PART Ⅰ THE BASICS 2
1 INTRODUCTION TO DERIVATIVE SECURITIES 2
1.0 Introduction 2
1.1 Forward Contracts 3
Formalization 4
1.2 Futures Contracts 6
Standardization 6
Clearing House 8
Settlement Price 9
Daily Settlement and Margins 10
Regulation 11
Why Standardization？Why Daily Settlement？ 11
Basis 12
Newspaper Quotes 13
1.3 Options 15
Call Options 15
Put Options 17
American versus European Options 19
1.4 Organized Option Markets 20
1.5 Option Newspaper Quotes 22
1.6 Interest Rates and Bond Prices 23
Zero-Coupon Bond Prices 23
Discount Rates 24
Simple Interest Rates 25
Discretely Compounded Interest Rates 26
Continuously Compounded Interest Rates 29
1.7 Summary 30
2 SIMPLE ARBITRAGE RELATIONSHIPS FOR FORWARD AND FUTURES CONTRACTS 34
2.0 Introduction 34
2.1 Definition of Arbitrage 34
2.2 Assumptions 35
2.3 Forward and Spot Prices 37
No Cash Flows on the Underlying Asset Over the Life of the Forward Contract 37
Formal Derivation （Cash-and-Carry） 39
Value of a Forward Contract 41
2.4 Known Cash Flows to the Underlying Asset 42
Formal Derivation 44
Value of a Forward Contract 46
2.5 Forward Contracts on Constant Dividend Yield and Interest-Paying Assets 47
Forward Contracts on a Stock Index 47
Foreign Exchange Forward Contracts 48
2.6 Forward Contracts on Commodities 51
Storage Costs 52
Convenience Yield 54
The Implied Repo Rate 55
Forward Contracts on Electricity 55
2.7 Forward and Futures Prices Compared 56
Equality of Forward and Futures Prices 59
Empirical Evidence 61
2.8 Summary 62
Appendix：Present Value of Dividends Over Life of Forward Contract 67
3 SIMPLE ARBITRAGE RELATIONSHIPS FOR OPTIONS 68
3.0 Introduction 68
3.1 Call and Put Options 68
3.2 Put Options 74
3.3 Relationship Between European Call and Put Options 79
3.4 Relationship Between American Call and Put Options 82
3.5 Summary 84
PART Ⅱ THE BINOMIAL MODEL 90
4 ASSET PRICE DYNAMICS 90
4.0 Introduction 90
4.1 The Lognormal Distribution 91
4.2 The Basic Idea （Binomial Pricing） 96
4.3 Formal Description （Binomial Pricing） 98
4.4 The Binomial Approximation to the Lognormal Distribution 99
4.5 Extensions 105
4.6 Stochastic Differential Equation Representation 105
4.7 Complications 107
Lognormal Distribution 107
Continuous Trading 107
Continuously Changing Prices 108
4.8 Summary 108
Appendix：The Expected Value of the Future Stock Price 112
5 THE BINOMIAL PRICING MODEL 114
5.0 Introduction 114
5.1 Single-Period Example 115
5.2 Multiperiod Example 119
5.3 The Binomial Pricing Model 123
The Binomial Model 123
Constructing the Synthetic Option 124
Risk-Neutral Valuation 126
Put Options 130
5.4 Hedge Ratio （Delta） 133
5.5 Lattice Parameters 133
5.6 The Black-Scholes Option Pricing Model 137
5.7 Forward and Futures Prices 138
Formalization 143
5.8 Replicating an Option on Spot with Futures 146
Formalization 148
Hedge Ratios 149
5.9 Summary 150
6 MARTINGALE PRICING 155
6.0 Introduction 155
6.1 Relative Prices and Martingales 155
The Money Market Account 156
Risk-Neutral Valuation 156
6.2 Martingales and No Arbitrage 157
6.3 Futures Prices 161
Formal Description 165
6.4 Summary 166
Appendix：Proof of the Proposition 171
7 AMERICAN OPTIONS 175
7.0 Introduction 175
7.1 Cum-Dividend/Ex-Dividend Prices 176
7.2 American Call Options 178
No Dividends 178
Dividends 181
7.3 American Put Options 183
Time Value 183
Dividends 185
7.4 Valuation 187
American Call Options 187
Computational Complexity 191
American Put Options 192
7.5 Options on Forward Contracts 195
Call Options 196
Put Options 198
Valuation 199
7.6 Summary 203
PART Ⅲ THE BLACK-SCHOLES MODEL AND EXTENSIONS 210
8 THE BLACK-SCHOLES MODEL 210
8.0 Introduction 210
8.1 Continuous Time Representation of Stock Price Changes 211
8.2 Interest Rates 213
8.3 Ito’s Lemma 213
8.4 The Equivalent Martingale Probability Distribution 215
8.5 European Options 217
8.6 Hedging 218
8.7 Properties of the Black-Scholes Model 224
8.8 Use of the Black-Scholes Model 227
Historic Volatility 228
Implied Volatility 231
8.9 Option Strategies 233
8.10 Partial Differential Equation 235
Derivation 235
Delta，Gamma，and Theta 236
8.11 Summary 237
Appendix A 245
Appendix B 247
Appendix C：Unequally Spaced Observations 248
Appendix D 249
9 EXTENSIONS TO THE BLACK-SCHOLES MODEL 251
9.0 Introduction 251
9.1 Known Dividend Model 251
9.2 Pseudo-American Model 255
9.3 The Roll Model 257
9.4 Constant Dividend Yield Model 258
9.5 Options on Futures and Forward Contracts 261
Futures Contracts 261
Forward Contracts 264
9.6 Summary 265
Appendix：Continuous Dividend Yield 271
10 REPLICATION AND RISK EXPOSURE WITH MODEL MISSPECIFICATION 272
10.0 Introduction 272
10.1 Problems with Delta Hedging 272
10.2 A General Approach 278
10.3 Delta Hedging 282
The Delta Neutral Position 283
Formalization 285
10.4 Delta-Gamma Hedging 286
Formalization 289
Theta Neutral 289
10.5 Delta-Gamma-Vega Hedging 290
Formalization 293
10.6 Model Misspecification 294
10.7 Summary 294
11 FOREIGN CURRENCY 302
11.0 Introduction 302
11.1 Foreign Currency Derivatives 302
Foreign Currency Options 303
Options on Foreign Currency Futures 305
Uses of Currency Derivatives 308
11.2 Single-Period Example 310
11.3 Multiperiod Extension 315
11.4 Formalization 318
Martingale Pricing 320
Risk-Neutral Valuation 320
Replicating Portfolio 320
11.5 Lattice Parameters 321
11.6 Closed Form Solutions （Modified Black-Scholes） 325
11.7 American Options 330
11.8 Options on Foreign Currency Futures 334
Closed Form Solutions 336
11.9 Replicating Options on Spot with Futures 338
11.10 Summary 342
Appendix：The Quadratic Approximation 349
12 STOCK INDEXES AND COMMODITIES 350
12.0 Introduction 350
12.1 Derivatives on Stock Market Indexes 350
12.2 Stock Market Index Futures 352
Futures Prices 352
12.3 Spread Trading 355
Same Index 355
Different Indexes 356
12.4 Index Options 357
12.5 Pricing Index Options 361
Binomial Pricing Model 362
Closed Form Solutions 364
Discrete Dividends 365
Synthetic Options 366
Circuit Breakers 368
The Wildcard Option 369
12.6 Options on Index Futures 369
Pricing Index Futures Options 371
12.7 Commodity Derivatives 372
Futures Prices 372
Futures Options 374
12.8 Summary 377
PART Ⅳ INTEREST RATE CONTRACTS，THE HJM MODEL，AND EXTENSIONS 386
13 INTEREST RATE CONTRACTS 386
13.0 Introduction 386
13.1 Zero-Coupon Bonds 386
Discount Rates 387
Simple Interest Rates 388
Continuously Compounded Interest Rates 389
13.2 Coupon Bonds 389
Pricing 390
Yield-to-Maturity 391
Quotes 393
Floating Rate Notes 396
13.3 The Term Structure of Default-Free Interest Rates 397
Forward Rates 397
Formalization 399
Par Bond Yield Curve 401
Computing the Zero-Coupon Yield Curve 403
13.4 Traditional Measures of Interest Rate Risk 404
Duration 404
Convexity 407
Limitations of Analysis 409
13.5 Treasury Bill Futures 410
13.6 Eurodollar Contracts 411
Eurodollar Deposits 411
Forward Rate Agreements （FRAs） 413
Formalization 414
Futures Contracts 416
13.7 Treasury Bond and Note Futures 418
The Delivery Process 420
13.8 Treasury Bond Futures 421
Conversion Factors 421
Cheapest to Deliver 423
Wild Card Option 424
Timing Option 425
13.9 Summary 425
Appendix：Duration and Convexity Correction for a Semiannual Coupon Bond 428
14 SWAPS 431
14.0 Introduction 431
14.1 Interest Rate Swaps 431
Pricing Schedules 433
Warehousing 434
Valuation 434
Par Swaps 437
Variants of Interest Rate Swaps 439
14.2 Foreign Currency Swaps 439
Valuation of Currency Swaps 441
Variants of Foreign Currency Swaps 444
14.3 Commodity Swaps 444
Valuation of Commodity Swaps 445
Variants of Commodity Swaps 446
14.4 Equity Swaps 447
Valuation 448
Variants of the Basic Equity Swaps 450
14.5 Summary 450
15 INTEREST RATE DERIVATIVES 455
15.0 Introduction 455
15.1 Construction of the Lattice 455
15.2 Spot Rate Process 463
Normal Distribution 463
Lognormal Distribution 468
15.3 Valuing Options on Treasury Bills 473
Put Options 474
A Replicating Portfolio 475
Call Options 477
Put-Call Parity 477
15.4 Treasury Bill Futures 478
Pricing 478
Hedging 480
15.5 Summary 482
16 PRICING TREASURY BILLS，TREASURY BONDS，TREASURY FUTURES，AND HEDGING WITH MODEL MISSPECIFICATION 487
16.0 Introduction 487
16.1 The Heath-Jarrow-Morton Model 488
The Forward Rate Evolution 488
The Spot Rate and Money Market Account Evolution 490
Zero-Coupon Bond Price Evolution 491
Arbitrage-Free Restrictions 492
Mean Reversion/Volatility Reduction 493
16.2 Hedging Treasury Bills 494
Hedge Ratio （Delta） 496
Gamma Hedging 500
Gamma and Convexity 501
16.3 Hedging Treasury Bonds 504
16.4 Treasury Bill Futures 506
Hedging 508
Hedge Ratio （Delta） 509
Gamma Hedging 512
16.5 Treasury Bond Futures 512
16.6 Summary 516
Appendix：Proof of the Results on Pages 492-493 520
17 PRICING INTEREST RATE OPTIONS AND HEDGING WITH MODEL MISSPECIFIGATION 522
17.0 Introduction 522
17.1 Options on Treasury Bills 522
Pricing 523
Hedging 525
17.2 Caps，Floors，and Collars 527
Caps and Caplets 527
Floors and Collars 531
The Black Model for Caps and Floors 533
17.3 Options on Treasury Bonds 537
Swaptions 539
17.4 Options on Treasury Bill Futures 544
17.5 Options on Treasury Bond Futures 547
17.6 Summary 551
18 CREDIT RISK 556
18.0 Introduction 556
18.1 Pricing Credit-Risky Bonds 557
Lattice of Default-Free Interest Rates 559
Risky Debt 560
Credit-Risky Debt 561
Formalization 568
Interpretation of Expression （18.10a，b） 571
Value of the Claim in the Event of Default 571
18.2 Pricing Options on Credit-Risky Bonds 571
Formalization 573
Hedging 573
18.3 Pricing Vulnerable Derivatives 575
Formalization 577
Hedging 580
Risk Management 581
18.4 Valuation of a Swap 581
Fixed Payment Side 581
Floating Payment Side 582
18.5 Credit Default Swaps 583
18.6 Regulation 587
Bank of International Settlement （B.I.S.） 1988 Accord 588
B.I.S.1996 Amendment 590
Limitations of the 1988 Accord and 1996 Amendment 591
18.7 What Can Go Wrong？ 592
Recipes for Risk 593
18.8 Summary 596
PART Ⅴ EXOTICS 602
19 NON-STANDARD （EXOTIC） OPTIONS 602
19.0 Introduction 602
19.1 European Digital （Binary） Options 602
Digital Call 602
Digital Put 603
Hedging 606
Gap Options 610
19.2 Paylater Options 612
Call Option 612
Put Option 615
19.3 Compound Options 616
Options on a Call 616
Options on a Put 618
19.4 Chooser Option 621
Special Case （T1=T2，K1=K2） 621
General Case 622
19.5 Options on the Minimum or Maximum of Two Risky Assets 623
Formalization 625
Call Option on the Minimum 625
Put Option on the Minimum 626
Call Option on the Maximum 627
Put Option on the Maximum 628
19.6 Summary 630
20 NON-STANDARD （EXOTIC） OPTIONS：PATH-DEPENDENT 635
20.0 Introduction 635
20.1 Barrier Options 635
Down-and-Out Options 635
Down-and-In Options 641
Up-and-Out Options 643
Up-and-In Options 646
20.2 Lookback Options 647
Standard Lookback Options 647
Options on Extrema 650
20.3 Average Options 653
20.4 Summary 659
GLOSSARY OF SYMBOLS 664
GLOSSARY OF TERMS 667
INDEX 677