Optimal portfolio and utility-indifference pricing and hedging in a regime-switching model

In this paper, we first derive the solution of the classical Merton problem, i.e. maximising the utility of the terminal wealth, in the case when the risky asset follows a Black-Scholes model with switching coefficients. We find out that the optimal portfolio is a generalisation of the corresponding one in the classical Merton case, with portfolio proportions which depend on the market regime. Then we analyse the utility-indifference pricing problem via the classical approach with the Hamilton-Jacobi-Bellman equation. First we show that the pricing system of reaction-diffusion PDEs for the exponential utility given by Becherer (2004) can be obtained with our approach. Moreover, we use the same technique to extend our approach to more general utility functions. Finally, we show that the marginal price obtained with an exponential utility satisfies a linear system of PDEs, which is obtained by linearizing Becherer’s PDEs.