Numerical methods on sparse grids applied to option pricing in finance.

Summary This thesis gathers several works related to the numerical solution of partial differential equations and integro-differential equations resulting from the stochastic modeling of financial products. The first part of the work is devoted to Sparse Grid methods applied to the numerical solution of equations in dimension greater than three. Two types of problems are addressed. The first one concerns the valuation of vanilla options in a jump model with multi-factor stochastic volatility. The numerical solution of the valuation equation, posed in dimension, is obtained using a sparse finite difference method and a collocation method for the discretization of the integral operator. The second problem deals with the valuation of products on a basket of several underlyings. It requires the use of a Galerkin method on a wavelet basis obtained with a sparse tensor product. The second part of the work concerns a posteriori error estimates for American options on a basket of several assets.