The Adjoint Method Formulation for an Inverse Problem in the Generalized Black-Scholes Model

A general framework is developed to treat optimal control problems for a generalized Black-Scholes model, which is used for option pricing. The volatility function is retrieved from a set of market observations. The optimal volatility function is found by minimizing the cost functional measuring the discrepancy between the model solution (pricing) and the observed market price, via the unconstrained minimization algorithm of the quasi-Newton limited memory type. The gradient is computed via the adjoint method. The effectiveness of the method is demonstrated on an European call option.