IiI'IIk to the bank of this payoff. As a retailer, an issuing bank typically does not accept the market lisk of issuing the CD. Banks offering products like this often hedge the option exposure by buying call options from an invstment bank or dealer. The bank itself need not have option expertise in order to offer this kind o t pro duct The CD is offered by a bank that wants to earn ommissions. Th originating bank will hedge the CD, and must either bear the cost and risks of deltachedging, or else buy the underlying option from another source. Retail custoprs may haye trouble comparing subtly different products offered by differeni bnnks. Customers who have not read this book might not understand option pricing, and hence will be unable io catculate the theoretical value of the CD. On balance, it seems reaspnable that we would find the value of the CD to be less than its retail cost by at least several percet. Here 're some other considerations:

Tlzis is

.

* It woul d have been cos tl y for retail customers to duplicate tl'lis payoff, particularly since s-yearoptions were not readily available to public ikestots at the time of issue. lnvestors buying this product are spared the need to learn as much about options and, for example, iaxeson options, as they would were they to yeplicate this payoff for themselves/ * The price we have just computed is a ballpark approximation: It is not obvious what the appropriate volatility and dividend inputs are for a 5.5-year horizon. Any specic valuation conclusion obviously depends entirely on the interest rate, volatility, and dividend assumptions. However, Baubonis et a1. (1993)suggest tat fees in this range are cofnmon for equity-linked CD products.

4It t'urnsout that the tax treatment in the United States of an equitplinked CD such as this one s fairly tcontingent interest complicated. A bond with a payment linked to a stock index is considered'to be debt-'' The bondholder must pay tax annually on imputed interest. and there is a settling up procedure at maturity. Issuers of such bonds frequently recommend that they be held in tax-exempt accounts.

$

485

allow

lt

1/(1

-

a)

=

MPv + c

J-')Pti + yBSCall(5',

K,

tr, ?',

(15.16)

F, J)

= 1

In the above example, a

$1243.61

OPTIONS

for issuer profit, a, as a fraction of the issue plice, a general expression for the value of a CD issued at par is

The two pieces together, assuming they could be purchased without fees or spreads in the open markt, would cost

$934.60 + $309.01

EM BEDDED

structuins example:

0.043 and J/

=

=

1300.

Many issues arise when designing an equity-linked CD. For

the product

@What index should we link the note to? In addition to the S&P, posgibilities include the Dow Jones lndustrials, the NYSE, the NASDAQ, sector indexs such as hightech, and foreign indexes, with or without currency exposure. How much participation in the market should the note provide? The CD we have been discussing provides 70% of the rettlnl (ifpositive) over the life of the CD.

@ Should the note make interest payments? (The example CD does not.) e How much of the original investment should be insured? (The example CD fully insures the investmento)

Alternative Structures Numerous other vadations in the structure of the CD are possible.

Sone examples

follow:

* Use Asian options injtead of ordinal'y options. * Cap the market participation rate, mrningthe product into a collar. * Incorporate a put instad of a call. * Male the promised payment different from the price. We will consider the

cover the other two.

srsttwo alternatives

in this section. Problems 15.9 and 15.11

The payoffdiscussed above depends on the simple over a period could 5.5 We the based the of instead comptge retul'n years. on avrage of year-end tlian an otherWise 14, As Asian option is worth less we saw in Chapter an P rices. ordinary option. Therefore, when Asian option used, the participation equivalent is an will be than with ordinary call. greater rate an Suppose we ;base the option on the geometric average price recorded five times ' . over the 5.5-yea.r hfe of the option, and set the strike price qual to tlie rrent index level. The value of this Asian call is $240.97as opposed to $441.44for an ordinary call. Assuming the equity-linked note pays no coupon and keeping the present value the same, the participation rate with this geometric-average Asian option is Asian options

'et'urn

0.7 x

44j.44 240.97

=

1.28

%.Fl NANCIAL

486

NG AND SECURITY

ENGINEERI

ENGI N EERED SOLUTIONS

DESIGN

If instead we base the option on the arithmetic average, the option price is giving us a participation rate of

441 273.12

=

Instead of shorting a forward contract, Golddiggers could issue a note promising to pay ap ounce of gold 1 year from now. Such a note is effectively debt collateralized by f'uture sales of gold. Ordinarily we would think a risk'y commodity like gold to be poor collateri for a debt issue. But if a gold-ltlillillg-firnt issues gold-linked debt, the lisk of the bond and the lisk of the collateral are the same.' Bondholders provide snancing as well as asorbing gold price risk. We begin with the information from Chapter 4: The current price of gold is $405/oz., the forward price is $420, and the effective annual interest rate is 5%. The effective annual lease rate is therefore 0.05 (420/405 1) = 1.296%. We wish to construct a debt contract that r atses $405 today (the cost of 1 ounce of gold), pays 1 ounce of gold 1 year from today, and if necessary, pays a coupon, c. We have already seen that the lease rate plays the role of a dividend. Thus, if the bond has a coupon equal to the lease payment on an ounce of gold, it should be priced fairly. Abond with these charactelistics should pay a coupon of 1.296% x $405 = $5.25. We can verify that such a bond is fairly pliced. The payoff to the bond in 1 year is $5.25 plus 1 ounce of gold. We lnow we can sell the gold in l year for $420 sine that is te folavardprice. The present value of the payoffis therefore the value of the coupon plus the prepaid forward price for gold:

1.13

The arithmetic Asian option has a higher price than one based on the geometric average, hence we get a lower participation rate. lncreasing the number of prices averaged would lower the price of either option, raising the participation rate.

-

Another way to raise the participation rate is to cgp the level participation of participation. For example, suppose we set a eap of k times the initial plice. Then the investor writesto the issuer a call with a strike of k.%, and th valuation equation for the

Capped

-

CD becomes

,ntl

-

a)

='

ste-rx' +

y x

pscalltsb,

u%,

c', r? t,

t)

Bscallt*(j,

-

kb, o-,

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))

j' Suppye we set a cap of a 100% retuw. Then the investor wdtes a !q' Example 15.6 t Ca 11with a strike of $2600to the issuer, and the valuation equation for th CD becomes tj ijr; 1300(1 0.043) 1300, + y x (sscalltlaoo 13()0 () 3 0.()6 5.5, 0.015) j' -BSCall(1300, 2.600, 0.3, 0.06, 5.5, 0.015)) ltr )) . .jjlj The value of the written 2600-stke call is $162.48.The participation rate implied by t(. . .. k k.. this equation is 1.11. -().(ox5.5

-

=

,

,

.

,

,

$5 25

-

-

SOLUTIONS

FOR GOLDDIGGERS

return to the Golddiggers example from Chapter 4 in order to see show how Golddiggers'could have used structured notes in place of forwards and options in the

we now

hedgingscenarios

we discussed.

Gold-tzinked Notes

FVPL

x /)1 +

=

-

$5.25+ $420

$405

=

Because the lease rate is paid as interest, the bond sells at pm We should verify that the bond serves as an appropriate hedge for Golddiggers. Table 15.1 summarizes the payoffs to Golddiggers and the bondholders at different gold prices in l year. The table assumes that Golddiggers invests the $405at s-this yields the $425.25that is labeled tFvtgross bond proceedsl.'' The net cash flow is deterinined by adding prohts without consideration of bond payfents (column2) to the difference between the invested bond proceeds (column3) and the payment to bond holders

t

15.4 ENGINEERED

487

contract linked to the hedging instrument, in particular the fact that it is azero-investment price of gold, meant to serve as a hedging instrument and not as a inancing instrument.

$273.12,

.44

0.7 x

$

FOR GOLD DIGGERS

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In 5As discussed earlier, fonvard contracts and swaps typically have collateralization requirments. contract, even when there practice, a company must have capital to cover a large loss on a snancial is an offsetting gain. For example, in a well-lcnown incident n 1999, Ashanti Goldfields had sold fonvard eght times annual production. The company sulered a $500 million loss on its fonvard gold sales when the gold price rose signilicantly. Although Ashanti had gold in the ground, it did not have cash to cover this loss. Ultimately, Ashant was able to keep operating by jiving warrants on l5% of its stock to its counterparties on the forward sale. For details, see Cooper (2000)and the l'Pt7!/Street Jottnlal (October 7. 1999, p. C l

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488

k: FINANCIAL

EERI Iqc AN D SECURITY

Excltq

DEsI

ENGI N EERED SOLUTIONS

chl

. ... .'.' '' .'''. '''' ;'''''.' ''''. '' -.'.t' .'..' ''''--'-'-,,,'f

(column 4). ln this case, issuing the bond achieves the same result as selling a forward contract (compareTable 15.1 and Table 4.2), so Golddiggers is completely hedged. The chief difference between the gold-linked note and the forward contract is that former provides financing, the latter doesn't. lf Golddiggers seeks financing (in the order to construct the mine, for example), the issuance of a gold-linked note might be preferable to borrowing and hedging separately.

Notes with Embedded Options

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uj

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The bondholders have written a put option to Golddiggers and hence in 1 year receive the futtlre value of the premium. If the price o? gold is above $420,Golddiggers

makes

$405

=

$425.25

$434.46+ (u$'1$380)

-

-

$380

=

$49.21

$9.21 $380 -

Table 15.2 summarizes the cash :ows to bondholders and to Golddiggers from the issuance of this bond. You can verify that this is exactly the same payoff as obtained when Golddiggers hedges by writing a call. The commodity-linked bond achieves the same effect. Instead of having alow coupon and protection againstlow goldprices, bondholders might be willing to bear the risk of a decline in the price of gold in excange for a higher coupon. For example, Golddiggers could issue a bond in which bondholders sell a 420-strike put to Golddiggers. Golddiggers in turn .would have to pay greater interest to compensate bondholders for selling the put. The bond would be structured as

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-

$380 $9.21 -

$420,Golddiggers makes

If gold is below

$425.25

$434.46+ ($420

.ll

-

-

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+

$380)

-

$30.79

=

With this bond, Golddiggers in effect buys a 420-sike put. Table 15.3 depicts the net cash flow to Golddiggers from issuing this bond. The cash flows are identical to Table 4.3, where Golddiggers purchased a 420-strike put option as insurance against low gold

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* The initial bond price is $405. * The promised payment on the bond is $434.46,a 7.274% rate of interest.

$420)

whereas if gold is less than $420, Golddiggers will male 'S' +

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Solving for c gives c $11.04, which is a yield of 2.726%. Golddiggers thus issues a bond for $405, with a 2.726% coupon, with additional payments to bondholders if the price of gold exceeds $420. The difference between the 2.726% coupon and 5% is due to the value of the embedded call option. What is the result for Golddiggers from having issued this bond? If Golddiggers invests at 5% the $405 bond proceeds, then it will have $425.25cash in 1 year. Recall that costs are $380/oz.If the gold price in 1 year exceeds $420, Golddiggers will show profts of $420 + $9.21

489

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The valuation equation for the bond is

$405 + c + $8.77 1.05

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- A gold-linked bond leaves bondholders with the risk of a loss should the gold price drop. Golddiggers could instead offer a bond tqt promises bondholders that they will receive interest plus ppreciation of gold above $420. Such a bond implicitly gives holders a call opiion on gold with a strike price of $420. From Chapter 2, the cost of this option today is $8.77, with a f'ut-ur value of $8.77 x 1.05 $9.21. Let the promised payment on the bond be the $405 issue price lus the coupon, c: ln 1 year, the bond tswol'th $405 + c + maxto,

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stock position has been collared, the collared stock can serve as collateral for a lpan. For the example, suppose an executive owns stock and enters into a zero-cost collar where receive at least $90 put strike is $90. With this position, the executive is guaranteed to the stoek ptus put as using of value lend $90 bank present the mattllity. Thus, a can at collateral. lt is also possible to engineer a single instrument that hedges a stock position and 2003, Walt Disney Co vice-chainnn Roy Disney pays cash to the owner. In August sold a s-yearvariable prepaid forward (VPF) contract covering a large pereentage cif his Disney stock holdings. The contract called for Roy Disney to deliver to Credit Suisse First Boston a variable number of shares in 5 years. To quote from th Form 4 filed with

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The VPF Agreement provides that on Augst 18, 2008 (tettlement Disneyl will deliver a number of shares of Common Stock to Date''), glkoy the cash equivalent of such shares) as follows: (a) if the CSFB iLC (or WeightedAverage Price''l of the Common Stockfor average VWA.P the 20 trading days preceding and including the Settlement Date (&ISttle7,500,000 shares; (b)if the ment Plice''l is less than $21.751,a delivery of Settlelent Price is equal to or greater than $21.751per share CtDownside TllreshTlzreshold'') but less than or equal to $32.6265per share Threshold/settlement Downside old''), a delivery of shares equal to the Price x 7,500,000; and (c) if the Settlementpr e is greater than the Upside Threshold, a delivery of shares equal to (1 (10.8755/Settlement Pricel) x .

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Corporations may also wish to sell an asset without Tax deferral for torporations creating aconstructive sale. Acorporationhas an alternative not open to mostinvestorsnamely, the issuance of a note with apayofflirtked to theprice of the appreciated stock. A well-known example of this is the 1996 issuance by Times-Mirror Co of an equity-linked note, where the note was linked to the stock price of Netscape. In April 1995, Times Mirror had purchased 1.8 millio shares of Netscape in a private placement. The price (adjustedfor subsequent share splits) was $2/share. The

Disney Reaps $125 Million in a Novel Stock-sale Package,'' gRandall Smith and Bruce Orwall, I.I/// Sreet Jtpllr/lc/, p. el August 21 2003. ilkoy

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stock was restricted and could not be resold publicly for 2 years even if Ntscape were to go public. ln order to sell the stock, Times Minor would have had to 5nd a qualified buyer (a wealthy or professional investor) and sell the shares in a private placement. l () ln August 1995, Netscape issued shares publicly. In Mtrch 1996, Times Mirror had approximately $85 million in capital gains on the stock. tf the shares had been sold on the open market, the tax liability would have been approximately 0.35 x $85m $29.75 milliorf. Although tflx law did not at that time desne a constructive sale, a suciently wide collar seemd likely to avoid challenge by the tax authorities. Times Mirror's :

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Under U.S. tax law, interest payments on corporate debt are tax-deductible, whit' .. ( ( ividend payments on equity are not. The distinction between debt aljd equity itkt may t seerri clear-cut, but Fith inancial engineering it is easy to blur th distinction. Fof suppose a firv,l issues equity-linked notes that promise ctsfkt '('.,. P yments tlt ( q y debt) but have a payment at maturity contingent on the firm's stock price (like equityjt' i.)( i ls such a Enancial cla im debt or equity for tax purposes? t ;) rt. ln practice it is possible to design financing vehicles that have a sijlcant equity component, yetforwltichthepayments are atleastpartially tmvdeductiblfpythe E.ii, In fil'm. '' . this section wewill examinein more detailtheMarshall &llsley .( j . . . . j.. Col'p.(Mljonvertible bonddiscussedin Chapter3, withthepayoffdepicted inFigure3.l6. j j.j..j . As weditlssed, the ... bond payoff resembles a stock coupled with written and t purchased calls. Nekfthelesja the bond o n tax-deductible are partially payments for the issuer. Moreover, M&I bankholding compalm and the bond provided capital for regulatory purposes.lz ) is a With a structure like the M&I convertible, mytiad details hinge on complex ix, accounting, and regulatol'y considerations. Our purpose in discussing the bond ij .t . .( i( understand at a general level the kinds of issues that financial engineering can addres. We ignore specic details that aren't needed in conveying E a sense of the transaction. The M&1 convertible bond actually consisted of two components: an ownership stake in a tnlst containing M&I bonds, and stock a purchase contract requiting that the convertible bndholder make a future payment in exchange for slaares. These two components are separable, in that an investor could in theoly hold one without the other. Here are some of the details on these two pieces: . ..

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E.

n this chapter we look at some contexts in which srmsissue derivatives, either explicitly First, Black and Scholes (1973)observed that common debt and equity or imyiitly. .. . . .; be vlewed as options, with the 'lirln s assets as the underlying asset. We show how can this insight can be used to price debt sublect to default, as well as the implications for determining how leverae affects the expected return on equity. We also examine warrants, convertible debt, and callable debt as examples of securities that explicitly

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EQU ITY, DEBT,

Appl-lcvloxs

of the firm's assets, they must pay the debt-holders X at time t, > B, equitpholders will pay Wto the bondholders since ellty will then be If Ap r. worththe value of the assets less the payment to bondholders, or Ar B > 0. However, if Aw < W,equity-hotders would have to inject additional f'unds in order to pay off the debt.In this case equity-holders would declare bankruptcy, permitting the bondholders take possession of the assets. Therefore, the value of the equity at time F, fw, is

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2A bond with a payoff specilied as in equation ( 16-2) is a debenture a bond for which payments are secured only by the general credit-worthiness of the company. Such a bond is said to be ltnsectlred. It is also possible for bonds to be secured by speci:c collateral. For example, lenders to airlines may have an airplane as collateral for their bond. .

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1.fwe assume that the assets of the 1114.11 are lognormally distributed, then we can payoffs the Black-scholes model value the to to the frm's equity and debt, equations use (16.1) and (16.4).For purposes of option pricing, the frm's assets are the underlying asset, the strike price is th promised payment on debt, W,the volatility of the rm's assets?o', is volatility, and the payout rate from the firm becomes the dividend yield. Y rate is 1' and the debt matures at time F, we have the lisk-free Et

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Asset value at mattlrity

debt with a The interpretation of equation (16.4)is that the bondholders own lisk-free the with strike plice option W.2 assets on Pay 'off equal to W,but have written a put

)r Example 16.1 supposea firm has issued zero-coupon debt with ''). W $6000 The maturity value of the equity is given by equation

$6,000

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Equation (16.3)says that the bondholders own the firm, but have written a call option to the equitpholders. This way of expressing the debt value explains where te call option in equation (16.1)comes from. Suinming equations (16.1)and (16.2)gives the te total plus debt-as Av. value of the srm--equity A different way to write equation (16.2)is the following: Sz

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t This model of the 51-111 is vel'y simple, in that there are no coupons or dividends, no refnancings or subsequent debt issues, etc. lt is possible to create more complicated models of a srm'scapital structure; nevertheless, this model provides a starting point for understanding how leverage affects returns on debt and equit'y and determines the yield on lisky debt. -

% CORPORATE

506

EQu ITY, DEBT, AN D WARRANTS

APPLICATIONS

Viewing debt and equity as options also provides a f'ramework for thinking about credit risk. Equation (16.4)shows that defaultable debt is eqtlivalent to owning defaultfree debt plus a put option on the assets of the frm. An investor owning a cm-porate bond could buy such a put; the result would be economically equivalent to owning a defaultfree bond. Thus, the value of the put is the value of insurance to protect bondholders against default. We will examine credit risk more in Chapter 26. ' t@'(' $90, ?' Example 16.2 Suppose that W $100, zito qt!) 'il, have We 5 and F mtkes no payouts), years. =

=

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=

where rzt is the expected return on assets, 1' is the risk-free rate, and Qs is the elasticity of the equity. Using equation (16.5),elasticity is AtLE

Qs

=

(16.9)

Et

where As is the option delta. We can compute the expected return on debt using the debt elasticty, f2,:

l-e

1-

=

( 16.10)

r) x f)s

(rzl

+

-

The elasticity calculation is slightly more involved for debt than for equity. Since we compute debt value as Bt = A/ E:, the elasticity of debt is a weightd average of the asset and equity elasticities: -

$100,0.25, 0.06, 5, 0)

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late. .

Leverage and the Expected Return on Debt and Equity Example 16.2 shows that, because of the possibility of bankruptcy, the yield to matul-ity investor in the bond earns the yield on debt exceeds the risk-free rate. However, arl Accounting for the possibility of bankrupt. does if the not go firm to maturity only less than the yield to matulity and return will eal'n a bankruptcy, the investor on average default bears some of the risk of that efeet, In can debt risk-free rate. greater than the lisk with equity-holders. sharing the this the assets, We can compute the expected return on both debt and equit'y using the concept elasticity of an of optioll e/t'/l//cf/)/, vhich we discussed in Chapter l2. Recall that the tlnderlying expected asset and return the the between relationship the on option tells us expected equity the return 12.10), eqtlation compute on option. we can Using ( that on the

The elasticity of any asset with respect to itself is one, so we have Qzt = 1. Using equations (16.8)-(16.1l), you can verify that if you owned a proportional interest in the debt and eqpity of the firm, the qxpected rettll'n on your portfolio would be the expected return on the assets of the firm: rE4

+

(toebt

x rs)

=

'E

=

r +

(?u

-

r)

1 tzEquity

=

r +

t?u -

r) x Qs

rz

(16.12)

( 16.8)

(16.13)

This is the familiar Modigliani-Miller expression for the expected retuna on levered equity Equation (16.13)can be obtained from equation (16.8) bf assuming that the delta of the equity is one, which implies that the delta of the debt is zero. Viewing debt and equity as options, by contrast, allows us take into the account the effects of possible bqnkruptcy. Equation (16.8)assumes that debt- and equity-holders share the risk of the assets, so equation (16.3)will give a higher l'E than equation (16.8).

as rs

(16.11)

fzs

It befkrs emphasizing that this relationship requires that ?'s represent the expected ?'c?Ifr?? on debt, not the yield to maturity. Using the option model, we can also see how the dollar values of debt and equity change whn there is a change in the value of the assets. In other words, we can ask what the deltas of debt and equity are. From equation (16.5),if assets infease in value by $1, equity will incree by the delta of the all option. From equation (16.6),debt will increase by one less the equity delta. It is instructive to compare the expected ret'urn calculation for equity in equation (16.8) with a common alternative calculation. lf we asstlme the debt is risk-free, the; expected return on equity isa

ln(100/62.93)

t) lij 'j'(j The debt yield of 9.26% is 326 basis points gleater than the risk-free =

=

ttzEquity x

tE)

).)

A/

.

value of the debt is

'rhe

;E

l

507

'

jl::EEE

. ;j: y

%

sTh-ls expression is also sometimes wn-tten as rs

=

rz + (rz

-

?-)

x D/E.

508

%.CORPORATE

EQU ITY, DEBT,

Appl-lcvloxs

100, 0.25, 0.06, 5, 0)

0.735

=

The debt detta is 1 0.735 0.265. Thus, if te asset value increases by $1, the value of the debt increases by $0.735and the value of the debt increases by $0.265. Using equation (16.9),the equity elasticity is =

-

Q

509

At the same time, debt also becomes liskier as leverage increases and there is increased chance of default on the debt. Figure 16.2 graphs the debt-to-asset ratio (computedusing equation 16.5) and the expected tet'urn on equity (computedusing equation 16.8) as a function of the asset value (f the fit'm, using the assumptions in Example 1.3. For very low asset values, the debt-to-asset ratio is almost 1 and the expected return on equity is almost 40%. As the asset value exceeds $200,the expected return on equity is about 12%. For purposes of comparison, Figure 16.2 graphs the expected retarn on equity, computed assuming that the debt is risk-free. For asset values close to $200, the

Use the same assumptions as in Example 16.2, and suppose that the Example 16.3 expected return on assets, rA, is 10%. The equity delta is BSCallDelta(90,

AN D WARRANTS

90 x 0.735 = 2.443 27.07 The expected return on equity is therefore l

'

=

'

'!' pf .(('.y! i'lff ,'(' F::qrqrr!'rrf y'y r' t';'t' 1q'' 5* p' (lqzjif qs7i)i:f 1E1.!)4E)1* frpif 7* 1* F' ilf (' y'. y,jyyjgy(j'. )' q' @' ,'t-' j'y' j!f ;ff g'. yyyjf ''r' ;j))'. .y'. l'y t':1l'. jjtryjjy'll.ty 1111k* p' )' j')'y' y'. y' j)jj))..j' jy'j ('4111:44* iill i7. : ).!8! liftrii'(i ' !ii q( (;' 7i:',. r(y'tjjyjyyj.' )' )!)-),' ty'. til'tf tjf t'j jgj,yf t:;yf . . ii' ' j-''li'il.E.'i 't!.' .' (i'' ' : l;k'11ltli gg. ); ( tk;yy l.. tE ryjj jg y..jj.(j.y.. y jjy:. j ..li j j 44jjjjjjj,:4 .. ) y . j : l t;sy t j, .. j . :j..j.j j j . . j Iljj:iiilitl !1II:::::;k:ty. y. . . . !y. jj.g.j jyj.yyy yr . 'tlr));.tr--.:-.-t:...-..... jr . )-----jj).. ... ... ,. . . ,4* (IyI4'jjjjjj,,jjjj;E yq'jE F' '!t' !(' (' (',LLLLjj,L.-..L y' pf (k' 'h'f'lq'qq: f' (;E' j'gjyt'. ' .'..'

0 (% +

(0.1 0.06) -

.

.'

x 2.443

IL;'plE

E . .. ;

= 0.1577

:

:1E E

. E .

.

. .

,,-i

--..

-.-......

-

E ! ''.' . @

: j.jk!

.. - .

:

.

.

.

........jk.,-

-......,.

.

. ....,.

! : :: E EE

: : i 5* ! :q!'ltE ! : . . : .yjEfjz.( : ! jk.jjjjggjj;;;,.

: i

E E;.

.

......

E

:. ::. ..E

'

-

.

.

,.

.

. -.

-.

.

.

.

. -.

.

pq!E '

.

-

..,...,....- .-.-..,.--...

''' -

r

.;-jjjjj.

E- .

i

;.... iE... E.E.E E. E . E : . .. .!.. .jE. :yi.y ).L y y y jy y (,y. :.2:.,grj. . yq-...,- - -g -.. y. -.g.2-., , . . . -.

-

.

-

.... ,

.

.

,,

.

Debt-to-Asset Ratio 1.0 0.8

The expected return on debt is therefore

?-s

=

0.06 +

0.6

(0.1 0.06) -

x 0.3793

0.4

= 0.0752

0.2

Note that the 7.52% expected return on debt is greater than the risk-ft'ee rate (6%)and less than the yield to maturity on debt (9.26%). If we owned equity and debt in the same proportion in which they were issued by the firm, we would have a return of

90 27.07 27.07 0.1577 + x 0.0752 x 90 90

100

50

150 Asset Value

150

0.1000

Since 10% is the

expected return on assets, this illustrates equation (16.12). Finally, ifwe were to (erroneously)assume that debt is risk-free and use equation (16.13) to compute the expected return on equity, we would obtain

t

1O0

200

3O0

250

($)

(%)

Expected Return on Equity

-

=

1

.

0

-

'

-

Equity return, risk'y debt return, risk-free debt

t,qu j ty

h 'i

50

''.

x. .

'E

=

1 0.06 + 27.07/90

(0.1 -

0.06)

0

= 0.1929 This calculation

gives an expected retul'n on equity substantially greater than 15.77%.

%

This example computes expected returns for a particular leverage ratio. As the becomes tirm more levered, equity-holders bear more asset risk per dollar of equity. lf assets have a pos itive beta the expected return on equity 'will increase with leverage. ,

I '

0

50

N.

.

1

100

I '

150 Asset Value

I

200

'

l

250

'

I

300

(s)

The top panel graphs the debt-to-asset ratio as a function of the asset value of the firm, using the Black-scholes formula to compute the value of the dbt. The bottom panel graphs the expected return on equity as a function of the asset value of the firm, using equations (16.13) and (16.8). Both graphs assume that there is a single zero-coupon debt issue with maturity value $100 and 5 years to maturity, and also assume that r 6%, (y' 25% (forthe assets), and t 0. =

=

=

510

k. CORPORATE

ArapLlcvlorqs

EQU ITY, DEBT,

difference is less than 20 basis points. For a very highly levered (low asset however, the difference in Figure 16.2 is dramatic. Conflicts

between

debt and equity

value) 'lirm,

511

Multiple Debt Issues The option-based model of debt accommodates multiple issues of zero-coupon debt with different seniorities, as long as a1l debt expires on the same date. By de:nition, more senior debt has priority in the event of banlruptcy. Suppose that there are three debt issues, with mamrity values of $30, $30, and $40, ranked in seniority from highest to lowest. We will refer to each distinct level of seniority as a tralche. The value of equity will be the sarize as in Example 16.2, since it is still necessary for equity-holders to pay $100 t receive ownership of the assets. However, the option pricing.approach permits us to assign appropriate yields to each level of debt. Senior debt-holders are the first in line to be paid. They oFn the 'lirm and have written a call option permitting the next set of bondholders to buy the firm from them by paying the mattlrity value of the senior debt, $30. lntermediate debt-holders own a call option permitting them to buy the rm for $30, and have sld a call option permitting the junior bondholders to buy the firm for $60. Junior bondholders in turn own the call option to buy the 5rm for $60, and have written a call option permitting the equity-holders to buy the finzl for $100. The values of these ptions

The idea that equity is a call option on the 'lirm

and that corporate bonds are risky provides insights into relations between debt- and equity-holders. Since equitpholders control the firm, bondholders may be concerned that equity-holders will tke actions that would harm them, or may fail to take actions that would help them.

There are two decisions equity-holders make that affect the relative value of debt equity-holders can affect the volatility of assets. Equity-holders can - increase asset volatility either by increasing the operating risk of existing assets, or by xasset substitution.'' replacing existing assets with riskierassets. An increase in volatility, other things equal, increases the va fue o f the equity-holder's call option and therefore reduces the value of debt. In Example 16.2, the vega of the equity is 0.66, so an increase in asset volatility of 0.0 1 leads to an increase in the market value of quity of $0.66, which is 0.66/27.07 = 2.4% of equity value. Debt value woul decline by $0.66. A second decision that equity-holders can make is the size of payouts to shareholders, such as dividends and share repurchases. To see why payouts are a potential problem forbondholders, suppose that the 51711makes an unexpected one-time $1 payout to shareholders. This payout reduces assets by $1. The delta of the equity with respect to assets is less than one, so the value of equity declines by less than $1. Since the value of debt plus equity equals assets, the vtp/lfc ofthe debt inl/.? declilte !?,),olte less the delta oftlle Unanticipated payotgts to equity-holders therefore can hurt bondholders. Bondholders are well aware of the potentially harmf'ul effects of asset substimtion and dividends. Bond covenants (legalrestrictions on the frm) often limit the ability of the firm to change assets or pay dividends. Viewing debt and equit'y as options makes it clear why such restrictions exist. Bondholders also encounter problems from actions that shareholder fail to take. Suppose the has a project worth $2 that requires shareholde's to make a $1 investment. If shareholders make the investment, they pay $1, the valtle of the assets increases and the value of the shares rises by 2 x A. The gain to shareholders is less than by the increase in the value' of assets. The difference of 2 2 x 2. goes to the bondholders. In making a positive NPV investment, shareholders help bondholders. The shareholders in this example only will make the investment if the value of shares goes up by more than the $1 they invest, which will only occurif A > 0.5. ln order for shareholders to be willing to invest, the NPV must be great enough that shareholders gain after allowing for the value increase that is lost to debt-holders/ Thus, because of debt, the shareholders may fail to make positive NPV investrpents. A relatd problem is asset substitution: Shareholders might make negative NPV investmets ihat increase asset risk, thereby transfening value from bondholders to stockholders.

$

AN D WARRANTS

and equity. First,

re

$30, 0.25, 0.06, 5, 0) BSCa1l($90, $60, 0.25, 0.06, 5, 0) BSCa1l($90, $100,0.25, 0.06, 5, 0) BSCall($90,

$67.82 $47.25 $27.07

=

=

=

(16.14) (16.15) (16.16)

Table 16.1 summarizes the value, yield to maturity, and expected return of eacfl tranche of debt. The junior tranche has a yield to mattlrity of 13.69%, very close to the required ret'urn on equity. The senior tranche, according to the model, is almost risk-free. The expected returns in Table 16.1 are computed using option lasticities. To illustrate the calculation, consider thejuniorbond, which is created by buying a 6o-strike call on the assets of the tirm and selling a loo-strike call. The two option elasticities are and 2.4432 tloo-stlikel. Using the fact that the elasticity of aportfolio 1.7875 t6o-sttikel

'tzllf/y.

'lirm

J

'$2

-

.' ''.'

i1iI:Eqq):lIi'. -.IEp!!!:' ;''(' E .11::)* tf j'; 1. .iIItEIIII::I'. t')' ?'. :liilpi;;d rtf #.(771I1iE-77q.' Jiti!!ijif y' ytt'@'.' tlk:tEljli.d ii;f 11* Ii:::::.'' j'yk;jjr'y.'j;.)f);'1''')' . y()j ': '('!.((q: ' .y (Ili)'. lll' 1* tyy'yjytk,,'. y)' 71*r)-yt. lrqqrr!:q t'). t'y '.11)::* y' .y' yf q'ql'. . . (!;),,'. ttf ) .j y j-. q.y ;.y ). . r. y.. j;yg ;. .yyyj jj.y - i..,. ,q-, (. y. jy g y 1. . 1177y y j jy y y g . j y .jy ,y y gy . yq j .yrjk (.... y ' .. .. ! y . . yj j,,,.. .. j ..y jg.-::y,a ii!:.. .!q;.);. j t.i .. y . 2 . . j,. jj-------,-,!jj r r : .jty.,-t:.$(:jj; yj. ;y..... . . :'r'-' -;' :'y'y' ;'. j'-' j;:y,,f -.:!i!!!f -y' '('jrf 5(44r5/rg* ljf (' 114tqq* 11:::::11* il::::::f ')'. .1:::::11* ,'lt'll ij-''/'yi 'Y ;')))' :,.::.1.*. ';'.lll...ll...:llh'. t'j,k' k))f :!!!;;' i;jj;,jf :!i!i;' y' :'q'(IE'TE')X );' (ilIr:II1.ti!iiI;'. (.,11:--,(111:*. (:1(E!11..(411)*. tI;:i!r:'. j'(')':11122* E: : . E!EE'''E y' li.,,..lf :!!E;;' .j1::y' .-* -.EE!,,' .y''q'y )' )'(qq-. 'q'' '))' '(E!E'iE.; IE : (j(E'!g('(. ! (!: ;! .EE:-''('(.''' ':';(':'E'.(E:.(E((((jE.E jq '.E.- :j :' EE). ' ):.(: l(i(! .' . .(... ' .(.. .i ( (i. .'g. . '. r . . ( .. E . . -i - . .. ; - - -; :. . E'i . ;;. .. . , -. -, y. ,-) ... ; .. . .:,jjjjjy. ,--q-)jjr .-111-..111k;,-.* .111,::,1:.* '.-1111.-;;'-.'*. :'.'''.,' -' .',-k-,,-,,f ,'

'.' ijli(,-'. ry'. yyyjj(y'; t't))-'. tlillkjppl:;d L'.-6iiLLkibLL-, .'kkjjjjjjjj;'.g . .,j .- .-.,j 1.. iiiii!!iili.

.' :

'

......E

.

.

. -y;;jjjjy. . .

:,...;..:

l

-

.

-. .

...

..

..

.

-.:

. .

.....

E

. .. .

-

. -' . ' . E - E E. E. .

-. .

'

.

-

. '

.

(:' . E' !.'. : . g. yjE( .

.

.

.

.. . .

.

.

:

.

.

'

-

. . .. : :. . . ... .

'

'

-

-

' .

.

: :' :.

.

.

Senior

.

.

.. .

-

.. .

.

. .

- '. . . .. .

.

LE E ('

i .

-y

.

2.

-

..

. .

t

.

-

.

-

.

-

.

.

. ..

.

. . ... .

. .

-.

Equity

:

. ..

.

.

.

bnds

'

. .

-

.

. ..

: ' . : .:

.

. .

lntermediate bonds

-.

.

-

.

-

:

-y-r - -.

- -. . ..

-

' . . . . -- . -. . . . :. ' : ...

''

' . . . .. . . . .'

-

:

y

Junior bonds 'lrf'heidea that the debt may harm investment incentives is developed in Myers (.1977).

g

-. - -

-.-

2

. -...----

.......

'

.. . (......EE.E'. ;E.... q... .'. E. . E. '-; 'E E q.. . ('(.(j.' k ( (..... yg.j' .'y'.(g.' j' .... ;.y((..r'g.y..'..'..j.' !.. .'Ej...y...'(j.. ...... (. j jj jj ( k. y ( j ( ; q... ,.-...yj.. j j,. qg-. .;.......;.;E... .. i....;j. rr r.r .. ..rr . ....k).......... ......,11!. . .r ....... .... .. .. .... ... ... .... ...

.7iy.'..'..l'

.

. -..

.., .. ...........

..

..

.

'(

' '

:

:

.

.

.

:

:'

:

:

.

.

,,'.

tranches.

'-.. i... ....(. (

g

.

E

e 11( a I.....llie k;,-'k::. jlq::::,,, e ll:... e e .11)::. k:!:,,--,.

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.-,.

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. ' - E . . . . . E. . - .. . . -. - .. '' ' . . .. . . .. '' ' . : . . .. . . . : . . .i ' .. . r . . . .

. -. . - . .. . . . ..

-l,s!i,,r Il..,,,l

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.

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E

'.'-!!ii1(ir-''.

r. ..

.

.

. .

-

y .. : ...

. ..

;

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'

-.

''

... E.g.. ......... ( .y ... .. . .

. .

.

-

. .. . .

. .. :.

. . . .. : j. . .

E :

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-.

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.

.

. .

. . .

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.

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. . . . . . .

jtj jyj ;(. jjr gy

.

.

.

.

.

' ' . . . - . - . . . . : .. :( . .

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yj

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j...... . .

6.042

C(30)

22.18

C(30)

C(60)

20.57

7.54

7.03

C(60) C(100)

C(100)

20. 18

13.69

9.63

6.04

g. .j. . ..

:

..

.. . .. .

y..

.

(y; j.....

......... ... . ........ . . . . ......

Assets

27.07

t.ls!il .... y

'' r::lii;i:E 11iEi!i: . 'E. E' .111;2:)1,.

-

.

. . .. .. . .. g . j... .. . . yyjuyjjyyy

. . .

15.77

%

CORPORATE

EQU ITY,

Appl-lcA-rlolqs

is a weighted average of the junior bond is

elasticities

47.25 x 1.t875 47.25 27.07

of the portfolio components, the elasticity of the

-

47.25

x 2.4432

27.07

-

=

0.9077

The expected return on the junior debt is therefore

nunior 0.06 + (0.10 0.06) x 0.9077 =

-

0.0963

=

Table 16.1 makes it clear why debt cannot be treated as a single homogeneous when class sl'mswith complex capital stzucttzres enter bankruptcy. The interests of the most junior debt-holders may well resemble the interests of equity-holders more than those of senior debt-holders.

Warrants Firms sometimes issue options explicitly. If a 'Iirm issues a call option on its own stock, t'warrant'' is used here to denote options on a it is known as a warrant. (The term practice 111-111 the term includes traded options issued issued by the firm itself, though in the warrant-holder pays the firm the exercised, in lixed supply.) When a warrant is strike price, K, and receives a share wort.hmore than K (or else the holder would not have exercised the warrant). Thus, the act of exercise is dilutive to other shareholders in the sense that the 111411 has sold a share for less than it is worth. Of course, existing shareholders are aware of warrants outstanding and can anticipate this potential exercise. The problem is how to value the wrrant, and how to value the equity given the existence of warrants. This valuation problem does not arise with ordinal'y options, because they are traded by third parties and their exercise has no effect on the fin'n. To see how to value a warrant, suppose the firm has 11 shares outstanding, and that asst value the outstanding warrants are European, on nt shares, with stlike plice K. is A exercise the warrants, they pay K #er share and At expiration, if warrant-holders :4 ??l After the + ,?7#, shares. wvants are exercised, the firm has assets worth reeive worth exercised warrants are hence 11 A A + /71:7 K K = (16.17) rfhe

-t-

11

lll

-

-

ll

??l

The expression A/?2 is the value of a share of equity in the absence of warrants. I'hus, eQuation (16.17)suggests that we can value a warrant in two steps. First, we compute price, ignoling an option plice wit Ajn as the underlying asset and K as the strike/7/(?1 correction dilution factor, + ??/). This dilution. Second, we multiply te result by a exercise the changes second step accounts for the fact that warrant numer of sltar:s price of K. The watrant issued at a outstanding, with the new shares Black-scholes formula: can be valued by using the E:

(16.19)

/2

This expression is different from equation ( 16.12) for warrants, because rather tha injecting new cash into the 'Iirm when they convert, the bondholders instead avoid taking cash out of the 'lirm. Conversion occurs if the assets increase suciently in value. lf the assets decrease, the firm could default on the promised maturity payment. Assuming the convertible iq the only debt issue, bankruptcy occurs if assets are worth less the promised payment to all convertible holders, or Aw < I'tM. The payoff of the convertible at maturity. time F iS '

,

???#.- mllx (0,IIIM

-

-

-l-

p7

AN D WAR RANTS

Convertible Bonds In addition to issuing warrants directly,

27.07 -

D EBT,

Bond

'Written put

z4rl

+ nlq x

12 ??+

nlq

0,

x m:tx ,32:/

Aw

M -

pl

q

/1

+ nla ,?

(16.20)

Purchased warrants

Thus, owning ??7 convertibles can be valued as owning a risk-free bond with maturity paymept mM, selling a put pn the firm's assets, and buying nlq warrants with strike Mjq x n + ??2t?)/n. Equation (16.20)can be rewritten as

max min lllM, Awl

nlq

,

-1-

?2

lllq

Av

(16.21)

This version of the convertible payoff can be interpreted as follows: Shareholders give bondholders the least they can lrninlN, Aw/??2)); if it is optimal to do so, convertible

k

514

Co

Rpo RATE APPLI

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holders can then exchange this amount for the conversion value, which is their proportionate share of the assets lnqjfn + lnqj x Aw) .

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AN D WARRANTS

D EBT,

515

Suppose a firm has assets of $10,000with a single debt issue consistingof six zero-coupon bonds, each with a maturity value of $1000and with F = 5 lisk-free = rate is r = 6% The yearsto maturity. The asset volat ility is o' 30% and the makes no payouts. 51-111 If the single debt issue is not convertible, the price is

Example

16.5

.

1* 11* )1 Example ''

Suppose a firm has issued ?n = 6 convertible bonds, each with 16.4 = mamrity value has ,7 400 M $1000and convertible into q = 50 shares. The 111-111 ttl ') common sh ares ou tstanding. Figure 16.3 shows the maturity payoff for the aggregate j 7,;) j',yvalue of the convertible bonds, comparing it with the matulity payoff of otherwise y', identical nonconvertible bonds issued by the same firm. The six bnds have a total ) promised mattu'ity value of $6000,so default occurs when assets re belw that level. Eqton (16.19)implies that conyeron occurs when assets exceed $1000x 700/50 = . I The slope of 4heconvertible payoff above $14,000is lnqln 3/7, + nlq) i itr .tijj . . because convertible investors share gains with existing )t) less than the slope . in default, . 1;) 't,')shareholders, but once in default, convertible bondholders bear additional losss alone ' )j (in default, shares are already worthless). %. 1j :

=

E

.

t14,000.

=

.

.

x $1000 .

d-0'06x5

i!!l..f .li..;.ltf )' (s'-li'':jq)f .--L'' ;'!!' ld t'r' 51:. :,ii;;' j't);f j(' )1* )' !i'1i:::;ki..' y' r' (' ).()' tf ..' !' y' jt'. (,111E!.;* )' 'Iii::-k.IIi'. j'j)' yyf tyf (' ki.'IEiit;C'. --':EiE!!!i-'..' ''ti!Ii'. '''jkd likEili:.f IEIt:'ili::'. 111:-* y' '1* ' (E E E' !E'' E E E' : t' !i '' i1: (r5r( ':..plE;ljq..(j'('E.E'.E(.(.'(..(.'E..(......k..'r(i./' g.iq.E.(j.. ..'(. .'...! .(('.('..' .yqq)''1:S'j(.(''':..'g'.(q.(g.yri'.('!' .(!( E. E; (!!jtE . j. E! E 'E E iI..(;p: E .q ;(. .EE ; E (.. j.r.E..... .q '.. E.( ; qE.. r. E ( E , . qq( . !i' @ . .tj. r.. (..j y..!. !...-';y j) ; ! . jy jy jj jj).yy..jEqyj.. y i!--'1IEI:Fp:. ;q(.. y..y.y. j -. j ..-y.r.j.q.....y.yy;.yy.-y.y E. . . .-. . . t-r:..-. .)......E.yy ..; ....'...;...--... E'..-;,...i r :(. . ..-. ..'... . .. . . . . . . . ;')' :j' -j' -1:::11.* .--)'' :r' E' j';'!' d.:ii!,f :'-'-k:;'. '.' j'-'''-.'F' !' l':d f' 'E' (' )'

(rlhf r'. (' y' t'. t'yyj'. '' tyf ttyf ( t .

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.

.

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-

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14:2.

18,000

value of an aggregate bond and a ordinarybond, using

convertible

.

..

6 x $1000 x + 6 x 50 x

e-0'06x5

-

Bsput

400 zo()

-y.

y x 50

14,000

z

12,000

conversion (Slope

=

$1000,0.30, 0.06, 5, 0

6 x $1000,0.30, 0.06, 5, 0) ($10,000,

x Bscall

$10,000 $1000400 + 6 x 50 0.30, 0.06, 5, 0) zo() zo(j

(

,

=

3/7)

'

8,000

!

4,000 2,000 0

!

Default

(slope 5,000

=

1)

Assets

514,000

=

!1

10,000 15,000 20,000 25,000 30,000 35,000 40,000 Asset value at maturity ($) .

#

,

5:

$1000 x (400+ 6 x 50)/50

=

$14,000

Convertible bonds are typically issued at terms such that a signitieant increase in the stck price is required for conversion to be worthwhile. In Example 16.5, each bond gives tlle holderthe right to convert into 50 shares, so the jtrike price is $1000/50 $20. the ratio of te strike price to the stock price, which is Since the stock price is $11.809, 69.4%. In practice, conversion called the conversion prenzinm, is $20/$11.809 l premiums are most commonly between 20% and 40%. Why do hrms issue convertible bonds? One explanation is thit convertible bonds resolve one of the conoicts between equity- and debt-holders. Shareholders can tale value from holders of ordinary bonds by increasing volatility, even if this action has as a whole. However, convertno benefcial effect from the perspective of the 111311 ibles are harmed less than ordinary debt by an increase in volatility and may even be helped. Financing with convertibles instead of ordinary debt thus reduces the incentive of shareholders to raise volatility. In practice, valuation of convertible bonds is more complicated than in this example. First, convertible bonds are typically American options, convertible for much of the life of the bond. Second, convertible bonds typically are not zero-coupon. The p

=

-

6,000

$701.27

.

# z

=

= Each bond has a price of $879.39. The yield on a bond is 1n($1000/$879.39)/5 bond because otherwise equivalent nonconvertible is below the yield on an 0.0257. This of the conversion option: The bondholders have a call option for which they pay by t:: accepting a loweryield on the debt. The value of a share is ($10,000-$5276.35)/400 $11.809. Bondholders will convert if at mattll'it.y the assets are worth more than

M x n + mqllq

Ordinary bond Convertible equity

'

16 #ooo

the parameters in Example (16.5).

..

6

'

,

= 0.0710, greater than the The yield on the nonconvertible bond is ln(1000/701.27)/5 lisk-free rate because of the risk of bankruptcy. Suppose the debtisjue is instead the convertible bond described in Example (16.4). Using equation (16.20),the value of all convertible bonds is

= $5276.35 Just as we valued ordinary zero-coupon bonds with the Black-scholes formula, we can also use the Black-scholes formula to value a bond convertible at mattlrity.

$10,000

ssput

-

=

516

%.Co RPORATE

Appt.l CATI olqs

EqulTv,

payment of coupons complicates the analysis because banltruptcy becomes possible at times other than expiration and the reduction in assets stemming from the payment of the coupon (or any other cash payment to the 5rm s security-holdrs). This coujori feduces assets but is also paid to the bondholders. Third, many companies pay ciividnds. If the dividends that can be earned by converting the bond into stock exceed the bnd coupon, there can be a reason for bondholders to convert before maturity. Finally, intefest rates and volatility can change, and the circumstances of default may be more conzplicated than we have assumed (for example, when there are other debt issues, default could occur prior to maturity). The value of the bond can then change for reasons other than stock plice changes. It is possible to value convertible bonds incorporating early exercise, dividends, It is also possible to incorporate interest rate risk using techniques and callabiliys described later in the book. ''

'''

''

'

'

*

Callable Bonds Many bonds are callable, meaning that, prior to maturity, the company has the right to give bondholders a predetermined payment in exchange for the bonds. The idea underlying a callable bond is that the bond issuer can buy the bond back at a rlatively cheap price if itbecomes expnsive. Abond can become expensive becguse interest rates have fallen. in which case the issuer would like to xchange the existing bondj for new bonds carrying a lower coupon rate. A bond can also become expensive if the market percives that the company is-less likely to default. Again, in this case the company would like to exchange the existing bond for a newly issued bond with a lower default premium. A tirm can always buy bonds back at a market price, but there is generally no advantage to doing so. Callability permits the company to buy bonds back at a relatively low price. Of course the company pays for this light by receiving a lower price when it issues the bond. The predetermined price at which a company can call a bond is specifed by a call schedule. Bonds are typically noncallable for several years after issujmce, a period which the bond is said to have call protection. Callability is, in effect, an option ' that investors sell to the bond issuer. In return, the company issuing the bonds pays a higher yield on the bonds. 'duling

Convertible bonds are typically callable.

When the issuer calls a convertible

bond, the holder of the bond has the choice of surrendering the bond in exchange for the call price or converting. Callability is a way for issuers to shorten the life of the bond, potentially forcing holders to convert prior to maturity. The issuer--acting in the interests of shamholders-follows a call strategy that ntillintizes the value of the bond. Bondholders, by contrast,

bond.

follow a conversion strategy that nlaxinlizes the value of the

DEBT,

AXD WARRAXTS

Since bonds can be called prior to expiration, we cannot value the call provision

using Black-scholes. However, we can value a callable bond binomially. The call strategy for an issuer is like the exercise of an American option: The issuer calls if it is more valuable to do so than to leave the bond outstanding for another period. Figure 16.4 presents binomial valuation examples for four bonds: an ordinary bond, a convertible bond, a callable bond, and a callable convertible bond. We examine noncallable bonds as benchmarks in order to better understand the effect of callability. We value these bonds using the binomial pficing model outlined in Chapters 10 and 11. In each case, the assets of the firm are the underlying asset, pnd the value of the bonds is determined by the value of the assets. To perform the valuation, we move backward through the tree. as in Chapter 10. The assumptions in Figure 16.4 are the same as those in Example 16.5, so the results in the example and figure are comparable for the ordinary and convertible bonds. bonds We first consider the binomial valuation of an ordicallable nonconvertible bond (Panel B noncallable Figure 16.4) and its callable countelpart (Panel D in in nary, Figure 16.4). The yield on the callable bond (7.57%)exceeds that on the noncallable bond (7.29%)because of the option that bondholders have written to the f-m. How are bond prices in Figure 16.4 computed? For the ordina:y bond, the value in year 5 is Bs

Il'tintl/l

=

A'f, A5)

=

min($6000, A5)

There are six $1000bonds outstanding, and shareholders will pay to these bondholder's the value of the fil'm or $6000,whichever is less. Note that default occurs at the bottom node in year 5 since the value of assets is less than the required bond payment, $6000. Prior to mamlity, the value at each node is calculated as in a typical binomial valuation,

with

-r/;

p

?

=

e

f+?, Eps+

..y

(j

-

#)y-f+?, g

For a given node at time /, Bt*+ is the bond value if assets move up from that node, anct Bt-+iti's the bond value if assets move down from that node. The binomial valuation of the callable bond is straightforward. Let A7all de'note the call schedule. The call plice in this example is set to equal te price of a bond yielding 6.75%. At time 1.67, for example, the company could call by paying the bondholders $4791.10, which, for a bond maturing in 3.33 years, is a yield of 6.75%. We assume that rate is fixed; tus the only reason for the company to call the bond is if the te lisk-free company could issue replacement bonds at a lower coupon due to a decrease in default

risk.

Shareholders wish to minimize the value of the bonds, hence their value is Bt

=

min ( Leave the bond outstanding,

call j

-r/'t

'

(1977)and Brennan and Schsva.rlz(1977)discuss

the valuation of convertible bonds.

A'callj

p)B- J+/, ) t :+1, + (1 P B+ (j6 p,2) = min 11'you compare the binomial trees in Panels B and D, it is apparent why the callable bond has a lower price at issue than the ordinary bond. At the top node in year 1.67, -

Slngersoll

%.

'

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518

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Rpo RATE AI7PLI CATI

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Year:

A: Firm assets

0 10000.00

1.67 16279.12

7502.88

3.33

5 43141.27 19883.36

26500.98 12214.03

5629.32

9164.04 4223.61

the noncallable bonds are worth $4912.38,for a yield of 6%. (1fassets reach that node, default will not occun) The ;rm calls the callable bond at that node since it is now possible to issue default-free debt. The prospect of this call prevents the bondholders from receiving a capital gain. This in turn lowers the initial plice of the bond. Problem 16.15 asks you to compute share prices at each node so that you can see the effect on shareholders of the different bonds. Callable

4166.82

4912.38 4396.40

5429.02 5429.02 4471.64

6000.00 6000.00 6000.00 4223.61

N/A

4791.10

5361.58

6000.00

D: Callabl bond $684.86 Price Yield = 7.5t%

4109.14

4791.10 4371.73

5361.58 5361.58 +471.64

6000.00 6000.00 6000.00 4223.61

E: Convertible bond

5324.34

B: Ordinary bond Plice = $694.47 Yield = 7.29%

C: Call schdule

=

Price Yietd

=

$887.39

=

2.39%

11357.56

8521:44 6000.00 4223.61

4471.64

F: Callable convertible Price = $818.14 Yield = 4.01%

bond

4908.85

6976.77 4371.73

Binomal valuation of a callable nonconvrtible

11357.56 5361.58 4471.64

18489.12 8521.44 6000.00 4223.61

and a callable convertible bond. The

assumptions are the sam as thos in Example 16.5. The binomial tree for assets in d Parlel A is generated using equaton (10.1 0) (a forward tree) with u 1 0.7503, p* 0.4044, T= 5, and three binomil time steps (hencethe time between 1 In each case there are 6 bonds outstandinj with a binomial periods is h 5/3 total maturity value of $6004. Convertible bonds conveft inyo 50 shares. The yleld for each bond is coniputed aj In(600/&)/5, where Bois the time 0 value of the six bonds. The price is Bo16. The call lcheule in Panel C is the price of a zero-oupon bond maturing in year 5 and yielding 6.75%. Callable bonds are call-protected until year Prices in italics denote calls of the bond; prices in bold deriote conversions, and 1 prices in bold lt'tzllcx denote conversions in response to a call. .6279,

=

=

=

lt + nlq

A5)

In Panel E in Figure 16.3, bondholders convert at the top two nodes in year 5, receive the mamrity payment in cash at the next node, and the firm defaults at the bottom node. Prior to maturity, the convertible investor values the bond as the gfater value of letting the bond live one more period:

of its

conversion value and the =

18489.12

6351.59

lltq

maxgmintplf, A5),

B: 7578..78 4733.96

We now consider noncallable and callable convertible

bonds

convertible

bonds, Panels E and F in Figure 16.4. Note 'Iirst thar, as in Example 16.5, the yield on the convertible noncallable bond (2.39%)is lower than that on the ordinal'y bond (7.29%) because convertible bondholders receive a call option and pay for this wlth a lower yield. Using equation (16.21),the year 5 value for the convertible bond is

=

max ( Continue to hold, convert max

e-r

h

+ (pG+y,+ (1

)

p4Bt+), -

-

IIIC + 11 nlq

(16.23)

AJ

Bt is the total value of the bonds. You may recognize this expression as almost identical to equation (10.10)for valuing an American call option on a stock. The difference is that the payoff is the conversion value instead of St K. -

When the bond is both convertible and callable, there is a tug-of-war between

the lirm and the bond investors. We can imagine the bond value being determined as follows: The bondholders decide whether to hold or convert (maximizingthe bon valuel.. Given this decision, the ;rm decides whetherto call (minimizingthe bond value).

If the 'Iirm calls, bondholders revisit their decision about whether or not to convert (again maximizing the bond value, conditional upon the behavior of the srm).This chain of reasoning implies the following valuation equation:

.671.

=

=

Bt

=

max

(mingmaxtContinue to Hold,

= mlx min

+ e-r It LpBt-gt +

max

(1

Hold

.67.

nlq 11+

lllq

XJ

Convert), p) B:y),j, -

-

Call ) Covert

)

,

lll'q

11+ mq Convert

z4;

call

,

Kt

,

Call

(16.24)

Convert

As you would expect, the abilit'y to call a convertible bond lowers its price. raising its yield from 2.39% to 4.01%. By comparing Panels E and F in Figure 16.4, you can see

% CORPORATE

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APPLICATIONS

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why this happens. In year 1.67 at the top node, it is optimal to wait to convert if the bond is noncallable. This gives a bond value in Panel E of $7578.78.However. if the bond is callable using the eall schedule in Panel C, the fil'mwill call at the top node in year 1.67. In response, the bondholders convert, giving them 50 shm'es worth $6976.66.The bond is worth less because shareholders cannot delay the conversion. The fil'm does not call at the lower node in year 1.67 because the credit quality of the bond deteriorates at that node.

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An alternative approach, often used in practice, is to base valuation of aconyertible bond on a binomial tree for the stock, rather than on assets. A standaydbinolial tree converttle bond for the siok, hwever, will never reach a zero stock price. and thus a This raises the default-free. question: How valued on this tree will be prlced as if it were j risk plicing procedure into the can we incorporate bond default Tsiveriotis and Fernandes (1998)suggest valuing separately the bond income and thestockincomefrom an optimally managedconvertible bond. Theirprocedure accounts for default by discoting bbnd income at a rate greater than the risk-free rate while the component of the bond income related to conversion into stock is discounted at the .

It is possible to issue bonds that are considerably more complex than those we have considered. Conversion and call schedules can vary with time in complicated ways. Bonds can be pltttable, meaning that the investor can sell them back to the 51711at a predetermined price. Bonds can pay colttillgellt interest, meaning that if a particular event occurs, the interest rate on the bond changes. As we saw in Chapter 15, particular structures are often a responje to tax and accounting considerations. Another example is the use of contillgelt ctulvcrrf/p/d bonds, alSO known as Firms report earnings on a per-share basis. For a til'mthat has issued only shares and ordinary debt, computing earnings per share (EPS) is straightforward since there is outstanding. However, if a :rm has issued no ambiguity about the number of shares should the 5rm use in computing EPS? shares eonvertible debt or warrants, how many the firm must compute the worstin the United States, Under linancial accounting rules issued a convertible seculity, the firm has When earnings share. per case fully diluted interest after-tax earnings back adding on the converible, and to this generally means outstanding. convertible shares shares to adding the number of ''

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The box on page 521 discusses the reaction of one company to the FASB'S ew rule, and makes it clearthat the accounting treatment of a seculity is important for issuers.

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When shares are used to pay employees (as for example with compensation options), there is an increase in the number of shares outstanding. Companies making heavy use of share compensation frequently buy shares back from other shareholders (a share rcpl/?'c//t7lclso there is no netlncrease in the number of shares outstandinps Many companies that repurchased shares during the 1990s also sold put options on their own stock; a commonly tated rationale for issuing such put warrants (see, for example, Thatcher et al., 1994) is that the put sales are a hedge ajainsi tfi cost of repurclasing shares. lntel, Microsoft, and Dell, forexample, all sold signifcant numbers of puts, with Microsoft alone earning well over $ 1 billion in put premiups duripg the 1990s. Here is a quote from Microsoft's 1999 l0-K desclibing the put prtgram: Microsoft enhances its repurchase program by selling put warrants. On June 30, 1999, l63 million warrants were outstanding with strike prices rangingfromssg to $65 pershare. Theputwarrants expire between September 1999 and March 7002. The outstanding put warrants permit a net-share settlement atthe Company's option and do not result in aputwarrant liability on the balance sheet. .

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16.3 THE USE OF COLLARS IN ACQUISITIONS

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A common financial transaction is for one firm (theacqltil'el') to buy another (the taqe by buying its common stock. The acquirer can pay for these shares with cash or by exchanging its own shares for target firm shares. Collarlike structures are frequently used in these tl-ansactions. Suppose that underthe purchase agreement, each target share will be exchanged for shares of the acquirer @ is the exchallge ratio). Once the target agrees to the purchase. -'r acquisition will generally take jme to complete, often six months or more.20 Target the will be concerned that the acquirer's stock will drop before the merger is shareholders in wlzich completed, case the dollar value of acquirer hares will be lower. To protect prie it is possible to exchange whatever number of shares have a fixed drop, against a value. (For example, if the acquirer price is $ 100, exchange l share for each target dollar If the acquirer price is $50, exchange 2 shares for each target share.) However, share. also shareholders wish to participate in share price gains that the acquirer target may xing the exchange ratio rather than the dollar value. There experiences; this suggests offer four structures that address considerations such as tlrlese--zl common are

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Figure 16.9 illustl-atesthese four types of acquisition offers. As this list shows, it is possible to modify the extent to which the target bears the risk of a change in the stock plice of the acquirer. More complicated structures are also possible.

The Nortlzrop Grumman

TRW merger

Northrop Grumman's 2002 bid for TRW is an example of a merger offer with a collar. ln ' Grumman and TRW aglee d thatNorthrop would pay $7.8billton for July 2002, Northrop TRW. News headlines stated that Northrop offered per share,'' but the offer actually resembled a collar. The number of Northrop Grumman shares to be exchanged for each '

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TRW share would be determined by dividing $60 by the average Northrop Grumman plice over the 'live days preeding the close of the lnerger, with the exchange ratio to be no less than 0.4348 ($60/$138)and no more than 0.5357 ($60/$l 12). Thus, if the price of Northrop Grumman at the merger closing was below $ 112, TRW sllareholders would receive 0.5357 shares. If the price was above $ 138, TRW shareholders would receive 0.4348 shares. lf the price, S, was in between $ l 12 and $ 138, TRW shareholders would receive $60/5, wllich is $60 worth of shares.22 The deal closed on December 11, 2002, Gruluman was $96.50-,TRW shareholders therefore when the closing price of Morthrop

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20Inmany cases- forexample, regulatory agencies examine the acquisition to see ifit is anticompetitve. 21Fuller (2003)discusses the kinds of offers and the Inotives for using alternative kinds of collars.

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I N ACQU ISITI ONS

22Tl)e acquisition terms changed between February and July. Intially, Northrop offered to exchange $47 worth of Nol-thropGrumman stock for each TRSV share: The number of shares to be exchanged was to be no more tllan 0.4563 ($47/$l 03) or less tllan 0.4159 ($47/$ l l 3). In May, tlle value of the bid was increased to $53 (no more than 0.4690 sllares ($53/$l 13) r Iess than 0.4309 ($53/$123)

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Suppose that TRW

shareholders were certain the merger would occur at time r, but uncertain about the future Norththrop Grumman stock price, S. TRW shareholders could then value the offer by noting that the offer is equivalent to buying 0.5357 shares of Northrop Grumman, selling 0.5357 llz-stn'ke calls, and buying 0.4348 l38-strike calls. ln adition, the TRW shareholders would not receive Northrop dividends paid prior to closing and would continue to receive TRW dividends. The time t value of TRW shares would then be

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Panel A graphs equaton (16.27) and the TRW stock price from Julyto December. Panel B graphs the absolute difference between the two Iines in Panel A.

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($) 241twas necessary for Northrop Grumman and TRW to receive regulatory approval from European authorities. the U. S. Department of Defense and the U. S. Department of Justice. Final approval from the Department of Justice was on December 10 and the merger was completed on December 11.

dividend yield of 1.2564, and volatility of 36+.

25Mitchell and Pulvino

(2001)examine the historical returns

earned by risk arbitrageurs.

.,

542

k

Conpo

RATE

Northrop. If the offer succeeds, tle position earns the difference in plice depicted in Figure l 6. l l ; if the oflkr fails, the difference should diverge and the arbitragetlr would

lose money.

Three col-porate contexts in which options appear, either explicitly or implicitly, are capital strtlcture (debt,equity, and warrants), compensation, and acquisitions. lf we view the assets of the firm as being like a stock, then debt and equity can be valued as options with the assets of the 'lirm as the underlying asset. Viewing corporate securities as options provides a natpral way to measure bankruptcy risk, and illuminates conoicts between bondholders and stockholders. Compensation options are an explicit use of options by col-porations. They exhibit variety of oluplications, some naturally occun-ing (earlyexercise decisions by riska employees) reloads, outperformance, and some created by the issuer (repricing, . averse multipliers). this they provide interesting For and context in wlaich to use the reason an pricing fonnulas from Chapter l4. exotic Oflkrs by one firm to purchase another sometimes have embedded collars. The Grunzman offer to buy TRW was an example of this.

FURTHER

READING

The idea that debt and equity are options was lirst pointed out by Black and Scholes ( 1973). Merton ( 1974) and Mellon (1977)analyzed the pricing of pepetual debt and deluonstrated that the Modigliani-Miller theorem holds even with (costless)batzltruptcy. Two principal applications of this idea are the detel-mination of the fair yield on risky debt and the assessnent of bankruptcy probabilities. Galai and Masulis (1976)derivrd the link between the return on assets and the return on the tirm's stock. The discussion of wal-rants and convertible bonds in this chapter assumes that the Op tions are European. .With merican wal-rants the optimal exercise strategy can be more complicated than with European options. The reason is that exercise alters the assets of the tirm. The problem of optimal American wal-rant exercise is studied by Emanuel ( l 983), Constantinides (1984),and Spatt and Sterbenz ( 1988). McDonald (2004)exam.t ines the tax ilnplications of wan'ant lssues, including put warrants. Complications also arise with convertible bonds, which in practice are almost always callable. Thus, valuing a convertible bond requires understanding the call strategy. Classic papers studying the pricing of convertibles include Brennan and Schwartz (1977)and Ingersoll ( 1977). . Hanis and Raviv ( l 985) discuss how asynzmetric information affects the decision to call the bond, and Stein (1992)discusses the decision to issue convertibles in the srstplace. Finally, there is a large empirical literature ol1 the converyible call decision-, for example, see Asquith (1995). Giintay et al. (2004)examine the decision to issue callable bonds. Papers on colnpensation options include Saly et a1.. (1999), Johnson and Tian (2000a), and Johnson and Tian (2000b). Replicing is studied by Chance et al. (2000) and Acharya et a1. (2000). Petrie (2000)examines the use of collars in acquisitions.

%

PRoBuEus

Appl-lcvlorqs

543

PROBLEMS For all problems, unless otherwise stated assume that the 'lirm has asjets worth A $100, and that o' = 30%, 1- = 8%, and the firm makes no payouts prior to the maturity date of the debt. =

16.1. There is a single debt issue with a matulity value of $120. Comjute the yield on this debassuming that it matures in 1 yeai, 2 years, 5 years, or 10 years. What debt-to-equity ratio do you observe in each case? 16.2. There is a single debt issue. Compute the yield on this debt assuming that it matures in 1 year and has a matulity value of $127.42,2 yearsgwith a matul'ity value of $135.30,5 years with a maturity value of $161.98,or 10 years with a maturity value of $218.65.(The maturity value increases Fith paturity at a 6% rate.) What debt-to-equity ratio do you obsel've in each case? 16.3. There are four debt issues with different priolities, each promising $30 at mamrity. a. Compute the yield on each debt issue assuming that a1l four mature in 1 year, 2 years, 5 years, or 10 years. b. Assuming that each debt issue matures in 5 years, what happens to the yield on each when you vary o'? When you vary r? 16.4. Suppose there is a single s-yearzero-coupon debt issue with a matulity value of $120. The expected ret'urn on assets is 12%. What is the expected ret'ul'n on equity? The volatility of equity? What happens to the expected ret'urn on equity as you vary A, c apd r? ,

16.5. Repeat the previous problem for debt instead of equity. 16.6. tn this problem we examine the 16. 1.

effect of changing te

assumptions in Example -

'

a. Compute the yield on debt for asset values of $50,$100,$150,$200,and $500. How does the yield on debt change with the value of assets? b. Compute the yield on debt for asset volatilities of 10% through 100%, in increments of 5%. For the next three problems, assume that a firm has assets of $100and s-year-to-mattil'ity zero-coupon debt with a face value of $150. Assume that investment projects have the same volatility as existing assets.

16.7. The 5rm is considering an investment project costing $1. What is the amount by which the project's value must exceed its cost in order for shareholders to be willing to pay for it? Repeat for project values of $10 and $25. 16.8. Now suppose the 51-111 financs the project by issuing debt that has Iower pliority tan existing debt. How muh must a $1, $10, or $25 project be worth if the shareholders are willing to fund it?

544

k

CORPORATE

AppLlcvlorqs

P Ro

16.9. Now suppose the firm finances the project by issuing debt that has higller pl4ority than existing debt. How mud must a $t0 or $25 project be worth if the shareholders are willing to fund it? 16.10. Assume there are 20 shares outstanding. the'shm-eprice for each of the following situations.

Comptlte the value of the warrant and

a. Warrants for 2 shares expire in 5 years and have a strike price of $ l5. b. Warrants for 15 shares expire in 10 years and have a strike of $20. 16.11. A tirmhas outstanding a bond with a s-yea.r maturity and maturity value of $50, 10 shares. also 20 There are shares outstanding. What is the convertible into price of the warrant? The share plice? Suppose you Were to compute the value of the convertible as a risk-fre bond plus an option, valued usinjthe Black-scholes formula and the share price you computed. How acctlrate is this? 16 12 Supp'ose ahrm has 20 shares of equity, a lo-yearzero-coupon debtwith amaturity value of $200,and wan'ants for 8 shares with a strike price of $25. What is the value of the debt, the share price, and the price of the wan-ant? 16.13. Suppose a firm has 20 shares of equity and a lo-year zero-coupon convertible bond with a matulity value of $200,convertible into 8 shares. What is the value of the debt, the share price, and the price of the warrant? 16.14. Using the assulnptions of Example 16.3. suppose you were to perfonn a bond plus 50 call options on the stock. valuation of the convertlble as a lisk-free cinaive''

How does the plice you compute compare with that computed

in the example?

16.15. Consider Panels B and D in Figtlre 16.4. Using the information in compute the share price at each node for each bond issue.

each panel,

BLEMS

%.

545

a. What is the value at grant of an option tlat will not be repliced? b. What is the value at grant of an option that is repliced when the share P

Iice

reaches

$60?

c. What repricing trigger maximizes the initial value of the option? 16.18. Suppose that top executives of XYZ are told they will receive at-the-money call optons on 10,000 shares each year for te next 3 years. When granted, the .opt ions have 5 ears to maturity. XYZ'S stock plice is $ l00 volatility is 30% and 1- 8%. stilnate the value of this prolnise. Hl-llt: Se Problelu 14.2 l ,

,

.)

=

16.19. Suppose that S 1, and J $100, c' 30%, 1- 6%, t 0. XYZ European put option on one share with strike price K $90. =

=

=

=

=

writes a

=

a. Construct a two-period binoluial tre for the stock and price the put. Compute the replicating portfolio at each node. b. If the lirm were sylzthetically creating the put (i.e., trading to obtain the same cash flows as if it issued the put), what tl-ansactions would it undertake? c. Consider the bank that buys the put. Wllat transactions does it tlndertake to hedge the transaction? d. Why might a firm prefer to issue the put warrant instead of bon-owing and repurchasing shares? 16.20. Firm A has a stock plice of $40and has made an offer for lirm B where A prolnises to pay $60/share for B, as long as A's stock price remains between $35 and $45. If the price of A is below $35, A will pay l 14 shares, and if the price of A is shares. The deal is expected to close in 9 months. above $45,A will pay l 40%, 1' 6%, and Assume t:r 0. .7

.333

16.16. As discussed in the text, compensation options are prematurely exercised or canceled for a variety of reasons. Suppose that conpensation options both vest and expire in 3 yeary and that the probability is l0% that the executive will die in year l and. l 0% in year 2. Thus, the probability that the executive lives to expiration is 80%. Suppose that th stock price is $ l 00, the interest rate is 8%, the volatility is 30%, and te dividend yield is 0.

=

=

=

a. How are the values 1 14 and l b. What is the value of the ofkr? .7

.333

an-ived at?

c. How sensitive is the value of the offer to the

volatility

of A's stock?

16.21. Firm A has a stock price of $40, and has made an offer for firm B where A promises to pay l shares for each share of B, as long as A's stock price remains between $35 and $45. If the price of A is below $35, A will pay $52.50/share, and if the price of A is above $45,A will pay $67.50/share.The deal is expected 0. to close in 9 months. Assume o' 40%, 1- 6%, and .5

a. Value the option by computing the expected time to exercise ging this into the Black-scholes fonnula as time to luaturity.

and plug-

b. Conpute the expected value of the option given the different possible times until exercise. c. Why are the

answers

for the two calctllations

different?

16.17. XYZ Cot-p.compensates executives with lo-year Etlropean call options, granted at the money. lf there is a signiscant drop in the share price, the company's board will reset the strike price of the options to equal the new share plice. The mattlrity of the repriced option will equal the remaining mattlrity of the original option. Suppose that

tr

=

30%,

1-

=

6%,

=

0,

.and

that the original share price is $ 100.

=

=

=

a. Hosy are the valtles $52.50and $67.50arrived at? b. 'What is the value of the offer? c. How does the

value of this offer colupare with that in Problem 16.20?

16.22. The strike price of a compensation option is generally set on the day the option is issued. On November l0, 2000, the CEO of Analog Devices, Jerald Fishman, received 600,000 options. The stock price was $.44.50.Four days later, the price

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rose to $63.25after an earnings release: Maria Tagliaferro of Analog said the timing of the two events (option grant and earnings release) is irrelevant because company policy is that no option vests until at least three years from its granting date. ttWhat happens to the stock price in the day, the hour, the year the option is granted is not relevant to that option,'' she said. stock price only becomes relevant after that option has vested.''z6 ltrf'he

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a. Using the inputs from the annual report, and assuming no dividends, estimate the vlue to the CEO of an at-the-money option grant at a stock price of $44.50.

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d. Do you agree with Maria Tagliaferro? Why or

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In its annual report for 2000, Analog Devices reported that the expected life of its options granted that year was 4.9 years, with a 56.6% volatility and 6% risk-free rate. The company paid no dividends until 2003.

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E' ' :' : E i : i . ' : .EE:' E: . 1?(j' E: E E' (E' ! E' ' E( :' ' iEE( (' !q iIjj:'E!! i ' !. i. (((' !'j(' E qrj: ::Ei E (E LLL. E :.E ;'j( '.(.: ('E. . ; '. i((;'5'! ( E EE' E . ( !. . (j ( E ' : : y' ('i.' j!Ij j)g''yykkkjk(((;k;y''. r'jyyyj (jtj -jk!. ljj.gyj: jjky:tj (EE.EE'(. -,-'ii): j.'.ij:iEk(('(E'E'j'Ej.(j':jEjjj'-.E.E--: Ej.: .kiEE.j'('j(EE'.E:ji--:' (E'(gEgj.!E: (..-Ej'-.j'g '. j ' : j.jy: jj j ' ). j jjEj.E..!j'r ..'i..(..jr'. (j.k j.j j.yg(j.(.'j'y-.' E'E;E.j.(..(k.y:j.;j(gjE-.ykj'i(';jj:jj' y'' jjj jjj (II(E''j'jijy ';'yE.j-;jj-;-ji (gi !-;r'j- -j (j -; ij g-. 'jjjjj jj;:kE( ' -!jjj jj(j j; (-j;-.(('. jjj)j' . .-- . ' E!1(- .1::-- ;-E ('( j'.iiy-t;'.-i-: j. -.',jjljjj), . -. ( : -.k j jg li --. j:'g jjtgtyyg '. ij. Ei;)kI! --;;-( ji(!. )k-iqi . .-.j.'j.-- -.-.-. (('i' j j'k'.j --'! .g g.j'jg(j..jj..j.gr . jq . .q'yj ; !-- ' j j.-(.r--..(i j--j ;'j.j-j(jjjjj jyjj'. -. . . -. -.,:(r -. k('. . jjjj.tr -.. -- -j,j-. -.;. -t. i. ..i - ;)'.--. il .-i -jr ; -. - -.- --. .. - -.. - i -i. - -.- --'p-'j. E.E. ---.. -- ;- .j; ;;@--.--E.-. -. . . : g.---.gE,-..-!.: -.. -...g.-..-. -'-----.-.'-. g.--E.----i -(.---.. !. - .;- . - ;--.,'-(--. -- -'--.-- ... ..-j'. - .- ;-.-t'k;-. ..-... ... -. - .. .;-.. ..-...-... .;:.- ;.-;!E-:. ;.-...--'-- .!..-. -..-..-q- ..i..-. !.-'.-j'..;-;-.-. .-jj:--k.; - -:.:. ---- -ti): L.-i,j :tt. .. . -'t.jy; ky'-k -''. -. );k. -)--'; (,-jt'. i,-.- -.(rtt'. ..-- . -... $ytil'-ttijti. E)-.'-)'--. -:'i -,i- -:; .,t, .-:'' :..'.i:-! -; jk.:rp ,,;. q,-.;-....: , --- ,j;'?,,;);:. ..LLLL'LLL ,-.. (. q'' ;- - . -' . .-. .... (;'. -. - p -'. ! - -... , - - ;.-.-- (k-. :...-..;-. --.;-. ;.E-.. E --;'.)':-:t'. .. . i ..E :;;,'. . . (!,'.-.p'.lii ;i -:-r-:; E.. . .j.': )''E.'EEi . E . gE. i : r .g .-!.' . . k),. (;;;Ij;IjjjjIIkjIjqE . .E ' .t'i tE' 'j!' . . . .. .' i;E . . . . : :). . . . -'-'. ::qtj.' -'.,)'. F:'. i . i;:: . . . E g.!rE t'E ttE -C'. -'. j i ! r; . i . .' Ltjkt'qLiqjqLi;. y':''( . . . . : ! ( E E . . . : ; :.. . : . . ' . pj.!i i .:tti ; ' I yyE. ; i ' ry'I )@(.t)(: i j.yj ,'. . E jE : : : :. . : : y)r' j'' (j; ! E.j !.. ;LLLL, f q E . ! . . : .:.. ,. . . . . E!, . :t....... . . ytt.ijjkkj'. .tit..r ' j'' .))'' !E ;''. ..k Ey':;E . (i();.i 1g: , . jj):. ( . ;jy ((lj'yy i .:E. . j j.j: ki ())ryy(. E;g. ;((. Lg (. . . jjr'i)i ! .ityjy:' j(q ( .gy)j!( ( js'kj ' ..' j . . :..: i?ry;'' .j. jlqllj. (q.:Ei..;. ( (ij p.;q)! j;.J.jrr!:ti , (('' . (,' k E .. ''.r'' jjjj'' .;'jj:.i .i.j yj jr'pj)r(rj!ris..r.. : :. . j.!(qy'. (yr);)gjg);r);;j)jj)pj.. ): t: 2:::Iy': . jlyikyjijig. (r. jr(;);. ( j)j:. jj.,gj: . ( (j; q! ijtj j)j'. .!'i(@; jr. ))y';g. 'j)(;)'. r..r( jl. ;'. j ... .'.Ej..'. E.(;.!qiE;;'.ijEijjjj l ....)jjjj(;y)))'. ( j)jg'. ..k i 2 jj)'. .r .q ( . . gi . jjj''. . . ;j'. . . L.j)L'. . . .jjy)));)'. . jj'. jjjjy)'. . . j)j'. . LLjLL'. . ()j;'. .. . . ..(j)r;)'. . jjjjy'. . ' jjL'. ' lr)'. . . . .. . . . . . .... j;)L'. . E ;q :. . .i ' . .' .'...(....,1.E:.l.'q(.(l.ti:,1.r .q q j y ... jjyry'. . j.;. : .. . y. . E - i q :!E -E ; . . E- :!;@I !( .... !.i r.jj-j'.y ij j qy -j.: . . i --'.- . --..-E y'E. Et,y.E.. !:-.-.k-ygyrr:. - -y- :-.;. E. .. ;- '- t'E r-!- - .y y ....,.,)..r); . . y ...q..,.!.'.....qy.,.j.yq.y.g.;.,..jq(q!j.jjijjkg!yy. 'y ..(.ylj,.y,...

-?--r;;;j,,.

!48

INVESTM

t. REAL op-rloxs

17.1 INVESTMENT

AND THE

NPV RULE

We first consider a simple investluent decision of the sort you wotlld enounter in a basic hnance course when studying net present value (NPV). Despite its simplicity, the example illustl-ates the issues that will arise again later in this chapter. Suppose we can invest in a maclzine, costing $ 10, that will produce one widget a year foreven ln addition, each widget costs $0.90 to produce. The price of widgets will be $0.55 next year and will increase at 4% per year. The effective annual risk-free rate is 5% per year. We can invest, at any tilne, in one such machine. There is no

Static NPV A natural first step is to compute the NPV if we invested in the project today. We obtain Npvlnvcst today .042

= $0.55 x

l + 1.05

.04

.04

l + + 1.053

.052

l

l

- $0.9 x

$0.55 = 1 x

l

l

+

.05

-

'

-

.04

1.05 a

$0.9

l l ()5 I

l

+

-

() ()5

j

.

.

.

1 1.05 a

$10

=

.+

.

.

.

$0.55

-

=

$27

This calculation tells us that if widget production were to start next year, we wotlld pay $27 for the project. For reasons that will become obvious, we call this the project's static NPV. Notice that in the early years, the project has an operating loss. If we activate the project today, then next year we will have negative operating cash flows, spending $0.90 to produce a $0.55 w'idget. ln addition, at a 5% rate of interest, the opporttlnity cost of the $10 investment is $0.50/year. Why is NPV positive if we will be producing at a loss? Although the initial cash flows are negative, the widget plice is growing. The project vill beconle prostable in the future. This eventtlal protability is why NPV is positive. This analysis stlggests that we might consider waiting until later to invest. Suppose we wait 5 years to invest instead of investing inuuediately. NPV is then Npvwait

1 (1 1.05s

.04)

5 years

=

:.5

$0.55 0.01

-

%.

(1 +

.04)''0.55

=

549

1.40

23.82.1 After 23.82 years, the widget price will have reached Solving for n gives us n value today of waiting that long to invest in the project is level. The break-even a =

(1.04)23.82$0.55 $0.90 - 0.05 0.01

-

$10

1 =

(1 + 0.05)23.52

$35.03

Problem 17.4 asks you to verify this result. You will discover that 23.82 yers is not exactly optimal. Rather, waiting approximately 24.32 years-not 23.82 years--rmxirnizes NPV. At this point the widget price will be about $1.43. We will see the reason for this slight difference in Section 17.4. It occurs because effective annual interest and growth rates of 5% and 4% are not the relevant rates the decision the since to put off the investment is made on a day-to-day basis. It is instead continuoltsly ct??n//t/llnt-/ct rates that matter. equivalyt the This example demonstrates the important point that making an investment decision requ'es thinling careftllly about alternatives, even under certainty. We ate lef4 with (at least) two questions:

The Correct Use of NPV The NPV rule worked correctly in the above example. The NPV ment decisions entails two steps:

l'ule

for making invest-

1. Compute NPV by discounting expected cash Qows at the opportunity cost of capital. 2. Accept a project if and only if its NPV is positive and it exceeds #?e NPV of p1-0jeCtS.l eXCIIISiYE fl/ldrplfllid ??7IfllfflP)?

Thus, it is better to wait 5 years than to invest today. 'What is the maximum NPV we can

all

When we computed the widget machine's NPV in equation (17.1),we neglected to into take account the NPV of alternative mumally exclusive projects, namely investing in the project tomorrow or at some other future date. Static NPV--'-NPV if we accept the project today-ignores project delay. Because static NPV measufes the value of an action we could take, namely investing today, it at least provides a lower bound on the value of the project.

$28

= $30.49 attain?

RULE

@ How do we approach this ldnd of problem in general? * Why didn't the NPV rule work? Or did it?

$10 $28

-

0.01

.

NPV

Common sense points to an approximate answer: We should notinvestuntil annual widget revenue covers marginal production cost ($0.90)plus the oppo>nity cost of the project ($0.50).,i.e., cost is at least $1.40. The widget price will be $1.40when ?? satisses

tlncertainty.

Before reading further, you should try to answer this qklestion: 'What is the most lights would to this project? pay to acquire the you

ENT AN D TI,I E

lerheprice must 23 82.

increase by a factor of 1.40/0.55 to reach $1.40, so we have ln(1.4O/-55)/ 1n(1.04) =

zlntroductory hnance textbooks state the NPV rule correctly. but in casual discussions it is sometimes stated incorrectly.

550

%.REAL

Op-rlolqs

IN V ESTM

In this example it would be correct to invest in the project today

ifllotclcrvt'lr?,jr

the

nautually project 1474/47.)'nleant #l7r It/d b,oltld Iose ?./b?-cT,t??: Under this assunption, exclusive alternative (nevertaking the project) has a value of 0, so taking it today would be correct. theE

To dvcide whether and when to invest in an arbitrary project, we need to be able to compute the value of delaying that investment. As stlggested at the start of the chapter, option pricing theory can help us to value delay.

'

Present value of costs

=

'

$10 +

0.05

=

$28

As we discussed earlier, we can view this present value as analogous to the exercise price in an option valuation. In returfl for paying $28, we receive a cash flow with a present value of Present-value of widget reventle

=

.+1 0.01

where .S'+1is the widget price the year after we make the investment. When $0.55, the present value of cash flow is $55. This present value of widger revenue is the price the revenue stream would have if it were traded separately. It is analogous to the stock price in an option valuation, and therefore it is sometimes called the the twin security or the t'raded present value of the project. Now recall the discussion in Sections 9.3 and l l about the three factors governing exercise of a call optio: the dividends forgone by not acqtliring the asst today; the arly interest saved by deferring the payment of the strike price; and the value of the insurance that is lost by exercising the option. It turns out that the same three considel-ations govern the decision to invest in the widget project. ,ol

=

.1

First, by delaying investment, we lose the cash flow from selling widgets. The cash

flow we do not receive is analogous to stock dividends we do not receive by holding an option rather than the underlying stock. The first period cash flow is $0.55. The present value of future cash Pows is $0.55/0.01 $55. Thus, the dividend yield is approximately l %. (We can also think of the dividend yield as the difference between the discount rate (5%Jand the growth rate of the caslz llows g4%j.) Second, once we begin widget production, sve are colnmitted to spending the present value of the marginal widget cost, $ l 8, along with the $ l 0 initial investment. =

The annual value of delaying investment is interest on the total ilwestment cost, or ' 0.05 x $28 $ 1 per year. .40

=

$

and therefore no

We can compute the value of the widget project option using the perpetual call calculation discussed in Section 12.6. The formula ajsumes continuously compounded 4.879%, and for the dividend yield we rates, o for the interest rate we use 111(1 difference compounded the continuously between the interest rate and growth rate, use 0.9569%. or 111(1.05) ln(1.04) .05)

=

'

=

-

=

$55 (thepresent value of revenue), K

=

$28 (the present value of

costs), '1- 0.04879,+ 0,a and t 0.009569, equation (12.19)gives an option plice of $35.03and investment when the widget price equals $ 1.4276. We will call this price the investment trigger price. We reach this price after about 24.32 years, which velifes the answer we discussed earlier. The example in this section illustrates the iluportance of thilzking dynalnically about a project and shows how this speci'lic problem can be modeled as an option. =

The decision to invest in the project involves a comparison of net present values. In what sense is this an option? We can interpret equation (17.1)so as to make the option analogy more apparent. 'When we take the project, we pay $ 10 and we commit to paying So.go/yearforever.The present value of this stream of costs is

UN C ERTA I NTY

Third and finally, in the widget project, there is no uncertainty

insurance value to delaying investment.

With S

The Proiect as an Option

ENT U N D ER

17.2 INVESTMENT

=

UNDER

=

UNCERTAINTY

In this section we discuss the valuation of rel investment projects when cash flows are uncertain. With the widget project in the previous section. waiting to invest was optimal because project dividnds were initially less than the interest gained from defelning the project. If we add uncertainty about project cash flows, the value of insurance (the implicit put option) also influences the decisiop to delay the project. ln such a case, waiting to invest provides infonnation about the value of the project. In this section we will use a binomial tree to value a project with uncertain cash flows. As before, the decision to invest in such a project is like exercising an American option: We pay the investluent cost (strikeprice) to receive the asset (presentvalue of -

future cash tlows).

A Simple DCF Problem We first examine a particularly simple valuation problem in order to better understand the link between discounted cash tlow (DCF), real options, and linancial options.

Suppose an analyst is evaluating a project that will generate a single cash flow, X, occulning at time r. As with many investment projects, it is not possible to observe market characteristics of the project. There is no way to directly estimate project returns, project volatility, or the covariances of the project wittl the stock market. Instead, suppose the analyst considers the economic fundamentals of the project and makes educated inferences about these characteristics. The ahalyst might also look for ptlblic firms with a business resembling the project. The nalyst could then use information about these public f'irmsto infer charactel-istics (suchas beia) of tlze project.

3lt is necessary to set o' to a small positive number stlch as 0.0000 1 to avoid a zero-divide en'or.

%.REAL

552

Opl-lolqs

INVESTM ENT UN DER UNCERTAINTY

After examining al1 available data, the analyst estimates that the cash flow will be Xu if the economy is doing well-an event with the probability p-and Xd if the economy is doing poorly. The project requires expenditures of % at time 0 and fw at time F. The analyst determines that projects with comparable risk have an effective annual expected cateof remrn of a. We can use this description to compute the value of the project, J/. The standard discounted cash flow methodology calls for computing the expected cash flow, and using as a discount rate the expected return on a project of comparable risk:

pxu + (1 pjxd (1 + J)F -

J/

=

Assuming that we either take the project now or never, we invest in the project if F 2: r)W. % + J/(1 + '

Example

17.1

Suppose that the risk-free rate is ?' 6%, the expected return on the )(l market is 0.60. 1.25, project is F = 1, Xu 10%, the beta rju $120, and' p p ).jjj )'t Xd gp $80. The expected return on an asset with the same risk as the project is =

=

E .E

hy (.j

a

.

.

t') The ('E)

('. q'ii :

q 'El'' E

=

0.06 + 1.25 x (0.10 0.06) -

EX)

j'.

=

(2.60 x

$120+ 0.40 x $80

=

p*Xu + Thus, we have

$104

t Using (17.2),the present value of the project cash flows is !( i j.l! $ 104 J/ p)( $93.694 ?).j

IZ

=

(1

involving options, suppose

.F

(1 + l )

p*)Xd

-

4 p

i

need to make f?2order

probability must therefore satisfy

0.11

=

expected cash flow is

!

.

lisk-neutral

=

EEtIIE

'

F0,T

=

=

. .

To see how to perform capital budgeting calculations

1p'

553

that if we invest Io to start the project, the subsequent investment of fl is optional'. We make this further investment only if the project at time 1 has suflicient value. Since the project has a good and bad outcome, it is namral to think about using binomial option valuation. ln order to do so, we need to lnow the risk-neutral probability of the high outcom. 5 Fortunately, we can easily compute risk-neutral probabilities for this project. Recall from Chapter 10 that the expected risk-neutral price is the fonvard price. We have computed U, which is the price of p.n asset paying a single cash flow.at time T. The forward price is The

j'('j'j' y' j '

valuing the project Jv' have already ppt-l#c alI r/?easswtlptions pt'I/l/c derivatives related to #?c plnject. to

$

*

Fo p

-

:

=

Xu

-

#(),w

=

Xd Xd

probability of a high This gives us the binomial tree xu and Xd) and the lisk-neutral using risk-neutral project distribution, then Notice that if value the the (p*). we outcome by c/n.lrldcrtwl we will obtain the original project value, F.

:

=

:.jyj

E ( 1.E

=

1 + 0. 11

i

t,) Suppose that Ifj $10 and f l $95 and that the manager commlts at time 0 to paying ti'r $95 at time 1. Then net present value is t.1 gqr .

).(the

=

=

Example 17.2 Consider the same parameters as in Example 17.1. The forward price for the project is Fo, l

E)

ijJ l..

The

Iisk-neutral

=

$93.694x (1.06)

=

$99.315

probability of the good outcome is

99.315 80 120 80 -

ps

=

=

0.4829

-

Valuing Derivatives on the Cash Flow

If we value the project using the risk-neutral probability, we obtain

0.4829 x $120 + (1 0.4829) x $80 = $93.694 1.06 Now male the same assumptions as in Example 17.1, except that we decide at -

The calculation in Example 17.1 is standard but it is nevertheless based on strong assumptions: We have specised the future cash flows in different states, the probabilities of those states, and the comparability of the project to a traded asset/ It t'urns wt that in

time 1 whether to

'lef'helast assumption in particulardeserves some additional comment. We are assuming that the returrls of the project are spalllled by existing traded assets-, in other words, th addition of the project to the universe of assets does not materially change the opportunities available to investors. If this were not true, we would have to know more about the preferences of investors in order to evaluate the project-

5In general we also need to know volatlity to value an option. As we discussed in Chapter 10, volatility detennines te vertical distallce between binomial nodes. Thus, in specifying the tree, we implicitly specined volatilitj?.

'incur

the

$95 cost. We will choose to produce output in time 1 only

)'554

% REAL )', when the

y(.

j

OpTloNs

INVESTMENT

cash flow is

$120, since

.. the output sells for $80. The value )t, ).). maxg () x ps) gI j ( j (,. q!)) p ?l + $i)) l.lytt :

E

)EE)

,y

.j.

-

,

y)j ljj

-

,,

.0.4829

$l5 by paying $95 to produce

we would lose

when

-

,

x

$25 + ( l j

Jj j -

=

10

0.4829)

.(j6

x 0

- $10

=

$1.389

. Given the risk-neutral probability and the cash flow distribtltion, we can value projects with options or other nonlinear cash fIows.6 You may be thinking that there appears to be little difference between standard discunted cash flow valuation and real options valuation. Recognize that ally snancial valuation entails assigning a dollar value today to a (possiblyuncertain) cash flow that occurs in the future. This descliption applies to the valuation of a project, as well as to valuing a bond, a stock, or an option. When we value an option on a stock, we rely on the market to have already pelormed part of the valuation naluely, valuing the stock. When we value an option on a project, we have to estimate the value of the project since we cannot observe it. This is p-llc vlletller or llot the p/r.#c/ colltaills optiolls. This discussion illustl-ates the point we made before in Section l l that riskneutral pricing and discounted cash flow are altelmative means of valuing a futtlre cash :ow. If done using the same assulnptions, the two methods give the same answeni ln practice, of course, it is common to make silnplifying asstlmptions for tractabilitjt Answers may differ because the simplifying assunptions for different valuation methods are inconsistent.

?-risk-free

-

0.07 + 1 0 15

.33(0.06)

PV

stcyj)

=

'-project glrwth rate $l 8 0.15 0.03 -

=

-

= $150 Static NPV is theyefore $150 $100 $50. Suppose we have 2 years in which to decide whether to accept the project; at the epd of that time, we either invest in the project or lose it. (Imagine, for exalpple, that the licensing rights for a technoloqc will revert at that time to the oliginal owner). The static NPV rqle will apply after two years because further deferral is not possible. However, at tilpe 0, we must evaluate the option to wait. The forgone initial cash flow (thedividend on the project) is $ 18 and the interest saving is $7 (7% x 100). Thus, considering only dividends and interest, it makes sense tl srart the project imrpediately. However, the project also has irnplicit insurance that we need to know the project we lose by investing in the jroject. To value the insrnce -

.2,

=

volatilityr

for project value Suppose that cash flows are lognormally distlibuted with a 50% volatility. Figure 17. l uses the Cox-lkoss-lkubinstein approach to construct a

A tre

,'?' .--.-' ' -' -' ,'--' !(j'

j'p' ,j,(' j''7,:'*(1* (' cd T' (jd ;'!' jfd t'.y' q-' )' iiii2-:-1I1E't,f j'ly-'t ((''l ()(-'-. ;(' tfd )' .-' !iIi:t:;;;k'. k' t.!)f r' rqd t)'. j'. y'l'y .y'.y' )' ttyf ))' tyy'.

.'iiiii:#!;. .' . .'p'jjjyj r: ; y ;. r .' . tjj ..lllii:tli!t. . yy . 1l1i. IIIEEEEE:. .yj.y.. jj(. ...-j.j j. j'.'j y yj --.. r, . .. ... .kj .,r. j t-.;.r..;...r. .(.. r.. E E EE( Ei ' EE:' ;' ! E' EtE : i; ' ' (l !E!( (... ..' (''.(. E E..i.. (.E'E!.iE (ll;Ej:EE'...'.''EE iq (E.'.yi'.: . jF . . .. !'(.EE. (..' (. ' : ;''7::.'(( . (' ((..!( q ' ()(j.. y.jjqjjj E .( . y... .y ;..qjgjy g j .. t.. yy g.yyyjy. ( .( q...q.yyy.j..qttjj .y..y y j y.yj.. y. yyy. yj. y.y..y.yy,, E yj y. . .y . i!. -. . - - j- - -.- . . -y . -. ( . :i -. - k;. . . . . .. . . . .. ''

.

Eyyjl -

E ::

..

. .

.

:IE! :

(iE

.

.

.

. .

:

.

E

'!

'

.

.

::. :

E . .. . . g

-.

:

' .

:'

...

E

: '

. .

: . :' .

-: .:.. :

.y.

:

.

.

- . ..

.... .. .

-

-

-- - -

-

.

.

-

.:

-.

y -. y y

r

-. . .

.

E'. . .

. .

'-;11111.

'

-

We now consider the problem of when to invst in a risky project. As before, the decision to invest in such a project is like exercising an American option: We pay the investment cost (strikeprice) to receive the asset (presentvalue of future cash flows). The widget project in the previous section had cash flows that were certain. Suppose a project costs $ 100 and begins prodtlcing an insnite stream of cash Cows l year after investment. Expected annual cash Pows for the srstyear are $ 18, and are expected to grow annually at a rate of 3%. Suppose further that the risk-free rate is 7%, the risk prelnium on the market is 6%, and the beta of the projet ls 1 Using the Capital Asset Pricing Model (CAPM), we compute the discount rate for the prject in

y

.

.

-

.

.

.

.

. y ., . -

. .

-. .

..

.. . . ....

Binomial tree for project cash flows, assuming binomial distribution with 50% volatility.

.31.

tiproblem 17.9 asks you to value a project paying the squared cash flow.

555

To value the project, we perform a standard discounted cash flow calculation. Since the project lives forever, we treat it as a pelmetual grwing annuity. The present value is

k

Evaluating a Project with a z-Year Investment Horizon

$

Jl(/-market'-risk-rree)

-1-

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Chapter 10. We can then solve for the value of the investment option just as we solve for the price of an American call option. The risk-neutral probability of the project value increasing in any period, p*, is given by:

The act of investing creates the project, therefore the value at each node in Figure 17.2 is the value of the project ifwelt/crd to ?lvl'r at l/?t7r?l(?#'. Figure 17.2 describes the evolution of the project's present value. The project does not exist prior to investment, but the tree provides the information we need in order to decide whether to invest. The tree in Figure 17.2 is exactly the same tree we would construct for the stock price of a company that had the project as its only asset and that paid dividends equal to the cash flow of the project. Such a stock would have an initial price of $150and a 50% volatility. being It ray trouble you that in valuing this project, option pricing formlas used in a context where literal replication of the option is not possible beause the twin lpnt/pptx/p/tpcLflf?'d also security does not exist. As we saw in Chapter l 1, however, the 1/t'l/i/t'#k't7?? Isl/?-/t'- f?? settiltg 'kW?c?-' ltsilg ntodel. J/' pelfonn #7' CAP.H or tprcrprfcn,g a notarbitrage-h'ee plices. Thus, we are using option plicing formulas to ceztefairprices,

The inputs are initial project value, S = $150.,investment cost, K = 6.766%) volatility, o' compounded risk-free rate, ?' = 1n(1.07)

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expiration, t 2 years. Since the market value of the project today is $150and the cash flow in a year would be $18 if the project were developed, the dividend yield is l2% ($18/$150). Since project value is proportional to next year's cash flow, the dividend ln(1.12) yield never changes. The continuously compounded dividend yield is 0.1133.

binomial tree for the evolution of cash flows with a binomial period of 1 year. If we wait to take the project, initial cash flows in l year will be either $18c0.5 $29.677or $18e-0.5 $10.918. Since the project value is proportional to cash 'Ilows, the value of the project is also lognormally distributed with a 50% volatility. ln l year, project value will be either $29.677/(0.15 0.03) $247.31 or 0.03) $10.918/(0.15 $91. lf we will continue to learn about the project at the same rate over time, we can build a binomial tree with constant volatility that shows the evolution of project value. This tree, constructed by discounting at each node the cash flows in Figure 17.1, is in Figure 17.2. -

ENT U N D ER

7efhis is not necessarily incorrect; as a logical matter, a constant discount rate for the option could be correct #- the tree for the underlying asset had discount rates that varied across the nodes. However, in decision tree analysis in practice, the issue of discount rate determination is often glossed over.

558

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Op-rloxs

REAL OPTIONS

return is constant; instead it tells us how to perform valuation so that the assuluptions about the project and the asstlmptions about the tree are consistent with each other.

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The above example assumes that we must start the project by year 2 and that we evaluate it annually. Suppose instead that the project can be stafted at any time and then will live forever. The project is then a pel-petual call ojtion that we can evaluate using the perpetual option pricing formula. Using continuously compounded inputs, we

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When the project valu is $ 150, the option value is $63.396qnd tlie optimal investment trigger is $2.45.71. In other words, we invest whep the project is worth $245.71, more than twice the investment cost. If we invest immediately, the project ls wrth $50. The . ability to wait increases that value by $ l 3.396.

17.3 REAL OPTIONS

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In Chapter 6 we explained that electricity forward prices can vary over the course of day. They also varj? seasonally-. ln the United States, electricity forward prices are high in the summer and low in the winter. In addition to this pl-edictable variation, electricity prces can be volatile. On extremely hot days, for example, prices can spike to l00 times their average price. A peak-load plant, as the name suggests, produces only when it is protable to do spikes in the price of electricity. Stlch plants are designed so that they can exploiting so, when idled the price of electricity is less than the cost of fuel, but they can be qtlickly be online brought to produce power when the price of electricity is high or when thq price of declines. Because it is tulmed on only when prolitable, owning a peak-load plant is fuel

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sate Forward price (top panel) and volatility (middlepanel) curves for electricity and natural

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with the g p so-s secausevolatility changes Equation + H x cc ti where ;f = vo'latility time subscript. 1,7.3laas expression in a t ime horizon in pgure 17.4, of optionality. We could also value account the tnklng of plant the value provides (17.3) this would be a static xpv calculation. we the plant assuming o'peration at all times; the true value by using put-call parity to rewre cart see how the static value relates to 2

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.

I N PRACTICE

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Section 14.6). The difference This is the payoff to a European exchange option (see generation, H x Sgas,is called the between the price of electdcity and the cost of spread is the variable spark the spark spread. There are operating costs besides gas, but component of marginal profit. and volatilities for electticity. ln order to value the option Fe need forward prices ' ' Figure. 17.4 jhows . and gas and the correlation between the two. The top panel in has a shape representative folavard curves for electricity and gas. The price curve for gas electricity peaks. curve, by The winter familiar from Section 6.9, exhibitinc seasonal implied by spread spark shows the eontrast, exhibits summer peaks. The bottom panel the prices in the first panel. provides. Let Fz.ti and The value of a plant is the sum of the operating options it and electricity gas delivered at time t If Fs t; represent the time 0 forward prices for .j operating plant is z value the of the we ignore other marginal operating costs, then

..

1')t'' li l

.

REAL OPTIONS

% REAL op'rlolqs

560

E

.

(

E: 'E ( .-t..

$..E

q.tt, r j qj,tj -....

Ey)! ,

.................................

well, sinceaplantmay buthousto-houras aboutday-to-dayoperations, p erators will thinknotonly night. and not at be operated during the day x-strikeprice-lt'ln is sometimes used to referonly to nongas variable costs. pracuce, tlae term BTUS 11, )I. required to produce one kilowatt/hourof electricity. The delinition of heat rate is tlzenumberof of Thennal Units (BTUS) to produce one MWh. number British the 1000 is The heat rate times = 9m BTUS is required to produce. one For example, if the heat rate is 9000, then 9000 x 1000 or producing electricity. If the price of natural gas is $3/MMBm, then the gas cost 9o

megawatt/hour of one Mwh of electricity is $27.

the underlying msset is a forwm.dprice, the risk-free rate, This is the Black fonnula, equation (12.7). Iznecause

?-,

is used as the dividend yield.

( 17.4)

562

%.REAL

REAL OPTI O NS l N

Op-rlolqs

paxfselec

=

-

(S

This payoff is that of a spread option, since the payof'f is positive if the spread

Research and Development Research and development is a capital expendimre like any other, in that it involves paying R&D costs today to receive cash flows later. lf R&D is successftll, a project using the new technology can be undertaken if its NPV is positive. This fnal option is a call option, just like the other projects we have analyzed. The R&D leading up to this project is therefore like an option premium: We pay R&D costs to acquire the project. R&D can be thought of as acquiring fumre investment options. Dnzg deveiopment by phannaceutical lirms provides a particularly clear example of the options in R&D since the drug development process has cleady delineated points at which there is a decision to abandon or continue development. Figure 17.5, based on a descliption in Schwartz and Moon#2000), summarizes the process, along with the probabilities of progressing from one stage to the next. In practice, stages sometimes l'un together, but Figure 17.5 re:ects a standard description of the process. As R&D costs are paid over time, phnrmaceutical frms are able to resolve uncertainties about their technical ability to produce and market the product. Specifcally, they answer the questions: Will the project work, and, if it works, will anyone want it? At a1l times, project managers have the option to continpe or stop the research. In effect, each ongoing investment purchases an option to continue development. Figure 17.5 shows that most potential drugs are abandoned before Phase 1 trials. As with pealoload electricity generation, value arises from what is not done. A pharmaceutical company that pursued a11potential drugs, nl matter how unpromising, would reap f'ull rewards from successful drugs but would be bankrupted by the unsuccessf'ul drugs. The put option to abandon a drug is what creates value for the finn.

s'elec -

-

13

When we include other costs, the static NPV of a peak-load plant is typically negative. Adding the shutdown option, however, makes NPV positive. One implication is that, in equilibrium, after the optimal number of peak-load plants have been constructed, electricity prices will continue to be variable. Otherwise, the marginal peak-load plant would have negative NPV. Thus, the c-vu/wcc ofpeak-load l'c/??ltl/t7gl? will ?7t)/ eliminate prices. ct-/lfflibrlfp/lvariability in d/clrl'/y volatility curves in Figure 17.4 are declining oyer that the final point, note As a of Year standpoint From the 0, a z-yetrvolatility is less than a l-year volati tyk time. which for with stocks, in is contrast This we typically assume volatility is constant over time. To understand the behavior of volatility for electricity, recall the discussion of stock prices in Section 11.3. The assumption that a stock price follows a random wnlk implies that volatility increases with the square root of time. Thus, volatility enters the Black-scholes model as o' T r; this expression measures the volatility of the stock 'price over the horizon frortl t to T. By contrast, we do not expect electricity prices to follow a randc?m walk. When the electricity price is high, users consum less electricity and produeers increase production. 'When prices are low, usrs consume more and producers produce less. Thus, the price of electlicity reverts to a level re:ecting the cost with T at a rqte less of production. When prices revert in this fashion, volatility F r/?tw7 t. '

Etd

'

-

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-

To consider a specisc example, suppose it is January. From this perspective, the July price this year and the July price next yeltr have similar distributions; ske won't lenrn much about the July price this yar or next until we approach July. tTl'lis is not strictly true because economic activity and even weather can follow long-tenn cycles, but suppose that it is a good approximation.) To compute an option plice, we require annualized volatility, c', which the option pricing formula transforms into volatility over

l3lrlaug (1998,p. 59) discusses

approximations

that can be used to value spread options.

563

-

H x S :,s exceeds c. An option with this payoffcannotbe valued using the Black-scholes g formula because neiter Saec H x SgasnOr S x has+ c is lognormlly distributed. Equation (17.3)is rherefore an approximation once nonfuel costs are added to the strike

price.

$

-

Sgas+ c), 0J

X

CE

the life of the option, c' F t. If you believe the uncertainty this July and next July is the same, the annualized volatility will be lower for next July since a given amount of uncertainty, when annualized, is spread across a greater period of time. If o' T. t is the same for two different 6.s, the cs will be different and the volatility curve will declie with horizon.

This way of writing the plant's value makes apparent th difference between a static (NPV calculation and the real options valuation. The static ' alculation assumes operation at all times', we can value the plant by simply discounting the spark spread computed using pc/llc of a peak-load plant forward prices. Equation (17.3)also makes it clear that the Il/pt?/? prices 47?-: high-all ,1471 pDnloperating stenhntn operate Ip/ltv? prices t-frc does bvhe'l 7?'' Iov. shutting down prices high-but m//?d?'/r??7 ln reality, equation (17.4)is overly simplilied. There are marginal distribution, operation, and maintenance costs associated with an operating plant. Represent these costs as c. When we take these into account, marginal prolit is Profit

P RACTI

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process for a new drug. Probabilities are the percentage of Pharmaceutical drugs proceeding from one stage to the next. For example, 74% of drugs submitted for FDA approval receive it.

. -

-

1% Preclinical Tesdng

7O% Phase I Trials

Phase 11Trials

47%

Phase III Trials

82%

FDA Approval

-r-lj-:--

74A

Marketing and Sales

Sottrce: Schwartz and Mpon

(2000).

k

564

CoM

op-rloxs

REAL

How do we evaluate pharmaceutical

investments?

The underlying asset is the

AS AN OPTI ON

$

565

P.eak-load pricing and research and development

are examples of how option

techniques are used in making investment decisions. In the next two sections we develop an extended example of commodity extraction, which is yet another area in which real option considerations are essential.

Assuming that all of these inputs are known, we can evaluate the sequential investment as in Figure l 7.6. The figure presents an example in which, in each period, row) to keep the it is necessary to pay an investment cost (shownin the Iinvestment''

.' .'i1iij:i:;;..

EXTRACTION

project alive for another period. The static NPV of the project is negative, since the initial value of the developed investment is $ 100, but the present value of the investment costs at a 10% rate of interest is $108.60. This static calculation ignores the staging of the investment, which pernzits maling later-year ipvestment costs only if the project shows prolnise. With staging, the value of the development option is $2.812. Schwartz nd Moon (2000),building on work by Pindyck (1993a),developed a general valuation model, with staging, which is applicable to phm-ntaceutical R&D.

value of the drug if brought to market. How do we find the value of this asset? With the gas) and peak-load electricity plant, we have forward prices for both the input tnatural the outptlt (electricity).We can estimate volatilities from market prices. However, in plaarmaceuticals, we must estimate development costs, potential revenues, volatilities, and correlations without the benetit of observing market prices. Project payoffs witl lisk, which must also be vary with the state of the economy and, hence, have systematic estimated.

y' :''' q'' ;yt' ;';'ylf r' (' Jjj'ql jjf )' ''' j')' q' y' yjf jljjjd (' ::Lji''. y' r' ykf Iijt!!!kd ;'. ;'yyf (k'-. .y'. (' (''(('' ''' : q'l' ('y' j. (.'(''( j!. 1:*. y' ).((.gg'.'(.((j';( (.((j rj((i.' (' j '...'. j.. E l . .j r;.....(;. !y.7.q);q.(...q.. )q(.;..)(.. j i; . ((j.yj q j.... (. .y. ...yry .jy.yyyy.jjj . . iyj.j: . .jljg::;jj:jy . y.. .j..... j.jy .q..jq . ( (y yy . iiij;-,;jIIj. . .iy..(. .)k..... jjj;iisq ' jjr... j..)).. :(yjq. j.y y.yy.qk....(.y(q.....(. . . y.. . . . yyjyk.y.y.t:y-

MODITY

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$100

EXTRACTION

AS AN OPTION

Natural resources investlpents are an important application of option techniques to inflom the ground exhibits many vestment decisions. 14 The extraction of a lesoulce siluilarities to the exercise of a Iinancial option. resotlrce has a value that can be realized by paying an extraction cost. The marketYorthe resotlrce is typically competitive so that the behavior of one producer does not affect the price. ln this section we will consider the problem of extracting oil frolu the ground. There is an initial cost to sink a well to columence prodtlction, after which we assume we keep producing forever. ln Section l 7.5 we introduce the possibility of shtltting down production when it is unprotitable. Our goal in studying the oil extraction problel'n will be to understand the ecoltonlics of this problem. The analysis is an exmuple illustl-ating the costs and beneEts of defen-ing investment and stopping and starting production. The specilic formulas do not pply in every sittlation. -

-

-

'he

5385.743 $285.743

$245.960 $96.216

$141.907

5156.831 $22.869

$41.907

$90.484

$100.000 $2.812

0.000

$52.205 50.000

557.695 $0.000

Single-Barrel Extraction under Certainty

$33.287 $0.000

Suppose there is a plot of land tlzat contains one ban-el of oil. The cun-ent price of a barrel of oil is $ l5, the oil forward curve is such tlzat the effective anpual lease rate, is 4% (constantover time and across matulities at a point in time), and the effective annual l'isk-free rate, r, is 5% (alsoconstant over time). There is no uncertainty about the future price of oil. The bal-rel can be extracted at any time by paying $13.60, which we denote ,

205 $0 000

$19

An example of staged investment. The value of the project, if developed, is in the top Iine at each node. The value of the option to develop the project is shown below the value of the project. In each year, it is necessary to pay the amount in the lnvestment row to keep the project alive in the next period. The tree is generated as a forward tree 0.15. ' 0.50, r 0.10, and f o' assuming S

E i

X. Finally, to make matters simple, suppose that the land is completely worthless once

the oil is extracted.

.$100,

=

=

=

=

l4see in particular Brennan and Schwartz ( 1985), McDonald and Siegel ( 1986), and Paddock et aI. ( l 988).

566

k: REAL

CoM

Op-rlolqs

If the price of oil at time 0 is

&, the time 0 forward price for delivery at time F is

given by

S 'T

=

1+

S0

1,

MODITY

$

AS AN OPTION

567

This expression shows that we defer investment as long as l't /

S

F

I

-


(x

=

g.

=

fr

().000()1 0.10

Valuing

0.3

=

571

%

AS AN OPTION

invest 1 in order to turn the undeveloped reserve into a developed reserve, Exactly l year after that, the reserve will begin to produce one ban'el of oil a year forever at a cost c per baln-el. We solve this problem by working backward. We hrst compute the value of the firm supposing that it is already producing, and we then sttldy the decision abotlt when to invest.

.

.:

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:.

.

.

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EXTRACTION

MODITY

e'

w.'>

,

.a..-

the producing

Once the lirm ha's invested, it will continue producing

firm

foreveltsince the price of oil is always rising. Recall from our discussion of commodity forwards in Chapter 6 that the lease rate is the discount rate linking the future commodity price with the current commodity price. Thus, the time ? valtle of a barrel received at J)F-J F),w/( 1,+ r)TStj l + time F is P P)(F;,w) The value the producing firm at time 0 is tlerefore

e'.v

#*

'of

=

a' P

V

=

X

J

. . ... . .

*;:.

e)

i= I

!,,,,,::,;,,,j,,,,,:,,,j,,,,.. .

.

>awoomx

/ t!t?7. hf

parea.

f

1

40 35 30 25 Extraction Trigger Price (5)

20

'

45

50

.796.

.$1

Valuing

-.5

=

$3.786.

the option

(1+ $3.7856

We can verify this calctllation by exploiting the insight that for option pricing purposes the lease rate is the dividend yield, and use the perpetual call formula:

Callpel-petualls l5, $13.60, 0. 15, 117(1.05), 111(1.04)1

=

The perpetual call calculation also gives s l3otlld occur.

=

($3.7856,$.25.3388)

$25.3388as the pcice

tt which exercise

'j'

Valuing an lnEnite Oil Reserve that can be Now suppose that the land contains an inhnite number of ban-els of oil firnz the that kYewill can at any time asstlme extracted at the rate of one ban-el per yeal-.

1-

sv (!

-

c

-,-

-

--i

-

1

=

1

l -j

(I +

,.)

v

(

Sw

-

g 1) c

-,.

+ 1

This is the value of the ltldeveloped oi1 reserve. Note the similarity with equation ( 17.6). If in equation (17.6) we replace Sv with the present value of oil extracted, Spj, and replace the extraction cost, X, with the present value of all extraction costs, cjl' + 1, then we have a problem that appears the same as in the single-ban'el case. We want to select F to maximize equation (17.9).The right-hand side of equation (17.9)expresses ban-els in the ground does the value on a per-barrel basis. times l /. Having luultiple not change anything fundamental about the problem if there is certainty and the oil price grows indeinitely.

ltI =

-

If the lirm invests at the price Sp, the value of the land at well less the investment cost, 1, or Sv/l cll' 1.

to invest

l r)T

$l 5 $13.60) 25.3388

c

-

that time is the valu of the producing The valge of the land today is

-

-

So

You might wonder why the present value of a ban'el of oil a year forever is So/. We know that a perpetual-coupon bond paying c/year is worth c/?' (the second term on the right in equation ( 17.8)). We saw in Chapter 15 that the lease rate on a colnmodity bond is analogous to the interest rate on a cash bond. The operating well is like a bond paying one unit of the comluodity forever. so the lease rate is the appropriate discount rate for a bond denominated in a commodity, and &)/ is the value of the well.

Value of Iand containing one barrel of oiI as a function of the trigger extraction price, = 0.00001, the maximum is -.S', for four different oil volatilities. For the curve where c of value For the curve where corresponding with 6.918 at 51 a attained value of corresponding = with attained $25.3388 at a c = 0.15, the maximum is

($25.3388

i= I

>==.=

( l + r4i

( l + &4

j

..f7

15

'-

=

()

10

?-)f

'>...

WMVQ

c

ua

=

xzyox

#:

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X

c

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*>''>-.>>

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.

Suppose So ti!)Example 17.3 )') ''jt the producing well is $15/0.04

i! .y'.

) l

$ l5, 1' $8/0.05

4%, c = $8, and the value of $215. lf the investment cost, 1, is $ l 80, then bleper-barrel extraction cost is cjr +. 14 = 0.04 x ($8/0.05+ $180) $ 13.60. The problem is the same as having 1/ options to extract at a cost of $ l 3.60; hence, the =

-

=

5%, J

=

=

=

572

k

REAL

Op-rlorqs

CoM

solution is exactly the same as in the single-barrel case. To appreciate the similarity, use the option pricing formula: Callperpet'ual =

jtl.t),yj $8

$l5

,

().:5

+ 180, 0.000001, ln(1.05),

ln(1.04)j

$422.956. Thus,

extraction

k

=

$15 $8 ,

+

$180,0.15, ln(1.05), ln(1.04)

=

($94.839,$633.4691

The well is wort.h $94.639and we invest when it is worth 5/0.04 = $633.469,or when S = $25.3388. On a per-barrel basis, the well is worth 0.04 x $94.639 = $3.7856. With these assumptions, the solution is the same as in the single-arrel case. k

ln the absence of any shutdown opyions, the single- and infrlite-barrel cases differ only in scale. The interesting difference arises when it is possible to avoid operating losses by shutting down, which matters pnly in the multiple-barrel case.

AN D RESTART OPTIONS

k

rjf F' j'(' y' p' t';')' EEL.)(.r.' .EE E'.'( 'y'yf yjf i'?'.'jE)'E(!E '''''E7'''7>/ES''':rqTE.i:' 'Eil''(.r''.'y ' .'.':.' .E):.E t'q'$'j'k' ).':;'' ...'E. '. . E .1.'L.. ..E.E:.. E' E. . . I !j.: . .. . .. .. l E : ;!.. !IEi. i..E!.E. . ... ... E. ;i... E.. E i.;E: : '. ..j.j(.y.kj..E.. E .. E .. . .E .E .. EE . . :..(; .... qq ..;. j(....g.q . . .. .j.. . . E.. .. .. jjjjjg,E) jq:.j..Ej @. E (y .E..g...j!(:2 .'. . .)gjy:'yIjy. g.yjE. .qF(!;EE..:.rjjj. . tjEl.iyyjj j q;i.j(...j.. .y.j . . .)j:.. . ;y . . ,j. .t..t.;.:t yy, .t . . . q. .j(j::ygjy......@ .;)Cgjjjj;i ..,...tk..q j..j .( jlj:,gjj .(. L,.LL ..L., . 1:::: jjE:;j:,. .EE.:. jg(:q;' :33433). 4(:(2:). ........,....-...'..3,,....-------.-,.1L-'.L.L.i.LL?L.-b'.,--'.3. .jgj .... ji.!I.1Iq2'' .....q......E)..... lrrb ..E.. ji. 4:5:,4:k11141, ;'!' L'

.' '''

:'(i

.X''.)' ':.:E;'( .'. :..

;''.:(.j)k(k:'.':.;E''!'(.;)1rjq.y)))ijjEy: :' .... .'.

..

:. .

' . ... :.

-

.

. .

. .::

. .

. .

:. :.. .

:

. - . . .

L .' ''':'.'S: : ..' . . .. .

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. .

.. . .

.

.

.

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:::.

:

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.

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:

-t--.... ...,.4.

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................

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.

.

Investment and Operating decisions for oiI well. lnitial an investment

OCCLICS

:

2),

.5'

..--.--.-..---...-....--.-.-............--.-.

== ....-. .-.

A .

z

-.

,.z==..==

-

Oil Price . lnvestment Trigger Restart Trigger shutdownTrigger

...-..-....-.--,,-.--.,--..---..-..........-..-...

..

..

. ..

when the oil price crosses the investment trigger price, at point -5'

17.5 COMMODITY SHUT-DOWN

EXTRACTION AND RESTART

WITH OPTIONS

With production occurring over time and uncertainty about the price of oil, we face two new operating decisions'. 'Whether to keep the well operating, or, if it has been shut down, whether to reopen it. There are thus three stages of production'.

nitial investment in the well: We begin with an empty feld containing oil. This is an undeveloped well. At what point do we drill the well and begin extraction?

We answered this question in Section 17.4, assuming that the well operates until the resource is exhausted.

573

Thus, thewell can be in one of three states'. Undeveloped, producilg, and shutdown

Make the same assumptions as in Example 17.3, except suppose 17.4 the price of oil lognormally distributed with a constant lease rate and volatility is is that 0.15. The land value anct optimal extraction decision is given by

0.04 0.05

SHUT-DOWN

(developed but not producing). Figure 17.8 shows a hypothetical price path of oi1 over time and possible operating rules. Investment occurs the lirst time the oil price reaches the investment triggerprice, S. Prodution is shut down at the shutdown trijgerprice, S., and restarted at the restart trigger plice, S*. Thus, before pointAthe well is undeveloped. Between points A and B, and after C, the well is producing. Between B and C the well is shutdown. Key questions are: How do we determine the investment, shutdown, and restart triggers, S, S., and S*' and what is the value of the land on which the oil is located? Once again, we have to work backward, as in the binomial valuation of a stock option. Before we can decide the rule forinvesting (determining), we have to determine the value of a producing well (thisis the present value of future cash flows at the point w

Example

Callperpetual

WITH

Restarting an operating well: Having shut doFn the well, if the oil price rises agin, it may be pojlble to pay a cost and l'estat't the well, turning it back into a producing well.

occurs

With uncertainty, we could have the ability to shut a producing well. We will assun.e for the moment that production continues forever. ln tat case, the prolem is the same as in the single-barrel cse.

o'

EXTRACTION

Once we have made the investment in an oil well, we Contintling to produce: say that the property is developed. However, a developed well may or may not be producing. lf we are extracting oil from the ground, we have a developed twzt'/ producilq well. However, if the oil price drops below extraction cost, it may make sense to pay a cost in order to shut down the well and avoid future operating losses. Then it is a shlttdovll well.

$422.956) ($z1.4.914,

The value of the well at which extraction occurs is when S = 0.04 x $422.956= $16.918.

MODITY

A. Shutdown of production occurs

the shutdown trigger, & (point B). Production is restarted at the restar-t at point C. trigger,

c S*

.*,

n B Time

..

.

.

..

k. REAL op-rlorqs

CoM MODITY

invest). ln order to value a producing well, we need to understand operating decisions, specifcally how S,v,and S* are determined.

It is helpful to analyze shutting down and restarting by considering three separate

3. Production

once, this time permanently.

.%

..'.

Suppose that we are operating the well. If the current price is S and we ignore shutting value of the operating well is simply

down, the

P'operating, no

shutdown

.

=

S -

-

-

&

c

(17.10)

?-

Suppose that we can at any time pay a cost of kx, abandon the well, and never produce again. S,vis the price at which we shut down. What is the value of shutting dowh? There are three considerations:

Once we shut down, we no longer sell oil. Thus, we give with present value Sj8. 2. We no longer pay the exaction

up the revenue stream

cost, so we gain te present value c/?'.

-

c

-?

.

-

ks

=

c

( ) -p

.

-

ks

.% -

.,,..-.ty

'

,'

, .04)1

=

''''

''''

''''''

''''

-$53.

-

t) !

li.

E

ltt

When ks

) jy

PutPerpetual($10/0.04,

$8/0.05

) ltt The

-$69.863.

=

-

25, 0.15, 0.04879, 0.03922)

0.04 x

$1.8j

: .

..'

.'

q

.

'. .

E

( 17 11) .

This is #7' payoff to a pl// option wl/l strike price c/?kx and asset price ,S/1. If value this operating and the price is detenmine the value of th S, put to we can are we option to shut down, as well as the trigger price, %, for shutting down.

($5.778,$90.137)

=

$90.137 $3.605.We pay $25to avoid losses =

%.

=

=

'

-

The value of the producing

well Given that shutdown is possible, whatis the value of a producing well? The answer is that the value of the well is the value of the perpetually producing well plus te value of the shutdown option: optiont) Voperaungts')= Fkoshutdownts') + Vshutdown

l-/ef'hemodel with inhnite shutdown and restart was lirst analyzed in Brennan and Schwartz (1985)and subsequently by Dixit (1989).

. '

Ci

To interpret the shutdown resultsa there are two nattzral benchmark prices to consider. The flrst is tlze price at which te NPV of the operating will become zero, wlzich $6.40. The second is the price equal to the parginal ost occurs Fhen S 8 x c/r of production, c $8. lf S > $8, then we are mnking money by operating and we wilt not shut down. If $8 > S > $6.40,then we are losing money on an operating basis but the NPV of operating the well is positive. In this case we would consider shutting down if we could later restart production, but, because the NPV of operation is positive, we would not shut the well if the shutdown were pelanent. Finally, if S < $6.40,it may make sense to shut the well even if the shutdown were pennanent. Note that in a11cases, te initial investment is sunk and therefore irrelevant. In Example 17.5, shutdown is pelwanent so tll zero NPV price CS $6.40)is' natufal the benchmark. The usual option exercise logic applies: We won't shut down as soon as present value is negative, because the decision is Aeversible. In fact, we wait price might subsequently increaset by shutting down we until te plice is below $5. unable to benest from this reversal. This is the counterpat't to not investing as soon are as NPV is positive. =

.

.....

$53.17.

$25, the shutdown solution is

=

.

,..

($9.633,$10.301

'

' .'jji)By shutting down production, we avoid losses of (

. '').. '(t... i..; . .

.....'

'l'he

Thus, the value of shutting down at price Ss at a cost of ks is +

..

..

=

3. We give up k, the shutdown cost. sS.

$8/0.05, 0.15, ln(1.05), ln(l

shutdown trigger is then S lilt 1 with a present value of

Permaneqt Shutting Down

.'

shut down production when 5'/ = $106.83or when s = 0.04 x point, the present value of continuins to produce is $4 2-/3 $8.00 = 17 0 04 0 05

E.

j

..y

..

. Putperpetuallslo/o.ozl,

), ) nus, we lp $4.273,.At this i'

y tt' t)E)

. .

'

)

Each case layers a new option on the previous case. ln qddition to allowing additional shutting down and (estarting, we can impose costs of doing so. We focus in this section on the case where the well can be shut down never or once after the initial investment. Appendix l7.B adds restarting to the analysis.

..

Ejy

'i

an infnite number of times.l?

can be shut down and restarted

$

j')'pj Example 17.5 Suppose the oi1 well is operating and the oil price is =:u $10. We '(7't)i = = also ' have 0.15, and effective annual r gnd 8 are 5% and 4%, fjjeiikely. o' $8, c (? ))1If ks = $0, the value of the option to shut down is . .. )

.

2. Production can be shut down once, then restarted

AND RESTART Ol7Tlolqs

t'j' i'

tEj

1. Production can be shut down once permanently. After the well has been shut, the land has no additional value.

SHUT-DOWN

WITH

)'

) .ltr

C aS e S :

EXTRACTION

% REAL

576

Opn-lolqs

CoM M ODITY

E''' 7E jjjf j'. j.jf yrf )' r' *' t'y' j't;;;'. jIIjjj!(-'. 1* jlllrf . 'fL;;''q''. )' .' r:-' i. .. EE i fi. r ! E....k..jjjjy qk ; : ; .( r. ! . q.qy. q ....j( .... ;. qljljjk;pl!t. ).;jyyj. .... y.j... .qy..... g.y...jq..I. y.y t t Iillk:,,jjjl . jlj;::;ii::. . ig.)j.yyjlj;q:;:r .(il.. r . .'i(.t; j(t.-!. . ( .yj.jg .g...j j..jj.y.. ., j.y.j;y.j.y. . . yj lyj.yyy y (. . . . .... . . .. . . .'a,pikj-pip-.'-;j:jjs '''t' .F' s' ;';' !' 'Eyf :'r' -'tyyj -' :;' ''''' s' 7)* t'f.tf ;' jj:ygr:jjf ,j2jj,!' y' j'''y'' t)' t?f

..' .'

jsE' : : ' ' 'Ef't.rE ' j't' ''!' q' t';y' ('.' :E y' .y;..;jjy' rytf qty'i yf )' j'. f'. q' ' ;' i' .:''' . E '::' . '. ' : . E

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.

.

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.

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. ... -y rE.:, E .:.-.i.E ..... E i.E ... E. E.:..'..: .: ,. E . . . .i E .y : .. : E: y y L:.'....... !!..:.... .. .: . , j. ; . .......... .

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-

Value of a producing well that can never be shut down (eqation 17.8), and that can be shut once wth no shutdown cost (equation 17.1 2). The shutdown triggers for 'ihedifferent volatilities

are $6.40 c 0.00001), $5.01 (c 0.1 0), $4.27 (tz 0.15), and $2.68 (c 0.30).

' ' : '' E' E .

-

-...

.. .

.

;.

'' '' ' ' ' .. . .. ' .

. .E. .y-...

E

.

kjjjy.

y

Developed A'Vell($) Weil withotlt Shtltdown Option 0.00001 tr

--

150

$

AN D RESTART OPTIONS

Value of well at time of investment

=

P'p

=

)

1

-

-' '

100

..*--.

o'

=

(r

=

0.15 0 3

..---c-:.r

.

.::.3

50

-!.-,

....-,,.

---1'%V.'''--''

o

=

:.

.

.

..

'

sr r

w

t

'

.:.z .r...*,--.

-ql,

To solve for we need to find the present pa/lfd of equation (17. l 3), and then choose maximize this egyation to present value. For a givn (l7.t3) tells us the value of lf the oil price today is S < S, we can compute the present value investing when S of equation ( 17.13) using equation (17.14).The value will depend upon the current oil price (thelower the price, the longel- it will take to hit S4,so we denote it as nvesttz; 'X). This present value is 'j

vp%

.

.

=

..e-

.....

The variance of a lognonnal random variable ij .jj

() 8

r

'

j)j 1n()z) () 7 j

.5).

-

,

=

-

zkr,

1)

While

(gr2

(jg.jy)

we can compute the variance of a lognormal variable, it is much more convenient only the variance of ln(y), which is normal. We will not use equation (18.14)in to use the rest of this book.

.

yls) 0.6 ). ) jt. j 0.5 r 4A j21* t't 0.4 y j

elnt-vl

val*(c. K4 1 L- Probtuf < #), so =

18.4 LOGNORMAL

PROBABILITY

CALCULATIONS

If St is lognormally distributed, we can use tlzisfact to compute a number of probabilities and expectations. Forexample, we can compute the probability that an option will expire in the money, and, given that it expires in the money, the pxpected stock plie. ln this section we will present formulas for these calculations.

larobt-,> K)

=

x(Ja)

(18.24)

The expression Ni) contains the true expected return on the stock, e. If we replace l the risk-free rate in equations (18.23)and (18.24), with a we obtain the rik-neutral probabilities that St is above or below K. lt is exactly these risk-neutral versions of ,

% THE

600

LOGNORMAL

LoG N O RMAL PRO BAB I LITY CALCU LATI ONS

DISTRIBUTION

equations (18.23)and (18.24)that appear in the Black-scholes formulas.

call andput option plj ing

$

601

have

Lognormal Con6dence lntervals tlwhat is the range of prices We can now answer questions about future prices, such as will be in that range 1 year from such tat there is a 95% probability that the stock price today?'' To answer this question, we compute the 95% con:dence intervals for a numbr of different time horizons. StL < Stj and StU such that Pobvf Suppose we would like to lnow the prices Pobkhu > St4 //2. If the stock price is %, we can generate StL and StU as p/2 and follows. We llmw from equation (18.23)that

similarly,for

Stu we have su

s te

=

J

=

(ct-t-j(rz)?+rrx/j-1.96 -

$256.40

=

=

Probt.

Example

18.7 Suppose we have a lognonnally distributed $50 stock with a 15% compounded expected rate of return, a zero dividend yield, and a 30% continuously -L). The monthly continuously comVOlatility. Consider a horizon of 1 rponth t Ia pounded mean ret'urn is

5',f-)= N-z)




K. The

PS,

K,

>

(r,

t,

)

/,

)

e-&tvsNdLj

=

e-rt E*K

=

K)

Ke-rtNd

-

2)

5'lK

-

>

S) x Prob''tf-

>

stock

S4

We can rewrite this as

and using (18.24)and (18.28),with P ( S K .?..- r t J)

0.5G2)/

,

,

,

,

a

r, this becomes

=

=

Ke-rtN-dz)

e-bv%N-dkl

-

(18.29)

18.5 ESTIMATING

THE PARAMETERS DISTRIBUTION OF A LOGNORMAL

ln this' section we will see how to estimate the mean and variance of lognormally distributed price data. When stocks are lognormally distributed, a price St evolves from the previous observed price at time t h, according to -

(18 30)

uf

=

Stne

Suppose we have daily observatjons. deviaion? We have

Formula

(18.23),(18.24),(18.28),and (18.30),we

K) x Prob*tk

which is the Black-scholes formula. Similarly, the fonnula for a European put option on a nondividend-paying is derived by computing

.

Nz)

15'>

r, this becomes

=

.)

K)

-

As before, except for the fact that it contains the expected rate of return on the stock, a, ins'teadof the risk-free rate, th second term isjust the Black-scholes expression, Ndk ). The conditional expectation is

=

e-rE*K

Using

>

(18 28)

N-,)

11(S0)

=

E (s, I.St

e-rt /(7*(u$'15* > K) x Prob*tu

=

..1

For a call, we are interested in the expected price conditional on S partialexpectation of St conditional on & > K is K

t)

C(S K c

E (s? I-C t
ix data points, %ve would have assigned quantile ranges of 0 16.67, 16.67 the order statistics would then be assigned to the quantles 8.333, 25, 41.67, etc.

-

33.33, etc-, and

9The straight line can be htted in numerous ways. For example, Matlab connects the quartiles. ln the case of the sample data, the 10% and 30% quantiles are 3 and 4, so by interpolation the 25% quantile is 3.75. Similarly, tlze 70% and 90% quantles are 7 and l l , so by interpolation the 75% quantile is 8.

(' )' 9* (j;t5j'.' j'?' fL'. jslrllrqillillllrllrid..ijtdl.ll.r ltd''rrrlilE.lltr''ljlllyl.''p);;l;l .E'' 1* ''f' F' Fl?'. Ejd t'. ;';1:''. '5j' (C'. * jI).'1j;E@E.' yjd jjj:d.tjr't.rsd jrljs';t j))d ;'(rL'trjj(j!'..(j(; f'. krtik::th:ljd 1* )' ld q' SIIII'!;X L'. 3L'. )j' jrd q' Ijd (()' ()'j(' (jr' Ir';)))ljl (7)5* )J' f.jk(jjj;j.(tgjr;))))' j'();' Lf'L (j' 'jl)d (' /'y' )' (i''d (j(' jjyd (rd jjd qi':l' 1:* q)llj)d jrj.j;rd iydtli ()'(iiJ(gj(!jt' (jIEjJIX t).)gd (' lrd )' (iIIZIIIJIESI.SIE;;I;)II;III!i;IIIL))X t'))' tttd rjd r' r.jlkd ' ;'. .'C' 'fl ;'9)* F' 1* 7* t..'t'..'..' )' :'.... T'..E'.. qjsr';kE

612

% THE

LOGNORMAL

DISTRIBUTION

.*:65* .'..' j' Eq.i..!.'kt!.I; q' . ..

..

E .( jjj' (F;(jIi . '

..

:

CHAPTER

iI

'

'

:

(

jI!'' '' ;'

i

75?:.

'

.

'

k' ;. 1'. ; :); :

'

E.

FURTHER i i Eljk))j;j)jg);)k;jl ijlljgjlj!tl' !'q.;fE t)l tjt)l)q (j)( : .

SUMMARY

E

E' :

.!(

' .' .';' :i i:(' ... ... . . ..

.'. '

.

:

(' li ;j(i.);'E j ;()(: lij(t; ''):(.$( . .. . . ' . :

E

.

:E

ii .. EE .

...).

E

%.

READIN:

' ' 1. E('q I' j.(E'i'jri '(k( (r?( 'iiE(q; 1' 1((;'jlllqi:''l;t. . 'rj' ri ('E.(()((.2.(!.(' (. ((.'; ;.((i':: .;(i. '.'(' .'.!. (2jl. , .r. . .

1. ;

: :. :

'

'

..

:

'' .

.

:' E . .

('

E:

jlE

:

:

. .

E

E

'

(jE E

' '

..

,j'

!j' (q()' ;'' ('t' ..

' ;

613 t:.r'ji('; : .

The norm?l distzibution has these characteristics: and left sides are mirror images of each other. possible (albeitperhaps unlikely) * ltruns to plus and minus infinity, which means it is the distribution. from draw when could occur you that alty number the mean. * lt is unimodal', i.e., it has a single hump, which occurs at

* It is symmeic;

@Sums of

i.e., the

light

normal random vmiables are normal.

The lognormi distribution arises from assuming that continuougly compounded characteristics: remrns are normally distributed. The lognormal distribution has these * It is skewed to the right. that negative outcomes are impos@lt runs from zero to plus insnity, which means sible. single hump), which occurs to the left of the mean. * It is unimodal (i.e.,it has a Products of lognormal random variables are lognormal.

o

The Black-scholes fonnula arises from a straightforward lognormal probability calculation using risk-neutral probabilities. The contribution of Black and Scholes was in th formula. not the particular formula but rather the appearance of the risk-free rate and weekly daily probability plots for From examining hijtogratns and normal large returns continuously compounded returns, we can see that there are too many compounded continupusly returns relative to normally distributed returns. Although do not appear to be exactly normal, the Black-scholes model and the accompanying assumption of lognormality is used frequently and we will continue to use ard develop this model in the rest of the book. We will also explore extensions that are consistent with departures from normality we have seen in this chapter.

FURTHER

READING

ln Chapter 19 we will use simulation to price options assuming lognormal stock prices. We will also extend lognormality by allowing stock prices tojump discretely. In Chapter 20 we will introduce the continuous time model of stock retrns used by Black and Scholes, which is the basis for modern option pricing and which, with their assumptions, generates loglrmal stock prices.

Both the hlstogram and normal probability plot verify that continuously compounded retulms in practice are not normally distributed. The question is whether this and pricing matters for Xi'cing, and if so, how to modify theassumed price distributions fonnulas to obtain more accurate derivative prices. Two modifications we will examine in later chapters are to allow tlae stock to jpmp discretely and to permit volatility to be

stochastic.

614

k. THE

LOGNORMAL

DlsTRl

18.A:

AppElqolx

Bu'rlolq

An excellent discussion of the basic characteristics of stock remrns is Campbell (1997, chs. l and 2). The history of the normal distribution is entertainingly recounted in Bernstein (1996). (See in particular the accounts pf DeMoivfe, Gauss, and

et a1.

Galton-)

EXPECTATION

A LOGNORMAL

oy

VARIABLE

%

615

12 How does this probability change when 18.8. What is Probtu? > $ 105) for t change does when you change (r2 it you change t How =

18.9.

What is E St l& yOu

c

IAOXVC

r

,

G'

18.10. What is Probts't change t

PROBLEMS

THE

$ 105) for t

>

$98) for


Kv) for a variety 18.12. Let Kp of Fs from 0.25 to 25 years. How do the probabilities behave? How do you reconcile your answer wit.h the fact that both call and put prices increase with time?

-3,

-

-0.02,

=

.85).

18.3. Suppose pV(-2, 2). The covariance between .V(1, 5) and is 1.3. 'What is the distribution of + xz What is the distribution of .r1

ew

-n

.x-

-'t1

and

.;t1

.:2

=

=

'

=

.;ta?

A'l

-

18.13. Consider Probtu;

K), equation (18.23), and E (&15)< K4, equation (18.28). Verify that it is possible to pick parameters such that changes in t can have with very short and long times ambiguous effrcts on Probts'f < K4 (experimeht 0.5&2). < K4 ambiguous? > Is the effect of t on Est Iu% to mattllity, and set a

and 18.4. Suppose Jrl .V(2, 0.5), and x?. X'(8, 14). The correlation between distribution of is What is the What is the distribution of + .::2 ? l -'t ew

x

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18.5. Suppose xl .V(1, 5), .V'(2,3) and Z M(2.5, 7), with correlations 0.1, and pjj.s 0.4. 'What is the distribution of Jr1 + + pj.2 0.3, p l.a a:2) ? + + x2 + (0.5 x -'t1+ (3 x -'r2

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Xl, 5), what is E (c' will se: why risk-eutral valuation is important for Monte ;(. E of Monte ))!g tjtjjltlti'y Cado, see how to produce normal random numbers, discuss the eciency EEE E EIl'. iE; !iiq )' (r ' (rjii !g!kiIil.l! I !))q.,tty.tl.,q i ''.'..'.E(.(i( E;.E!''.'.t.'.);,;j,),ryry.y.y.i;r,.I,pI,.j.j#tjI ' ! . nonlognormal patterns in .( distribution to help account for inaoduce the Piss ;g! i . '(.(' 'E.E.(.) I)j. j !(ll I (jgjjjijjjyjryjty, g .t ....i kq;ip; jji. . jyjjtjiy . yt$'. (uta crrelated plices. randm stock and see how to create tj)e ;;:j;jgy .;;j.,.j.yjE..,,..(y!.yg.;.:jiIjjij.jy.yy.4yj.y)jygri.y r. qj r..(;.jj.;'. yj..j'.rjqgj jj,yj;jg.y ..'...t. . trr,tjjjjjy t',jjytjjy. t,,yjys.jtsy j jyj;y;'. . ,.....E.'.E'..i;,k@(Irj.g(.yj.ky.(jyrr.().j;gj.yyk..t) ytjjjjjtyy .......'E..';'.'.iE,EIi.,!!E,.EiE,I..i1.i,y.yiiji;)jtyjgrjt.yjyy)yI4 . . c.. '' ' :'E' '. : '! yyjjyyyjjjyiy (11::2(4q22:,),:t.. ' . : E.;!;l) i,, jp:,!j44g22),,. jqil!ii:;.((ijIqgF).,: -'.rgjII((y..-((jIIrqgq:11.jljllll. lj ; i ;;.;.i (gir:!'''.-k:j!r.' tgjjlliiiir:. qqljlll.., . Ett))j'., jtj ((j11:4. (jllrqg;,l,. gi)). .711r((,,,j;;r..'. j tqjll!lq..jillrl' ' . E i i :(..q . ....i.EI j!;l!i.!.(iq@ !;;(..r(. !;. (ggyy yjr'' j yyj (,-jjj'jj rr 411:yygggg2:,,,,. jj;:gggllllr:j jj;yj...q 411:24gg4)():: . . . ; '))!.E'.'.; qyy(j;i(.r;jyrk;j.;;j!;('. l i . y)g. ('. yjjj:!!.',,k2kjj... jyjjjllgyyqr)jj, yy r yyjjjjggjg:j, j ((jj(ygq,l,'. yljjjjjjig:,, yyy::yj;,jj::llj j . y j y! y j yy (jjjjgllygy:jj). yjjjjygj g (jjjjjjjjj:;r y( y y ..r.j.r.jjjjj( t'l'. jttyyjy .;j.j j.y gyt'jj. . .. ..(EErilEEiiljiEr,krtttthq! yg . j'. jyjj . y. . w i ; ; ; .1.. y jyljjj, ;:! q E. EE'.',,t..';q'p),ryEitytylgt'yy'yr.$tjjtjq'jri)gljtpjg; , j jy t!. ' g j r ( j j . . jjrq))4j)k'.. . y r . ltjrrjll./. '.'E.q.i.i...!.((r;k.j..jyjq!j;j jljjj. concept of risk-neutral valuation is familiar from earlier Chapters 15. We saw that ! :: yyk t'l yyrj.. y. .. rjj..jyy rjjj'qj .. . EEl'l,ilikEEEigEiylyllt,'.kiittjyriyyjty.yjyyy,,jjly yytjitili)! . . . ' .t)'tjjjj'. option valuation can be performed as if al1 assets enrned the risk-free rate of return and kjj(;;yiq;Ej.y-.g..:..y..(y . '.... tljp)t !!)g(j). j jtgg;jgjyy jtyg;j jkyyri gggjjggjy,,jjjj jtyjjksCrj!lyllj, jgyy;j :jgjjjgjs gjyy jkggji jkygrrjtryyjgyy 4::j1k,4:22:2 gj.kjks gjyy gjgy gtygjj $,g,j, gjjs (jjjjj;yjjyygjyg;j.tjj glrrrjljr'dlrg;j (jj, jkz:r g;yj;ys qlrjll, gjyg;jj ::i1t, jl;, jlr:lk .rgj. .' '.'E'E.'EEl7'.'-ii.l:.i-'Eii.jjyj rjy;j (jEr. $:E,t. (jyjjjyjrjjys jij. jlr:jg (jjygj.tjjj 42j2:: ql.;lr. 4(:)), :EE;, :j;; ,;:4:). jjy gjgr.glr)lr:)k, yjjy;jj 4k12:; (jL jiktjlrrjk. :i;,lrl l::EE j j q2 4j44,4t41 tljgkq jjkprjl tbj;q y .g y tq jt'y. tjjj ..'....','E','.'r::il!iir;'jt1)jy:yypEE!!yr).y !..jyj!.;yy jjjyjj..y.y yjg. j j jji j jyyry i ; ( . y , j j jj..y. their simulate and of remrnst. risk-free the .'... .. .. .y.j..q;yg j jyq return rate tyj,jtltiktt; uight. We assttme that assets earn ..E: ) jjj yy.y.iyyy,q;..!.,,..(.j,,...j .'. ;q....(j..q.y.,y(gj,.yj.j.jrjyjj,.j.j.jgjj.y,y.(gjr,yy.,yj, . jyy .....y jjg yy ;.jj..j.yjyj.yjiy,.,.g:y..y.j. . .,' . . s., jj-yyy(. tjt-). . . ; i i('j!'j j'j.(;-')yy(.. jggjjjj.,, rl:tllsrdtt'jyjg'jyj'yytrl j. j.j. tk g(jj.t'jj ytg '...q..('.'...'.i;(yyj.y;yE(;;(((g;!(jryj..jjjt-t'jl'jj. jjI.ji!jjj. tj .r...(.q.;.iiiq. j!jjj!(-@. j!jjqLLL., ( jj(j'..yyjg ) y'(t'kkjj y, . .j .q. ; y y)yyj'yyy'; yy;j.q(yj yj('.gj'i yy.. k.rjijjjrj'sjr')r g;jy.ji ..(y. ... ;;rjy..;(j.jj.jjtj': ..... .y.yjysyyjjj,jyjj yy j.... .......;i;.i.y..(.;. !ygE.jr.jkjryj.y(yj', ijj jyg(()'. tk. yjtkjj. tk;. j jjy).yj.... j yjyj'. y(;tyjjy.'yy'j'. t;(j..jjjy-yyr. yjj)jjjjj4. rq.qy ..i. .;...r.jj;y(;;;)(. kjjr'1jgj(j),'.. : jjy,,-'.jr(y',. . j!.....E.(j yyj,'yjjjy'. . . i;;E';i i).jl.ytt;;yjy :! . :

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lt:lqlzyr ,(::)) (21()(1E:jl. Cjqlk.lll. :dI:E ,11:4((2: . ,::2::$4E31. llEr'llls jlg 4(:2), 4:22:; 41:2,), qlrr)k. 41::,). :11:21L :EEIL.3I:..IIL. Cjzlk qltj,lL qdlgrjiL (11r:1L, 41E2,), ,1EE!, :jlk. :y-4222:, :!;. ji:!;. (jrlddf/rjlEr qlrr)k, rl:!ik??rjlq Ctlmpllte PIVeS Stock and the Se these imlllatedplices to v aj ua tsu we sjmujate f'uttlre kpctd pyoffof the option. Th ida that an option pdce is a discounted the discoutd . . .. . value is familiar .f'rom our discussion of the binomial model in Chapter. .11 and expected . . formula in Chapter 18. tjje sjacsscholes fisk-neutrtl distributicm Where we Monte Cado valuation i peffblrled using tlie discount the expected assume that assets earn the risk-free rate on average, and we thenIisk-neutfal . plicing is a this chapter that P ayoff using the risk-free rate. We will see in valuation', using the acmal distlibution instead would create Cado cornerstone of Monte . . . .. a complicated discounting problem. f'uture values of the sectlrit'y'z as you simulate the possible sincewith Monte . Carlo . ..'f'he Of payoffs. distribdtion can be extremely a b roduct you generate the distribution :!iE4;. wjjen you watzt tb ompare t'wo investment strateocriesthat have differynt distri. . sj . . . . . butiors of utctjmes. Computing value-at-liskfor complicated portfolios is a cornmon

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probabilities of reaching each of the 10.5, with the addition of the total lisk-neutral terminal nodes. Figure 19.1 demonstrates that the option can be priced by cpmputing the expected payoff at expiration using the probability of reaching ech 'Iinal node, and then discounting at the lisk-free Figure 19.1 rat. You an verify that the option price is the same as that in Figure 10.5. We can also use the tree in Figure 19.1 to illustrate Monte Carlo valugtion. Imagine gambling wheel divided into four unequal section, where each section has a probaa bility corresponding to one of the option payoffs in Figure 19.1: 9.5% ($34.678),34% ($12.8 14), 40.4% (0), and 16% (0). Each spin of the wheel therefore selects one of the linal stock price nodes and option payoffs in Figure 19.1. lf we spin the Fheel numerous times and then average the resulting option vlues, we will have an estimate of the expected payoff. Discounting this expected payoff at the risklffee fate prvides an estimate of the option value. It is easy to compute the actual expected payoff for the option in Figure 19.1 without using a gambling wheel. However, the example ill.ustrates how random trials can be used to perform valuation.

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t using true probabilities, we need to compute the discount rate-/r eh #Ir/3: 'Thf . eight possible paths for the stock price, four of which result in a positive .optio pyfft annualized continuously compounded discount rate Au of these paths have a srst-period discount The subsequent 35.7%. rates depend on the path the stock takes. Table o 19.1 verifes that discounting payoffs at path- (je p endent discount rates gives the correct osyou jsyx. so tjjja just tjw jjrst rowy the disctmnte (j expected option payoff for that 14::2j11.. :lijg;:j;:. ()1(, .1j,,)!It. ::!j:Il., (jlqr-. (ii. (41(,:!EE;:. (jjgl::j)-

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622

k: MONTE

CARLO

VALUATION

S l M U LATI N G

Using Sums of Uniformly Distributed Random Variables

LOG N O RMAL STOCK

PRI CES

$

623

distributed: I.f you draw from a distlibution, by desnition any quantile is equally likely to be drawn. The algorithm is therefore as follows:

One standard technique to comput normally distribpted random variablej is to sum 12 unifonn (0,1)Tandom variables and subtract 6. Thus, we compute the X(Q, 1) random variable 2 as

1. Generate a unifonnly distributed random number between 0 and 1. Say this is 0.7. 2. Ask: What is the value of z such that #(z) 0.7? The answer to this question is computed using the iltverse cll?nlf/lrvc dtriblltiol htllction. ln this case we have #-1 (0.7) 0.5244. This value is a single draw of a standard normal random variable (0.52+M. =

=

lli are distributed uniformly on (0,1). This technique works because the variance of a variable that is uniformly distributed between 0 and 1 is 1/12 and the mean is 1/2. Thus, if ytj sum 12 uniformly distributed random vaziables and sqbtract 6, you get a random variabl Fith a variance . of 1 and a mean of 0. The sum of 12 Jmiform variables is not precisely nonnal, ut it is close.This technique is an apptication of the cential limit theorep.

where the

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lt is also possible to draw a single unifonnly distributd'random nmbr and convert it N(0, 1) and z M(0, 1). to a normally distributed random number. Suppose tat lt denoted Uw) for the As we saw in Chapter 18, the cumulative distributiolstnctil, uniform and Ny) for the normal, s th probability that If < m o- z < y, i.e., ew

Uw) X()')

=

Probf

=

Probtz S )?)

.

This procedure simulates draws from a normal distribution.

To Simulate a log-normal

random variable, simulate a nol'mal random variable and exponentiate the clraws. This procedure of using the inverse cumulative probability distribution is valuable because it works for any distribution for which you can compute the inverse cumulative distribution.

.

Using the Inverse Cumulative Normal Distrlbutioq .. .

3 Repeat. .)

x.

S wj

As discussed in Chapter 18, w is the Uw) quantile and y is iheX(y) quantile of the two distributions. If we randomly draw a uniform number tt, how can we use 11 to construct a corresponding nonnal random numbe', z? lt mrns put that the same idea we used to conjtnlct nonnal plpts in Sction 18.6 i ts perm' us to geperate a pormal tandomnumberom a unifonp rapdom npmber. Instead of interpreting a random draw from the uniform distribution as a nltlnben we intqrpret it as a qltantile. So, for example, if we draw 0.7 from a N(t, 1) distribution, we interpret tllis as a draw corresponding to the 70% quantile. We then use the inverse distribution function, #-l (If), to find the value from te normal distribution corresponding to that quantile.3 This technique works because, for any distribution, quantiles are uniformly

19.3 SIMULATING

LOGNORMAL

Recall from Chapter 18 that if Z

Sje (a-&-)o.2)w+gxfz

=

written

( jq a) .

Suppose we wish to draw random stock plices for 2 years from today. From equation (19.3),. the stock price is driven by the normally distributed random variable Z. Set 2, a T and o' 0.30. tf we then randomly draw a set of standard 0.10, normal Z's and substitute# the results into equation (19.3),the result is a random set of lognormally distributed Sz's. The continuously compounded mean ret'urn will be 20% (10% per year) and the continuously compounded standard deviation of 1n(u%) will be 0,3 X 'Ui 42.43%. .0

=

=

=

=

=

Simulating a Sequence of Stock Prices 'I'IJere is another way to create a random set of prices 2 years from.now. We can also price. This will give generate tw??7ll/f random prices and compound these to get a z-yea.r exactly the distribution for plices. Here how do is it: to same us z-year .

* Compute the l-year plice, 3The Excel function Nornlslnbt computes the inverse cumulative nonnal distribution. Unfortunately, there is a serious bug in ths function in Oflice 97 and Office 2000. ln both versions of Excel, Nol'nlSInj'Q.999%9%6j= 5.066, and NonnSIn3,Q.99%9991) = 5,000,000. Because of this, Excel will on occasion produce a randomly drawn nonnal value of 5,000,000, whch ruins a Monte Carlo valuation. I tank Mark Broadie for pointing out this problem with using Excel to produce random normal numbers.

PRICES

.V(0, 1), a lognormal stock price can be

x

Sp

STOCK

SL

.

as

S3

l+o../fz(l)

x Sve (0.1 J0.32) -

=

* Using this 5'1 as the starting plice, compute Sz.. Sz

x 1+0.3xf(2) 5'l e (0.1 $0.32) -

=

-

ln these expressions, Z(1) and Z(2) are two draws from the standard

normal distribution.

'

624

k

MONTE

CARLO

VALUATION

MONTE

If we substititute the expression for 5'1 into Sz, we get Sz

(0. l Soe -

=

x2+0.3Yi1Z(

J0.32)

The difference between this expression and equation ./iZ we have (Z(1) + Z(2)). Note that

l )+Z(2)1

as Vv.,

(19.3)is that instead of the term

V

and

.

.

.

2

Therefore, equations (19.3)and (19.4)generate with the same distribution. If we really want to simulate a random stock prie after 2 yearsj there is no reason to draw two random variables instead of one. But if we want to simulate the pathof the stock price over 2 years (forexample, to price a path-dependent option), then %ve can do so by splitting p the 2 years into multiple periods. ln general, if we wish to split up a period of length F into intervals of length /7, . the number of such intervals will be 11 = r/ /7. We have qz's

S

Szl, = She @-J-Jo.2)+o.Xz(2)

F, zzz: -!7,,

-

I )/, t?

a-t-qcrljlt-o.xfltj

l stock prices can be intepreted as equally spaced points on the stock price path between times 0 and F. Note that if we substitute % into the expression for %h the expression for Szh into that for Ss, and so on, we get

maxto,

,

.usxge(a-:-Jo.2)F+gx/igj7.j (a-&-)o.2)w+(rx/Tg.J.;w

= Ste

=

v

-

-

(0,

=

max

E;'-l ztfj

-

-N

.,

i

=

1,

.

.

.

,

x

i maxto, Sv

-

K4

i= 1

This expression gives us an estimate of the expected optiopayoffat time /. We discount the average payoffback at the risk-free rate in orderto get an estimate of the option value:

'' p' $'

=

c-rT-

N

1

maxto,

N

= l

i

sv

-

K)

Suppose we wish to value a 3-month European call option where p Example 19.1 )1 the stock price is $40, the stnke price is $40, the risk-free rate is 8%, the dividend yield ltl lt1is ljj? zero, and the volatility is 30%. We draw random 3-month stock prices by using the t.yexpress kon l' )). 53 months Ste (0 os-o.32/2)x0.25+0.3x/(EB'z .

.

In Monte Carlo valuation, we perform a calculation similar yo that in equation (19.1). The option payoff at time F is a function of the stock price, Sp. Represent this payoff

r. We generate random

N

1

.

VALUATION

=

Average the resulting values:

ztflj

Since x/v E''i=z l Zi) -wA'(0,1), we get the same distribution at time F wit.h equation (19.5) as if we had drawn a single X0, 1) random variable, as in equation (19.3). The important difference is that by splitting up te problem into 11 draws, we simulate yhe path taken by S. The simulation of a pat id useful in computing the value of pathdpendent delivatives, such as Asian and barrier options, the value of which depend on ie path by which the pce arrives at Sv

CARLO

.J(.)

uaetr-d-o.so.zlrio'xffzi

K)

l

19.4 MONTE

(19.6)

standard normal variables, Z, substitute them into equation (19.3),and generate many random futtlre stock prices. Each Z creates one trial. Suppose we compute N trials. For each trial, i, we compute the value of a call as

These

sr

w)

Monte Carlo Valuation of a European Call We assume that the stockplice follows equation (19.3),with a

and so on, up to -i2-(?,

vs

i=1

rl'he

Sue(e-J-Jo.2)/,+o.Wz(l)

=

11

,

,

=

--e-rr

=

S) are ?) randomly drawn time-r stock prices. For the case of a call where option, Jorexnmple, Fts'w F) max(0, Svi K). Bot.h equadons (19.1)and (19.6) use approximations to the time-r stock price distribution t9 compute an option price. Equation (19.1)uses the binomial distribution to approximate te lognormal stockplice distribution, while equation (19.6)uses simulated lognormal prices to approximate the lognormal stock price distribution. As an illusation of Monte Carlo techniques, we will Erst work with a problem for which we already lnow the answer. Suppose we have a European option that expires underlying asset has volatility o' and the risk-free rate is r. We can in F periods. use the Black-scholes option pricing formula to price the option, but we will price the option using both Black-scholes and Monte Carlo so that we can assess the performance of Monte Cado valuation. ,

z(2)1

625

31

1

z(,%,0)

=

Var(z(1) +

k

vALuvlolq

F). The time-o Monte Carlo price, F(.a, 0), is then

(19.4)

VartWzzl 2

CARLO

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=

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626

CARLO

VALUATION

Mo

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N'rE CARLO

VALUATI

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E. . E.; ..,j,y .:

:

yrj

y

-

-

-

.

--- --. -. . .

..

.

...

.

To assess accuracy, we need to ltnow the standard deviation of the estimate. Let Cvki) be the call price generated from the randornly drawn v lf there are 11 trials, the Monte Carlp estimate is

'

:

:

:

.

.

'.

.

'

.. .

-

- . --

.

..

of European call wth 8%, t 91 days, and & 0. The Black-scholes price is $2.78. Each trial uses 500 random draws. =

=

=

.

..

=

=

=

) E'(E

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. !. E. .

. . .

.

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E

E (lljlEEE)'Ii. (ki ( l!1811F2:111:::11k.: EE ( 111:::)1,' ' ' tIiriiiIiI..' E.E ;i ' ;.k1II:!E1:(E:!!. ' (jllpliil,E'. E. .11Ii'. . .. ' )... . . . . .. .. . .. .

(..'. ..

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-

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.

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.

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.

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2. 2.75 2.63 2.75 5

1.91

Average

2.804

=

E( E . . .

-

-

.

;' .j

Option payoff

t.)

=

maXto,

c(l) iz=l

Let c'c denote the standard deviation of one draw and o'tt the standard deviation of draws. The variance of a meh, given independent and identically distributed Si is

. .

1,

's,

2

Gtt

=

1 a

-Gc lt

''''''

c/)

=

1

W

cc

Thus, the standard deviation of the Monte Carlo estimate is inversely proportional to the square root of the pumber o? draws. ln the Monte Carlo results reported in Table 19.2, the standard deviation of a draw iS a Ot1 t $4 05 tTllis value is computed by taldng the standard deviation of the 2500 Price estimates used to comput the average.) For 500 draws, te standar deviation is Ss mnths

-

.

$4.05

$40)

500

.i...

' tjt We repeat this procedure many times, average the mjulting option payoffs, and discount ' IJ ' ' tjl the average back 3 months at the lisk-free using 2500 draws, rate. With @.single estimate jl' t') we get an answer of $2.804(seeTable 19.2), close to the true value of $2.78. % In this example

11

.

it' For each stock price we compute i l..

11

-

..

.

j'''j'

j

'&?,

we pliced a Europearsstyle

option. We will discuss in Section

19.6 the use of Monte Cado simulation to value American-style options.

Accuray of Monte Cado There is no need to value aEuropean call usingMonte Carlo methods, butdoing so allows us to assess the accuracy of Monte Carlo valuation for a given number of simulated stock price paths. The key question is how many simulated stock prices sufce to value an option to a desired degree of accuracy. Monte Carlo valuation is simple but relatively inecient. There are methods that improve the efciency of Monte Carlo; we discuss several of these in Section 19.5. To assess the ccuracy of a Monte Carlo estimate, we can run the simulation different times and see how much variability there is in the results. Of course in this case, we also lnow that the Black-scholes solution is $2.78. Table 19.2 shows the results from running 5ve Monte Carlo valuations, each containing 500 random stock price draws. The result of 2500 simulations is close to the correct answer ($2.804is close to $2.78). However, there is considerable variation among the individual trials of 500 simulations.

= $0.18

Given thatthe correctprice is $2.78,a $0.18standard deviation is a substantialpercentage of the optionprice (6.5%).Wit 1500observations, the standard deviatin is ctt $0.08, suggesting that th $2.80estimate from akeraging the five answers was only actidentally close to te correct answer. ln order to have a 1% ($0.028)standard deviation, we would need to have 21,000 trials. .

.

.

'

.

.

'

.

.

.

Arithmetic Asian Option ln the previous example of Monte Carlo valuation we valued an option that we already could value with the Blk-scholes formula. In practice, Monte Carlo valuation is usef'ul under these copditioj: * Where th nmber of random elements in the option valuation problem is too great to permit direct numrical solution. o Where underlying yariables are distributed in such a way that direct solutions (lifitl tllt. @Where options are path-dependent, i.e., the payoff at expiration depends path of the underlyinj asset price.

re

upon the

For the case of a path-dependent option, the use of Monte Carlo estimation is straightforwrd. As discussed above, we can simulate the path of the stock as well as its terminal vlue. For example, consider the valuation of a security that at the end of 3 months makes a payment based on the arithmetic average of the stock price at the

)' j'

E ((' . . .. . . E ': .' .'.. . g.j

M o NTE

end of months 1, 2, and 3. As discussed in Chapter 14, this is an arithmetic average plice Asian option: tlAsian'' because the payoff is based on an average, and average price'' because the arithmetic average stock price replaces the actual stock price at expiration.

s2

.,j,

s3

.q$

2

%

19

.

E

Tgble 19.3 lists plices of Asian options computed using 10,000 Monte Carlo trials each/ The first row (wherea single terminal plice is averaged) represents the price of an ordinary call option with 3 months to expiration. The others represent more frequent averaging. The Asian plice declines as the averaging frequency increases, with the largest plice decline obtained by moving from no averaging (thefirst row in Table 19.3) to monthly averaging (the secod row of Table 19.3). Note also in Table 19.3 that, in any row, the arithmetic average price is always above the geometric average price. 'Ihis is Jensen's inequality at work: Geometric

g(r-J-o'2/2)F/3+o.x/r/Jz(2) .(r-J-(r2/2)zy3+o.x/r/Jz(3)

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where Z(1), Z(2), and Z(3) are independent draws from a standard normal distribution. We repeat the trial many times and draw many Z, The value of the security is then computed as 's.

CAsian

Example

=

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.' .'ptp.:.'...

.%z4/3

K, 0))

-

( 19.7)

19,2 Let r 8%, o' 0.3, and suppose that the initial stock price is F'igure19.2 ompares histograms for the actual lisk-neutral stock plice distri-

$40. tyd rtyd tt'. $111/(*, t'ttd tgptd .-'yy' .'

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averaging the three month-end stock prices during the same period (bottom panel). Assumes S r .$40,

=

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.40c(r-1-(r2/2)F/3+(rx/F/Jz(I)

1

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HoF will the value of an option on the average compare with an option that settles based on the actual expiration-day stock price? Intuitively, averaging should reduce the likelihood of large gains and losses. Any time the stock ends up high (in which case the call will have a high value at expiration), it will hav traversed integnediate stock prices in the process of reaching a high value. The payoff to the Asian option will reflect these lower intermediate pcices, and, hence, large payoffs will be much less likely. We compute the l-month, z-month, and 3-month stock prices as follows: =

VALUATI

)t bution after 3 months and that for the average stock price created by averaging the !@ l') As expected, the nonaveraged distribution has signifcantly .: three month-end prices. j(' . higher tail probabilities and a lower probability of being close to the initial stock price tjk ).1of t.j $40. k.

tialithmetic

s'j

CARLO

i' ..

'.. ..g

4A trial in this case means the computation of a single option price at expration. When 40 prices are averaged over 3 months, each tlial consists of drawing 40 random numbers; hence, 400,000 random numbers are drawn in order to compute the price.

'. . . .

.-.

(

.. ...

630

k

MONTE

CARLO

EFFICI ENT MONTE

VALUATION

CARLO

$

VALUATION

631

This is a control variate estimate.u Since Monte Carlo provides an unbiased estimate, E'-) = G. Hence, S(A*) = '(A) = z4. Moreover, the variance of A* is

averaging produces a lower average stock price than arithmetic averaging, and hence a lower option price.

*) vartW) + var ('&) zcovtW vartz't

'&)

=

19.5 EFFICIENT

MONTE

CARLO

,

As long as the estimate 'J is highly cormlated with the estimate W,the variance of the :stimate A* can be less than the variance of W. method can be dramatic. hpractice, the variance reduction from the cntrolvariate Figure graphs the results from the frst 200 simulations in pricing an arithmetic Asian option. The control variate estimate converges in just a few trials to the correct value of about $1.98. For example, the very first draw in the graphed simulation gave an arithmetic option plice of $0.80and a geometric price-using the same random pricesof $0.75. The correct geometric price is $1.94. Correcting the estimate gives a price

VALUATION

Monte Cado, making no attempt We have been describing what might be called for givn nulpber of trials. There are a the variance of the simulated reduce answer a to valuations.D . number of methods to achieve faster Monte Carlo

'naive''

.19.3

Control Variate Method We have seen that naive Monte Carlo estimation of an arithmetic Asia option requires many simulations. ln Table 19.3, even with 10,000 simulations, there is still a standard deviation of several percent in the option price. In each row of Table 19.3, the same random numbers are used to estimate the ' option price. As a result, the errors in the estimated arithmetic and geometric prices are cofrelated: Whe te estimated price for the geometdc option is high, this occurs because we have had high returns in the stock plice simulation. This should result in a high arithmetic price as well. Tlzis observation suggests the control variate.method to increase Monte Carlo estimate underlying this method is the The idea to error on each trial by using accuracy. the price of a related option that des have apricing formula. 'I'he error estimate obtnined f'romthis conol price can be used to improve the accuracy of the Monte Carlo plice on each trial. Asian options provide arl effective illustration of this idea.6 Because we have a formula for the price of a geometric Asian option (seeSection 14.2), we know whether the geometric price from a Monte Carlo valuation is too high or too low. For a given set of random stock prices, the arithmetic and geometric prices will typically be too high or too 1ow in tandem, so we an use information on the error in the geometric price to adjust our estimate of the adthmetic price, for which there is no formula. To be specisc, we use simulation to stimate the arithmetic plice, W, and the Let G and A represent the tnle geometric and arithmetic prices. The geometric price, error for the Monte Carlo estimate of the geometric price is (G V). We want to use this enor to improve our estimate of the arithmetic plice. Consider calculating

-

of

Control variate plice

=

$0.80+ ($1.94 $0.75) -

This example illustrates that if the con-elation control variate method works very well.

=

$1.99

between the two estimates is high, the

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Price of Arithmetic Average Price Call 40 #

3.5

-

-

Naive Monte Carlo Cont-rol Variate

3.0 ;

2.5

'-.

1.5

-

*

:4

=

W+

(G

-

1.0

'-)G

sExcellent overviews are Boyle et al. (1997)and Glassennan (2004). See also Judd (1998,ch. 8), which in turn contans other references, and Campbell et al. (1997,ch. 9).. ti-fhisexamplefollows Kemna mzdvorst (1990),whoused thecontrol variate method topricearithmetic Asian options. '

0.5

0

20

40

60

80 1O0 120 Number of Trials

140

160

180

200

''naive'' Comparison of Monte Carlo estimate of arithmetic average option price with method. Graph depicts first 200 simulations for an option with variate control $40, (7' 0.3, r 0.08, T= 0.25, f 0, and the final price computed with three # .

=

,$40,

=

averages.

=

=

=

---.*,'k1-.

)'i '

--

632

% Molq'rE

CARLO

Boyle et al.

VALUATION

VALUATI

(1997) point out that equation (19.8)does not in general provide

*

=

W+ p (G

-

'-)G

( 19 10) .

The variance of this estimate is Var(A*)

=

'J) var(W)+ p2var('J) 2pCov(W,

(19.11)

-

Var(X*) is minimized by Setting p = COVIX, V)/Var(V). One Way to of trials, perform small nunber Monte Cado obtain p is to nm a regression of equation a remaining trials. The optimal value of p will obtain and then (19.10) to use p for the p, depending the application. on vary

The

Valiance

19.6 VALUATION

Other Monte Carlo Methods

-0.5.

Tere canbe an efciency gain because the two estimates are negatively correlated; adding them rdces the variance of the estimate. ln practical tenns, this means that if you draw an exeme estimate from one tail of the distribution, you will also draw an extreme estimate from the other tail, balancing the effect of the srstdraw. Boyle et a1. (1997) nd modest benefits from using the antithetic variate method.

,

.

.

.

,

,

,

,

-

$

OF AMERICAN

OPTIONS

American options. We will discuss pricing a 3-year put optin with a stlike of $1.10, the example used in Longstaff and Schwartz (2001). In order to analyze early exercise we need to consider times before maturity, so we must simulate stock pjcepaths. Figure 19.4, taken from Longstaffap.dShwartz (2001),illustrates eight hypothetical simulation paths, with intermediate stock prices generated annually. The in-the-money nodes (thosefor which

calculation.

=

O PTI ONs

It is generally more dicult to value American-style options than to value Europeanstyle options, and this remains true when using Monte Ctrlo valuation. Standard Monte Cado entails simulating stock price paths fonvayd,then averaging and discounting the matulity payoffs. ln American option valuation, te difliculty is knowing when to exercise the option', this requires worlcing backvard to determine the times at wltich the option shouldbe exercised. Recently, Broadie and Glasserman (1997)andtaongstaffapd Schwartz (2001)have demonstrated feasible methods for using Monte Carlo to value

The control variate example is just one method for improving the efficiecy of Monte Carlo valuation. The antithetic variate method uses the insight thatfor every simulated realization, there is an opppsite apd equally likely reylization. For eyampl, if we drpw By ujing a random normal number of 0.5, we could just as well have drawn the opposite of each normal draw we can get two sipulated outcomes for each random path we draw. This seems as if it would help, since it doubles the number of draws. But drawing a random number-is often not the time-consuming part of a Monte Carlo

Another important class of methods controls the region ln which random numbers ar'e generated. Stratitied snmpling is an example of this kind of method. Suppose 100. With naive Monte Carlo you have 100 unifonn random numbers, IIf i = 1, (IIf). #-1 This calculation each random number as would treats compute you zi representing a random draw from the cumulative distributon. However, because of random variation, 100 uniformrandom numbers will notbe exactly uniformly distributed and therefore the zi will not be exactly normal. We can improve the distribution of the Iff and therefore of the zi if we treat each number as a ?w??#(??n drawh'om wc/? percentile of f/?c ttniform distribtttion. Thus, tale te first draw, If 1 apd divide it by 100. 'I'he Take the second draw, divide it resulting I11 is now uniformly distribtuted over (0,0.011. by 100, and add 0.01. The resulting I12 is uniformly distributed over (0.01,0.02).For the This value is unifonnly distributed over the fth draw, compute i = i 1 + IIf)/100. th percentile. Proceeding in this way we are guaranteed to enerate a random number for each percentile of the normal distribution. You can select a number of intervals

AM ERI CAN

different from 100, and you can repeat the simulation multiple times. A generalization of this technique when the payoff depends on more than one random variable is Latil santpling, discussed by Boyle et al. /?lwd?'cal?c (1997). There are other techniques for improving the efciency of Monte Carlo. The approach called importaltce samplillg concentrates the generation of random numbers where they have tle most value for pricing a particular elaim. futnp discrepancy seqltences use carefully selected deterministic points to create more uniform coverage of the distribution. Boyle et al. (1997)provide an excellent summary and comparison of the different methods. lf you are performing a one-time calculation, the simplicity of naive Monte Carlo is appealing. However, if you are pedbrming a Monte Carlo valuatin repeatedly, you may achieve large efliciency gains by analyzing the problem and using one or more variance reduction techniques to increase efciency.

the minimum variance Monte Carlo estimate, and in some cases can even increase the variance of the estimate. They suggest that instead of estimating equation (19.8),you estimate A

ohl o F

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E' ' ' (. .( ((i.i'(; E!i !; q' ! q72q'(i.';.' E.. ;''.('.'(;E.(('.'g'(('. .'j. (.. ( !.(..' 7..'.'. (.'' !.. ..' E'E('' . j EEEE j . .. . . . E...E E.. '- ! p rE ! E EE E E .. ; E(I!!i:1I:::;pkkE. !.yjEy EE....Ey q q .jgj.rq EiiiI:--k1I1I. j.. . . Ej .yq;yr y . . y 1...:.E' ! .. . . i .'.i.:i. . . . . . . :' ' ..

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.jj... .y.y.yy(;qq.qyg(..(y q. r y( j.t; jjjjj:jrj. . t.r-;..y.gy.) jr..r..y..q. r....j.j ..y iqy . y lllI:i;iIi(( :!!-.. ; . .. ).(.;)(. . ... .r... .r.. ..... . ./1!ik.1i!; 2!5;2 1115!, 1L..lI lr''1I''11 11E;* () .y4:j4 # 1 and r 6%. Prices in bold are nodes where early exercise might be optimal. .

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(llI(q:'I,' . (jii.-1Iii,,rlll:.:llt. :IIt(: c ((111E(4:.1,rlll:r''. ;:.E!I,L. 1 t = 2 t 0 t 3 t 1.00 1.09 1.08 1.34 1.00 1.16 1.26 l 1.00 1.22 1.07 1.03 1.00 0.93 0.97 0.92 l 1.11 1.56 1.52 1.00 0.76 0.77 0.90 l 1.01 0.92 0.84 1.00 0.88 1.22 1.34 =

=

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Soltrce: Longstaff and Schwartz (2001).

634

%.Morq'rE

CARLO

VALUATION

VALUATION

Continuation value at time 2

-1.07

+ 2.98 x S

-

1.81 x S2

(19.12)

( 10.10).

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Exercise

0.l 88 0.085

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Exercise

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Exercise

Wait

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which exercise is optimal (in bold) for the paths in Figure

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?For example, this is the comparison in the binomial valuation in equatinn

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where S is the time 2 stockprice. Now foreach node where exercise could be optimal, we insert the stock price at that node into the regression equation and obtain an estimate of continuation value. By comparing this to intrinsic value, we decide whether to exercise 1.08 in row 1, the esimated value of waiting to at that node. For example, when S exercise is

635

=

-

early exercise is optimal?

=

$

.08

.l0

.10)


??7 observations. the covaziance matrix estimated from theje data will be positive-desnite. However, if different covariances are estimated from different data sets, positive-desniteness is not assured. The results of a simulation based on such covariances may produce nonsensical results.

-

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.

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pairwise correlatios we have to take into account. between kariables i and j as pf,y. .

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We denote the original uncorrelated random The correlated random variables tre Z(1), Z(2), .

F(Z(f)Z(.j)) We can generate the Zi)

We will denote the correlation ..

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Xo, 1) variables as

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.

,

.Z(?l), with

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c2,

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pf,y

as i

Zi)

aitjej

=

;=l

FURTHER

selected to make sure the pairwise correlations lre correct. where the ai,j are coefcients Creating the coefficients i,j has a recursive solution. That is, we construct Z(1), the'n Z(2) using the solgtion to Z(1), and so n. The formula for the ai,.j is Cli'j

j-t

1

= a

9i

j.j

'

j

Monte Carlo methods entail simulating asset rettlrns in order to obtain a futtlre distribution for prices. This distribution can then be used to price claims on the asset (for example, Asian options) or to assess the risk of the asset (we will focus on suh uses in Chapter 25). Performing simulations requires that we draw random numbers from an appropriate distribution (for example, the nonnal) in order to generate future asset prices. There are adjustments, such as the control variate method, which can dramatically increase the speed with which Monte Carlo estimates converge to the correct price. It is possible to incorporate jumps in te price by mixing Poisson and lognol'mal random variables. Simulated correlated random vatiables can be created using the Cholesky decomposition.

Lvkcli'k

-

READING

The rst use of Monte Carlo meths to price options was Boyle (1977)and the technique is now' quite widespyead. An excellent survey of the use of Monte Cado in valuing derivatives is Boyle et al. (1997).We will see how Monte Carlo is used in value-atrisk calculations in Chapter 25. Bodie and Crane (1999) use Monte Carlo to analyze

k.1

( 19 18) .

(' i .-. l

ai

9

;

1

=

*

-

a;2k I .,

lzsuppose there are three random variables, a, b, and c, each with a variance of 1. Suppose that a is perfectly correlated wit.h b pa /) l ) and b is perfectly correlated with c pb.c 1). It must then be the case that c is perfectly correlated with a. lf pa c z 1, the matrix of correlations is not positive-definite. To see this, suppos that pa.c 0, then compute Vartc b + c). You will find that the variance is -1, which is impossible. To take a different example, suppose ps.c. 0.9. You will then hnd that Vartc which is again impossible. 2b + c) If a matrix of correlations is not positive-definite, it means that there is some combination of the random variables forwhichyou will compute anegative variance. (Formanycombinations the variance will still be positive.) The ntemretation of a negative variance is that you had an invalid correlation matrix to start with. =

=

=

'

For the case of two random variables, this reduces to equatlon (19.17). The matrix of azj's is called the Cholesky decomposition f the original corgive coefcients, matrix. order for equation the set of In relation correct (19.18) to which positive-definite, that the correlations be correlations must must be such means a' variables and negative variance. is random This is that'there no way to sum compute

-

=

-0.2.

-

=

646

% MONTE

CARLO

VALUATION

retirement investment products. Schwartz and Moon (2000)use Monte Cado to value a by simulating f'uture cash flows. 1117.n The papers by Broadie and Glasserman (1997)and Longstaffand Schwartz (2001), which present techniques for using Monte Carlo to value American-style options, have clear discussions of their respective methodologies.

1 e. slok

19.8. Assume that the market index is 100. Show that if the expected return on the market is 15%, the dividend yield is zero, and volatility is 20%, the probability of the index falling below 95 over a l-day horizon is approximately 0.0000004.

jumps. Naik and Lee (1990)illustrate option plicing in the presence of systematicjumps.

19.9. Suppose that on any given day the annualized continuously compounded stock ' return has a volatility of either 15%, with a probability of 80%, or 30%, with a probability of 20+. This is a mixture of normals mdel. Simulate the daily stock returfl and constnlct a histogram and nonal plot. What happens to the normal plot as you vary the probability of the high volatility distribution?

Risk aversion affects the option price in such cases.

PROBLEMS 19.1. Let

N(0, 1). Draw 1000 random Iff and construct a histogram of the results. What are te mean and standard deviation? 'w

Iff

l Ef=I 2

19.10. For stocks 1 and 2,

19.3. Suppose that of c'' and e'''2

ew

.x1

.h (0 1) and

.h:2

c'K2?

19.4. The Black-scholes price for a European put option with S 0.08. J 0, and t 0.25 is $1.99. Use Monte o' 0.30, r price. this Compute the standm'd deviation of your estimates. need do you to achieve a standard deviation of $0.01for your

=

=

=

19.5. Let 1' 0.08, S $100, 0, and tion, simulate 1/u$'1 What is EjzLl =

=

=

.

Paying 1/,1 ?

tr

=

.

0.30. Using the risk-neutral distribuWhat is the fonvard price for a contract

=

=

$40, K = $40, Carlo to compute How many trials estimates?

=

100, r 1l 6 Supp ose Sj 0.4 and 0.06, o's prices for claims that pay the following: .

=

0. Use Monte Carlo to compute

a. 5'l2 b. 5'1 SI c. -2 -0.3

=

'

,

a. 5'l :1 b. VSL/QL c.

SL

:c2

=

=

19.11. Assume

$100, r = 0.05, o' = 0.25, = 0, and r = 1. Use Monte Carlo valuation to compute the plice of a claim that pays $1 if Sv > $100,and 0 othenvise. ('rhis is called a cash-ormothing call and will be further discussed in Chapter 22. The actual price of this claim is $0.5040.) us'n

=

a. Running 1000 simulations, what is the How close is it to $0.50402

:1

=

.

,

=

.

=

estimated

price of the contract?

lour

Monte Carlo estimate? What is b. What is the standard deviation of the 95% consdence interval for yot!r estimate?

c. Use a l-year at-the-money call as a control using equation (19.8).

variate and compute

a orice -

'

d. Again use a l-year at-the-money cgll as a control variate, only this time use equation (19.10).'What is the standard deviation of your estimate? 'Forthefollowingteeeproblems, assumethat Sz $100,r 0.08, c4 = 0.20, c = 0.30, and J = 0. Perfonn 2000 simulations. Note that most spreadsheets have built-in functions to compute skewness and lturtosis. (1nExcel, the functions are Skew and KulKl For the normal distribution, skewness, which measures asymmetly is zero. Kurtosis, discussed in Chapter 18, equals 3. =

and that s'a $100, 19.7. Suppose that ln(5') and 1n(:) have correlation p 0 06 o' s 0 4 and o' c 0.2. Neither stock pays dividends. po $10t l Use Monte Carlo to find the price today of claims that pay =

$40, Sz

,

Why? and a. What are the means of b. Cfeate a graph that displays a frequenc'y distribution in each case =

?'

=

correlation.

M(0.7 3). Compute 2000 random draws E'

c'VI

=

Let

-

x

,

SI

$100, and the return coryelation is 0.45. 0.08, (7' l = 0.30, (7'2 = 0.50, and J1 = (z 0. Generate 1000 l-month prices for the two stocks. For each stock, compute the mean and standard deviation of the continuously compounded return. Also compute the remrn

lti 19.2. Let lti. 6, 1000 times. (This will use 12,000 N(0, 1). Compute random numbers.) Construct ahistogram and compar it to a theoretical standard normal density. What are the mean and standard deviation? 'x'

647

d. 1/(S'1 Q l )

(1976)derived an option pricing formula in the presence of idiosyncratic

Merton

k

PROBLEMS

=

=

1/52. Simulate both the continuously compounded actual return and 19.12. Let /? the acmal stock plice, St-v What are the mean, standard deviation. skewness, and kurtosis of both the continuously compounded return on the stock and the stock plice? Use the same random normal numbers and repeat for /2 l Do any of your answers change? Why? =

.

=

.

j'

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.ggyjyj.yj.yyjj j j gjy.j . .. ( q.!...@ jgjyyy jyy, jyyy:. yy .......g.....q...).qy.j.gj;j.(;.g;jjryqg;.jjy.,j.ryj.yjyjyjjjrj .g jjjjyy, yy .g yy (...j.jry.;;.jy);jjy jyy j)jrjr j jg jyjy . y , y -. j r ; y;j y jy . y ; jgjy .. ....y.;.yy.j.;.jy.)jjy.jjgjj.yyyy;jyy jyygy.jjjj.ygyjg yj yjy yy .,y-ygy.... ...((... !(. (i(..iiy;qj;.jjyy;yjgy..jyj.jj.gg.ygj yyyjy yyyyy yj.yy ...j . ...... .(.;. (.y)(jjjj..jyygyjyjryy. .y. g yygy.g. jy:j!. y.jyj.jrjjjy.. jyjj j.y. g j . .qy. '.E'E.y..!. iyq@ ;;iiE g . j . y. @ j j.j(y. q. )j yi E l!jjjjjj..jy ; j j . . . . . i.j.!yj.;jrq.jg.jr;!)lg j.j. (iy ' ' .q..q..;r;@ . ; j i ''.'''(q.y!;y,..,...y.(;yi,t ...(!.(j.!jI!... j!yjj.j. q. y .jjj.yy..... jjrrggjjgjggry yjjj..jjjjjgjj; .jjjtyyjy, . E :.!! : j gyjjjjjj;jjjjjjgjj,,.,y .j;jryjjjj,,,rgjjjjj, jjjjjjg qj;ijrI.t yjy ty gjjgyyjjjjj;;jj,.. rEjq,y j j jt. j ;q y ! ; y j l E yr;, .'.....'r..'...y!.(;..i;Ii..;jly.(I.j.;:i ij . gg.j.jj.qj. jj,:ygyyyyjr:. . i p . iljjjjjj;;, . jjjjljjjjr;;(.ji;!!g.l.,k j.. k yyj jk! y jyjj. ...E.. !.''iililiirlittjjy. ; . yjjjjgyg;jrr.., y j .. j r rljjjj;;i . j j j ).l.;!(@ . ; j,yjjjyy..yj ).jIyr . j yjpir . j . y j tyj .. . , tyjyy. yy, gjy.jjyy ... i....g...k..(q.E!y(k.jyg(j;.j.jqr.q!r.rrj yy .j.j vast magority of technical option pricing discussions, including the original j .(.'E''lEEiilgllEjljllkjlijiiiiilytiliyyrjlytylltkyjryjyiijtijjtk.r'jyyqry.d,p,r,.yq ; yj .yj.g.j.g( jgy .. q.;j !j!yk jy.gj!;jjyjy paper jg r .' E (yj( kjy! by Black and Scholes, beesn by assuming that the price of the underlying asset follows j!yy.. . ..E..q'... !j;.;.j;(j:;i;;:jl@ r. liji jjyj. p .rj.iyi:jy.jjyy.. jr '....' ..;).(.@;..;..!yjjy!r!jly:yj.rjgj-i :yyjyjj,s :!jjj;; irjjj;irjjj.jyjg:;;yy, kjjyrr;jjs yjjysyjjys (g j ..(. tjjj.s.1,y24jjkyyr),,. yjjjjy:j,, . .. j(y.ijj..j.yyy)r. ) jjr:yjI(jji. yy.g. ......('..q..E..('.!...(Ej.q.iEEq;q.jryjjyj;I;jjj jj rjr !jj, t! y- ; (.(. . i . . i !y i jj:jjy . ; ( .. ( q!!jjItjjj . j g).j.jjg.y. ... ....'. E...EE..; jjy)jjj::;,r!, jjr. jjj j-j. E. jtyy i; jjy,)ttr ;jr. j j ''..('.i.q;.ii;r!.!.iqi-.y!.;;.j;.(..;jlyi.yj.y!lj;j.)p. j;jj jj:yy. ' ... E..-!(.'qr;r.y(,!jjj);jjrjr.:jkIj:y.jjti yy()-jyy(...;yy;qyj(. jjy)jj . (.;. !.y(y.;;yj;jjjyjjj.j;y;y. p j jjjq, . j.j.(g.,.. ,( jy j j jj ryjjjy r j rjjyy ;jjg yjj,jytj jy. i q ir ln this expression, is the stock price, dst) is the instantapeous change in the stock (.i.. (jp I ;; j .jjtitttyti. y4 qy jj ..' . E''!;).''p':;iE!.qjiti.i', r;.;;j..;!;.)j;)j!jg!yjqjyj yjjy.yj(. j.j ( ; j q price the continuously ; : compounded expected rettlrn on the stock c' is the continyjjtyyj; is E'E y a ; !j;! ( ( ;..jjyj y '.''...E;..kE( yp.yyjjyjjky y!yy yy. )jL)j,;;, .j.;.yj.jj y.jy)j j ;jj(y jr uously ( (... .(..Iry compounded standard deviation (volatility), j (. and Zt) is a normally distributed .. .ytllty,t, .. . jjjg g.j yj .. .E'E,E.'i!Egr!j!'E!j'i.I;jj);t;;..t.;i. jgjjg . y. .j!iijllyjt. yarzom (q E.E..;i;; i;..rji.,ji.;.,yq!j .;;l i;. variable follows that called l motion. The variable dzt) y ; j Brownian ;y,y.. y g j y jj y a process . ; t) y yj yj .(..yy.. ... '.;...-...;;..;.kL-L;--j.q3.-;LkLL.;qj;.j-)3.)L;j.;L.L;kkLj3L,qb,bb-b.)bb;..b-()j y,y.yy.. )j());;j.)k.yj-).) .::11!. Elq. dt:::r ::E1..:1::,.. 1(:q dt::)p ,!:E dl::l, (jgr,,kzrr ;!'. 4(::),. j,E,. (lr)l. (1(7.,!:: l:.:ii. d:::)p glrr'.kiz,i ,i!;.ql:fql:E)l. 1!:: ,!;;e l::)k. ':'' '. i. iEE!(!k: ;jjr; :;)k'.I jiz. tiL. (I:.)'r:lk. '(:E),. qlg:l.. ,;;Ei!!:' 'IE::,klii!;q ll.;,l il::)k. 'I::,jjy;:,. tiL e i. e n (2,tio n e a ! s qlp:,. e e (li/,e ;jj . s 1t:(d:::::lpt;:: !ii!;' e d::llk. a ql:: (jr yjyjjttjyjppgyr i ..pE!)-.jrt-..yj...-.( ':.yj.jjr(!. .E.''-. rjj ljgrj. . . ;jj ,(:j11. g ty il:: 4(::2,, ,(:::) ,:Ejil. d:ql . 1k2:: .4::;,. dt:i!iir'.t:F,' . ilr'hir .4:22(ld;:L d::zr .i::;1!4;, )k(jy.t), d!:y;, dt::;,dl:E)pCEEIL. :1!;; :i! ilr'j!r ..i!Eiil' :5E;; 1!5;-SEEI.IIL. :::1!: l:'',(E2)p jiL. CIEiI.iL. 21r:1L ;qrl''dtz:z,. (12. (1L ll. (1L. 4:E(11;!' ipr ;!! ;pr :Ir. j''.Lt ipr ;!r d t:E;p :i!;, :sk:. l:''lkiEr: dkiE:i (1!111!2 ( ( IL. il. jllr:p' ' tilk t!;i (zli iz )y. 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FOR GEOMETRIC

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APPENDIX 19.A: FORMULAS AVERAGE OPTIONS

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b. Repeat for an at-the-money put. 19.14. Repeat the previous problem, only l-year at-the-money straddles.

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a. Use Monte Carlo (with1000 simulations) to estimate the expected return, standard devtation, skewness, and kurtosis of the remrn on the call when it is held until expiration. Interpret your answers.

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BRowxlAx

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651

These properties imply that Z(l) is a martingale: a stochastic process for which

ELzt

ln(5'(r)) ''w.Y1n(kS'(0)j+ (ct 0.5c.2)7* o'3T) -

,

The assumption that the stock follows geometric Brownian motion thus provides a foundatin for our assumption that the stock price is lognormally distlibuted.

Lognormality tells us about the distribution of the stock price at a point in time. For many purposes, however, we are interested notjust in the distlibution at a terminal point but also the path the stock price takes in getting to that terminal point. With barrier options, for example, the price of the option depends upon the probability that the asset price reaches the barrier. Geometlic Brownian motion allows us to desclibe this path. Our goal is to provide a heuristic, rather than technical, understanding of equations like (20.1).

IZ(l)) = Z(f). The process Zt) is also called a diffusion process. To represent this process mathematically, we can focus on the change in .Z(l), which we model as binomial, times a scale factor that rriakes the change in Zt) small over a small period of time. Brownian motion is then the limit of a sum of insnitesimal increments over a period of time. Denote the short period of time as h, and let F41) be a random draw from a binomial distribution, where i'(/) is +1 with probability 0.5. Note that F(i'(r)1 0 and Var(F(/)) = 1. We can write .)

+

=

Z(l + hj

Z(l)

-

ROAVNIAN Mo-rloN

?1

Z(F) =

-

Z (0)

-

* Zt +

,)

Zt)

is normally distributed with mean 0 and variance s. Z(!) is independent of Z(f) Zt * Zt + sz), where 'I n > 0. In other words, nonoverlapping increments are independently distlibuted. * Zt) is continuous (youcan draw a picture of Brownian motion without lifting your Penc il) -

F i ll)

/?

11

1 Yih) W J-l ,-I

./j7

z(r) z(0)

=

To understand the properties of Z(r),

rst understand the properties of

we must

the term in square brackets in equation (20.3). Since E (l'(f/7)) 0, we have =

?1

1

E

n.r

,*G

Also, since Var(1'(/7))

=

Yih)

0

=

i= l

1, and the l''s are independent, we have Va.r

0.

=

.

1)/?q) =

zg(f

-

i= l

-

?1

1 11

Lt

* Z(0)

11

.

(Z(f /7)

=

F/?7, we can also write this as

DeOition of Brownian Motion Z. t) represent the value of a Brownian motion at time r. Brownian motion is a continuous stochastic process. Z(l), with the following charactedstics:

(20.2)

=

Since 11

A stochastic process is a rndom process that is a function of time. Brownian motion is a stochastic process that is a random walk occurring in continuous time, with movements that are continuot!s rather than discrete. It is basic building block for standard derivatives plicing models. A rapdom walk can be generated by flippig a coin each period and moving one step, with the direction determiiled by whether the coin is heads or tails. To generate Brownian moiion, we would flip the coins infinitely fast and take innitesimally small steps at each point. Since all steps are inlinitely small, movements are continuous. As you might guess, a careful mathematical formulation of this concept requires mathematics beyond the scope and purpose of this book.l

Ftf + hj.vl'

Over any period of time longer than h, Z will be the sum of the binomial increments specifed in equation (20.2).Let 11 Tj /?be the number of intervals of length /7between ( 0 and F. We have

i= 1

20.2

=

i' ih)

1

=

11

1=

-

11

i=u I

1

i=c I

Thus, the term in square brackets has mean 0 and variance 1, since it is the sum of independent random variables with mean 0 and variance 1, divided by V-/l. By the Central LimitTheorem, the distribution of the sum of independentbinomial random variables approaches normality. We have

11

'1)

-

-

-

,

.

l'l'lle development in this section draws heavily on Cox and Miller (1965,ch. 5) and Merton ( 1990). For a more abstract approach to this matelial, see Dufie (1996)and Karatzas and Shreve

(1991).

lim ?/-.s co

11

l

g-. py

l'(f)

./'(0,

''x-

1)

y.j

./'7

The division by in this expression prevents the variance f'rom going to infnity as lt infnity. to goes Remrning to equation (20.3), the multiplication by F on the right-hand side multiplies the variance by F. Thus, in the limit we have Z(F)

-

Z(0)

-->

X'(0, r)

Q BROWNIAN

AND IT?s

MOTION

LEMMA

BRowlq

To summarize, we have verised that the Z(F) we have constructed has some of the characteristics of Brownian motion: it is normally distlibuted with mean zero and variance F, and increments to Z(F) are independent. We have not verified that the Z(F) dened in equation (20.3) is a continuous process, hence we have not demonstrated that it is tnle Brownian motion. However, it is plausible 't hat Z(F) is continuous because the magnimde of the increments is X-h F/?), and 13 0 as ?1 x. It is common to write down expressions denoting the Brownian increments. As 11 becomes small, rename /? as dt and the change in Z as dzt). We then have

hence

11

1lm

?1..-.:(13

=

F(r)

dt

i

Z(F)

Z(0) + lirpuVF lle

!j

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Z(0) + u

.

j(j

The integral in expression (20.5).iscalled a stochastic integral.3

11

l

(z(i'/?.) zgt -

JI'-FIXI

i= l

-

1)/2.1)2 =

1im

/?Iy,

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i= l

2

17 =

/7F2f/,

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/1-:X

l

lim

i=l

1

F

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i= I

1, qnd hence

=

11

X-11 IYilll =

/7IYiltl

X

11H

c-

F lim

=

Qll

f=I

N-'YX

V'

=

(x)

Arithmetic Brownian Motion o

The Brownian motion process described above is a building block for more elaborate and realistic processes. With pure Brownian motion, the expected change in Z is 0, and te variance per unit time is 1. We can generalize this to allow an arbitrary vtriance and a nopzero mean. To make this generalization, we can write

i= 1

X t + /7) This equation implies that XT)

zshould we think of F(1) as being binomially or normally distributed? For any t and E:, Zt +c) Zt) is te sum of innitely many dztlks. Therefore. we can think of i'(?) as binomial or normal; either way, Z(/) is nonnal for any finite interval. 3Because dztj is a random variable, considerable care is required in dehning equation (20.5). See Neftci (2000,ch. 9) for a discussion of stochastic integration. Karatzas and Shreve (1991)provide a more advanced treatment.

lirn

=c

This means that the absolute length of a Brownian path is inlinite over any finiteinterval. I order for a path to hate innite length over a firtite interval of time, it must move up and down rapidly. This behavior implis the ilfinite crossing propel-ty, wich states that a Brownian payhwill cross its starting point an in:nite number of tipes in an interval of any finite lengt.

Properties of Brownian Motion

lim

1)/11)2 = F

-

i= l

11

?l'-+(x7

is

h l'f/, l

Again, treating l$. as binomial, we have IFf/;l

1im

We now use equation (20.3) to understand some of additional properties of Brownian motion. The following derivations will be informal, intended to provide inmition rather than actual proofs. In particular, we continue to use the binomial approximation to the Brownian process. The quadratic variation of a process is defined as the sum of the squared increments to the process. Thus, the quadratic variation of the Brownian process Z(/)

Z((f

-

11

lt-

(20.5)

#Z(l)

F

=

l

=

12

lirn(X)

2*

/%

1 and

Surprisingly, the quadratic valiation of a Brownian process from time 0 to time F is not a random vatiable, but it is inite and equal to F. An important implication of the fact that quadratic variation is finite, is that higher-order variations are zero. Thus, for example, the sum of the cubed inrements is zero. The snitequadratic variation of a Brownian exiemely important fesult that we will encounter again. Process t'ul'ns out to be art The total variatlon of the Brownian process is

itover

-->

w 11 '

I

=

(Z()

lim

(20.4)

11

-

31-yco

i= I

JI'-y())

-

ar'

'''

=

l

This representation of the Brownian process is mathematically informftl but surplisingly useful. Equation (20.4)is just like'equation (20.2),except that Zt + h) Zt) is now called dzt), and X-his now dt. 2 Equation (20.4)is a mathematical way to say: small periods of time, changes in the value of the process are normally distributed with a valiance that is proportional to the length of the time period.'' Although expressions like equation (20.4)appear in the derivatives literattlre, it is ' mathematically more convenient to deal with sums of increments rather than increments. These sums are written as integrals, for example: =

.

11m

42,

In other words,

-->.

dzt)

/1

2 = /?F;?,

.

%. 653

Mo-rlolq

Since we are treating Yi as binomial, taking on the values +1, we have

=

-.+

IAN

-

X t)

=

ah + o' F t + 11) X-h

is normally distributed. Since /7

'

11

-

x(r)

-

x(0)

=

J'q a- r izcj

l

+ cvih) ll.

-

r 11

i' ih)

-ar+. zf;)'-lW i= l

=

F/?7, we have

654

k

BRowrq

IAN

AND I'r's

Mo-rlolq

LEM MA

GEOM ETRIC BROWNIAN

We have seen that as ?) x, the term in parentheses on the right-hand side has the distlibution .V(0, F). We can write -->

X(F)

X(0)

-

=

aT + O'ZCT)

The differential form of this expression is

dxt)

(20.8)

This process is called arithmetic Browninn motion. We say that a is the instantaneous (7'2 is the instantaneous variance per unit time. The variable Xt) mean per unit time and is the sum of the individual changes dX. An implication of equation (20.8)is that XT) - is normally distributed, or

x')

=

x(O)

x

M(aF,

cr2r)

As before, there is an integral representation of equation '

F

-Y(r)

=

7(0) +

drif't term: dxtj

+ o'dzt)

gc Xtldt

=

-

(20.9)

When a = 0, equation (20.9)is called an Ornstein-uhlenbeck process. Equation (20.9)has the implication that if X rises above a, th dlift, l(y X(?)), will become negative. lf X falls below a, the drif4 becomes positive. The parameter measures the speed of the reversion: If is large, reversion happens more quickly. In the lon nm, we expect X to revert toward a. As with arithmetic Brownian motion, X can still become negative.

20.3 GEOMETRIC

ln general we can write both the dlif't and volatility as functions of X (or other variables):

(20.t):

dxt)

o'dzt)

and volatility, o', depend on the stock price, is called

Suppose we modify arithmetic Brownian motion to make the instantaneous mean

(20.8):

and standard deviation proportional to X(r): dxt)

drkft into the process. Adding

adt

has the

e

Being able to adjust the drift and variance is a big step toward a more useful model, arithmetic Brownian motion has several drawbacks: but

o There is nothing to prevent X from becoming negative, so it is a poor model for stock prices. @The mean and variance of changes in dolla.r tenns are independent of the level of the stockprice. ln practice if a stock doubls, we would expectboth the dolla.r expected ret'urn and the dollar standard deviation of remrns to approximately double. We will eliminate both of these criticisms with geometlic Brownian motion, which in Section 20.3.

we consider

Process

Another modiEcation of the alithmetic Brownian process permits mean reversion. It is natural to consider mean reversion when modeling commodity prices or interest rates. For example, if the interest rate becomes sufciently high, it is likely to fall, and if the low, it is likely to rise. Commodity p ces may also exhibit this value is suciently tendency to revert to the mean. We can incorporate mean reversion by modifying the

=

+ o'xtldkt)

axtldt

This is an 1t process that can also be written

dxt)

-

.1(0).

(y,

(20.10)

an It process.

* Xt) is normally distributed because it is a scaled Brownian process. * The random tenn has been multiplied by a scale factor that enables us to change variance. Since dztj has a variance of 1 perunit time, o'dzt) will have a variance of tr :t per unit time. The adt term introduces a nonandom ffectof adding a per unit time t

ce(X(r))#l + (rEX(l))#Z(l)

=

This equation, in which the drift,

0

This expression is equivalent to equation (20.7). Here are some of the properties of the process in equation

MOTION

BROWNIAN

T

dt +

0

The Ornstein-uhlenbeck

655

-

adt + c#Z(r)

=

$

MOTION

=

Xtt

adt

(20.11)

+ o'dzt)

'Ihis equation says that the dollar mean and standard deviation of the stoct price change are aX t4 and O'X (r), and they are proportional to the level of the stock price. Thus, the Malue is /l&rDlflPlldistlbtlted 'kWl/iilstcllltclneolls Nldflpl t:t pel-centage ct-lnqd ill tlle JIJJV/' c2. equation The in variance instantaneous and process (20.11)is lnown as geometric will of the book, motion. the For frequently rest Bro&vnian assume that prices of we equation 1). follow and other stocks' assets (20.1 The integral representation for equation (20.11)is XT)

-

X(0)

r =

'

axtjdt

o'xtjdzt)

+

0

0

Lognormality We now circle back to our discussion of lognprmality because of this fact: A variable that follows geometric Brownian is lognormally distributed. Suppose we strt a process at xY(0) and it follows geometric Brownian motion. Because the mean and variance at time t are proportional to Xt), the evolution of X implied by equaton (20.11)generates compounding (the change in X is proportional tp X) and, hence, nonnormality. However, while X is not normal, ln(X(/)) is normally distributed:

lngA-(l)J M(ln(X(0)j x

+

(a

-

0.5(r2)/, o'ltj

(20.12)

656

%

BROWNIAN

MoTlohl

AND I'r's

:'. ''s' r'. y' jjjjjy,;jjjd ,'j,,rgr:,jf j,:yyyryf j,::ry;,,f j,::yyyyf j;:ggg,,f jjyyjg,,f jj:ryy;,,f j,:yyrr,,f ,'''';' :'71:,7*,,)'7:'*755q* j.j..,,'. F' 'j' : iE .jj:g!jjj' jlE j'$' @;' ''' 1* p' 4jj:jjjjjj.'. (E . E (iE ( i E E

:,jjj;' j,,.,,,,y,,.,,f :';'t''. ,!jg;;' q' i'tlf .':' (1): (' -jjjIr'j;j!r. j'jjyE,,f .y' ,;,,,,,j' .jjjyy' .jjyyy' )' 'jjjjj,.' q' y' (.' ('''i' q'l'' '!'l' (' ' (.!'qq' !( i i r! (' ((4( (lqjjjtj t'r' ggjjjjjj;yljjjjryd q: .'j.'.i('. (.'j.''(.' q( ;.ii i('' ( /;j jy'. jjf jyf jjj,,,!,;f j'jjj.,,r,jf )' y' j'-jk,j (' (jljj.jjjj.d (qjjjljqy,,jj'. rjjijd .. . )'(. (). )' 1* y'. .' yt('; t'ilq''jy ttf ; .' ;yj (j;' !ti .yyy.y, '( (lj ' ry .''tlitlffk yq .y,t.-,. '...'.. ... gy'- ;. jrj;,jjjjy. (yjyy ..y'y..y ygjty jj(.:(y ... yy ; y . . y . . y . lit-.it.t.i . jjrjjgjjjky. . .j.. . gyjyyjyyyyg .. .t.yy (j j j ---j.;:;;Lj;;j.j,:j. .,,,jjj,,,.' .;jj::' ,4,::)* ,,:,' '''q' ;' .',,,,,' '....,)4,6''.jjj4qjjjj..'.

As a result, we can write X(0)d (e-0.5(r2)?+(rs'z

=

(atl .

jg;

where Z X'(0, 1). This is the link between Brownian motion and lognormality. lf a variable is disibuted in such a way that instantaneous percentage changes follow geometric Brownian motion, then over discrete periods of time, the variable is lognormally distributed. Given that X follows (20.11),we can compute the expected value of X at a point in the f'uttlre. lt follows from the discussion in Section 18.2 that (a-0.5(r2)f

,E'(.Y(?)q= Xje

= Xle =' X (0)eJ

(20.11)is the expected,

Relative Importance

(e-0.5o'2)J

g.()(go'Wz) e

0.512/

continuously

(20.14)

compounded ret'urn on X.

-. !E E.--..)y.

. .

:!.-...

'x.

Thus, a in equation

.' E.

Xt)

.

:

-

......

'

:

:

-

.

q

...,.

. . ...

.j..) :-

......

..

l

..( ' ' ' . .

E ..

E '' ' ('

EEE E:E' '' EE.Ei :'' '':!! E' '' ' EE '' ' ( .. . i(' j E(g . i . . . ' E . . ' . .. -.;.. -. -

:

- . - - .- .E.. .E - . .. .-

t..-....,..,i-.....-,J!. 1?-.!-:-.

i:' : .

:

.

!E -.

.. ........ . . ..

'

( .

( r.

.

...

.

..

-.

(..--

.. .

( '

:

'.

.

.

'.

...

'

.

.

.

g

.

'. .

.

.

' ' ' '' : . . . .. .. .

. .

' '..(.' . . E.EE...'....'... .'. .... . ......

J

.

. . .

.

(' . . r . .. .

.

.'.

Flve years One year

.

.

.

. .

'

..

.

g .

.

.. . .

j(

.

.

..'.

' .. .....

X(r)

=

+ o'xtlytl.-

axtlh

Overashortinterval of time, there are two components to thechange in X: adetrministic An important fact is that and a randop component, o'xtlYtlx-h. component, axtlh, J/zt/?-? oftime, the c/zt-f?wcldr ofthe S?-(n$/?)n process is #'/'!?p??k7W allnost periods over entirely by the ?-t'I/7#tp/n component. The drift can be undetectable amid all te up and down movement generated by the random term. To understand why the random term is important over short horizons, consider the ratio of the standard deviation to the drift: tr X (t )XLl

ax

tlh

tr

=

5

1

This ratio becomes innite as h approaches dt. A numerical example shows this more concretely. Suppose a 10% and tr = ' 10%. Over a year, the mean and standard deviations are the same. Table 20.1 shows that the ratio increases as the time interval becomes smaller. Over a period of 1 day, the standard deviation is 19 times larger than the mean. This is important in practice since it means that when you look at daily returns, you are primarily seeing the movement of a random variable following pure Brownian motion/ The deterministic drift tthe expected remrn) ig virtually undetectable. =

are other considerations when you look at prices over short periods of time, including the bouncing of prices between the bid and the ask, and the effects of trades such as large bloks tbat may temporarily depress prices. Brownian motion implies that even in the absence of these effects,' prices would still bounce around significantly. .

. .

. .

.

:

.

(

:

.

..

. .

.

.

( j( . yy. (. .j y.g. i ..... . .. . . . .. . .

y

.

j. .

.

:j.

. .

.

.

,6'..

-

-. .

-

.

0.0833 0.0027

One minute

0.000002

jj:yyy:jjj

j,y,., ,,y,., jj, jjryry:,,. jjyyyyr,,.

,jj:yy.

(ll

(

.

.

j

gAj

g xa ( r .j q'.. . . . .. . . . . . . .. . ( 'r . .. . ...,j,,,.jj,' j. . j (. . .('j ('.. j IjLLLLjjLLL,L j. j (.j.g.j j .. j y . . j . . . . g . j . j . 'k:vvbb,j j . y . .... .g. y j j . .... ... y y.... . y. j. . . ..yy..y ..j. .. .. . yg..y . y. y ..y y. ( j. j . j g g.. .. . .. g y.............. y.......... y. ... y .j .. . . '... . .. . . ... . . .. . . ...

.. .

.. .

. y.

: g

......'

..'.. ..

-

-

.

:

.

. . .

:

. .

. ..

.

.

.. . .

. .

'.'.jj.. g... . .. :

.

. y . ..

.

.

.

.

.

. . . .

.

g .. .

l(

' y

..

:

.

..

.

.

..,...:...

E

'.. j.( j. .. .

.

,

-

y.

.

.. . ,,,,4,j,,,,.,,j1,,.j,r..-.

. .

.

.

.

.

.. g. .

g

g

y

g

..... .

,

y

..

.

g

. y :

..

.

.

.

. . .

.

g y

. . ..

. .

. .

.

..

..

.

.

.

..

.

:y g

k j .jj

. -

:

. . . :. . . . .. . . . . . . . . .. . . . . . . .

0.5 0.10

0.2236 0.10

1.00

0.0083

0.0289

3.464

0.00027 0.0000002

0.0052

19.105

0.00014

.

.

.

0.447

724.98

As the time interval becomes longer than a year, the reverse happens: The mean becomes more important than the standard deviation. Since the mean is proportional to /? while the standard deviation is proportional to VV,the mean comes to dominate over longer horizons. Since we take the ratio of the instantaneous stfldard deviation to the instantaneous mean, Table 20.1 also holds for arithmti Brownian motion.

Correlated lt Processes Suppose that

we have the 1t process =

(20.15)'

ct(2E:(r)1#r+ o-pE:(r)1#Z'(r)

where Z'(/) is a Brownian motion. The Brownian motion Z'(/) in equation (20.15) can be correlated with Z(l) in equation (20.10)kFor example, if X and Q represent stock plices, X and Q will typically be correlated. Let F1 t) and Wi(r) be independent Brownian motions. We can then write Z(r) Z't)

=

Fl

=

pFl

(r) (r) +

1

(20.16)

p2W:?(r)

-

You may recognize this is as the Cholesky decomposition, which we discussed in Section 19.9: Using equation (20.16),the correlation between Ztj and Z't) is '(z(l)Z'(?)1

Verhere

..

One day

One month

dotq

,/-#

a

. .. . .

jy .. ..

.

Consider the discrete counterpart for geometric Brownian motion: -

jjyy!g,,r jjyyyy,,r jj.,,,jjy kyjg,j, jjj j,,,,,,,j ,,.,,,,,,,,,,,, ,,,,.. jjyryyy,j-

jjyyg,,r

:. '. ( .. g ' : . . . . . . . .. ( ( ( . :. . ' .( ( ' '. .jjj; y j. . . . y. . .. ... . . . .: . .. . ... . g. E. .. . .. . j jjiii!!:. . .. .g . . :. . E . . . . . gj . . .. .q q.. g. y;. :. ... ..g . . q.y.. q y: y .E..(. E. j jjt:::ljt!:. .y.yg y. iyq(. q.. .y. ., . y j.. :.j....g .... y. .g. . .. . . y. . . . . . . . . .y . . .. . y y . . . . .. . y . . . . . . . . . y. . y.;y : '' ' E ' ' ' ' E ' '' ' : : .E .. . . E EE.'. (. ... .. . . .. . ... . .. .. .. . ..' .. .' . '. ..... '. . ... .. E . E.. ... .... ... .. . '.....'. E . .y . . . . . ........... . ....... ........... . . . . ....... .. . '

-

.jjjyy, k;jjg,,i jj,,,,,jj

standard deviatin to yhe per-period mqan for different time intervals. The ratio becomes infinite as the time interval goes to zero.

. (.j rjjijr.,. tjjjjkrjlky -: (jjjitgg:llll .. . . ((.. ! .. (' ' ll ( ! '

'' y.... ''.

,jjjj!i:

657

' .

:

-

.

'

'. .

.. .

' :

-y.. -y. -- E.- - .. -.yE -. -! . y- - .g y y .. . yy,yyl

.....--try

) '

' . . . . ' g. . :

'

'(' E' . - .

(

'

qy. iiyjjjj-

. .

of the Drift and Noise Terms

Xt + /7)

$

GEOM ETRIC BROWN IAN MOTION

LEMMA

=

=

pF(F1

(/)2) +

1

-

p2F(FI

(r)W(/))

pt + 0

The second term on the right-hand side is zero because FI We then say the correlation between JZ and #Z' is pdt.

t4 and lP2(1) are independent.

658

k

BRowlqlAx

MoTlox

Axo

I'r's

LEMMA

THE SHARPE

Multiplication Rules

(#z) 2

The dominance of the noise tel'm over short intervals has another implication. Since the ehavior of dx is dominated by the noise term, the squared return, dX4l, re:ects primarily the noise term. We have + hj

(Xt

-

.Y(/)j2

Laxtlh + cxt)

=

1:(/)./jfj2

Expanding this expression and simplifying, we have

Lxt + h)

2

.;t-(/')1

-

=

a 2X(/)2/?2 +

dz

Suppose that h is 1 day. Then /7 0.00274, /1 = 0.000143, and /l2 = 0.0000075. lf 11is 1 hour, then /? = 0.000114, /11.5= 0.0000012, and /?2 = 0.00000001. Clearly, the relative magnitude of the term mltiplied by /1is much greater than the other terms aj It becomes vel'y small. ln addition, if we tllink of (' as binomial, ten i'(r)2 = 1. Tis leads us to write .Y(/)) 2 ;kJ o'lxtllh

-

or E#X(/)1 2 = c zxtjzdt

,

We can also consider terms like

Rewriting this expression

jx(r)/?

-

Xtjh

gives us

+ axtljrtl/i

F,

ax(?)2

-

+ o-xtjvtlh

''5

Since the smallest power of /7 is 1.5, this entire term vanishes relative to 12as /7 goes to ZeFO.

dt

(20 17c)

pdt

(20.l7d)

.

J.f asset i has expected return ai the

lisk

premium is desned as

Risk premiumf

=

ai

-

1.

where r is the risk-free rate. Abasic idea in nance is that th return on an asset is lirtked to its risk, where risk is measured as the covariance between the ret'urn on the asset and the risk that matters is the investor utiliys In the Capital Asset Pricing Model ICAPMI, covariance between a stock and the market ret'ul'n since investor utility depends on the market rettll'n. There are other models of lisk, but for our purposes we need not take a stand on a particular model. The Sharpe ratlo for asset i is the risk premium, ai r, per unit of volatility, cf : Sharpe ratioy

=

i

1*

-

i

(20.18)

The Sharpe ratio is commonly used to compare well-diversilkd portfolios and is not intended to compare individual assets. In particular, if diversisable risk is different, two assets with the same c can have different Iisk premiums (and hence different Sharpe ratios) if they have diffejnt covariances with the market. However, we can use the Sharpe ratio to compare two perfectly correlated claims, such as a derivativ and its underlying asset. The main point f this section is that two assets that are perfectly correlated will have the same Sharpe ratio. To seethattwopefectlycoaelated assets musthave the same Sharpelptio, consider the processes for two nondividend paying stocks:

.

Suppose we have two differeptlt processes such as equations (20.10)and (20.15),. We can write dz ' t) Y'XX where E (1'(r)1''(r)) p. Thus, p is the correlation between Yt) and Y't). We have =

f (E.X(/ + = E

=

.Y(r)1(P(r+ ) P(l))) LLaxtlll + o'xtlyt + hlxlLac :(?)/7+ )

-

-

o'c

Qtlkr't

+

hj.Xj

:(r)/7 +

dt

x dz

(#/) 2

=

0

(20.17a)

=

()

(20.17b)

dS3

=

cl Stdt + cr I S3dZ

d%

=

azbdt

+ o'zhdz

(20.19) (20.20)

Because the two stock prices are driven by the same dZ, it must be the case that ,-)/c2, (al ,')/c l = ag oy else tere will be an arbitrage opportunity. Before we examine the arbitrage, let's explore the intuition. For example, in the CAPM, the lisk premium of asset i, ctf r, is -

(termswith power A: 1.) = a One way to make these calculations mechanical is to use the following so-called rules'' for tenus containing dt and #Z: o'o'cp Xt)

'Rmultiplication

=

=

-

essentiily ignoling all terms that are higher powers of h. This equation tells us that if we lpok at the squared stock price change over a small interval, gl1 w are seeing is the effct of the variance. (X(/' + /2)

659

The reasoning behind these multiplication rules is that the multiplications resulting in powers of dt greater than l vanish in the limit.

,

l'5

Lxt + /7)

%

20.4 THE' SHARPE RATIO

gagxLtjlyLtjh.s + o'lyLtjlyyjzjt

=

x

t'/z'

RATlo

-

-

ai

-

r

=

piaju

-

r)

(20.21)

660

k

BnowxlAx

Mo-rlox

AND

I'r's

LEMMA

TH E RISK-NEUTRAL

where au is the expected return on the market portfolio. The beta of asset i is

//

=

Pf, A:rGi

(20.22)

G' A;

where py,,v is the correlation of asset i with the market, o-i is the volatility of the asset, and 0-51 is the market volatility. Using equation (20.22),we can rewrite equation (20-21) aS i

1-

-

J

k,/

= p ; wj

1*

-

(20.23)

'

c Thus, if two assets have the same correlation with the market pi,5g), they will have the same Sharpe ratios. In equations (20.19)and (20.20),the fundamental uncertinty driving theprocesses forboth 5'1and Sz is #Z. Thus, both assets have the same correlation with the market, and in the CAPM would have equal Sharpe ratios. We will now demonstrate an arbitrage if the Sharpe ratios in equtins (20.19) and (20.20)ure diffrent. Suppose that the Sharpe ratio for asset 1 is greater than that ) for asset 2. We then buy 1/(c' 1 S1) shares of asset l and short l/to-iual shares of asset 2. These two positions will generally have different costs, so we inkest (orborrow) the bond, which has cost difference, l/c' 1 1/c2, by buying (orborrowing) the lisk-free the rate of return rdt. The return on the two assets and the risk-free bond is 11 ?1 1 1 dSL cej az rdt d% + dt (20.24) (7' l c'zxs'a Sj I o'z o- 1 o-z o-

t)-i

yv

-

-

-

-

=

-

-

.

This demonstrates that if the Shyrpe ratio of asset 1 is greater than that of asset 2, we lisk-free return. Therefore, to can construct a zero-investment portfolio with a positive preclude arbitrage, assets 1 and 2 must have the same Sharpe ratio.6 This link between volatility and lisk premiums forperfectly con'elated assets arose in Chapter 12 when we discussed option elasticity. There we saw that the Shrpe ratio for a stck and an option on the stock are the same. The reason is that the stock and option have the same underlying source of risk-the same #Z. They do not have the volatility of a call option is greater than that of the stock-and, same volatility-the hence, they do not have the same risk premium. Nevertheless, they do have the same Sh e ratio.

PROCESS

$

661

lisk-neutral pricearn therisk-freerate and discounting expected payoffs at that rate. The ing method alises from no-arbitrage pricing, but there is underlying economic intuition that we discuss in this section.

The notion of a

lisk-neutral

process for the stock arises with 1t processes,

just

as in te binomial tree. We saw in Section 20.2 that Brownian motion is a process for

+ J)) = Z(/). In other words, Brownian motion is a martingale. ln order male such to a statement, we must specify a probability distribution, or nleasltre, for Z(/ + s), conditional on Z(r). This point requires emphasis: in order to say whether a stochastic process is a martingale, we need to specify a probability distribution for the Which Efzt

PFOCeSS.

Suppose the tnle price process is dst)

Stt

=

(a

-

sldt +

(20.25)

o'dzt)

where is the dividend yield on th stock. This process represents the stock plice we observe in the world, with #Z(r) the unexpected portion of the stock return. On average J, and the deviations from this ret'urn we expect the stock to appreciate at the rate a words. should be a martingale under te true Z(l) should have mean zero. In other probability distribution. -

investor assesses the stock price process. Now let's consider how a lisk-averse defnition risk-averse investor by values $1 of gain less than $1 of ln utility telnns, a receives such investor the loss. When return on an asset, the portion of the return an negative expected value in utility terms: The gain from an martingale will have that is a worth less than the loss from an equal decrease in Z(l + /?). in /7) will be increase Zt + offset expected utility is by the asset paying a risk premium, equilibrium, this lower In (Z

--

1-.

This description suggests that there might be another way to write the rettirp on the asset. Suppose we create a new process ktj such that (/) does generate a martingale investor. (Note that this implies that Qt) is not a in utility terms for a lisk-averse the with mmingale respect to true probability distribution.) In order to offset riski modi the distribution for Zt + ll) so that positive outcomes are aversin, we must negative However, than if we want to substitute dkt) for dztl outcomes. more likely transformed will the equation not give us the correct stock in (20.25), process drift for the reduce the tenn to compensate greater expected value of #Z(/). unless we It turns out that if 2 (r) is a martingale when evaluated in terms of the investor's utility, the appropriate offsetting change in the drift is to remove the lisk pmmium. This gives lisk-neutral plice process us the ,

arice

20.5

THE

RISK-NEUTRAL

PROCESS

We saw in Chapters 10 and 11 that we can interpret the binomial model either as representing a risk-neutral stock plice process, in which case we stprobabilities so that assets ).., earn the risk-free rate and we discount cash flows at the risk-free rate, or as rejresenting the true stock price process, in which cas we use true probabilities and discount at an interest rate. In valuing options using Monte Carlo simulation appropriate (notIisk-free) in Chapter 19, we again used the risk-neutral approach, assuming that stocks on average

dst) S(r)

= (?'

-

jdt +

o'dkt)

(20.26)

In this equation, we subtract the lisk premium, a r, from the dlift a &,and we replace dzt) with dkt). As in the binomial model, when we replace a with ?', we must also modify the probabilities associated with stock price movements. Also here and in the binomial model, when we switch to the risk-neutral process, the volatility remains the -

Same.

-

662

k

BRowrqlAx

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LEMMA

The probability distribution associated with the lisk-neutral process, d2(l), is pricing, we will said to be the risk-neutral measure. When we perform lisk-neutral implicitly assume that we are using the risk-neutral measure. With this revised price lisk-neutral, because we have process, we can perform valuation as (/' the investor were transformed Zt) to (l) in order to make the investor behave risk-neutrally with resgect to the revised process. Thus, using equation (20.26),we can perform valuation as if Zt) were a martingale, because a risk-averse investor will treat it as such in valuation. It is important to emphasize that in the applications we have discussed to this point, we need not do any extra work in order to use equation (20.26)for pricing.? In P articular, we do not need to worry about how to construct (/) from Z(l). For example, when we perform Monte Carlo valuation of a derivative with a stock as the underlying asset, we draw mean zero random variables to simulate kt) and we use those random variables for pricing--discounting e'xpected payoffs at the risk-free rate-without any further adjustment. The technical result underlying the transformation from equation (20.25)to equation (20.26)is called Girsanov's theorem-s Girsanov's theorem states that a Brownian ' process, Z(/), can be transformed into a new process, d 2 t) = dzt) + qdt, that is a martingale under a transformed probability distribution.g Girsanov's theorem explains how to create the transformed probability distlibution. ln the case we are discssing, the transformation is based on the Sharpe ratio: set ?? = a ?')/tr. Thus, we have

((z

ldt +

-

o'dzt)

(r

=

-

l

=

ldt +

-

. -

gjvjt

dzt)

o'

+

---

/e

G

20:6

IT'S

dividend yield of Suppose a stock with an expected instantaneous return o f follows geometric Brownian motion: ,

'

on Girsanov's theorem include Karatzas and Shreve (1991,sec. 3.5), Baxter and Rennie

(1996) and Neftci (2000.ch. 14).

gAsimplihed version of Girsanov's theorem states that if Z(l) aBrownian motion underthe probabils = Z(J) + qt, is a Brownan motion under the probability density ity density ft), then (tlf (J), 0.5p2/). The probability density f'unction for a nonnal variable with where ( (l) = exp(-r/Z(?) .2(?)

-

variance t is

dvtj

=

l lt

CXP --Z

l

::!

2;.r1

Writing out (tj and

a

l lz.t

=

exp

l --

21

(Z + qt) a

l

zz.t

The Iast expression is the probability density for a nonnally distributed vmiable with mean

-.r)t.

f)

-

(S(l),

J1)

dt +

(20.27)

(S(r), r1#Z(r)

=

ztj,

(

Recall that d. is a geometric random walk with drif't. Suppose for a moment that the drift is zero, in which case S obeys a geometric random walk with equal probabilities of uP and down moves. Now look at Figure 20.1. Notice that equal changes up and down in S do not give liseto equal changes in Vs, t). Since 7 is an increasing convex function of S, a change to S + 6 increases JZ by more than a change to S 6 decreases 7. Thus, if the expected change in S is zero, the expected change in J/ will not be zero. The actual expected change will depend on the curvature of J/ and the probability distribution of S, which tells us the expected size of the up and down moves. -

.' ..' .'.';

)' t'kjj:j;ij:'. :''','(r',jjjk,,jjjjj :5:7:7* ;'p:' r' ,',qjL..jlk..' if @' :'r' ;'$' (' ()' k' y' i7'#E-' 'F''E 'T'ql'7qlf q'E'' '' ! '''q' )' llllitlji:)f (' k)' ''( ; ( :' ( ' ('' E E((' (! 7* j'ij 'j,;yj;jj'. ;'y' )'. rrqf ''1*(';(71jk*)(41* ('' t',))',' /'. yyf '#'t')if' 'g' ; (ii'(.(i'1* '(.'E('q k' )' ' 'tEi. E;(.!.yq r' t'q)' ttf .('( E.' (';'' (.(i!(i..; (k(ii'r! .(:i' (@ .!':@l'(' (q1));( ..'y. ' ..T''' :.E .i!iI !j.! : ')' ' .q.,y .rr; ...,.r!;. 'j.q .. ; i!ir )!,jjj;:jjj;j;:. r..y . ... i. .;y.(lyy q . lki )E. ; y .. y yy ( : y r tr j . '.r . . . .. g y!rjLj;. . yq....jy(.yj.yrr . ikllr .. . ykk-, 2.; ).j. ;....,... r.,...rgy ) L.L...L..L .y tjg y):-. 1.L.....:...2,.,,,,.L',j...... . .... ..... . ..

ilii!i!lf (1)*. ';jj'tti ilkl:;q ...r. 5* ;--'' '';'-' ,.jjjj::jjj,jF' 57* qE:f 7* )' E '' (E: E jE

:

'

.

-

. ---

-

.

-

-

-

E

! :

E' . -E

-

.

-

-

'' :' i ..

E.-... -..

.

.

CE ' E'.

:'

'

!

''

-

'.

E-

E :.E ' '

-.

-

...';:

...........

.

.......

......

........:...E.............

. . ....... .

' ' '' ;!E EE.. E '

'

.-

-

- -

-

-..)-y);

.

y-. .

'

:

-

.l1EE..

'

''

E E'' E E ' '

- . -.-. -E -

.

.

-

-

.

-

. . .

E.-

-

-.

.t)jk... .

-kyj.y-yj;. -

.

.

.

.,. .

.

.

------

-

.

-

Illustration of Jensen's

inequalty. The function 7(5)) is convex.

Equally spaced changes give rise to in unequally spaced changes ln /(.(02. In particular, 7(5'(t) + E1 yg5'(t)j> 7g5r)) 7(5'() E') because 7 is an increasing convex function of S. -

-

1 It) exp --Z(/) 2/

t3ES(l),

Functions of an lt Process

-

You can see how the Girsanov transformation works by examining Itlftj. combining terms, we obtain

and

ES(/), J) = Gstt. Now suppose that we have a derivative clqim that is a functio of the stpck price. Express the value of this claim as JZ(5'(l), /). Givtn that we know hw the jtockbehaves, how dos the claim behave? ln particular, i'tow can we describe the behayior of the claim in tenns of the behkior of S

.(t)

.(/)

=

=

The economic intuition behind the change of measure is that investors in equilibrium are indifferent to a small change in the allocation of their portfolio between risky

Slkeferences

,

ln this equation, tr, J, and c can be functions of the stock plice. 'When Stj follows ast), and eometric Brownian motion, we have (5'(1),1) g5'(/),/'1

#r

7We will see in later chapters. particularly when discussing hxed income, that in some contexts we need to take account of the risk premum on the underlying llsset when moving from equation (20.25) to equation (20.26).

LEMMA

instantaneous volatility

o'cjkqtj

..y

663

and risk-free assets. The Sharpe ratio appears in the transformation because it measures the equilibrium remrn investors require to absorb additional lisk.

-

(

k

LEMMA

-

-

-

. .

.

.. .

-

.

.

664

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In Figure 20.1, the second derivative is positive', that is, the slope of 7 becomes greater as S increases. As is evident in te sgure,the expected change in 1/ will then be E (1/(u)) if J/ is convex positive. The sgureillustrates Jensen's inequality: F(' (')) Appendix C). (see Using a.rfaylor series expansion (seeAppendix 13.A), we can see how JZ depends S. on We have Lvssd'jl

+

+2

CLst + h) t + hj

CLSt)

-

,

/.(1

,

+ ll) (u$'(/

+ terms

The multiplication rules aheady discussed in Section 20.3 tell us that since S is an 1t process, the tenns dtll apdtfu xdt vnish, alongwith all higher-orderterms. lntuitively, since the interval of time is shott, the noise tenn dominates, and the squared noise term This result stemj from the quadratic is the same ordr of magnimti as the drift. vadatio p'ropertywe dijussed in Sction 20.2. If we integrate the Taylor expansion w'ith respect to time, then the term containing squared changes will be pffrtional to time. Higher-order terms will sum to zero. This calculation is the basis forlt's Lemma.

r

Prosltion

Lemma)

20.1 (It's

(20.27).If CLSt), in C, dCLSt4, ?), is 6ICLS' t)

=

=

1)

Let the chage i the stock price be given by is a twice-differentiable function of Stj, then the change

Example 20.1

ast) + Ctdt

t5',tjjcs

-

(20.28)

J(5',tllcss

+

)/

+

c,

)

dt +

t5',tlcsdz

lt's

=

a

=

8

-

-'.5

1

o

2

=

r,

A, and 0

Ct

(20.29) Lemma to

=

p5'trl avst) dzt) dt + pg(?) nt

-= (a(alstjdt -

J

gt(v -

8)SCs +

jclslcss cij dt +

+ O'SCSCIZ

If there is no uncertainty-that is, if o' O-then It's Lemma reduces to the calculation of a total delivative familiar from ordinary calculus: =

dCCS, t)

=

Csds +

o'st);

=

p2u(r)

.

-

=

)Z(?)2

o'=St)

is given as

.)

,

t)

+ 0LSt), tjh

5'(0)c*--lo.2)'+':rZ(')

avt) oztl

,(/);

-

-

-

dCS,

J'j

dSs and recall that

-.>

The expression for a lognormal stock price is

Lemma states that dst)

dstj

=

=

5'(l))2Pg5'(/'),

-

0

Css ;

/.)

The stock plice is a function of the Brownian process Z(l). We can use It's characterize the behavior of the stock as a functipn of Z(r). We have

lCj&S1, and Ct = pC/j/.) The tenns in We use the notation Cs = DCIOS, Css braces are the expected change in the option price. ' In the case where St) follows geometric Brownian motion, we have &LSt), r) = ast), (St) r) = stj and (u(/), rj = o'stl, hence ,

-

,

-

Csds + lcssdl 2

(E(5', t)

665

The delta-gamma approximation over a vel'y short period of time is lt's Lemma. We can use lt's Lemma to verify that the exprssion for a lpgnormal stock plice satisfies the equation for geometric Brownian motion, equation (20.27).

st) equation

>

) St)

-->.

in dt43ll and higher

5'(r)) Ag5'trl

-

+ J(5'(/

Male the substimtions /? dt and St + /7) partial derivatives of the option price: are just

+ Vstdsdt

2

%

The extra term involving the variance arises from dS)l and is the Jensen's inequality correctin due to the uncertainty of the stochastic process. naming it-when we discussed deltaWe encountered lt's Lemma-without approximations in 13. Equation Chapter (13.6)stated gnmma

+ 1V)J(#l)2

LEM MA

tr2j

+

1 alStl

i

pz(?) a

g#z(r))z .j1

tldt + cutjdztj

+

o'lsstjdt

+ o'vtldzt)

In going from the second line to the third we have used the fact that dztjl calculation demonstrates tat by using Ito's Lemma to differentiate equation recover equation (20.27).

=

dt. This

(20.29),we

%

Cdt

Multivariate It's Lemma ';

losee Karatzas and Shreve (1998)or Merton (1990)for more details.

.

So far we have considered the case where the value of an option depends on a single It process. A delivative may have a value depending on more than one price, in which case we can use a multivariate generalization of It's Lemma.

666

%.BRowlqlAx

MoTlohl

Proposition It processes'.

29.2 (Multivariate

LEMMA

l'r's

Axo

vALul

lt's

dSi t) aidt + St) =

Lemma)

o'idzi

Suppose wehave

i

,

Denote the pairwise correlations as Edzi x dzj) twice-differentiable function of the Si we have

1,

=

.

.

FP

(5,(F):'1

.

.

&, tj is a

.,

The forward price for

s'la

,-rFJ(0)r',Er'(r-)+l,(c-1),2)T

=

0,z' ,

dc (k1 ,

.

.

Sn t)

,

.

,

Cs d uf +

=

i= 1

)

The lease rate for a claim paying

1

11

E

Ldc .

,

.

.

.

,

& r)1 ,

/1

ai

=

&Cs +

i= I

qi1Example 20.2

j..j jjj

Suppose

CCSL

,

Sz)

:.

=

J

it

o-i c'yyfy

zh

Csih + Ct

uy

i = l juzzI

5'1Sz. Then by It's

4

!( @

7 This

E E LLLLL ..

t5'jXj

=

uat/xj

+

s'jdy +

dSL

j

Edcj,::zz

py

,

(al +

(r

-

)

-

1c(7

1)c'2.

-

2

%.

az + pc

equation (20.30).

jtycluj

dS'

sg =

=&

%.

0.

=

m,

aS'-$dS

)N(t7

+

#u s'l s%+

1t7(/

L)9-lLo'vS)1dt

-

1)5'f'o' 2 dt

-

2

we get

dsa sa =

(tz

)+

-

JJ(7

-

1)o'2j dt + aadz -

aa

-

?')

+

Suppose we have a. claim with a payoff depending on S raised to some power. For example, we may have a claim that pays ST)l at time F. In this section we examine this claim with two goals. First, we want to compute the price of such a claim. Second, we want to understand the different ways to approach the problem of valuing ihe claim. The following proposition gives us the forward and prepaid fonvard prices for this claim.

(20.33)

.

'.

.

..

.

-

ST) Using equation

a

-

1-

r). premium is aa Thus, the There is another way to obtain the drift term in equation require the use of lt's Lemma. We can wlite

Sd

(20.32)

1)c2 and Thus, S'' follows geoletlic Brownian motion with drift aa ) + 1:1(/ 2 14ska dZ. Hence, if a is the expected return for S, the expected retul'n of a claim with price S6lwill be

lisk

ON

a

-

We obtain

Sz) does not depend explicitly on time, Ct

A CLAIM

?-

=

=

Example 20k2 is interesting because we know that the product of lognormal variables is lognonnal. Hence, we might expectthatthe driftforthe productof two lognormal variables would just be the sum of the drifts. Howeyer, Example 20.2 shows that the drift has an extra term, stemming from the term dSL dsz, due to the covariation between the two variables. The intuition for this result will be explored further in the discussion of quantos in Chapter 22.

VALUING

*

is

SCl

To prove this proposition, we will first use lt's Lemma to determine the process followed by S'. We will then use three different arguments to obtain the pricing formula,

Dividing by

20.7

.

ST)a. If S follows Consider a claim maturing at time F that pays C(5'(F), F) equation (20.27),then we can use lto's Lemma to determine the process followed by S'.

lu'a

implies that

) i). )y )1 .t) )) Note that since C'j

(20 31)

The Process Followed by S'

Lemma we have

j. 1E(

d

S(0)fldL&trYJ)+f'(J-1)G21F

=

i = I jcuzl

The expected change in C per unit time is

dt

(20 30)

lt

dSi dSj Cs h + Ct #/'

667

is

F 0,T. (5'(F)''1

lt

%.

-

's,

Jl

.5/

ox

20.3 Propositlon Suppose S follows the process given by equation (20.27).The the prepaid forward plice is value at time 0 of a claim paying ST)'l

11

.,

pi,-idt. If Csj

=

11 correlated

A CLAI M

xc

'

(20.32)that does

not

y(g)cgtI(a-(-0.5t7'2)W+co'Z(r)

=

(18.13)to compute the expectation of a lognormal variable, we have

'. ...

ELSLT) a)

=

S(0) t7 c

() a = 5'( ) e

c(tz-J-0.5o'2)F+0.5f2&2T (J(a-J)+0.5:(f'J-

1)c'21F

668

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Thus, the expected continuously compounded as in equation (20.32).

return on

is aa

SLl

-

t)

+ Q.5ap

1)tr2,

-

Proving the Proposition Given equations (20.32)and (20.33),there are three arguments we can use to compute the time-o value of a claim that pays ST)t at time F. All three methods will confinn Proposition 20.3. Risk-neutral

aLa

-

?'),

for #k&:

First we use lisk-neutral pricing. Subtract the lisk premium, prlcing from the drift, aa obtain the following ), to as the risk-neutral process -

dsa g

a;

=

-

1

J) +

-

j.aa

-

1)&2 dt + aadz.

E* (5'(F)r'4

()

5'(0)&c''tr-J)+lt'(,-1)o.21T'

=

(20 35)

We can- also value the claim on S1 by discounting the true value. To do this we must compute te expected value of expected (nonrisk-neutral) claim and discount expected payoff appropriately. From equation (20.14),the the the ST)Ct value of is expected cash flow

,j.a

) z::z =d =

e-(r+tl(a-r))T' -(r+t'

-

'

-

.

=

=1

-

t7(?-

-

ta

-

-

&)

-

-

1(7(t7

-

-

1)4y2

-

2

The value J* is the lease rate of the claim paying S'. The prepaid forward price is then FP

&

0,F

CS )

=

SQ)'e-&'T u$(O)flc(-r+fl(r-t)+JJ(fl-

l)(r2)F

=

This is the same as equation

(c-r)1F

g

ST)Cl

at

We now examine four special cases of equations and 2. Claims

on J'

First, suppose a

=

ty(rl(ct-()+.).c

(t'J,

l )o'2jF

(20.30)then gives

1. Equation =

(20.30)and (20.31):

Se

us

-:z'

This equation is just the prepaid forward plice on a stock. Claims

(u(y.)c) (())t?

,

(20.30).

Examples

Specc

5'0

=

If a = 0, the claim does not depend on the stock price; rather since on J0 1, it is a bond. Setting a c.=: 0 gives us V(0)

y(0)Jc-rTe(J(r-J)+rl(c-l)c2)F

=

e-rT

-

Note that the risk premium on the stock, a r, drops out. The forward price for 5l, which we will denote F9,w S'), is just the future value of the prepaid forward: -

F0.r( S'' )

-

J/(0)

The discount rate is expression (20.33).The price at time 0 of a claim paying time F is the prepaid forward price, which we will denote Foppm4l w(

%. 669

Finally, we value the claim by nding its lease rate. We ask the lease rate the claim cash would have to make in order for us to willingly hold it, or what payment what equivalently, payment we would have to make to short-sell it. We can then treat lease dividend yield and compute the forward price. the the rate as Fromequation (20.32),the claim has risk ao'dz and so must be expected to earn r) + r. Equation (20.32)also tells us that the acmal expected a rate of remrn of aa 1)(r 2 capital gain on this security is aa ) + 1zaa In order to hold the security, we would need to eana the difference btween the expected return and expected capital gain as a cash payment. Thus, the payment would have to be . ?-) 1)g2j &) + lzutt,I + r (* aa

J(0)r'cIr'(e-J)+),(,-1)G2JF

El (5'4F)''(1 =

FoP

.c

ox

.

We saw in Section 10.1 that the expected price under the lisk-neutral measure is the forward price. Thus, equation (20.35)gives the fonvard price. Discounting this expression at the risk-free rate gives us the prepaid forward price, equation (20.30). Discounted

A CLAI M

Finding

(20.34)

Using the drift term in equation (20.34),the expected value of the claim at time F under *, 'the risk-neutral measure, which we denote E is

xc

=

=

e

r?- FP 0,r(

5'r')

which is te price of a F-period pure discount bond. Claims

price is

.$2

on

When a

=

2 the claim pays 5'(F)2. From equation

(20.31),the forward

y(0)rle(fl(r-J)+JJ(&-1)G21F

The use of a single discount rate works in this case because the payoff to the claim is simple. In general, computing a plice as a nonrisk-neutral discounted expected value is more difctllt than this. -

(20.36)

670

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LEMMA

vAuulxc

Thus, the forward price on the squared stock plice is the squared forward price times a variance term. The squared forward price is intuitive, but the variance term requires some discussion. One way to think about equation (20.36) is to perform the following thought experiment. Suppose that we have an ordinary stock with a price denominated in dollars. Now imagine that we have a second stock that is identical to the first except that instead of receiving dollars when we sell the stock, we receive one share of ordinary stock for each dollar in the quoted price of the second stock. This conversion from dollars to shares is what it means to have a squared security. With the squared stock, when the stock plice goes up, we not only receive the extra - dollars a share of stock is worth, but we also receive the appreciated value of each share we receive in lieu of dollars. We therefore receive an extra gain when the stock price goes up. The effect works in reverse when the plice goes down. ln tat case, we receive fewer dollars per share, and each share received in lieu of dollars is wfth less as well. However, the lower price per share hurts us less because we receive fewer shares! Thus, on average, the extra we receive when the plice goes up exceeds the loss when the price goes down. This effect becomes more important as the variance is greater, since large losses and large gains become more likely. The result is that we will pay extra for the security, and the extra amount we pay is positively related to the variance. This example provides anoter illustration of Jensen's

Claims

with a

on 1/J -1,

=

Finally, 1et a we get

=

-

F0,r(1/&

1, so the claim pays I/S.

(l/u(0)J

=

-1 = F 0,T e

Using equation

dQ

W

Now we generalize the previous example by having two prices. Consider a claim paying ST)t' QTjl' where S follows -

tsldt

cldt +

-

o'adzc

+ o'sdzs

(20.37)

671

(20.38)

hlt

=

Proposition 20.4 Suppose that S and Q follow the processes given by equations (20.37) and (20.38). The forward ptices for S6land Qbare given by Proposition 20.3. The forward price for S' Qbis the product of those two folavard prices times a covariance correction factor: Ft,Ts

a Qb )

:::

(sa) s f,F

F),F

Lcbleabpo.so.cr-Ij

'l'htr variance of S Qb is ooiven by 2 2 a 2o'. + yl c c + gaypcscc

%

.2,

Note tat te squared security, is a specitl case of Proposition 20.4. When S = Q, a = b = 1, and p = 1 (sincea vadable is perfectly correlated with ittelt), th covariance term becomes

abpcso'c same result as equation

=

tr. 2

(20.36)for

the forward price for a squared

Using multivariate Ito's Lemma, the process for St

valuing a Claini on Saob

as S =

ac

dzsdzc

stock.

czp

sc k

where

This gives us te

As with the squared security, the forward price is increasing in volatility. The payoffs for bot,h the Sl and 1/S securities are convex; hence, Jensen's inequaltells ity us that the price is higher when the asset price is risky than when it is certain. In both cases the forward price contains a volatility term, and in both cases the price is increasing in volatility. lf we considered a concave claim, for example X-S,the effect of increased volatility would be to lower the value of the claim. End-of-chapter problems 20.5-20.8 provide examples.

61S

=

(20.31)

etJ-r)Fecr2F

ox

and Q follows

'

inequality.

A CLAIM

d (5'cp') sa :d? =

tay -

$s) + bac

Qbis

1)c2y + kbb 8c) + 147(47 2 2

-

-

+ abpaso.c

dt + acsdzs

-

1)0,2

(2

+ bccdzc

The expected ret'urn on this claim depends on the risk premiums for'bot S and

r + aas

-

r)

+ bac

-

(20.39) ::11

r)

As before, there are tltree ways to find the price of a prepaid forward on tllis claim. Here we use risk-neu.al pricigk Problem 20.13 asks you to use the discounting and lease-rate methods to find th'e answer.

llproblem 20.11 aslcs you to velify that this expression gives the expected return.

672

%.BROwNIAN aas

-

MoTloN

I'r's

AND

LEMMA

JuM ps llq 'I'H E s'rocK

'Fhe risk-neutral process for d5l Qb is obtained by subtracting the risk premium, r), from the drift in equation (20.39).This gives r) + bac -

saQbunder

The expected time-r value o

:(r)bq E *(5'(r)c

dt + aasdz)

abpaso'c

+ bccdz*c

the risk-neutral measure is

Wecan write a stockpriceprocess

(20.31),this expressiop can be rewritten

Using Prpposition 20.3, in particular equation ZS

F0,v CSa

/)

Q)

(z/pyltryo'tj

=

F0,w(S a ) Fz,v (Qb )c

T

(gtj y(;; .

The expression on the light is the product of the forward prices times a fpctor that accounts for the covariance between the two assets. This is an important result: The price that results when we multiply two prices togetherrequires a correctionfor the covariance. We will see this result again in Chapters 21 and 22. Proposition 20.4 can be generalized. Suppose there are 11 stocks, each of which ' follows the process

(20.41) where dzidzj

=

(1976)proposed modeling the stock price as lognormal

=

=

-

Probuump) Probtno

dutljst)

(20.42)

The forward price for 7 is then F0 z.(F)

1-+' (F0,r(&)1f''c E;'-i (2'! -'

i

20.8 JUMPS IN THE STOCK

=

1

-

kdt

@

=

hkldt + o'dz

-

(20.44)

+ dq

=

0 1'

-

if there is no jump if there is a jump

1

drift term contains for the reason discussed in Chapter kkdt. and Edq) dq tel'm has a nonzero expectation, so we subtract kkdt in order to preserve the 19: interprtation of a as the expected return on the stock. We have 'l'he

-kkdt

=

E dSjS)

11 =

=

rl''he

i= l

'

kdt

where

11 SC

jump)

=

We can then write the stock price process as

dq =

withjumps as follows. With the Poisson process,

the probability of ajump event is proportional to the length of time. Furihennore, for an insnitesimal interval dt, the probability of more than a sipgle jump is zero (this is part of the desnition of the Poisson process). Let qt) represent the cumulative jump and 0. dq the change in the cumulative jump. Most of the time, there is no jump and dq When there is ajump, we let the random variable (' denote the magnimde of the jump, 1 is then the expected percentage change in the stock price. If is the and k E (F) expected number of jumps per unit time over an interval #r, then

pijdt. Let V (l)

with an occasional dis-

crete jump. One way to model such jumps is by using the Poisson distribution mixed with a standard Brownian process, as we did in Chapter 19. As discussed in Chapter 19, the Poisson distribution counts the number of jumps that occpr in any interval. Conditional on ajump ocurring, we assign some distribution to the change in the stock price. lt is convenient to use the lognormal density to compute the price change if the jump occurs.

s(0),:(0):eEr'(r-Js'+(r-:c)+l,(c-1)&+?,(-1)&$+a:'c,c:1r

=

%, 673

in the lognormal model. On a smaller scale, consider the stock price of a company that reports unexpectedly favorable earnings. To account for such nonlognormal behavior,

Merton

+

PRICE

/% o.o.iaajl-

(20.43)

l

PRICEIZ

=

(a

-

kkldt + E (tr#Z) + E dq)

,

adt

=

Thus, for example, if there is on average a downward jump, then k < 0, and, when no > 0 to compensate for the occasional jump is occurring, we need extra drift of bad times due to the jump. The upshot of this model is that when no jump is occurring, the stock plice S evolves as geometric Brownian motion. When the jump occurs, the new stock price is fact that it is straightforward to modeljumps does not necessarily mean that it is YS. easy to plice options when there arejumps. We will discuss this further in Chapter 21. -kkdt

'l'he

A practical objectio to the Brownian process as a model of the stock price is that Brownian paths are continuous-there are no discrete jumps in the stock price. In practice, asset prices occasionally do seem tojump; a famous example is October 19, 1987, when the Dow Jones index fell 22% in one day. A move of this size is exceedingly unlikely

Proposition 20.5 Suppose an asset follows equation (20.44).lf CCS, t) is a twice continuously differentiable function of the stock price, the process followed by C is dCCS ' t)

=

Csds + lcssclfdt 2

+ Ctdt + XEyLCLSY, t)

-

CLS,

/)j

(20.45)

674

% BRowlqlAlq

Mo-rlolq

AlqD

I'r's

LEMMA

The last tenn in equation (20.45)is the expected change in the option price conditional on the jump times the probability of the jump. %.

'I'he last term in equation (20.45)accounts for the jump. That term is not present version of It's Lemma for a stock that cannotjump, equation (20.28). the in

PROBLEMS

%

675

PROBLEMS For the following four problems, use It's Lemma to determine the process followed by the specised equation, assuming that St) follows (a)arithmetic Brownia motion, equation (20.8).,(b) a mean reverting process, equation (20.9)',and (c)geometric Brownian motion, equation (20.27).

20.1. Use It's Lemma to

evaluate

#(ln(&).

20.2. Use lt's Lemma to evalugte dSl. 20.3. Use It's Lemma to evaluate dS-6 HAPTER

U

ARY

A stochastic process Z(l) is a Brownian motion if it is normally distributed, changes independently over time, has variance proportional to time, and is continuous. The .change in Browpian motion is denoted #Z(r). 'I'he process Z(r) and its change dzt) provide the foundgtiop for J'nodel.n derivatives pricing models. The Brownian process Zt) by itself would be a poor model of an asset price, but its change, dzt), provides a model for asset risk. By multiplying dzt) by a scale factor and adding a drif't term, we can control the variance and mean, and thereby construct more realistic processes. Such processes are called It processes or diffusion processes. Black and Schols used just such a process in their original delivation of the option pricing model. Given that a stock follows-a particular It process, lt's Lemma permits us to compute the process followed by an option or other claim on the stock. The pricing of claims with payoffs St' and St3:, where S and Q follow geometric Brownian motion, illusates the use of It's Lemma. An important objection to Brownian motion as a driving process for a stock is the continuity of its path. It is possible to add jumps to a Brownian process, and tere is a version of It's Lemma for such cases.

READIXG

We will use the concepts in this chapter throughout the rest of the book. In the next chapter we will directly ajply the concepts of this chpter, in particular lt's Lemma, showing that prices f delivatives must satisfy a particular partial differential equation. ln later chapters we will use these concepts to discuss the plicing of ekotic options (Chapter 22), options bqsed on interest rates (Chapter 24), and risk assessment (Chapter 25). Many books cover the material in this chapter at a more advanced level. Merton (1990) in particular is an outstanding introduction. Other good sources include Neftci (2000), Dufie (1996), Wilmott (1998), Karatzas and Shreye (1991), and Buter and Rennie (1996).

t(,/J1.

20.4. Use It's Lemma to evaluate

20.5. Suppose that S follows equation (20.37)and Q follows equation (20.38). Use 1t ' s L emma to 'lind the process followed by Sl :0.5 .

20.6. Suppose that S follows equation (20.37)and Q follows equation (20.38). Use It's Lemma to 'lind the process followed by lntkol. 20.7. Suppose 0.4 apd $100, r 0.06, o's compute prices for claims that jy the follWing: u&(0)

=

=

=

0. Use eqation

=

(20.30)to

a. Sl b. S ,s'-2

c Compare your answers to the answers ypu obtained to Prpblem 19.6:

20.8. Suppose that 1n(5') and ln(:) have correlation p and that $100, :(0) $100,?' 0.06, o's 0.4 and (y c 0..2. Neither stock pays dividends. Use equation (20.40)to find the price today of claims that pay -0.3

u(0)

=

=

a. b. FURTHER

.

c. d.

=

=

=

=

SQ S!Q SQ

j.(SQ4 s1Q e.

Compare your answers to the answers you obtained to Problem 19.7.

20.9. Suppose that Xt) follows equation (20.9). Use lt's solution to this differential equation is

.;' l

=

xe-k

+ a

(1 -

e-')

Lemma to verify that a t

+ c.

eks-tsdz

.

0

(Hint: Note that when t increases by a small amount, the integral term changes by dZt).)

676

%

BROWNIAN

AND IT'S

MOTION

LEMMA

PRoBLEus

20.10. The formula for arl infinitely lived call is given in equation (12.19).Suppose that rVdt. Use S follows equation (20.27),with a replaced by ?- and tlat f'*(tf7) lt's Lemma to verify that the value of the call, J/(5'), satises this equation: =

1atr25'27

(r

ss +

20.11. Suppose that the processes for

l.V

-

=

0

=

ajzkdt

=

azubhdt + o'zszdzz

+ o'js'l#zl

=

(2

I#Zl Qdt + Q (?7

+ ?7a#Z2)

Show that, to avoid arbitrage, ac

-

r

=

Pl

cl

(a j

.

-

?

)+

P2 0-2

(tz2 ? ) .

-

(Hint: Consider the strategy of buying one unit of Q and shorting Qlj /,5*1& 1 units units of %. Finance any net cost using risk-free bnds.) of St and :722/5'2c.2

20.12. Suppose that S follows equation (20.27) with follows the process

dQ/Q

=

What is ac, expressed in terms of a S and Q that eliminates risk.)

acdt

-

=

0. Consider an asset that

ldz

llint'. Find a zero-investment position in

20.13. Suppose that S and Q follow equations (20.37)and (20.38).Derive the a claim paying u(F)& Q'lb by each of the following methods:

value of

a. Computing the expected value of the claim and discounting at an appropriate rate. llint: The expected ret'urn on the claim can be delived using the result of Problem 20.11.) b. Computing the lease rate and substituting this into the fonuula for the forward price. 20.14. Assume that one stock follows the process dS/S

=

adt + o'dz

(20.46)

Another stock follows the process

(IQ/Q

=

acdt

+ c#Z + dq3 + dqz

(20.47)

(Note that the o'dz terms for S and Q are identical.) Neitherstockpays dividends. dq3 and dqz are both Poisson jump processes with Poisson parameters l anct 2. Conditional on eiterjump occurring the percentage.change in the stock price is F1 1 or F2 1 -

.

Consider the two stock price processes, equations (20.46)and (20.47). 0), what would be the 2 a. lf there were no jump terms (i.e., l and between lation aq a re b. Suppose there is just onejump term (2 0) and that 1'I > 1. In words, what does it mean to have F1 > 1:/ What can you say about the relation between a and ac =

=

'

c. Write an expression for ac when both jump terms are intuitively why ac nzight be greater or less than a.

Note that the diffusions #Zl and dZz are different. In this problem we want to lind the expected ret'urn on Q, ac, where Q follows the process

do

677

=

and Sz are given by these two equations:

5'I

dSf d%

jl5'ys

-

k

nonzero.

Explain

y' ('.;';' j'kt(y'.' t'(jp((,j' f' j.-pd (jd '('jkd jlr.d 4(:2j1(* jjd lr' j'yyd :jryg' !j' '1111:44))(4(::* f'f' y' f' 41222:::* pjytjyjj.,!d ' )' !(' '(' t.,d (' jjjj:yjjjjjyyrq'd ;'(jtlp'fjjq)))d 1* $' E2' j';.j;' yjd jid )g' k' gtd ((' )y*.* 'j.j);' )' yyjjjjjj'.' j';'j' yjd p' yj..-jd .j' yjj'jjjjjjjgyjyyyd ld sfiyjtd 411:qjjj1;;* (' '/' (tEj,j()4yj..' (jj' (jk' )' 2pj' $jk:* kd 'jd itd jsd lj' !:' j'jkjd ;y' )t' )' (IIE.)j)X 'jj))jj'.' .(t':)' @)' j'jd (jd 'j' (()j' k' t)' )y' y' i)jd jjjjd jyt'jj;d ;jy' jyy..' iyt!rd jy';tjjy'.' tppird j'1('1(' jyqd ;r:t' jjd j;';' yj,yr,,d jjjyjd jqpd jypd (4!g.:),.' y-yd rrd p)' J' j'.' jjd ?p' tjd gyk)y)j(jjj()k'.' kjd j)j)j'.' jqd y' '''(' ltd t)y.' @' r' jt' (jj' .(j)' j.jjd j',' jyyjjjjj';j))d ;')' glli'ilkht'.rrqqd y)j'..jyjjj$yj'.' ():id (' q' y' .jyjk.' jyyyytjy'yd j'jd yjjjd jyjd jy.,;ry)'(yjj);)',' gt.t.jyjjjjj'.' jyjgj,..' jjpljd !' (1)* rpjjjy'j)jtd g' 1@* jjjrr'jtd jtyjjy!-' yytyytd jy'rjy.y)d yrd ).j!t' jjd y')d (ppd )k' jjyj)yd p' l'jlt'jd pd j.jd l'1* lyttd jgk.' jj'trrd (1* 14* j'jtj'gyjd $44j* j.;d )' @' jktjd y;hd ))'yy' ''Eyj'IX y' .));' j.yjd j)' jy'jjyjyyy.,' r' j'jd yyd (' jjj;.' j,' ;;' jjd j(' ;'j'1* g' k' jjjd !' yrd j;yd jjjjjjyjjjqd )' p' /jjj((ytyj.' j'yd J' y.yjd )y' 'i)d (jy('.' )jkjjjq;;y,;' kjjjjjj((.' 'y'

(' yy'jjj' p)' jjy,'.j(.tk'jj(tjyy'.' $' yyy'yrttjjj!y;d )' t'jjjjjyyyjjtd y' (rd g' 'jjj)j,',' t'yjd lq'yd yjd 1* tyjyttj..sp/;d ))' J' )!trd )(' ()'tr' q' ;;' (;tjd qj)d yy-t!yyd y;jd 7)* p' y.jd jyjjtd (' f' )' 4* )jj' l'y' h' yyd 1* )'jjj' $*j44y* y')d (' ;'t'?' qjd @' k' tjd ''y' jj'yjjjy'tjd yjjjjyidjtijkyjjjjjz'--d t't' tyl' ytjjyjy'.' rjd t,kjrd t'((' tyd jljjjqyd jqjr:!',,..kkjj''d jljjljj;;l::,..d jjjjljjjjq:,d t.tjd trj)d ttj)jd ryd ttd t.)' yqjl:!''''!kkj!''d (ljjjjljqlg:jj)d (jjjjjg;;l::...d t.jg)))d t?d t't'yyd 1*1* t--' r'tyd kyjggjj(jjj(jjjj(tj'...' r' tyj...gd 'ts.s.-yjgd yjjy;d rjd rr-.)'

'j' -' .?' -,td -',')' -'.',.' -.' .-' .,' ..' ...' .'' --' -'.y;';' ''''!!jll(t!'''' ..'y;yy' ..jjjt!!!!jjk..' ''11jjj.,,jkjj''' 'j' ')' '.'.,,' 'j' -i.))' -''jjjjj!jjj(r'' '''jjjj!y,,,,' ''.''j' :';'...!' ..9* ;',-',r.-jyyry,j(yyjjjgyy'.ygjyyyyjjjyjyjyjjjjyyy.(kyyy;..' -y' -'-.',';' 't!kj' il'ss;tpj;d (t'jyjgd r.)qd !g' Eyd yyd -()' -t' (' ,.jjkt' E(' -j' j',';?' y.y.d '?' 5,* :.' y' E;' :r' C.;' Ejd r' :'' k' --;it.' .r' ijd t'jd j'-'''' ;'yytEj.ld ':j'jjzrzr!:d j'!':,,' ,'y' yjd iy'jd ?' g' y' jkd j'Ejd y'sd .;' j)jjy)y)jy(E.' 5* y;.t(yyjjjjy;'

.'.' .'.'

.'' ''.' ......' ' '', . . .. . . . . . . . ' ' . ')tr'(.1E.E.'

i 'E E. ; ! ii E E E j(' :((.j ( !.jE j j:j' .E.y'i .yj.E jj'' i.g': i'i i' E! 41 ' E. ( '!j ' EE'' j(y ' 'E rj' qq ' Eljjyj. ( q;'. '.j;. jyg(j@g'' j.' !'jj(;jjj' '.yj 'jjj tjE jp' '. E.L/LLLjI. !q !Ej;ytt j(ijjl (j;jrj jii i jyEtkjyj;jjjjjjjk)t'.. j jy ;jg. ( ;).gLLLLLLLL. (('j ;!1!:. ; ' )jj(LL:LL jy'.j,'' g'g ......jLLjL(jLLLL))(j?. LLLLLLLLLLL.. kg .('.'. .jjjjjjjjLjj. j((y. g . 2 ...j.(.j. 'LL.L ;rjrjj'.j. ', . (.. .. (t(. yk(. (.' t.)(:. ) .;jj. yjj)yj(j'. j)j). jjk. jkkljj. (.j.j..j)y': . (((j.'(y, .(i. kj. j.i jjjjjjtydyjjjjjjjjjtjgdzrkjjr.:). yj'. ..(... kj)jj'. .. jj)L''jj()L)L'. .... yj.. .(j'i..)'' )jky. . tj . (yjjkjlljj. . r', jk)rk@'. ....ry:.g. (j(jjj('. . . . . (jj'. ..... . tyt)gyy((.. . jj))j.(... .y( )jj'. . .))jj'. . jyjjkt())g'. ..j,.'.,,: .' k:... j j)j)t)))(. .r! k...j'. ('. . jjyjtjjj)(. ())j'. .... ()gjj)'. y(yy);jjj!j'. .. .. tjyj(. . r: .LLLLjjLLLLL)LLj(.. . tjl. 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'. lIply;.(g. . j . jr.pJ; 71j t! . .'r;r ;.l (t.. t-')): tr j(;( j 13 the market-maker who sells delta-hedging market-maker. As in Chapter we saw a a i )k.. jj y;.jr)jjj r jg ;). . . jj option buys shares to offset the lisk of the wlitten call. To analyze this situation it . . 2yr'. . ( y j ( ; .. . t.j . js necessary to characterize the lisk of the position as a function of the share price. It s ', pi ltdlj i k (pjl t'i.y j; ; . j jj . E. ..''.E.EE..'-!EE';Ei.(-. E.! ; ijiy,'y.. Lemma, discussed in Chapter 20, provides a tol that permits us to see how the option gyjy j; y. yg.; g'.qrr!; y.g.yyy.yE. .-'. y.yjjjyyyyj. yjl.rjtitjlty rtlyj-, j.,j(!.();y...-.. . yy.yttijj gjgy.jjg gjy.yjjs jtyy:r jyggri jyggry jtyg;j j:yrr jtyg;j jkk,rigjry jrrg;jyjy, jkj:rr j;::.jkygry j;yy:y$,yj(. jkygri gjyy j..:4;; gyy.jj :jg; :jj;i . E!(E',E,;qq.E.E,',iskiEllsjyji.i.gj.yjj.y.iijj.gljjt :jj;ijjjs gjrr;js gjy., gjrr)js gjyr;js gjyr)js (jjky;, ) (tjr iI tjjjj-.-,.. 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l Robe,.t Merton also contributed

fundamentally to understanding tlae option pficing problem. As a ' result, the analysis in this chapter can also appropriately be referred to as the tBlack-scholes-Meron jkjgr,. ,,22r),.,,22::,:j(..,,y.:l; (jj.,,j. jj,(,222),, g:jj,;,. -pjjjI;k ,j,

-'.q4;;;p,,.

:!!4(((g,1,-

680

% THE

BLACK-SCHOLES

TH E BLACK-SCHOLES

Equvlorq

shows that ST) can equal $1 only if A

The Valuation Equation

S(J)

A farniliar equation from introductory 'linance is the following:

St)

Dt + 11411 + St + /?) . 1L (q ? 2)

=

--h-

-/,

This equation says that the stock price today, St), is the discounted value of the future stock price, St + /1), plus dividends paid over the period of length /7, Dt + hlll. The discount rate over a period of length h is ?v, We can also interpret St) as the price bf a bond and D(/) as the coupon payment. Whatever the interpretation, we can rewlite equation (21.1)as .

St + /7)

-

+ Dt + Illh

S

?v

Change in Stock Ice

St)

= Retum

Cash Payout

(21

on Stock

-.>

e-rT-'t

The cpndition ST) = $1 is called a terminal boundary condition because it sets the u$'(0), bond price at its maturity date. If instead we lnew the bond price today, we could set z4 so that the equation gave the correct value for 5'(0). That value would be an initial ' boundary condition. Tfle solution consrms what you already lnow: The price of the bond is the present value of $1.

Dividend-paying Stocks We can interpret St) as the price of arisk-free stockthatpays acontinuous lixed dividend of D and has a pdce of J at time F. Equation (21.3) then says that at evel'y time, t, dividends plus capital gains on the stock provide the risk-free rate of return. Since we know the value at time r will be we also have the boundary condition .,

sT) Equation

(21.3)with

,j

=

this boundary condition haj the solution F

+ Dtj

=

l'st)

St)

(21.3)

Equation (21.3)is a differential equation stating the condition that the stock must appreciate to earn an appropliate rate bf return. The transformation from equation (21.1)to equation (21.3)illustrates the sense in wlzich an equation describing the evolution of the price is linked to valuation.

De

=

-r(.-/)

t'j s

+

#g-r(F-l)

l

The stock price today is the discounted value of dividends to be paid between nw and time F, plus the present value of the stock at time F. Again, the discrete time version of this equation is the standard present value formula taught in every introductory finance C aSS.

The General Structure

Bonds Let Stj represent the price of a zero-coupon bond that pays $1 at time F. Since the bond makes po payouts, the evolution of the bond price jtisfes equtipn (21.3)with D = 0. The interetation is that at every time, t, the percentage change in the price of equals the interest rate. This is a familiar condition tht the bond the bond (dst dt jstl satisfy is if it fairly priced. The general solution to this equation is j should

st)

=

Ae-rT-'s

(21 4) .

where A can be any number. You can check that this is in fact a solution by differentiating it to be sure that it satisses the differential equation.

The differential equation describes the bond's behavior over time but does not tell what A is. In order to price the bond we also need tp ltnow the bond price at some us particular point in time. This price is called a boundary condition. Ifthe bond is worth $1 at maturity, we have the boundary condition ST) $1. Examining equation (21.4) =

Undercertainty, abond orstock will bepriced so that the ownerreceives aliskrfreereturn. The differential equation in these examples desclibes how the secul'it.y change from a given p'oint. The boundary condition desclibes the price at some point in the security's life (suchas at a bond's maturity date). By combining the differential equation and the boundary condition, we can detenuine the plice o t t e on d at any point in time. By analogy, if at every point you ltnow an automobile's speed and direction, and if you know where it stops, you can work backward to gure out where it started. Essentially the same idea is used to price options: We lnow the price of the option at maturity (for a call it is maxgo, S #1), and we then need to ltnow how the opti (jn p ri ce changes over time. -

21.2 THE BLACK-SCHOLES

EQUATION

Consider the problem of owning an option and buying or selling enough shares to create a rislless position. Assume that the stock price follows geometric Brownian motion: ds &jt'lt + o'dz T = tce -

2You might wish to verify that St)

681

.2)

Written in this form, the equation says that the change in the stock price plus cash payouts (such as dividends) equals the return on the stock. Equation (21.2) is written to emphasize how the stock price should evolve over time, rather thap the value of the stock at a point in time. Dividing by /2 and letting /? 0 in equation (21.2),we obtain

dst) dt

$

$1. Thus, the bond price is

=

$1 X

=

EQUATION

Ae-rl--tt

+ a satisis

the differential equation only if a

-

0.

682

%.THE

BLAcK-5cHoLEs

TH E BLACK-SCHOLES

EquATloN

where a is the expected return on the stock, c is the stock's volatility, and is the continuous dividend yield on the stock. The option value depends on the stock price,

and time, t, so we write it as 7(5'(f), l). Also suppose there are risk-free bonds that pay the return r. tf we invest 1,1/in these bonds, the change in the value of the bond St),

position is

#W'

rWdt

=

(21.6)

in the option, stocks, and the risk-free

bond. Let 1 denote the total investment Suppose that we buy N shares of stock to hedge the option and invest F in risk-free bonds so that our total investment is zero. Then we have

1

J/(5', t4 + NS + F

=

=

0

The zero-investment condition ensures that we keep track of snancing costs. It imposes the requirement tat in order to buy more of on asset we have to sell somethipg else. To buy stock, for example, we can short-sell bonds. Applying It's Lemm' a to equation (21.7),we have

dl

=

dv + Nds

+

sdtj +

#JP

10'2,:27 dt + NLCIS+ ss = 1/1dt + J/, dv + a

+ rwdt s.%dtl

(21.8)

If we own the physical stock, we receive dividends; this accounts for the N&sdt term.3 As in Chapter 13, we delta-hedge the position to eliminate risk. The option's delta (A) is F-. We delta-hedge by setting N

=

i

-

Holding this number of shares has two results. First, the ds and, hence, dZ terms in equation (21.8) vanish, so the portfolio is no longer affected by changes in the stock price-the portfolio is risk-free. Second, because we are also maintaining zero invest21.7) our holding of bonds is whatever is necessary to fnance the net ment (equation sale of the option and the hedging position in stock: Purchase or F

Substimting N

=

dl

-Ji =

=

I/,s.

-

(21.9)

J/'

and this expression for J/I/into equation

Vdt + lclfvssdt 2

Vs8vdt +

-

l-vss

(21.8)gives -

Vjdt

(21.10)

With a zero-investment, zero-risk portfolio, we should expect to earn a zero ret'urn or else there is arbitrage, so that #f = 0. lmposing this condition in equation (21.10)and dividing by dt gives y'ss + (?. Jl5'J& X + 1c25'2 2 -

-

rv

=

0

EQUATION

$

683

This is the famous Black-scholes partial differential equation (PDE), which we will call (We will refer to the formula giving us the price of a the Black-scholes eqltatiol? European call as the Black-sclaolesfo?'p??lf/rl.) Appendix 21.A derives the generalization of equation (21.11) when the value of &' depends on more than one underlyig asset. Yhesignicance of eqtlation (21.11) is that the price of an option must satisfy this equation, or else there is an arbitrage opportunity. In fact you may recall this equation from Chapter 13. There we examined delta-hedgin'g and saw that the delta, gamma, and theta ofa fairly priced option had to be related in a certain way. Since IZy.C is the option's gamma, JZ. the option's delta, and P) the option's theta, equation (21.11) describes the same relationship among the Greeks. We started this discussion by supposing that we owned an option that we wished to delta-hedge. Nothing in the derivation uses the fact that 1/ is the price of a call option or indeed any particular kind of option at all. Thus, equation (21.11) descl.ibes #?' challge p?p/'.'5 7J-ll???/.#kb??To be ill vt-l/lfc ofally contillgellt c/clkn./br vhich J'c ltllderlyillg are assumed deal: That underlying follows geometric have the great asset (a) a sure, we Brownian motion with constant volatility, (b) the underlying asset pays a continuous proportional dividend at the rate J (thiscan be zero), (c)the contingent claim itself pays no dividend and has a payoff depending on S, (d) the interest rate is fxed, with equal borrowing and lending rates, and (e) there are no transaction costs. These assumptions are unquestionably violated in practice. There are transaction costs, volatility and interest rates change over time, asset prices canjump, etc. However, our goal is to have a thorough understanding of how derivatives plicing and hedging works in this basic setting. This is a starting point for developing more realistic models.'

Verifying the Formula for a Derivative We can now answer the main question of option pricing: Given that asset prices follow geometric Brownian motion (equation21.5) what is the correct fonnula for the price of an option? As discussed in Section 21.1, there are two conditions. The pricing formula must satisfy the Black-scholes equation, (21.1l), and it must also satisfy the appropriate boundary conditions for the option. ((f we satisfy both conditions, we have the conect option plice. Almost all of the nonstandard option formulas we looked at in Chapter 14 solve the Black-scholes equation.6 The pricing formulas seem different, but they differ only

4When the den-vative claim makes a payout, Dtjd,

then equation

(21 1l ) becomes .

(21.11) sEquation (2l 1) holds for unexercised Amercan optons as well as for European options. 6The exception is Asian options. Since the Asian option payof is based on the average stock price, prices of those options solve a different partial differential equation, in which there is a term reflecting the evolution of the average. -1

t

684

%.TH E BLACK-SCHOLES

TH E BLACK-SCHOLES

EQUATION

in the boundary conditions. Appendix 21.C discusses a general set of solutions. Here, we discuss severtl particular solutions in order to convey the basic idea of how the Blackcholes

equation

works.

Let's begin by considering

two familiar calculasimple present value tions: the price of a zero-coupon bond and the prepaid forward contract for a stock. Suppose the bond matures at tim F and pays $1. The boundly condition is that it must be worth $1 at time r. ln addition it must satisfy the Black-scholes eqpation, equation (21.11).Consider this formula for the price of the bond: ce-rr-t)

J/ 1 t, y,)

(p,j

First, tis satis:es the boundary c6ndition since &'l (F, F) = $1. Second, the price of the bond does not depend on the price of a stock. Thuq, 7.9 = 0 and J/.- = 0. Equation (21.11) then becomes J$l

=

l

.

l

,

f.j

=

) #(:,/1)

#l

lntu$'/ffl

=

o' T

dz

t./1

=

-

-

o'

t

stle-r-'t

We will verify that equation

(21.13)solves 2

J1

=

2 JG

=

e

rj

=

$1. the

ST)

-

F

+ 1.o. a1 '

(?' -

'

are both equal

-

t

c'

t

-

-

-

-->

=:> 1n(5'/ff)

>

knS!K4


K.

Equation (21.13)thus satisfes the Black-scholes equation and the boundary condition. Notice that for both claims, /&. = 0', their gamma is zero. We already saw in Chapter 5 that we can replicate a prepaid forward by buying a tailed position in the stock. No further trading is necessary. This static hedging strategy works because gamma is

maxgo, S'j

-

K if S > K, Thus, at expiration the Black-scholes formula for a call evaluates to S and 0 if S < K, so it satisfes the boundary condition, equation (21.14).The call formula also satisses equation (21.11), but we will not verify that here. Puts can be analyzed just like calls. European puts have the boundary condition

&stte-'1TMl

=

and e-rr-t

-

-

Fg5'(F), F)

Ndzj

.

substitutingthese into' the Black-scholes equation gives $stle-bTMt jst) x e-t--tb + 12 G lSt)l x 0 + r

Call option

-r(F-J)

As t approaches F, the difference between #1 and dz goes to zero, since the term F t goes to zero. Moreover, t he term (r 8 + 1&2) r t also goes to zero. -c c F Thus, both #1 and dz are governed by the term lntu$'/ffl/o' t. If S > K, then the option is in-the-money and LnSjK) > 0. lf S < K, the option is out-of-the-money and ln(5'/ff) < 0. Thus, as t r, we have

(21 13)

O

5=2

+

F

-

,

Ke

-

'

Since this contract pays a share at maturity, the boundary condition is that it is worth a share at matulity: J/ 2(ST)

685

-

v.

Equation (21.12)satisfies this eqtlation, with the boundary condition Fl (F, F) = Now cosider the prepaid frward contract for a share of stock. W lnw value is

7 2(st)

-J(W-f

For an option at expiration, since t = r, the terms e-'T-'s N(#I ) and A'(#2):? to 1. What happens to the definitions of tlt and dzL slightly will rewrite We

ja)

.

$

We Let's velify that the Black-scholes formula does satisfy the boundary condition. approaches l option expiration F, the date. formula as can examine the behavior of the From equation (12.1),the value of the call is Se

talculations

EQUATION

/)

e

=

=

.---tts

e-r

x N

w-,)x x

1nES(r)/A-1 +

-

F

c

1nES(r)/ff1 +

8 + 0.5o-2)(F

-

-

c

on its own is a legitinate price of a delivative. What

0.5c2)(r

-

F

-

r)

t

-

Er

-

-

?q

t

are they the prices of?

(21.15) (21.16)

686

%

THE

BLACK-SCHOLES

TH E BLACK-SCHOLES

EQUATION

#F

A cash or nothing option pays $1 at time F if ST) > K, and nothing otherwise.s has the same value at mattlrity as a cash-or-nothing option and satisifes Equation (2l the Black-scholes equation. Thus, equation (21.16)gives us the time-l value of a cashor-nothing option. Both asset-or-nothing and cash-or-nothing options are examples of all-or-nothing options, which pay a discrete amount or nothing. A European call option is eguivalent to buying one asset-or-nothing option and selling K cash-or-nothing options, both maturing at time r. The price of a European call is the cost of this strategy:

-o'

=

-

-

2

S a J/&&+

(J

In the foregoing derivation of the option pricing formula we required that a delta-hedged position earn the risk-free rate of return. A different approach to pricing an option is to impose the condition that the acmal expected retul'n on the option must equal the

IZJ

'

dt +

S FJITT/Z Unexpected

retum

Thus, the instantaneous expected return on the option is Edvj

1

1

7 v

-

H

)F

F,g. + a g1:0'25*2

l5'li + Jjj dt

-

(2 1.17)

v

Jopton

The unexpected portion of the ret'urn is E (# JZ)

# J/

-

J/'

S J/y

=

J/

J/

ln interpreting this expression, have

recall that

SVs(Jf is the option's

o'option =

'

(21.18)

o'dz

(21.19)

c'option#z

H

elasticity,

f2. Thus, we

(21.20/

S-ZG

This is a result we presented in Chapter 12. We know from Chapter 20 that two assets with returns generated by the same #Z must have the same Sharpe ratio. Thus, we have (Z

tz option

1*

---

1*

*--

=

coption

G

(2l

Using equation

.20),

we can rewrite equation Joption

The Black-scholes Equation and Equilibrium Returns

l5'k%+

-

Expected return

K x J/4g5'(/'), lj

You should verify that this is in fact the Black-scholes formula. (See Problem 21.7.) The fact that 73 and J/' solve theBlack-scholes equation gives us pricing formulas for two new derivatives, asset-or-nothing and cash-or-nothing options. Also, however, because74 by itself solves the Black-scholes equation, we could have sold any number of cash-or-nothing options and still had a valid price for a derivative claim. ln order to create a standard call, we buy one asset-or-nothing option and sell K cash-or-nothing options. The options. However, suppose we had instead sold Q.5K cash-or-nothing 0.5# if ST4 > K and 0 otherwise. This is a resulting claim would have paid 5'(F) gap option, discusqed in Chapter l4. This analysis verifies that equation (14.15)gives the correct pric for a European gap call.9 The boundary conditions we have considered thus far are all tenninal boundary conditions, meaning that they are satisied by an option at expiration. American options and some nonstandard options have a boundary condition that must be satisfed prior to expiration. For example, barrier options have boundary conditions prior to expiration rlated to lnocking ip or out. Nevertheless, their price still solves equation (21.11).

l

2

.16)

lj

-

1-

=

S 7. F

(21.21)to give (J

-

r)

lisk

elastictty.

in a particular way.

is also called digital cash.

9In practice, all-or-nothing Chapter 22.

and gap options are difficult to delta-hedge. We will discuss this further in

(21.22)

premium on the option is the risk premium on the Stock times the We can interpret equation (21.22)as stating an eqltilibriunl cpdition option ln other words, if we view the option as just another asset, it option obey. the must that its expected d that return is related to the expected return on the stock be must p rice so

ln words, the

7-rhis claim is also called a digital share. S-rhis claim

687

We can decompose the return on the option into xpected and unexpected components. UsingIt's Lemma, we have

option.

,

k

equilibrium expected return.lo As we saw in Section 11.2 in the context of the binomial model, the expected return on the option changes as the option moves into or out of the money.

option pays one share of stock if ST) > K, and nothAn asset-or-nothing K, and ing othelavise.? Examine 73 closely. We have F3(5'(F), F) = 0 if ST) 73g5'(F), Fj = ST) if 5'trl > K. Thus, at time F, 73 has the same value as an assetor-nothing option. Moreover, because 73 satises the Black-scholes equation, it gives the correct value at time t for this payoff. Thus, 73 is the value of an asset-or-nothing

F 3ESt)

EQUATION

'oBlack and Scholes also used this method to solve for the option price in their original paper.

688

k

THE

Bl-Acx-scuot-Es

Using equation

TH E BLACK-SCH

Equvlorq

(21.17),substitute for Joption in equation (21.22).This gives

If you compare equation

O LES EQUATI ON

$

689

(21.26)with (21.1l), the dividend yield, $, has been replaced

with I1, the difference between the equiliblium expected return and the actual l expected return on noninvestment widyets.l e Let y.. We can interpret J as follows: Ix is the return you get from holding and is the return you must expect if you Jtre to voluntalily hold a widget. id t w g a -

1

gy

+ (a Jl5'k', g147'25*2/.$, JZ

+ y;j dt

-

2

us'gy

l zz:

(a

.

-

JZ

r)

-

(21.23)

When we multiply both sides by J/ and rearrange terms, the expected return on the stock, a, vanishes: We once again obtain the Black-scholes PDE, equation (21.11). Thus, an intelpretation of the Black-scholes equation is that the option is priced so as to earn its equiliblium expected rettlrn. When we equate expected and actual returns, we can interpret the result as giving gfair price for the option, as opposed to a no-arbitrage price. This is eqttilibriltln us j'ricing. The no-arbitrage and equilibrium prices are the same. The equilibrium approach makes clear that determining a fairprice for the option using the Black-scholes equation does not depend upon the assumption tliat hedging is actually possible.

What If the Underlying Asset ls Not an lnvestment Asset? So far we have been discussing option plicing when the underlying asset is an investment asset, meaning that the asset is pliced so as to be held by investors. Stocks and bonds are investment assets. Many commodities are not (seeChapter 6, especially Sections 6.3 and 6.4). Suppose that widgets generate no dividends; and that the price of widgets, S, follows the process dk lxdt + c#Z (21.24) S = From this equation, widget price risk is generated by the term #Z. Let $ represent the Sharpe ratio associated with JZ and let represent the expected return for an asset with ?')/ty, adz risk. Since the Sharpe ratio is $ ( we have =

&

-

-

Thus, in order foryou to hold a widget you wotlld need an additional return of = Given the expected widget price change, y,, the only way to receive the extra return is through a dividend. This is the reason that Jz replaces the dividend yield in the Black-scholes equation. is the Iease rate for the widget, or We have encountered this concept before: more generally the lease rate for an asset with expected capital gain g, and risk c#Z. When you lend a commodity, you receive its capital gains. The lease rate is the extra income you need to make you willing to buy and lend the asset. In the same way, J is the extra income you need to make you willing to hold a widget as an investment asset. In practice, a widget-linked bond could be used to hedge the l4sk of a widget option. If the widget bond were constructed so that its price equalled the widget price today and at maturity, we saw in Chapter 15 that the bond would pay the widget lease rate as a coupon. This coupon, being a cash payment on the underlying asset, would play the role in the option pricing formula of a dividend on the underlying asset. This idea of a hypothetical lease-rate-paying, widget-linked security is also like the tbvin Jcclfr/y mentioned in Chapter 17. It provides an investment vehicle for owning the risk #Z. If such a twin security existed, we could use it to hedge the risk of the option, and its dividend yield, would appear in the option price. An equivalent way to write equation (21.26)is to replace with 1' + $)'. We then -y..

-

,

obtain

2 2 W+ 1a tr S liy + (/2

In this version,

the coefcient

-

$G

)S V

(aj L.;)

.()

.v

-

l

.

on the 5'Fs term is th e d r ift on the widget less the risk

premium appropriate for widgets.

r + o'#

=

=

'

= Note that when the asset is an investment asset, a and /2 = a and reduce equation Both equations to (21.26) (21.27) (21.11). $c To summarize, the Black-scholes PDE, equation (2l 1), also characterizes derivative prices for assets that are not investment assets. ln the case of an asset that is not an investment asset, the dividend yield, f, is replaced with the lease rate of the asset, =

The important characteristic of an investment asset is that Jt = What happens if an < is and investment ? not asset asset an /2 which says that the expected return on the option Consider again equation (2l equals te actual rettlrn on the option. When we delive this equation again using as the equilibrium expected return for an asset with risk #Z and Jz as the acmal expected remrn for widgets, we obtain .

.20),

$+

1G25'27

z

ss +

gSVS

q/y

-

J/'

?-

=

(& -

1/

r)

(21.25)

r+

-

-

.

.1

.

To see how to use equation (21.26),suppose we have an option for (') Example 21.1 ))i whih the maturity payoff is based a upon the stock price raised to a power, S vus type ! .

Rearranging this equation, we obtain J/t + lczvvss 2

+

y

-

(& yljvs -

-

rv

=

()

(21.26)

ll'rhis modification andSiegel ( 1984).

to the Black-scholes

equation is discussed in Constantinides

(1978)and

McDonald

%

690 j'y'

TH E BLACK-SCHOLES

EQUATION

RISK-NEUTRAL

For example, we could have a call option with a

a power option.

) of option is called !i payoff of

.j'. (! .y .

The Black-scholes

equation,

(21.11), can therefore be 1 dt

:

!((i

mftxt

v)a

-E*dV)

Ka g;

t: tj pitwe have already seen in Proposition 20.3 that the lease rate on an asset paying 51 is t1i 1zaa 1)(r 2 From It's Lemma the volatility is ao-. Thus, using J) r ar t) lti equation (21.26),we can plice the option by using St, as tjje stock price, K as the strike jlt ''ilprice, as the dividend yield, and ao' as the volatility. Q j

. ' j...

.

,

.

*

=

-

-

Under the risk-neutral

rewritten

PRIcI lqc

691

as

1. J/

=

%.

(21.31)

process, the option appreciates on average at the risk-free rate.

-

-

.

*

21.j Rlslt-xsuvRac

The Backward Equation Closely related to equation (21.31)are the following equations, which charactelize both the actual and risk-neutral probability distributions:

PRlclxc

does not appear in the Black-scholes equation, return on the stock, when delivatives on investment assets, only the riskpricing equation (21.11). Thus, expected free rate matters', the acttlal return on a stock is irrelevant for pricing an option bipomial pricing formula (see Chapter l0) also dpends only on the on the stock. The

The expected

,

risk-free rate.

This observation led Cox and Ross (1976)to the following important observation: Since only the lisk-free rate appears in the Black-scholes PDE, it must be consistnt with possible world in which there is no arbitrage. lf we are tlying to value an option, we any world in which it is easiest to value the option. Valuation that in the we are can assume risk-neutral world, in which (if it actually existed) all assets would easiest in will be a lisk-free of and the return rate we would discount expected future cqjh flows at the earn options and other derivative claims by asstunillg that value Thus, risk-free rate. we can risk-free of stock the the rate return and calculate values based on that premise. earns We assume that the stock in this world follows the process

ds

fldt

= (?-

+ o-dk

(21.28)

As we keep emphasizing, the risk-neutral distribution is not an assumption about investor risk preferences. It is a device that can be used when pricing by arbitrage is possible (see Appendix l 1.B fol: a discussion).

lnterpretingthe

Black-scholes

The actual expected

change in the option plice is given by

1 Edv)

J?

/)

=

Equation

+

y c a u a J ss +

.

@

(ylqyu

-

(21.29)

Let E* represent the expectation with respect to the risk-neutral distribution. Under the risk-neutral distribution, the expected change in the stock price is E*dS) (?' ldt. The drift in the option price can thus be written =

1

,y

L-.E (Jz) dt

j (#7) 0 #r l -f*(#F) 0 dt quation (21.33)is -E

=

v',+

=

c a s a p'ss +

(? .

-

j;

s j/j

(21.32)

=

(2l

=

For the risk-neutral process,

F) + 1 zo' lSl

/

ss +

(?. Jju/j -

=

0

(21.34)

E quation (2134) is called th Klmogorov backward equation fpr the geometric Brownian motion process given by equation (21.28).Whraj the Black-schols PDE characterizes prices, the backward equation characterizes probabilities. The backward 12 equation is just like the Black-scholes PDE except that there is no ? term The Black-scholes equation can be interpreted as saying that the expected return on the option must equal the risk-free rate. The backward equation pertains to probabilities of events, such as the probability that an option will expire in-the-money. To understand how such probabilities should behave, suppose we decide that the probability is 0.65 that the stock price 1 year from today will be greater than $100. We lnow today that if the stock price goes up tomorrow, we will then assign a greater probability to this event. If the stock price goes down tomoirow, or estimate of the probability will go down. However, we should not expect our estimate of the probability to change on average: Our expectation today, of toltlolanwl probability, must also be 0.65. lf today's estimate of tomorrow's probability were not 0.65, then 0.65 could not have been the correct probability today. Thus, whereas the price of a financial asset is expected to change over time, the expected change in the probability of an event is zero. This is why the backward equation does not have the l'V term. If fsp; St4 is the probability density for Sv given that the price today is St, both of these expressions would satisfy the backwtrd equation: .

-v'

.

'

-

(21.30)

.33)

l2The backward equation is covered in detail in standard texts (see,forexample, Cox and Miller (1965), and Karlin and Taylor (1981)). Wilmott (1998,ch. l0) contains a particularly clear heuristic derivation of equation (2l .34).

f

692

% THE

Bl-Acx-sclqouEs

C HAN

EcuvloN

K

Sv ; 5'?)#q%

Sr

f

Sr ; St l#kr

,

The Erst is th probability a call is in-the-money at time F. The second is the partial The e-kpectatiollof the stock price, conditional on Sv > K. Both are undiscounted. equation holds for both the and risk-neutral distributions generated by It backward true PFOCCSSCS.

21.4 CHANGING

The solution to equation (21.31)is equivalent to computi. ng an expected value of the delivative paypff under the risk-neutral probability distlibution and discounting at the risk-free rate. The specisc form of the integral depends upon boundary conditions and payouts. We can see how this works with our ssumptions (in particular a constant lisk-free interest rate) by considering a simple European call option on a stock that pays continuous dividends at the rate In thatcase, equation (21.11),along with the boundary A-), is equivalent to the condition that the option at expiration is worth maxlo, ST4 expectation discounted .

-

,

l-,

T

-

t,

(5)

=

e

-r(F-J

X

) K

(SLy')

zy -

ju (w;

g

,

,

y.,

j.

ggqtjjcjy ,

(y,;

where f*LST), c', ?-, J; S(1)) is the risk-llettral probability density for STj, conditional In general it is possible to write the solution to equation on the time-l price being (21.11), with appropriate boundary conditionj, as an explicit integral-l3 lf a probability 1'P(5', t4 satisises the backward equation under the lisk-neutral distlibution, expression (21.33),then J/'(5', 1) e '-rl--tt Ft5' t) the present value of u$'(r).

=

F(5',

will satisfy the Black-scholes

/'),

,

equation, equation

that W#(56,t) satisfes the backward equation, i.e., 1 l)) .

E (#1'P(5',

since 1/

=

e-rT-tlw

=

,

(21.31).To see this, suppose

0

we have j dt

Es

(#JZ (S,

l))

j =

dt

=

This is the Black-scholes

?'

Es

(#

.mqv..t

; j,jy

qs

,

j;y

y

J/

PDE, equation

(21.11).

13seefor example Cox et aI. (1985a,Lemma 4).The integral fonn of the Black-scholes equation is alsocalled the Feynman-lac solution. See Karlin and Taylor (1981 pp. 222-224) and Dufhe (1996, ,

Chapter 5).

.

THE NUMERAIRE

derivative conNow we consider what happens when the number of options (or other adset price. This determined by random, expiration is some tracts) that we receive at example. the following Consider payoff is cornmon. odd-sounding

' Derivative Prices as Discounted Expected Cash Flows

CLSt), K, c.

693

plnbabilities t7?l# partial expectarisk-neutral probability or partial expectation Thus, any tions are prices of derivatives. of this, we saw in Chapter exaymple price. derivative As corresponding an also has a that an option is probability risk-neutral N(#2) is the 18 that the Black-scholes term -r(r-l)x(#a) is therefore the probability, discounted The e in-the-loney at expiration. expiratton. option at if the in-the-money is plice of a derivative that pays $1

X

K

$

This result means that discoltltted riskmeutral

X

.f

G l N G TH E lN1U M ERAI R E

j' ')' y-'-' )' The priee today of a nondividend-paying stock is $100, and the ll Example 21.2 j .tjj forward price is $106.184.Joe bets Sarah that in 1 year the stock price will be greater loser . than $106.184.Joe wants the loser to pay one shm'e to the winner. Sarah wants the qj) q . to pay $106.184to the winner. The share received by Joe would be worth more than $106.184if he wins. Sim.., she wins. ti ilarly, Sarah's desired payoff of $106.184is worth more than one share if i bet valuable side of the if it is ltt)Are either of these fair bets? tf not, who has the more ') denominated in shares? Who has the more valuable side of the bet if it is denominated ll.

-'.

!

;g Ei# .

.E

lj.. . EE .

E!).

k

k..in cash?

will If Strah wins (i.e.,the share price is below $106.184),a payment of $106.184 thap $106.184), shae is the greater price exceed the value of one share. lf Joe wins (i.e., worth more than $106.184. However, it ip not obvious a payment of one shltre will be whiehbet has a greaterfair value given a current stock price of $100.Assuming no inside bet information about the stock, would an investor pay a greater price for Joe's desired ,

or Sarah's desired bet? nllmeraire or We can describe the two fonns of the bet as each having a different desired whereas Sarah's denominatedin shares, desiredbetis unitofdellomilation. Joe's bet share-denoininpted paying the interpret as You can bet is denominated in dollars. of price dollar dollars of vatiable number (the either a lixed number of shares (one)or a of number a ($106.184) dollars or bet fxed dollar-denominated pays a one share). The general The value with shares the number of $106.184). variable number of shares (the denomination) for question we want to answer is how a change in the numeraire (unitof derivative. of plice the a derivative changes the Here are some other examples where a change of denomination is relvant: oliginating in yen (forexample) can be valued o Currency translation: Acash flow in some other currency. We will discuss this example in depth in Chapin yen, or

ter 22

694

k. THE *

BLAcu-scHoLEs

clqAxcllqc

EQUATION

An agricultural producerwho wants to insure production of total revenue-the product of price and quantity-rather hedge an entire Eeld must alone. than quantity

Quantityuncertainty:

* All-or-nothing options: All-or-nothingoptions, which webrieqy discussed earlier, can be stiucttzred either to pay cash if a certain event occurs (suchas the stock price exceeding the stlike) or shares. The payoffs to the stock price bets above are in fact all-or-nothing payoffs; thus, the bets can be valued as all-or-nothing options. To see what happens when we change the denomination of an option, suppose

is the price of an asset that follows #Q = (ac

cjdt +

-

-

Q

ty cdzc

Q

=

Jzg5'(F) K c., r,

Q(F)

,

,

r, J)

(21.36)

Qb,of claims, 7. The value of this payoff

Equation (21.36)represents arandom number, is given in the following proposition.

k

695

(21.16)we have that

equation

IZ -

I()

u%(0) ,

K c ,

,

T J)

1',

,

=

e-rT

F+g5'(0) K

-

,

=e

,

S(F), Fq

RE

This expression is the discounted risk-neutral prob.ability that the bet pays off; it is also the second term in the Black-scholes formula. If you hold both a bet that pays $1 when ST) % K and a bet that pays $ 1 when ST) < K then for certain you will receive $1 e-rT and we have at time r. Therefore, F+ + J/- =

(21.35)

Let F(5', t) represent the price of an option denominated in cash, where S follows the process in equation (21.5). The correlation between dZc and dz is p. Suppose we receive the time-r payoff 1'(:(F).

therefore from

THE NUMERAI

-rT

(1

,

tr

s

1.,

T Eq ,

Ndzlj

-

Now consider 4he bets denominated in shares. Let F+ denote valpe a bet that pays one shar when 5'(F) > K and Y- the value of a bet paying one share when ST) < K. Holding a share-denominated bet is like having a random number, ST), of cash bets. By Proposition 21.1, the value of the share bet is obtained by multiplying J/ by the &2 l and p = 1 since S forward price for S, and replacing $ with (we have b multiplies a claim basd on S). Maling these substitutions, the value of the share bet is 5'(0)e(r-J)TF(5'(0) K o', ?', r, J tr2j, or =

-

-

Suppose the process for S is given by equation (21.5) and the 21.1 for by equation (21.35),with p the correlation between d. and dQ. Let Q process 7(u$',K, tr-, r, F-r, 8s) represent the price of aEuropean derivative claim on S expiring at time F. The price of a claim paying Qb7 is given by

Proposition

QU)ber-&*j--lj

yg,$t/.l K as, ,

,

$c) br r here,7 8 bpo'o'c and valueQbclaims, each with value 7, we replace the Q(/)'dtr-*)(F-r) mu ltiplythe resulting price by W

=

-

*

=

-

-

-

.

t.,

lbb

r

t,

-

(p,j

;yj

.gey)

1)(r2

In other words, to dividend yield on S, (!, by ?7, and 2

-

q

.

k.

The proof is in Appendix ZI.B. Equation (21.37)is quite important and deserves further comment. We encountered J* in Section 20.7, in Proposition 20.3., it is the lease :(?)@'dtr-J*)(T-J) is the forward price for a claim paying Qb. The rate for Qb. Thus value of a claim paying QbJ/ is thus the forward plice for Qbtimes 7 evaluated at a modifed dividend yield. We know from Section 20.7 that if Q and S are correlated (in which case Q and F are correlated), there must also be a covariance term. Ttle term ?? replaces the dividend yield 8 to account for this covariance. We will now value the share-plice bets described in Example 21.2. t! Example 21.3 liqLet J/ + denote the value of bet that pays $ j at time r if vST) > K, and 7- the value a y)) il', of a bet 7,7,,;

that pays

$1 at time r when ST4


K, and also a bet that pays one share when ST4 < A'. then you will for certain receive a share e-Tvo, and we therefore have at time r. Thus, 1'+ + Y- = .15),

- (.S'(0) 1-,

K c .

,

1',

T

,

t1

=

SQleMT

= Sloje

-

FV(5'(0)

-

y(;;e-JF

,

-&T

K c', r, ,

r, Jq

,y(#j)

In the case of Joe and Sarah's bet, suppose that the share-price volatility is 30%, compounded lisk-free rate is 6%, the iime to expiration is 1 year, and the t.he share price will be above $106.184.The dividen Joe bets that ls. the share pays no cpntinuously

This argument link-ingthe two terms in the Black-scholes 14' inatonis due to Geman et a1. (1995).

equation by changing the units of denom-

696

%.TH E BLAcx-scHoLEs

EquATloN

an empil-ical Ultimately, theimportance ofjumps andtheirsystematic of jumps. effects the into insights useful provide Merton's formulas issue. Nevertheless, componentis

value of this bet is Value of Joe's Bet

=

l'+E100,106.184,

The opposite side of Joe's bet-receiving

0.30, 0.08, 1, 0q =

one share when ST)

value$100 $55.962 $.44.038.


nd Merton (1973b). Merton (1976)extends the Black-scholes model to allow diversi/able jumps in the stock price, and Naik and Lee (1990)develop a model to price options when

There is a different boundary condition when the stockjumps

to zero:

y)

-

cra

2

21.3. Use the Black-scholes equation to verify the solution in Chapter 20, given by Proposition 20.3, for the value of a claim paying S'l. 21.4. Assuming that the stock price satisses equation (20.27),verify that Ke-rT-'b + vtle-nT-t' satises the Black-scholes equation, where K is a copstant. What is the boundary condition for which this is a solution? 21.5. Verify that Stje-&l--sNdk 21.6. Verify that e-rT-tsNdz)

) satisfits the Black-scholes equation. satisses the Blck-scholes

equation.

21.7. .'Use the answers to the previous two problems to verify that the Black-scholes formula, equation (12.1), satisses the Blak-scholes equation. Verify that the boundary condition V(5'(F), T) maxlo, ST) A') is satished. -

21.8. Consider Joe and Sarah's bet in Examples 21.2 and 21

.3.

a. In this bet, note that $106.184is the forward price. A betpaying $1 if the share price is above the fomard price is worth less than a bet paying $1 if the share price is below the forward price. Why? b. Suppose the bet were to be denominated in cash. If we want the bet to pay if S > Jr, what would have to be in order to male the bet fair? .v

ISFor a put option, the solution does not entail replacing eve:'y occurrence of r with r + 2.. The reason is that the PDE for the put option is different from the PDE for the call option in the case of ajump.

2(?

.

.j.

-

(7 z

=

ln Chapter 22, we extend the Black-scholes 24 to studying interest rates.

-47

c. Now suppose that we pay one share if S in this case to make the bet fair?

>

x. What would x have to be

21.9. Consider again the bet in Example 21.3. Suppose the bet is S $106.184if the price is above $106.184,and $106.184 S if the price is below $ 106.184. What is the value of this bet to each party? Why? -

/7(0, tj rather than 0 in the case of a call.

=

Ke -rF

What is the

PDE for -

tr a

equation.

-

700

%.THE

EQUATION

BLACK-SCHOLES

21.10. Suppose that a derivative claim makes continuous that the Black-scholes equation becomes 1 2 2 kJZ. P) +

(r

c S V.s. +

)

-

payments at the rate

r.

Show

/1

lM

+

?

-

to

11.11. What is the value of a claim paying QTj1ST4 Proposition 20.4.

Check your answer using

21.12. What is the value of a claim paying Q(F)-l ST) Propositipn 20.4.

Check your answer using

(P(F)

21.14. An agrieulmral producer wishes to insure the value of a crop. Let Q represent the quantity of production in bushels and S the price of a bushel. The insurance payoff $50. is therefore QT) x J/g5'(F), FJ, where F is th price of a put with K Wh at is the cost of insurance? =

1 2

?1

11

o-ic'ypf,y Si Sj Vsh i = l j= l

Consider a elaim for which the payoff depends on the where ai

ijdt

-

11

asset prices,

u1

,

%,

.

.

.

,

Sn,

,

.

.

??

.

% t, F) be the assets, and bonds, ,

,

OF PROPOSITION

21.1

-

=

-

t

2

By the multivariate Black-scholes equation described in Appendix 21.A the claim Yz, Q, tj must satisfy ry (2) Pl% + 1 + o'l :2):: + gpcccvoyscj jv''s + +

(r

(r

-

-

2

Vlslyss

=

Q

Ac-(r-J-)l QbF where A is determined by boundary conditions, Guess the solution 1' J* is to be determined, and lJ; satisfes the same bounda.ry condition as F. Compute the derivatives of this guess and substitme them into equation (21.44).After simplification (in particular, the (h'multiplying evel'y tenn divides out), this yields

(*

-

r + la(?' -

+ 1ty2Qbb

t5(2)

+

-

1

1)

-

2

(J'JS+

p'

-

(J

2 bpo'c (2))5'W- + 1aca S J,I/'.u) = r

-

Using the multivariate version of lt's Lernma (Proposition 20.2, Multivariate lt's Lemma), the change in the value of the portfolio is dl

=

$dt +

1 Z-sdSi + 2

?1

11

11

dSi dSj

-

f=l

f=l

j=3

/-y

sj

:/1

Ni t5'f + d J,I/

+ i=1

:% Hold bonds to snancethe residual such that In order to delta-hedge 7, set Ni 1 = 0. The same analysis used to derive equatioh (21.11)leads to the following PDE =

(2l

.46)

F The tenu in braces is the same as equation (21.45),except that is replped with 77 = W' is the same as P' except that $ is replaced by ?2. With this - bpao'c. replacement, fromequation (21.45),thetermin parentheses equals rF. Equation (21.46)

becomes -

1'

+ br

-

8Q) + 1(y2 bb 2 Q

-

1) +

1-J!/' =

-

.

I,y

)o'Lbb

?'

1). Thus, with the 32 and $a) 1' bl' This equation is satislied if The parameter z4 J* in Proposition 21.1, the candidate solution solves equation (21.z14). er-*s? = For a European option, set is set so, at the point the option is exercised, X c(r-J*)r = condition. boundary ternainal solve the to yt 'h

=

APPENDIX 21.C: SOLUTIONS PROBABILITIES

Ni Si + F

+ = l

tt

(21.44)

.

-

-

-

-

11

/

=

?- IZ

J.n this section we will verify the solution in Proposition 21.1. We begin by assuming that we have a derivative plice IZ4,5', t7', ?', F t, J), that satisses ?./' l5'J/y + 1c25'2 J/'SS J/ + (r (21.45)

t*

(21.43)

+ o'idzi

F(5'I Sz, The pairwise con-elation between Si and Sj is pq. Let the claim, of consisting the value of this claim. Consider a portfolio J'P, such that

I

=

'rhus,

APPENDIX 21.A: MULTIVARIATE ANALYSIS BLACK-SCHOLES

St. =

-

=

K?

-

(Note that this payoff can be negative.) Should you accept the offer?

dSi

J'k +

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21.1.3. You are offered the opportunity to

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where dz is dened in Table 22.2. Equation (22.3),multiplied by the strike price, #, is the second term in the Black-scholes formula for a call option. If you were to be paid if S > K, you could value this as cash-or-nothing options'. -'r

To see how the naming scheme works, considerthe cash-or-nothing option, aclaim thing option pays the holder that we introduced in Chapter 21. One kind of cas h conditin under wlzich it pays off, The K. stock price than is greater F if time the $1 at Sp > K, is like that for arl ordinal'y call option, but it is not an ordinaz.y call because it call'' cashcall), pays $1 instead of S. ff.We will identify an option like this as r. t cond all-when i Sp > k. the cagh under c on as a t that same i.e., a contract pays Some options make payments only if multiple events occur. Forexample, consider Sz has not been $1 only if Sv > K and the barrier H > tcash'' a cash-or-nothing call thatkpayst:cash because call'' and this refer out will We to casluocallll up hit. as a rises to the stock price if the does and out'' because not occur payment it pays $1, ttcall'' because payment requires Sp > K. Similady we will use the terms barrier, and options to refer to options that pay off that pay off in shares and refer to ttasset'' to only when Sp < K. To simplify the formulas in this chapter, we will ude the notation -or-no

lash

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You could also have a seculity that pays $1 if S is le-s ///tw? K. This is equivalent to a security that pays $1, less a security that pays $1 if Sv is greater than K. Such an option is called a cash-or-nothing put. The value is Cashputtk,

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in Table 22.2.

/ xamp Ie 22 1 Suppose S $40, K $40,o' 0.3, ?' 0.08, F t 0.25, and =.a0. The value of a claim that pays $1 if S > K in 3 months is $0.5129,computed using equation (22.3). The value of a claim that pays $1 if S < K is $0.4673,using equation (22.4).The combined value of the two claims is ,-0.08x0.25 $4).9802. k. .

=

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Cash-or-Nothing Options Recall from Chapter 18 that the risk-neutrftl probability that Sp > K is given by Ndz) lisk-neutral from the Black-scholes fonnula. We ltnow f'rom Chapter 21 that discounted probabilities are prices of derivatives. Thus, the pricefora cash-onnothing call-which

We lnow that equations

(22.3)and (22.4)are correct since, as discussed in Chapter

21, both formulas satisfy the Black-scholes equation (equation(21.11)) artd the appropriate boundary conditions.

706

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ALL-OR-NOTHCNG

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An asset-or-nothing call is an option that gives the owner a unit of the underlying asset if the asset plice exceeds a certain level and zero otherwise. As discussed in Chapter 21, Propostion 21.1, the price of an asset-or-nothing call is obtained from the price of a cash-or-nothing by replacing the dividend yield, &,in the cash-or-nothing fonnula with tr2, and multiplying the result by the forward plice for the stock. The result is

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OpTlohls:

11

ALL-OR-NOTH

BARRIER

22.2 ALL-OR-NOTHING

OPTIONS

Banier options, introduced in Chapter 14, are options in which the option comes into ban4er.z There are or goes out of existence if the price of the underlying asset hits a stock pl-ice hits a barrier price if the down-and-out options, which become worthless up-and-in options. down-and-in, and below the initial stock plice, as well as up-and-out,

/M?'?'cr options. We can construct options such as these using all-ol'mothg > paying Sp K, cash-or-nothing call but modify it by $1 if Suppose we take a expiration requirement will if the stock has it only that pay $1 at adding the additional < .S'(0), this is a If the option. H the life sometime during of also hit the barrier H this be Table 22.1, would notation the in - down-and-in cash call. Using a CasI2Dlcall. digital options, options ordinary from Just as we were able to construct we will also be barrier options. options from'digital able to constnlct barrier We will examine three different kinds of barlier options:

o A contract that pays reached

on either a barrier having or not having been barrier optiollsj.

$1 contingent

cash-or-llothilg

o A contract that pays a share of stock worth S contingent on either a barrier having l75l'?-Ik?' optlls). or not having been reached asset-ormothq e A contract that pays $1 at the time a barrier is reached rebate optiolls) or that pays $1 at expiration as long as the barrier has been reached during the life of the option options). ' deferl'ed ?z'Jnd#c By valuing these pieces and adding them together we can price any standard barrier option. The assumption that the stock follows geometric Brownian motion malej it possible to delive relatively simple formulas for tese options. There are 16 basic linds of all-or-nothing barrier options. First, consider cash-ornothingbanieroptions thatpay $1 atexpiration. Such options canknock-in orltnock-out; they can be calls (pay cash if Sv > K4 or puts (pay cash if Sv < A-); and the barrier event can occur if the barrier is above the price (up-and-insor up-and-outs) or below the plice (down-and-insor dosyn-and-outs). This gives us 23 8 basic cash-or-nothing barrier options to value.. By the same reasoning there are also 8 basic asset-or-nothing barrier options, for a total of 16 all-or-nothing barrier options. =

Cash-or-Nothing Barrier Options options. To anticipate the We srst consider the valuation of barrier cash-or-nothing value how will section, in this first to we results one particulltr barrier cash-orsee formula this call. From down-and-in cash option, we will be able to value one nothing a cash-or-nothing deferred rebate options. options and remaining the seven

(1991a),Rubinstein

%.

711

Assume that the option is issued at time 0 and expires at time F. Let t denote the greatest stock price between times 0 and t (wheret S F) and let j denote the lowest stock price between times 0 and 1. Suppose the barrier is below the initial stock price, i.e., H t < &. A cash down-and-in call casllDlcalll is an option that pays $1 if two conditlons are satished. First, at some point prior to maturity, the stock price drops to reach S, i.e., a.w (! S. Second, at expiration, the stock price is greater than the strike plice, K. probability that Wecan analyze this option by frst examining the lisk-neutral this joint event (a.wS H and S.r 2: K) occurs. This probability should satisfy three conditions:

1. Once the barrier has been hit Sv

K

(the barrier at this pint

2. If at time F, 3. If at time F,

j.v .w

jh

:s

S) the probability eqals

:j

H and Sv A: K, the probability equals 1.

>

H or Sv




K)

=

S

IT

-

-

l

Nd4l

(22.6)

The terms #1 through ds are defned in Table 22.2. In Appendix 21.C, which is found we saw that an expression of this on the book's Web site (www.aw-bc.com/mcdopald), fonu solves the backward equation. We also want to verify that it satisses the three boundary conditions described above. First, at the point where St = S, equation (22.6)collapses to Ndz), which is the risk-neutral probability that Sv > K. (rfhisoccurs because when S = St, #1 = ds. You should exnmine equation (22.6)to verify that this happens.) Thus, once we hit the bnrrier, the barrier value H drops out of the expressiotl because it is irrelevant. Second, if at expiration j.w :G S and Sv > K, ten equation (22.6)equals 1. Yhereason is that the probability equals 1(46) once the barrier is hit, and if Sp > K, N(#a) = 1. Finally, if lw> H, i.e., St never reaches S, then at expiration Hl < SpK (recallthat S S K) and equation (22.6)collapses to 0. Thus, equation (22.6)both satisfes the backward equation and obeys the appropriate boundary conditions. Equation (22.6) assumes tat H S K. Why is this important? The answer is that if H > K, the boundary conditions may be violated. Consider the case where at expiration $55, K = $45, and H = $54 (thus violating the condition H S K), and the boundal'y has not been hit. In this case a correct expression for the probability 111(542/45 x 55) = 0.164, will evaluate to zero at expiration. However, 3nHljSvK) equation mattlrity will equal when the 1 has event not occurred. (22.6).at so equation (22.6), you might ask why it is necessary to As a snalcomment on (S/5')2(r-O/J2-l The answer is simply that #(#4) by itself mult.ip 1y Ndz) by the tenu does not solve the backward equation, whereas equation (22.6)does solve the backward kw

=

=

.

z-fhree comprehensive discussions of barrier options are Rubinstein and Reiner and Reiner ( I991b), and Derman and Kani ( 1993).

I NG BARRI ER Op-rloNs

equation.

,

k. Exo-rlc

712

To handle the case where H (22.6). When H > K, we have Probtiw S S and Sv

K4

>

version of equation

K we need a more complicated

>

X(#2)

=

-

Nj

s

+

zp

1(#8)

y

c, r, F

e-r(r-?) d

,

t, $, H)

-

jx

-r(F-J)

=

1 (#,)j tj )2V x

s s

K

y

K

(22.8)

-

N Ldoj+

Ldz) -

>

Equation (22.8)gives us the value foracash down-and-in call when uo > S. There three elosely related options we can now price: Cash dnwn-and-out calls casllDoare Call), cash down-and-in puts casllDlplttl, and cash down-and-out puts caslzDopkt. We can value each of these using only the formulafor the cash down-and-in call, equation (22.8). ln addition, we can value a defen'ed down rebate option. We first value a deferred down rebate, which is aclaim the barrier has been hit over the life of the option. The time F as long as that pays $1 at depend claim this does not payoff to on a strike price: It pays $1 as long as the banier rebate'' claim hit. We will call this has been a deferred do&vnrebate. lt is a because and if reach the barrier, it is the payment is at becaus it pays $1 we vtlue obtain of this claim barrier. the the We time reach than rather the expiration at we > always Sv result is a equation Since have the setting K in 0, by $0 (22.8). we have3 Thus, Sv S. that F long :j claim we as pays $1 at as Deferred

down rebate

ty,

r, F

K, o', r, T

cash put lf you buy a down-and-ip cash and Sv < K. If you buy reached bnrrier is the receive $1if reached and Sv : K. Thus, if bnrrier is receive $1 if the long receive as the barrier is hit. $1 as call and put, you have thus, defen'ed rebate; we Down-and-in

CashDlputts',

option

K, c.

r,

T

-

t,

S)

(,

=

DlkDeferredtu,

t,

-

r,

-

(5,

t5)

(22.10)

H)

put with strike price K, you a down-and-in cash call, you you buy both a down-and-in This is the same payoff as a c

,

-Casl1DICa1l(5', K, a,

?-,

F

?-,

T

r, 8, H)

-

(22.11)

t, (T,S)

-

Buying down-and-in and down-and-out eash puts creates an ordinary cash put. Thus, the value of the down-and-out put is Down-and-out

(.t)2S-l NLdj s

Cashcallts', K,

=

(22.7)

Equations (22.6)and (22.7)give us expressions for the probability that the barrier is hit and Sv > K. What is the value of a claim that pays $ 1 when this event occurs? To answer this question we can use the result from Chapter 21 that discounted risk-neutral probilities are prices of derivative claims. Discounting equations (22.6)and (22.7),we have

#,

H)

,

-CashD1Call(5',

cash call

CashDlcalltu,

t,

-

j

-

Problem 22.3 asks you to verify that this equation satisfies the boundary conditions. = Ndbj', the formula again reduces to Ndz). Note that when S = H, N) Down-and-in

#, o', r, F

713

a down-and-out cash call is

pay $1 if S,;m> K. Thus, the value of CashDocallt5',

opa-loxs k.

BARRIER

Auu-oa-rqo-rlqlxc

11

OpTloxs:

Casloputtq,

cash put

K,

o', r, T

t,

-

,

H)

=

Cashputtu, K, o',

K, c', r, T

Casllllllptltti,

-

r, T

-

/,

-

t,

(!)

(22.12)

J, Hj

As a linal point we can compute the risk-neutral probability that we reach the banier. The deferred down rebate option pays $1 at expiration as long as the barrier is hit. Thus the price of this option is the present value of the risk-neutral probability that the barrier is reached. Therefore, e

rtf-llDlkDeferredtu,

0, o., r, F

is the risk-neutral probability that the bnrrier is

-

t,

(22.13)

()

during the life of the option.

reached

ftdown

Edeferred''

=

DltDeferredts',

c, r, F

-

t,

(!,

S)

=

CashD1Call(S, 0, o', r, T

-

t, 8, Hj

(22.9)

0, the value of the deferred down rebate does not depend on Note that since we set K strike price. the Now we can compute the value of the remaining three options.

Suppose S $40, c 0.3, 1, 0.08, Example 22.4 value of a cltim that pays $1 if the stock hits the barrier H computed by setting K $0 in equation (22.8):

We can create a synthetic cash call by buying down-and-in cash calls with the same banier; this combination is guaranteed to

cash call

and down-ad-out

3In peforming this calculation, to avoid a zero-divide error it is necessafy to set K to be small, such as ' K = $0.00000 I rather than exactly $0. ,

=

t = 1. The 0, and F the next year is $35 over -

=

CashD1Ca1l($40,$0.0000001,0.3, 0.08, 1, 0, $35) -r(T'-/)

=

Down-and-out

=

=

=

=

)

d

-

NLdoj+

S

=

2(r-J)/c2-

I

N Lds)

-'

=

$:.5,74

probability that the stock will hit the barrier is the undiscounted value e0'08 = 0.622. of this claim, or 0.74 x 'rhe value of a daim that pays $1 if the stock hits the barrier, $35, and then is also greater than K = $35 at the end of the year is

The risk-neual

e

sr(F-/)

S -

2(r-t)/o'2-

1

NLd4) $0.309 =

714

%.ExoTlc

ALL-O R-NOTH

11

OpTloIqs:

tk This is the value of CashDICall($40, $35, 0.3, 0.08, 1, 0, $35). The risk-neutral proba() (s rli) 0.335. x. r) bility of hitting the barrier and being above $35 is 0.309 x e jk

'

=

This example illustrates an interesting point. The value of the claim that pays $1 at expiration when the stock at expiration is greater than $35 and has hit the $35 barrier ($0.309), is approximately one-half the value of the claim that pays $ l at expiration as long as the stock has hit the $35 barrier ($0.574).The reason is that once the stock has hit $35,it subsequently has about a 50% chance of being above orbelow that value. This observation suggests that the probability of being above $35 conditional upon having hit $35 is 0.5 x 0.622 = 0.31 1. The actual probability is greater than that, however. The 0.5&2 = 0.035, which is positive. Thus, after reason is that the lognormal drift is 1' dfifts higher. stock the having hit $35, on average verify this To intuition. suppose we set the lognormal drift equal to zero. We can setting the risk-free rate to 0.045, which gives us 1' 0.5&2 = 0.045 0.5x do this by 0.32 = 0. We might expect that the value of a claim paying $1 at F if the balnier is hit jtti price at is one-half the value of a claim paying $1 at F if the barrier is hit and th = 0.5c2, probability 1* the differently, When barrier. Put than the expiration is greater of hitting and ending up above $35 is half the unconditional probability of hitting $35. The next example shows that this intuition works. -

0.3, ?' = 0.0X5, = 0, and r t = 1. The Suppose S Example 22.5 $40, = $35 over the next year is value of a claim paying $1 if the stock hits the banier S (y

=

t?-r(F-J)

j

-

2(r-J)/o'2-

H

NLdo)+

I

NLd)

T

=

.

Given equation (22.14),the procedure for obtaining the prices up rebate the cash-or-nothing three options when H > Sv is analogous to that before. of other First, by setting K = cxl in equation (22.14),we obtain the price of a claim paying $1 at expiration as long as the banier is reached..'f Deferred

UlkDeferredtu, o',

r,

r

t, J, S)

-

Cashulputtu, x,

=

c, r, F

t, &,H)

-

(22.15)

With this equation, we can solve for the price of the other cash-or-notlzing options. Up-and-out

Buying up-and-in and up-and-out cash puts gives an ordinary

cash put

cash put; hence, Cashuoputts', K,

o', r, F

-

t,

S)

t,

=

Cashputts',

- Cashulputts', Up-and-in

K, c, r, F

K,

(r,

r, F

r, )

-

/,

-

S)

,

(22.16)

Buying an up-and-in cash call and an up-and-in cash put yields

cash call

the same payoff as a deferred up rebate. Thus, we have

$0.6274

Cashulcallts',

K, c, r, F

t, J, S)

-

,0.045

The corresponding risk-neutral probability is x 0.6274 = 0.6562. The value of a claim paying $1 if the stock hits the barrier and is then greater than K $35 at the end of the year is =

This is one-half of 0.3 137 = 0.328 1

715

-

-

=

Q

If you compare this formula to equation (22.8),you will see that Ndz) is replaced with #(-#2), Ndz) with N-d4), and so forth. We know from Appendix 2l.C that these terms also solve the Black-scholes equation. The effect of these changes is to reverse the effect of the di terms. As a consequence, equation (22.8),which prices a down-and-in cash call, is transformed into an equation pricing an up-and-in cash put. Problem 22.4 asks you to verify that equation (22.14)solves the appropriate boundary conditions for an llp-an d i n CaS h P ut

-

-

I N G BARRI ER OPTI ONS

$0.6274. The corresponding risk-neutral probability is

,0.045

=

UlkDeferredts', o', r, T

- Cashulputts', Umand-out

cash call

cash call; hence, Cashuocallt,.

K, c, r,

-

r

-

t, J, H) t, 8, H)

(22.17)

Buying up-and-in and up-and-out cash calls gives arl ordinary

K, o-,

F

?-,

-

t,

,

H)

=

Cashcallts',

- CashUICall(5', K,

x

K, o-, r, F tr, r,

T

-

-

t,

t, J) ,

S)

(22.18)

.

Asset-or-Nothing Barrier Options

cash put Now we consider cash-or-nothing options when the barrier is stck price. First, consider the following formula for an up-and-in the above current which when Sv > H and Sv < K.' cash put, pays $1 Up-and-ln

Cashulputts', K,

c, r, F

-

r, $, H)

=

(#.)2S-l x(-#4)

e-r(r-?) s d

-r(w-J)

jx(-#a) -

s-ds)

H yy K +

x(-#s)j t.?slzr;-i-l

u


S is

-->.

22.3 BARRIER

OPTIONS

At this point it is easy to construct the barrier option formulas from Chapter 14 using the preceding formulas. A down-and-out call, for example, can be valued as CallDownoutts',

K,

(r,

r, T

t, J, S)

-

ty,

r, F

t5,

t,

-

111

H

S)

=

S

a

gyg

X(Zl)+

-

-

S

X(Z2)

=

LnH/S)

Zz

=

qn(S/5')

1-

9

1

?-

-

hz

=

G'

-

')

-

')-

+

tjj/w

T

tjj/w

T

?-

-

2

2

&

-

t.o'

(22.20)

1 2 -

-

1 2

-

-

o.

-.

o A rebate call, which pays the $20 when the stock hits $120prior to * Aknock-out call with a strike of $100,which knocks out at $120.

t

-

-

2

')

+

+

--,r

o,-

2?-

(,.2

-,

-.>.

=

0, d'ssil

.%T c'

t)

r

-

t

'''$

w

k!ll'

-,%T

?)

-

:::::::

'''''''''''

o'

F

-

Thus, the formula evaluates to 1 when the banier is hit.

t

S)

t,

t,

S)

11.'

expiration.

If the stock reaches $120 prior to expiration, the rebate is triggered and the call knocks out. lf the stock has not hit $120 prior to expiration but is above $100, the

2 1-

This formula satisfies (as it must) both the Black-scholes equation and the boundary conditions for a rebate option. Suppose that 4he barrier is not hit over the life of the F. At the point when the barrier option. Then H < S and both terms go to 0 as t = distribution is symmetric around normal the Because ln(S/& S and 0. H is hit, -

(,

-

e)

-

')

2

t

-

o'l

1

ts

o,

1-

+

-

2

-

-

gT

-

-

=

-

+ gT

Zl

j

-

-

c, r, F

t,

down-and-ins, and so forth are all constructed analogously. See Table 22.3 for a listing of formulas for barrier calls and puts. As another example of the use of all-or-nothing options as building blocks, capped options are single options that have the payoff of bull spreads, excepy that the option is exercised the first time the stok price reaches the upper strike price. An example of an American capped option is an option with a strike plice of $100 and a cap of $120. When the stock hits $120 the option pays $20. If the option expires withut the stock 100, 0). This option can be priced as the having hit $120,then the payoff is maxts'w sum of the following two options:

2

1 l.

=

-

-

Up-ad-outs,

where, letting g

K, o', r, F

- K x CashDOCall(5', K,

.

DR(5',

AssetDocalltu,

=

;''''''.'''. -'j.;r' -,' r' ''''''. j' (' t'''k'''rd -.--'.. 15)4,* ('' ,;' y' j'(' t';y---' ).1* q' ' jd yrd y' j'q!jjtqjd (' )'

.'''

.'..' ...

t''f' q'-'tf ,'---LL;' )'. y' tt)f t'-. r' ..';:.' ii2iL:-,k-.' )-' r;d i'. (jjjy'r ''t''l'; t't-' '-' qlkihitslik--.'

t'tf ''.' (i( j:k(qik((.y ' ('.' ;)(.''.. .('. k. g.( ;( 1)* ytj(y7yty-y.--;yyj-yyy..y-y-.-.y--y..y.y.y.-..--j.g.-..-:-...;yt.:q.--,)...-.t-'yy.yj.yy.yyj..y qq---ttyty yy-.j...yr'..-jjjjjg(.:r ',)-y-.jjjjjjjy)). li).'t.i ) yyyy.. r..-)-:-; t-:i2kpiiir'7.. .', r iiIItIiI...-. Ii.IIiEii:. ; !. i.l'i tlr:t!!;i tl:::!!;: t '. .(..: Ir..I:::,. ,r,. jr..,r..,j jy::y I..ll lr..,r..lk Ir..ll Ir..llea I..1.as Ik..,r pr.,,r 14 11 1/..,,a Ir,.!g((2I .It:. . )..;r k.... ... a Ir..Ir..i; Ir..Ir::,p a Ir..ll iiil.11 a ,!!;.jg:-,t:::,r ; r; s IL . i . ', '

.

;EE:. E

-!

.

-

i .

.

.

'

'

EEj'j ((i

i. -

(

.

E''' ' (' ( (j (g i.'.'( . .,:.-k: k.- -

-.'

. -

..... ... .. .

E

' ' ': ' g. .. .

jE :

;

-'.

--

...

E EE

j j .(jgi('g !(E : g. . .:

. ..

.

:

: : . .. . '

'

'

-

. .

.

'

:

.

.

....

. . .. . .

. . : .

'

.

7)4 ' :

'

. . . .. .. . .

.

-

.

..

.

:.. --

--.--

(- -.

- -.. .. .

-- .

-...

.

. . . ..

.

.

-

..

Up-and-ln

Up-and-out

Down-and-ln

Down-and-out

'.l

'

:

.

'

'

.

.

.

.

y...((. ' y.'(.E......E.... ' . '.E.''.' '.... ... . . . . ..

. .

'. -

'

.

:

. . . .. .

.

..

.

.

: ' . '. . . . . . . . ...

:'

.

.

.. ..

Assetulcall

Assetuocall AssetDlcall

AssetDocall

'

:

.

. :

j'. . ..

)36'.i.i.'3. 411::1k,. '''.'''''''''.

' ' - . . . .. . ..... ....

-

-

-

-

.pIj! ' .. .(E.. .. E'. ... . . .1111 . .' . .. EE. . '' . E. E

l ... ..... ..

''Ik''''''.''''k'. .

-

. . . .. . .

.

... ... .

-

.

.

'::::.

..

for the functions are (.S,#, o', r, T

(.. E (.. .' .... (.' E... .. ..E...: .(. j)...:. (..(q(. . . E .. ' . . .. .. . '. . ' ' .' ' . : E . . E. E . . . . ..

: . . . ': . : . . . . . . . g . . ' ' . ' E E ' ' . E. E . .. . . . : . ' .. . . .. . : . . . .... . . . . .. . . . .. . . .

d:::jll ,::::

d!!,r

j:gy,j

-.

' . . ( . ..; (. y(. E. .. . (. .. . . ... .. . .'.. . . ( ; ' EE E. .. ' ....'. ... . '' ........ . .. . . . . . ...

' ' 'E ' ' : . . . . :

(E E' ' j'q' q ''r(7(''EqF'qqt(Fq' . E((E. j(.E(j( E;k .

: ' ' ':'' E' E( E.. .g'(-.,-3.3,,,,.,..-3.)3333.. ..''. .' (.

'rljjyyyt-tjjjjjj.

-

-

t,

dt:::

. .. ' (. . .. .. . . ' . . ( E.. .q. . .. :. . .E . : .. .. .. .. .;. . .. E. ... : . . .E ... E... . q ' .. ..:. .. . ((. ' . . . . .. . . .. . . . . . . . '' .. . ': . '..' '... E . ' E' ' ...... . .. ..' . .- . . ' E. '' .E . .' .. . ; ... : . ... . ... . : . . . E E ..... : .... ..'.'E. ...EE .......'.. E.E- .i. .. .. . E E. E... .. : : i' .: . .. . ..

.. .. . : . . .' ' ' E ' 'E - . . . : . .. : ... .. . . . . . . . . . . . . . .. . ..

'.. ''. ....

'

.

'.'' ''l ......' .

'

. .

:

.:,11rt::111:

' .. .

.

-;. . .

.

K x Cashlxcall

K x Caslrtput K x CashDoput

. .

. .

.

.

.

.

.

1.1.''.1...- ' ' ' . :: . . - . .

.

K x Cashtrut K x Cashuoput

'

'

:

:

K x Cashulcall K x Cashuocall K x CashDocall

d:i,r

--.

d!!!l

' rjjr... (. (.. (1j,.11,2. . E EE. E( ':'. (. l l '. . . . ; .. . . .... . .

-

dli!,h

J, 1h.

. . . ..

.

E .

: . .. .

. . : . . . . j(. . . j ..... . . .. ... . .. . . .. .E . .. . E ' y. . . -. . ' q.. . (.;.. ... (:.. EE'. '..'.-. . !'...'.'. .. ... . .. :

. ;..' .

.

. . - .

'

'

'

..

i...(. ......

.

.

.

.

. '

-

.

.

.

-

Assetufput

-

Assetuoput

-

-

'. ' .. .

AssetDlput AssetDoput

'

.

k. Exo-rlc

718

QuAxvos

11

OpTloNs:

Nikkei by directly investing in the cash Nikkei index or by investing in yen-denominated futures, such as a Nikkei futures contract trading in Japan. Both strategies have a payoff denominated in yen. lf the Nikkei appreciates but the yen depreciates, the f//l/srt??' call lose ???tw?el' despite being correct abtt #?' ntovelllel't ofthe Nikkei. You could try to reduce the problem of exchange rate risk by hedging the Nilkei investment using currency futures. However, the quantity of yen to be exchanged is high when the index has a high return and 1ow when te index has a 1ow return. Thus, there This price uncertainty creates is ?2t? wJy to 177/:$7i'?7advance /251:/many )?e??to quantity uncertainty with respect to the yen exposure. We could imagine a synthetic Nikkei investment in which the quantity of currency fomards depended upon the Nilkei's yen rettlrn. Such a contract would permit an investor in one currency to hold an asset denominated in another currency, without exchange rate lisk. This contract is called an equity-linked forward, or quanto. For reasons that will become clear belom a quanto is also sometimes defined as a derivative having a payoff that depends on the product or ratio of two prices. The Nikkei 225 index futures contract, traded at the Chicago Mercantile Exchange discussed in Chapter 5, is an example of a quanto contract. This futures contract is and marked-to-market daily in dollars, even though it settles based on a yen-denominated price.s There is also a yen-denominated Nikkei futures contract that trades in Osaka. Both f'umres are based on the Nikkei 225 contract but they differ in currency of denomination. We will see in this section how their plicing differs. The box on page 728 discusses Nikkei put warrants, which were another example of a quanto contract. Table 22.4 lists the symbols and specific numbers used throughout the examples in this section.

knock-out call pays S $100. The table below illustrates the payoffs, assuming that the option stlike is K, the cap is S, and the option expires at F: -

H Hit

H Not llit

Purchased Knock-out

0

m?lxto, Sp

Rebate

H

Total

H

-

-

-

K at Hit

0

K at Hit

maxto, Sv

-

K)

.5-/70?-1.

Kj

The table shows that we can price the American capped option above as a straight applieation of the rebate formula together with a lcnock-out. The hplder owns a 100strike call with a knock-out of $120,and a rebate call with a $20rebate fayablet $120. Note that a European capped option is much simpler to price. Since the payoff does not this option is just an ordinary vertical spread (buy a loo-strike occur until expirtion, call and sell a lzo-strike European call with the same times to expiration).

t' ) Example 22.6 t)

.'

. . yjythat S

=

$100,ty

)') , an up-an d -Ou

=

Consider the capped call discussed in the text above and suppose 0. We can compute the price of 1, and 0.3, 1' 0.08, F t =

t call as

lt) jty )i'i'( Callupoutts', K, y.'). , t)

.. .. . .

c',

?-,

F

-

t, J, f)

=

! :

j)E

Assetuocalltu, K x

li.

lE. j1iI i'l'E

=

=

-

K,

o-, r, F

cashuocallt.

-

t,

K c r F

,

S)

-

H)

t .

''C'' :';' q' 77* :'' 't' );' i t)E );'q!fq(E' $' )1* (' q'' )' r' .'y' (r'!q))i)' '(' ! (i Eii.;'ti(Eql(E' t'qf '')'

'

)

tt: E

..,....ttbi?bbi.. .-

-

$100,0.3, 0.08, 1, 0, $120) callupouttsloo,

20 x $0.5649+ $0.4298 .) l.q.l) The price of a European bull spread for the same parameters would be j

=

=

$11.73

.

.

)i.l )t) BSCall($100, $100,0.j,

jj

ljtj ttij

) The capped .t. jlt tjt and end

.

'

0.08, 1, 0)

-

BSCall($100, $120, . 0.3, 0.08, 1. 0) =

$15.7113 $7.8966 $7.8147 -

=

. call is more expensive because all of the stock price paths that cross

$120

up lower result in the maximum payout on the capped call but a lower payout 17lil, spread. bull the i % on

' : ''

E: . .

..j.;.

LL(?.1.f()Lj36-

-

.jEE.

'''

'

!

.:

(EE.

.

@

('.'(. ( (':7: . .' (' ' ;.;; ;(' i(i((;. (.) (. k;i (.') ';)((( ;.jL.....'..; .y;y.yy.y..(y..y...y-yyj.'-'.y. .. . ltiqr). yytyy-y yyyt.'y.yyj(.yj..,y....y.j.: j-y .tyj IjI--l. .jj....).j...y yjtjllrriiprilllytgfitfitjlyilp lllilili y jr..jr..jj.,4g, jj..jr kyyjj jr.j kjjjs jr.,tjy;r tjyy lq.ti:i1i!:.. iliihl k.,yyyry......-y?j...q.......;q.... yj j y .yy ( jy:;j. iltj.li .(.##!r..:.,-, j-:.,.;;.jjL)..j;,.j.. gjgjjgr;. .

The plice of the capped call is 20 x UR($100, 0.3, 0.08, 1, 0, $120) +

k

..

EE: : E :)E !''E''.:''y.-..E.-jjyj..j'y.y.yy'..yyyqj...E.-.-yj . iE i:.'':' ' E': E'':. .-..)..(.()y...kE.Ey.g..)t.ty:.yy-.....'y,.y.yy''y(jjyyy.yjy'.y''y'y). E' : r.': ' E' ' :' E'' :''' '' E.i.E ( :(i. .: ( .. E .E'.... : y .. y yjy.syvys,yysssss-y( y yj y .,...y.... .y .. 'q!q'q'7('7;':

.

-y.(

'

:

'

.

.

,jkjjjj;.

-

...'

Dollar-denominated

A U.S. investor wishing to invest in a foreign stock index can purchase the foreign index ciirect1y or hold futures based on that index. However, the investor then bears two risks: The Iisk of the foreign index and currency (exchangerate) risk. For example, suppose that a U.S. investor wishes to invest in the Nikkei 225 index, expecting that it will increase over the next month. The investor can tale aposition in the

jtjj,r jkyj,rjyjjjjtj jjj;y jjj;y jj jy jjjr..;j jj..;j

j:ygjj jzyj!r

yjjj.tyjjj,,

.jjy

interest rate

.jyg,

,,

0.08 rg

0.04

Current Nikkei index

Ql

720,000

Nikkei dividend yield

$c

Nikkei volatility (Y)

(a'

0.02 0.15

Current exchange rate

($/Y)

jj jyyg;j

jyygyy jr..jjjgygyj jgjgjs jr..yjjrgy;jjjkygyg jr..jy..jj jyjryy yj,.jy k;,jgy kyj;j

Yen-denominated interest rate

-p

c

0.t100

Exchange rate volatility Nikkei-exchange Timp to expiration

22.4 QUANTOS

jkyg,l jkyyjr

rate ($JY) correlation

p F

1 year

5To illustrate dollar settlement, suppose the Nikkei 225 is at 22,000. Under the terms of the CME contract, a one point move corresponds to $5, so the notional value of one contract is 22.000 x $5 = $1 10,000. If instead the Nikke had been 22. 100, the notional value of the contract would be $1 10,500, a difference of $500.

%.Exo-rlc

720

itf t'. jji;L3j3L'. .'t'y' -Lj:;iL3t'.

Op'rloNs:

QuAxTos

11

.'.') yg'i.l. !''';'r' -y'.

: E:. . : : . . :

-

' .

'

'

.

.

. -.

'

E ;

-. -

:

: : .: . . . .

' : i. :

:. '

-

.

.

'

...

' ::

:

:

.

..........

.........

.

.

:

-

.

. ..

E

.

:

E

.

.

-

.

.

.

.

-..

.

..

-. . . . .

, ..

-

''

E' .

'E .

-

. .

--

. .................

:' ' '

;. .

k illi!!t!tyyty '

E E.

.Binomial .

......;...;...,z

.

E

i

;:.

:

-

.

:

-

-

-

.

:' :

.

y-

-

.

jy...j -

-- -.. . -. .

. . .

.

721

and that for te exchange rate is

4* )' ,'j' :'':' '17''17( ' trjf '!'f' 7* E(E ;'p' E .r' p'. @' )' 4* l'''lqr'' 71* i'iif (' (E(E(()!EE!E(Ei(E(qTE : !pi)i ( ; ( i'''(E.EE (.'.E(iEE F(E.Eji(EE... i'(..'E.E'.'E'...'('E.'.'... (((E 17r'('' jqf j)'r.r:' y' )'kk;., ;'(' jlf (r7l' ((' ($: jtf ''jyd ?' 4* rqf .q . .;.''.;.((''.'('(' . (.i'j y' t-ft-ftl'-yy). )f j'(y' t')'q' tyryyyyf )'. k',j. ..' jjd lllif 1IIiEiqq'. t'. )' r'l'f . ;fEE; .E..'..''. .(.. . y.. E.i. . .j.;.. . yyy.. '.g(r.Ey p. y ( qri E. .. EiE ( (;i(ii 'i'lr:iriik.y. E. E'. . ( (! . ( y.( r )- yg(.. . y. y y .k ..jj.(yyy.jy '(.j!(!ti . -j''Iii-L:LIII!' j qkj. j';.....(. yy y y . yj j y.. ty. jj EEi . .. j j.yj . j.i .. ).tt;jilp;,.. jy. . . ! ---E 1i .Ei.E .). : .. jjjj; ,rt;. y.. .,..qy;.q.... (q q .y '.. .. '. . r Ii'!!:pili: ..... .. .,.,. y.:( ;j--. . r . . ..... . .., . .. . . . . .. . ...' ,'

%

#0,l (1/1)

trees for the dollar and the Nikkei index from the perspective of a yen-based investor. Both are forward trees constructed using the parameters in Table 22.4. The risk-neutral probabilities of up moves are 0.4750 in the dollar tree and 0.4626 in the Nikkei tree.

=

Y100d (0m-0.o8)x l

We can also compute the folavard prices on both tree we have Binomial

Binomial

Tree for the Dollar

Tree for thr Niltkei ,

:23,706.10

$106.184 y2(),o(o

$1(0

Y17,561.91

$86.936

F(),I

'

(1/1)

=

0.4750 x $106.184+

(1

-

v96 (j,79

=

.

ees as expected values. For the dollar 0.4750) x

$86.936 $96.079 =

For the'Nikkei tree we have6 Fa,I

(:)

=

70.4626

x

Y23,706.10 + (1

-

0.4626)

x 117,561.91

=

720,404.03 $

'I'he Dollar Perspective Now we consider yen and Nikkei investments f'rom the perspective of a dollar-based investor. yen fonvard price is given by rfhe

Exchange rate forward ($JV): Fo r(a')

The Yen Petspectiv The yen-based investor is interested in the yn price of $1 and, hence, faces an eyhange rate of 1/-v()= 100V$.Because the Nilckeiindex and the yen price of a dollar are both denominated in yen, we use the usual formulas to find fonvard prices fpr the yen and Nikkei. For the Nikkei, we have Nilkei forward (Y): Fo w(:)

Qzerf-&otl-

=

,

For the exchpge rate, the dollar-denominated forward prie is

(22 22) *

interest rate is the yield on dollars, so the

Exchange rate forward (%$): Fo

--erl'-r

w(1/-Y)

#'

*

x

*

=

1

)w

< . ..

-,'tr (Tl

These will be the forward prices observed in Japan. A yen-based investoi would construct binomial trees for the yen and Nikkei in the usual fashion. Figure 22.3 depicts trees for' the dollar and Nikkei. The nodes on the dollar tree are constructed as (1/-Yz')

=

and on the Nikkei tree as

Ur Example

22.7

=

Quer.-sct'zko'cn

Given the parameters in Table 22.4, the l-year yen-denominated

(:1224) .

However, from the dollar perspective, the forward price of the Nikkei is not so straightAs discussed above, any Nikkei investment entails a combination of currency and dollar-based investor buys e .scv un y(s ojtj)e xj.kkei and holds it for F years. The actual steps in this transaction are as foltows:

index risk. To se why, suppose a

1

Exchange :oa:()e-d(!W the index). e-QT

-vf

doars

into yen

(this is enough dollars to buy

units of

units of the Nikkei index and hold for F periods.

3. Dividends are paid continuously over time and reinvested in the index; after T years e&QT shares. we have an additional 4. After T years sell the index and convert back into dollars. The time-F viue of the investment, denominated in dollars, is F(F)

(1/ab)d(r/.-rlrclaxff

xoer-rfb-

fonvard.

2. Buy

(22.23)

=

=

xvov

The payoff is a combination of yen and Niklcei lisk; we will call this investment the currencptranslated index. Here is a point tat is crtlcial for understanding what follows: From the perspective of a dollar-based nve-ftpl; the dollar-trallslated price ofa asset, Yv, isjust like L-ln)? other #t?//t7r-#v?t7??/f?)5l?# asset. However, yw-#d??t7??2?7t7lc# of is plice dollar-based investor, because there is no simple not for the asset an a Qv lisk. risk of without also bearing obtain the to currency Q way

forward price for the Nilkei index is F0,l (U)

=

1 V20,000/ (0.04-0.()2)xc:Favpozm4 ()3 ,

.

6The calculation shown here uses rounded numbers and therefore does not exactly equal Y20,404.03.

.

722

k

ExoTlc

QuANTos

Op-rlolqs: 11

If you are not convinced that Yv really is like any other dollar-denominated asset, consider the following thought experiment. Suppose you leftrn of a new stock, traded The price on a U.S. exchange, called ttA.s Amel-ican as Apple Pie, lnc.'' IAAAPII. investigate the stock decide stock. You to like other domestic is in dollars, just any American Depositary actually is the surjrise, lenrn company that an and, to your you Receipt (ADR) for the Nilkei index.? The sole asset of this company is shares of the Nilkei, translated into dollars at the current exchange rate, and held in trtlst. The value at any time is i'; = xt Qt and it has a dividend yield of 8 j This ADR trades just like a dollar-denominated stock, because it is a dollar-denommated stock. Had you not investigated, you would never have lnown that you were holding a cun-ency-translated yen-denominated stock. This thought expeliment is important because it tells us that, for a dollar-based investor, the forward price for i' is given by ) .

1') F0.w(

yctr-dt'lr

=

(22 25) .

Since we can trade shares of AAAPI, we can undertake arbitrage if the folavard price is anything other than equation (22.25). Similarly, the prepaid fonvard price on the cun'ency-translated index is Fo# (l') w

=

-

vtlptlc-t?r

(22 26) .

In order to obtain Nikkei risk without currency Fe need to combine the dollarlnmitively, we want to invest position with forward contracts. in Nikkei translated yen a risk. exchange Let V) w(F: a'; ; F) represent hedge and rate 1/ eontracts to in use currency of the Nilkei. The boundary dollar value the maturity, of claim that, price at the pays a security is this for condition

.:

This giveslo

# 1.20(1/1)

Vv,vLYv xv ; ,

r)

Fz -'ltr :!-

1

z)(w-,)

Ab

1 dt

--Ldx

'(#x))(#F

-

-

E (#F))

=

pso.q + s

Lemma, tat

:t

(22.28)

hen'ce, when we compute te forward ln apply ing Proposition 21.1, we have b with $c replace bc price for Q we (-1)(p'c c + -1.,

=

.2).

-

Putting thls all together, using Proposition 21.1 and with J r ,

(

I'J ,

xt - F) ,

1

xt

=

1 we have

e(r-J(?-p.(r:-x2)(F-r)

2)(r-?)

.-erf-r''s

=

.k

Qt x #

(22.29)

r.f-&o-pscrcjlr-)

= :,c The dollar-denominated fprward price for the Nikk'ei index is the snme as the yendenominated prepaid quanto index fomard plice, With a covarimlce correction. rfhe

(22.27)

where is an arbitrary exchange rate determined in advance. The boundary cpndition says that we will receive Yv, the dollar-translated Nikkei, convertit to yen by multiplying (Since the only purpose of by l/-tr, and then convert. back t dollars at the rate nd is arbitrary the value is to convert yen to dollars, we can set it to 1. ln practice Prior rate.) exchange to time F, the parket-maler the contract can call for any fixed -i

-?.

.

7In simplest terms, an ADR is a claim to a trust containing a foreign stock- ADRS are a common means for investors in one country to buy a stock trading in another countly SWe saw n Chapter 20 that 'if S and Q are asset prices, then the fonvard price for SQ containd a covariance term. You may be wondering why there is no such covariance term in equation (22.26). ln general, if an asset with price sb is traded and can be held by investors, then its fonvard price is s'actr-dlr This is a no-arbitrage result. ln the discussion in Chapter 20, we assume that S and Q are bolh prices of assets that can be held directly by inveslors. ln that case SQ does not represent the price of an asset. ln the discussion here, a-Qis the price of an asset that can be held directly by investors, but Q cannot be held directly. In general, S, Q, and SQ cannot all simultaneously be the prices of traded assets.

'-erl-r'

=

We also need to know the covariance between x and Y. Using It's covariance is

,

.

723

hedges the value of the security using both the dollar-eanslated Nikkei and exchange rate coneacts. Equation (22.27)makes it clear why quantosare said to be derivatives that depend on the ratio or product of two assets.g We can price this cotract by using Propbsition 21.1. That proposition implies that we obtain the fonvard price for Ypjxp by multiplying the fonvard price for i' by the forward jrice for 1/x, and adjusting the dividend yield to take account of the covariance between and F. To apply Proposition 21.1 we need to know the forward price of 1/x for dollarinvestors. We can obtain tlle forward price for 1/.x using Proposition 20.3. #c?ztp?/lfntzrc#

lisk,

=

%

f OI'War d PIice

is tlms p F(j,z' (Qj =

U0dfry-c-pcs-rjp

(gg go .

The role of the covariance term in equadon (22.29) is inmidve. Consider an onverts yen back to dolinvestor who buys the cash Nlkkei index and ultilately lars. Suppose the index and the exchange rate (measuredin dollars/yen) are positively ''

'

' '

'.

.

91 the market-maker hedges a claim with value / using and F, the claim value must satisfy the mulitvariate Black-scholes equation. ln this case, the equation is .)7

l0In this calculauon we start with the fonvard price for f'romthe dollar perspective and convert it to the fonvard price for l/-r. As a result, the fonvard price for 1/-r given here is different from that given by equaton (22.23),which is the appropriate fonvard price given a yen perspective. -:'

i

k

724

ExoTlc

Op-rlolqs:

QuAxTos

11

correlated. When the index does well, there are many yen to exchange. If p is positive, on average the exchange rate is favorable when there are many yen to exchange. When the index does poorly, there are fewer yen to exchange so the decline in the exchange rate does not matter as much. Thus, other things equal, the positive correlation systematically benests the unhedged investment in the Nilkei relative to a contract with a sxed exchange rate. Consequently, if the exchange rate is fixed, as in a quanto contract, the price for the index settling in dollars will be lower in order to compensate the buyer for the loss of benescial correlation between the index and exchange rate. )) E-xample 22.8 forward price is (1)j!) )i) j 1)

.

Using equation

.j((jj

-'r

wand Q are lognormal,

jjjj

''!jj (.( The .

=

-

her-rf?

jjj.js-jyjjyj

0.0 1Sj''e

=

.p,

('(

E

: ..

.

i

F t) I

(( Finally, )( t'.( E.

(jyjE..j

),,.)

'tEEE

yjyj

@.t lt'j t)

't This

-

.

lsij

=

-v()

=

QIt

e

(r-ryl-fxa//k-zl

(Lvjga)a) .

U0t?r-scllt-vcrtxlliz

=

(ar,jajjn) .

and Z2 binomially, so that However, we want to induce corfelation between Zj and Z2. We can create correlation by using the Cholesky decomposition discussed in Chaptef 19. Begin by rewriting equation (22.31)using equation (19.17):

ln the siandard binomial model, we simply approximate =

ZI

+l.

-

v0 :()e(r-'(?)?=

using equation

=

jyjj j jyyjyyyys .

Nikkei is

forward pr ic for the currency-translated

1',' '(

they evolve like this:

'

F 0, I

()

(22.29)and the information in Table 22.4, the yen

725

same answer either way. Here we will model the yen and Nikkei. Problem 22.17 asks you to jointly model the yen and dollar-translated Nikkei. The basic idea underlying the joint binoluial model for and Q is as follows. lf

Zi

..' ('

%

,

-

:=:

-

(22.29),the quanto forward price is

F(),l ($200,0.01$/ y

)

=

=

is lower tlaan the yen-denominated -'

()(oe(0.04-0.02-0.2x0.l x0.15)

vpp ,

=

Qh

=

.Y0d

(:u aaa) .

x/i/l

U0dry-cjh-bo'c

ZI

=

E:1

I -/J2)

p-ez

(g)g,jggj;; -

-

-

Y20, 342.91 in Example

-

k.

u 22.7.

-Yc

=

w't.

=

Jr D

For eacla yen move, tlaereare two

A Binomial Model for the Dollar-Denominated lnvestor

Vzt

=

As another way to understand quanto pricing, we can construct a binomial tree that simultaneously models the currency-anslated index and the exchange rate. In addition to this prticular applicatipn of tWo-variable binomial trees, some options have prices that depend on two state vriables. In Figure 22.3 we costructed separate binomial trees for the yen-based investor. For the dollar-based investor we need to construct a tree that takes account of the con'elation between the Nikkei and the yen. We can do so by hrst modeling the behavior of the yen in the usual way, and then, collditioltal 1f/p(?n the J'd?l, model the movement in the Nildei.l l Since for each yen move there are two Nikkei moves, the tree will have four binomial nodes. We will denote these as (I/lf 1/#, #1/, ##), with thesrstletter denoting the yen move and the second the Nikkei move. We have a choice of constructing ajoint tree for the yen and Nilkei or the yen and dollar-translated Nikkei, but we obtain the ,

l lrhis is similar to the two-variable binomial model in Rubinsten (1994).See also Boyle et al. ( 1989) for a procedure to generate a-asset binomial trees.

-

-

-

Nikkei forward price of 120,404

(r-ry)/,+,x4kl

1 p2c2. By construction, this ZI and Za have and Z2 = p/ l + correlationp. Now we construct the binomial tree by setting cl = +1 (theexchange There are four rat moves utj or down) and ez = +1 (ihe index moves up or down). :i= :z = = = 1', l), C 1), B 1', will label which A 2 (/1 (/l we possibl'utomes, z:2 :i= (E = = and E::t 1). D l 62 1 1), I ; (6 1 ; Fr a dollar-basd investoc the jossible yen rovs are

Thus,

$212 367 v2o o0()e(0.0s-0.02)

() 01$/v x

Xh

QB

=

QC

=

QD

=

B

-Y()e

=

tr-ryl/l--xxff

=

(r-?.y)-,x/J' -Y0d

=

=

lt

(22.33a)

vY0d

(22.33b)

.10

xilckeimoves..

Qze(ry.-J(?)/7+(reW(#+ l-p2) Q0c(ry-5:)+(r(?W(pry-slll-ovqxll-p,. 1-p2) U0d l-/a2) ry-&cjlt-o.cxll-pU0d l

-p2)

=

=

=

=

g 0z Q0 p U0c Q0 p

(jg g4a) .

tgg.jxgyj); f

(p,p,g4cl .

tgu.g4tp

probabilitie: associated with the nodes. Finally, we have to determine th As in Chapter 10, the risk-neutral probability fr an u!j move of the currency is lisk-neutral

p

=

e

r

J

tt

-

-

#

t'J

(2:,.35)

where If and# are impliedby equation (22.33a)and (22.331$.The risk-neutralprobability for an up move in t.he currency is 0.4750.

Recall that the risk-neutrtl probability arises from the requirement that an investment in the asset earn the risk-free rate. Speciscally, for the cun-ency, we consider an investment in the yen-denominated risk-free asset, hedged to remove currency risk when the investment is t'urnedback into dollars. This investment earns the dotlar-denominated risk-free return if the probability of an up move is given by equation (22.35).

%.Exo-rlc

726

CURRENCY-LI

11

OpTloNs:

'Fhe forward price for the urrency-lanslated

i ' i?f j''k' Ef (lt:jsjyjjyjsjyjyjjjjjjtylyl'. t'jlf (' j,,j!.jj,r'!l))ty)jt!'.. jiitfhppjrkjgr.ty.jjtjtj'jt!!yk. .-,111111::k*.* y' E E'E i 7': ;'(Eq! E :Eq! q' (' i'q?'.qL?1Ib'6t'11L'LLLLiqqkj6k. (r'tji k' r' 92* 774:* ,!!!F.1I1Ii,t', jtr...'t):kyj(jjj'.' grd Ep;: ( E q' E ;'pl'rqjqqiji'r:qi Eqf qt' jjjjyrjjtj('. lr' (q;7r71r* ('qq' ,.,..tb;;Ljjtjj:'.j)jjj. 11117-11112)* !1I1i....IIp'' .;' )' j'y)'. '(.i.'.. .i(;:.'... t:..E (Ei ! jj'i!). EEp' @pi (;(li jii!qjjf r' rf jr:' j')':' y' t(;'@(i 'j ' t.')t.' jjf )t'jyjy'! j'('! 'y' )j)'. . q! . .. . ;'.7t. ttf @(. (tjjjyj)jtj'. tq'jy'lk t')'.('('t'. (E E E EiE @. ! !y . j jj( r. .i rj j.i.'(..q.( .E(q ;r.ty. yyjj. j: ..jy.j(:),;.kjyj.E.;;.gyj).qj;.jt;yj y yjyy. y )E. i ! .jj!. y).. ...t;, ! t'...-.......1111::::kE q 11111* jy r yt.(jr 'Tpr.. ljjy .33..31. y( '. .,', . y ..' .. k . ':77i??q'J/i' .yjjjjjjj'))r.f ..' .'-,-3:36633333*1 -;'' --.'(' -r' ,,,jkjj,,;' r' ,,,,,,,k,,-'. s'jj ;':',.....'. '')''' t'j' didr'. ;)t'. ;'$8,(44/;*.1t12*:@t64* r.'' F' y:)' yyk'..y)jjj'. )' :'-'2.* 6* ;'!

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Risk-Nual

Time 1 Nikkei, Q1

Jttl

%23,706.10

$272.68

0.2194

Y18,647.88 722,325.56

$214.50 $210.25

0.2556 0.2,425

717,561.91

$165.39

0.2825

Fo

We need a similar argument for the Nikkei. Since we cnnnot oFn the Nikkei index without bearlng currency risk, we model an investment in the dollar-translated Nikkei. Let #* denote tlle probability of arl up move in te Nlkt-ei, condidonal on te move in the wesnent earn the dollar-denominated yen. Werequire that the dollar-anslatedNikkei lisk-free rate. This gives us zl

pp* + xu

Qs#(1

kdPc(1

#*) +

-

1

xdon

+

ptp'

-

# )(1

-

-

#*)

.X0

=

Qfter-8osh

(22.

g6)

Solving for p* gives x,, psp xoptyetr-dt?l'' -

p

*=

#Af(Qz er-sqlh

QBt

-

-

+

jjsp

-

=

pjxdoc

(1 dpLj -

xdvstl

-

-

-

p)

Po)

(22.37)

p)

-

D) '#lf (A B) + (1 pjdc Tls expression is a generalization of the ope-variable formula for a risk-neual probability, taldng account of the two up and two down states for Q. Figure 22.4 depicts te blomial tree constructed using equations (22.33)and (22.34), and probabilties of each node constructed using equations (22.35)and (22.37). 'l'he quanto forward price can be constructed as the expectation E (FI/.x1). -

-

('

-

Using Figure 22.4 we carl compute forward prices for the yen, the j Example 22.9 1' dotlar-translated Nikkei, and the quanto Nikkei. risk-neutral probabil f an up yj fomard price is 0.4750. the yen is yen ytqmove in 'l'he

'l'he

)!'.') .

tlt) F(),1@)

=

$212.367

Finally, the quanto forward price is

The binomial process for the dollar/yen exchange rate (x)and the Nikkei (Q).The Iast two columns contain the value of the c'urrency-translated Nikkei and the risk-neutral probability of each node, computed using equations (22.35)and (22.37).

Q

0.2194 x $272.68+ 0.2556 x $214,50 + 0.2425 x $210.25+ 0.2825 x $165.39

=

I

(:)

=

0.009418$/Y

AI

Nikkei is

0.2194 x

$272.68 $214.50 + 0.2556 x 0.011503$/Y 0.011503$/Y

+0.2425 x

$210.25 $165.39 + 0.2825 x 0.009418$/1 0.009418$/7

'

Y20,000 0.01$/Y

=

=

Probability

AIQI

0.011503$/Y

Qo

727

'

Yen, xl

=

$

-

trq!)--!1!F5q...t.-..,... ,. '''-?':''

N KED OPTIONS

0.4750 x 0.011503$/Y + (1

-

.. 0.4750) x 0.009418$/Y

=

0.010408$/Y

720,342.91

=

All of te prices computed f'rom the tree match those in Example 22.8.

'l'he eee in Figure 22.4 can be extended to muldple periods. Rubinstein (1994) shows that in general, wit.h n steps, there are (n + 1)2 nodes; for examplet WithtWo steps there are rtine nodes. To see why. if we add another binomial period to the tree, DD). The order there are 42 16 combinations of the up-down moves (AA, AB, BA. This equivalence e nziates of the moves is irrelevant, so, for example, AB CD. Because n x (n 1) 6 nodes. leaving 10. Further, f'rom equauon (22.34),AB 1)2 9 unique nodes. this leaves + 2, (n n =

.

.

.

,

=

-

=

=

=

=

22.5 CURRENCY-LINKED

OPTIONS

There are several common ways to constnlct options on foreign assets, for which the retllrn has an exchange rate component.'z 'I'he different variants permit investors to assume different amounts of cuaency and equity risk. ln this section we exnmine fotlr variants and their pricing fonnulas. We will continue to use te notation and numbers from Table 22.4. recall theresultfromchapter Before wediscuss pndl'culrcuaency-liAedoptions, 12, equation (12.5),and Chapter 14, equation (14.16),that an option can be pdced using only the prepaid foeard prices for the underlying asset and strike asset, and the reladve volatility of the >0.13 The intuion forthis result is that a market-maker could hedge an option position using the two prepaid forwards, neither of which, by definidon, makes any payouts. In the discussions to follow we will use this result to sifnplify the valuadon of seemingly complex options.

section draws heavily from Reiner (1992),in palticular adopting lzel'11is kinds of options. different

Reiner's terminology for the

l3To convince yourself of this. note that

sscallt,s

K.o.,r,

r.&) sscallt-e-dr Ke-rv

This equality will hold for any inputs you try.

-

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k. Exo-rlc

728

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playis. rE ; yE( y y y y y in th lte Japanese instttinat cfedii ntti itit itij Vis eiikitif,t # tl) j. tkfk.kk wer j ( 1980sbought Nikk-el utt notej. jj jj j jj(( j ( gj ( ( (y:j tEy j ((( yj j jgyyj j j j j soVfig ( yy j yji y fijjt r E jy yg jjgy nd yq yj kq bonds that cafried a high cotlp y ttkkktttj.; y, nttplf ;( ( g,itiljt' l Jkri.kttjEtki , q. . TI,4 an ernbeddd wriuv ppt zkrel: l ( k ( EE 'E tltkittihltl/ The note pripcipal wws npt p/t ip iltl.i? te y y wqj tt j'pk EyCqIE iht ti Vkit Vtittitruki,tti ktis'i. ijsrt ft Nilkei fell betmv V32,tU. trttikt: , jok . tntiita'p tj fi, ikiltiaiu tii : notes ( ) ; wtptd WaS a Ellfopean a tliat y k y yyyiq..yE,yl,g y (Eyy :. ,E iiltjyfk/ p' q. put spreal to a inkestrtmt bakr ryj,. N'llsi ;; t .. yfrtstwfj ,# j ;. embedded . . . . . . . .. . . pt) ij lty..,t.' .. . ,' p .', t ktkkenterd into ?. cufrnty k>'p t , j , ; t' I)it..:i:, r ! .i!iiL.. and .t)I). 1iq.',y..t y'j.... ;1L.iri1:, ii.dEj,' jrriL JElf.!lr)L 1E;8. ,iIE, jlgjp .14!:... ,::1: d:Ejk yjir .t:r.1., lrrlErltiil itl: ii).'$tki;' !r).i:tI2., 1r2 .....,......g..y.....y'..i...'..., ,LCIL ),4::.. ill I::1(:)2 1:2 #..iEri.. d?r lrrlt ,::)'jr. 4::)1 lrr)k !::: hl:llrkt lrr)t IEt d::lE .....'....,'y.,y. jl:EIt(l:.... )).......... ,11). t.). Lt,';. q a.lrrlL lii!i iil . CE t. . ... 4:21: i E Nkarint tv ... .E.. E .. . lldElle 1(7. i ... .. . wilng ... .. t isk Japanese buyers t: il r. ir!'p i were . . ..-..')..t. ..-...E.t ..yE.. ..-..) ..'.-.j'..t'.y '.'...- )..' ..) .y''.E.. ' ..-. .i...r.... ryt......-...'..-.. ..' ' ...'. !k:i: Elqq''l:::li .'.Ej....) IIEE 1!t' ..... . 11:7. 'kErk CEEIL. '...'E.....''..t. '(..''y..) . .....' il::li il)lk. (11. . ytt......y.. t-)...... .. .31q11. .t .t ).. i. t ' ! .... . E ' dl:::illt. ' .E qE'... ... i.' E'' ...q '..E;... ... . i. : .. yyg...yy '.'E'. l ' '.).' i.. .; ;. ;. r (..jy Id:irr:. 'i2ii,r:. '1IiE::!,illiii!t;''.'. E. . j . . Eli:':l.:. ;l!EE!I.. ;IE(:l'.. '11:'711.. . ' .. U :llEr-' :'EE!'...EIIE::!.:. (lll?t::::. y . (Elliisiiiz;. . . . y y 't iii.:1551;. j . 'tli:r:!l:. j .( .y y . Ei

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Using the parameters in Table 22.4 and assuming strike price of ' ' . . . . wepdceiecebyusingeBlak-scholesfo=ulaudseing S = Y20,000 t)v19,500, l! volatility in Nikkei index price), K = 719,500, o'c = 0.15 ttheNikkei )i1 .q . tthecurrent = = = yield theNikkei). dividend and ?.y 1, We obtain 0.04, r n itt'yen), sc 0.02 tthe $tk acall price of BsCa11(Y20,000;719,500., = dollar 0.15.,0.04., 1', 0.02) Y1632.16.

q Example 22.10 jij

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'l'he

y1

l

PZce iS $16 32 .

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%.

-. . - . .. .- . . . . y

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-

Foreign Equity Call Struck

Suppose we have a call opdon to buy the Nikkei but we denominate the strike, K, in dollars. If we exercise the option, we pay K dollars to acquire the Nikkei, which is worth xp Qp. Thus, at expiration, the option is wrth

FtxwQr, F)

(

j. .. . j . . j. ..

. .. . y..g.y..... y... ...

Domestic Cllrrency

l

:

maxto, xvov

=

-

K4

In orderto price tllis option, recognizethat F(r) xp Qv,the currency-eanslatedindex, prepzd forward price for the currency-anslated is priced lile any domesic asset. Qze-&QT. The prepaid forward price for the strike is index is, from equation (22.26),n:tl Ke-rT. The value of the option will depend upon the distribution of xv Qw',thus, the volatility that enters te opon pricing fonnula is that of the currency-anslated index. =

'rhe

Foreign Equity Call Siruck in Foreign Currency If we want to speculate on a foreign index, one possibility is to buy an option completely denominated in a foreign currency. The value of this option at expiration is

Z(Vw,F)

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...

.

' : . ' ' r . . . . . . . . . ' .' ' . ( . ' ' ( . ( ' '. ' . '. . ' . (' . E (. .' E.' y. . .. . . (.. . : g . .. E. .. ( . : . . . j....(. . .. E .: . ' . ..E .. : . . ... . E .y . . E . . . E .. . . ' ' E. : q . . . . .' .q . ( . E.... :...g .y.. . . . . . .. . q .. . . . .. . . j.i : : . y.E (. . . . q . . . . .. . . . j .... . . .q . . ;. . . . . yy .. . .. . .: . .. .y... . . . . .. y. .y.. ;..y yy. y.. :.. ..yyyj.. y ... . . .. .. y.. . jy ..g. g. .g.. ...y .. .. .. . y .. . . .

,

'j

. .. ... .. . .. .. ., , ..E..y''....y'........-'...'.'..i.

-i'

'

=

,

'.

'

c

cq-t

yy

.

.......

.

' '

:

:

:

'

'ti'

...k...

,r.

...........

:

!1ii!:: (;;iI12:(

y

y

-lsk,,p.ti.jtqtiik,.t

.

4t

o

ylyy

E

tl

.y

y

tliti,l E

:

.

,,

r

,

.

,

.

..

,.,(,

,

E

'h

..

.

.)

EE

contained

tc

y:

(EE'.

'tse

(22.38)

+ 1(y2 t :

d-rJJ)

E

.

.

s

.

.f

Thus, we price this option by using the Black-scholes formula with inputs appropriate for the agset being denominated in a different cuaency.

'

..t.t.'...)

E

1n CQ ()e-botjK

l

(f2

E)

itkkttjfd.''. ilttt-tjfitltlalzittl .y...t .... ...... ... ('loittltiirit t. (.. . kitttt- .t.,... Wii kttl t q Et y,'yE @ pf kpr if ttl,fl'my k s littiritiiiatt jp/itijc : iiitii rit jj y gj r (Th dhktiotkrt,iovt t itk iil ) j jt i 'ikki.'jttikitj

dc

: .. .

.

.

Kge-rttNdz)

-

..

(.... r..

.

Qze-&otNdjj

729

-

. L... ju.. . ..y....j. .o( . k(... oougj.jk tu . ( y y. t. .. 'E, ...g

th

....

,

.

-y;

.

'

i

,..

.

..

Cxs =

E'

.

;

-

: : . . . ' ' .: . . ... .' ' ' ' . . . ' ' ' . E E . . . ': . . . . .. . . g.. ........y . . . . ...

. .

E .' :.' ' ' '.:' E . . ' E ( '. ... . :. . . E .. .. . . E' E' . : : . E . . . r . .y.... . . . y. g .y..( y...yq. . .. .. : . .

E

..

- .;t

'

..

i .

''

i ' !. ' E E ! (:. g . j -. E. . E- -. - --y. . - - . -y

--

-.

-

...; y. .0.. y .(..j.y.. yy j... ... . .

.

.

-.

-

i y

.y..

.

.

-

:

'.j

g

.

...

-.

. - -

E . ' .( i ' : .' . . q i .E..E . .: .. y . .: ,4,...g q q ...

.

.

..

.

t. Syjpy '

. .

.

EE'

:

.j

-

-.. -

-

..

.

:i '

!

.

..

-

'

::

:.

. . . .. .

y .

. ..

'

'

.. ..

.

.

....y '....

'

.

.

-j.

--

.'.'..

) .....

E. .. '

. . .. .

--

-

q.

?'

.

.

. ..

. .

pj jvestjnt pt kjid, hd jtri ekkj It .jold olliiioerititdlktkt: Wii jkflk Eqfp:n

.

.

r r r

ttwesto

....

'

:'

E' g' j E --

..

.

-

.

glbal

kripj

.

.E . ..

..

.

ad

.. ...

. . . .

' y. .... (g.. . .. ..' ( . j. . . . . . . . . ... ..E. . j Ey( t E .( . .

kample pf quanto options is th ti . 225put warrants .that. traded on the rtlerican Exchange beginning in 1990. Ryan ad stock E Granovsky (2000)provid an intrjtir .

....

: :' . . g. . : j . . : . g . .

:E

:. .

. . ..

%.

rqKED Op-rloxs

CURRENCY-LI

11

OI>'rfoNs:

=

maX(0,

Qr

-

where Kf denotes the strike denominated in the foreign

#yl

The voltility

of xt Qt is

v

+ sl + kpo'cs

Using this volatility and the prepaid forward prices we have

C

=

artlpod

-sT

Ndj)

-rt

-

ln @0 Qfje-bTje-rK)

currency.

we might have a l-year call option to buy the Nilkei index by paying 119,500. An investor based in the foreign currency would use this kind of option; thus, it can be priced using the Black-scholes formulafrom the perspective of the foreign and the Nikkei volatility currency. Only yen inputs-the yen-denomipated intezest rate dollar price The pricing the formula. can be obtained by and dividend yield--enter rate. exchange the current converting the option price at

&2

=

As an example,

#rt

=

d1

-

tl

Xt

e

sxt# :t )

(22.39)

+ 1p2f :?

vJ't

You can interpret tllis formula in terms of prepaid forward prices or as the Black-scholes formula wit.h ab Po as the stock price, &Q as the dividend yield, w as the volatility, the domestic interest rate ?- as the risk-free rate, and K as te strike price.

i

% Exo-rlc

730

Ol7TloNs:

11

)' ) Black-scholes fonnula. We obtain

%' :'

j'q'. 14*

1q Example

ri

22.11

Using the parameters in Table 22.4, the volatility is

.j.j'.; (

(

E t'El

:

,tt ).. : '. ijrqand q

p

assuming

2

0.15a + 0.1 + (2 x 0.2

=

-sv

t,

lt1 jjj

=

:

x 0.15 x 0.1)

$195,we price the call

a strike price of

)(j kjt'y B SCall (.'t% Qje

0.1962

=

.E;

using prepaid forwards as

() g, BSCall(0.01$/Y x Y20,000c-0'02 $195 x e-0.08 () 1962 0, 1, ()) Ke

p

,

,

,

5

@

'

3

(p)

(),..r

=

$24..:719

k

j.

t) l!j Problem

Suppose we have a foreign equity call denominatd in the foreign currency, but with the option proceeds to be repatriated at a predetermined exchange rate. This is a quanto option, analogous to the quanto fonvard, with the value of the option translated into dollars at a sxedexcliange rate. Let J represent this rte. The payoff to this ption with stlike price K.f (denominatedin the foreign currency) is =

x maxto,

.

= maxto, kov

QT -

-

K.f)

kKy)

Once again we an construct the pricing formula by thinking in terms of fonvard prices

for the underlying and strike assels. From a dollar perspective, the underlying asset, .#Qv,is a quanto index investment. The sile asset is simply a fixed number of dollars, 1. Since is just a scale factor, we can set translated at the rate -i

.i

=

v.

yhe volatility that affects the value of the option Because the exchange rate is sxed, that of the foreigp-cuaencyvenoenatedforeign indx, o'c. We can obtain the pricing is by using the prepaid forwards for the underlying and strike asset: formula C =

FP 0,r 1n(FP

.dt

=

dz d I FvPpQ)is given =

-

(:)1(:/1 ) 0,? o.

-

e-rTK

f #(#2)

,

o' Q

O

r

,

,

,

0)

=

() (uozxlxj

x().15+().os-a.()m

'

,

'

(22.40)

Q)!e-rTK 1 j + 1(y2 F 2 q

(; ijyyv x vjq, 5(se-().08 .

,

0. 15, 0,. 1 0) ,

y

Foreign Exchange Call

. $15.3 187

=

22.6 asks you to verify that you obtaih the same answer with . an appropriate choice of the dividend yield.

ltljtjjy. ! . .. underlying asset and

Equity-toked Fixed Exchange Rate Foreign Equity Call

V(P?',F)

Kye-''T ,

(

o

,

Bscall(FP

tj) (ti BSCMl1(0.0l$/Y xY20, 000 x eljj!j. (ytt

-rp

,

op-rloxs k.

cuaaExcy-t-lxxEo

-p

.

Qo as the

%

.

lf we invest in a foreign asset, we might like to insure against 1ow exchange rates when we convert back to the domestic currency, while still having the ability to profit from favorable exchange rates. Buying an exchange rate put is insucient because the quantity of curreny to be exchanged is vncertain.What we want is an option that guarantees a minimum exchange rate bvhen ;$?' convert l asset VJJIIC bqck to the fftppid-rccll?-re/?cll.Such an option must therefore protect a variable quantity of currency. '..t-c/?p??,gE' This is an eqltitplinkedforeigl option, which is another example of a quanto option. Let K be the minimum exchange rate. Thetl the payoff to such an insured position would be

Qvxv+

QT

maxto, K

.Yr)

-

=

QvK +

Qpmaxto, xv maxto, Qpxp

K)

-

(22.41) QvK) = QvK + Theexpression to the leftof the equal sign in equation (22.41)is te unprotectedcun-encytranslated Nikkei investment plus Qv exchange rate puts with strike K. The equivalent expression on the right is a quanto investment with the Exed exchange rate equal to K, plus Qv exchange rate calls. Either way, the protection entails receiving the payoff to a random number of options. All cash flows are denominated in the home currency. There are at least two ways to value the payoff in equation (22.41):(1) By using Proposition 21 1 and (2) by using the prepaid forward approach. Using Proposition 21.1, we value Qv x maxto, xv K) as a currency call with a change of numeraire. We will pursue that approach', Problem 22.7 asks you to derive the same formula using prepaid forward plices The forward plice for Qvis the quanto folavard price, equatioh (22.29).Th put is standard currency call denominated in the home currency. Since it is a curlreny option, a the dividend yield is the foreign interest rate. To change the numeraire we subtract po's, the covariance between Q and -Y, from the dividend yield, ry. Thus, we price the payoff K) by multiplying Qjerf-bo-pn'T the quanto forward price, times Qv x maxto, te Black-scholes currency call formula with rg replaed by ?'y po's. This gives -

'

.

&c Wf

-

c ,/F

.

in equation (22.30). Note that all values are dollarThe formula for denominated since implicitly multiplies all prices. By substittlting for FvP:, equation (22.40) is the Black-scholes formula with Qtlas the stock price, q + pso.c + r r.. as the dividend yield, the domestic interest rate r as the risk-free rate, K.f as the stlike, and o' q as the volatitity. .

-

-vw

-

-

c

))1Example

22.12

Using the parameters in Table 22.4 and assuming a strike price of 0.0 l$/Y, we price the call by using the t' Y19,500 with a fixed exchange rate of -:i

=

u:z

=

goelr; .0

-p(r.) -(j?

-J Q F

O()c

F

g

X(#1 )

ye-lj.-poj'T

-

N tt./j ; .e-rT

-(r+Jc+pcJ-ry)Fx(#

K O()d

S' y 2

)

tyzjj

(22-42)

k: Exo-rlc

732

Op-rloxs:

OTHER

11

ln@a/#) + (r

1Y IL:::=

ry + po's +

-

First, consider the ptice of an option with

0.5,2)r ''''''''''''''''''''

snc-r

s VF Tltis is the price of a call option with ab :0 as the stock price, K Qoas the sttike price, liskfree rate, 8c as the dividend yield, and s as the volatility. ?-+ 8q + po's ry as the lt is perhaps surprising that only the volatility of the exchange rate matters. This occtlrs because the underlying option is a cuaency option and the change of numeraire does not affect the volatility.

d2

#l

=

-

J*

andthe

=r

+ J

J

()2

= q +

-

c

-

2

(22.44)

(7'

2

g p o.g :

(:2:.45)

-.u

Its plice is

-&sT-t)NLj1)

=

S/QE

Jl

=

ln(&:) + (r

e-rtNLj2 )

-

J *+ 0 5.2)(r

-

F

Jz

-

-

JI

=

T

-

(pa.46) tj

-

.

(22.47)

t

-

Etitt

(22.48)

t

-

''

-'

.

l). To do tltis we apply Now we want to value the opdon with payoff Q x WCSIQ, (r2:. covariance betveen Q and S1Qis pco'p applying Proposidon 21.1. Proposition 21.1, we replace 8 with rfhus,

'l'he

-

BSCa1l($200, $195,0.10, 0.063, 1, 0.02)

=

$15.7287

%.

Dh

J* Finally, multiplying J7 by

22.6

underlying asset. We can

c

c WCS/Q,r).

by

WCSIQ,?)

.xopt)

l,j

733

c2 + pao'c

'using

t,)

$

variance is

Denote the value of this opon a stlike price The price of = = 0.01$/Y underlying is strike the l) Y20,000 the price is $200, asset x t.(4 = risk-free replaced tlae by 0.00975$/1 x v20,000 $195, rate is r + c + po's r.f = = = li () 08 + () 02 + o.2x 0.15 x 0.1 0.04 0.063, and the dividend yield is ta 0.02. ii):t. )l) The vlue of the option in equation (22.42)is

SjQ as the

OPTIONS

price tltis opion using the Black-scholes formula with the lease rat: for SjQ in place of the dividend yield and the vaziance of jnsjoj as the variance. Proposiuons 20.3 and 20.4 imply that the lease rate for SjQ is

-

j Example 22.13 Using the parnmeters in Table 22.4 and assuming 11of 0.0097$/Y, we price the call by the Black-scholes formula.

MULTIVARIATE

OTHER

MULTIVARIATE

Quantosare aparticular

kind

OPTIONS

V(S,

claim with apayoffdependent on tlle price of two assets. 'l'here are many ot her op tions for which the payof depends on two or more assets. In this section we exnmine several kinds of multivariate options that can be priced either by modifying the Black-scholes formula or by using the bivariate normal distribuion. We will also see how to price some of these options binomially. Throughout this section, we assume that the assets S and Q fotlow the processes given by equadons (22.1) and

-

-

:e(r-&)(7--J)

Q, tj

=

)f

(22.2).

pco.c

Jj

=

(7.2

)

+ &

r

=

ln(5'/:) +

&Q

-

-

Qe-sqT-t,yLdgj 8 + 0.5 F

=

sc

gives

Se -&T-t)yLd j)

z Jl

-

F

-

-

2 p )(

-

(22.49)

tj

-

t

This is the formula for an exchange option f'rom Section 14.6. 'I'he risk-fre rate is replaced y Je nd the volatility by American exchange opuons can be valuedusing a two-state variable binonzial tree, 22.4. However, it is also possible to value an American exchange option Section in as using a one-variable binorizial tree. Rubinstein (1991b)shows that a standard binomial tree can be constructed setting the volatility equal to the dividendyield equal to 8, and the risk-free rate equal to 8c. This result can also be demonsated by using argments based on Propositions 20.4 and 21.1. .

Exchange Options We saw in Section 14.6 that exchange options, in which the strike price is the price of a risky asset. can be priced with a snple modification of the Black-scholes formula. Here we will use a change of numeraire to see why this is so. At mattlrity, an exchange option with price Vst Qt, t) pays

,

,

VCST,

Pr, F)

=

maxtur

-

PF,

= Qp x mzxspjop

0) -

1, 0)

(22.43)

This payoff is like receiving a random number, Qp, of options in which the underlying asset is Svjov and the strike price is $1. We can pricq tis option by applying Propositions 20.4 and 21.1.

Options on the Best of Two Assets Suppose an investor allocates a portfolio to both the S&P index and the currencytranslated Nikkei. Mocating the portfolio to the index that the investor believes will obtain the highest remrn is called market-flming. A petfect market-timer would invest in the S&P when it outperformed the Nikkei and the Nilkei when it outpedbnned the

734

% Exo-rlc

Ol7-rloNs:

11

OTH ER

S&P. What is the value of being able to infallibly select the portfolio with the stlperior Pedbrmance? We can answer this question by valuing an option giving us the greater of the two returns. This option would have the payoff maxts'w,Qv),where S is the S&P index and the Q the Nikkei index. Note that

maxts,

Qv)

Qv +

=

maxts'w

N

then

a,

z2 < b', p)

b; p)

(22.50)

where zl and 7:2 are standard normal random variables with correlation coeficient p. You may recall that we used the bivqriate normal distlibutionin Chapter 14 to value compound options. The formula for a rainbow option is Rainbowcallts',

qNdsc)

pc-Jc(F-?)

LNdcsj

= +

Q, A-, t7', s, p, &, tt?. F .c-J(F-J)

t)

-

(u), dsc, (po': c'l/J.jj (y())/j) NNL-t'IL (p), dcs, tpo' pj + Ke-rT-bNNL-dzsj, NNL-dL

-

-

-

where

#1(U)

LnS/K) =

+

@ 8 + 0.5c2)(r -

o' T LnQjK)

+

=

(r

(u$')

d,

=

dk (Q) = c' Q

#2(5-) #2(V)

-

o-

T

t)

This is equal to maxtk, Rainbowcallts,

((

+

r 2

z

o' + o'c

-

-

t)

t

-

lpo'o'c

.

Q, K) K, Whichhas q, K, o-, o,c- p, s, tc, -

-

ft'q

the value r,

w

-

,)

-

Ke-rv-',

S ome op tions that seem as if they migt be valued using the fainbow option forlul, howevr, cannot be. For example, in Chajter lt we discussd the valuation of peak-load electricity plants and encountered spread options, which have the jayoff

maxto, S

-

Q

K)

-

While there are approximations for valuing such an option (seeHauj, 1998, pp. 59-61), more exact solutions reqire Monte Carlo or two-state binomial trees.

-

-

Basket Options Basket options have payoffs that depend upon the average of two or more asset prices. Basket options are frequently used in currency hedging. A multinational (11711 dealing in hedglng currencies, for example, might only about the multiple care average exchange rather than each exchange individpally. As another example, rate rate, an option on S&P off if might only the S&P outperfonns of the index the pay an averace currency' translated Nlkki and Dax (German stock) indice. With equal weights on the Nlkkei and Dax, the payoff to such an option would be -

-

t)

t

t

-

(22.51)

k.

o

8c + 0.5cn) (F m

-

T

-

t

-

cc T =

ln(:/5')

t)

-

-#:?(:),

#l(S)

=

t $Q + 0.2)(F

-

-

-

mflxtu, Q) maxEo,

NNa,

=

&+ 0.5 :.2)( r

-

this daunting formula by recognizing that, at maturity, the option You can.understand worth either be S, Q, or K. By setting t = F, you can verify that the formula must satisfes this boundary condition. The fonnula for an option that pays minlu, Q, ffl-a rainbow put-is obtained by putting a minus sign in front of each EW''lrgument in the normal and bivariate normal functions. Certain related options can be valued using the rainbow option formula.ls For example, consider an option on the maximum of two assets with the payoff

:w)

This option, called a rainbpw option, has no simple one-variable solutiop. Instead, this opi)n requires the use of the bivariate normal ltstriution.l'l The bivariate ltlipg yq normat distlibution is desned as


H and K


S, and knocks out at Hz > Sl Suppose knok-out'' that K < S2 and the option expires at F. Call this a option. Here is a table summarizing the payof'f to this option (notethat because Sa > Sl it is not po'ssible to hit Hz without hitting Sl ): .

J(z.-?)

t: &, rz?) = olstej e-rLT-t)

#()#0) j

x(u)#?) 6

2/-

t

2/'

-

N-oldsj

J)

-

,

st 1-2V NLcodsj a.. St

2

G

2

-

G

-

equation (22.29).

22.9. ln this problem we use the lognormal approximation (see equation 11.19) to draw one-step binomial trees from the perspective pf a yen-based inyestor. Use the information in Table 22.4.

739

22.12. Covered call writers often plan to buy back the wlitten call if the stockprice drops and, suciently. The logic is that the written call at that point has little written the position could sustain loss stok the call. from if the recovers, a

-

t

k

PROBLEMS

11

8j

(22.52)

t

Where

d

=

Lknstjt) + (r

h

=

d

dq

=

+ (?' (ln(J?/.%)

Jj

=

dq

(y

-

c'

-

F

F

-

-

-

J + 0.5o'2)(F

-

&+ 0.5c'2)(F

-

tjjo-

T

tljc

T

-

t

t -

-

t

t

The value of a lookback call is obtained by setling lookback put is obtained by setting Jt = S: an d ) =

jh

=

t -

and

jl

=

1. The value of a

1

.

GEknock-in,

H Hit

H3 Not Hit 0 What is the vale

maxto, Sv

K)

-

a. What is the value of a lookback put if St gives you the same answer.

22.11. Suppose the stock price is $50, but that we plan to buy 100 shgres if and when 1, and 0.3, 1* 0.08, F / the stock reaches $45. Suppose further that c =

-

..v.

22.14. For the lookback put:

0

of this option?

=

a. What is the value of a lookback call as St approaches 0? Verif'y that the formula gives you the same answer. b. Verify that at maturity the value of the call is Sp

Hz Hit

S2 Not Hit

22.13. For the lookback call:

-

=

= 0. This is a noncancellable limit order.

a. What transaction could you undertake to offset the'risk of this obligation? b. You can view this limit order as a liability. What is its value?

=

0? Verify that the formula

b. Verify that at matulity the value of the put is v

-

Sv.

22.15. A European shout option is an option for which the payoff at expiration is maxto, S K, G K), where G is the price t which you shouted. (Suppose you have an XYZ shout call with a strike price of $100. Today XYZ is $130. If you shout at $130, you are guaranteed a payoff of max($30, Sv $ 130) at expiration.) You can only shout qnce, irrevocably. -

-

-

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.

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''''''

80

90

100

110

120

130

140

150

160

.

,:

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.

The top panel rdplcts the oId VIXfrm 1986 to 2004. The bottom . panel compares the new and old VIXindices during 2004. .

1

o.270

170

volallity (%) lc() 80

60 40

20 0

1987

1990

1993

1995 Date

1

s..

Implied Volatility

1 year g yazry

VXO

18

0.6

16

0.4

14

80

90

100

110

120

130

140

150

160

! '

l

170

02/23

see volatilities plotted in a three-dimensional graph, with time tp maturity on one axis and strike prices on a different axis. Such a plot is called a vola ti ty su rf ace. The pattrn of ilplied volatilities g:nerally ij referred to as the olatility skew. However, specisc patterns are frequently observed. If you pse your imginadon, the pattern implied volatility plot ip Figure 23.1 rejembles a lppsided grin or a smirk. When the ptot of implied volatility in the kure is someiims called a voltility agaist strike jrics toks like a smile, it is clld a volatilit smile. Volatilityhnwns may also be observed. rf'he

.

kolatiliWmay

'

.p?if/'/c.

.

seem like a namral way

'

'

.

io measure the volatility tat is

expectedtoprevail overafuttlreperiodof fime. However, thefactthatimpliedvolatilies are not constant across strike prices and over time raises at least two issues. First, it is using the Blakuxchols model, whlch ssuGes common to meastlre implied volatil that volatility is constant. The volatility skew may reflect pricing or specificadon enor in the Black-scholes model, which raises the question of whatimplied volatility acmally measures. Second, since there is no single measure of implied volatility (for the same asset, implied volatilities differ across strikes and across option mamrites), how should we interpret the implied volatility numbers? Should we look at volatility at a pnrticular t'moneyness''? ls there some way to average the different volatihties? We will see later in this chapter that some theoretical pricing models are able to account for implied volatility patterns such as those in Figure 23.1. .

.

.'

.

.

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.

12

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Soltrce: Optionmetrics

Implied

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2001

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vojatuity

.

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j)):!5;[email protected]!.(qylll;l?j'?@tr'1/ll!itt))r7(:i@(!

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VOLATILITV

09/10

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;

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source:cOE ln addition to exnmining thepattern of impliedvolatinties atapointintime, itis also possible to ack implied volatilities over time. Since 199 1 th e Clzicago Board Options Exchange (CBOE) has reported an index of implied yplatility for near-term S&P 100 indx options. This index is called the ICVlX,''afterits tickersymbol. Usipg this peasure, rijinally, te csci tqmputed over tiuae in implied volatitity. can tiackchags extracting volatility f'rom near-the-money options, much by implied implied'volatility This is called the V1X,'' with ticker symbol discussed in Secon 12.5. index as we rld volatility beginning 2003, the CBOE began computing implid based in VXO. However, on a new fonzpla that we will desclibe later ih this chapter. Figure 23.2 plots the old '71X index f'rom 1986 to 2004, and comples te new and 'WX for one yeqr. The spike in the VIX in 1987 occurs on fotlr days, October 19, old 22, and 20, 26, when the WX exceeded 100%. Tlzis period corresponds to the October decline over 20% on one day.l 1987, market crash, in which the Dow Jones 19 'I'he bottom panel in Figure 23.2 compares the new and old VIX during one year, showing that while the two measures are not idetical, they are generally quite close. At

we

tndex

lplotting implied volatility over time raises the question of whether implied volatility is an accurate forecast of acmal future volatility. Generally, implied volatility is a biased forecast of future volatility (e.g., see Bates, 2003).

'

:'j' (' ;y'. t'1* !' $' (li.l.qiqllql'llhissqsst:l (!(5r' (!' '1* J' q' pd tiid ii)d ''.-' )yrk'k' r' ljd ;).jjjL'. @1* 't' tqd .'1I1Ij:illIi.. 1111:*. -'''1IIi::;2;i-'-' ).g).))'. ;r,$)'.)' :Ei5!1l:-'. j,,r'. yyjt'ttjjd jyd .jj' p'y y'. #(j?'. tEEijdIll'; j;trd i'. i' lt'y

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r' tj)j'j);;)jj,'. t'

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744

k. VOLATILITY the scale in the top panel, it would be difficult to detect any difference at a11in the two series.

th We will see in Section 23;3 below that the new VW measure natly snesses o? considering prices aII pr0blem of which particula.r option implied volatility to use by out-of-the-money options with a given time to mamrity.

23.2 MEASUREMENT

AND

BEHAVIOF

OF VLATILITY

(. !

dst/'t

(a

=

8jdt + o'st,

-

.YI

,

tjdz

(23.1)

where a is the continuously compounded expected return on the stock, is the conXI t) is the instantaneous volatility. tinuously ompounded dividend yield, and o'st loking equation is subtly different than the process we assumed in eadier This tainililr chapters. In equation (23.1),volatility at a point in time can depend on the level of the stock price S, other variables, X'. and time. By comparisqn, the standard Black-scholes Equation (23.1) is an example of (ro, a constant. assumption is that o''t Xt 1) which in volatilit'y, instantaneous volatility can stochastic with stock price process a randomly.z change Given a series of stock prices that we observe evel.y 11periods, we can compute continously cmpounded returns, 6: ,

,

,

,

=

fr+/,

=

1nSt+I,IS;)

We will assume throughout this section tat 11is small, and therefore that we can ignore the man rettlrn

I'Iistorical Volatility The namral starting point for examining volatility is historical volatility, wlzich we compute using past stock returns. Suppose that we observe 11 continuotlsly compounded stock returs over a period of length r, so that 11 = T/?l. Under the assumption that volatility is constant, we can estimate the historical annul variance of returns, ir2, as

4.2 H

=

-

1

1

l

(?? 1)

t

?

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i= 1

(23.,,)

E' . ' E': E .. .. : . :. .: : . : .

M EASU REM

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ENT AN D BEHAVI OR O F VO LATI LITY

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E.E'' IEE tE (E'EE' E' E' ( (. E . i'i)! . li(: 'i'!Ey'..!.'i(.'(E.('.i.(.:EE.'ii i. ''( (.(.((E(i .. (.'...(. (.'y(i!..'EqEE.;!.(::76:1. . . ti. '! . E E ( .E . !. . i. iE ! :E:. E . E E .EE .yE .

i:.i.;. !k!.t : . . ;. !.) yI E. Ii ; ! t:y:.y..8.j; . :y.jyy. . jyyy.jjy. . 1iI!EiEi:.. j..((y...rq.yk.ii:!..y!.j;.r.jjy. .. .; j1lIi-:-.iIl'. ))j . jjj.jyjj. :' .IIIE:qdq.. . ..E; .q. ;; riji.yi)! ..i . ;tj. . '-)): -----LLj;;, .. !... ..;F.. !.)-. '. i . . . r . .. .. ........ . . r. (. ..r . .. rl . . . . . ... . . . Sixtpda# volatility estimtes for IBM nd the S&P 500 indexk The top pane I j hows volatilityestimates using equation (23.2), whereas the bottom panel uses equation (23.5), with b = 0.94.

rttj

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k .

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1 v.

1994

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e xl.l yj r I s .J ! sv

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,,

1998

1 1

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Year Volatility

O.8

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t' -

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S&P 5O0 Index IBM Ii

,

111,

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1996 Year

l

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1998

2000

Ssuppose

?-f is computed daily. lt is important to annualize aher computing the standard devation of the daily returns, as opposed to annualizing the dailyreturns and then computing the standard deviation. To see why, let (?n l be the set of nonannualized daily returns. The annualized standard deviation is

252 x

Varlt/-f))

If instead we Iirst annualize the returns by multiplying them by 252 and then compute the standard deviation, we obtain va(252 x

zNote that there are no jumps in equation (23.1).and we will assume that the volatility, CLS, X, t), also does notjump. It is possible to permitjumps for both; for example, see Dufse et al. (2000).

......

.

The multiplication by 1//) annualizes the varimce estimate in square brackets.3 T calculation differs f'rom the usual formula for variance since it assumes the per-perio'/plyt (.( (. . ' E' ' '; (ii. i'. This assumption makes little differene'' iil . mean is Zlr. 1!811I:E1i5:1L ElEi'' $rE11. 11L t!i;i lL. yy(.,y,. ( ( j The top panel of Figure 23.3 displays the historical 60-day volatility for IBM and t the S&P 500 index from 1991 to 2002. Each day, the preceding 60 trading days are used (( ; to compute the standard deviation of the continuopsly compounded daily return. Since ( ( (' '

E 11.,; 'E.l :E'i ';i)f;'.fE 1( ( !

ln this section we exanzine different ways to characterize and measure the behavior of volatitity using only historical information about the asset price. In our examples we will concentrate on stock plice volatility. We take as a starting point the lognormal model of stock prices. Suppose the stock price follows this process:

t(l)Eyii

l' g((' EE.(i(.''.E.(.i . '. . ' ..'' .'. .'.(.('( g'.'.iEE@ E .i (.'' '.. .' ' ..'. . . ...j.E.j . . . .). . .r ..E, E..k...(.(.:'j.:.(..).yi..g'.g.(.q.(r'.(y..'..... . .. E.EL..(.., .. . ...tljjt4j .

(rf))

=

252 x

Varttrf ))

Annualizing returns before computing the standard deviation creates a return series that has too much . volatility-

746

k. VOLATI

MEASU REM ENT AN D BEHAVIOR

Ll'rv

there are approximately 252 trading days in a year, the resulting standard deviation is multiplied by 252 to produce an annualized standard deviation. The use of overlapping 60-day intervals induces smootlmess in the series, since each day's returfl affects the next 60 days of voltility calculations. Even so, there is a great deal of valiability in the standard deviation. For both selies, volatility appears to have risen toward the end of the 1990s. lt is natural to estimate volatility using daily returns for a11trading days. It is important to recognize, however, tat not all calendar days, and not even a11 trading days, exhibit the same volatility. For examplej if all days were te same, ret'urn volatility over a weekend (fromFriday's close of trading to Monday's close of trading) should be ,/-3 times the weekday volatility. However, French and Roll (1986)showed that returps

fromFridaytoMonday were signiscanyly less volatilethanreturns ovetllreeconsecgtive weekdays. More recently, Dubinslcy and Johnnnes (2004)showed that individual stock t.11:.11 price volatilities are greater on the days when firms make enrnings nnnouncements nnnualized provides estimate of volatility, wltile equation other days. Thus, (23.2) an on the volatilities on individual days can vary.

OF VOLATI LITY

$

beconaes 11

2 ENVMA,I

j.. j (1 b4b (1 = b)b ''-l./,-1

..

-

=

?1

f==l 11

=

(1

b4b j

-

...

j

.j.-

j

c2 t

-f

C2

(23.5)

-i

1

i'= l

bn '

-

= Becaus E''izzzl (1 @p)!?f-l 1 bn the weights again sum to one. There is also a simple updating formula, analogous to equation (23.3),in the case of a moving window estimate. Each peliod we add the latest observation and drop the oldest observatio. Equation (23.5)is equivalnt to -

-

;

2Ewxlx-,

:, EwMA,,-I

=

Example 23.1 Suppose b term in equation (23.6)is then

=

1

+

0.94 and

-

b

z b g6,-t s,, -

j

11

-

=

,,

60. We have 1

6,-1-,,1 z

-

(rqa6) .

b'l

=

0.9756. The first

(1 0.94) 0.0615 0:9756 = This compares wit.h a Weight of 1/60 0.0167 for each observatio in the equalweighted estimator in equation 23.2. Subsequent (earlier)observations hake weights of . C 0 0578 0 0543 0 051 1 etc -

Exponentially Weighted Moving Average Because volatility in Figure 23.3 appears to be changing over time, it is natural to try to take tlzis variation into account Fhen estimating volatility. We rnight reason that if volatility is changing, we want to emphasize more recent obselwations at the expense of more distant observations. One way to do tlzis is to compute an exponentially wer/dtf moving average (EWMA) of the squared stock returns. 'Fhe E formula computes volatility at time t as a weighted average of the 2 estimate, sqvMx,i-j and the time t 1 squared stock plice change, time t 1 E Thus, we have 62 t 1 -

-

,

.

-

2EvMx,/

=

(1

-

2 y)e?-l + y).zEyvMA,?-l

(p,aa) -

We can lgg qpgtipn Fhere b is te weight applied to the previous EWMA estinpt. 2 right-hand side of the substitute resulting expression and for into (23.3) te mvvxr-j esimator as a weighted equadon (23.3).Continuing in this way, we obtnin the E average of past squared remnzs:

=

The bottom panel in Figure

13.3displays

the E

stimptefor b

2

ENVMA,I

=

J-'lg(1 !7l#-lj -

-i

(23.4)

i=l

'Fhe term in square brackets in equation (23.5)is the weight applied to historical returns. The weights decline at the rate b, with the most recei fttlt'n receiving the peatest weight.Because E=fz.j (1 !7)!7f-' = 1, the weights on past squared rettlrns sum to one. volatility. For lt is also possible to use a moving window when estimating E ?1 of equation might previous days data. In this case, use only the (23.5) example,we -

0.94 and

standard histodcal volatility estimate. 'his addttional vaiiailit'y oecurs because the most recent observation has four times the weight in the E estimator as in the standard estimator. Thus a pgrticulayly lyrge rettlrn will crate a large effect on the estimate. This eflct will then decay at the rate b. %. There are two problems with the E estimator, one practical and one concepFirst, if we use the EWMAestimator in equation (23.3)to forecast future volatility, t'ual. a constant expected volatility at any horizon. The reason is that the forecast of we obtain 2 E t2 i S t j S o that all forecasts of future volatility would equal (72 Thus the E t j estimator does not forecast patterns in future volatility. Seotzd, the EWMA estimator is not derived from a formal statisticl model in whih volatility can vary over time. ARCH and GARC, which we discuss next, address both problems. -

,

.

-

62 t

=

,

Time-varying Volatility: ARCH A casual examination of data, such as looking at historicl volatilities (Figure 23.3), or looking at tlie behavior over time of implied volatilities (Figure 23.2), suggests that

748

%.VOLATILITY

M EASU IIEM t)'.'yj )' ('El' !' tlf 77* (' r-k;'j','jj'' ;:y5,j.' j.jyf y' r' !j' 7r:7r'* 'q' yyqyj;f ;')' y' 1* rsf r(' j','j;f r,,::.f y' trf r'' r' .(:::jI(.' jj'' ytt'j,',)('. .'('i' :E ('. 'ii E!.' ij;.lj'i(!1i .I..t..j'. (.( i (.'(.'.' q((!'.(((q'.( (t.jr.E;. '... ('.;'.' q'' i(''r7'(7.qr:' .jjjjjjkkjjjj't)' jj'. tt,f j'kkEI!iii;;;;'. ?' y,.kf ( (.(.((('!'j'( );'. tf ijji!j!f . .'y .. ' .7(4'rq .r' tl-). . . p.''y (l'' ,(tt'-!i:.. qqy.ry i.; ' (''.(j.qy, 'q. j,. . ;;.q. y j;jj'.j jy.. q . . jy E:. ) . r yjyyyj.q..jj..yyyy...qyy.y.(j(.. . j. ..jliti:tlkii. . j jy j. .jy . j j ;q. ). iijiiiq. .jyyg... jj;;kyjjjj;. y.( . .. y jj; . .j.j.yy.jyg . .. . . qy y y yy y j r y . y y y . . y . ... . . ....... .... ....

.'.'

ENT AN D BEHAVIOR

oy

VOLATI LITY

$

749

-'.' --' -' --y' :':'' 5* .(' F' ;-'

volatilities are not constant/ What do we do once we formally accept that volatilities change over time? Ideally we would have a statistical model that permits volatility changes to occur. Such a model could serve both to provide better estimates of volatility and also to provide a building block for better pricing models. Research bn the behavior of volatility shows that for many assets, there are periods of turbulence and periods of calm: high volatility tends to be followed by high volatility and low volatility by low volatility. Put differently, duling a period when measured volatility is high, the typical day tends to exhibit high volatility. (High volatility could in principle also arise from an increased chance of large but infrequent price moves.) Figure 23.4 displays squared daily rettlrns for the S&P 500 index and mM. At a casual level, this figure exhibits tis effect, with peliods in which many of the daily squared retul'ns are large, and periods when many are small. This is called volatilit.y clustering.

tk;kjjjjjjyk'. @q' ijiji!;j;ff ) iillr ' ' CE ' . : : : : :: E

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. .. .-. .

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Sqtlared daily returns the S&P 500 index (top panel) and IBM (bottom panel) from january 2, 1991, tp October 24, 2001.. Or1

sap s()()Index

0.01 0.008 ().(06

0.004 0.002 0

1992

1994

XL K.-.A,'A 1996 1998

-.-..-.=

..

2000

E

If volatility is persistent, a volatility measure should weight recent remrns more

difference in weighting is exactly how an E heavily than more distant remrns. volatility estimie differs from the ordinary equally weighted volatility measure. ARCH and GARCH models also give more weight to recent returns. rfhis

Squared Ret'urn

0.03 0.025 0.02 0.01j

The autoregressive conditional heteroskedasticity (ARCH) model and the subsequept GARCH (Generalized ARCH) model of Bollerslev of Engle (1982) models thatattemptto capture statistically andwidelyusedvolatility important (1986) are volatilitys fact Engle ebb and in flow of the won the 20013Nobel Prize ip econornics for is that 750). The basic idea motivating.Rcl-l the box this in is Work area (see on page Engle be high high today it is likelier than volatility to tomorrow. (1982) if is average provided a statistical framework for modeling this effect. A statistical modl for asset prices could take the form

The ARCH model

Lnst/vt-h) In this specifcation,

(ce

=

0.5c 2)11+

-

-

E:t

(23.7)

the error term would have variance

var(6;)

=

c lh

(23 8)

4We can perform a back-of-the-envelope calculation by assuming that continuously compounded returnsare normally distributed. ln that case, the ratio of variances drawn from independent time periods hasthe the F distributin. If %ve estimate two annual volatilites usng 252 observations, the ratio of thetwo estimated variances is distributed F@, 251 251). The 99.5% and 0.005% confidence lekels F-l (p, 251 251), where p 0.995 or p 0.005. At a l % signilicance levelr ihe are obtained from two bounds for the ratio of estimated variances are 1.386 and 0.722. corresponding to volatility ratios of 1.177 and 0.849. Thus, if volatility in one year is 15%, a subsequent measured volatility outside the range 12.74%-17.66% rejects the hypothesis of constant varance at a 1% signi:cance level. ,

,

sBollerslev

=

=

the literature on ARCH and its vafiants. Two recent accessble introductions were written as a result of the 2003 Nobel prize: Diebold (2004)and Royal Swedish Academy of Sciences (2003).Nelson (1991)proposed exponential GARCH, which models the behavor of 1n(o-2). et al.

(1994)surveys

.

IBM

0.01 0.005

1992

1994

1996

1998

2000

If (7.2 is constant over time, we say the error term, 6:, is homoskedastic. Based on Figure 23.4, however, a more reasonable specitication would be to assume that the variance of varies over time, in which case it is heteroskedastic. t lf the time interval in equation (23.7) is short, then (as we saw in Chapter 20), 0.5c2)/1 is small, and /f2 is essentially the squared return. We will the mean, (a continue to assume that 11is short enough so that we can ignore the drift in equation -

-

(23.7). Let kfq denote the information that is available up to and including time t, and therefore information that we have available at time t. The idea behind Engle's ARCH model is that squared remrns have a variance that changes over time according to a statistical model. Speciscally, let qt be the conditional (upon information available at time t 1) value of the rettlrn variance, i.e., -

qt

EEE

The ARCH model supposes that we cap

2Itp ;-I E (6?

)

write

(23.9)

;

k

750 (y'(j. jjjjry,rf j'' j'y ;'(' ;;;jj!r(('. j.' j'y' (:' )' t't'1)* )' j('(' y' jjjjjj,,'. jjjjf jjjjyg:jj;j'. y'. j(jy'. j.' j'yj'.y'. jjj'. j'y (j'y,'jjj'. jjjjyi:jjj,f )'. ( E!(E. EEjj; y' t'5*)'('y'..'.jgjjjj'. ('..' jtj'g k .; .!jg(y:(.. .j; j E

voLATl

MEASUREM

LlTv

..jjjjr:j' :'. .1!jjj;jjjjp,' .'...jjjj'' ;.' r'. :','j' :..jy'tjj y' (' jjjj(yrjjj;'. j,)jjL,'. y' '!' ;';'I.'E ',' :'5* t','.;'L'

tgjjjjjjjy,f tjjjjyjjj('. .;jjjrjgjjj;'. t'.' .(,,'g. jjjjj;jjkjjjj .jj.j. . . ... jjjjj . .. ':. .. .

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jjjjjkyjy:. ( .,

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.

:

.

2003 Nobel Prize in econrfticg was awardedto Robert F. Engle and Clive Grager their work in statistical naethods in economics.Engle was cited for his work in studying the behavior of volatility. This from the Royal Swedish Academy o? quote, Science press release announcing the awtrd ot the2003 economics prize, describes Engle's '

'

.

.

fluctuations over time volatility are particularly signiscant options because the value o and. other linancial instruments depends on their risk. Fluctuations can vary considerably over time; turbulent pe1i o ds with large fluctuations are followed by calmer periods with small fluctuations. Despite such time'olatility, in want of a better VIW)ing 5 resafthefs sd to Svofk altefnative,

Random

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,

conditional hetAt this point we cap understand the meaning of depends point time value in at a on past eroskedasticty.'' Alltoregressij'e means that the valtles. Hetelnsketlasticity means that variances are not equal. The unconditional varicollditional variance is ance is the variance estimated over a long period of time. The upon'') taking into variance recent estimated at a point in time, account (ttconditional the volatility. Thus, alttol'egt'essive collditional hetelnskedasticity essentially means that the level of variance depends on recent past levels of variance. This is the behavior captured Gtautoregressive

by eqtlation

(23.9).

forecasts An important practical question is how many lags we need ARCH volatility equation estimate order (23.9). To better understand the behavior of an ARCH in to 0, i > 1. The volatility where we set /71 > 0 and ai single consider lag, let's model,

a

::s

(t'I()

+ a I :2,-

(u

=

+

ao +

??-period-ahead

tRI

)

j

E(c2,+l j+i-jl

t7.1

+ a + a (,,0 gt-o l

j

:,2-

j

)j

forecast, we have /1

1+J l ) .-

-

av

ai

1+

l

+ anel l t

-

I

(23.11)

j .. .

l . .. .

.

Equation (23.11) implies that unconditional volatility long-run average) is a

(thevalue we would estimate (23.12)

,

.

+ aI

(/?a.1'l-I)

squared return today implies larger squared returns at al1 futtlre dates, but the effect decays per period by the factor t7l Shocks to volatility are expected to die off at a constant rate. RS

???. Equation (23.9)is an ARCH(m) model, signifying 1 where at > 0, ai a: 0, i that there are /?? lagged terms. ln order for volatility to be well-behaved, we must have S)'=jai < 1 This model states that volatility at a point in time depends upon recent observed volatility.

(23.10)

This predicted pattern of volatility persistence is vel'y specifc and inflexible. A large

'2

j

E@'' .

' ' (.' ' '.g . .

2 t'l16,-)

i= l

i

'

'

.' ' .. E' '

,

Continuing in this way, for an

E

'

:-11E

J.-.I..?'?

( .

.

ri '

'''''

'''''''

''' '''' '''''' 1:;E;7 ''''''''

)

=

,

(E

l

.

+

Similarly, for a two-period ahead forecast,

,

,,

lilitikiiwEf

lflkaftl .

.. .

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,

i

E

.

.'

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i

q E,

'E Ik k$ fk wuj itd f'ok-fithcz a staiijtial ttftditi.'

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t'In

a. E (6,+1 1k1-1)= az +

EE

E

,

7 51

where aj > 0 and t?I < 1. Equation (23.10)is an ARCHII) model. Suppose we forecast volatility at time t + 1, t + 2, etc., using only the information we have at time t. Equation (23.10)implies that for a one-period ahead forecast of qt we have

'EE

.

Q

equation is then

'

..

:

curtelf ? cpmrs t. tbfttj rpalty yini siij ttddklpd ifith.z t 5E I'ridtijs ods for jtaiitlt tti- ( ; t klfility, , itctt kjtk'y .. '(' . ' ' ' ' iijlijikl tk E.( r.q q' i hpyebfn ! ',.r' zty fi iedfcfifk, bt ti. t.( y qyyjr E anit irk, lyst tij

E

.

jg

.

. . . y. . g . . gg . .. . : . : .. . .

.

.( ' . '. liletti.' .. . j . .( . g'. . y. .. j.j.'. jy...yq'E . . . l thiE' . djltl . t . . ....... With sttlsiial . r i. jy,l ( voltillikt , kbbeii cstant slos Ethfttf )'E Engle-s dikcverk wj fiikthipj, majf i' !''. . ii .!..t (( .) E1( . . .... pitjltkit ittjt#tji.jyy . ejt the .

-

.

::

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tinil htfskedjtliv

shares,

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. .. . y

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,;

for

contribut j o n.

..

.

..

ENT AN D BEHAVI OR OF VOLATI LITY

Thus, with

stipates of ao and tz1 we can compute the unconditional volatility. ln prctice, if markets become more turbulent, they may remain more turbulent

for a p'eriod gf time. Equation (23.9) with a single lag cannot account for a period of sustained high volatility. As you might guess, more than one lag-generally many lags-are necessary for ARCH to fit the data.

The GARCH Model The GARCH model, due to Bollerslev (1986), is a variant of ARCH that allows for infinite lags yet can be estimated with a small number of parameters. The GARCH model has the fonp

(23.13)

752

MEASUREMENT

%.VOLATILITY This model states that volatility at a point in time depends upon recent volatility as well as recent squared rettlrns. Equation (23.13)is a GARCH(m,n) model. GARCH(1,l) is frequently used in practice. The GARCHII,I) model is

qt

+

t70

=

a +

J11 6;- l

bL

ql-

(23.14)

l

lt is instructive to eompare the GARCH(l,1) model to the ARCHII) model, equation (23.10). To do this, we can rewrite equation (23.14)to eliminate tzf-l on the right-hand side. Lagging equation (23.14)gnd substituting the result for o-1 on the right-hand side of equation (23.14),we obtain 62,-, + t-lj c2/-:t+ Jpjo-2,l + bk az + 51

(c()

qt =

AND BEHAVIOR

OF VOLATI LITY

$

753

For a GARCH(1,1), qi is a function of e, 471 and bj The maximum likelihood stimate the of is set pazameters 5a, a I and 871 that muimizes the probability of obsrking the remrns we actually obselwed. Typically it is easiest to maximize the log of the liielihood function, in which case maximizing the likelihood is the same as maximizing ,

.

,

l

g-0.5 ln(o)

0..5c/Iqij

-

(23.18)

i= l

We omit the term ln(2;r) since it does not affect the solution. The muimization of equation (23.18)can be performed in statistical packages or even using Solver in Excel (see the chapter appendix). -0.5

Continuing in this way, we obtain We can forecast volatility in the GARCH(1,1) model as we did volatility foretasts ARCHII) the model. To understand the calculation, recognize that since qt in f(c2ItpJ-l), fr j k: 1. Thus, using equawe then have E qt 1%-.j) E (c2I%-.j) / I tion (23.14),we have =

=

(23.15)

E (:/+11lJ-1 A GARCHII,I) model is therefore equivalent to an ARCHIXI model in which the lag coefcients decline at the rate /71 Notice that the last tenu in equation (23.15)can be rewritten in terms of an EWMA volatility estimator (23.5):

)

=

aft + 51 1 bt 1 bI -

-

2

swM.,k

(23.16)

in the EWMA expression in equation (23.16) (23.3),but is estimated as part of the GARCH

bL

It is important to note that the parameter is not arbitrarily chosen, as in equation estimation procedure.

= aj +

Iillihood

estimation

varian'ce q: and mean zerp, we an estimate a GARCH model using maximum likelihood.6 The probability density for t conditional on q;, is ,

fEt

; qt4

1 e lz'qt

=

-fj.szifqt

lt

11

i= l

) + bkEqt l%-jl (f71+ bj )f qt I%-l ) + /71)(e + ctl6?-2

(JI

+ b 1:1-1

)

The goal in this calculation is to express the forecasted value of t.n+I in terms of what we can obselwe at time t namely, 6,-1 and (h-1 Following the same procedure, we obtain .

E (t.n+2I +,-1

)

az +

=

e(t?I

+ :1) +

It.n) -

i= l

l eln'qi

k

E (t?,+k 1+/-1) = az + az '

(t7l+

bL

)2(:70+

(7I

cf2-j + fplth-j

)

t

-

As we let k go to insnity, we obtain an estimate of unconditional GARCHI1,l ) model:

volatility in the

X

.

() seg: '

(t71+ lall + (t?l+ l1l)'(t7l62 j + pplo-j )

f= l

(p,y jg)

Since the e? are conditionally independent, the probability of observing the particular set of 11 returns is the product of the probabilities, which gives us the likelihood function: fEi

E (6,2Itp ?-l

t'll

For a k step-ahead forecqst, we have

of a GARCH model Given the assumption that conditionally normal, with distribution that is continuously compounded returns have a Maximum

+

afl +

=

.

qt

t?0

=

'2

(t7I

aj

=

i

+ bj )

:70 =

1

f=0

Using equations

in terms of the E

(23.16)and (23.19),we can estimate

-

aI

.z

express

bj

(23.19)

the GARCH(1,1) equation

of volatility:

'

q:

Jt'i

2 + (j

-

J)

4.2 EwxjA

(za po) .

where a (1 t71 bj )/(1 bq). Thus, the GARCHII, 1) expected volatility at a in time int is a weighted average of the unconditional valiance, #2,and the current po 2Ewua. estimated EWMA volatility =

6Alexander (2001) discusses the estimation of GARCH models and is replete with examples.

=

-

-

-

!111)* (jj' (.j4* j'((jy'. k' 't' t'(' ttd tl.ij:llllid

k

754

''.' .' ''.

VOLATI LITY

j' ))' Example 23.2 t'(r'r'

r' E' ''t:.

''' .'

Estimating a

GARCHI1,1) model for IBM using daily return data

1 .) from Janum'y 1999 to December 2003 yields the GARCH volatility estimate '#jj :.,r ') 0.000001305 + 0.0,4466/-1 + 0.95520-j qt tEj ljjtit The implied estimate of the unconditional annualized volatility is .)) .t .qgrjy 0.000001305 tyt 1.5318 x 252 )t) 1 0.0446 0.9552 ( yj The historical volatility during this period was 39.85*. An estimated unconditional 1') 'i..'olatility of 153% suggests that the GARcH(1,l) model has trouble fitting the data. In lt t')(tllis case, it ttlrns out that the problem is caused by large returns on days during which y

=

'

=

-

-

-

'

E ' .

:

.

l DM announced enrnings. .j.jj

'

MEASUREM

!: 'E E E (E' E :' i'Eq:' E: .' ' )(i ' ' ' ' ' E' '' '' ETE S'' : i ('E EE gi. . ' '::... ' :E(..'(.i ';i(i!(E! .!!( .'..('.: (''.(..((''@ p'' ' 'q i5.'(' E(.'.(.'(. .7;.Ji .'. ' (..i(( :Ei .E.!!'. .. . (' (..(.'.@(: . ( '( T' jl(' (: E('; EEj.(Eg ('!:j(( jrT: (. ..:' ' .. 'g':.E'E.'E:. :. ( E! !EE! ( (E . E ;EE ; j.j.yjj j jj .j g(.j j yj g r j ( .( . ( . . . y . (.. . t' j jj (j.:;rjjjjjj;jji . (iI:EI::::Ik.E -. j y j j y . . . j j g E g . E...jj E i- ,E. !E ;.; j.g.. . g.E.jE.j. .j.-j.. j .j. . jjyy-jy i.gg .. .. E E .E.. .k . --.. -,,t. ,.i (-2 g.- (.-. .. E..., .-. .... . .- -. '' :

'''' 'd'''''''' '''

'

'

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.

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:

.

--

.

.

.

.

-

-

Comparison of GARCHII,I) volatility and EWMA volatility for IBM, January1999 to December 2003. The EWMAvolatility . estimate set.s b = 0.94, while the GARCH are parameters '

estirnqted.

..

.

E (

:It:q. i:!:,.. ii.ii.

0.8

-

-

GARCH EWMA

,

.

..

.

&

. 't

v

,

4

j

E

''

.

;

j

.

tj

E

@

E

.

. g. :

/

,

'.

j 'hk.y, E

t

.

:

!tj

%.iii e;j ,j

=

Mar99 Oct99

Aproo

.,.

Novoo

Mayol

Dte

Decol Jt1n02 Jan03 Aug03

0.4229

-

k

E E

y

QuadraticVariation

We saw in Chapter 20 that the quadratic variation (the sum of squared increments) of a Brownian motion from t to F is F t. That is, suppose we frequently sample diffusion a process', aZt). Letting lt (F tl/ll and Zi) Zt + ih), we have -

=

(23.20).

7In order to obtain an unbiased estimate for nonearnings announcement days, the correct procedure wouldbe to eliminate all earnings announcement days, notjust those which, after the fact, exhibited highsquared remrns.

k.

-

=

/1

(c'Z(f + 1)

=

17 tion equa

i

Quadraticvariation

over time.

1

li

.

y

0

=

j!

j

jt ..

.

:

x

.

;

z

-

.

:z '

'

.

'

((.jEi .

''

. .y.

. .... . jg .

;

.

..

'''''''''

.

r

-

0.4

;L,.L.?.

Ettj

.

'

=

'l) tion to eliminating earnings announcement days, one could permit a fatter-tailed rettlrn t.)t distribution (e.a., see Bollerslev, 1987). ),!) l estimate 1tj Figure 23.5 compares the GARCH volatility estimate with an E lt ' although estimate EWMA different, the )) where b 0.94. The two are not dramatically 1ilexhibits 0.9462 would be even more extreme behavior. An EWMA estimate with b )) tj ll! closer to the GARCH estimate, because of the relation between GARCH and EWMA in

..

.

-

0.1

Realized

,

.

.

.j The other parameters do not change much, and this unconditional volatility estimate of tlj t )tt42.29% is more reasonable. This example illustrates that a GARCH model estimated li) ') using normally distributed returns can be sens itive to extreme data points. In addi-

-

g

:

.

''

=

.

.

0.5

.

-

-q,

-

i i

.

0.6

.

j . These parameters imply an unconditional volatility of j( ' . tjlit ().()()41()2:/3 i x 252 '@7 1 0.0507 0.9462

. .

.

,1

ov

:

'

.. .

t

.

'''

.

.

It

.

)'E'

; y;j. 11 ( jr( .!. q t. j, ...'... '.. . ...' ).j....q.. j.lj . .

. .

j:

(1, 1)

:;45 1:2/

'fhe

:

pj:rrr,,-

(() (i4

1. 1 During the 1999-2003 period there were four days on whih the abtolute one-day (j t return exceded 12%. On each of these days (April 2l, 1999; October 20, 1999; July 19, k....y . 153% volatility illustrates lij 2000: and October 17s 2000), DM announced earnincs. (.. . the GARCH model's dificulty in explaining these large magnitude returns under the '' ion that returns are nonmally distributed. If we omit these four days from the .j assumpt ' (1,sample, the estimated GARCH model is7 )t a + 0.9462:,-1 0.000002203 + 0.05076,-j qt ) .

.

'

.

..

. .

' 755...

' ' .

%

OF VOLATI LITY

.:

l,Et

'E(. .

ENT AND BEHAVIOR

:

.

j(k . .j@Ijiii:q. 1Iji!Ei!k.. j tj((jjjjg j ..j. )-. ... . r .-.. (tt,.ljj)-. . .,. r.r.. (.,...r... .. . '

.

'

.

=

-

c'z(j)j2

;4s

0'2(w

-

t)

1

therefore provides an estimate of the total variance

of the process

Suppose stock retulms are generated by equation (23.1),in which volatility varies time. Consider what happens if we compute the quadratic variation of the log stock over price. In ordef t do this, se would need to observe the stock price at high frequency, a for example, using multiple prices over the course of the trading day. Suppose we observe continuously compounded rettlrns from time r to T every 11periods. To simplify notation, let St + ih) = Sij, and c' i4 = o' L&+ihXk-vii t + /l). The realized quadratic variation of the stock price from time t to time F is then the sum of squared, ,

,

.

756

k. VOLATILITY

$

HEDGI NG AN D PRICI NG VOLATI LITY

757

''''j!.

#'

cpntinuously compounded returns:d

.

:

::.

():

l

1)) 2

jnLzil/ui

:

.

.....

E :

E

!

' E'E''

Ijl::z 121

...

:

'

.( :

.

14;::: .....r.. .

..

.

...

.

*

a -lI ri,a for IB anua .

(23.2 1)

'

.....:;(;.....

.

!!! i!;

.s... : ... . . i : ' ; ' r: ' : : j

:.. ::

.: : ... . . : :

a tili 1999 to ece . er 2002, Io ed a ainst 1 1 volatll. lt R .. esti ate . . I-I;r(?

i= l

'

.

...,.....; ,(

...

.

:.....:.

:..(

!

V )a t.j)j

r

!/()1

-

-

1

-

-

-

:

.

j '

..'j.

(

j

: r :. p'l f..Ei

.'!

Kt

h

.!1 '!.

::Q

Q

When 11is small, the drift term in the sumination is small relative to the diffusion term (see Section 20.3.4), j that the squared. change in Z(/) dominates the summation.g The light-hand side is an estimate of the total stock price variance from time t to F.

.

! ':

,

Etz

1E)tii .

7 F, l

S.A! 3 ;1.. ux

.1

1.

t p .tJ .y g . %)! .@ I Z)SI'C pg p i eq j; IF . 9*4. 1E... 1t i.

!

! ! r.l

l

rr

.

.4

One well-lnown difculty with using high-frequncy data is that some observed Price movements' occur simply because yransactions gltemate between customer purchases (made at the dealer's offer price) and customer sales (made at the dealer's bid bounce.'' pric). The resulting up and down movement in the price is called intraday when they with data derrinstfate deal use curAndersen t al. (j003) a way to data to compa.re reiktized quadratic variation with other vazipnce stirpptes, such rency 3o-minute as GARCH(1,1). They calculate realized quadratic variation as follows. At following the and immediately ask preceding prices intervals, they observe the bid and and'ask pliees impute price bid the averaged at to a 3o-minute mark. They interpolate contin3o-minute the price this tcl measure imputed 3o-rninute intervals. They then use uously compounded return, from which they construct realized quadratic variation. In compm-ingforecasts based on realized quadratic variation with other methods of forecasting volatility, both in- and out-of-sample over one- and ten-day horizons, they find that realized quadratic variation is generally at least as good as other estimates.

-. . jt e aljzed vlolatlll R (1 1)

1.5

,

c ;';j-

J) :k).

.r

.:

ilE!; .fvy

R

.

..ti ...q

j

.

.

. f.il '!j ?Jg jC''i . l f. .-/F.4yr. -.if?'i%L. v

.'?..

;,

.g

.,

j!

'j

r.-c

6-

i:

t't' tj. ' R!

: kkJrr ! Jly : . . )? $ .;,r, 4 q . . .JJ:k.....#.>g lkti2 $LAX. ' qlx ! .s3?tSjuvrlhsk'' I $ 5 r tr C?'mz 1.''.. ' T@ 4(q. #% ;; :p!: To j ! 1ETirs . .sl,.l./jij q/tk. ? k ;ty i tf' tj tvflkr:k ? .t' t. y ..j.... :J9)'; .c 1.t.jy. s rr 17 ; y .1. ik';D 1 p(j, t. :';j' x&l4t . t; :k:;;....J! !;r$'' y h. 1 pgr ok. :f ;k'S .k .t) 7r qi F:5' q ! : t x. r Ii .1. . :f13.p : A' : # 4./1 ' k'F .z y .q ?T Q m''. 1!Z .I'. ssCsr%s.rosru 4 jj! qt(? j $:: # t..J w 'F' . 1 :! w/y: ; 9.?1 u : 9.j : I: AJ jrikrk!r sbrs v qtth...tr. ;d w a : gk ,. t.j)1)) & r;h rlf j: fjzrl i7'.y:jthr rj''gtTs :.f :.- j' r V:'? '.'l ! J4u > y%' a k(.L.' .;#4't.: ZJ@ t.!. f.1 :.y.Ir Ln' I tb3T.Cg j,, i.h@. ' .. 23. trs:. J't1.jj' TaJ1 tf/xt x!prly,u E 'a Jk 4: w py :; s.:.. ; t$ lp:dl.J. v ;rj . Ju :').14 XSCCJ. hk Ey). (krrqtrly.. t rsj.t;. k ': ft j y lhqak'):sz. ?x 12 $.)y':> 7rsYJ7-'y?;. . . sr e. k jqijtjg. x xtw k 2 g! j!! tjt jr jytitjqv . a x yc . :ky. 4T. glkrjrl:.>;a.. ;r' y j j bujvnq.. 1 j; g. . au sI . . . tj tr yyyjk . 0. r,'jn s z. a rx1+ -

11.$.) ' .'.r

.

....

.E.ry

.$,-.1j

.w

rg

.

:.(jj4..

.

.k

..

''

Ey

ry .

-.

.

..

..

''

>.

.

.*

-

.

.

.

..

, jE!

.

-(t'i'

+1

*

'p'

-

-)sg:'

-:

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.

.

Gtbid-ask

ar99

ov00 ate

ayol

ecol

Jul02

Figure 23.6 plots daily realized quadratic variation for IBM (usingreturns computed at five-minute intervals from open to close) against the GARCH(1,1) volatility estimate using the parameters computed in Example 23.2. The GARCH estimate is a forecast of volatility, while reatized quadratic variation measures actual volatility on a day. Thus, realized quadratic variation is considerably less smooth than the GARCH forecast. (Realized quadratic variation is sifiother, on the other hnd, than the squared daily returns depicted in Figure 23.4.) In interepreting the graph of realized quadratic variation, it is important to recognize that if the priee bounces around a great deal during the day, then realized volatility will be high even if the rettll'n for that day is not high. The empirical question is whether the magnitude of this intraday bouncing around of the price predicts volaiilit'y on subsequent days. The evidence in Andersen et a1. (2003)is tllat it does.

'volatilitl'-''

e'realized

'trealized

Aproo

Soltrce: ndersen et al. (2005)-

.

gUnfortunately, there does not appear to be a standard terminology for the realized quadratic valiation lt would seem consistent to refer to the sum of of an asset price'. lt is common to call (y' the volatility'' would then be the square root of variance,'' qnd tlze squared returns as the of squared the returns is sometimes called (he however, In realized variance. practice, sum the volatlity'' See, for example, Andersen et al. (2003).Moreover, the realized quadratic vmiation only c'2, under certain regularity conditions. Thus, for measures the variance of the stock price diffusion, quadratic variation,'' which s unambiguous, albeit clumsy. clarity, we will use the term 9You may wonder about the difference between estimating historical volatility and realized quadratic variation. Recall the histo/ical variance estimate, equation (23.2). Nlultiplying by 1//1 in equation (23.2) is the same as multiplying by nl T. Thus, the historical variance estmate can be rewlitten as

c199

4trealized

ltrealized

aH

1

= r

1

J1

:?

-

(?2 lj'

'

-

farl

Apart from the ?7/(/) 1) tenn, this appears to be the same as annualized realized quadratic variation. thistorical volatility'' usually implies the use of daily or. less frequent data over In practice, the term tquadratic variation'' implies the use of intraday data over short medium to long horizons, while -

horizons.

'

23 3 HEDGING *

AND PRICING

VOLATILITY

In this section we discuss delivative claims that have volatility as an underlying asset. We begin by discussing volatility and variance swaps (includingone contract based on the V1X). We then look at an example of pricing a variance swap. Finally, we discuss the constnlction of the histol'y of the VLX volatility index reported by the Chicago Board Options Exchange (CBOE). ln this section we will let JZ denote measured volatility and 72 measured variance.

- t).))) ! ,..). j.) ..qj .j

758

k

VOLATILITY

H EDGI

Variance and Volatility Swaps

l,eprevents

,

,

(

-

# 0,F

(1S))

X

1

by

i

l 1 l tt

volatility

j

N

'i

where F(),z(F) is the forward priee for volatility

* Whether returns are continuously compounded or arithmetic ( the valiance is measured by subtracting the mean or by simply squaring whether . the retulps .. .

noct of' time over which variance is measured e ' How to handle days on which, unexpectedly, trading does not occur The

Most of these design issues are straightforward, but the last deserves some comment. Most futures contracts settle based upon a final observable price. A variance contract, by contrast, settles based upon a series of prices. Therefore, failing to observe a price (for example, because the market is unexpectedly closed) creates a problem for closed on day r, then measuring the realized variance. If the market is unexpetedly will will have which be tvo-day measured return, the next ret'urn a greater expected a how following example shows The than one-day variance return. one contract deals a this issue. with

)j Example 23.4 The Chicago Board Options Exchange volatility index, the VJX, is li(lr the basis volatility for f'utures contract that trades on the Chicago Fylmres Exchange. jtt a ('' (( Unlike the variance futtzres contract, the volatility fumres contract settles . based upon the ('. VLX index. The payoff is .)() jE

j

'

E

.

E

j't ji

'

)

'

1000 x FlXz.

-

Fo,p (F)j

1 ).

ln comparing the volatility f'uttlres contract (Example 23.4) with the variance futures contract (Example 23.3), note that the two contrcts are based on volatility measured over different periods of time. The variance contract settles based realized on quadratic variation overthe periodfromo to and thus thefumres pricereCcts volatility r, expectations from time 0 to time F. The VLXcontract, since it is based tti VI.Xindex on e measures volatility expetations f'rom time F to time F + 30 dys. Thus, the kplatility contract measures volatility going forward from the settlement date, Whil the variance contract looks backward from the settlement late. There are at least two rasons that the valince contract is in some sense more ''natural'' than a volatility contract. First, we will see below that it is possibleto plice andhedge avariaceforwardcontracttgiven someassumptions) using optionprkrs. The pricing of a volatility forward contract is more complicated dtle to Jensen's inequality. Because the volatility is the square root of the variance, Jensen's inequality implies that the volatility fonvard price will be less than the square root of the variance forward price of the vazianc. Second, variance swaps arise namrally from dealers hedging theiroption positions. Recall from Chapter 13, in particular equation (13.11),that the prost of delta-hedging a dealer depends on the squared stockprice change. Dealers can hedge this lisk in realized variance by using variance swaps. For example a dealer with negative a gnmma position could enter a swap that pays the dealer when the stock has large price change. a ,

Three-pwnth S&P 500 variance futures traded on the Chicago FuExample 23,3 tures exchange are an example of a variance swap. The payoffis based on the anpualized 2. sum o f squared, continuously comgounded daily returns over a three-month period, . as IZ a x 10,000, and by desnition a one-unit change in The masured price is quoted

this number (calleda varialtce pt?/?2r)is worth $50. For simplicity, we treat the payoff as if it were a fonvard contract, settling on one Qi be the continuously compounded return on day i. The payoffat expiration is

day. Let

?1:, l

Ei

i= l

??c 1

:?

---

$50 x

l0, 000 x 252 x '

-

-

Fo,w(JZ

2

)

?lr, is the acttlal number of S&P plices used in constructing 72 (hence 1 returns), and lle is the number of expected trading days at the outset of there are the contract. Thus, in the event of an unexpected trading halt, the sum of squared returns will be divided by a number ltrger than the number of squared returns. The reason for this is that the trading halt will not necessarily change the total variance over the period (if the trading halt is on a Tuesday, for example, the Monday to Wednesday return will

In this formula, la

-

l

t j

k

'

* How frequently the return is measured .

a

this.

A volatility swap is like a vmiance swap except that it pays based on rather than variance. The payoff is

-

where N is the notional amount of the contract. There are numerous measurement details that we have to specify in order to write the contract for a variance swap:

.

. 759

%.

typically reflect two days of volatility). Dividing the sum of squared returns by na would mechanically increase the measured variance when there is ading halt. Dividing

A variance swap is a forward contract that pays the difference between a forward plice, 2, F 0,T (72) and some measure of the realized stock price variance, over a period of payoff variance is notional The to a time multiplied by a amount. swap w(J/2)j 72 Ftl x N

(

N G AN D PRl CI N G VO LATI LITY

Pricing Volatilie We will see in this section one way to determine the fair price for a forward contract on

variance.

:

760

k. VOLATILI'I'Y

HEDGI Nc

Consider a variance contract that pays the sum of squared price changes from time 0 to time F. If price changes are measured over an interval of length /1 F/??, the contract would have the payof'f =

lt

Payoff

$lt Sih 5'(f+1

:!

-

=

(23.22)

Slt

f= l

Note that, since arithmetic and continuously compounded returns are close over small intervals, this is the same as the realized quadratic vmiation measure discussed in the previous section. As /? gets small, this equation (23.22)becomes F

Payoff

o'st

=

.

Xt

,

1) ldt

,

(23 23) .

0

where o's, .Y, r) is the diffusion coeficient in equation (23.1). We want to answer two closely related questions. First, how is it possible to replicate the payoff to such a contract? Second, how should the contract be priced? As you might suspect, replicating the' contract yields a wayto price it. In principle, the price of a forward contract on variance will be the expectation of equation (23.23)under the liskzneutral measure. ln practice, how do we compute such an expectation? The following section will provide one answer. The Iog tontrau

# euberger (1994) pointed out

that a'forward contract that pays

(23.24)

- knupjb)

could be used to hedge and speculate on vmiance. A claim with the payoff in equation (23.24) is a log contract. As of early 2005, there is no exchange-traded log contract in existence, but for the moment, suppose such a contract does exist. Assuming the stock price follows equation (23.1), we can use It's Lemma to charactelize the process followed by the log of the stock price. (Note that we assume that the stock price does notjump.) Equation (23.1)permits a wide range of processes for the volatility, but the prospective volatility over the next instant is known. Applying lt's Lepma, Fe hay tgln

&)j

1 =

y

ds

05

-

.

)

y.

df

Thus,

0.5c(r)2#r l ds =

Intepating

T

jv

c2(r)#?

=

j

T

j

ydS

%.

-

j

T tglnts'f

ljdt

The integral on the left-hand side is quadratic vmiation over 0 to F. Let 42 denote

761

annualized realized volatility from time 0 to time T We can then rewrite this as r j 0.5F 2 IS jnsvjh) (23.25) 0 The right-hand side of equation (23.25) is the cumulative return to an investment in .

-6

=

-

1jS shares, less the return on a contract paying the realized continuously compounded return on the stock price from time 0 to time F. The left-hand side is anntlalized realized quadrqtic variation, which fromequation (23.23)is also the payoffori variancecontract. Equation (23.25)demonstrates the connection between the payoff to the log contract and

volatility.

Take expectations of both sides of equation (23.25)with respect t the lisk-neutral expectation of 6IS/S under the risk-neutral distribution is

stock price distribution. The rdt. Hence we obtain

T

j2j

*

0.5Ff

1

E*

=

y#5'

LnSp(%)

-

0

= l-T E* llntu$'w/uVll (23.26) This expression seems to be of little help in pricing volatility. There is trick, however, a for pricing the log contract using other instruments. -

Demeters et a1. (1999)and Cal-rand Madan (1998)indevaluing the Iog contract pendently showed that it is possible to use a portfolio of options to replicate the payoff on the log contract. Note frst that

j,

-%U

.-ll/

K

g,

sv4dlc

In(r)

=

-

+

=

In(,)

In(u) +

-

a

.-J.S -

&

t'

Use this to obtain the following identity, for any Sv (see Demeterfi et al., 1999):10 % l X ST ST S+ 1 ln maxtft0):/A- + + maxts'w K, 0)#:a S. S. K K1 o -

=

.%v,

-

-

-

.

s.

Notice that if we take expectations of both sides with respect to the risk-neutral distribution for 5'w,the integrals on the right-hand side become undiscounted option plices, and the expected stock price is the forward price. We can add and subtract ln(&) to the

#glnt&ll

-

this equation, we have 0.5

AN D PRlcI NG VOLATI LlTv

l0To interpret the expression on the right-hand side, notice that for any given Sv, the value of the first integral is zero for K below Sv, and the value of the second integral is zero for K above Sv. Thus, the effective integration bounds are not 0 and co, but they instead depend upon Sv. For example, if 5N Sv m K : S the equation becomes =

*

1

t-

- ln Ss

t=

2

-

&

25--

+

.9

l Ka

mxtc

-

S, Q)dK + 0

;

762

k

VOLATILITY

EXTEN DING TH E BLACK-SCHOLES C' 'yy'. $ii1:tt:::kr'. S' F' (' f' )' 1'* 1:* yjjd !..))' (pyyjj.d (:' (' t'jytj)'. E' j'ytryd J)' (i' ipjjjy;yd @' yjf jlljjitqqd y' jj'. !' )' r' t';' @1ijtj--jIj('. j'k'yjf (' y' E y'y j;;qj,:?;;'. t'. t'tyd rij!!!j:i,d i (i;(g'!' E('' '((E..((((;' i EE'E..EEE;.E. . EE(jg ' E.EE'.. .' (('':'.:!' q. E(('EEE.:'(q '':( '!i;tj.'.i' i(i'.i !ij:'I.Et!(!)' .(E ri. . '... .... .. . ( . . . !.(... r'.' 'i.l..1'7. (: ..jjtjltjr:y ').i.r' . . ...j! ((i.(:!i)EJ:E)j;i(j:(.(.;(jj..() . ..(lly.'j.y... .r. y.(. (.-Ey.yyEyEgy....t..(-j.y...y..yygy.yjyj(jyy...y.jy.yyy..i,t..;.yjjj gk ..j.j.jy yy...y.yyyjyy .yy...y y g . yy.y . y j ; j y jy Ci EEE. j . jj y j . .. . y :. .jL3LL3j3:. jiji!!!!jt (j, .g. .j. yj .gj(. lE' )y' j'1111: ..jjyy ' ... . jljl:i;lil:, .y.;;:yjj;E... .jyyj,. r . ik. .jr.yyr..y..qy). y.y.yyyjyy..q...g..y.jy.y.. . . .jy . .., .. . . .

.'

Q

MODEL

763

'r:(' -k;-'. ..14.,,;,-.* -;' ,':)' j')':''r' 7* :'C' ',yj'.

lft-hand side and then take expectations, to obtain '.r

E* ln -

s

.

'f;

,

=

.

F 0, F

-

E

- .

S*

+ ln

'

' :

s) s. -

S.

.v

+ e

%

?

()

x

l K :j

PKjdK

+ s.

in equation

Us this expression to substitute for F*glnt5'w/&ll

1 K2

CK)dK

(23.26).The result is

(23.27)

:'

' ;

.

-

:

. . .

. -.. -.

.

-.

-

,. . .

gy

y

''

' ' : . ..

' : ': .:

..

,-

:

. . ... . . . . . . . . .

.. ..

:

'tq

':

!

rj

-

y .

. .. ... .

-

..

- ..

. .. ....

..

--- -. , .

''' :-.

( ((( .y-.

.

,

y

solid Iines depict vegas of options with (from Ieft to right) strikes of 25, 30, 31, 40, 45, 50, and 55. The dashed Iine is the weighted sum of the vegas, with each divided by the squared strike price, times 600. The calculations assume ()- = 0 30 r = 0.08, T 0.25 ? and 0. .

.

.

. E . .. . y . --- . . . ..

.. . ..

..

-. .. . .- . ..

'

.

.

.

-

-

.

'i

.

, ,

-

.

=

F0,w(5').the first three tenns on the right-hand side of

.

:

-

. .

' r. .

:

'

'

-

- -.. . . ..

jjzEq!r

() y.:a '

0.1 ..---.-

2

Fr

oz.enrT

=

w

()

K

:t

s

l

0.06

N

Ao

tr

sk

,

f:

0.04

,

'

Nx

R$

,''

xN

:

=

N. 4

0.02

X

N.

X,

cxa

1*

xs

?

Finally, note that if we set S. equation (23.27)vanish, and we have =

z

0.08

PLKIIIK +

,.w

) K :j

0 10

(rs g.jg)

C( K )#w.

.

Remarkably, this formula gives us an estimate of expected realized variance that we compute using the observed piices of out-of-the-money puts and ealls! (lrut of the money'' here is with respect to the forwtrd plice rather than the current stock price.) It is important to note that we have not assumed that options are pliced using the BlackScholes formula or any other specifik model. One important characteristic of equation (23.28)is that the variance estimate can be replicated by trading options. It is possible to buy the qtrip of out-of-the-money puts and calls, weighted by the inverse squared strike price, to create a portfolio that has the

value of 2 .

To get a sense of why equation (23.28)works, we can examine the vega of a portfolio of options held in proportion to 1/#2. Figure 23.7 graphs vegas for a set of options and also displays the vega of a portfolio where the option holdings a'e weighted by the inverse squared strike price. The resulting portfolio has a vega that is not zero andis constant over a wide range of stock prices. lf you hold such a portfolio, you make or lose monef depending on volatility changes. CBOE'S

Computing the VIX We can now explain the formula used to compute the volatility index. The calculation is based on equation (23.28). ln practice, option

new stlike prices are discrete and there may be no option for which the stlike plice equals the index forward price. The actual formula used by the CBOE is a discrete approximation to equation (23.28): 2 Fo,r LKi rwcaljtsj; 2 2 Ll-i a e + e Puttfl = T T 14% F Kz A'z K >Kz KsKh (23.29)

20

30

40

stockPrice .

g

.

.

. .

.

y.

50

60

70

.

where Kfj is the flrst strike b:low the fonvard price for the index and LKi (A-/+1 f-j)/2.1l The last term is a con-ection for the fact that there may be no option with a strike equal to the forward price. =

23

*

4 EXTENDING

.

.

'

'

.

TIIE BLACK-SCHOLES

In thi! section we examine three priing

MODEL

models that are capable of generating volatility

skew patterns restnbling those obserkd in ontion markets. The coal is both to understand how the Black-scholes lodel can be extended and also to gain a sense f how these xtensions help us to beqer understand the data. We consider three models: (l) the Merton jump diffusion model, which relues the assumption that stock price moves are continuous', (2) the constant-elasticity of vaziance model, primarily due to Cox, which relaxes the assumption that volatility is constant', and (3) the Heston stochastic volatility ntodeli which allows voltility t follow an 1t pmcess that is orrelated with the stock price process. These models have al1 been siynifiantly generalized, but we can use them as touchstones for better understanding the economics of departures from the

Black-lcholes

lognormality assumption. Atthe outst nbtethattheBlk-scholes model easily accommodates tilelkarying ! volatility if the volatility pattern is deterministic. Speciscally, Merton (1973b)showed

.-1

,.w

.j

-

-

.

-

o

1lFor the highest strike, lh'i is the difference between the high strike and the next highest stlike. LKI for the lowest strike is defined as the diflkrence between the low strike and the next highest strike.

gj.)' j'yjd y' (.jd i,jlj!ilrkid td t';'ytj)d (' j' rlll)liqqd 1**

% VOLATILITY

764

.'.' ''

that ifvolatility is a deterministic function of time, then it is possible to price a European T 2 t) in the option with r t periods to maturity by substituting j t:r sjt' l s for c2(F Black-scholes formula. We can think about this result in the context of a delta-hedging market-maker. As long as the market-maker knows the volatility at each point in time, the delta-hedge will work the same as if volatility were constant. What creates a problem is a l'altdom change in volatility. -

-

.....l..;,,..:r;i;,..!):qyrji.k;.j(.;y!.q..q

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unchanged, th put will clearly be mor kaluable Whehth stock canjump to zero. Thus, by parity, the call must be more valuable as we1l.12 The standard implied volatility calculatio uses the Black-scholes formula without ajump adjustment to compute the implied volatility. The no-jump prices allhave implied volatilities of 30%. Whatvolatility in the no-jump model is requiredto generate theprices in the jump column? For the 4o-st14ke option, vega is 0.0781, so a change in volatility of approximately 0.0257/0.0781 0.323 percentage points, or a volatility of 0.30323, will generate the higher price of $2.81. For the 3o-strike option, however, vega is only 0.0083. Thus, a change in volatility of apprpxirflately 0.0359/0.0083 4.3 percentag points is required in order for the no-jump Black-scholes model explain to the plice of $10.6679. The actual implied volatility in this case is 0,334. When vega is lower, a largerchanje in yolatility is reqired to explain a gike change in the option price. Tlzis exqlpple is at most suggestiv. Ip prctice, jurppscafl be positive or negative of and tpcertat fpagnitude. Ifjumps cn occur in bot diretions, then we would expect to see higher implid volatilities for .both in-the-money andu out-o f the-money options. Furthermore, jump risk is unlikely to be purely diveisisable since there can be marketwide Ioves. exaple does, however, illustate impriant intuition for why jumps can generate volatility smiles. .

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766

EXTEN Dl Nc TH E BLAcl(-scHoLEs

%.VOLATILITY Constant Elasticity of Variance Cox (1975)proposed the constant elasticity of valiance (CEV) model, in which volatility varies with the level of the stock price. Speciscally, Cox assumed that the stock follows the process d.

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The CEV pricing formula for a European call is different for the cases p < 2 and p > 2. Qa, b, c) denote the noncentral chi-squared distribution function with b degrees of freedom and noncentralit'y parameter c, evaluted at c.14 The CEV call is given by 'price

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From equation (23.30),the elasticity of the instantaneous stockprice variance with respect to the stock plice is a constant ;:

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topstant lasticity of valiance'' comes from. This is the name CEV model was the Ending in Black (1976b)(see also One motivation for the volatility inreass when stock prices fall: One potential explanation Christie, 1982) that thinldng about the effect on equity lisk from a fll in leverage. As for this stems from the stock piice decrees, the debt-to-equity ratio rises, and th: equity volatility hould therefore increase. Thus, the negative coqelation btween stock prices and volatility is ciled the leverage etfect. Adrawback of th CEV model may already have occurred to you. If the stockprice In declines and p < 2, the CEV model implis that volatility will inceasepennanently. relationship volatility stock and the detenninistic between and in such theoly practice a price level seems implausible. Howevr, such a relationship seems more reasonable when modeling interest rates. The interest rate in the Cllk model (seeSection 24.2), for example, follows a CEV process. 'where

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are lmplied volatilities computed usng the Black-scholes formula strike maturity, different different to times model for three computed using the Heston that volatility, The panel assumes of o'v. top volatilities qnd prices, two different panel assumes that cv 0.5. In both panels, v* 2.0, o.v 0.1 0, while the bottom 0. v(t) 0.32, k* 0.25, r 8%, and =

=

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The main challenge for an option pricing model is to match the observed volatility skeml; The literature investigating ways to do this is too large to adequately summazize ' tead we will sketch the nature of fndings in the literature and highlight issues here. ns that arise when trying to match models to data. The pricing models in this section illustrate ways il which modifying the BlackScholes assumptions can enable a pricing model to better fit observed option prices. For example, al1 the pricing models we have discussed are cappble of gnerating higher implied volatities for tn-the-money (low-strike)calls. The Merton jupp model and the CEV model in the examples above both generate implied volatility curves that are :atter incfeases.l8 of the models, juch as alleston model that combinations astime to maturity also allows jumps, seem able to rproduce qlalitatibe fatpres of Figure 23.1. However, matching models to data is a more involved exrcise than just a visual comparison of ,

womo.

0.26 1

l

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light-hand panels of Figure 23.10 arises from assuming a negative con-elation between volatility and the stock price.

Evidence

0.3

. 0.26

=

implied volatility curves. To illustrate the issues, suppose you want to match the Heston stochastic volatility modl to data. There are a number of ways you might proceed. First, on a given day, you could find a set 9f model inputs that best matches the volatility curves for that day. This entails tindina a retulm. variance (tl(r)), a volatility of volatilitv (cu). a mean reversiol rate (v*),a long-run risk-neutral variance (J*), and a coaelation between kolatility and the stock return p), t flat ma tch the data for a particular day. Matchipg implied yolatilities across a set of options on a given day is a crosssectional test of the model. Once you admit mpltiple days of data, the model has time-series implications as well. Equations (23.35)zand (23.36)imply that volatility evolves over time in a specific way. lf you look at the evolution of volatility over time, does it match equation (23.35)2Are the pararpeters that enable the model to 'lit the crosssection consistent with those implied by ihe vo l a tility tim series? W'hn there is a lisk premium in the equilibrium pricing model (asin the Heston model), it is potentially easier

l'Cerhe

that the stock price then use Blaek-scholes to compute implied volatilities. We assume ranging from $30 pl-ices strike with is $40 and compute implied volatilities for options months to l year. to $50 and with maturities from 3 values of o'v Figure 23.10 shows the result of this experiment for two different volatility, current the than less and p. In the ligure the lonprun volatility, 5*, is 25+, with time to volatility decreases 32%. Because volatility reverts to the mean, implied = is almost = there no skew, maturity in evel'y case. In the panel where o'v 0. 10 and p 0. and 50% p = 0, the although the mean reversion in volatility is apparent. When o'v asymmetric skew in both The exhibits both symmetric skew and mean reversion. E

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MODEL

true model should give equal implied volatilities for options at different strikes and maturities. For example, if the Heston stochastic volatility model were true and option prices were conkistent with equation (23.37),then Black-scholes implied volatilities would exhibit jkew, but if the Heston model were used to compute implied volatilit'y, then the options in Finure 23.10 would all have implied volatilities of 32%. lSlt

is not obvious how to interpret the change with respect to maturity in implied volatility curves such s those in Figures 23.1, 23.8, 23.9, and 23.10. The issue is that the economic distinction between a$50 strike and a $55 strike is greater with one week to epiration than with one year tp expiration; therefore, other things equal, we might expect to see iatter volatility curves as time to expiration increases. Bates (2000) corrects for this effect by scaling strike plices by o' F. effectively measuring distance between strikes in deviation unit-s-'' tstandard

'

772

% VOLATI

FU RTH ER READ l NG

LITY

to mconcilethe behavior of the stock with option prices because there is an additional parameter. However, as Bates (2003)emphasizes, a risk premium must be plausible. Bakshi et al. (1997)and Bates (2000)both asked whether option pricing models incoporating jumps and stochastic volatility can generate realistic volatility skew for options based on the S&P 500.19 Both studies find greater volatility skew at short maturities than at long maturities. lf you compare Figures 23.8 and 23.10, you can see that this pattern is generatedby thejump model but not as obviously by the Hest6n model. This explains why, although Bakshi et al. (1997)foundthatthe stochastic volatility model provided the best overall explanation of prices, they added jumps to account for skew at short maturities.zo They also found that permitting stochastic interest rates (which can be added in the same fashion as stochastic volatility) helped explain prices at longer maturities. Bates (2000)found thatjump models (as in Figure 23.8) tit near-terr ption prices betterbutfound thejump parameters implausible: the stockprice does not appeartojump also as often as implied by the estimates necessary to explain implied volatility. Bates voltility of explain skew, the volatility model orderforthe to stochastic that in concluded developed Duffie al. However, implausibly large. et had be plicing (2000) volatility to procedure that permitted jumps in both the asset price and the volatility, and noted that allowing jumps in volatility potentially addressed the problem of an implusibly large volatility of volatility. with jumps in returns clrastically Broadie et al. (2004) conclude that in offer a significant plicing overall pedbrmance jumps improve (andq in withjumps model cross-section unless the remrs ij llowed in improvement a pricing words, In other in at least some volatility-of-jump risk premium.'' jumps have a to dimensions are important, and risk premia can be important as well.21 To add one more layer of complication, casual observation suggesis that in some enrnings, cases volatility changes deterministically over time. When a flrm announces for example, volatility will be higher than on ordinary days. You can show that this is true by comparing the volatility of returns on earnings announcement days against that on other days. Dubinsky and Johannes (2004)show that this effect is also apparent in ption prices, which imply a higher volatility before an enlmings announcement than after. 22 This Ending suggests that in addition to the use of increasingly sophisti'tmodels

'volatility

.

.

.

'

lgBakshi et al. (1997)examined European options on the S&P 500 index while Bates (2000)examined optons on the S&P 500 futures contract. zocarrandWu (2003)formalize the intuition thatjumps mattermore if there is ashort time toexpiration. 21It is possible to measure volatility Iisk premia directly by looking at the remrns on portfolios that lind negative retums are hedged against stock price risk. For example, Coval and Shumway (2001) written straddle loses the S&P indexes. S&P and Since straddles 500 written the l0O zero-delta on on m' attributable to a volatility risk money when volatility increses, this finding ay be at least in pal't premium- Bakshi and Kapadia (2003)lind smallerrisk premia associaled with delta-hedged individual stocks than with index options. 22111fact, %%'e saw the effect of earnings announcements in Example 23.2 when we estimated a GARCHII l ) model for IBM and found the estimates sensitive to the inclusion of four earnings announcement days. ,

Q

773

cated mathematical pricing models, careful option pricing requires data sets that identify anticipated days Of tlnklstlal VOlati1ity.23

For options on a given underlying asset on a given day, implied volatility generally varies across.option strikes and mamrities. Implied volatility also varies over time. As a result there is great interest in measuring volatilities and in plicing options when volatility can Valy

Methods of measuring volatility using past data include historical volatility, exponentially weighted moving average volatility, ARCH, GARCH, and realized quadratic variation. ARCH and GARCH estimates are based upon a formal statistical model in wilich volatility is random. Realized quadratic variation exploits high frequency data to obtain a reliable volatility estimate using data from a short time holizon. Both varance and volatility swaps permit hedging and speeulation on volatility. The valiance forward price can be obtained as a weighted sum of the plices of traded European options, acalctllation that is the basis for the VIX measure of implied volatility. The Black-scholes model does not perfectly explain observed option prices; there is volatility skem which means that implied volatility varies with the strike price and time to expiration. Two modiscations to the model are to permitjumps in thestock price and to allow volatility to be stochastic. Both changes generate option prices that exhibit volatility skew and that better tit the data than the unmodised Black-scholes model. ' Attempts to explain prices of traded options suggest that it is important to account for jumps in both the asset price and volatility, and that risk premiums on one or both jumps may be importantk

FURTHER

READING

Early studies of stock remrns (e.g., Fama, 1965) found that continuously compounded rettlrns exhibit too many large rettlnas to be consistent with normality. In recent years, research has focused on speciing stock price processes that give theoretical option prices consistent with observed prices. lntroductions to GARCH models include Royal Swedish Academy of Sciences (2003),and Bollerslev et a1. (1994).Alexander (2001)is a readable text for less technical readers. Realized quadratic variation as a measure of volatility is presented and applied in Andersen et al. (2003). Demete

et al.

(1999)present a clear and well-written discussion of volatility

hedging, and the paper also develops the volatility measure VIX. See also Chicago Board Options Exchange (2003).

used now to construct the

23This is not just an issue for individual firms. Governments make economic announcements on prespecihed days at set times. aftd these announcements sometmes generate large moves in pfkes. For example, Hanweck (1994)shows that implied volatility in Eurodollar futures options is greater on days when the government announces aggregate employment.

(t' (

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. -. . y... .

774

% VOLATI

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Ll'ry

PROBLEMS %.

tirst papers to suggest alternative assumptions about the stock price for option pricing were Cox and Ross (1976),Cox (1996),and Merton (1976). Merton noted in his paper that the jump model had the potential to explain volatility skew patterns noted The

by practitioners at the time. The first stochastic volatility models were proposed by Hull and White (1987),Scott (1987),and Wiggins (1987). The Heston (1993) model has been generalized by Duflie et al. (2000),who develop a prieing framework that can accomodate jumps in volatility as well as in the stock price. The empiricl literamre examining the ability of option pricing models to fit observed prices is rapidly evolving. Well-lnown papers include Bakshi et al. (1997),Bates (2000), and Pan (2002). Current research (whichinclude citations to numerous other papers) include Andersen et al. (2005) and Broadie et a1. (2004). Dubinsky and Johannes (2004)examine deterministic volatility changes, such as those due to earnings

775

a. Verify that the Black-scholes price is $50.0299. b. Verify that the vega for this option is almost zero. My is this so? c. Verify that if you compute the option price with volatilities ranging f'rm 0.05 to 1.00, you get essentially the same option price and vega remains about zero. Why is this so? What happens if you set tr 5.00 (i.e., d. What can you conclude about difculties in computing implied for very short-term, deep in-the-money options?

23.7. Use the same inputs as in the previous problem. Suppose that you observe a bid option price of $50 and an ask price of $50.10.

PROBLEMS For many of the srstffteen problems you will need to use data on the CD accoinpanying t is book. The b contains stock prices, option prices, and interest ratej for' kariety of maturities. If an interest rate you need is missing, use the rate fpr the nearest available

c. (Optional) Examine the code for the Bscalllnlpvol function. Explain why changing the starting volatilty can affect whether or not you obtain an

maturity.

23.2. Compute daily volatilities for 1991 through 2004 for IBM, Xerox, and the S&P 500 index. Annualize by multiplying by 252. How do your answers compare to those in Problem 12 23.3. For the peliod 1999-2004, using daily data,

compute

the following:

IBM'S volatility using all data. a. An EWMA stimate, with b 0.95, of 0.95, of IBM'S volatility, at each date b. An EWMA estimate, witl b of data. days previous 60 only the using =

=

Plot both estimates. How different are they?

23.4. Estimate a GARCHII, 1) for the S&P 500 index, using data from January 1999 to December 2003. 23.5. Replicate the GARCHII,I) estimation in Example 23.2, using daily rettlrns from estimates with on IBM from January 1999 to December 2003. Compare your four without the largest returns. and 23.6. Use the following inputs to compute the price of a European call option: S $100, K = $50, r = 0.06, o' 0.30, F = 0.01, J = 0. =

L

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interpreting =

$50,

=

a. Verify that the Black-scholes plice is zero. b. Verify that the vega for this option is zero. 'Why is this so? c. Suppose you observe a bid price of zero and an ask price of $0.05.What answers do you obtain when you compute implied volatility for these P

d. Why would market-makers set such prices? e. What can you conclude about difticulties in computing and interpreting implied volatility for very short-tenn, deep out-of-te-mney options? 23.9. Compute January 12 bid and ask volatilities (using the Black-scholes implied volatility function) for IBM options expiring January 17. For which options are you unable to compute a plausible implied volatility? Why? 23.10. Compute Janual'y 12 bid and ask volatilties (using the Black-scholes volatility function) for IBM options expiring February 21. a. Do you obselwe a volatility smile? b. For which options are you unable to compute a plausible implied ity? Why?

h '

t

23.8. Use the followinginputs to computethepriceof aEuropean call option: S K $100,?- 0.06, o' 0.30, F 0.01, J 0. =

'

1

ansWef.

What can you conclude about difculties in computing and implied volatilities for deep in-the-money options?

(q

volatility

a. Explain why you cannot compute an implied volatility for the bid price. b. Compute an implied volatility for the ask price, but be sure to set the initial volatility at 200% or greater. Explain why the implied volatility for the ask price is extremely large.

23.1. Using weekly price data (constructedWednesday io Wednesday), compute historical allnltal volatilities for-lBM, Xerox, and the S&P 500 index for 1991. through 2004. Annualize your answer by multiplying by 52. Also compute volatility for each for the entire peliod.

.

=

announcements.

-

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implied

volatil-

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776

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APPEN Dlx

VOLATILITY

23.11. Compute January 12 implied volatilides using the average of the bid and ask prices for IBM options expiring Febl'uary 21 (usethe Black-scholes implied volatility function). Compare your answers to those in the previous problem. Why might someone prefer to use implied volatilities based on the average of the bid and ask prices, rather than the bid and ask volatilities individually? 23.12.

ln this problem you will compute January 12 bid and ask volatilties (usingthe Black-scholes implied volatility functiop) for pne-year IBM options expiring the following January. Note that IBM pays a dividend in Marclt, June, September, and December. volatilities ignoling the dividend. a. Compute implied b. Take dividends into account using the discrete dividend correction to the Black Scholes formula, presented in Chapter 12. For simplicity, discount future dividends at a 2% continuously compounded rate. all How much difference does this correction make in implied volatility? .observed

adividend yieldformM based c. Take dividends into accountby computing this dividend yield on its annualized dividend rate as of January 12. Use in the Black-scholes model. How different are the implied volatilties from those you obtain in the previous part?

d. Do you observe a volatility smile? 23.13. For this problem, use the implied volatilities for the options expiring in January 2005, computed in the precedlng problem. Compare the implied vlatilties for calls and puts. Where is the difference largest? Why does this occur? 0. Use the Black30+, F 1, and 8%, o' 23.14. Suppose S $100, 1' ranging from with strikes the and call prices put generate formula to Scholes $40 to $250,with increments of $5. Compute the implied volatility from these 13.29).What happens to your prices by using the formula for tle VIX (equation estimate if you use strikes that differ by $1 or $10,or strikes that range only from $60 to $200? =

=

=

=

=

23.15. Explftin why the VIX formula in equation (23.29)overestimates implied volatility if options are American. The following three problems use the Merton jump formula. As a base = 8%, o' = 30%, F = 1, and = 0. Also assume case, assume S = $100, r and cg = 0.30. that = 0.02, a.l =

c. How is the implied 0.50?

volatility

plot affected by changing

23.A O'J

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777

to 0.10 or

23.17. Using the base case parameters, plot the implied volatility curve you obtain for the base case against that for the case wher there is a jump to zero, with the same 0'.20, and in part (b) consider expected 23.18. Repeat problem 23.16, except let a.t alternate jump magnimdes of 0.10 and 0.50. The following two problems both use the CEV option pricing formula. Assume in both that S $100,?' 8%, o'o 30%, F i= 1, and 0. .

=

=

=

=

=

23.19. Using the CEV option pricing model, set p 1 and generate option prices for strikes from 60 to 140, in increments of 5, for times to mtgrity of 0.25, 0.5, 1.0, and 2.0. Plot the resulting implied volatilties. (Thi should rproduce Figure 23.9.) =

23.20. Using the CEV option pricing model, set p 3 and generate option prices for strikes from 60 to 140, in increments of 5. for times to maturitf of 0.25, 0.5, 1.0, and 2.0. Plot the resulting implied volatilties. =

APPENDIX

23.A

Here is one way to set up aspreadsheetin

likelihood using Excel.

orderto estimate aGARCH modelby maximum

1. Enter daily prices in column B, beginning in B l0. 2. Compute continuously compounded returns for a period in column C, begins-yea.r ning in row 11. Leave cells A1:C7 empty. 3. In column D compute the squared continuously compounded be 6 2

return.

4. ln cell E1 1, enter the variance of the continuously be your starting value for q.

returns. This will

This will

.

compounded

5. ln cell E12, enter the formula =$B$1+$B$2*D12+$B$3*E1 1. Be sure to pay attention to which cells are absolute and which are relative references. Copy this formula down the length of your data.

-0.20

23.16. Using the Merton jump formula, generate an implied volatility plot for K 150. 50, 55, .

.

=

plot affected by changing a.i to

-.-0.40

or

-0.10:7

b. How is the implied volatility plot affected by chapging

=

7. Suppose that

.

volatility a. How is the implied

6. In cell F13, enter the formula lntf 13) D13/f' 13. Copy this formula down. This is your log-likelihood function for each observation.

to 0.01 or 0.052

-

-

'your

SUM(Fl3:F1200).

last return is in row 1200. ln cell B4, enter the formula This is the log-likelihood function for your data.

=

8. ln Solver, set up the following constraints: B 1 R: 0.0000001, Bl k) 0.00000001, #3 > 0, Bl S 0.99999999 S3. -

'

)' j;'. ;y' j;)'LLL' r' )I(t' (11(:-,* (y' .j' yj'jy'. j'jtd 'y' 1j1j::jjjjjjj2:7.* yjd LL-)';3'))LL,, y:'-y-': j)-.' jp'' y;',' kt4'. ...''.' ..'j'-')j.'.' q,,'.' (j)'. ..'()))jy)(jtt-(;;'. p.' yjllyydj.iy.' ;j'' t'k-2! y,' !j,t' gjjjj'tti )q'(rjr(yr' i',q'-'qjiL'. r' jjd (j'j @' ?' y)jd '-' $' )'(jd y,',--. y'I .-.-.,-f,j1Lff,?f)LL'jj)j, y'. :-'E-.?,'---'))'.' (jk-'. q' tyrtd k' j)jtd (.;(jjjj' (qtyrrd yyy::'. ).s)jjd .,k;;L)LLqL.');q;L)L1'. )y' 4* jjjd r.....ty;(jj$$kj.yyypyjjy'.i$)'t;)j$Ij('. p))kd p,jd .yr'.' .'L;(L)jLq;j;Lf:),--.',' tljikqjttg'tttttjpl'tttlgk j.jyt-tpy'r))d (r' ,!'. $y'jyy'jj)'. L'.'. ,'p'';(jry'. jtd j,','j)' y.,),'. t'p'; q' jj': jrd jy'. jyd (' j')' j'. jkryj'j jj'yyyy-')'jj' jy.jd )' j', jjy'tl r' .y'q j(' j'' j'. )'. y'i 'y El'f( 1* yjd ;;' (j'.' (ti.' ;'jyj:. ','' )' jjjd 'k'' p@'' )(.,.')-'. .yy'. jyr.' )-,' )('. r.y')j'.j ('..'. ..'j#.tj;. p'p' )'jjj... (llr'. j''. );,y'. j'.;,' j'.' ;')#)p'.' y'))qj)krrp).'. ()('. )))):),)*. q'ty;, ('(' l'y l,'yd ;)y;.' j)d .j'y tj('. tj'!. j', pllirs'rtrgj'trgj' :'(' ;)' k)'. rt'g yjyjd jyy'yyyd j'-jy'j.jd y', r' y' yy'k iq'tyr j,'j': y-'q),k:.' )''-'-' jj'. .j' )'rrptyrd jjryjttjtd jrd jqrqrtjpjyzrlt.tj'pg ,j)'. y)d tjp'. yyd .,' rqd ;'. #','.-' y'rg l'j );yj'yyy' yrgd ;'ygtd k' tjd .j' g' y'j q' 7* y'. (' j't' 4* t'tjd '4* t'. )'. )q' jjd f' y'! l'j )q)' qjpt'.. jf'. )r'(k'(.'. !(p(('. @' 1*. ipt'd j'''. k'. ('. @)' .(y'' p' .j@(.h))j'-! ljd yqt@;j'. ;'jjjq)'. j'y.' jjy' .;'' ;'' .)'' t', j'j' ..'p )'.'' ;#j)'. (('. kjd l'kd k)'. yj')jd ,j:.-.ykktt;...' ly'j $r-,(t' ))'. t'! y';-j' ?'-t' )y', ,;' ).!-j-)': ?'. ?' j'. jyjd l'y ))));.' ''jd y'gd j'j!j! jj')'-' y'j p'' ('k' ('; ):' ((' jj'. ).j:d j'y'. j'j)y)j'. )jy'. rtt'jrdqjqjtylk.tjjjpjjd rj'. y(jygt4'. 44* ))j.' yttd $,* :rt'. t')'rj' yyd y' y-' ('. (', yyd )' y'. t:jd yj'. y' (' y': )'..(pr))))t'. (4t4@*.* yj')d f3L'kq3L3L3qLq3q(LLLL((L1. .t@jjp('. '(' .(r'k' jjy-y'. ..';tt'.' ttd.ti. llyjjl' ..@'. tjt'.t;i )'. q' jj))('. ;''rd .;-'j )1(;))(j)))():)r*,* $$' j,r'y:r'. yyjjd ,),' ' pr.yd tjd ?:'. tt.j--.)' 'yjrjyg'tr'. t;d t'(t' tlttjd )yt'@.t)q(jpjry));,' ytd ;-';-'. r' ,',r'-' )t'-' t'. #'t'. tyd t'; t)' t'q 1* tyd ljylly' yjrk'.' )-'. ..yt'r;'. rpq.'i$(llpti.' ..'. t)'. t'.' t.j)'.'. t)i)pq;j'. )jjt))'(' tt'. ;))kt()))))):jy('. tjd tj)))'-.' iLLjkLLkLLkLLkl'' LL;(L3));)));('.f,f,fqLq(kb3LL((;. t'--' t')tq.' jjjjjjt'td t'' t'-.. ttd tjj'. tjj!jttrd t(' tf'. ';-.)' r'

llsjl':. )yy)(y'. ty)'. ';.t' rt('( )y--;--)'. r'. ..'' rtd rtd rz'z' r'I'I' t))j'rt':r r(' r' .jjjj!ry)gjr-'

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% VOLATI

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9. In cell B5, enter the formula B 1/(1-B2-B3). This is your unconditional estimate.

vmiance

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This is your unconditional annualized

11. Set up Solver to maximize cell 84 (the likelihood) by varying cells B 1:B3 (the parameters).

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'.. ('j!li.j((jj.jikj.. j ( .y ... ' (. .(.(!!(.. ;.@. )(r j y(jqj.;.yjjjljy. !j;.(q. . ; jy yr.y..jy ..........;..,.,.....,yj(.j..,yyj(,,.,.j.jyI(y!:.y;.kj!jyj,!t;;..! :... r . .. ; . y j jy j . . j y . . . ;j..j .( E' ( .jyyyyj I g.j ' ' ('q(ji k jjjj.yjj(Iyrjjjgj! .yj,y.sy.jsy. ' ' ...(.. (jj.jj..j!gjj.!(!yy ; y .jj;.jy.g. y y . j . . ' E(((.' 1. EE. ! ('E '. )jij(.!ii y .. . '. (gy(!y;y'.lij.j(jjy!ryjq;iyy y j Ejj . jy'y i .; E..tj: E.. ' ..(' (.i.E!('.@..!j.g(@ i; i;jj!.(iy!! i ijk y! (j;j(.'.. y .l..(.@ .j).j.. ''...'.;..)..E'.(.......k('i!;ij;;i;)yjjyjj!j...yyy(!jjyy!ip.jyy;q.jyjj;yy .. . .E .j . ;.; ; j.q .....i....' (.i(iE.(iiil:ij.!!)I!!(;.jiE.jiq;rkj jgjjjy.y.y .' ..'.E..i.r'(.((.(('.;yg j;;.! .;yyj. jj.)j( .. yj yj.j y j.jj y y.yjjj jy jj))t j yj. .yy!,jl.. E 't' EqEiE,iiiElik'qi,iyg'jil.trilljyry -

.-

-.

.

E'- E- EEE 'E- ; ! E. E.E iE yy.. . @! ! ! EE :E . E-- E : qi i -E . E.E. . E : ; -y -.j;: . ! -. ' j ; ; : ' . ..q(. E..i::E;!q;;.E,i iyq;gy; q( yqy . qq . . ' ' y:.Ei.. . yi !r!.. iyg!Eyy !:- .y ' . @ i . E ! . i ('q(' ('EEjl(yjE.;I.q.kq!;E E..E : (. j jyyr .y j. ('':'( .k j;E: (.,E,..E..;(E.;E..'E',j.q.i.@')..lE,iEE;!ilj(E!,!.;y;).k.!i.i;l;).iE; j . .

.

.

-yyy

.

.

.

''

-

-y iy

:.!.

.

.

.

E

.

.

.

--

.

--

- E --:.-. . ?;E'E ; - ;i E-E ;!.!E q. .'j..q. q. . E . ''''y ---''k i)tr2. --. j.ty-E ii. F.yF.E ! . : I ( . r. I'ijy . ,.. .j --y y -'------,;;---. -. -.. -. - -- -. E - --- - - - -. ..--. -. (;:.);));)))jjj)))r. . Etq):. j)E'tlqi' :.-. i.' :. : E EE,. -E - :E . ! i !..,!E. E j! LE f.iE ..:.; t-- it.?.. . .- EE . k. .:.; . ! ! . ' ...

--).

-

-

:.

12. Solve!

; . ,: . , 'iyj.. . ''ttyjjjr:. -(y; . i;':

;

.

'

-

'.

10. In cell B6, enter the formula SQRT(B5IR252). standard deyiation.

.

.. - . : E ..; : !; i jE ' yy..y..-.. i)li E'' ' E E EE ii ..- E. . EE -'- --'-'. - -.-.- E''.: -:.-- ;E:. - - -

-

7 y!!jgjjljj)l

.;,yy;;;jp,.

y

t0

k. IXVEREST

RAVE MooEus

MARKET-MAKI

The Behavior of Bonds and Interest Rates Before discussing how a bond market-maker would delta-hedge, we Erst need to specify how bonds behave. Suppose we try to model a zero-coupon bond the same way we model a stock, by assuming that the bond price, Pt, F) follows an lt process:

dp

ar,

P =

(24.1)

t jdt + q (r, tjdz

In this equation, the coecients a and q cannot be constants and in fact must be modeled bond satisfies its boundary conditions. For example, that the carefully to ensure rather maturity Also, the volatility of the bond pric should be worth must bond the $1 at maturity-a given change in interest rates affects the the bond approaches decrease as thr of a short-lived bond. Neither of thse than price lopg-lived bond of price a more re:ected in equation is automatically (24.1). ln order to acommodate such restrictionj of the interst riit ad time. specised functions carefully and be ptttst behavir a An alterhatite to beginning with equation (24.1) is to model the behavior of the interest rate and solve for the bond price. If we follow this approach, the ond price will alttonlatically behave in an approp' liate way, as long as the iterest fat jiocess is

reasonable. Suppose we assume that the short-term interest rate follows the It process dl- t7 rldt + c' l-tdz (24.2) =

This equation for the behavior of the interest rate is general, in that the dlift and standard deviation are functions of r. Given eqpation (24.2),what is the bond plice? We will see that different bond price models arise from different versions of this interest rate process.

An Impossible

Model

Bond Priclg

We will srstlook at a bond pricing model that is intuitive, appealing in its simplicity, and widely used informally as a way to think about bonds. We will aisume that the yield curve is tlat; that is, at any point in time, zero-coupon bonds at al1 maturities have the same yield to maturity. lf the interest rate changes, yields for al1 bonds change uniformly so that the yield curve renaa'ins flat. Unfortunately. this model of the yield cul've gives lise to arbitrage opportunities. It can be instructive, however, to see what doesn't work in order to better appreciate what does. To analyze the flat-yield curve assumption, we assume that the interest rate follows equation (24.2).The price of zero-coupon bonds is given by .e-,.(w-J)

p

(p,y a)

t, w)

,

=

N P t Fl ) + P t F2) + F ,

By It's Lemma, and using the fonnula for the bond price equation (24.3),the in the value of the portfolio is Nd P t Fl ) + d P t F2) + d J'P' dl =

,

= N + .

(F2

,

-

-

tlpt,

FI)#r

j (Fl 2

+

t) P t, F?,)#?- +

-

t) atr z Pt,

-

=

change

Fl ldt +

?'#(r,

Fl ldt

Tzldt +

l-pt,

Fc)#/

(24.6)

'

j

(F2

-

2

,)2c'2#(/

-

,

+ rvdt

We pick N to eliminate the effect of interestrate changes, 611., on the value of the portfolio. Thus, we set (F2 1) P t Fa) N = (24.7) (F, t) P t, Fl ) The delta-hedged portfolio has no tisk and no investment', it should therefore earn zero: -

,

-

-

dl Combining equations

=

0

(24.8)

(24.4),(24.6),(24.7),and (24.8),and then simplifying, gives us 1 (F2 Tl )c'2 O (24 9) -

2

=

.

This equation cannot hold unless FI = T2. Thus, we conclude that the l7tp?/Jvalltati' ilnplied %'eqltations (24.2)alld (24.3)is ilnpossible, in the sense that arbitrage ???t7#c! possible if the yield curve is stochastic and always iat. is This exnmple demonstrates Jhe diculties of bond pricing: A casually specifed model may give rise to arbitrage oppo>nities. A crucial feature of bond prices is the' nonlinearity of prices as a function of interest rates. a characteristic implicitly ignored in equation (24.3).The same issue arises in pricing stock options: The nonlinearity of the option price with respect to the stock price is critical in pricing options. This is another example of Jensen's inequatity. The example also illustrates that, in general, hedgillg a bond polfolio based on #lt/'drt?ll does ?lt?l result l a pelfect hedge. Recall that the duration of a zero-coupon bond is te bond , s time to mattlrity. hedge ratio, equation (24.7),is exactly the same as eqution (7.13)in Chapter7. The use of duration to compute hedge ratios assumes that the yield to matul'ity of a1lbonds shifts by the same amount, which is what we assumed in equation (24.3).However, this assumption gives rise to arbitrage opportunities. The use of duration to compute hedge ratios can be a useful approximation', however, bonds in equilibrium mllst be priced in such a way that duration-based hedeing does not work 'l'he

W

exactly.

=

0

An Equilibrium Equation for Bonds Let's consider again the bond-hedging problem, only this time we will not assume a particular bond pricing model. Instead we view the bond as a general function of the short-term interest rate, r, which follows equation (24y2).1

Since J1/ is invested in short-term bonds, we have

d I/P

781

,

-(Fl -

%.

-

In this specifcation, every bond has yield to maturity ?'. We now anlyze the delta-hedging problem. lf we buy one bond maturing at time by buying N bonds maturing at time FI and fnace th difference at the hedge F2, short-term interest rate, the bond portfolio has value I

N G AN D Bo N D PR1 CI N G

?' Wdt

l'rhediscussion in this

sect-lon follows Vasicek-(1977)-

782

k

INTEREST

RATE MODELS

MARKET-MAKI

First, let's see how the bond behaves. From It's Lemma, the bond, wlzich is a function of the interest rate and time, follows the process 1)# 1 lP l# dt d? + #?. + dpr, t, T) = ( a )/' 2 alol.)2

-

=

t7(?')

p# pr

1 :2#

+

cll'j z +

-

)?-Z

2

-

ar,

t,

r)

q (r, /', F)

=

a (r)

P (r, t, F)

#

l

=

G

P (h.t, T) 1),-

We can now rewriie equation

dt +

ot

p# Pr

(24.10)

ctrllz

'P 1 02# .2 + cr (?) + l)/' nl- 2 l- n

p'

(24.11)

-

(24.12)

(r)

' #(r, t,

'

(24.13)

By using equations (24.11)and (24.12)to define a and q, equations (24.1)and (24.13) are the same. Note that a and q depend on both the interest rate and on the time to maturity of the bond. Now we consider again the delta-hedged bond portfolio, the value of which is given by equation (24.4). From It's tzemma,we have

ln order to eliminate interest rate risk, N=Note that by using the delition

we set

P (r, t F2) q (r, t F2)

rl ) q (r, t, ) of q, equation (24.12),this can be rewritten J'r (?',t, ra) N=Pr r, t, Fl ) FI

P (?-,t,

lf you compare this expression to equation (7.13),you will see that #r(r, t, F) replaces duration when computing the hedge ratio, N. Substituting equation (24.15)into equation (24.14),and setting dl 0 (equation (24.8)), we obtain =

-

,

,

Substituting equations

(24.11) and (24.12)for

1)2# 1 2 l ) + Et'l(? ) -o. )?-:t 2

.)4(/.

t)- l

,

(24.17)

a and q then gives us

pP

/.)j

pP

..).

-

)?-

ot

rp

.()

(24.18)

When the short-terminterest rate is the only souree of uncertainty, tlispartial ltf/rnzlf eqttation ???I/./ be satisjed /J.)/ al' zero-coupon bond. Different bonds will have different maturity dates and therefore different boundary conditions. All bopds solve the same

PDE, however. The Black-scholes equation, equation (21.11),characterizes claims that are aftmction of the stockprice. Equation (24.18)is the analogous equation forderivative claims that are a function pf the interest rate. A diference between equation (24.18)and equation (21 1) is the explicit appearance of the risk premium, c (r, /)/(?', t), in te bond equation. Let's talk about why that

?-

=

a

(1-,t,

T2)

-

r

(24.16)

q (r, t F2) ratio the /5$// bollds is equal. Since both bond This equation says that the 5'/7t7//, random prices are driven by the same term, #Z, they must have the same Sharpe ratio if they are fairly priced. (We demonstrated this proposition in Chapter 20.) ,

w/b?-

,

happens. ln the context of stock options, the Black-scloles problem entails hedging an option with a stock, which is an investment asset. The stock is expcted to earn its risk premium, which we will call $2c. Thus, for the stock, the dlift term, which is analogous to c(l-), eqtmls '' + 4/(r The Black-scholes delta-hedging procedur eliminates the risk premium on the stock. By subtrqcting the risk premium, we are left with the lisk-free rate, r, as a coeficient on the ) V/OS term in equation (21.11). Fc ltet-est rate, /J.: contrast, is ,1:)1 the price of (kpl westment asset. The interest rate is a characteristic of an asset, not an asset by itself. The risk-neutral process for the interest rate is obtained by subtracting the risk premium from the drif't. The lisk-neutral process for the interest rate is therefore .

dl-

,

,

FI ) a (r, t Fl t (r, ) q

783

.1

et?-, t, Flt'/r + t.?(?-, t, F)#Z

=

F)

$

-

'

-

(24.10)as

dpr t F)

Denote the Sharpe ratio for #Z as /(r, l). For any bond we then have ar ' t F) ?' = 4(?-,t) qlh F)

.

(24.1),but we can desne terms so that it does.

This equation does not look like equation Let

1

-

p#

NG AN D BON D PRICI NG

=

(t-l(r) -

(7-(?-)4(/-,

tljdt

.#

o'rldz

(24.19) .

The drift in this equation is what appears in equation (24.18).You can also confrm that equation (24.18)is the same as equation (24.17). Given a zero-coupon bond (whichhas a terminal boundary condition that the bond is worth $1 at mattlrity), Cox et al. (l985b) show that the solution to equation (24.18)is

PLt T ,

'

r(/.)q

=

E*J

-&tJ'rlj

(24.20)

where E* represents the expectation taken with respect to risk-neutral probabilities and Rt, F) is the random variable representing the cumulative interest rate over time: T

.R(/, F)

(24.21)

=

I

Thus, to value a zero-coupon bond, we take the expectation over all the discount factors implied by these paths. We will see the discrete time analogue of this equation when we examine binomial models.

784

%.INTEREST

EQu I Ll BRIU M SFI ORT-RATE

RATE MODELS

the bond payoff Keep in mind that it is ltot correct to value the bond by discounting

by the average interest rate,

#

=

E* g.R(/, F)):

Pt, T,

?-)

+ e-k

on single-variable models in this chapter.

Delta-Gamma Approximations for Bonds lt's Lemma, One interpretation of equation (24.18)is familiar from Chapter 21. Using of the interest distribtion risk-neutral the under bond plice in the change the expected rate, equation (24.19),is DP p/7 82# 1 l ?)j o-ll-jlr, (J(r) a + E*(dPj = + )?-Z )?-f 2 nt .

-crf?-l

-

-

-

'

-

Equation (24.18)therefore says that 1 -f*(##) dt

=

MODELS

Q

785

We discussed bond duration and convexity in Chapter 7. For a zero-coupon bond, dtlration is time to maturity and convexity is squared time to maturity. Conceptually it seems as if duration should be the delta of a bond and convexity should be gamma.

However, this is true only in the bond plicing model of equation (24.3). For any correct bond plicing model, duration and convexity will be different than Prjp and Prr/P. We will see examples of this in the next section. 'impossible''

different Beeause of Jensen's inequality, this seemingly reasonable procedure gives a equation (24.20). bond price than details Different bond price models solve equation (24.20),differing only in the lisk premium. the of modeling and the of how ?' behaves with apodel of To summarize, a consistent approach to modeling bonds is to begin differential equation that the interest rate and then use equation (24.18)to obtain a partial eqation), Blackcholes the the really is equation as same describes the bond price (this conditions, boundary with together the PDE Using time-varying interest rate. but with a lf this seems familiar, it should: It is exactly the we can determine the price of the bond. procedure we used to price options on stock. of The derivation of equation (24.18)assumes that bond prices are a ftlnction a allow bond prices single state variable, the short-term interest rate r. lt is possible to empirical for having bond support is and there variables, additional state to depend on estiprices depend on more tlan one state variable. Litterman and Scheinkman (1991) that a three-factor model typically mate afactor model forrfreasury bond remrns and find bond's variability return. They identify the three in of the than a 95% explains more single most imporfactors as level, steepness, and curvature of the yield cuve. The of the movement in l'ates, 90% almost for interest accounts of level tant factor, te of interet rates explains why level of the ovemhelming importance The bond returns. duration-based hedging, despite its conceptual problems, is widely used. We will focus

-

BON D PRICE

'

'

-

(24.22)

/-#

lisk-neutral distribution, This is the same as equation (21.31)for options'. Using the lisk-free rate. bonds are pliced to eal'n the The fact that bonds satisfy equation (24.22)means thaq as in Chapter 13, the deltajp/creur approxnatiollfor #7echaltge n a bondprice holds awc//y gtw?7//lcl-lpM exactly the bond for are not a rate ???tp1?,u one standall dejtiation. However, the Greeks convexity. same as duration and 'fthe

24.2 EQUILIBRIUM PRICE

SHORT-RATE

BOND

MODELS

ln this section we discuss several bond pricing models based on equition (24.18),in which all bond prices are driven by the short-term interest rate, r. The three pricing models we discuss Rendleman-Bartter, Vasicek, and Cox-lngersoll-lkoss--differ in teir specifcation of J(?'), 0'(?'), and $r). These differences can result in very different P

Iicing

implications.

The Rendelma' n-Bartter Model The simplest models of the short-term interest rate are those in which the interest rate follows arithmetic or geometric Brownian motio. For example, we could write

#?'

=

adt + O'CIZ

ln this specihcation, the short-rate is normally distributed with mean o'lt. There are several objections to this model: ance

(24.23) ?'a

+ at and vari-

shorto The short-rate can be negative. lt is not reasonable to think the llonlillal would cash be since if it investors negative, prefer holding under can rate were, a mattress to holding bonds. . @The drift in the short-rate is constant. J.f a > 0, for example, the short-rate will, drift up over time forever. ln practice if the short-rate lises, we expect it to fall; i.e., it is lneal-revertillg.

@The volatility of the short-rate is the same whether the rate is high or low. ln practice, we expect the short-rate to be more volatile if rates are high. 'l''heRendleman and Bartter (1980)model, by contrast, assums that the short-rate follows geometric Brownian motion: dr

=

at.dt + o'rdz

(24.24)

While interest rates can never be negative in this model, one objection to equation (24.24) is that interest rates can be arbitrmily high. ln practice we would expect rates to exhibit mean reversion', if rates are high, we expect them on average to decrease. The Rendleman-Bartter model, on the other hand, says that the probability of rates going up or down is the same whether rates are 100% or 1%.

%:INTEREST

786

'.

EQUI LI BRI UM SHORT-RATE

RATE MODELS

--jjjiilijkky' --' -' :'')' '''''--?'''';;iii..f -l::::::f (S'E -y)' dl!!!!,,f ,j:::::' j;:!,,f ;'-y' '(y' qjf )' E ill:jilll.d llij!i:d dl:!i!,f dl:!!!!,f (' ' EE. : (E jjE E' j').' yl):f ,'y'-' y' y..y' ;'y' y'. r' qltlE: i E .:. ' E ''':'E (::ziiij-..:-'''!jlll);lp.d .iI222!:' t'':';'''q' Ir..'f y' dl:::E::f t,f (' yy' E (E.' 'E . .: .'. ' tE ''.;;!;' (q' ..kyi (E.(E;(;'j!( jq.;.(.''.(.'.'(..E''''-'.'(.''..q.((': .. : ii.. :rr':.E(..k:((. ... ..! 'y.(E..!.E. .. . r(('.' ( .. . - i - ' -. -E.. . . ..E-. .- .jE E . .E . .:;.; ,. . q.

E( ' '. ' E ( '7* jyf r' r'rf .11:::* jjjjjj!jf kj'i ygf ''r( q(Eyj jtf jyf jjq:yg:,,f j'(' 117-$::.* jjjkg!j;,j;f llr...:l.f tjf rs'yyj ;'tr'(yyy' )' l'tt-r y.;')' ( . (EE.. j (..g(.' ' @' (;;;'. 'yk-kj:jjjjrj'. ;'rt-r;(j(r:jr'. 'j qq ; .q -..y. E.;-.y.'.-. E.'. 1. ' :'r'jy.' )ii.(r. .?-).. '. . 'y.j2jj;jjjjj;. 'ii..('EErq;li1 ; g.r . . y- ...(y j. (kjj-yr y()r. .jyy ..yyyjy.jy....y.yjjyy..jrjyy-y.. yj.. ((. : j y..y . jjjjj-k;k,. ;yyyy jjyyts l.'.' yy;y,jy. .. .jt.j.. ... .y.rt;,ijjjj. . .. , ..... . . ....j.. -t;'' '(' '''

t')'r'i

.' . -

.

.

. - -

-

........

'' '

.

. .

' '

-

.

. .

.

.

- .

. . -.

.-

...

. .

. .

''y; '( ''

'

'.

.

..

.

.

. .

.

:11.

..

''r

:

. .

.

:

'. '

'

.

.

. .

.

.

.

...

.

.

. .

..

,

....

...

.- -

,

-

.

-,y .

-

-, .

-

-

. --.

-

.. -

-. . ..

.

., -

''j ... .j

.

.

-

- . ..

.

-

,1:1,,,. jjyyyg, Ir'... Il'...pl.s iiie .1Ir::. .1It::. ,!(((2II .tIk::.ii'Ir...!.l .1It::. k:,,,,:y, .,1t2:, ii'Il'...rli ,I::::rI1. li....:ll e e I!iii;. a e iiiIr....,'. a I;....,l-

model. Msumes a

=

0.2, b

=

0.1, and

o'

=

0.01

1 a lP tr )?-a + Lab

.

i'

(IIi2r:1IL. iliEi!j,r ljjijj!;;q i:jq!ilt. 'rlllj'illi. !. (.... y'. (i.. qq (. (...y.. .. (qllliil::;. . lljjtk- :tii. 1i iEgik qigl. tt.'l.. (k kkk .. ;.) ..' !li) 1lj qlj IjEFjZI .j iz ilsj:.'. 11) )' #..klr)l tj. kjiii w.'' 0.01 5% E E' ' .. -. i .. ;. q....- . .( . . ..E. -

E .

.

.

'.k

10% l5% 20%

;' .'. .. .... . .y .

. ... . . .

-. .

.

E. .. -

.

.

't

' E ' . . . .. E ... .- . : . .- g. E . . . . 't..

.E

!11!E!2E'

,111:(((((2:2.

..

.

.).

r)

-

p.p

c'/)

-

p?-

')... ... E .

P g/,,F,

0

-(/.))

?

At

=

$

787

bP

?'#

-

at

=

0

subject to the boundm'y condition

r),-f('.F)r(?)

(24 26) .

where

-0.01

-0.02

At, F)

=

F)

=

Bt

e

FBt,Tt+t-T)-Blo'lj4tt

(1

e-*l--blja

-

-

0.5c2/472

with being the yield to maturity on an infnitely lived bond. FJ

The Vasicek Model kasicekmodel

MODELS

,

P = b + c4/c

The

+

The bond price fonnula that solves this equation #(F, r, r) = 1, and assuming a /: 0, is2

.E - .E E .

. .

). E.( E.

PRICE

(24.25), equation (24.18)becomes

E

-. -y-

-

..

-. - -.y .

BOND

incorporates mean reversion:

dr

=

t7

b

-

rltlt

(24.25)

+ o-dz

rldt terpz induces This is an Ornstein-uhlenbeck process (see Chapter 20). The ab = = These parameters 1%. 10%, and. 20%, b o' mean reversion. Suppose we set a basis points. The is 100 short-rate for the one-standard-deviation imply that a move > short-rate th If b, interest which shortlterm level revert. rates r to parameter b is the illpstrates rise. Table is expected 24.1 < 1the short-rate If to /p, is expected to decrease. mean reversion. The parameter a re:ects the speed with which the interest rate adjusts to b. lf = 0, then the short-rate is a random walk. lf a = 1, the gap between the short-rate a 20+, we exlect the rte to decrease in and b is expeeted to be closed in a year. If a 20% of by the the first year gap. N,ote also that the term multiplying #z is simply o', independent of the level of the interest rate. This fonaaulationimplies that it is possible for interest rates to become negative and that the variability of interest rates is independent of the level of rates. In the Rendleman-Brtter model, interst rates cold not be pegative because both (jj the mean and variance in that model are proportional to tjye level of e j n (erest rate. Thus, as the short-rate approaches zero, both the mean and variace also approach zero, and it is never possible for the rate to fall below zero. In the Vasicek model, by contrast, rates can become negative because the variance does not vanish as 1' approaches zero. Why would anyone construct a model that permitted negative interest rates? Vasicekusedequation (24.25)to illustrate the more general pricing methodology outlined in Section 24.1, not because it was a compelling empirical description of interest rates. The Vasicek model does in fact have some unreasonable pricing implications, in pallicular negative yields fot long-term bonds. We can solve for the priee of a pure discount bond in the V.asicek model. Let the Sharpe ratio for interest rate risk be /. With the Vasicek interest rate dynamics, equation -

The Cox-tngersoll-Ross Model The Cox-lgersoll-lkoss rate model of the form

(C1R) model (Cox et a1., 198518 assumes a short-term interest

=

=

dr

ab

=

rldt

-

+ cvflz

(24.27)

The variance of the interest rate is proportional to the square root of the interest rate, instead of being constant as in the Vasicek model. Beause of this subtle difference, the CIR model iatisties all the objections to the earlier models: 0, the drift in the rate is @It is impossible for interest rates to be negative: If ?positive and the variance is zero, so the rate will become positive. =

* As the *

jhort-rate

rises, the volatility of the short-rate also rises.

short-rate exhibits mean reversion.

'l'he

The asjttmption iha t the variance is proportional to V-/also t'tlrfls out to be convenient analytially Cox, Ingersoll, and Ross (C1R) derive bond and option plicing using this model. Th lisk prrium formulas in the Cm model taked the form

4(,-,r) /,/--r/(y =

z'When a

0, the solution s equation (24.26),with 0.5rz4(r-J)2+tr2(r-J)3/6

:4 B

When a

=

e -

F

-

t

0 the interest rate follows a random walk; therefore, F is undelined-

(24.28)

788

%.INTEREST

RATE MODELS

EQU I LI B Rl U M SH O RT-RATE

With this specification for the risk premium and equation (24.27),the C1Rinterest rate dynamics, the partial differential equation for the bond price is 1)2P 1 -0-2,+ g/t/p :/-2

.

pP

-

?-)

-

2

-

r/)

+

Dl-

pP -

)r

?-'

=

0

The Cllk bond price looks similar to that for the Vasicek dynamics, equation with A(/, F) and Bt, Tj dehned differently:

#(/, F,

?-(r)j

=

At, T)e

(24.26),but

'j' j'(' .t'. t'))!:(T(' 'yjf t;f Eptjrtf (' .jjyy' gjjyy;jj,.;jjd y'g r' jgf jqf jjj'jjy (td y:lf );' .j' j.' j'y'y' jjjd jjyyjj---r'yjjy. yjj.d (' )' ytjf r' tqf ;)' p' k' 'y('. jt(jt' yjf j))'(j;'. t';',,t#'g;'. q'. y' (tf l'q q' E'''' ( ' E (.(j' y ( ' E' ' . E'' '. ; (.' E(. Ejj (g;'ij' yq!(qj: (E'.(ij 'E.''' . . . .(( r . . j.. j g. . . y)( ('qy('j7y(('!((q( . . tj 'E.(.'' j(g.'.j)(j .' j. !' ql'l!l!ryrq :'E E..Ei jEEi! . jE. E.ii.. E-.. EE.E E!. . . !.1.E. yE .. E . .) .f.; . )! . . . .I.. . . .... ..qq.gj.!...ri .. .q.E;. . .j: ! j ; . !E(gq. .jy. qy.iq! .(.$. E. E . y j., q y..y q 2 ; . r . . 1I1Ik,--1III. . y.( . E E gy... yyg.yyj.yg ..jyy. . jy. .)j.. E g.yjg ;..y . .. .LLkk ).. ;..y ....... .: . .. :...

.'.'y. ' .. . . . ..

y''l!I::::11kr-. )(' (' );' t('. tt;'r''ilII:.-' 11IEiEiE,'. 1111F14!7*t''' tjg.' r' ttt'. .. ri . ttj!. . y yr . k-y@.. :-1:::1E!1:... .j y.rqj !IIi::;lit:. . 1111). ;;;kq))-)))-. . . ' ' . . . . . . .. . . . .. . Yieid curves implied by the Vasicek and CIR models, assuming that 0 * 05 a 0 2 r b 0.1 In the top panel, c 0.02 in the Vasicek model and c 0.02/ 0.05 = 0.0894 in the CIR model. In the bottom panel, o' 0.10 in the Vasicek model and o' 0.10/ 0.05 0.447 in the ClR model. In aII 0. cases, $

'. *(). ''::

:

:

:

.

-

: !'

i

'

:. ..

.

(,4. r,lj)) .

where

: ' ' ' ::

.

- E.

.

.

.. : .. . . .

:

.

'

.

:

.

-

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.. . . . .

-

.:

:

..

.

-.

. :

' :...

'' :

:

.

-

.

.

. .

'''

.

-

..

..

:'

:. --.. -. . .

.

---.

. . . . .. .. .. . .

--

.

:.

: .- : .- E . . y

-. ..

zablo.l

A t, F)

=

2Y -

a+ $+

)z)(dvtT-')

p,(,1z(T'-/)

Bt, F) ?=

=

-

a+

$

-

1) + 2/

+ 1z)(c7?(7'-''

.

789

.

- . .

.,,jj(j,,.

jjjs

.. .

.

.jj)rggyj;,,,, kyyyjjj rjry,.

jjrjyrr

clR

() (s '

.x.

0.08

vasicek

xv..=x>xx>>.x,,>>'*'*'.* -.--w.-..

.-........,1

'q:

EE E: E .'.: '

-i.I I

-

.

-

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:

.

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'..'

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.

y.-

E.

.,

.

-

.

- .yy-

--. .

.

.,..E. ...

vear t)

vear 1

bnd Tree of 1 implied by the prices tree in Figure 24.5. Tiie price at each node is 1/(1 + r), where r is the rate at the corresponding node in that figure.

vear 2

vear a

:'

:

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,$0.8321

,

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.

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SE

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EL

%

803

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y,.

i .

.

.

-

M OD

jjryyygjjg jjkjyrry iijyyjj;j

.

,

.

y -

ikjggjj.s ryyjj,.s $0.8832

$0.8644

$0.8798

$0.9091

-' .q-li.lir-..,lisi,,!f ..'ljilsj''.'ll.i,,. '' -' ;',.' -.' il'jyj :'' -.--y-..,,.......(k,j'. S' ;'--' j'774* ,'-)y'(' y'E j'i'' .

.' .'..'

j'(' p' 1* )' Illi)f pIi1k-iiIr'. ().k)-'-'. jy'jf j'! r' jjjjjjf )'. ip!piip4!py'. Ilii!!i!:f rti''i y..tr,'yjj)y'' (q' t'. r,-'.' t)jf 'y' ti-f t'. illi:)f . ' jj jy.(;ti. '.'(. L-jj,jLj,j, ';j((k,. jg .j..'.yjj..y r'.).j ytyyy .tj t,. ;jy..ty . )LLfLL-. . 1. t r . . t-. ' ... .... . .... . ... . r. . Tree illustrating the evolution of the 3-year zero-copon bond, based upon the prices in Figure 24.6.

j'!' (' :';f .* .y' E);tf )' q--,' t'1;* r.'.)' 'jy',,,j,jjjjjj,.-. ,'y' q' ''qqq'll!f j'p' ;')' (' jtE E i i ' ''E (t'' r' (' y' (y' ;j($,-'.. E ' E 77!4: j'(Il.')tt)t' r,lr:r,f r,lr:r-f ::;!!:.' r,lr:r.f tl?f pf t',111:::122;.F*. 7(((q (r.r.i';qgl i')((. (' q;)i 11)444))4! y: !t)jpl ( 'j.'r(.'E. j.EE.(.'(.(..(((E.ti.q.E'!'q(:i.(.;t..((.'IE'.' q( ' 'rtqj. .''( ( i('(!'( ( '(( j (. ( '@ . y;)l . 'E!- E . i. E. EE: E E i; E: !.i .p; rtEE;jj. !.E jjjjjijjjjy,. !. :y klE' . ... iyjy(qgyj . .; .rl gjjy t q j. - y rEyq.y.-(.; y.j.;!y.qqy-.(yyy.jqgy; k: fl r . . i . i (.E :. i .. E; !.. ;E !- .-..-.-. - ---; . (r.. k, .Eil. E . . ' -

E

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.

-'.

.

.. .. .........

$0.8906

$0.9153

-

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.

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i.

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$0.9023

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-

. -

:

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.

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.

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-

.

..rliptjr-...lk!,,-

a

dll::q:,,-

a r:Ell..

a

$0.8321

50.7560

Vertfy'ing Yield s The rate at the first node is 10%, which corresponds to the current l-year yield. We can compute the price (and thus yield) of the z-yearzero-coupon bond by starting in year 2 and worling backward. It is slightly more convenient to use the tree of will be worth either $0.8832 l-yearbond prices ilzFigure 24.6. ln year 1, tlw z-yearbopd 10.8219. of Thus, discounted expected plice yield the yield of 13.22%) (a or $0.9023(a time is 0 at

$0.9091x (0.5x $0.

83 2 + 0 5 x .

$0.9023) $0.8116 =

Figure 24.7 illuseates the eee corresponding to this calculation. The price of the 3-year zro is cbmputed in a similar way. Worling backwards from the year-3 nodes we have

$0.9091 y

$0.8832)nd repay the loan at t the contract payoff in year t is -

i

-

rg)

b. The fonvardinterestrate, rc, foraEurodolla-styleforwrdrateapeement that settles at the time borrowing is initiated. That is, if you borrow at t 1 at the l-year rate and repay the loan at 1, tlze contract payoff in year t 1 is '',

year zero-coupon bond price gen-

24.9. Verify that the l-year yield volatility of the erated by the tree in Figure 24.6 is 0.14. 24.10. Verify that the price of the l2% interest rate

cap in

24.11. Using a binomial tree like that in Figure 24.10, rate 3 years hence in Figure 24.5 is 14.0134+.

Figure 24.9 is $3.909.

velif'y that

the l-year forward

For the next four problems, here are two BDT interest rate trees with efective interest rates at each node.

Tree #1 0.08000

0.07676

0.10362

0.08170 0.10635 0.13843

0.07943

0.07552

0.09953 0.12473 0.15630

0.09084 0.10927 0.13143

0.15809

annual

-

-

c. How is the difference between r.f and re affected by volatility (you can compare the two trees) and time to maturity?

/

24.15. %ouare going to borrow $250mat a floating rate for 5 years. You wish to protect yourself against borrowing rates greater than 10.5%. Using each tree, what is the price pf a s-yearintlrest rate cap? (Assume that the cap settlos each year at the time you repgy the borrowing.) APPENDIX 24.A: THE HEATH-JARROW-MORTON The Black-Dermgn-rlby

MODEL

model illustrates one particular way to construct a binomial

tree from data. There are other ways to construct trees, such as the Ho and Lee (1986) model, which we do not discuss here. The Heath-larrow-Morton model (Heath et a1., 1992) is notable for proposing a general structure for interest rate models, one which contains othermodels as aspecial case. Theirbasic insightis thatno-arbitrage restrictions

.',?' ,.,' .-' .j' ;',',',,jjjj:!pjjj-..' .-.' -.'(' '.,r' -j' -.' .'' :','';'' .'.'t)' ..f' :t.' ..;:' '',)2' .'-' ,',' -.j' ,k' -','-y,:,' -'-y' ,'.t' -'-' -'.yjy' -ty'

.,' ;'kt't,,d !(' ijd j'q' g' F' 'y' (' ;'j)t,zd ,'!' J' j))sd .jgEik' i'@' ';j' :';jj)t))d t)d j'CE;ZS;!J,X )' f:tt.qd t'Ej5' td g' '' ())' f' ttd 5* ;'Eyd jj' ?;' r''d )' '''jjj' Egq)d ijd F')' yl))t:jj)kld C' f' y' j.)' jyd jyig'ig' F' jgd ?' ytd jjy.:jjd ,:gyj(.' ,j,,rjr' 'jj:ygq:jpd (' ...IIji;jj,;;;p,..' :''1* k' f' td i'd 'd4* (tt' (y' E.' (' )' ,(.j!,' q;,' ld )(' f:' :'k' )2;' ;';r' t'6' irlrllick)klkz)d ljd ljtltttjli..ld Err' :(j' #' ).r' ;yy' '')' ;;' y' yrd f)d Etd (-tggd y' i.'qt$$t,' kji''pd j;yyd )' igjyrjj,d '(' )' .j' y' j'jrd @' t:t:tsd tri';d ;','r' )y' k' ljd ,'ktd r;d ti)d ;(t(' f;' :;' :'y' rjjd tli';gd h' (jtlttll-'ryyd t''ij.td 'r'4* Ijt' ijd jjd j)d kjyyd jtyd jytd t'jjdty'jjy't)d tj,tjd yjjj.d j'g' q' rjd ;t' jyd 1* ;r4' yyd j'r'(y,t'-q' jrjd g' r' j)d ;');(' 'j' ).;)' (' @' j)tjd j-' ygd q)'.' jydgiyy'pyd ))' jttd k)jd )(t.k' j)jd itd j.qd jpd yjd ji'y;jd tld ytd jjd ;)' rtd :'q' (' jjd j-)jd yjd j'j-(' yd p' jyd y' ipd ))'(' (jj' t'ytd )' ijjjjjj;;d j't,,d y' ljd qkd )' l!jd tjd y'y' jkd ;'(j' jjd ;kt' td jrd (' j''p' yyrtttgd j.jd )i' ;(' jjj)'j)' .j:' ll).)yyrd IF'I);-;X );tjd jjtjd j)td j'td g' y' jyy)d j)'t),'(.f(.(' r)' .j' jyjd j'yd jtd k' jyd yyyd gtd j'j'yyj)F'j1:' t'f' (y',td jt't,.)';jjy',' j'yyjd p'

1* gyd y' j:ijjj!!;;d )' j.jd 4* (t' j'lqd (' liyd yd jgr;;jl!rj,d y' yyd jyd (t..' tyd )' t'$' /(jjjr.' )t;t(' r' )(' ('(t' (' jj)d )p' )jd l'-y' (td jd )t*.* #' )y' )tj.-' 'r' y' .j' yr'yttd .p)')' Lq')jj' ))' j)' j(q'j.jyj' @' (j' ytd (q' jpd ytyd j'yyyd pyd g' 1*4jr* j;rtd yjd 'yyllti'jd ;-'#' 1* ((' tjjd jjjgrd ltd yjjj'gjyjy;'jjjyy'yy;j.y' )'t' yjt'yyjjd yjj.jd jyyiyyjjjd i'ltd ;'t' yrd y' t'('jg:jj,,,,d )' jjjjjjjjjly,d 1* q' )..' qjd p' (' )j' y';)d (jqjd ljd jyjjjjjjtk-kjjd.jjljtjdjtkllyjjjjjkijtljjjyylrj'tjd )jjjj)jjjjjj)j;jj)t)'..' )' y.' t''' tjd y-ry,jj'.jd t'yjd i)')' tyjgjjty'jjd tyy'y',' yy,-dyjjjgysjyr'yyyjyyjj.d tjd t'j'.y' t;d tt'.jy' trd t'd t)d t')d t;yyd .(y' ryd 'y(5;jryy'.' t(' ttd td .r' .t' yj)';' ty'yyd tyjjjjjd tyjjjjjj:;;:jjyd rr)jd rqd rqyd r' r.rl)d r' r.;d

tjyy'jd ryjjjjj,..d -y' Etd k,,d -'-' ;'g' E(' ,';''yyd ;kL' (','-' --.,--'-' j'-,',-':y' ,--.t,' Sllljljktrrtji.;lrjtytt!?d ,E..j;tjj' '-' f)d y' L:' S' :'pt.:)d ;'r' ?' ..';' ;)' f''' -rd ' -'-' k)d -(' Ejd

812

% INTEREST

RATE MODELS

require that the evolution of fonvard rates (or equivalently, forward bond prices) hinges in a specife way on bond price volatilities. When you adopt a specisc volatility model, forward interest rates. you implicitly adopt a specisc model for tlae evolution of IiI'IIC between volatilities and forward rates, suppose the singleTo understand the and we have two zero-coupon bonds with state valiable is the short-term interest rate, r, FI ?') implied forward zero-coupon bond > The r) nd Pt, F2, with F2 plices Pt, Fl price between F1 and Fa is .

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have a complex portfolio containing stocks, bonds with mamutjes and with various embedded options, and instruments denominated ouaeyjcus. The form of these instruments could be simple notes or complex kn (jkjsreyjt Op tions. Value at risk (VaR) is one way to perform risk assessment for such a pprtfolio. vjje.jtjea yjjjjof value at risk is to estimate the losses on a portfolio that occur with a given .jjyyy .jjyyyyjj, .jjyrggjj, yjjyj jjjl;:r,jrjjgr.. with an estimate of the distribution of outcomes we can either ask about the projnoilit.y otzlosing a given sum (e.g., what is the chance our loss exceeds $5m?) or ask for a given probability, how much nzight we lose (whatlevel of loss do we exceed with a 1% probability)? For example a derivatives market-maker could estimate that sr a given portfolio there is a 1% chance oy josses in eycess of $500000. The amount $5:: (s() js then the l-day value at risk with a 99% level of conidence.l In general computing value at lisk means fnding the value of aportfolio such that there. is a specised probability that the portfolio will be worth at least this much over a ocriven horizon. The choice of horizon and probability will depend on how VaR is to be use (j jj.e g ujators have tjkjjueyu

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proposed assessing capital using the 99% lo-day VaR (seethe box on page 815 fpr more details). lkiskmetrics'' (seeJ. P. Morganmeuters (1996:,developed by J. P. Morgan in the mid-1990s, is one comprehensive proposal for a value at risk methodology. Much ofa the discussionin this section, especially forbonds, follows thevskmetrics methodology. Before we discuss how to compute value atrisk, recognize thatthe ideas underlying liskiness of bank portfolios. risk assessment matter in contexts other than measuring the and capital can pursue one of two investment For example, suppose a firfn has $10min million. One year, investment A rettlrns $12 million for opporttmities, each costing $10 :

$0 sure, whereas investment B rettlrns $24 million risk-free and the B is idiosyncratic risk of the investment probability one-half. Suppose the same rate is 10%. Standard investment theory will assess both projects as having 11r111 will its entire lse time half of B, the the with investmpnt positive NPV. However, additional make inkstments, ordr capital. In of to its investment and therefore all must raise additional capital, a costly process. Once we accout for the costs the (11-111 associated with 'losing a11capital, A and B may no longer seem equally attrctive. More much of a Erm's capital is at risk with a generally, managers will want to lnow how affect project selection.z terefore given project. Risk assessment can level as well. Suppose you are the personal of Distributions outcomes matter at both how much to save nd how to decide will need You to planning for retirement. For any stratejy, a key other and assets. stocks, bonds, allocate your savings among this strategy probability thatby following you will fail to achieve the question is: Whatis retire'p This is not the the time savings by retirement level of you a desired minimum pnnilegs--t of probability leaving lzigh with you ask, but a only question to a sategyconsideration. careful We call grounds-should for other desirable on no matter how lisk underlying but chapter, the ideas planning in tltis personal will not discuss snancial decisions. assessment can be used in making personal decisions as well as corporate regulators mentioned, First, risk. of value can at as There are at least three uses Second, instimtions. Enancial managers requirements for capital use VaR to compute risk-management decisions. Third, can use V:.R as an input in maldng risk-taking and bank's models. For example, if of quality the the also use V:R to assess managers can trading operation will lose $lm that particular 5% is chance there that a the inodels say a 20 days (5% of the t'ime) the ading then l-day horizon, on average once evel.y over a frequently, the models are this size of lf losses should more lose occur $1m. operation such lf losses activities. bank's risk the little occur less frequently, the to assigning too much risk. assigning too models are Most of the examples in this section use lognormally distributed stocks and linear normal approximations to illusate VaR calculations. Currencies and commodities can be modeled in this Fay as well. Although for long horizons it might not be reasonable with

with probability one-half and

2See Stulz(1996) foradetaileddiscussion of the linkbetween

nvestmentdecisions

andriskassessment.

3Bodie and Crane (1999)for example use Monte Carlo simulation to examine return distributions the suitability of financal products for retirement savings. assess

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to treat commodities as lognormally distributed, for short horizons this is generally a reasonable assumption. We ignore the possibility ofjumps. We discuss bonds separately.

Value at Risk for One Stock

is the dollar return on a portfolio over the holizon h, and f @, /?) is the Suppose of distribution retunas. Define the value at lisk of the portfolio as the rettlrn, x (c), that Probty such S xh (c)) c. In other words, xIl (c) is the c quantile of the ret'urn horizon h. the distribution over .%

=

Value at risk measures the loss that will occur with a given probability over a specified period of time. Notice that the defnition of value at lisk requires that we specify both a holizon, /7, and a probability, c. Suppose a portfolio consists of a single stock and we wish to compute value at risk the horizon h. If the distribution of the stock price after h periods, S, is lognormal, over

.

816

k

VALU E AT

VALU E AT R1s K

Rlsu

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we have

With 95% probability, the value of the portfolio over a l-weekhorizon

;(' E( . tgt ()..k

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As we saw in Chapter 18, if we pick a stock price Plice will be below u is Probts

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0 (if you multiply the risk by 1O,the risk measure increases by 10). -

SIf there is no risk premium on the stock

putprice.

(25.21)

i.e., a'

=

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this calculaton

produces the Black-scholes

838

% VALU E AT

PROBLEMS

RlsK

at the 99% level for the writer of the two options is therefore $1 VaRt-A

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This expression has the opposite inequality as equation (25.21).so V:.R is not subadditive lisk, as measured by 99% VaR, in this example. In words, te institution can eliminate by undertaling the two activities in separate entities. As adifferentexample that illustrates this point, suppose you are comparing activity C, which generates a $1 loss with a 1.1% probability. with activity Dj which generates a $ lm loss with a 0.9% probability. Any reasonable rule shpuld assign greater risk (and mquire more capital) for activity D, but a 1% VaR would be peater for C than for D. These examples highlight an intuitively undesirable property of Vak. aj a risk large measure: a small change in the VaR probability can cause VaR to change by a ,-PA, amount. For the written cash-or-nothing call in this example, ;t 0.81% VJtR is while the 0.79% VaR is $ 1 PA In contrasi with VaR, tail VaR and the cost of insurance are both subadditive. lntuitively, tail VaR takes into account the distribution of losses beyond the V:.R level, so it does not change abruptly when we change the VJI.Rprobability. To qee that it is subadditive in our examples, the tail VaRs at the 99% level for A and B are -

.

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With the same consdence

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ggjkjkjjj.'. j'y' q' 1* f' :( r'r7'!llqr)'r'r?!' ?' tjk;,,,d )' (' 1* j'1* j' !' 'i!'(q('(' ( ( (; @(: ; (jj1j::jj12rjj1(.* jjjyjy(jjj,'. (r' .' (liti. ;(iy( tj.pil.r (i. ('lii -r,'ll?'rt'C j'(;jjg.'. r' t---' t'-':r--f .( (()* jjjgjjjjjj.d tktli.lij-'. t'(jjj(g!,,'' 7h)t7'. ih'('i !' t')' 'Sjjqf .lll)lql;-r(). 7'7 rql' iilli!!iilf y( t'ti'iiiiii@#7'-.. tir.'tf . r! ... g...;r .-g-jiy-jj.tt.tjjjjjjj.y ,-.-jy(j( .rjjj)ytyyj.. 1lt);. . yy(-s))d:::)ldt!si yytjyytjyy ,:yyggy,::r(2;),jr.,,j Idiit,--l. dl::)p rlt:. ,k.,,, Ir..,. i,..Ir.. j!..l jr..lg..,r Ik..,l ) ' '. jr.. ,::E,Ir jr:;jiIrlt::!!;i jg jr,r,.,j ,r., .qI!a t!;r,d;.,L..,I irlr..,l irtgylgjy ig jr::y !(g(I. e lgllyj jryyr:)y a ,It:il:::,, e.irr-;i,k i q e .). . .. ... :.t . .. uncorrelated. Promised payoffs to the bonds are $140 (senior), $90 (mezzanine),and $70 (subordinated). j':j' q' (' f' ;')'

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%

IXSTRUMENTS

Plice Yield Default probability

Average recovel'y rqte

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40

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0

131.828

83.403

50.347

Price

129.963

76.283

59.331

0.07613

0.3296

Yield

0.165

0.165

0.0280

0.2710

0.074 0.1

0.1 0

0.1 0

0.0601 0.0010 0.857 1

0.4286

Default probability Average recovery rate

0.1281

and three bonds default (0.103 0.1% probability). To' compute the price of a CDO default 'anche, we can compute the expectedupayoffof the tranche using the lisk-neutral probabilities. The CDO pricing is illustrated in Table 26.7. Note that the senior tranche is almost risk-free. The only time the senior tranche is not fully paid is in the unlikely (0.1% probability) event that all three bonds default. In that case, the senior tranche receives $120, arecovery rate of 85.7%. Since itis almostpaid in 111,investors will pay $131.828 for the senior tranche, which is a yield of 6.02%. The mezzanine tranche is fully paid if there is one default, but it is not fully paid if thef are two or three defaults. The yield is 7.61% and the average recovery rate is 0.4285. Finally, the subordinated tranche receives less than 40/90 x 0.027/0.028 f'ull payment if there are any defaults. Consequently, it is priced to yield 32.96%. Note that the sum of the prices of the three tranches is $265.58.As you would expect, this is the same as the price of the three bonds put into the asset pool. This example might remind you of the discussion of tranched debt in Chapter 16, especially Table 16.1. The idea is exaetly the same, except that instead of valuing claims on corporate asset (thebonds in Chapter 16), we are valuing claims on a pool of corporate bonds. =

0.857143

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90 90

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10

if at maturity, either a11linns default or none do. A compalison of Tables 26.7 and 26.8 shows the importance of default correlation in the pricing of CDOs. Given the strucmfe of the CDO and the assumptios about therecovel'y rate, with perfect correlation of defaults, the mezzanine and subordinated eanches have te same yield. The senior tranchq becomes riskier because the probability of three defaults-the only circumstance in wlzich the senior tranche is not f'ully paid-is greater with perfect con-elation.

'

=

ln the preceding example we assumed that the bonds A CDo with correlated defaults uncorrelated. guessed, this is an important assumption. Table might have As you were

26.8 shows how the CDO tranches are priced if the bonds are prfectly correlated-i.e.,

The previous examples showed that it is possible to pool lisky baskets bonds and create rislier and less lisky claims. A particular variant of this strategy is the Nth to defaltlt basket. Consider a CDO that contains equal quantities of N bonds. Over the life of the CDO there can be anywhere between 0 and N defaults. lt is po'ssible to create tranches where particular bondholders bear the onsequences of a particular default. The owner of the first-to-default tranche bears the most risk: If any of the bonds in the asset pool default, the srst-to-default owner bears the loss from this default. The owner of the last-to-default generally bears the least risk, since allbonds must default in order for this claim to bear a loss. Table 26.9 shows the pricing that results from this structure, assuming that the defaults are uncorrelated and occur only at time F. By comparing Table 26.9 with Table 26.7, you can observe a similarity between te subordinated tranche and the srst-todefault on the one hand, and on the other, the senior tranche and the third to default. Nth tp default

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