Abstract:
In this talk we consider 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.
Spatially inhomogeneous market models often arise in the context of commodity modeling due to mean-reversion. 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 combined with a small jump approximation.
Besides the spatial inhomogeneity, also the assumption of temporal homogeneity of the coefficients of the partial integrodifferential equations is weakened. Temporally inhomogeneous models often need to be considered for the calibration over various maturities. The well-posedness for pricing equations with degenerate coefficients in time is shown via a weak space-time formulation. We present a class of numerical methods for PIDEs arising in such models. The use of appropriate wavelet bases for the discretization in the space-time domain leads to Riesz bases for the ansatz and test spaces. Besides, a class of variational timestepping schemes is discussed and their exponential convergence is proved. For the spatial domain a novel low rank approximation using the tensor-train format is employed.
Numerical experiments in multiple space dimensions confirm the theoretical results.