A trinomial tree method for pricing path-dependent options in a eneralized jump-diffusion model

Abstract

Option contracts have become increasingly important in the field of financesince they possess characteristics that are attractive to speculators and hedgers.One important problem is determining the ”fair value” of an option efficientlyand accurately.In this thesis, we review some basic option pricing theories. After discussingsome popular numerical methods for option pricing, we focus on dealing withpath dependent options. We first propose a generalized parabolic integro differentialequation (PIDE) model for pricing path-dependent options with jumps.Since the PIDE model does not have a closed-form solution, in order to knowthe approximate solution, we present a trinomial tree method instead of thetraditional binomial tree method and show its consistence with our proposedPIDE model. We also give an explicit finite difference scheme and show itsequivalence to the trinomial tree scheme. Therefore we prove the uniformconvergence of the trinomial tree method for European-style path dependentoptions with jumps. Further, comparison studies are performed to demonstratethe advantages of the trinomial tree method over the binomial tree method forpricing European put options computationally.