In complete frictionless securities markets under uncertainty, it is well-known that in the absence of arbitrage opportunities, there exists a unique linear positive pricing rule, which induces a state-price density (e.g., Harrison and Kreps in J Econ Theory 20(3):381–408,
1979). Dybvig (J Bus 61(3):369–393,
1988; Rev Financ Stud 1(1):67–88,
1988) showed that the cheapest way to acquire a certain distribution of a consumption bundle (or security) is when this bundle is anti-comonotonic with the state-price density, i.e., arranged in reverse order of the state-price density. In this paper, we look at extending Dybvig’s ideas to complete markets with imperfections represented by a nonlinear pricing rule (e.g., due to bid-ask spreads). We consider an investor in a securities market where the pricing rule is “law-invariant” with respect to a capacity (e.g., Choquet pricing as in Araujo et al. in Econ Theory 49(1):1–35,
2011; Chateauneuf et al. in Math Financ 6(3):323–330,
1996; Chateauneuf and Cornet in
Submodular financial markets with frictions,
2015; Cerreia-Vioglio et al. in J Econ Theory 157:730–762,
2015). The investor holds a security with a random payoff
X and his problem is that of buying the cheapest contingent claim
Y on
X, subject to some constraints on the performance of the contingent claim and on its level of risk exposure. The cheapest such claim is called
cost-efficient. If the capacity satisfies standard continuity and a property called
strong diffuseness introduced in Ghossoub (Math Op Res 40(2):429–445,
2015), we show the existence and monotonicity of cost-efficient claims, in the sense that a cost-efficient claim is anti-comonotonic with the underlying security’s payoff
X. Strong diffuseness is satisfied by a large collection of capacities, including all distortions of diffuse probability measures. As an illustration, we consider the case of a Choquet pricing functional with respect to a capacity and the case of a Choquet pricing functional with respect to a distorted probability measure. Finally, we consider a simple example in which we derive an explicit analytical form for a cost-efficient claim.