Introduction to stochastic calculus applied to finance

Damien Lamberton, Bernard Lapeyre, Nicolas Rabeau, Francois Mantion0412718006, 9780412718007

In recent years the growing importance of derivative products financial markets has increased financial institutions’ demands for mathematical skills. This book introduces the mathematical methods of financial modelling with clear explanations of the most useful models. Introduction to Stochastic Calculus begins with an elementary presentation of discrete models, including the Cox-Ross-Rubenstein model. This book will be valued by derivatives trading, marketing, and research divisions of investment banks and other institutions, and also by graduate students and research academics in applied probability and finance theory.

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Table of contents :
Обложка ……Page 1
Титульный лист ……Page 3
Аннотация и сведения об издательстве ……Page 4
Contents ……Page 5
Options ……Page 7
Arbitrage and put/call parity ……Page 8
Black-Scholes model and its extensions ……Page 9
Acknowledgements ……Page 10
1.1 Discrete-time formalism ……Page 13
1.2 Martingales and arbitrage opportunities ……Page 16
1.3 Complete markets and option pricing ……Page 20
1.4 Problem: Cox, Ross and Rubinstein model ……Page 24
2.1 Stopping time ……Page 29
2.2 The S nell envelope ……Page 30
2.3 Decomposition of supermartingales ……Page 33
2.4 Snell envelope and Markov chains ……Page 34
2.5 Application to American options ……Page 35
2.6 Exercises ……Page 37
3.1 General comments on continuous-time processes ……Page 41
3.2 Brownian motion ……Page 43
3.3 Continuous-time martingales ……Page 44
3.4 Stochastic integral and Ito calculus ……Page 47
3.5 Stochastic differential equations ……Page 61
3.6 Exercises ……Page 68
4.1 Description of the model ……Page 75
4.2 Change of probability. Representation of martingales ……Page 77
4.3 Pricing and hedging options in the Black-Scholes model ……Page 79
4.4 American options in the Black-Scholes model ……Page 84
4.5 Exercises ……Page 89
5.1 European option pricing and diffusions ……Page 107
5.2 Solving parabolic equations numerically ……Page 115
5.3 American options ……Page 122
5.4 Exercises ……Page 130
6.1 Modelling principles ……Page 133
6.2 Some classical models ……Page 139
6.3 Exercises ……Page 148
7.1 Poisson process ……Page 153
7.2 Dynamics of the risky asset ……Page 155
7.3 Pricing and hedging options ……Page 162
7.4 Exercises ……Page 171
8.1 Simulation and financial models ……Page 173
8.2 Some useful algorithms ……Page 180
8.3 Exercises ……Page 182
A.1 Normal random variables ……Page 185
A.2 Conditional expectation ……Page 186
A.3 Separation of convex sets ……Page 190
References ……Page 191
Index ……Page 195