Introduction to stochastic calculus applied to finance

Since the publication of the first edition of this book, the area of mathematical finance has grown rapidly, with financial analysts using more sophisticated mathematical concepts, such as stochastic integration, to describe the behavior of markets and to derive computing methods. Maintaining the lucid style of its popular predecessor, Introduction to Stochastic Calculus Applied to Finance, Second Edition incorporates some of these new techniques and concepts to provide an accessible, up-to-date initiation to the field.

New to the Second Edition
Complements on discrete models, including Rogers' approach to the fundamental theorem of asset pricing and super-replication in incomplete markets
Discussions on local volatility, Dupire's formula, the change of numéraire techniques, forward measures, and the forward Libor model
A new chapter on credit risk modeling
An extension of the chapter on simulation with numerical experiments that illustrate variance reduction techniques and hedging strategies
Additional exercises and problems

Providing all of the necessary stochastic calculus theory, the authors cover many key finance topics, including martingales, arbitrage, option pricing, American and European options, the Black-Scholes model, optimal hedging, and the computer simulation of financial models. They succeed in producing a solid introduction to stochastic approaches used in the financial world.

Introduction	p. 9
1 Discrete-time models	p. 15
1.1 Discrete-time formalism	p. 15
1.2 Martingales and arbitrage opportunities	p. 18
1.3 Complete markets and option pricing	p. 22
1.4 Problem: Cox, Ross and Rubinstein model	p. 26
1.5 Exercises	p. 31
2 Optimal stopping problem and American options	p. 37
2.1 Stopping time	p. 37
2.2 The Snell envelope	p. 38
2.3 Decomposition of supermartingales	p. 41
2.4 Snell envelope and Markov chains	p. 42
2.5 Application to American options	p. 43
2.6 Exercises	p. 46
3 Brownian motion and stochastic differential equations	p. 51
3.1 General comments on continuous-time processes	p. 52
3.2 Brownian motion	p. 53
3.3 Continuous-time martingales	p. 55
3.4 Stochastic integral and Ito calculus	p. 58
3.5 Stochastic differential equations	p. 72
3.6 Exercises	p. 80
4 The Black-Scholes model	p. 87
4.1 Description of the model	p. 87
4.2 Change of probability. Representation of martingales	p. 90
4.3 Pricing and hedging options in the Black-Scholes model	p. 91
4.4 American options	p. 96
4.5 Implied volatility and local volatility models	p. 101
4.6 The Black-Scholes model with dividends and call/put symmetry	p. 103
4.7 Exercises	p. 104
4.8 Problems	p. 108
5 Option pricing and partial differential equations	p. 123
5.1 European option pricing and diffusions	p. 123
5.2 Solving parabolic equations numerically	p. 132
5.3 American options	p. 138
5.4 Exercises	p. 146
6 Interest rate models	p. 149
6.1 Modelling principles	p. 149
6.2 Some classical models	p. 158
6.3 Exercises	p. 169
7 Asset models with jumps	p. 173
7.1 Poisson process	p. 173
7.2 Dynamics of the risky asset	p. 175
7.3 Martingales in a jump-diffusion model	p. 177
7.4 Pricing options in a jump-diffusion model	p. 182
7.5 Exercises	p. 191
8 Credit risk models	p. 195
8.1 Structural models	p. 195
8.2 Intensity-based models	p. 196
8.3 Copulas	p. 202
8.4 Exercises	p. 205
9 Simulation and algorithms for financial models	p. 207
9.1 Simulation and financial models	p. 207
9.2 Introduction to variance reduction methods	p. 215
9.3 Exercises	p. 224
9.4 Computer experiments	p. 225
Appendix	p. 235
A.1 Normal random variables	p. 235
A.2 Conditional expectation	p. 237
A.3 Separation of convex sets	p. 241
Bibliography	p. 243
Index	p. 251

Available:*

On Order

Summary

Summary

Table of Contents