We address the analysis and implementation of fast, deterministic pricing schemes for general contracts on assets driven by general Levy processes. Our approach is based on finite element solution of the associated deterministic differential equation. This equation involves the Dynkin operator of the semigroup generated by the price process which for Levy processes is an integro-differential operator. Upon discretization, the linear systems to be solved in each implicit time step have dense and ill-conditioned stiffness matrices. We propose here a spline wavelet basis for discretization in the (logarithmic) price variable which allows to compress these matrices to sparse, well-conditioned ones that can be inverted by iterative schemes in almost linear complexity while not affecting the accuracy of the computed prices. For American style contracts, the wavelet basis allows to precondition the iterative solver for the associated Linear Complementarity Problems (LCPs) in each time step. The algorithm allows to treat any Levy price process, even pure jump processes with infinite jump intensity. Moreover, general pay-off functions are admissible allowing in particular to handle compound options, with contracts of European or American style. This work was done together with Christoph Schwab (ETH Zurich), Tobias von Petersdorff (University of Maryland) and Andrej Nitsche (ETH Zurich).