Introduction to Stochastic Calculus Applied to Finance book cover
2nd Edition

Introduction to Stochastic Calculus Applied to Finance

By

Damien Lamberton

,

Bernard Lapeyre

Copyright Year 2007


ISBN 9781584886266
Published November 30, 2007 by Chapman and Hall/CRC
254 Pages
USD $99.95

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Book Description

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.

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    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