Computation of the Unknown Time-Dependent Volatility of American Options from Integral Observations

In this paper we consider a model for American call price option with unknown time-dependent volatility. In order to determine the volatility, we impose an integral overdetermination for the option price. A front-fixing transformation is applied to the Black-Scholes equation and the overdetermination, which leads to a non-linear problem that involves homogeneous boundary conditions, independent of the free boundary. The derived explicit difference scheme is positive and monotone and its solution is easily decomposed with respect to the unknown volatility. Then, after solving the arisen subproblems, the volatility is calculated from the discretized overdetermination. Computational test examples show good performance of our method.