A Simple Approach for Pricing Equity Options With Markov Switching State Variables

We provide a polynomial time lattice algorithm for pricing options with Markov-switching volatility. This parsimonious model bridges the computational gap between constant volatility models of Cox et al. (1979), which are easy to compute, and fully stochastic volatility models (e.g. Heston 1993), which are hard to compute and do not handle American options. By increasing the number of states in the Markov chain, we can move from the former class of models to the latter. It complements the literature on trees with stochastic volatility (see e.g. Bollen 1998) as well as the normal mixture diffusion class of models (see e.g. Alexander (2004), Mercurio (2002), Brigo and Mercurio (2002) and Brigo et al.(2004)). By making an adjustment to transition probabilities, we are able to implement skew models whereas the standard class of mixture models tends to have smiles with lowest implied volatilities at the money. The algorithm is Oðkn2kÞ, where k is the number of states volatility can take and n is the number of periods on the lattice. We show that the model runs very fast and also converges to the exact price when n becomes large, without a substantial increase in run time. The model has been implemented differently in subsequent work by Balsara (2004) using a finite-difference approach, and binomial trees by Florescu and Viens (2004)