A SIMPLE OPTION PRICING MODEL WITH MARKOVIAN VOLATILITIES

Abstract

This paper develops a stochastic volatility model, which overcomes the Black-Scholes model tendency to overprice near at-the-money options and underprice deep out or in-the-money options. In contrast to the previous literature that assumes diffusion volatilities, this paper assumes that the volatility follows a Markov chain on a discrete state space. This intuitive approach has easier mathematics and, by taking limit, the diffusion results can be obtained. By generalizing the binomial model to the Markovian volatility model, a recursive pricing scheme is first developed, under a particular assumption on preference to the volatility dynamics, and then a continuous time result by taking limit. The both discrete and continuous models give general conditions under which the call value is increasing in the current volatility. Also, based on the local convexity and concavity of the Black-Scholes equation in volatility, we explain why the deficits of the Black-Scholes equation take place. Some numerical experiments are also given to support our results.