DFG-Research Centre "Mathematics for key technologies - MATHEON": Adaptive FE Algorithm for Option Evaluation (TP E 6)

The aim of the research project is the development of an a posteriori error analysis for American options in the Black-Scholes setting. We will first consider the method of lines to obtain a system of ODEs which are solved efficiently with a proper ODE solver. If it is assumed that no or only small errors occur by solving the ODE only space discretisation errors have to be considered. Using finite elements (FE) for space discretisation, we will develop a posteriori error estimates for the space discretisation in order to obtain an adaptive algorithm. Elementary time discretisation such as the Euler method or discontinuous Galerkin schemes of order 0 (dG(0)) and FE discretisation in space for the corresponding obstacle problem will be considered and a posteriori error estimates and implementation of an adaptive algorithm will be applied. Time discretisation with the Crank-Nicolson method or discontinuous Galerkin schemes of order 1 (dG(1)) schemes will be used where the space discretisation is performed with FE. Appropriate error estimates will be used for obtaining an adaptive mesh-refinement algorithm.