Polynomial affine approach to HARA utility maximization with applications to OrnsteinUhlenbeck [formula omitted] models.

Abstract

This paper designs a numerical methodology, named PAMH, to approximate an investor’s optimal portfolio strategy in the contexts of expected utility theory (EUT) and mean-variance theory (MVT). Thanks to the use of hyperbolic absolute risk aversion utilities (HARA), the approach produces optimal solutions for decreasing relative risk aversion (DRRA) investors, as well as for increasing relative risk aversion (IRRA) agents. The accuracy and efficiency of the approximation is examined in a comparison to known closed-form solutions for a one dimensional (n=1) geometric Brownian motion with a CIR stochastic volatility model (i.e. GBM 1/2 or Heston model), and a high dimensional (up to n=35) stochastic covariance model. The former confirms the method works even when the theoretical candidate is not well-defined, while the latter illustrates low errors (up to 8% in certainty equivalent rate (CER)) and feasible computational time (less than one hour in a PC).Given the potential of this method, we investigate a relevant practical setting with no closed-form solution, namely when assets follow an OrnsteinUhlenbeck 4/2 stochastic volatility (SV, i.e. OU 4/2) model. We conduct sensitivity analyses of the optimal strategies for DRRA and IRRA investors with respect to key parameters; (e.g. risk aversion, minimum capital guarantee and 4/2’s parameters). In particular, the efficient frontier for the IRRA case is presented. A comparison to important sub-optimal strategies in terms of CER is performed, indicating low CER performances due to ignorance of stochastic volatility for CRRA investors, i.e. a myopic strategy would be even better than ignoring SV. The analyses highlight the importance of efficient and precise numerical methods to obtain substantially higher CERs.