Exotic Option Pricing and Advanced Lvy Models

Exotic Option Pricing and Advanced Lévy Models......Page 1
Contents......Page 8
Contributors......Page 14
Preface......Page 16
About the Editors......Page 20
About the Contributors......Page 22
1.1 Introduction......Page 26
1.2 Lévy processes......Page 27
1.3 Examples of Lévy processes in finance......Page 29
1.3.1 Compound Poisson processes and jump-diffusions......Page 30
1.3.3 Meixner processes......Page 31
1.3.4 Generalized tempered stable processes and subclasses......Page 32
1.3.5 Generalized hyperbolic processes and subclasses......Page 34
1.4.1 Path variation......Page 35
1.4.2 Hitting points......Page 37
1.4.3 Creeping......Page 39
1.4.4 Regularity of the half line......Page 41
1.5.3 Meixner process......Page 42
1.5.4 Generalized tempered stable process......Page 44
1.5.5 Generalized hyperbolic process......Page 48
1.6 Conclusions......Page 49
References......Page 51
2.1 Introduction......Page 54
2.2.1 Modelling with Lévy processes......Page 55
2.2.2 Lattice methods......Page 56
2.2.3 Simulation methods......Page 57
2.3 A basis for a numerical approach......Page 58
2.3.1 The subordinator approach to simulation......Page 59
2.3.2 Applying the subordinator approach......Page 60
2.4.1 Stratified sampling and bridge methods......Page 61
2.4.2 Bridge sampling and the subordinator representation......Page 62
2.5 Valuing discretely reset path-dependent options......Page 64
2.6 Valuing continuously reset path-dependent options......Page 65
2.6.1 Options on extreme values and simulation bias......Page 67
2.6.2 Bias correction for Lévy processes......Page 68
2.6.3 Variation: exceedence probabilities......Page 69
2.6.4 Application of the bias correction algorithm......Page 70
References......Page 73
3.1 Introduction......Page 76
3.2 CGMY model details......Page 79
3.3 Estimation details......Page 82
3.3.1 Statistical estimation......Page 83
3.3.2 Risk neutral estimation......Page 84
3.4 Estimation results......Page 85
3.4.3 Results on gap risk expectation and price......Page 86
3.5 Conclusions......Page 88
References......Page 90
4.1 Introduction......Page 92
4.2 The models......Page 93
4.2.2 The Heston stochastic volatility model with jumps......Page 94
4.2.3 The Barndorff-Nielsen–Shephard model......Page 95
4.2.4 Lévy models with stochastic time......Page 96
4.3 Calibration......Page 99
4.4.1 NIG Lévy process......Page 103
4.4.5 Path generation for time-changed Lévy process......Page 104
4.5.1 Exotic options......Page 105
4.5.2 Exotic option prices......Page 107
4.6 Pricing of moment derivatives......Page 111
4.6.2 Moment options......Page 114
4.6.3 Hedging moment swaps......Page 115
4.6.4 Pricing of moments swaps......Page 116
4.7 Conclusions......Page 118
References......Page 120
5.1 Introduction......Page 124
5.2 Model and assumptions......Page 125
5.3 General description of the method......Page 130
5.4.1 Symmetry......Page 131
5.4.2 Valuation of European options......Page 136
5.4.3 Valuation of American options......Page 138
5.5.1 Symmetry......Page 139
5.5.2 Valuation of barrier and lookback options......Page 140
5.5.3 Valuation of Asian and basket options......Page 142
5.6 Margrabe-type options......Page 144
References......Page 149
6.1 Introduction......Page 154
6.2.1 The Heston stochastic volatility model......Page 156
6.2.2 The Barndorff-Nielsen–Shephard model......Page 157
6.2.3 Lévy models with stochastic time......Page 158
6.3 Static hedging of Asian options......Page 161
6.4.1 Characteristic function inversion using FFT......Page 163
6.5.2 Performance of the hedging strategy......Page 165
References......Page 170
7.1 Introduction......Page 174
7.2.1 Notation......Page 176
7.2.2 Consequence of the modelling choice......Page 178
7.3 The real option characteristics......Page 180
7.4.1 Optimal discount rate......Page 181
7.4.2 Average waiting time......Page 182
7.5 Robustness of the investment decision characteristics......Page 183
7.5.1 Robustness of the optimal time to invest......Page 184
7.5.2 Random jump size......Page 185
7.6.1 Error in the optimal profit–cost ratio......Page 186
7.6.2 Error in the investment opportunity value......Page 188
Appendix......Page 190
References......Page 192
8.1 Introduction......Page 194
8.2.1 One-step trinomial market......Page 195
8.2.2 One-step finite markets......Page 197
8.2.3 Multi-step finite markets......Page 198
8.2.5 Power-return assets......Page 199
8.3.1 Lévy processes......Page 202
8.3.3 Power-jump processes......Page 203
8.4 Enlarging the Lévy market model......Page 204
8.4.1 Martingale representation property......Page 205
8.5.1 Equivalent martingale measures......Page 208
8.5.2 Example: a Brownian motion plus a finite number of Poisson processes......Page 210
8.6 Optimal portfolios......Page 211
8.6.1 Optimal wealth......Page 212
8.6.2 Examples......Page 213
References......Page 217
9.1 Introduction......Page 220
9.2 First-passage distributions and other results for spectrally positive Lévy processes......Page 223
9.3 Description of the model, basic definitions and notations......Page 227
9.4 A renewal equation approach to pricing......Page 229
9.5 Explicit pricing formulae for American puts......Page 232
9.6 Some specific examples......Page 234
Appendix: use of fast Fourier transform......Page 238
References......Page 239
Further references......Page 240
10.1 Introduction......Page 242
10.2 Formulation of the problem......Page 243
10.3 The result and proof......Page 245
10.4 Remarks on numerics......Page 256
Appendix......Page 258
References......Page 259
11.1 Introduction......Page 262
11.2 Review of the backward free boundary problem......Page 264
11.3 Stationarity and domain extension in the maturity direction......Page 267
11.4 Additivity and domain extension in the strike direction......Page 270
11.5 The forward free boundary problem......Page 272
11.6 Summary and future research......Page 275
Appendix: Discretization of forward equation for American options......Page 276
References......Page 282
12.1 Introduction......Page 284
12.2 The CGMY process as a Lévy process......Page 285
12.2.1 Options in a Lévy market......Page 286
12.3.1 Discretization and solution algorithm......Page 288
12.4 Numerical experiments......Page 295
Appendix: Analytic formula for European option prices......Page 296
References......Page 300
13.1 Introduction......Page 302
13.2.1 Static no-arbitrage prices......Page 304
13.2.2 No-arbitrage price processes......Page 307
13.3 Convertible bonds......Page 311
References......Page 314
14.1 Introduction......Page 318
14.2 Formulation of the problem......Page 319
14.3 Solution of the free-boundary problem......Page 321
14.4 Main result and proof......Page 324
14.5 Conclusions......Page 327
References......Page 329
Index......Page 332