Chapter 5. Computational Methods for Derivatives with Early Exercise Features In this paper we consider various computational methods for pricing American style derivatives. We do so under both jump diffusion and stochastic volatility processes. We consider integral transform methods the method of lines operator splitting and the Crank Nicolson scheme the latter being used to generate the benchmark solution. Overall we find that the method of lines approach is quite competitive with other methods for the problems considered in this paper. As one goes to higher dimensions it may be necessary to use methods such as the sparse grid approach.