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PART Ⅰ THE BASICS2

1 INTRODUCTION TO DERIVATIVE SECURITIES2

1.0 Introduction2

1.1 Forward Contracts3

Formalization4

1.2 Futures Contracts6

Standardization6

Clearing House8

Settlement Price9

Daily Settlement and Margins10

Regulation11

Why Standardization？Why Daily Settlement？11

Basis12

Newspaper Quotes13

1.3 Options15

Call Options15

Put Options17

American versus European Options19

1.4 Organized Option Markets20

1.5 Option Newspaper Quotes22

1.6 Interest Rates and Bond Prices23

Zero-Coupon Bond Prices23

Discount Rates24

Simple Interest Rates25

Discretely Compounded Interest Rates26

Continuously Compounded Interest Rates29

1.7 Summary30

2 SIMPLE ARBITRAGE RELATIONSHIPS FOR FORWARD AND FUTURES CONTRACTS34

2.0 Introduction34

2.1 Definition of Arbitrage34

2.2 Assumptions35

2.3 Forward and Spot Prices37

No Cash Flows on the Underlying Asset Over the Life of the Forward Contract37

Formal Derivation （Cash-and-Carry）39

Value of a Forward Contract41

2.4 Known Cash Flows to the Underlying Asset42

Formal Derivation44

Value of a Forward Contract46

2.5 Forward Contracts on Constant Dividend Yield and Interest-Paying Assets47

Forward Contracts on a Stock Index47

Foreign Exchange Forward Contracts48

2.6 Forward Contracts on Commodities51

Storage Costs52

Convenience Yield54

The Implied Repo Rate55

Forward Contracts on Electricity55

2.7 Forward and Futures Prices Compared56

Equality of Forward and Futures Prices59

Empirical Evidence61

2.8 Summary62

Appendix：Present Value of Dividends Over Life of Forward Contract67

3 SIMPLE ARBITRAGE RELATIONSHIPS FOR OPTIONS68

3.0 Introduction68

3.1 Call and Put Options68

3.2 Put Options74

3.3 Relationship Between European Call and Put Options79

3.4 Relationship Between American Call and Put Options82

3.5 Summary84

PART Ⅱ THE BINOMIAL MODEL90

4 ASSET PRICE DYNAMICS90

4.0 Introduction90

4.1 The Lognormal Distribution91

4.2 The Basic Idea （Binomial Pricing）96

4.3 Formal Description （Binomial Pricing）98

4.4 The Binomial Approximation to the Lognormal Distribution99

4.5 Extensions105

4.6 Stochastic Differential Equation Representation105

4.7 Complications107

Lognormal Distribution107

Continuous Trading107

Continuously Changing Prices108

4.8 Summary108

Appendix：The Expected Value of the Future Stock Price112

5 THE BINOMIAL PRICING MODEL114

5.0 Introduction114

5.1 Single-Period Example115

5.2 Multiperiod Example119

5.3 The Binomial Pricing Model123

The Binomial Model123

Constructing the Synthetic Option124

Risk-Neutral Valuation126

Put Options130

5.4 Hedge Ratio （Delta）133

5.5 Lattice Parameters133

5.6 The Black-Scholes Option Pricing Model137

5.7 Forward and Futures Prices138

Formalization143

5.8 Replicating an Option on Spot with Futures146

Formalization148

Hedge Ratios149

5.9 Summary150

6 MARTINGALE PRICING155

6.0 Introduction155

6.1 Relative Prices and Martingales155

The Money Market Account156

Risk-Neutral Valuation156

6.2 Martingales and No Arbitrage157

6.3 Futures Prices161

Formal Description165

6.4 Summary166

Appendix：Proof of the Proposition171

7 AMERICAN OPTIONS175

7.0 Introduction175

7.1 Cum-Dividend/Ex-Dividend Prices176

7.2 American Call Options178

No Dividends178

Dividends181

7.3 American Put Options183

Time Value183

Dividends185

7.4 Valuation187

American Call Options187

Computational Complexity191

American Put Options192

7.5 Options on Forward Contracts195

Call Options196

Put Options198

Valuation199

7.6 Summary203

PART Ⅲ THE BLACK-SCHOLES MODEL AND EXTENSIONS210

8 THE BLACK-SCHOLES MODEL210

8.0 Introduction210

8.1 Continuous Time Representation of Stock Price Changes211

8.2 Interest Rates213

8.3 Ito’s Lemma213

8.4 The Equivalent Martingale Probability Distribution215

8.5 European Options217

8.6 Hedging218

8.7 Properties of the Black-Scholes Model224

8.8 Use of the Black-Scholes Model227

Historic Volatility228

Implied Volatility231

8.9 Option Strategies233

8.10 Partial Differential Equation235

Derivation235

Delta，Gamma，and Theta236

8.11 Summary237

Appendix A245

Appendix B247

Appendix C：Unequally Spaced Observations248

Appendix D249

9 EXTENSIONS TO THE BLACK-SCHOLES MODEL251

9.0 Introduction251

9.1 Known Dividend Model251

9.2 Pseudo-American Model255

9.3 The Roll Model257

9.4 Constant Dividend Yield Model258

9.5 Options on Futures and Forward Contracts261

Futures Contracts261

Forward Contracts264

9.6 Summary265

Appendix：Continuous Dividend Yield271

10 REPLICATION AND RISK EXPOSURE WITH MODEL MISSPECIFICATION272

10.0 Introduction272

10.1 Problems with Delta Hedging272

10.2 A General Approach278

10.3 Delta Hedging282

The Delta Neutral Position283

Formalization285

10.4 Delta-Gamma Hedging286

Formalization289

Theta Neutral289

10.5 Delta-Gamma-Vega Hedging290

Formalization293

10.6 Model Misspecification294

10.7 Summary294

11 FOREIGN CURRENCY302

11.0 Introduction302

11.1 Foreign Currency Derivatives302

Foreign Currency Options303

Options on Foreign Currency Futures305

Uses of Currency Derivatives308

11.2 Single-Period Example310

11.3 Multiperiod Extension315

11.4 Formalization318

Martingale Pricing320

Risk-Neutral Valuation320

Replicating Portfolio320

11.5 Lattice Parameters321

11.6 Closed Form Solutions （Modified Black-Scholes）325

11.7 American Options330

11.8 Options on Foreign Currency Futures334

Closed Form Solutions336

11.9 Replicating Options on Spot with Futures338

11.10 Summary342

Appendix：The Quadratic Approximation349

12 STOCK INDEXES AND COMMODITIES350

12.0 Introduction350

12.1 Derivatives on Stock Market Indexes350

12.2 Stock Market Index Futures352

Futures Prices352

12.3 Spread Trading355

Same Index355

Different Indexes356

12.4 Index Options357

12.5 Pricing Index Options361

Binomial Pricing Model362

Closed Form Solutions364

Discrete Dividends365

Synthetic Options366

Circuit Breakers368

The Wildcard Option369

12.6 Options on Index Futures369

Pricing Index Futures Options371

12.7 Commodity Derivatives372

Futures Prices372

Futures Options374

12.8 Summary377

PART Ⅳ INTEREST RATE CONTRACTS，THE HJM MODEL，AND EXTENSIONS386

13 INTEREST RATE CONTRACTS386

13.0 Introduction386

13.1 Zero-Coupon Bonds386

Discount Rates387

Simple Interest Rates388

Continuously Compounded Interest Rates389

13.2 Coupon Bonds389

Pricing390

Yield-to-Maturity391

Quotes393

Floating Rate Notes396

13.3 The Term Structure of Default-Free Interest Rates397

Forward Rates397

Formalization399

Par Bond Yield Curve401

Computing the Zero-Coupon Yield Curve403

13.4 Traditional Measures of Interest Rate Risk404

Duration404

Convexity407

Limitations of Analysis409

13.5 Treasury Bill Futures410

13.6 Eurodollar Contracts411

Eurodollar Deposits411

Forward Rate Agreements （FRAs）413

Formalization414

Futures Contracts416

13.7 Treasury Bond and Note Futures418

The Delivery Process420

13.8 Treasury Bond Futures421

Conversion Factors421

Cheapest to Deliver423

Wild Card Option424

Timing Option425

13.9 Summary425

Appendix：Duration and Convexity Correction for a Semiannual Coupon Bond428

14 SWAPS431

14.0 Introduction431

14.1 Interest Rate Swaps431

Pricing Schedules433

Warehousing434

Valuation434

Par Swaps437

Variants of Interest Rate Swaps439

14.2 Foreign Currency Swaps439

Valuation of Currency Swaps441

Variants of Foreign Currency Swaps444

14.3 Commodity Swaps444

Valuation of Commodity Swaps445

Variants of Commodity Swaps446

14.4 Equity Swaps447

Valuation448

Variants of the Basic Equity Swaps450

14.5 Summary450

15 INTEREST RATE DERIVATIVES455

15.0 Introduction455

15.1 Construction of the Lattice455

15.2 Spot Rate Process463

Normal Distribution463

Lognormal Distribution468

15.3 Valuing Options on Treasury Bills473

Put Options474

A Replicating Portfolio475

Call Options477

Put-Call Parity477

15.4 Treasury Bill Futures478

Pricing478

Hedging480

15.5 Summary482

16 PRICING TREASURY BILLS，TREASURY BONDS，TREASURY FUTURES，AND HEDGING WITH MODEL MISSPECIFICATION487

16.0 Introduction487

16.1 The Heath-Jarrow-Morton Model488

The Forward Rate Evolution488

The Spot Rate and Money Market Account Evolution490

Zero-Coupon Bond Price Evolution491

Arbitrage-Free Restrictions492

Mean Reversion/Volatility Reduction493

16.2 Hedging Treasury Bills494

Hedge Ratio （Delta）496

Gamma Hedging500

Gamma and Convexity501

16.3 Hedging Treasury Bonds504

16.4 Treasury Bill Futures506

Hedging508

Hedge Ratio （Delta）509

Gamma Hedging512

16.5 Treasury Bond Futures512

16.6 Summary516

Appendix：Proof of the Results on Pages 492-493520

17 PRICING INTEREST RATE OPTIONS AND HEDGING WITH MODEL MISSPECIFIGATION522

17.0 Introduction522

17.1 Options on Treasury Bills522

Pricing523

Hedging525

17.2 Caps，Floors，and Collars527

Caps and Caplets527

Floors and Collars531

The Black Model for Caps and Floors533

17.3 Options on Treasury Bonds537

Swaptions539

17.4 Options on Treasury Bill Futures544

17.5 Options on Treasury Bond Futures547

17.6 Summary551

18 CREDIT RISK556

18.0 Introduction556

18.1 Pricing Credit-Risky Bonds557

Lattice of Default-Free Interest Rates559

Risky Debt560

Credit-Risky Debt561

Formalization568

Interpretation of Expression （18.10a，b）571

Value of the Claim in the Event of Default571

18.2 Pricing Options on Credit-Risky Bonds571

Formalization573

Hedging573

18.3 Pricing Vulnerable Derivatives575

Formalization577

Hedging580

Risk Management581

18.4 Valuation of a Swap581

Fixed Payment Side581

Floating Payment Side582

18.5 Credit Default Swaps583

18.6 Regulation587

Bank of International Settlement （B.I.S.） 1988 Accord588

B.I.S.1996 Amendment590

Limitations of the 1988 Accord and 1996 Amendment591

18.7 What Can Go Wrong？592

Recipes for Risk593

18.8 Summary596

PART Ⅴ EXOTICS602

19 NON-STANDARD （EXOTIC） OPTIONS602

19.0 Introduction602

19.1 European Digital （Binary） Options602

Digital Call602

Digital Put603

Hedging606

Gap Options610

19.2 Paylater Options612

Call Option612

Put Option615

19.3 Compound Options616

Options on a Call616

Options on a Put618

19.4 Chooser Option621

Special Case （T1=T2，K1=K2）621

General Case622

19.5 Options on the Minimum or Maximum of Two Risky Assets623

Formalization625

Call Option on the Minimum625

Put Option on the Minimum626

Call Option on the Maximum627

Put Option on the Maximum628

19.6 Summary630

20 NON-STANDARD （EXOTIC） OPTIONS：PATH-DEPENDENT635

20.0 Introduction635

20.1 Barrier Options635

Down-and-Out Options635

Down-and-In Options641

Up-and-Out Options643

Up-and-In Options646

20.2 Lookback Options647

Standard Lookback Options647

Options on Extrema650

20.3 Average Options653

20.4 Summary659

GLOSSARY OF SYMBOLS664

GLOSSARY OF TERMS667

INDEX677