Robust preferences and robust portfolio choice

Financial markets offer a variety of financial positions. The net result of such a position at the end of the trading period is uncertain, and it may thus be viewed as a real-valued function X on the set of possible scenarios. The problem of portfolio choice consists in choosing, among all the available positions, a position which is affordable given the investor’s wealth w, and which is optimal with respect to the investor’s preferences. In its classical form, the problem of portfolio choice involves preferences of von NeumannMorgenstern type, and a position X is affordable if its price does not exceed the initial capital w. More precisely, preferences are described by a utility functional EQ[U(X) ], where U is a concave utility function, and where Q is a probability measure on the set of scenarios which models the investor’s expectations. The price of a position X is of the form E∗[X ] where P ∗ is a probability measure equivalent to Q. In this classical case, the optimal solution can be computed explicitly in terms of U , Q, and P ∗. Recent research on the problem of portfolio choice has taken a much wider scope. On the one hand, the increasing role of derivatives and of dynamic hedging strategies has led to a more flexible notion of affordability. On the other hand, there is nowadays a much higher awareness of model uncertainty, and this has led to a robust formulation of preferences beyond the von Neuman-Morgenstern paradigm of expected utility. In Chapter I we review the theory of robust preferences as developed by SCHMEIDLER [1989], GILBOA and SCHMEIDLER [1989], and MACCHERONI et al. [2006]. Such preferences admit a numerical representation in terms of utility functionals U of the form