Markov chains and the pricing of derivatives

A numerical method for pricing financial derivatives based on continuous-time Markov chains is proposed. It approximates the underlying stochastic process by a continuous-time Markov chain. We show how to construct a multi-dimensional continuous-time Markov chain such that it converges in distribution to a multi-dimensional diffusion process. The method is flexible enough to be applied to a model where the underlying process contains local volatility, stochastic volatility and jumps. Furthermore, we introduce a method to approximate the dynamics of the realized variance of a Markov chain and an algorithm to reduce the complexity of computing the joint probability distribution between the realized variance and the underlying