Daily trading volume in the foreign exchange markets often exceeds $1 trillion. Participants trade in the spot currency markets, forward markets, and futures markets. In addition, currency options, currency swaps, and other derivative contracts are traded. For simplicity, we focus on the spot currency market only.

A spot currency transaction is simply an agreement to buy some amount of one currency using another currency.^[1] For example, a British company might need to pay a Japanese supplier 150 million yen. Suppose that the spot yen/pound rate is 154.7733. Then the British company could use the spot currency market to buy 150 million yen at a cost of 969,159.41 (=150,000,000/154.7733) British pounds. A sample of today's cross-currency spot rates is given in Table 1.

To continue the example, suppose the company canceled the order from the supplier and wanted to convert the 150 million yen back into British pounds. From Table 1 the pound/yen spot rate is 0.00645. So the company could use the 150 million yen to buy 967,500 (=150,000,000 x 0.00645) pounds. Notice that the 967,500 pounds is less than the original 969,159.41 pounds. The difference is the result of the bid-offer spread: The price to buy yen (the bid price) is greater than the price to sell yen (the offer price). The bid-offer spread represents a transaction cost to the company.

Occasionally market prices may become "out of line" in the sense that there are arbitrage opportunities. In this context, arbitrage means that there is a set of spot currency transactions that creates positive wealth but does not require any funds to initiate, i.e., it is a "money pump". When such pure arbitrage opportunities exist, supply and demand forces will generally move prices to eliminate the opportunities. Hence, it is desirable to be able to quickly identify arbitrage opportunities when they do exist and to take advantage of them to the greatest extent possible.

Question

Formulate a decision model to determine whether there are any arbitrage opportunities with the spot currency rates given in Table 1. Note that an arbitrage opportunity could involve several currencies. If there is an arbitrage opportunity, your model should specify the exact set of transactions to achieve it.

		To
		US Dollar	Pound	FFranc	DMark	Yen
	US Dollar	-	0.63900	5.37120	1.57120	98.8901
	Pound	1.56480	-	8.43040	2.45900	154.7733
From	FFranc	0.18560	0.11860	-	0.29210	18.4122
	Dmark	0.63610	0.40630	3.42330	-	62.9400
	Yen	0.01011	0.00645	0.05431	0.01588	-

Table 1: Cross-Currency Spot Rates