Stochastic Volatility and Jumps Driven by Continuous Time Markov Chains

Abstract

This paper considers a model where there is a single state variable that drives the state of the world and therefore the asset price behavior. This variable evolves according to a multi-state continuous time Markov chain, as the continuous time counterpart of (Hamilton 1989) model. It derives the moment generating function of the asset log-price difference under very general assumptions about its stochastic process, incorporating volatility and jumps that can follow virtually any distribution, both of them being driven by the same state variable. For an illustration, the extreme value distribution is used as the jump distribution. The paper shows how GMM and conditional ML estimators can be constructed, generalizing Hamilton's filter for the continuous time case. The risk neutral process is constructed and contingent claim prices under this specification are derived, in the lines of (Bakshi and Madan 2000). Finally, an empirical example is set up, to illustrate the potential benefits of the model.

Keywords: Option pricing, Markov Chain, Moment Generating Function

JEL Classification: C51, G12, G13