Optimal investment under dynamic risk constraints and partial information

Abstract

We consider an investor who wants to maximize expected utility of terminal wealth. Stock returns are modelled by a stochastic differential equation with non-constant coefficients. If the drift of the stock returns depends on some process independent of the driving Brownian motion, it may not be adapted to the filtration generated by the stock prices. In such a model with partial information, due to the non-constant drift, the position in the stocks varies between extreme long and short positions making these strategies very risky when trading on a daily basis. To reduce the corresponding shortfall risk, motivated by Cuoco, He and Issaenko [Operations Research, 2008, 56, pp. 358–368.] we impose a class of risk constraints on the strategy, computed on a short horizon, and then find the optimal policy in this class. This leads to much more stable strategies that can be computed for both classical drift models, a mean reverting Ornstein–Uhlenbeck process and a continuous-time Markov chain with finitely many states. The risk constraints also reduce the influence of certain parameters that may be difficult to estimate. We provide a sensitivity analysis for the trading strategy with respect to the model parameters in the constrained and unconstrained case. The results are applied to historical stock prices.

Keywords:

• Portfolio optimization
• Utility maximization
• Risk constraints
• Limited expected shortfall
• Hidden Markov model
• Partial information

Acknowledgements

The authors thank the Austrian Science Fund FWF, Project P17947-N12, the German Research Foundation DFG, Heisenberg Programme, for financial support, and two anonymous referees for their stimulating comments and suggestions. The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of their employers.