Robust no arbitrage and the solvability of vector-valued utility maximization problems

TL;DR

This paper establishes a fundamental link between robust no arbitrage conditions and the existence of Pareto optimal solutions in vector-valued utility maximization within markets that have proportional transaction costs and no numéraire assumption.

Contribution

It proves that robust no arbitrage is equivalent to the existence of Pareto solutions in vector utility maximization and shows how to construct consistent price processes from these solutions.

Findings

Robust no arbitrage holds iff a Pareto solution exists.

A consistent price process can be derived from Pareto maximizers.

The results apply to markets with proportional transaction costs without assuming a numéraire.

Abstract

A market model with $d$ assets in discrete time is considered where trades are subject to proportional transaction costs given via bid-ask spreads, while the existence of a num\`eraire is not assumed. It is shown that robust no arbitrage holds if, and only if, there exists a Pareto solution for some vector-valued utility maximization problem with component-wise utility functions. Moreover, it is demonstrated that a consistent price process can be constructed from the Pareto maximizer.