American options in a multidimensional Black-Scholes model.

Summary The aim of this thesis is to study American options in a multi-dimensional diffusion model. Mathematically, this study is related to an optimal stopping problem with a finite or non finite horizon. The first part of the paper is concerned with the description of the valuation model of American options as a solution of a parabolic variational inequation and with the existence or not of a stopping region more commonly called the exercise region in finance. The first chapter provides a necessary and sufficient condition on the infinitesimal generator of the diffusion for the stopping region to be non empty. The following chapters study the properties of exercise regions associated with some types of options commonly traded on the markets: convexity, regularity and asymptotic behavior for infinite or near zero horizons. The second part concerns the numerical analysis of American options in dimension two. After recalling the different formulations using partial differential equations (solution in sobolev spaces or viscosity solution), two approximation methods of the alternate directions type are proposed and two convergence theorems are established. A comparison result between these methods ends this part. The last part studies the critical price of the American put in the vicinity of the maturity when the stock pays dividends. A result concerning the strict monotonicity of the critical price is proved as well as a framework of this price in the vicinity of the maturity.