  This paper presents the solution to a European option pricing problem by
considering a regime-switching jump diffusion model of the underlying financial
asset price dynamics. The regimes are assumed to be the results of an observed
pure jump process, driving the values of interest rate and volatility
coefficient. The pure jump process is assumed to be a semi-Markov process on
finite state space. This consideration helps to incorporate a specific type of
memory influence in the asset price. Under this model assumption, the locally
risk minimizing price of the European type path-independent options is found.
The F\"{o}llmer-Schweizer decomposition is adopted to show that the option
price satisfies an evolution problem, as a function of time, stock price,
market regime, and the stagnancy period. To be more precise, the evolution
problem involves a linear, parabolic, degenerate and non-local system of
integro-partial differential equations. We have established existence and
uniqueness of classical solution to the evolution problem in an appropriate
class.
