Financial markets have known from the studies conducted during the last three decades , a considerable expansion and an emergence of diverse and varied products. Among the most common , there are American options. An American option is by definition an option that has the right to be practiced before the agreed maturity T. The most basic are the American Put or Call ( respectively put option (K - x ) + or purchase (x - K) +). The first part of this thesis is devoted to the study of American options in exponential Lévy models . First, we consider the multidimensional framework and a pay-off function not necessarily bounded and we characterize the price of an American option using a variational inequality in the distribution sense . We study then the properties of the exercise region. These results are further refined by studying in particular the area of exercise of an American Call on a basket of assets as well as the exercise boundary region ( at maturity ). In a second step , we study the behavior of the critical price ( function defining the exercise area) and the behavior studying an American Put near maturity in one-dimensional framework. Specifically, we consider the case where the price does not converge to the strike K in a Jump- diffusion model and a model where the Lévy process behaves to an alpha-stable process (alpha stable like). The second part deals with the pricing of the Credit Valuation Adjustment (CVA ) . It presents a method based on Malliavin calculus inspired by the methods used for American options . A study of the complexity of this method is also presented and compared to purely Monte Carlo methods and to methods based on regression.
