An inverse problem for a parabolic variational inequality arising in volatility calibration 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 free boundary associated with this variational inequality can be interpreted as the price for which the option should be exercised. The calibration of volatility from the observations of the prices of an American option yields an inverse problem for the previously mentioned parabolic variational inequality. After studying the variational inequality and the exercise price, we give results concerning the sensitivity of the option price and of the exercise price with respect to the variations of the volatility. The inverse problem is addressed by a least square method, with suitable regularization terms. We give necessary optimality conditions involving an adjoint state for a simplified inverse problem and we study the differentiability of the cost function. Optimality conditions are also given for the genuine calibration problem.