Large Financial Markets and Asymptotic Arbitrage with Small Transaction Costs

TL;DR

This paper characterizes asymptotic arbitrage opportunities in large financial markets with small transaction costs, showing how these opportunities relate to measure contiguity and providing conditions under which arbitrage exists or is eliminated.

Contribution

It introduces new characterizations of asymptotic arbitrage in markets with small transaction costs using measure contiguity, extending frictionless market results to more realistic settings.

Findings

Asymptotic arbitrage is characterized by contiguity properties of equivalent measures.

Strong asymptotic arbitrage can exist without transaction costs, but not with small transaction costs.

Small transaction costs can eliminate arbitrage opportunities under certain conditions.

Abstract

We give characterizations of asymptotic arbitrage of the first and second kind and of strong asymptotic arbitrage for large financial markets with small proportional transaction costs $\la_n$ on market $n$ in terms of contiguity properties of sequences of equivalent probability measures induced by $\la_n$--consistent price systems. These results are analogous to the frictionless case. Our setting is simple, each market $n$ contains two assets with continuous price processes. The proofs use quantitative versions of the Halmos--Savage Theorem and a monotone convergence result of nonnegative local martingales. Moreover, we present an example admitting a strong asymptotic arbitrage without transaction costs; but with transaction costs $\la_n>0$ on market $n$ ($\la_n\to0$ not too fast) there does not exist any form of asymptotic arbitrage.