Abstract

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.

摘要

我们考虑一个投资者，其投资组合由单一风险资产和无风险资产组成。风险资产的收益具有严重的拖尾分布，因此没有较高的阶跃矩。因此，她旨在最大程度地提高以中位数收益定义的投资组合的预期效用。这样做的前提是要管理按照高阶分位数定义的风险价值（VaR）。回顾中位数和其他分位数始终存在，并呼吁它们的联合分布的渐近正态性，我们使用随机最大值原理来完全动态地阐述动态优化问题。通过适当的逼近技术可以解决目标函数的非光滑性问题。我们还将使用现实生活中的数据提供详细的经验例证。我们获得的方程式没有任何明确的解析解，因此对于数值工作，我们寻求精确的近似值来估计值函数和最佳策略。由于我们的校准策略本质上是非参数的，因此无需事先了解分布函数的形式。我们的结果表明与财务直觉非常一致。我们希望我们的结果将增加高频交易者的实力。