Financial Derivatives in Theory and Practice

Financial Derivatives in Theory and Practice......Page 3
Contents......Page 9
Preface to revised edition......Page 17
Preface......Page 19
Acknowledgements......Page 23
Part I: Theory......Page 25
1.1 Option pricing in a nutshell......Page 27
1.2 The simplest setting......Page 28
1.3 General one-period economy......Page 29
1.3.1 Pricing......Page 30
1.3.2 Conditions for no arbitrage: existence of Z......Page 31
1.3.3 Completeness: uniqueness of Z......Page 33
1.3.4 Probabilistic formulation......Page 36
1.4 A two-period example......Page 39
2.1 Introduction......Page 43
2.2 Definition and existence......Page 44
2.3.1 Limit of a random walk......Page 45
2.3.2 Deterministic transformations of Brownian motion......Page 47
2.3.3 Some basic sample path properties......Page 48
2.4 Strong Markov property......Page 50
2.4.1 Reflection principle......Page 52
3 Martingales......Page 55
3.1 Definition and basic properties......Page 56
3.2.1 Martingales bounded in L(1)......Page 59
3.2.2 Uniformly integrable martingales......Page 60
3.2.3 Square-integrable martingales......Page 63
3.3.1 Stopping times......Page 65
3.3.2 Optional sampling theorem......Page 69
3.4.1 Total variation and Stieltjes integration......Page 73
3.4.2 Quadratic variation......Page 75
3.4.3 Quadratic covariation......Page 79
3.5.1 The space cM(loc)......Page 80
3.5.2 Semimartingales......Page 83
3.6 Supermartingales and the Doob—Meyer decomposition......Page 85
4.1 Outline......Page 87
4.2 Predictable processes......Page 89
4.3 Stochastic integrals: the L(2) theory......Page 91
4.3.1 The simplest integral......Page 92
4.3.2 The Hilbert space L(2)(M)......Page 93
4.3.3 The L(2) integral......Page 94
4.3.4 Modes of convergence to H • M......Page 96
4.4 Properties of the stochastic integral......Page 98
4.5.1 Continuous local martingales as integrators......Page 101
4.5.2 Semimartingales as integrators......Page 102
4.5.3 The end of the road!......Page 104
4.6.1 Integration by parts and Itô’s formula......Page 105
4.6.2 Differential notation......Page 107
4.6.3 Multidimensional version of Itô\'s formula......Page 109
4.6.4 Lévy’s theorem......Page 112
5.1.1 Basic results and properties......Page 115
5.1.2 Equivalent and locally equivalent measures on a filtered space......Page 119
5.1.3 Novikov’s condition......Page 121
5.2.1 Girsanov’s theorem for continuous semimartingales......Page 123
5.2.2 Girsanov’s theorem for Brownian motion......Page 125
5.3 Martingale representation theorem......Page 129
5.3.1 The space I(2)(M) and its orthogonal complement......Page 130
5.3.2 Martingale measures and the martingale representation theorem......Page 134
5.3.3 Extensions and the Brownian case......Page 135
6.1 Introduction......Page 139
6.2 Formal definition of an SDE......Page 140
6.3 An aside on the canonical set-up......Page 141
6.4.1 Weak solutions......Page 143
6.4.2 Strong solutions......Page 145
6.4.3 Tying together strong and weak......Page 148
6.5 Establishing existence and uniqueness: Itô theory......Page 149
6.5.1 Picard—Lindelöf iteration and ODEs......Page 150
6.5.2 A technical lemma......Page 151
6.5.3 Existence and uniqueness for Lipschitz coefficients......Page 154
6.6 Strong Markov property......Page 158
6.7 Martingale representation revisited......Page 163
7 Option Pricing in Continuous Time......Page 165
7.1.1 A model for asset prices......Page 166
7.1.2 Self-financing trading strategies......Page 168
7.2 Pricing European options......Page 170
7.2.1 Option value as a solution to a PDE......Page 171
7.2.2 Option pricing via an equivalent martingale measure......Page 173
7.3 Continuous time theory......Page 175
7.3.1 Information within the economy......Page 176
7.3.2 Units, numeraires and martingale measures......Page 177
7.3.3 Arbitrage and admissible strategies......Page 182
7.3.4 Derivative pricing in an arbitrage-free economy......Page 187
7.3.5 Completeness......Page 188
7.3.6 Pricing kernels......Page 197
7.4.1 General payout schedules......Page 200
7.4.2 Controlled derivative payouts......Page 202
7.4.3 More general asset price processes......Page 203
7.4.4 Infinite trading horizon......Page 204
8.2 An economy of pure discount bonds......Page 207
8.3 Modelling the term structure......Page 211
8.3.2 Pricing kernel approach......Page 215
8.3.3 Numeraire models......Page 216
8.3.4 Finite variation kernel models......Page 218
8.3.6 Short-rate models......Page 221
8.3.7 Heath—Jarrow—Morton models......Page 224
8.3.8 Flesaker—Hughston models......Page 230
Part II: Practice......Page 237
9.2 The real world is not a martingale measure......Page 239
9.2.1 Modelling via infinitesimals......Page 240
9.2.2 Modelling via macro information......Page 241
9.3 Product-based modelling......Page 242
9.3.1 A warning on dimension reduction......Page 243
9.3.2 Limit cap valuation......Page 245
9.4 Local versus global calibration......Page 247
10.2 Deposits......Page 251
10.2.1 Accrual factors and LIBOR......Page 252
10.3 Forward rate agreements......Page 253
10.4 Interest rate swaps......Page 254
10.5 Zero coupon bonds......Page 256
10.6.2 Deposit valuation......Page 257
10.6.4 Swap valuation......Page 258
11.2 Forward rate agreements and swaps......Page 261
11.3 Caps and floors......Page 262
11.3.1 Valuation......Page 264
11.3.2 Put—call parity......Page 265
11.4 Vanilla swaptions......Page 266
11.5.1 Digital caps and floors......Page 268
11.5.2 Digital swaptions......Page 269
12.2 Futures contract definition......Page 271
12.2.1 Contract specification......Page 272
12.2.2 Market risk without credit risk......Page 273
12.2.3 Mathematical formulation......Page 275
12.3.1 Discrete resettlement......Page 276
12.3.2 Continuous resettlement......Page 277
12.4 Recovering the futures price process......Page 279
12.5 Relationship between forwards and futures......Page 280
Orientation: Pricing Exotic European Derivatives......Page 283
13.2.1 Model requirements......Page 287
13.2.2 Terminal swap-rate models......Page 289
13.3.1 The exponential swap-rate model......Page 290
13.3.2 The geometric swap-rate model......Page 291
13.3.3 The linear swap-rate model......Page 292
13.4 Arbitrage-free property of terminal swap-rate models......Page 293
13.4.1 Existence of calibrating parameters......Page 294
13.4.2 Extension of model to [0, )......Page 295
13.5 Zero coupon swaptions......Page 297
14.1 Introduction......Page 301
14.2.1 Affine decomposition of convexity products......Page 302
14.2.2 Convexity corrections using the linear swap-rate model......Page 304
14.3 Examples and extensions......Page 306
14.3.1 Constant maturity swaps......Page 307
14.3.2 Options on constant maturity swaps......Page 308
14.3.3 LIBOR-in-arrears swaps......Page 309
15.1 Introduction......Page 311
15.2 Implying the functional form D(TS)......Page 312
15.3 Numerical implementation......Page 316
15.4 Irregular swaptions......Page 317
15.5 Numerical comparison of exponential and implied swap-rate models......Page 323
16.1 Introduction......Page 327
16.2 Model construction......Page 328
16.2.1 Log-normal case......Page 329
16.2.2 General case: volatility smiles......Page 331
16.3.1 Spread options......Page 332
16.3.2 Cross-currency swaptions......Page 335
Orientation: Pricing Exotic American and Path-Dependent Derivatives......Page 339
17.1 Introduction......Page 343
17.2.1 Vasicek—Hull—White model......Page 344
17.2.2 Log-normal short-rate models......Page 346
17.2.3 Cox—Ingersoll—Ross model......Page 347
17.2.4 Multidimensional short-rate models......Page 348
17.3 Parameter fitting within the Vasicek—Hull—White model......Page 349
17.3.1 Derivation of φ, ψ and B·(T)......Page 350
17.3.2 Derivation of ξ, ζ and η......Page 351
17.3.3 Derivation of µ, λ and A·(T)......Page 352
17.4 Bermudan swaptions via Vasicek—Hull—White......Page 353
17.4.2 Specifying the ‘tree’......Page 354
17.4.4 Evaluation of expected future value......Page 356
17.4.5 Error analysis......Page 358
18.1 Introduction......Page 361
18.2 LIBOR market models......Page 362
18.2.1 Determining the drift......Page 363
18.2.2 Existence of a consistent arbitrage-free term structure model......Page 365
18.3 Regular swap-market models......Page 367
18.3.1 Determining the drift......Page 368
18.3.3 Example application......Page 370
18.4 Reverse swap-market models......Page 371
18.4.1 Determining the drift......Page 372
18.4.2 Existence of a consistent arbitrage-free term structure model......Page 373
18.4.3 Example application......Page 374
19.2 Markov-functional models......Page 375
19.3 Fitting a one-dimensional Markov-functional model to swaption prices......Page 378
19.3.1 Deriving the numeraire on a grid......Page 379
19.3.2 Existence of a consistent arbitrage-free term structure model......Page 382
19.4.1 LIBOR model......Page 383
19.4.2 Swap model......Page 385
19.5 Multidimensional Markov-functional models......Page 387
19.5.1 Log-normally driven Markov-functional models......Page 388
19.6 Relationship to market models......Page 389
19.7.1 Mean reversion and correlation......Page 391
19.7.2 Mean reversion and forward volatilities......Page 392
19.7.3 Mean reversion within the Markov-functional LIBOR model......Page 393
19.8 Some numerical results......Page 394
20 Exercises and Solutions......Page 397
Appendix 1 The Usual Conditions......Page 441
Appendix 2 L(2) Spaces......Page 443
Appendix 3 Gaussian Calculations......Page 445
References......Page 447
Index......Page 451