Periodic Sequences of Arbitrage: A Tale of Four Currencies

TL;DR

This paper explores the behavior of arbitrage sequences involving four currencies, revealing that such sequences can be periodic rather than converging smoothly, which differs from the three-currency case.

Contribution

It introduces the concept that four-currency arbitrage sequences can be periodic, providing new insights into arbitrage dynamics beyond the well-studied three-currency scenario.

Findings

Arbitrage sequences with four currencies can be periodic.

Such sequences may not converge to a balanced exchange rate set.

This behavior differs from the three-currency arbitrage case.

Abstract

This paper investigates arbitrage chains involving four currencies and four foreign exchange trader-arbitrageurs. In contrast with the three-currency case, we find that arbitrage operations when four currencies are present may appear periodic in nature, and not involve smooth convergence to a "balanced" ensemble of exchange rates in which the law of one price holds. The goal of this article is to understand some interesting features of sequences of arbitrage operations, features which might well be relevant in other contexts in finance and economics.