  This paper includes an original self contained proof of well-posedness of an
initial-boundary value problem involving a non-local parabolic PDE which
naturally arises in the study of derivative pricing in a generalized market
model. We call this market model a semi-Markov modulated market. Although a
wellposedness result of that problem is available in the literature, but this
recent paper has a different proof. Here the existence of solution is
established without invoking mild solution technique. We study the
well-posedness of the initial-boundary value problem via a Volterra integral
equation of second kind. The method of conditioning on stopping times was used
only for showing uniqueness. Furthermore, in the present study we find an
integral representation of the PDE problem which enables us to find a robust
numerical scheme to compute derivative of the solution. This study paves for
addressing many other interesting problems involving this new set of PDEs. Some
derivations of external cash flow corresponding to an optimal strategy are
presented. These quantities are extremely important when dealing with an
incomplete market. Apart from these, the risk measures for discrete trading are
formulated which may be of interest to the practitioners.
