The optimal hedging in a semi-Markov modulated market

TL;DR

This paper proves well-posedness of a non-local PDE in a semi-Markov market model, introduces an integral representation for numerical solutions, and discusses optimal strategies and risk measures in incomplete markets.

Contribution

It provides a new proof of PDE well-posedness, an integral representation for numerical computation, and insights into optimal strategies in semi-Markov market models.

Findings

Established existence and uniqueness of solutions without mild solution techniques.

Derived an integral representation enabling robust numerical schemes.

Presented derivations of external cash flows and risk measures for incomplete markets.

Abstract

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.