# Cost-Efficient Contingent Claims with Market Frictions

Mathematics and Financial Economics - 10(1):87-111, 2016.

24 Pages Posted: 21 Jul 2015 Last revised: 8 Aug 2017

Date Written: July 15, 2015

### Abstract

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 (1979)). Dybvig (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. (2011), Chateauneuf et al. (1996), Chateauneuf and Cornet (2015), or Cerreia-Vioglio et al. (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 (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.

**Keywords:** Payoff Distributional Pricing, Cost-Efficiency, Contingent Claims, Nonlinear Pricing, Bid-Ask Spread, Ambiguity, Knightian Uncertainty, Non-Additive Probability, Choquet Integral.

**JEL Classification:** D89, G11

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