PartⅠ Spot and Futures Markets 3
1 AnIntroduction to Financial Derivatives 3
1.1 Options 3
1.2 Futures Contracts and Options 5
1.3 Forward Contracts 6
1.4 Call and Put Spot Options 8
1.4.1 One-period Spot Market 9
1.4.2 Replicating Portfolios 10
1.4.3 Martingale Measure for a Spot Market 12
1.4.4 Absence of Arbitrage 13
1.4.5 Optimality of Replication 15
1.4.6 Change of a Numeraire 17
1.4.7 Put Option 18
1.5 Forward Contracts 19
1.5.1 Forward Price 20
1.6 Futures Call and Put Options 21
1.6.1 Futures Contracts and Futures Prices 21
1.6.2 One-period Futures Market 22
1.6.3 Martingale Measure for a Futures Market 23
1.6.4 Absence of Arbitrage 24
1.6.5 One-period Spot/Futures Market 26
1.7 Options of American Style 26
1.8 Universal No-arbitrage Inequalities 31
2 Discrete-time Security Markets 35
2.1 TheCox-Ross-Rubinstein Model 36
2.1.1 Binomial Lattice for the Stock Price 36
2.1.2 Recursive Pricing Procedure 38
2.1.3 CRR Option Pricing Formula 43
2.2 Martingale Properties of the CRR Model 46
2.2.1 Martingale Measures 47
2.2.2 Risk-neutral Valuation Formula 50
2.2.3 Change of a Numeraire 51
2.3 The Black-Scholes Option Pricing Formula 53
2.4 Valuation of American Options 58
2.4.1 American Call Options 58
2.4.2 American Put Options 60
2.4.3 American Claims 61
2.5 Options on a Dividend-paying Stock 63
2.6 Security Markets in Discrete Time 65
2.6.1 Finite Spot Markets 66
2.6.2 Self-financing Trading Strategies 66
2.6.3 Replication and Arbitrage Opportunities 68
2.6.4 Arbitrage Price 69
2.6.5 Risk-neutral Valuation Formula 70
2.6.6 Existence ofa Martingale Measure 73
2.6.7 Completeness of a Finite Market 75
2.6.8 Separating Hyperplane Theorem 77
2.6.9 Change of a Numeraire 78
2.6.10 Discrete-time Models with Infinite State Space 79
2.7 Finite Futures Markets 80
2.7.1 Self-financing Futures Strategies 81
2.7.2 Martingale Measures for a Futures Market 82
2.7.3 Risk-neutral Valuation Formula 84
2.7.4 Futures Prices Versus Forward Prices 85
2.8 American Contingent Claims 87
2.8.1 Optimal Stopping Problems 90
2.8.2 Valuation and Hedging of American Claims 97
2.8.3 American Call and Put 101
2.9 Game Contingent Claims 101
2.9.1 Dynkin Games 102
2.9.2 Valuation and Hedging ofGame Contingent Claims 108
3 Benchmark Models in Continuous Time 113
3.1 The Black-Scholes Model 114
3.1.1 Risk-free Bond 114
3.1.2 Stock Price 114
3.1.3 Self-financing Trading Strategies 118
3.1.4 Martingale Measure for the Black-Scholes Model 120
3.1.5 Black-Scholes Option Pricing Formula 125
3.1.6 Case of Time-dependent Coefficients 131
3.1.7 Merton's Model 132
3.1.8 Put-Call Parity for Spot Options 134
3.1.9 Black-Scholes PDE 134
3.1.10 A Riskless Portfolio Method 137
3.1.11 Black-Scholes Sensitivities 140
3.1.12 Market Imperfections 144
3.1.13 Numerical Methods 145
3.2 A Dividend-paying Stock 147
3.2.1 Case of a Constant Dividend Yield 148
3.2.2 Case of Known Dividends 151
3.3 Bachelier Model 154
3.3.1 Bachelier Option Pricing Formula 155
3.3.2 Bachelier's PDE 157
3.3.3 Bachelier Sensitivities 158
3.4 Black Model 159
3.4.1 Self-financing Futures Strategies 160
3.4.2 Martingale Measure for the Futures Market 160
3.4.3 Black's Futures Option Formula 161
3.4.4 Options on Forward Contracts 165
3.4.5 Forward and Futures Prices 167
3.5 Robustness of the Black-Scholes Approach 168
3.5.1 Uncertain Volatility 168
3.5.2 European Call and Put Options 169
3.5.3 Convex Path-independent European Claims 172
3.5.4 General Path-independent European Claims 177
4 Foreign Market Derivatives 181
4.1 Cross-currency Market Model 181
4.1.1 Domestic Martingale Measure 182
4.1.2 Foreign Martingale Measure 184
4.1.3 Foreign Stock Price Dynamics 185
4.2 Currency Forward Contracts and Options 186
4.2.1 Forward Exchange Rate 186
4.2.2 Currency Option Valuation Formula 187
4.3 Foreign Equity Forward Contracts 191
4.3.1 Forward Price of a Foreign Stock 191
4.3.2 Quanto Forward Contracts 192
4.4 Foreign Market Futures Contracts 194
4.5 Foreign Equity Options 197
4.5.1 Options Struck in a Foreign Currency 198
4.5.2 Options Struck in Domestic Currency 199
4.5.3 Quanto Options 200
4.5.4 Equity-linked Foreign Exchange Options 202
5 American Options 205
5.1 Valuation of American Claims 206
5.2 American Call and PutOptions 213
5.3 Early Exercise Representation of an American Put 216
5.4 Analytical Approach 219
5.5 Approximations ofthe American Put Price 222
5.6 Option on a Dividend-paying Stock 224
5.7 Game Contingent Claims 226
6 Exotic Options 229
6.1 Packages 230
6.2 Forward-start Options 231
6.3 Chooser Options 232
6.4 Compound Options 233
6.5 Digital Options 234
6.6 Barrier Options 235
6.7 Lookback Options 238
6.8 Asian Options 242
6.9 Basket Options 245
6.10 Quantile Options 249
6.11 Other Exotic Options 251
7 Volatility Risk 253
7.1 Implied Volatilities of Traded Options 254
7.1.1 Historical Volatility 255
7.1.2 Implied Volatility 255
7.1.3 Implied Volatility Versus Historical Volatility 256
7.1.4 Approximate Formulas 257
7.1.5 Implied Volatility Surface 259
7.1.6 Asymptotic Behavior of the Implied Volatility 261
7.1.7 Marked-to-Market Models 264
7.1.8 Vega Hedging 265
7.1.9 Correlated Brownian Motions 267
7.1.10 Forward-start Options 269
7.2 Extensions ofthe Black-Scholes Model 273
7.2.1 CEV Model 273
7.2.2 Shifted Lognormal Models 277
7.3 Local Volatility Models 278
7.3.1 Implied Risk-Neutral Probability Law 278
7.3.2 Local Volatility 281
7.3.3 Mixture Models 287
7.3.4 Advantages and Drawbacks of LV Models 290
7.4 Stochastic Volatility Models 291
7.4.1 PDE Approach 292
7.4.2 Examples of SV Models 293
7.4.3 Hull and White Model 294
7.4.4 Heston's Model 299
7.4.5 SABRModel 301
7.5 Dynamical Models of Volatility Surfaces 302
7.5.1 Dynamics ofthe Local Volatility Surface 303
7.5.2 Dynamics ofthe Implied Volatility Surface 303
7.6 Alternative Approaches 307
7.6.1 Modelling of Asset Returns 308
7.6.2 Modelling of Volatility and Realized Variance 313
8 Continuous-time Security Markets 315
8.1 Standard MarketModels 316
8.1.1 Standard Spot Market 316
8.1.2 Futures Market 325
8.1.3 Choice of a Numeraire 327
8.1.4 Existence of a Martingale Measure 330
8.1.5 Fundamental Theorem of Asset Pricing 332
8.2 Multidimensional Black-Scholes Model 333
8.2.1 Market Completeness 335
8.2.2 Variance-minimizing Hedging 337
8.2.3 Risk-minimizing Hedging 338
8.2.4 Market Imperfections 345
PartⅡ Fixed-income Markets 351
9 Interest Rates and Related Contracts 351
9.1 Zero-coupon Bonds 351
9.1.1 Term Structure of Interest Rates 352
9.1.2 Forward Interest Rates 353
9.1.3 Short-term Interest Rate 354
9.2 Coupon-bearing Bonds 354
9.2.1 Yield-to-Maturity 355
9.2.2 Market Conventions 357
9.3 Interest Rate Futures 358
9.3.1 Treasury Bond Futures 358
9.3.2 Bond Options 359
9.3.3 Treasury Bill Futures 360
9.3.4 Eurodollar Futures 362
9.4 Interest Rate Swaps 363
9.4.1 Forward Rate Agreements 364
9.5 Stochastic Models ofBond Prices 366
9.5.1 Arbitrage-free Family of Bond Prices 366
9.5.2 Expectations Hypotheses 367
9.5.3 Case of It？ Processes 368
9.5.4 Market Price for Interest Rate Risk 371
9.6 Forward Measure Approach 372
9.6.1 Forward Price 373
9.6.2 Forward Martingale Measure 375
9.6.3 Forward Processes 378
9.6.4 Choice of a Numeraire 379
10 Short-Term Rate Models 383
10.1 Single-factor Models 384
10.1.1 Time-homogeneous Models 384
10.1.2 Time-inhomogeneous Models 394
10.1.3 Model Choice 399
10.1.4 American Bond Options 401
10.1.5 Options on Coupon-beating Bonds 402
10.2 Multi-factor Models 402
10.2.1 State Variables 403
10.2.2 Affine Models 404
10.2.3 Yield Models 404
10.3 Extended CIR Model 406
10.3.1 Squared Bessel Process 407
10.3.2 Model Construction 407
10.3.3 Change ofa Probability Measure 408
10.3.4 Zero-coupon Bond 409
10.3.5 Case of Constant Coefficients 410
10.3.6 Case of Piecewise Constant Coefficients 411
10.3.7 Dynamics of Zero-coupon Bond 412
10.3.8 Transition Densities 414
10.3.9 Bond Option 415
11 Models of Instantaneous Forward Rates 417
11.1 Heath-Jarrow-Morton Methodology 418
11.1.1 Ho and Lee Model 419
11.1.2 Heath-Jarrow-Morton Model 419
11.1.3 Absence of Arbitrage 421
11.1.4 Short-term Interest Rate 427
11.2 Gaussian HJM Model 428
11.2.1 Markovian Case 430
11.3 European Spot Options 434
11.3.1 Bond Options 435
11.3.2 Stock Options 438
11.3.3 Option on a Coupon-bearing Bond 441
11.3.4 Pricing of General Contingent Claims 444
11.3.5 Replication of Options 446
11.4 Volatilities and Correlations 449
11.4.1 Volatilities 449
11.4.2 Correlations 451
11.5 Futures Price 452
11.5.1 Futures Options 453
11.6 PDE Approach to Interest Rate Derivatives 457
11.6.1 PDEs for Spot Derivatives 457
11.6.2 PDEs for Futures Derivatives 461
11.7 Recent Developments 465
12 Market LIBOR Models 469
12.1 Forward and Futures LIBORs 471
12.1.1 One-period Swap Settled in Arrears 471
12.1.2 One-period Swap Settled in Advance 473
12.1.3 Eurodollar Futures 474
12.1.4 LIBOR in the Gaussian HJM Model 475
12.2 Interest Rate Caps and Floors 477
12.3 Valuation in the Gaussian HJM Model 479
12.3.1 Plain-vanilla Caps and Floors 479
12.3.2 Exotic Caps 481
12.3.3 Captions 483
12.4 LIBOR Market Models 484
12.4.1 Black's Formula for Caps 484
12.4.2 Miltersen,Sandmann and Sondermann Approach 486
12.4.3 Brace,Gatarek and Musiela Approach 486
12.4.4 Musiela and Rutkowski Approach 489
12.4.5 SDEs for LIBORs under the Forward Measure 492
12.4.6 Jamshidian's Approach 495
12.4.7 Altemative Derivation of Jamshidian's SDE 498
12.5 Properties of the Lognormal LIBOR Model 500
12.5.1 Transition Density of the LIBOR 501
12.5.2 Transition Density of the Forward Bond Price 503
12.6 Valuation in the Lognormal LIBOR Model 506
12.6.1 Pricing of Caps and Floors 506
12.6.2 Hedging of Caps and Floors 508
12.6.3 Valuation of European Claims 510
12.6.4 Bond Options 513
12.7 Extensions of the LLM Model 515
13 Alternative Market Models 517
13.1 Swaps and Swaptions 518
13.1.1 Forward Swap Rates 518
13.1.2 Swaptions 522
13.1.3 Exotic Swap Derivatives 524
13.2 Valuarion in the Gaussian HJM Model 527
13.2.1 Swaptions 527
13.2.2 CMS Spread Options 527
13.2.3 Yield Curve Swaps 529
13.3 Co-terminal Forward Swap Rates 530
13.3.1 Jamshidian's Model 535
13.3.2 Valuation of Co-terminal Swaptions 538
13.3.3 Hedging of Swaptions 539
13.3.4 Bermudan Swaptions 540
13.4 Co.initial Forward Swap Rates 541
13.4.1 Valuation of Co-initial Swaptions 544
13.4.2 Valuation of Exotic Options 545
13.5 Co.sliding Forward Swap Rates 546
13.5.1 Modelling of Co-sliding Swap Rates 547
13.5.2 Valuation of Co-sliding Swaptions 551
13.6 Swap Rate Model Versus LIBOR Model 552
13.6.1 Swaptions in the LLM Model 553
13.6.2 Caplets in the Co-terminal Swap Market Model 557
13.7 Markov-functional Models 558
13.7.1 Terminal Swap Rate Model 559
13.7.2 Calibration of Markov-functional Models 562
13.8 Fiesaker and Hughston Approach 565
13.8.1 Rational Lognormal Model 568
13.8.2 Valuation of Caps and Swaptions 569
14 Cross-currency Derivatives 573
14.1 Arbitrage-free Cross-currency Markets 574
14.1.1 Forward Price of a Foreign Asset 576
14.1.2 Valuation of Foreign Contingent Claims 580
14.1.3 Cross-currency Rates 581
14.2 Gaussian Model 581
14.2.1 Currency Options 582
14.2.2 Foreign Equity Options 583
14.2.3 Cross-currency Swaps 588
14.2.4 Cross-currency Swaptions 599
14.2.5 Basket Caps 602
14.3 Model of Forward LIBOR Rates 603
14.3.1 Quamo Cap 604
14.3.2 Cross-currency Swap 606
14.4 Concluding Remarks 607
PartⅢ APPENDIX 611
A An Overview of It？ Stochastic Calculus 611
A.1 Conditional Expectation 611
A.2 Filtrations and Adapted Processes 615
A.3 Martingales 616
A.4 Standard Brownian Motion 617
A.5 Stopping Times and Martingales 621
A.6 It？ Stochastic Integral 622
A.7 Continuous Local Martingales 625
A.8 Continuous Semimartingales 628
A.9 It？'s Lemma 630
A.10 Lévy's Characterization Theorem 633
A.11 Martingale Representation Property 634
A.12 Stochastic Differential Equations 636
A.13 Stochastic Exponential 639
A.14 Radon-Nikodym Density 640
A.15 Girsanov's Theorem 641
A.16 Martingale Measures 645
A.17 Feynman-Kac Forrnula 646
A.18 First Passage Times 649
References 657
Index 707