Use of stable laws in finance.

Summary The author proposes a general alternative model to the classical Gaussian models. This model is based on alpha-stable processes or Levy processes. These processes have many important characteristics: much larger occurrence of large events (thick-tailed laws), asymmetry of ups and downs, self-similarity in agreement with statistical measures, taking into account the volatility smile. The thesis presents a complete set of analytical and numerical solutions allowing an effective implementation and computation associated with the use of these processes. It also proposes a closed-form pricing formula for foreign exchange options, extensible to equities and interest rates, and studies different strategies of use: arbitrage on the volatility smile, improvement of hedging management costs.