Derivatives markets 2e by robert l mcdonald

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Derivatives

Markets

Second Edition

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Library of Congress Cataloging-in-Publication Data
McDonald, Robert

L.

(Robert Lynch), 1954-

Derivatives markets, 2e I Robert

p.cm.

Includes index.
ISBN 0-321-28030-X
1. Derivative securities. I. Title.
HG6024.A3 M3946 2006
332.64'5-dc21
ISBN 0-321-28030-X
12345678910-HT-09 08 07 06 05

L.

McDonald.


Fo1· Irene, Claire, David, and Hemy



Preface

Chapter 1
1.1

XX1

Comparing a Forward and Outright
Purchase

Uses of Derivatives

Credit Risk

3

2.2

3

The Role of Financial Markets
Risk-Sharing

4

How Are DerivativesUsed?

Option

2.3

7

Short-Selling

37

38

39

Payoff and Profit for aWritten Put
Option

40

The "Moneyness" of an Option

11

2.4
14

Risk and Scarcity in Short-Selling

33

Put Options
Option

10

12

The Lease Rate of an Asset

32

Payoff and Profit for a Purchased Put

Buying and Short-Selling Financial

Assets 11
Buying anAsset

31

Payoff and Profit for aWritten Call
Option

6

Growth in DerivativesTrading

1.4

Call Options

Payoff and Profit for a Purchased Call

5

Derivatives in Practice

30

30

Option Terminology

FinancialMarkets and theAverages 4

1.3

28

Cash Settlement Versus D� livery

2

FinancialEngineering and Security

1.2

Diagrams

1

Perspectives on Derivatives
Design

Zero-Coupon Bonds inPayoff and Profit

Introduction to Derivatives

What Is a Derivative?

26

LongPositions 44

15

Short Positions

Chapter Sumn!al)' 16
Further Reading 16
Problems 17

43

Summary of Forward and Option
Positions 43

2.5

44

Options Are Insurance

45

Homeowner'sInsuranceIs aPutOption

45

ButIThoughtInsurance Is Prudent and Put
Options Are Risky... 47

PART ONE

Call OptionsAreAlsoInsurance

INSURANCE,

2.6

Example: Equity-Linked CDs

47

48

HEDGING, AND SIMPLE STRATEGIES

Graphing the Payoff on the CD 49

19

Economics of the CD 50
WhyEquity-Linked COs?

Chapter 2
and Options

2.1

An Introduction to Forwards

21

Forward Contracts

21

ThePayoff on a Forward Contract
Graphing thePayoff on a Forward
Contract 25

23

51

Chapter Sumn!al)' 52
Further Reading 53
Problems 54
Appendix 2.A: More on Buying a Stock
Option 56
Dividends

56

vii


Viii

� C O N T E NTS
Exercise

Insurance: Guaranteeing a MaximumPrice

57

with a Call Option

Margins for Written Options 57

4.3

Taxes 58

Strategies

3.1

10 3

Reasons Not to Hed ge

59

Basic Insurance Strategies

10 1

Reasons to Hed ge

Insurance, Collars, and Other

59
4.4

Floors' 59

Golddiggers Revisited
Other CollarStrategies

Selling Insurance 6 3

Paylater Strategies

Synthetic Forwards

66

4.5

68

1 13

70

QuantityUncertainty

Bull and Bear Spread s

71

Chapter Stmti11G1)' 119
Further Reading 120

Problems 120

Ratio Spread s
Collars

3.4

72
73

Speculating on Volatility
Strad d les

78

78

Butterfly Spread s

PART 1WO

81
82

Chapter 5

Chapter Summary 85
Further Reading 86
Problems 87

Management

4.1

Futures

5.1
5.2

92

Arbitrage

5.3
96

Basic Risk Management: The Buyer's
Perspective 98
98

129

129

Pricing Prepaid Forward s with
Divid end s

95

Hed ging with a Forward Contract

128

Present Value

93

Ad justing theAmount ofInsurance

127

Pricing thePrepaid Forward by

Insurance: Guaranteeing a MinimumPrice

4.2

Alternative Ways to Buy a Stock
Prepaid Forward Contracts on
Stock 128

Pricing thePrepaidForward by Discounted

Basic Risk Management: The
Producer's Perspective 91

Insuring by Selling a Call

Financial Forwards and

127

Analogy

91

with a Put Option

125

Pricing the Prepaid Forward by

Introduction to Risk

Hed gingwith a Forward Contract

FORWARDS,

FUTURES, AND SWAPS

Example: Another Equity-Linked
Note 83

Chapter 4

116

73

Asymmetric Butterfly Spread s

3.5

113

114

3.3- Spreads and Collars
Box Spread s

10 8
1 12

Selecting the Hedge Ratio
Cross-Hed ging

106

108

Selling the Gain: Collars

Insuring a Short Position: Caps 6 2

Put-Call Parity

106

EmpiricalEvidence onHed ging

Insuring a LongPosition:

3.2

100

An ExampleWhereHed gingAd d s
Value

Chapter 3

99

Why Do Firms Manage Risk?

13 1

Forward Contracts on Stock

133

Creating a SyntheticForward
Contract

135

Synthetic Forward s in Market-Making and
Arbitrage

136

·


C O N T E NTS
An ApparentArbitrage and

No-Arbitrage Bounds withTransaction
Costs

Resolution

13 8

Quasi-Arbitrage
140

6.5

5.4

14 1

Futures Contracts

142
14 3

Margins and Marking toMarket

144

Comparing Futures and Forward

5.5

Quanta Index Contracts

14 9

Uses of Index Futures

150

AssetAllocation

150

Cross-hedging with Index Futures

5.6

Currency Contracts

154

CurrencyPrepaid Forward
Currency Forward

Eurodollar Futures

155

Storage Costs and ForwardPrices

18 1
18 2

Gold Futures

18 2

184
18 7

BasisRisk

197

15 6

Weather Derivatives

Futures

6.1

19 9

160

19 9

Chapter SummGI)' 200
Further Reading 201
Problems 201

Chapter 7
Futures

7.1

Interest Rate Forwards and

205

Bond Basics

205

Zero-Coupon Bonds

20 6

Implied Forward Rates

Chapter 6

18 7

6.8 Seasonality: The Corn Forward
Market 188
6.9 Natural Gas 191
6.10 Oil 194
6.11 Commodity Spreads 195
6.12 Hedging Strategies 196
HedgingJet Fuel with Crude Oil

Chapter SumnWI)' 160
Further Reading 162
Problems 162
Appendix 5.A: Taxes and the Forward
Price 166
Appendix S.B: Equating Forwards and
Futures 166

Coupon Bonds

Commodity Forwards and

20 8

210

Zeros from Coupons

169

·

2 11

Interpreting the Coupon Rate

Introduction to Commodity
Forwards 169

17 9

Storage Costs and the Lease Rate

Evaluation of Gold Production

15 6

Covered InterestArbitrage

5.7

15 1

2 12

Continuously Compounded Yields

6.2

Equilibrium Pricing of Commodity
Forwards 171

6.3

Nonstorability: Electricity

6.4

Pricing Commodity Forwards by
Arbitrage: An Example 174

·

172

17 8

181

Gold Investments

Arbitrage in Practice: S&P 50 0 Index
147

Carry Markets

The Convenience Yield

6.7

14 6

Arbitrage

17 6

178

Forward Prices and the Lease Rate

6.6

The S&P 50 0 Futures Contract

Prices

The Commodity Lease Rate

The LeaseMarket for a Commodity

An Interpretation of the Forward Pricing
Formula

175

Pencils Have aPositive Lease Rate

13 9

Does the ForwardPricePredict the Future
Price?

ix

�

7.2

2 13

Forward Rate Agreements, Eurodollars,
and Hedging 214
Forward RateAgreements
Synthetic FRAs

214

216

Eurodollar Futures

218

Interest Rate Strips and Stacks

223


X

7.3

� C O N T E N TS
Duration and Convexity
Duration

223

Further Reading 275
Problems 275

224

DurationMatching
Convexity

227

228

7.4

Treasury-Bond and Treasury-Note
Futures 230

PART THREE

7.5

Repurchase Agreements

Chapter 9

233

Chapter Summary 235
Further Reading 237
Problems 237
Appendix 7.A: Interest Rate and Bond Price
Conventions 241
Bonds
Bills

Chapter 8
8.1

244

Swaps

248
250

255

304

Chapter Szmmzary• 305
Further Reading 306
Problems 306
Appendix 9.A: Parity Bounds for American
Options 310

Appendix 9.B: Algebraic Proofs of
Strike-Price Relations 311

260

26 2
26 3

267

267

268

The Commodity Swap Price 26 8
Swaps with Variable Quantity and

8.5
8.6

29 9

Exercise andMoneyness

264

Currency Swap Formulas

29 4

29 7

Different Strike Prices

257

26 1

Other Currency Swaps

29 3

Time to Expiration

258

Am ortizing and Accreting Swaps

29 3

Early Exercise forAmerican Options

Computing the Swap Rate in General

Why Swap Interest Rates?

29 0

Comparing Options with Respect to
Style, Maturity, and Strike 292

Prices

254

The Swap's Implicit Loan Balance

28 8
28 9

Maximum andMinimum Option

25 3

Pricing and the Swap Counterpart}'

Commodity Swaps

28 6

European VersusAmerican Options

254

A SimpleInterest Rate Swap

Price

9.3

250

The Market Value of a Swap

8.4

28 6

Generalized Parity and Exchange
Options 287

Currency Options

Physical Versus FinancialSettlement

Currency Swaps

28 3

Options to Exchange Stock

Why Is the Swap Price Not $20 .50 ?

Deferred Swap s

281

What Are Calls andPuts?

247

The Swap Curve

Put-Call Parity

Options on Bonds

247

Interest Rate Swaps

281

Options on Currencies

9.2

279

Parity and Other Option

Options onStocks

24 2

The Swap Counterpart}'

8.3

9.1

An Example of a Commodity
Swap

8.2

Relationships

OPTIONS

26 9

Swaptions 271
Total Return Swaps

Chapter Sumnzal)' 274

Chapter 10
Pricing: I

10.1

Binomial Option

313

A One-Period Binomial Tree
Computing the OptionPrice

272

The Binomial Solution

313

3 14

3 15

Arbitraging a Mispriced Option

3 18


C O N T E N TS
A Graphical Interpretation of the Binomial
Formula

3 20

Constructing a BinomialTree

3 21

Another One-PeriodExample

3 22

Summary

3 22

Many Binomial Periods

Option on a Srock Index
Options on Currencies

Forward

Pricing: II

330

3 32
332

3 34

3 35

Utility-Based Valuation

336

Risk-Neutral Pricing
Example

Formula

343

Understanding Early Exercise

11.2

Understanding Risk-Neutral
Pricing 346
The Risk-Neutral Probability

343

12.1

Valid?

12.2

The Binomial Tree and
Lognormality 351

Continuously Compounded Returns
The Binomial Model

3 55

Lognormality and the Binomial
Model

3 55

379

Applying the Formula to Other
Assets 379
Dividends

380

Options on Currencies

352

The Standard Deviation of Returns

3 75
378

Options on Srocks with Discrete

351

Modeling Stock Prices as a Random
Walk

Introduction to the Black-Scholes
Formula 375

WhenI s the Black-Scholes Formula

3 47

TheRandomWalkModel

The Black-Scholes

Put Options

Pricing an OptionUsing Real

11.3

372

375

Call Options
346

Options on Futures

3 53
3 54

12.3

37 1

37 1

Why Risk-Neutral PricingWorks

Chapter 12

11.1

3 69

Standard Discounted Cash Flow

Binomial Option

Probabilities

363

Chapter Summary 365
Further Reading 366
Problems 366
Appendix ll.A: Pricing Options with True
Probabilities 369
Appendix 11.B: Why Does Risk-Neutral
Pricing Work? 369

Chapter Summary 337
Further Reading 337
Problems 338
Appendix 1 O.A: Taxes and Option
Prices 341

Chapter 1 1

3 62

A BinomialTreeUsing the Prepaid

3 30

Options on Commodities
Summary

Tree

3 23

Options on Futures Contracts
Options on Bonds

361

Problems with the Discrete Dividend

323

3 26

Put Options 328
American Options 329
Options on Other Assets

3 59

Estimating Volatility 360
Stocks Paying Discrete
Dividends 361
Modeling Discrete Dividends

Two or More Binomial Periods
ATwo-PeriodEuropean Call

10.3
10.4
10.5

11.4
11.5

xi

35 8

Is the Binomial Model Realistic?

Risk-Neutral Pricing

10.2

Alternative BinomialTrees

3 19

�

Option Greeks

38 1

38 1

382

Definition of the Greeks

382

Greek Measures for Portfolios
Option Elasticity

389

3 88

3 73


Xii
12.4

12.5

� C O N T E NTS

Profit Diagrams Before Maturity

Theta: Accounting forTime

Calendar Spreads 3 9 7

Understanding theMarket-Maker's

Implied Volatility

Profit

400

Computing Implied Volatility
UsingImplied Volatility

12.6

Delta-Gamma Approximations 4 24

395

Purchased Call Option 3 96

40 0

13.5

Re-Hedging?

404

Gamma-Neutrality

13.6

432

433

Market-Making as Insurance

436

Insurance 436
Market-Makers

43 7

Chapter Swnmmy 438
Further Reading 438
Problems 438
Appendix 13.A: Taylor Series
Approximations 441
Appendix 13.B: Greeks in the Binomial
Model 441

410

Vega

43 1

Delta-Hedging inPractice

Chapter Summary 405
Further Readilzg 405
Problems 406
Appendix 12.A: The Staizdard Normal
Distribution 409
Appendix 12.B: Formulas for Option
Greeks 410

Theta

429
4 29

What Is theAdvantage to Frequent

Perpetual Calls 404

Gamma

The Black-Scholes Analysis

Delta-Hedging ofAmerican Options 43 0

403

Barrier Present Values 403

. Delta

4 27

TheBlack-ScholesArgument

40 2

Perpetual American Options

Perpetual Puts

4 25

4 10

410
411

Rho 4 11
Psi

Chapter 13

14.1
14.2

Market-Making and

Delta-Hedging

13.1
13.2

Chapter 14

411

41 3

What Do Market-Makers Do?
Market-Maker Risk 414

XYZ's HedgingProblem

413

ComparingAsian Options 447
AnAsian Solution forXYZ

Delta and Gamma a sMeasures of
Exposure

13.3

417

14.4

4 22

The Mathematics of
Delta-Hedging 422

Compound Options

453
454

Options on Dividend-Paying Stocks
CurrencyHedging with Compound

Delta-Hedging for Several Days 4 20
A Self-FinancingPortfolio: The Stock

Options 456

14.5
14.6

Gap Options

457

Exchange Options

459

European Exchange Options 459

Using Gamma to BetterApproximate the
Change in the Option Price

450

Compound OptionParity

Interpreting theProfit Calculation 4 18

13.4

448

449

Types ofBarrier Options

41 7

.Moves Onea

Barrier Options

CurrencyHedging 451

An Example of Delta-Hedging for 2
Days

14.3

41 6

Delta-Hedging

445

Options on theAverage 446

Option Risk in theAbsence of
Hedging 4 1 4

443

Exotic Options: I

Introduction 443
Asian Options 444

4 23

Chapter Summary 461
Further Reading 462

455


C O N T E N TS

Chapter 16

Problems 462
Appendix 14.A: Pricing Formulas for
Exotic Options 466

Applications

Asian Options Based on the Geometric
Average

16.1

Corporate

503

Equity, Debt, and Warrants
Multiple DebtIssues

46 7

Infinitely Lived Exchange Option

503

Debt andEquity as Options 503

466

Compound Options

xiii

�

Warrants

46 8

51 1

512

Convertible Bonds
CallableBonds

513

516

Bond ValuationBased on theStockPrice
520

PART FOUR

OtherBond Features S20

FINANCIAL

PutWarrants

ENGINEERING AND APPLICATIONS
16.2

471

ValuationInputs

Financial Engineering and

Security Design

The Modigliani-Miller Theorem

15.2

Pricing and Designing Structured
Notes 474
Zero-Coupon Bonds

Option Grants

473

16.3

Equity-LinkedBonds 476
Currency-LinkedBonds

482

483

483

486
48 8

Strategies Motivated by Tax and
Regulatory Considerations 490
Capital Gains Deferral·

490

Tax-DeductibleEquity

495

Chapter Summary 498
Further Reading 498
Problems 498

17.2

547

Real Options

Investment and the NPV Rule
StaticNPV

Engineered Solutions for
Golddiggers 486
Notes withEmbedded Options

15.5

Chapter 17
17.1

Alternative Structures 48 5

Gold-LinkedNotes

The Use of Collars in
Acquisitions 538

Chapter Summary 542
Further Reading 542
Problems 543

482

Valuing and Structuring anEquity-Linked
CD

53 4

53 8

Bonds with Embedded Options
Options inEquity-LinkedNotes

53 1

532

The Northrop Grumman-TRY Merger

478

48 1

Options in CouponBonds

52 8

Level3 Communications

CouponBonds 475
Commodity-Linked Bonds

52 7

Repricing of Compensation Options
Reload Options

474

523

525

AnAlternativeApproach to Expensing

473

15.1

15.4

Compensation Options
Whose Valuation?

Chapter 15

15.3

522

548

548

The CorrectUse ofNPV

549

The Project as an Option

550

Investment under Uncertainty

551

A Simple DCFProblem 551
Valuing Derivatives on the Cash Flow
552
Evaluating a Project with a2 -Year
InvestmentHorizon

554

Evaluating the Project with anInfinite
InvestmentHorizon

558


XiV
17.3

17.4

� C O N T E N TS
Real Options in Practice

Lognormal ConfidenceIntervals 6 0 0

558

Peak-Load Electricity Generation 559

The Conditional Expected Price 6 0 2

Research andDevelopment

The Black-Scholes Formula 6 04

563

Commodity Extraction as an
Option 565

18.5

Single-Barrel Extraction under

18.6

Certainty 56 5
Single-BarrelExtraction under

Histograms 6 0 8

Uncertainty 569
Valuing anInfinite Oil Reserve

17.5

Normal Probability Plots 6 09

570

Commodity Extraction with
Shut-Down and Restart Options
PermanentShuttingDown

Chapter Summary 613
Further Reading 613
Problems 614
Appendix 18.A: The Expectation of a
Lognomzal Variable 615
Appendix 18.B: Constructing a Normal
Probability Plot 616

572

574

InvestmentWhen Shutdown Is
Possible 576
RestartingProduction
Additional Options

Estimating the Parameters of a
Lognormal Distribution 605

How Are Asset Prices
Distributed? 608

57 8

57 8

Chapter Swnmal)' 579
Further Reading 580
Problems 580
Appendix 17.A: Calculation of Optimal
Time to Drill an Oil Well 583
Ap pendix 17.B: The Solution with
Shutting Down and Restarting 583

Chapter 19
19.1

61 7

Monte Carlo Valuation

Computing the Option Price as a
Discounted Expected Value 617
Valuation with Risk-Neutral
Probabilities 61 8
Valuation withTrue Probabilities 6 19

19.2

Computing Random Numbers

621

UsingSums of UniformlyDistributed

PART FIVE
THEORY

Random Variables 6 22

ADVANCED PRICING

Using theInverse CumulativeNormal

5 85

Distribution 6 22

19.3

Chapter 1 8
Distribution

18.1

The Lognormal

Simulating a Sequence of Stock

587

The Normal Distribution

Prices 623

587

Converting a-Normal Random Variable to

19.4

Sums ofNormal Random Variables

The Lognormal Distribution 593
A Lognormal Model of Stock
Prices 595

18.4

Lognormal Probability
Calculations 598
599

624

Call 6 25

591

18.2
18.3

Monte Carlo Valuation

Monte Carlo Valuation of aEuropean

StandardNormal 590

Probabilities

Simulating Lognormal Stock
Prices 623

Accuracy of Monte Carlo 6 26
ArithmeticAsian Option 6 27

19.5

Efficient Monte Carlo Valuation

630

Control VariateMethod 63 0
Other Monte Carlo Methods 63 2

19.6

Valuation of American Options

633


C O N T E N TS
19.7
19.8

The Poisson Distribution 636
SimulatingJumps with the Poisson
Distribution 639
MultipleJumps 6 4 3

19.9

Further Reading 674
Problems 675

Chapter 2 1

Simulating Correlated Stock
Prices 643

Equation

Generating11 Correlated Lognormal

21.1

Random Variables 6 44

Chapter Summary 645
Further Reading 645
Problems 646
Appendix 19.A: Formulas for Geometric
Average Options 648

Chapter 20
Lemma

20.1
20.2

Dividend-Paying Stocks 6 8 1
.
The General Structure 6 8 1

21.2

The Black-Scholes Equation

681

Verif ying the Formula f or a
Derivative 6 8 3
The Black-Scholes Equation and
Equilibrium Returns 6 8 6

The Black-Scholes Assumption about
Stock Prices 649
Brownian Motion

650

WhatIf the Underlying AssetI sNot an
InvestmentAsset?

21.3

6 55

Relative Importance of theDrift and Noise

DerivativePrices as DiscountedExpected
Cash Flows 69 2

21.4
21.5

Terms 6 56

69 7

Multiplication Rules 6 58

The Sharpe Ratio

20.5

The Risk-Neutral Process

20.6

Ito's Lemma

Chapter Summa/)' 698
Further Reading 698
Problems 699
Appendix 21.A: Multivariate
Black-Scholes Analysis 700
Appendix 21.B: Proof of Proposition 21.1
701

659
660

663

Functions of anIto Process 66 3
MultivariateIto's Lemma 665

Valuing a Claim on sa

666

TheProcess Followed by S"

66 7

Proving the Proposition 66 8
Specific Examples 669

Valuing a Claim on S" Q"

20.8 Jumps in the Stock Price

Chapter Sumntal)' 674

Changing the Numeraire 693
Option Pricing When the Stock Price
CanJump 696
Merton's Solution f or DiversifiableJumps

Correlated Ito Processes 6 57

20.4

690

TheBackwardEquation 691

The Ornstein-Uhlenbeck Process 6 54

655

Risk-Neutral Pricing
Equation 690

Arithmetic BrownianMotion 6 5 3

Geometric Brownian Motion

688

Interpreting the Black-Scholes

Properties of BrownianMotion 6 5 2

20.7

Differential Equations and Valuation
under Certainty 679
Bonds 6 8 0

649

Lognormality

The Black-Scholes

679

TheValuation Equation 6 8 0

Brownian Motion and Ito's

Definition of BrownianMotion 6 50

20.3

XV

�

670

672

Chapter 22
22.1

Exotic Options: II

Ali-or-Nothing Options

703

703

Terminology 70 3
Cash-or-Nothing Options

704

Asset-or-Nothing Options

706


XVi

�

C O N T E NTS
706

Ordinary Options and Gap Options

23.3

Options

22.2

PricingVolatility

707

Ali-or-Nothing Barrier Options

710

23.4

Cash-or-Nothing Barrier Options 7 1 0
Asset-or-Nothing Barrier Options
Rebate Options

Quanto�

TheHestonModel

717

Evidence

718

The Dollar Perspective

7 21

Dollar-DenominatedInvestor

22.5- Currency-Linked Options

7 24

727

Foreign Equity Call Struck inForeign
7 28

Foreign Equity Call Struck in Domestic

Chapter 24
24.1

7 29

22.6

Rates

Exchange Options

73 2

Bonds

24.2

73 5

Volatility

ComparingVasicek and CIR

741
741

23.2

Measurement and Behavior of
Volatility 744

Bond Options, Caps, and the Black
Model 790

24.4

A Binomial Interest Rate Model

793

79 4

Yields andExpectedInterest Rates

796

OptionPricing 79 7

744

746
Time-Varying Volatility: ARCH

788

24.3

Zero-CouponBond Prices

ExponentiallyWeightedMovingAverage
747

7 51

Realized QuadraticVariation

785

The Cox-Ingersoll-Ross Model 7 8 7

Implied Volatility

The GARCH Model

Equilibrium Short-Rate Bond Price
Models 785
TheVasicek Model 7 8 6

23.1

HistoricalVolatility

781

784

The Rendelman-BartterModel

Chapter Swnnial)' 736
Further Reading 736
Problems 737

Chapter 23

780

Delta-Gamma Approximations for

732

Options on the Best ofTwoAssets 733
Basket Options

780

A nEquilibriumEquation for Bonds

73 1

Other Multivariate Options

779

An ImpossibleBondPricingModel

73 0

Equity-Linked Foreign Exchange
Call

Interest Rate Models

Market-Making and Bond
Pricing 779
TheBehavior of Bonds andInterest

Fixed Exchange Rare Foreign Equity
Call

76 8

771

Chapter Summa/)' 773
Further Reading 773
Problems 774
Appendix 23.A 777

A Binomial Model for the

Currency

759

Extending the Black-Scholes Model
763
Constant Elasticity ofVariance 766

The YenPerspective 7 20

Currency

757

758

Jump Risk and Implied Volatility 764

715

7 16

Barrier Options
22.4

Hedging and Pricing Volatility
Variance andVolatility Swaps

Delta-Hedging Ali-or-Nothing

755

24.5

The Black-Derman-Toy Model
Verifying Yields

802

VerifyingVolatilities

8 03

Constructing a Black-Derman-Toy
Tree

8 04

Pricing Examples

805

798


C O N T E N TS

Chapter Szmzmary 808
Further Reading 808
Problems 809
Appendix 24.A: The Heath­
farrow-Morton Model 811

Chapter 25
25.1

81 3

Value a t Risk

Value at Risk

813

Value at Risk f or One Stock 815
VaR f orTwo orMore Stocks 81 7

VaR f orNonlinear Portf olios 81 9
VaR f or Bonds 8 26

Estimating Volatility 830
Bootstrapping Return Distributions 83 1

25.2

Issues with VaR

832

PART SIX

APPENDIXES

Appendix A

The Greek Alphabet

Appendix B

Continuous

Compounding

B.1
B.2

26.1

26.2

Related Mod els 845

Bond Ratings and Default Experience
847
Using Ratings toAssessBankruptcy
Probability 84 7
Reduced FormBankruptcyModels 8 5 2

Credit Instruments

853

Collateralized DebtObligations 8 53
Credit Def ault Swaps and Related
Structures 858
Pricing a Def ault Swap 86 2
CDSIndices 86 4

C.2
C.3

jensen's Inequality

881

·

Example: The Exponential Function

881
Example: The Price of a Call 882
Proof ofJensen's Inequality 884

Problems 884

Appendix D

An Introduction to

Visual Basic for Applications

D.1
D.2
D.3

885

Calculations without VBA 885
How to Learn VBA 886
Calculations with VBA 886
Creating a Simple Function 886

Recovery Rates 8 50

26.4

Problems 878

C.1

Def ault atMaturity 843

26.3

The Language of Interest Rates 875
The Logarithmic and Exponential
Functions 876

Appendix C

Default Concepts and Terminology
841
The Merton Default Model 843

875

8 78

Subadditive RiskMeasures 83 7

841

873

Symm etry f or Increases and Decreases

835

Credit Risk

871

Changing Interest Rates 8 7 7

VaR and the Risk-Neutral Distribution

Chapter 26

xvii

Chapter SumnWI)' 866
Further Reading 867
Problems 867

Alternative RiskMeasures 83 2

Chapter Szmnnmy 838
Further Reading 839
Problems 839

�

A Simple Example of a Subroutine 888
Creating a Button toInvoke aSubroutine
888
Functions Can Call Functions 88 9
IllegalFunctionNames 88 9
Dif f erences between Functions and
Subroutines 8 9 0

XViii
D.4

� C O N T E N TS
Creating a BinomialTree

Storing and Retrieving Variables in a
Worksheet 890
Using aNamed Range to Read andWrite
Numbers f rom a Spreadsheet

8 91

Other Kind s o f Loops

D.9

Reading and Writing Arrays
Arrays as Outputs

Reading andWriting to CellsThat AreNot
Named

892

Using the Cells Functions to Read and
Write to Cells

D.S

Arrays as Inputs

D.lO

892

Reading f rom within a Function

Debugging

Iter!:ltion

8 97

899

A Simplefor Loop

899

896

Glossary

907
921

Bibliography

Index

905

9 06

Creating an Add-In

8 95

D.6 -Checking f!Jr Conditions
D.7 Arrays 897
D.8

Getting Excel to Generate Macros f or You

Recalculation Speed

8 94

DefiningArrays

9 03

904

UsingMultipleModules

UsingVBA to Compute the Black-Scholes
The Object Browser

Miscellany

901

9 01

904

8 93

Using Excel Functions from within
VBA 893
Formula

900

9 01

935

906

9 05


De

riva6v ; , have moved to the cent« of modem coq>nmte finance, ;nve"men", and
the management of financial institutions. They have also had a profound impact on

other management functions such as business strategy, operations management, and
marketing. A major drawback, however, to making 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 finally have a derivatives text
that is a wonderful blend of the economics and mathematics of derivatives pricing and
easily accessible to MBA 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 neither overly sophisticated nor watered down, but rather
a mix of intuition and rigor that creates an inherent flexibility for the structuring of a
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 treatment of options then follows logically from con­
cepts developed in the earlier chapters. The heart of the text-an extensive treatment
of the binomial.option model and the Black-Scholes equation-showcases the author' s
crystal-clear writing and logical development of concepts. Excellent chapters on finan­
cial 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 provides an advanced treatment of the most important
concepts of derivatives discussed earlier. This part can be used by itself in an advanced
derivatives course, or as a useful reference in introductory courses. A rigorous de­
velopment of the Black-Scholes equation, exotic options, and interest rate models are
presented using Brownian Motion and Ito's Lemma. Monte Carlo simulation methods
are also discussed in detail. New chapters on volatility and credit risk provide a clear
discussion of these fast-developing areas.
Derivatives concepts are now required for every advanced finance topic. There­
fore, it is essential to introduce these concepts at an early stage of MBA and under­
graduate business or economics programs, and in a fashion that most students can un­
derstand. This text achieves this goal in such an appealing, inviting way that students

will actually enjoy their journey toward an understanding of derivatives.

·

EDUARDO 5. SCHWARTZ

xix



ThiTiy

yea.-' ago the Blaok-Scho!o' focmula wa, new, and derivative< wa, an especialized subject. Today, a basic knowledge of derivatives is necess ar}r to understand
modern finance. For example, corporations routinely hedge and insure using deriva­
tives, finance activities with structured products, and use derivatives models in capital
budgeting. This book will help you to understand the derivative instruments that exist,
how they are used, who sells them, how they are priced, and how the tools and concepts
are useful more broadly in finance.
Derivatives is necessarily an analytical subject, but I have tried throughout to
emphasize intuition and to provide a common sense way to think about the formulas. I
do assume that a reader of this book already understands basic financial concepts such as
present value, and elementary statistical concepts such as mean and standard deviation.
Most of the book should thus be accessible to anyone who has studied elementary finance.
For those who want to understand the subject at a deeper level, the last part of the book.
develops the Black-Scholes approach to pricing derivatives and presents some of the
standard mathematical tools used in option pricing, such as Ito's Lemma. There are also
chapters dealing with applications: corporate applications, financial engineering, and
real options.
In order to make the book accessible to readers with widely varying backgrounds

and experiences, I use a "tiered" approach to the mathematics. Chapters 1-9 emphasize
present value calculations, and there is almost no calculus until Chapter 1 8 .
Most o f the calculations i n this book can b e replicated using Excel spreadsheets on
the CD-ROM that comes with the book. These allow you to experiment with the pricing
models and build your own spreadsheets. The spreadsheets on the CD-ROM contain
option pricing functions written in Visual Basic for Applications, the macro language in
Excel. You can easily incorporate these functions into your own spreadsheets. You can
also examine and modify the Visual Basic code for the functions. Appendix D explains
how to write such functions in Excel and documentation on the CD-ROM lists the option
pricing functions that come with the book. Relevant built-in Excel functions are also
mentioned throughout the book.

PLAN

OF

THE

BOOK

This book grew from my teaching notes for two MBA derivatives courses at Northwest­
ern University's Kellogg School of Management. The two courses roughly correspond
xxi


xxii

�

P R E FA C E

to the first two-thirds and last third of the book. The first course i s a general introduction
to derivative products (principally futures, options, swaps, and structured products), the
markets in which they trade, and applications. The second course is for those wanting a
deeper understanding of the pricing models and the ability to perform their own analysis.
The advanced course assumes that students know basic statistics and have seen calculus,
and from that point develops the Black-Scholes option-pricing framework as fully as
possible. No one expects that a 1 0-week MBA-level course will produce rocket scien­
tists, but mathematics is the language of derivatives and it would be cheating students
to pretend otherwise.
You may want to cover the material in a different order than it occurs in the book,
so I wrote chapters to allow flexible use of the material. I indicate several possible paths
through the material below. In many cases it is possible to hop around. For example, I
wrote the book expecting that the chapters on lognormality and Monte Carlo simulation
might be used in a first derivatives course.
The book has five parts plus appendixes. Part 1 introduces the basic building
blocks o{ derivatives: forward contracts and call and put options. Chapters 2 and 3
examine these basic instruments and some common hedging and investment strategies.
Chapter 4 illustrates the use of derivatives as risk management tools and discusses why
firms might care about risk management. These chapters focus on understanding the
contracts and strategies, but not on pricing.
Part 2 considers the pricing of forward, futures, and swaps contracts. In these
contracts, you are obligated to buy an asset at a pre-specified price, at a future date. The
main question is: What is the pre-specified price, and how is it determined? Chapter
5 examines forwards and futures on financial assets, Chapter 6 discusses commodities,
and Chapter 7 looks at bond and interest rate forward contracts. Chapter 8 shows how
swap prices can be deduced from forward prices.
Part 3 studies option pricing. Chapter 9 develops intuition about options prior to
delving into the mechanics of option pricing. Chapters 10 and 1 1 cover binomial option
pricing and Chapter 12, the Black-Scholes formula and option Greeks. Chapter 13

explains delta-hedging, which is the technique used by market-makers when managing
the risk of an option position, and how hedging relates to pricing. Chapter 14 looks
at a few important exotic options, including Asian options, barrier options, compound
options, and exchange options.
The techniques and formulas in earlier chapters are applied in Part 4. Chapter 15
covers financial engineering, which is the creation of new financial products from the
derivatives building blo cks in earlier chapters. Debt and equity pricing, compensation
options, and mergers are covered in Chapter 16. Chapter 17 studies real options-the
application of derivatives models to the valuation and management of physical invest­
ments.
Finally, Part 5 explores pricing and hedging in depth. The material in this part
explains in more detail the structure and assumptions underlying the standard derivatives
models. Chapter 1 8 covers the lognormal model and shows how the Black-Scholes
formula is an expected value. Chapter 19 discusses Monte Carlo valuation, a powerful
and commonly used pricing technique. Chapter 20 explains what i t means to say that


W H AT Is N EW I N T H E S E C O N D E D I T I O N

�

xxiii

stock prices follow a diffusion process, and also covers Ito's Lemma, which is a key
result in the study of derivatives. (At this point you will discover that Ito's Lemma has
already been developed intuitively in Chapter 13, using a simple numerical example.)
Chapter 21 derives the Black-Scholes partial differential equation (PDE). Although
the Black-Scholes formula is famous, the Black-Scholes equation, 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
_
the Black-Scholes and binomial analysis apply to bonds and interest rate derivatives.
Chapter 25 covers value-at-risk, and Chapter 26 discusses the burgeoning market in
credit products.

WHAT IS NEW IN THE SECOND EDITION
There 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 vari­
ance swaps; and pricing models that incorporate j umps and stochastic volatility,
such as the Heston model.
Chapter 26 covers structural models of bankruptcy risk (the Merton model);
tranched structures such as collateralized debt obligations; credit default swaps
and credit indexes.

There are numerous changes and new examples throughout the book. Among the
more important changes are the following:
o

An expanded discussion of bond convexity

o

An expanded treatment of computing hedge ratios

o

An expanded treatment of convertible and callable bonds

•

•

•

Discussion of the new option expensing rules in FAS 123R and the B ulow-Shoven
expensing proposal
Discussion of a variable prepaid forward on Disney stock issued by Roy Disney
In-depth discussion of a mandatorily convertible bond issued by Marshall & Ilsley,
including pricing and structuring

•

The use of simulation to price American options

•

Additional discussion of implied volatility

•

Enhanced discussion of the link between discounted cash flow valuation and risk­
neutral valuation

:xxiv

� P R E FA C E
•

•

A n expanded discussion o f value-at-risk
New spreadsheet functions for pricing options with fixed dividends, CEV option
pricing, the Merton jump model, and others

NAVIGATING THE MATERIAL
There are potentially many ways to cover the material in this book. The material is
generally presented in order of increasing mathematical difficulty, which means that
related material is sometimes split across distant chapters. For example, fixed income is
covered in Chapters 7 and 24, and exotic options in Chapters 14 and 22. Each of these
chapters is at the level of the neighboring chapters. As an illustration of one way to use
the book, here is the material I cover in the courses I teach (within the chapters I skip
some specific topics due to time constraints):
o

Introductory course: 1-6, 7. 1 , 8-1 0, 1 1 . 1-1 1 .2, 12, 1 3 . 1-13.3, 14, 15.4--15.5, 1 6,

o

Advanced course: 13, 1 8-22, 7, 8, 15. 1-15.3, 23, 24, 25, 26.

17.

The table on page xxv outlines some possible sets of chapters to use in courses that

have different emphases. There are a few sections of the book that provide background
on topics every reader should understand. These include short-sales (Section 1 .4), con­
tinuous compounding (Appendix B), prep�d forward contracts (Sections 5 . 1 and 5.2),
and zero-coupon bonds and implied forward rates (Section 7.1).

A NOTE ON EXAMPLES
Many of the numerical examples in this book display intermediate steps to assist you
in following the calculations. In most cases it will also be possible for you to cre­
ate a spreadsheet and compute the same answers starting from the basic assumptions.
However, numbers displayed in the text are generally rounded to three or four decimal
points, while spreadsheet calculations have many more significant digits. This creates a
dilemma: Should results in the book match those you would obtain using a spreadsheet,
or those you would obtain by computing the displayed equations?
As a general rule, the numerical examples in the book will provide the results
you would obtain by entering the equations directly in a spreadsheet. The displayed
calculations will help you follow the logic of a calculation, but a spreadsheet will be
helpful in reproducing the final result.

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