Portfolio optimization managing value at risk under heavy tail return, using stochastic maximum principle

We consider an investor whose portfolio consists of a single risky asset and a risk free asset. The risky asset's return has a heavy tailed distribution and thus does not have higher order moments. Hence, she aims to maximize the expected utility of the portfolio defined in terms of the median return. This is done subject to managing the Value at Risk (VaR) defined in terms of a high order quantile. Recalling that the median and other quantiles always exist and appealing to the asymptotic normality of their joint distribution, we use the stochastic maximum principle to formulate the dynamic optimization problem in its full generality. The issue of non-smoothness of the objective function is resolved by appropriate approximation technique. We also provide detailed empirical illustration using real life data. The equations which we obtain does not have any explicit analytical solution, so for numerical work we look for accurate approximations to estimate the value function and optimal strategy. As our calibration strategy is non-parametric in nature, no prior knowledge on the form of the distribution function is needed. Our results show close concordance with financial intuition. We expect that our results will add to the arsenal of the high frequency traders.