Abstract. The modeling of financial markets by Lévy process has become an active area of research during recent years. This has motivated an equal amount of, if not more, research activity into the necessary numerical methods. Due to the large body of work in this area, we focus our survey on fast numerical methods for Lévy markets. Particular emphasis is placed on grid-based methods. 1. Introduction. Numerical

Numerical methods such as Monte Carlo method (MCM), finite difference method (FDM) and finite element method (FEM) have been successfully implemented to solve financial partial differential equations (PDEs). Sophisticated computational algorithms are strongly desired to further improve accuracy and efficiency. The relatively new spectral element method (SEM) combines the exponential convergence of spectral method and the geometric flexibility of FEM. This disserta-tion carefully investigates SEM on the pricing of European options and their Greeks (Delta, Gamma and Theta). The essential techniques, Gauss quadrature rules, are thoroughly discussed and developed. The spectral element method and its error analysis are briefly introduced first and expanded in details afterwards. Multi-element spectral element method (ME-SEM) for the Black-Scholes PDE is derived on European put options with and without dividend and on a condor option with a more complicated payoff. Under the same Crank-Nicolson approach for the time integration, the SEM shows significant accuracy increase and time cost