Markov-Modulated Models for Derivatives Pricing

The aim of this thesis is to investigate the mathematics of Markov-modulated models for derivatives pricing. We consider a model where instantaneous stock volatility and drift are driven by a continuous time finite Markov chain. We present a new derivation of an integral representation for attainable non-path dependent options’ prices in a twostate and three-state Markov chain model, and compute three of the Greeks in a two-state model. We find that occupation time distributions of Markov chains are required to obtain closed-form pricing formulae, and derive the occupation time distribution of a two-state continuous time Markov chain, by first deriving its Laplace transform. We also derive using this Laplace transform an apparently new closed-form representation of the characteristic function for the terminal log stock price in a finite Markov chain model. We then derive a pricing formula for a European call option by employing the Fourier inversion formula. We finish by implementing our two-state Markov-modulated model using these pricing methods, and via Monte Carlo simulation to confirm our results.