Lamberton D., Lapeyre P. Introduction to stochastic calculus applied to finance (2000)

Boca Raton, London, New York, Washington DC: Chapman & Hall/CRC. - 2000. - 195 p.

Contents:
Introduction.
Options.
Arbitrage and put/call parity.
Black-Scholes model and its extensions.
Contents of the book.
Acknowledgements.
Discrete-time models.
Discrete-time formalism.
Martingales and arbitrage opportunities.
Complete markets and option pricing.
Problem: Cox, Ross and Rubinstein model.
Option stopping problem and American options.
Stopping time.
The Snell envelope.
Decomposition of supermartingales.
Snell envelope and Markov chains.
Application to American options.
Exercises.
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.
Exercises.
The Black-Scholes model.
Description of the model.
Change of probability. Representation of martingales.
Pricing and hedging options in the Black-Scholes model.
American optionsi n the Black-Scholes model.
Exercises.
Option pricing and partial differential equations.
European option pricing and diffusions.
Solving parabolic equations numerically.
American options.
Exercises.
nterest rate models.
Modelling principles.
Some classical models.
Exercises.
Asset models with jumps.
Poisson process.
Dynamics of the risky esset.
Pricing and hedging options.
Exercises.
Simulation and algorithms for financial models.
Simulation and financial models.
Some useful algorithms.
Exercises.
Appendix.
A.1 Normal random variables.
A.2 Conditional expectation.
A.3 Separation of convex sets.