Numerical methods and probabilistic algorithms for the evaluation of exotic interest rate derivatives in the framework of Libor and Swap rate market models.

Summary This thesis addresses the problem of valuing exotic interest rate options in the context of Libor and swap rate market models. It consists of four chapters. The first chapter is devoted to the theoretical construction and calibration of the Libor and swap rate market models. The second chapter presents the valuation of exotic interest rate options using closed-form approximations. We also present a theoretical and numerical study of the error of these approximations. The third chapter presents the valuation of exotic interest rate products using Monte Carlo and Quasi-Monte Carlo methods. We numerically compare these methods using several sequences with low discrepancies. We also study numerically the speed of convergence of the Euler and Milstein schemes and the Richardson extrapolation method. The fourth chapter deals with the evaluation of American and Bermuda interest rate options. We introduce approximations to the stochastic differential equations that govern the Libor and swap rate market models. These approximations reduce the size of the market models and allow the valuation of American and Bermuda options by classical valuation techniques (partial differential equations, variational inequalities, the tree method, etc.). ). We evaluate, numerically and theoretically, and compare this approach to other methods, based on the Monte Carlo method, recently proposed for the valuation of American and Bermuda options in high dimension.