Option Pricing in a Regime Switching Jump Diffusion Model

Abstract

There has been extensive literature available in the theory and practice of option valuation following the pioneering work by Black and Scholes (1973). Contrary to subsequent empirical evidence from the dynamics of financial assets, the Black-Scholes model assumed a constant growth rate r and a constant deterministic volatility coeffcient . In subsequent studies, to overcome the demerits of B-S-M model, various option valuation models have been proposed and implemented in tune with realistic price dynamics. These include stochastic volatility models, jump-diffusion models, regime-switching models etc. The market in these models is incomplete where a perfect hedge may not be possible by a self-financing portfolio with a pre-determined initial wealth. In this thesis, we consider a regime-switching jump diffusion model of a financial market, where an observed Euclidean space valued pure jump process drives the values of r and . Further, we assume the pure jump process as an age-dependent semi-Markov process. I this, one has an opportunity to incorporate some memory effect of the market. In particular, the knowledge of past stagnancy period can be fed into the option price formula to obtain the price value. We show using Follmer Schweizer decomposition that the option price at time t, satisfies a Cauchy problem involving a linear, parabolic, degenerate and non-local integro-partial differential equation. We study the well-posedness of the Cauchy problem.