Introduction to Stochastic Calculus Applied to Finance (Hardcover, 2)

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 Ito 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.