A Neyman–Pearson problem with ambiguity and nonlinear pricing

Abstract

We consider a problem of the Neyman–Pearson type arising in the theory of portfolio choice in the presence of probability weighting, such as in markets with Choquet pricing (as in Araujo et al. in Econ Theory 49(1):1–35, 2011; Cerreia-Vioglio et al. in J Econ Theory 157(1):730–762, 2015; Chateauneuf and Cornet in Submodular financial markets with frictions. Working Paper, 2015; Chateauneuf et al. in Math Finance 6(3):323–330, 1996) and ambiguous beliefs about the payoffs of contingent claims (see Gilboa and Marinacci, in: Acemoglu, Arellano, Dekel (eds) Advances in economics and econometrics: theory and applications, tenth world congress of the econometric society, Cambridge University Press, Cambridge, 2013). Specifically, we consider a problem of optimal choice of a contingent claim so as to minimize a non-linear pricing functional (or a distortion risk measure), subject to a minimum expected performance measure (or a minimum expected return or utility), where expectations with respect to distorted probabilities are taken in the sense of Choquet. Such contingent claims are called cost-efficient. We give an analytical characterization of cost-efficient contingent claims under very mild assumptions on the probability weighting functions, thereby extending some of the results of Ghossoub (Math Financ Econ 10(1):87–111, 2016), and we provide examples of some special cases of interest. In particular, we show how a cost-efficient contingent claim exhibits a desirable monotonicity property: It is anti-comonotonic with the random mark-to-market value (or return, etc.) of the underlying financial position, and it is hence a hedge against such variability.