The continuous-time pre-commitment KMM problem in incomplete markets

Abstract

This paper studies the continuous-time pre-commitment KMM problem proposed by Klibanoff, Marinacci and Mukerji (2005) in incomplete financial markets, which concerns with the portfolio selection under smooth ambiguity. The decision maker (DM) is uncertain about the dominated priors of the financial market, which are characterized by a second-order distribution (SOD). The KMM model separates risk attitudes and ambiguity attitudes apart and the aim of the DM is to maximize the two-fold utility of terminal wealth, which does not belong to the classical subjective utility maximization problem. By constructing the efficient frontier, the original KMM problem is first simplified as an one-fold expected utility problem on the second-order space. In order to solve the equivalent simplified problem, this paper imposes an assumption and introduces a new distorted Legendre transformation to establish the bipolar relation and the distorted duality theorem. Then, under a further assumption that the asymptotic elasticity of the ambiguous attitude is less than 1, the uniqueness and existence of the solution to the KMM problem are shown and we obtain the semi-explicit forms of the optimal terminal wealth and the optimal strategy. Explicit forms of optimal strategies are presented for CRRA, CARA and HARA utilities in the case of Gaussian SOD in a Black-Scholes financial market, which show that DM with higher ambiguity aversion tends to be more concerned about extreme market conditions with larger bias. In the end of this work, numerical comparisons with the DMs ignoring ambiguity are revealed to illustrate the effects of ambiguity on the optimal strategies and value functions.