# Triangular Arbitrage

**Topics:**Foreign exchange market, Arbitrage, Exchange rate

**Pages:**18 (5523 words)

**Published:**October 24, 2007

Table of Contents0

1Introduction1

2Market Efficiency and Arbitrage Opportunities1

2.1Triangular Arbitrage without Transaction Costs2

2.2Triangular Arbitrage with Transaction Costs2

2.3Examples5

3Triangular Arbitrage Opportunities between Turkish, British and Euro Currencies7 4Can Triangular Arbitrage Opportunities Exploited in Real Life?8 4.1Artefacts8

4.2Slippage in Price Quotes9

4.3Stale Quote9

4.4Weekend effects and non-trading hours9

5Appendix10

5.1Spot Rates between 1/10/2007 and 11/01/200610

5.2Triangular Arbitrage Calculations12

6References13

1Introduction

Triangular arbitrage is a financial activity that keeps cross exchange rates consistent. Consistency' means that the cross exchange rate between two currencies calculated from their exchange rates against a third currency must be identical to the cross rate that is actually quoted. If this is not the case then the equilibrium condition precluding triangular arbitrage is violated. Consequently, arbitragers will buy and sell currencies in a sequence dictated by the nature of the violation of the equilibrium condition is restored as a result of arbitrage itself. At this point the cross exchange rates are consistent and profit from arbitrage is zero.

2Market Efficiency and Arbitrage Opportunities

According to the market efficiency theory, the minimum requirement that a market must satisfy is that no arbitrage opportunities exist. Consistent deviations from that rule, after accounting for market imperfections such as trading costs can be interpreted as evidence of market inefficiency in allowing such profit opportunities to go unexploited. Triangular arbitrage is in theory a type of risk less arbitrage that takes advantage of cross rate mispricing. Triangular arbitrage involves positions in three currencies. Let c1, c2, c3 be their names respectively. The relationship that should hold in an efficient market is the one that no risk less profit can be realized, therefore

(c1/c2) (c2/c3) = (c1/c3)

Figure 1

The above relationship can be realized in two ways depending on the trades. The forward and reserve triangular relationships are given below,

Forward arbitrage: (c1/c2)ask (c2/c3)ask = (c1/c3)bid

Reverse arbitrage: (c1/c2)bid (c2/c3)bid = (c1/c3)ask

2.1Triangular Arbitrage without Transaction Costs

To begin with, we assume for now that there are no transaction costs or bid-ask spreads. Suppose the three currencies of concern are the dollar ($), the French franc (FF), and the British pound (₤). If we observe Ss/FF = 0.2046 and SS/₤ =1.5876, then the cross-rate, SFF/₤, is 1.5876/0.2046 = 7.7595. This is the rate a bank will quote, if it offers the service of exchanges between the franc and the pound. Any other quote would imply an arbitrage opportunity that can be realized by a triangular operation. To illustrate, suppose a bank quotes SFF/₤ = 7.800, higher than the implied cross-rate. Then we would sell ₤, buy FF, sell FF, buy $ and sell $, buy back ₤. For each pound we will make an arbitrage profit of ₤0.005215, or FF0.0405. The amount of profit available is the size of the deviation: 7.800 7.7595 = 0.0405 FF/₤.

2.2Triangular Arbitrage with Transaction Costs

As shown above, in case of no transaction costs, given the quotes of two currencies vis-à-vis the same third currency, the cross-rate is no longer unique. As a matter of fact, even the bid and ask rates are not unique. All we can infer in this case is an allowable range within which the cross-rate's bid-ask can be quoted. Given three currencies, there are three possible cross-rates and as a result, we can identify three allowable ranges for bid and ask rates. To this end, let SA/askB and SA/bidB denote the rates at which the bank sells and buys currency B vis-à-vis currency A, respectively. We then have SB/bidA = 1/SA/askB. Furthermore, we assume...

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