An Inverse Problem for a Parabolic Variational Inequality with an Integro-Differential Operator

We consider the calibration of a Lévy process with American vanilla options. The price of an American vanilla option as a function of the maturity and the strike satisfies a forward in time linear complementarity problem involving a partial integro-differential operator. It leads to a variational inequality in a suitable weighted Sobolev space. Calibrating the Lévy process amounts to solving an inverse problem where the state variable satisfies the previously mentioned variational inequality. We propose a regularized least square method. After studying the variational inequality carefully, we find necessary optimality conditions for the least square problem. In this work, we focus on the case when the volatility is bounded away from zero.