Optimal investment policy for hyperbolic absolute risk averse utility function under the CEV model

This paper investigates the optimal portfolio choice problem for an investor with hyperbolic absolute risk averse utility over terminal wealth facing perfectly hedgeable stochastic cash flow under the constant elasticity of variance (CEV) model. The cash flow can be interpreted as an exogenous liability and is assumed to follow a Brownian motion with drift. Using techniques of stochastic optimal control, we first derive the corresponding Hamilton-Jacobi-Bellman (HJB) equation, and then reduce it into two parabolic partial differential equations (PDES) by directly conjecturing the functional form of the corresponding value function. Finally, by finding explicit solutions of the pdes we obtain the optimal investment policy. We show that the optimal non-self-financing portfolio choice problem is equivalent to a self-financing portfolio choice problem with its initial wealth equal to the sum of the endowment of the non-self-financing one and the cumulative discounted expectation of the stochastic cash flow with respect to risk neutral probability measure. Besides the familiar myopic and dynamic hedging demand, there is an additional components, the static hedging demand to hedge the risk of stochastic cash flow in the optimal strategy. The optimal policies reduce to previous results as the parameters in the model take special values. Finally, a numerical example is also provided to demonstrate the effect of parameters on the additional investment demands. © 2020, Editorial Board of Journal of Systems Engineering Society of China. All right reserved.