Non-parametric model calibration in finance.

Summary Vanilla calibration is a major problem in finance. We try to solve it for three classes of models: local and stochastic volatility models, the so-called "local correlation" model and a hybrid model of local volatility with stochastic rates. From a mathematical point of view, the calibration equation is a particularly complex nonlinear and integro-differential equation. In a first part, we prove the existence of solutions for this equation, as well as for its adjoint (simpler to solve). These results are based on fixed point methods in Hölder spaces and require classical theorems related to parabolic partial differential equations, as well as some a priori estimates in short time. The second part deals with the application of these existence results to the three financial models mentioned above. We also present the numerical results obtained by solving the edp. The calibration by this method is quite satisfactory. Finally, we focus on the algorithm used for the numerical solution: a predictor-corrector ADI scheme, which is modified to take into account the nonlinear character of the equation. We also describe an instability phenomenon of the edp solution that we try to explain from a theoretical point of view thanks to the so-called "Hadamard instability".