Utility Maximization in a Multivariate Black Scholes Type Market with Model Uncertainty on the Drift
It is a by now classical observation that in a (realistic) financial market (model) simple portfolio strategies can outperform more sophisticated optimized portfolio strategies. For example, in a one period setting, the equal weight or 1/N-strategy often provides more stable results than mean-variance- optimal strategies. This is due to the fact that a good estimation of the mean returns is not possible for volatile financial assets. Pflug, Pichler and Wozabel (2012) gave a rigorous explanation of this observation by showing that for increasing uncertainty on the means the equal weight strategy becomes optimal in a mean-variance setting which is due to its robustness. We aim at extending this result to continuous-time strategies in a multivariate Black-Scholes type market. To this end we investigate how optimal trading strategies for maximizing expected utility of terminal wealth under CRRA utility behave when we have Knightian uncertainty on the drift, meaning that the only information is that the drift parameter lies in a so-called uncertainty set. The investor takes into account that the true drift may be the worst possible drift within this set. In this setting we can show that a minimax theorem holds which enables us to find the worst- case drift and the optimal robust strategy quite explicitly. This again allows us to derive the limits when uncertainty increases and hence to show that a uniform strategy is asymptotically optimal. We also discuss the extension to a financial market with a stochastic drift process, combining the worst-case approach with filtering techniques. This leads to local optimization problems, and the resulting optimal strategy needs to be updated continuously in time. We carry over the minimax theorem for the local optimization problems and derive the optimal strategy. In this setting we show how an ellipsoidal uncertainty set can be defined based on filtering techniques and we demonstrate that investors need to choose a robust strategy to be able to profit from additional information.