Universal Arbitrage Aggregator in Discrete Time Markets under Uncertainty

TL;DR

This paper introduces a universal arbitrage aggregator in discrete time markets under uncertainty, linking arbitrage notions to martingale measures and providing dual representations and market feasibility conditions.

Contribution

It develops a universal arbitrage aggregator framework and characterizes arbitrage absence via martingale measures with various support properties.

Findings

Absence of Model Independent Arbitrage equals existence of a martingale measure.

Absence of Open Arbitrage corresponds to full support martingale measures.

Provides dual representation of Open Arbitrage using weakly open sets of measures.

Abstract

In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class $\mathcal{S}$ of significant sets, which we call Arbitrage de la classe $\mathcal{S}$. The choice of $\mathcal{S}$ reflects into the intrinsic properties of the class of polar sets of martingale measures. In particular: for S=${\Omega}$ absence of Model Independent Arbitrage is equivalent to the existence of a martingale measure; for $\mathcal{S}$ being the open sets, absence of Open Arbitrage is equivalent to the existence of full support martingale measures. These results are obtained by adopting a technical filtration enlargement and by constructing a universal aggregator of all arbitrage opportunities. We further introduce the notion of market feasibility and provide its characterization via arbitrage conditions. We conclude providing a dual representation of Open Arbitrage in terms of weakly open sets of probability measures, which highlights the robust nature of this concept.