Optimal Stopping For Markov Modulated Ito-diffusion With Applications To Finance

Abstract

Despite the outstanding success of the Black-Scholes model, it relies on the assumption that drift and volatility of the underlying equity remain constant throughout time. This inaccuracy has motivated a number of interesting and innovative refinements, one of the most natural being Markov modulation. In this dissertation we analyze a variety of financially motivated optimal stopping problems under Markov modulated Ito-Diffusions. In Chapter 3, we generalize and refine a technique developed in [13] pricing an infinite time horizon American put option and we present a rigorous proof of optimality. In Chapter 3 we use this generalized technique to discover an optimal selling strategy for an infinite horizon American style forward contract. In so doing, we extend the work done in [12]. Finally in Chapter 5 we price the infinite horizon American put using a non-traditional model of a mean reverting Ornstein-Uhlenbeck process, further illustrating the broad scope of applicability of the technique developed herein.