This dissertation is devoted to high performance computational methods for options valuation in Levy process and general Markov jump-diffusion process models. In the first part of the dissertation, a highly efficient method based on Hilbert transform is developed for pricing European and discrete barrier options in Levy process models. This method involves evaluation of Hilbert transforms which can be discretized with exponentially decaying errors. In addition to options valuation in Levy process models, this method provides a powerful computational framework for applications in applied probability and credit risk modeling.; For non-Levy Markov jump-diffusion models and for barrier options with continuous monitoring, numerical solutions of partial integro-differential equations (PIDEs) are typically needed. In the second part of the dissertation, a remarkably efficient method based on extrapolation for time discretization of these PIDEs is developed. Applications to options pricing in one-dimensional jump-diffusion models as well as in a two-dimensional stochastic volatility jump diffusion model are studied.
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