Efficient European and American Option Pricing Under a Jump-diffusion Process

When the underlying asset of an option displays oscillations, spikes or heavy-tailed distributions, modeling it with the lognormal diffusion process is inadequate. In order to overcome these real world difficulties, Merton proposed a jump-diffusion model, where the dynamics of the price of the underlying are subject to variations due to a Brownian process and also to possible jumps, driven by a compound Poisson process. There have been a lot of attempts to obtain a discretization of the Merton model with tree methods in order to price American or more complex options, e. g. Amin, the O(n3) procedure by Hilliard and Schwartz and the O(n2:5) procedure by Dai et al. Here, starting from the implementation of the seven-nodes procedure by Hilliard and Schwartz, we prove theoretically that it is possible to reduce the complexity of this method to O(n2 ln n) in the American put case. Our method is based on a suitable truncation of the lattice structure; the proofs provide closed formulas for the truncation limitations