Foreign exchange rate arbitrage using the matrix method

Abstract

This article focuses on finding arbitrage opportunities in the international foreign exchange market. The traditional methods of distinguishing arbitrage opportunity have low efficiency, and difficulties arise when we face multiple currencies. This poses a need for an efficient method of searching for a multiple-currency arbitrage path. Ma (2004) has developed a matrix method that efficiently searches for an arbitrage path in an N-currency market. Ma also derives a simple sufficient condition that will guarantee the presence of arbitrage opportunities. Ma’s work is based on the assumption that one can buy or sell each currency at zero bid-ask spread. However, the bid-ask spread represents an important component of the transaction costs, and therefore should be incorporated into the search for arbitrage path. In this article, the author modifies Ma’s matrix method to accommodate the case where the bid-ask spreads are nonzero. We then illustrate the use of the modified method, using a 65-day series of exchange rate quotes. We also observe that the sufficient condition derived in Ma’s work still holds even for the case of nonzero bid-ask spread.