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.
