Utility–indifference hedging and valuation via reaction–diffusion systems

Abstract

This article studies the exponential utility–indifference approach to the valuation and hedging problem in incomplete markets. We consider a financial model, which is driven by a system of interacting Itô and point processes. The model allows for a variety of mutual stochastic dependencies between the tradable and non–tradable factors of risk, but still permits a constructive and fairly explicit solution. In analogy to the Black–Scholes model, the utility–based price and the hedging strategy can be described by a partial differential equation (PDE). But the non–tradable factors of risk in our model demand an interacting semi–linear system of parabolic PDEs. To obtain the solution for the underlying utility–maximization problem, we use a verification theorem to identify the optimal martingale measure for the corresponding dual problem.