  This paper studies pricing derivatives in an age-dependent semi-Markov
modulated market. We consider a financial market where the asset price dynamics
follow a regime switching geometric Brownian motion model in which the
coefficients depend on finitely many age-dependent semi-Markov processes. We
further allow the volatility coefficient to depend on time explicitly. Under
these market assumptions, we study locally risk minimizing pricing of a class
of European options. It is shown that the price function can be obtained by
solving a non-local B-S-M type PDE. We establish existence and uniqueness of a
classical solution of the Cauchy problem. We also find another characterization
of price function via a system of Volterra integral equation of second kind.
This alternative representation leads to computationally efficient methods for
finding price and hedging. Finally, we analyze the PDE to establish continuous
dependence of the solution on the instantaneous transition rates of semi-Markov
processes. An explicit expression of quadratic residual risk is also obtained.
