Calibration of financial models by relative entropy minimization and models with jumpsbPrinted text.

Summary The implied volatility smile observed in the options markets reflects the inadequacy of the Black-Scholes model. With the need to develop a more satisfactory financial asset model, comes the need to calibrate it, which is the subject of this thesis. Calibration by relative entropy minimization has been recently proposed in the framework of the Monte-Carlo method. The convergence and stability of this method have been studied and it has been extended to more general criteria than relative entropy. For the absence of arbitrage opportunity to exist, the discounted underlying must be a martingale. The consideration of this necessity is absolved from the perspective of a moments problem. In the second part, we considered a simple model of the crash phenomenon by introducing in particular jumps in the volatility of the underlying. We computed the quadratic risk and performed an approximate development of the smile which constitutes a tool for the calibration. Finally, in the third part, we use the relative entropy to calibrate the intensity of jumps in a diffusion model with jumps and local volatility. The stability of the method is proved using optimal control techniques and the implicit function theorem.