Optimal claims with fixed payoff structure

Abstract

Dybvig (1988) introduced the interesting problem of how to construct in the cheapest possible way a terminal wealth with desired distribution. This idea has induced a series of papers concerning generality, consequences, and applications. As the optimized claims typically follow the trend in the market, they are not useful for investors who wish to use them to protect an existing portfolio. For this reason, Bernard, Moraux, Rüschendorf and Vanduffel (2014b) imposed additional state-dependent constraints as a way of controlling the payoff structure. The present paper extends this work in various ways. In order to obtain optimal claims in general models we allow in this paper for extended contracts. We deal with general multivariate price processes and dispense with several of the regularity assumptions in the previous work (in particular, we omit any continuity assumption). State dependence is modeled by requiring terminal wealth to have a fixed copula with a benchmark wealth. In this setting, we are able to characterize optimal claims. We apply the theoretical results to deal with several hedging and expected utility maximization problems of interest.