American-Type Options (eBook)

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Dmitrii S. Silvestrov, Stockholm University,Sweden.

Preface 5
1 Multivariate modulated Markov log-price processes (LPP) 11
1.1 Markov LPP 11
1.2 LPP represented by random walks 18
1.3 Autoregressive LPP 28
1.4 Autoregressive stochastic volatility LPP 38
2 American-type options 54
2.1 American-type options 54
2.2 Pay-off functions 57
2.3 Reward and log-reward functions 63
2.4 Optimal stopping times 73
2.5 American-type knockout options 82
3 Backward recurrence reward algorithms 86
3.1 Binomial tree reward algorithms 86
3.2 Trinomial tree reward algorithms 98
3.3 Random walk reward algorithms 110
3.4 Markov chain reward algorithms 116
4 Upper bounds for option rewards 125
4.1 Markov LPP with bounded characteristics 125
4.2 LPP represented by random walks 137
4.3 Markov LPP with unbounded characteristics 143
4.4 Univariate Markov Gaussian LPP 164
4.5 Multivariate modulated Markov Gaussian LPP 169
5 Convergence of option rewards – I 177
5.1 Asymptotically uniform upper bounds for rewards – I 178
5.2 Modulated Markov LPP with bounded characteristics 190
5.3 LPP represented by modulated random walks 204
6 Convergence of option rewards – II 213
6.1 Asymptotically uniform upper bounds for rewards – II 214
6.2 Univariate modulated LPP with unbounded characteristics 224
6.3 Asymptotically uniform upper bounds for rewards – III 230
6.4 Multivariate modulated LPP with unbounded characteristics 241
6.5 Conditions of convergence for Markov price processes 248
7 Space-skeleton reward approximations 251
7.1 Atomic approximation models 252
7.2 Univariate Markov LPP with bounded characteristics 261
7.3 MultivariateMarkov LPP with bounded characteristics 272
7.4 LPP represented by multivariate modulated random walks 285
7.5 MultivariateMarkov LPP with unbounded characteristics 304
8 Convergence of rewards for Markov Gaussian LPP 313
8.1 Univariate Markov Gaussian LPP 313
8.2 Multivariate modulated Markov Gaussian LPP 322
8.3 Markov Gaussian LPP with estimated characteristics 331
8.4 Skeleton reward approximations for Markov Gaussian LPP 345
8.5 LPP represented by Gaussian random walks 357
9 Tree-type approximations for Markov Gaussian LPP 367
9.1 Univariate binomial tree approximations 368
9.2 Multivariate binomial tree approximations 377
9.3 Multivariate trinomial tree approximations 389
9.4 Inhomogeneous in space binomial approximations 404
9.5 Inhomogeneous in time and space trinomial approximations 408
10 Convergence of tree-type reward approximations 423
10.1 Univariate binomial tree approximation models 423
10.2 Multivariate homogeneous in space tree models 434
10.3 Univariate inhomogeneous in space tree models 451
10.4 Multivariate inhomogeneous in space tree models 466
Bibliographical Remarks 475
Bibliography 485
Index 511