On the Construction of Optimal Payoffs

Abstract

The quantile formulation for optimal portfolio selection problems under increasing law-invariant objectives allows to reduce any such problem to an optimization problem on real functions under monotonicity conditions. We solve two basic types of these optimization problems, which makes it possible to solve in a unified way several portfolio selection problems of interest. In particular, we completely solve the optimal portfolio selection problem for an investor with preferences as in Yaari's dual theory of choice. Extending previous work we also derive a reduction result in general form when the payoff is required to have a fixed copula with some benchmark (state-dependent constraint). Specifically, we show that if one can determine the optimal payoff under a concave law-invariant objective, then one can also determine the optimal payoff when adding the state-dependent constraint. Finally, we identify market conditions which ensure attainability of the optimal payoff by means of a static portfolio of puts and calls.

Keywords: Optimal portfolio selection, State-dependent preferences, Conditional distribution, Yaari’s dual theory of choice, Incompleteness