Exotic options, infinitely divisible laws and Lévy processes: theoretical and practical aspects.

Summary This thesis has three independent parts. The first part deals with closed forms of the Wiener-Hopf factorization for Lévy processes. We survey the half-dozen cases for which the factorization can be written explicitly, and focus on meromorphic functions with poles of order two. The second part focuses on the inversion of the Laplace transform. Its goal is to present a new approximate method, in a probabilistic context. If the Laplace transform has an easily identifiable behavior in zero and if the associated density is bounded, then this method allows to obtain a uniform bound for the error on the distribution function. The efficiency of this method is tested on two non-trivial examples. Finally, the third and last part is dedicated to the pricing of exotic options in the log-stable finite moment model of Carr and Wu. In some cases, it is possible to obtain closed formulas in the form of convergent series for the prices of lookback and barrier options. For all other cases, we study various simulation techniques for the trajectories of the underlying process, with the aim of an evaluation by Monte-Carlo method.