Locally risk minimizing pricing of Asian option in a semi-Markov modulated market

We consider a regime-switching model where the stock volatility dynamics is a semi-Markov process. Under this model assumption, we find the locally risk-minimizing price of some Asian options with European-style exercise. The price function is shown to satisfy a non-local degenerate system of parabolic PDEs in dimension two with a terminal condition. We show this by deriving the F-S decomposition of the discounted contingent claim. The related Cauchy problem involving the PDE is shown to be equivalent to an integral equation (IE). The existence and uniqueness of the classical solution to the PDE are determined by studying the IE and using semigroup theory. To be more precise, we first obtain the mild solution of the PDE and then we show that the mild solution has sufficient regularity. The locally risk-minimizing hedging for the option has also been identified in this work. Finally, the computational aspects of Asian option prices have been discussed by solving the equation numerically.