Silvestrov American-Type Options

Stochastic Approximation Methods, Volume 1
1. Auflage 2013
ISBN: 978-3-11-032982-7
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, Band 56, 519 Seiten

Reihe: De Gruyter Studies in Mathematics

The book gives a systematical presentation of stochastic approximation methods for models of American-type options with general pay-off functions for discrete time Markov price processes. Advanced methods combining backward recurrence algorithms for computing of option rewards and general results on convergence of stochastic space skeleton and tree approximations for option rewards are applied to a variety of models of multivariate modulated Markov price processes. The principal novelty of presented results is based on consideration of multivariate modulated Markov price processes and general pay-off functions, which can depend not only on price but also an additional stochastic modulating index component, and use of minimal conditions of smoothness for transition probabilities and pay-off functions, compactness conditions for log-price processes and rate of growth conditions for pay-off functions. The book also contains an extended bibliography of works in the area.This book is the first volume of the comprehensive two volumes monograph. The second volume will present results on structural studies of optimal stopping domains, Monte Carlo based approximation reward algorithms, and convergence of American-type options for autoregressive and continuous time models, as well as results of the corresponding experimental studies.

Zielgruppe

Researchers, Graduate Students, and Aractioneers in Financial Mathematics; Academic Libraries

Autoren/Hrsg.

Silvestrov, Dmitrii S.

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Stochastik
• Mathematik | Informatik Mathematik Mathematik Interdisziplinär Finanz- und Versicherungsmathematik

Inhaltsverzeichnis

1;Preface;5
2;1 Multivariate modulated Markov log-price processes (LPP);11
2.1;1.1 Markov LPP;11
2.2;1.2 LPP represented by random walks;18
2.3;1.3 Autoregressive LPP;28
2.4;1.4 Autoregressive stochastic volatility LPP;38
3;2 American-type options;54
3.1;2.1 American-type options;54
3.2;2.2 Pay-off functions;57
3.3;2.3 Reward and log-reward functions;63
3.4;2.4 Optimal stopping times;73
3.5;2.5 American-type knockout options;82
4;3 Backward recurrence reward algorithms;86
4.1;3.1 Binomial tree reward algorithms;86
4.2;3.2 Trinomial tree reward algorithms;98
4.3;3.3 Random walk reward algorithms;110
4.4;3.4 Markov chain reward algorithms;116
5;4 Upper bounds for option rewards;125
5.1;4.1 Markov LPP with bounded characteristics;125
5.2;4.2 LPP represented by random walks;137
5.3;4.3 Markov LPP with unbounded characteristics;143
5.4;4.4 Univariate Markov Gaussian LPP;164
5.5;4.5 Multivariate modulated Markov Gaussian LPP;169
6;5 Convergence of option rewards – I;177
6.1;5.1 Asymptotically uniform upper bounds for rewards – I;178
6.2;5.2 Modulated Markov LPP with bounded characteristics;190
6.3;5.3 LPP represented by modulated random walks;204
7;6 Convergence of option rewards – II;213
7.1;6.1 Asymptotically uniform upper bounds for rewards – II;214
7.2;6.2 Univariate modulated LPP with unbounded characteristics;224
7.3;6.3 Asymptotically uniform upper bounds for rewards – III;230
7.4;6.4 Multivariate modulated LPP with unbounded characteristics;241
7.5;6.5 Conditions of convergence for Markov price processes;248
8;7 Space-skeleton reward approximations;251
8.1;7.1 Atomic approximation models;252
8.2;7.2 Univariate Markov LPP with bounded characteristics;261
8.3;7.3 MultivariateMarkov LPP with bounded characteristics;272
8.4;7.4 LPP represented by multivariate modulated random walks;285
8.5;7.5 MultivariateMarkov LPP with unbounded characteristics;304
9;8 Convergence of rewards for Markov Gaussian LPP;313
9.1;8.1 Univariate Markov Gaussian LPP;313
9.2;8.2 Multivariate modulated Markov Gaussian LPP;322
9.3;8.3 Markov Gaussian LPP with estimated characteristics;331
9.4;8.4 Skeleton reward approximations for Markov Gaussian LPP;345
9.5;8.5 LPP represented by Gaussian random walks;357
10;9 Tree-type approximations for Markov Gaussian LPP;367
10.1;9.1 Univariate binomial tree approximations;368
10.2;9.2 Multivariate binomial tree approximations;377
10.3;9.3 Multivariate trinomial tree approximations;389
10.4;9.4 Inhomogeneous in space binomial approximations;404
10.5;9.5 Inhomogeneous in time and space trinomial approximations;408
11;10 Convergence of tree-type reward approximations;423
11.1;10.1 Univariate binomial tree approximation models;423
11.2;10.2 Multivariate homogeneous in space tree models;434
11.3;10.3 Univariate inhomogeneous in space tree models;451
11.4;10.4 Multivariate inhomogeneous in space tree models;466
12;Bibliographical Remarks;475
13;Bibliography;485
14;Index;511

Über Autor(innen)

Dmitrii S. Silvestrov, Stockholm University, Sweden.