Hedging in incomplete markets with HARA utility

Abstract

In the context of Merton's original problem of optimal consumption and portfolio choice in continuous time, this paper solves an extension in which the investor is endowed with a stochastic income that cannot be replicated by trading the available securities. The problem is treated by demonstrating, using analytic and, in particular, ‘viscosity solutions’ techniques, that the value function of the stochastic control problem is a smooth solution of the associated Hamilton-Jacobi-Bellman (HJB) equation. The optimal policy is shown to exist and given in a feedback form from the optimality conditions in the HJB equation. At zero wealth, a fixed fraction of income is consumed. For ‘large’ wealth, the original Merton policy is approached. We also give a sufficient condition for wealth, under the optimal policy, to remain strictly positive.