Adaptive Finite Element Methods for Local Volatility European Option Pricing

We investigate finite element discretizations using functions that are discontinuous in time and continuous in space for European options with local volatility Black-Scholes models. We present an a posteriori error estimate where a user-specified functional of the error is controlled by the inner product of the finite element residual with the solution of a dual problem that involves the density of the target functional as prescribed data. Examples of error functionals are discussed in the context of either option pricing or volatility calibration from market data. The a posteriori error estimator is then localized onto the space-time cells of the computational mesh and implemented in the framework of an adaptive mesh refinement/derefinement algorithm which provides some form of optimal compromise between accuracy requirements and computational costs. Numerical examples illustrate the efficiency of the proposed methodology.