Optimal measure preserving derivatives revisited

TL;DR

This paper explores the link between pricing kernel monotonicity and stochastic arbitrage opportunities in complete markets, introducing a generalized optimal measure preserving derivative under certain technical conditions.

Contribution

It clarifies the relationship between pricing kernel monotonicity and arbitrage, and generalizes the optimal measure preserving derivative for distributional replication at minimal cost.

Findings

Pricing kernel nonmonotonicity relates to arbitrage opportunities.

Adequacy condition affects the equivalence between monotonicity and arbitrage.

A generalized derivative achieves minimal cost distributional replication.

Abstract

This article clarifies the relationship between pricing kernel monotonicity and the existence of opportunities for stochastic arbitrage in a complete and frictionless market of derivative securities written on a market portfolio. The relationship depends on whether the payoff distribution of the market portfolio satisfies a technical condition called adequacy, meaning that it is atomless or is comprised of finitely many equally probable atoms. Under adequacy, pricing kernel nonmonotonicity is equivalent to the existence of a strong form of stochastic arbitrage involving distributional replication of the market portfolio at a lower price. If the adequacy condition is dropped then this equivalence no longer holds, but pricing kernel nonmonotonicity remains equivalent to the existence of a weaker form of stochastic arbitrage involving second-order stochastic dominance of the market portfolio at a lower price. A generalization of the optimal measure preserving derivative is obtained which achieves distributional replication at the minimum cost of all second-order stochastically dominant securities under adequacy.