Advanced Variational Methods for Option Pricing

This talk will be a review of the possibilities of numerical variational methods such as the Finite Element Method for the partial differential equations of finance. We are quite aware that deterministic methods for option pricing may be hard to impose considering the wide spread usage of Monte-Carlo simulations. Yet deterministic methods offer several advantages for accuracy and speed of computations in many cases.

In finance, the finite element method is better than finite difference and finite volume methods for two reasons: mesh adaptivity and a posteriori error estimates. This is even more so for multidimensional problems. However for dimensions greater than 3 one needs to use sparse grid, a powerful yet hard to implement variational method.

In this talk we will compare the Monte-Carlo simulations and the Finite Element Simulations for European and American options, for greeks, for stochastic volatility models with and without jump processes. Large drift terms, as in the case of Asian options can be handled by the Galerkin-Characteristic method.

Finally Calibration is made possible by Dupire's identity which we will show to hold at the discrete level whenever the underlying model is linear. Examples of calibration will be given as well.

Olivier Pironneau, University Paul and Mary Curie, Paris VI, France

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