Determining risk neutral probabilities and optimal portfolio policies in a dynamic investment model with downside risk control in the presence of trading frictions

Abstract

This paper develops an approximate method for solving multiperiod utility maximization investment models with downside risk control characterized by the minimum attainable wealth among all possible scenarios. The stochastic control problem is decomposed into two subproblems: one is a static model identifying an " ideal" terminal wealth; and the other replicates the identified optimal portfolio by minimizing the downside replication deviation. The replicating portfolio coincides with the optimal solution to the investor's utility maximization problem for a market having general market asset return models. Multiperiod stochastic linear programming methodology yields an efficient test for the existence of arbitrage opportunities and for implementing the portfolio replication process. Instead of solving a dual programming model of a large scale stochastic linear programming for the required risk neutral probability, we decompose the problem to a sequence of deterministic linear programming models that characterize the conditional risk neutral probability at each node of a scenario. A numerical example illustrates the difference between the replicating result and the ideal portfolio, which statistically shows that including constraints can improve portfolio performance.