Periodic Sequences of Arbitrage: A Tale of Four Currencies
  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. 
Rod Cross, Victor Kozyakin, Brian O'Callaghan, Alexei Pokrovskii,   Alexey Pokrovskiy
q-fin.GN math.DS q-fin.TR