A Matrix Approach To Foreign Exchange Arbitrage With Bid-ask Spreads

Matrix arbitrage approach is a theory aiming at identifying risk-free arbitrageopportunity and realizing arbitrage profits in global spot foreign exchange market based ongold parity hypothesis and mathematical properties of exchange rate matrix. This paperintroduces a new analysis framework of matrix arbitrage approach, solves the identificationproblem of risk-free arbitrage opportunity considering the real existent bid-ask spread, andthen examines quotation behavior of market maker.Based on the related theory of positive reciprocal matrix and its consistency, this paperestablishes a new analysis framework of foreign exchange market matrix arbitrageapproach, pointing out that the identification method derived from maximum eigenvalue ofexchange rate matrix is flawed. The critical value of arbitrage indicator is a function ofmatrix scale (exchange rate) and matrix order (quantity of currency). This paper argues thatthe historical data can be used to determine reasonable critical value. On the basis ofinformed research on consistency of interval reciprocal comparison matrix, this papertransforms the issue of arbitrage opportunity identification with bid-ask spread to thatwithout bid-ask spread through a convex decomposition. After the transformation, therisk-free arbitrage opportunity can be identified by indicator without regard to bid-askspread.Based on the HS model, this paper finds that if either the market depth or the quantityof informed trader whose only information advantage is mastering the matrix approachescompared to noise trader is enough, the behavior of market makers will be very efficientand the foreign exchange quotation will be very close to exchange rate of arbitrage-freebenchmark matrix. Informed trader should submit order promptly after identification of opportunity.