Numerical methods for Volatility Estimation and Option Pricing

This manuscript presents a synthesis of my contributions during the years of my thesis that I defended in 2015. We have been interested in the applications of mathematics in finance such as option pricing and volatility estimation. First, we dealt with calibrating local volatility problem from market option prices, which is an inverse problem. We proposed an alternative approach based on the regularization method of Tikhonov, using the Dupire partial differential equation modeling the option price, and we tested numerically the proposed algorithms. On the other hand, we developed an adequate method of "DDGRK" Direct Discontinuous Galerkin, and Runge-Kutta of order two and three for the time discretization for pricing Option. Finally, we proposed the Runge-Kutta stochastic method in time for the evaluation of an European option with stochastic volatility when the volatility dynamics follows the CEV model.