Triangular Arbitrage

Table of Contents

Table of Contents 0

1 Introduction 1

2 Market Efficiency and Arbitrage Opportunities 1

2.1 Triangular Arbitrage without Transaction Costs 2

2.2 Triangular Arbitrage with Transaction Costs 2

2.3 Examples 5

3 Triangular Arbitrage Opportunities between Turkish, British and Euro Currencies 7

4 Can Triangular Arbitrage Opportunities Exploited in Real Life? 8

4.1 Artefacts 8

4.2 Slippage in Price Quotes 9

4.3 Stale Quote 9

4.4 Weekend effects and non-trading hours 9

5 Appendix 10

5.1 Spot Rates between 1/10/2007 and 11/01/2006 10

5.2 Triangular Arbitrage Calculations 12

6 References 13

1 Introduction

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.

2 Market 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

2.1 Triangular 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/₤.

2.2 Triangular Arbitrage with Transaction Costs

As shown above, in case of no transaction costs, given the quotes of two currencies vis-Ðo-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-Ðo-vis currency A, respectively. We then have SB/bidA = 1/SA/askB. Furthermore, we assume that all cross-rates as well as regular rates are quoted by banks. Thus, using the three currencies in the previous illustration, the allowable quotes for the three cross-rates, FF/₤, $/₤ and S/FF are summarized in the following three inequalities:

1) SFF/bid$ * S$/bid₤ ≤ SFF/bid₤ ≤ SFF/ask₤ ≤ SFF/ask$ * S$/ask₤

2) S$/bidFF * SFF/bid₤ ≤ S$/bid₤ ≤ S$/ask₤ ≤ S$/askFF * SFF/ask₤

3) S$/bid₤ * S₤/bidFF ≤ S$/bidFF ≤ S$/askFF ≤ S$/ask₤ * S₤/askFF

To understand the meaning of the inequalities, let us look at first one for the cross-rate FF/₤. The middle part of the inequalities simply says that the ask rate is higher than the bid rate. The first part of the inequality identifies the lowest bid on the pound, which essentially says, if an investor wants to sell ₤ for FF, he will not get a better deal by first selling ₤ for at S$bid/₤ and then selling $ for FF at SFF/bid$. He will end up with fewer Francs by going the roundabout way. In the same way, the last part of the inequality identifies the highest ask rate on the pound. It means that, if another investor wants to buy ₤ with FF, he will not get a better deal by first buying $ with FF at SFF/ask$ and then buying ₤ with $ at S$/ask₤. He will end up spending

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