Fast deterministic pricing of options on Levy driven assets

Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Levy processes satisfy a parabolic partial integro-differential equation (PIDE) partial derivative(t)u+A[u]=0. This FIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the theta-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for A can be replaced by a sparse matrix in the wavelet basis, and the linear systems in each implicit time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by O(MN(log(N))(2)) operations and O(N log(N)) memory. The deterministic algorithm gives optimal convergence rates (tip to logarithinic terms) for the computed solution in the same complexity as finite difference approximations of the standard Black-Scholes equation. Computational examples for various Levy price processes are presented.