Numerical Procedure for Calibration of Volatility with American Options

In finance, the price of an American option is obtained from the price of the underlying asset by solving a parabolic variational inequality. The calibration of volatility from the prices of a family of American options yields an inverse problem involving the solution of the previously mentioned parabolic variational inequality. In this paper, the discretization of the variational inequality by finite elements is studied in detail. Then, a calibration procedure, where the volatility belongs to a finite-dimensional space (finite element or bicubic splines) is described. A least square method, with suitable regularization terms is used. Necessary optimality conditions involving adjoint states are given and the differentiability of the cost function is studied. A parallel algorithm is proposed and numerical experiments, on both academic and realistic cases, are presented.