Numerical option pricing beyond Lévy

This work considers the numerical approximation of option prices in different market models beyond Lévy processes. The Lévy setup is extended in several directions. The arising partial integrodifferential equations and inequalities are solved with the finite element method. European as well as American type contracts are considered. Spatially inhomogeneous market models are analyzed, specifically certain Feller processes are considered. The well-posedness of the arising pricing equations is proved using pseudodifferential operator theory. The resulting pricing equations need no longer be parabolic and can exhibit degeneracies under certain conditions. Classical continuous Galerkin methods are therefore inapplicable for the numerical solution of the corresponding pricing equations. Thus we employ a discontinuous Galerkin discretization or alternatively a streamline diffusion approach. Convergence results are shown in both cases. Besides the spatial inhomogeneity, also the assumption of temporal homogeneity of the coefficients of the partial integrodifferential equations is weakened. The well-posedness for pricing equations with degenerate coefficients in time is shown via a weak space-time formulation. The main problem arising in the discretization of such equations is the non-applicability of classical time-marching schemes due to the possible degeneracy of the coefficients. Therefore two alternative approaches are considered. First, a continuous Galerkin method for the space-time discretization is used, in this case optimality of the solution algorithm can be shown. Second, a discontinuous Galerkin discretization for the temporal domain is studied, in which case exponential convergence of the algorithm can be shown. Numerical examples are given to confirm the theoretical results. Partial integrodifferential equations with spatially as well as temporally inhomogeneous coefficients are solved numerically. European and American options are priced.