Stéphane Crépey

Professor in Mathematics, University of Evry

Finance numérique

Les objectifs du cours

The course bears on the modeling and numerical analysis of financial derivatives. The objectives are:
1) Understanding the financial meaning of the related mathematics: model parameters, implied volatility, Greeks.
2) Learning how to derive a pricing equation based on the probabilistic formulation of a model, possibly with stochastic volatility and/or jumps,
3) Learning how to implement a theta-scheme of finite differences or a tree pricing method,
4) Learning simulation Monte Carlo pricing and Greeking methods: basic principles and variance reduction techniques, first in a set-up of random variables or vectors, then in a dynamic set-up of stochastic processes.

Plan du cours

1) Motivating examples: Black-Scholes and Dupire model, Realized volatility vs Implied volatility vs Local volatility,
2) Derivation of the Pricing Equations in various models,
3) Deterministic Pricing Schemes: Finite Differences methods and Tree Methods
4) Simulation Pricing Schemes: simulation of random variables and stochastic processes, Pseudo Monte Carlo versus Quasi Monte Carlo, variance reduction techniques.

Bibliographie

Crépey S., Computational Finance Lecture Notes, 2009 edition, 188 pages, available on http://www.maths.univ-evry.fr/crepey
Lamberton D. and Lapeyre P., Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall, 2nd revised edition, 2007.
Shreve S., Stochastic Calculus for Finance II, Springer Finance, 2008.
Hull J., Options, Futures, and Other Derivative Securities, Prentice-Hall, 7th edition, 2009.

Examen

Project (in teams of two to three students)