Introduction to Stochastic Calculus Applied to Finance

Summary

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.

Table of Contents

INTRODUCTION

DISCRETE-TIME MODELS
Discrete-time formalism
Martingales and arbitrage opportunities
Complete markets and option pricing
Problem: Cox, Ross and Rubinstein model

OPTIMAL STOPPING PROBLEM AND AMERICAN OPTIONS
Stopping time
The Snell envelope
Decomposition of supermartingales
Snell envelope and Markov chains
Application to American options

BROWNIAN MOTION AND STOCHASTIC DIFFERENTIAL EQUATIONS
General comments on continuous-time processes
Brownian motion
Continuous-time martingales
Stochastic integral and Itô calculus
Stochastic differential equations

THE BLACK-SCHOLES MODEL
Description of the model
Change of probability: Representation of martingales
Pricing and hedging options in the Black-Scholes model
American options
Implied volatility and local volatility models
The Black-Scholes model with dividends and call/put symmetry
Problems

OPTION PRICING AND PARTIAL DIFFERENTIAL EQUATIONS
European option pricing and diffusions
Solving parabolic equations numerically
American options

INTEREST RATE MODELS
Modeling principles
Some classical models

ASSET MODELS WITH JUMPS
Poisson process
Dynamics of the risky asset
Martingales in a jump-diffusion model
Pricing options in a jump-diffusion model

CREDIT RISK MODELS
Structural models
Intensity-based models
Copulas

SIMULATION AND ALGORITHMS FOR FINANCIAL MODELS
Simulation and financial models
Introduction to variance reduction methods
Computer experiments

APPENDIX
Normal random variables
Conditional expectation
Separation of convex sets

BIBLIOGRAPHY

INDEX

Exercises appear at the end of each chapter.

Reviews

The second edition of this book provides a concise and accessible introduction to the probabilistic techniques needed to understand the most widely used financial models. This edition incorporates many new techniques and concepts to be used to describe the behavior of financial markets. … the solutions obtained using SciLab for computer experiments are available at http://cermics.enpc.fr/~bl/scilab/ These experiments were well designed by the authors based on their teaching and research experience and were found to be effective in communicating these concepts and ideas and enhancing the understanding of readers. … a solid introduction to stochastic approaches used in the financial world. The authors cover many key finance topics … . The book can be used as a reference text by researchers and graduate students in financial mathematics. It also is ideal reading material for practicing financial analysts and consultants using mathematical models for finance.
—Technometrics, May 2009, Vol. 51, No. 2