Triangular arbitrage (also referred to as cross currency arbitrage or three-point arbitrage) is the act of exploiting an arbitrage opportunity resulting from a pricing discrepancy among three different currencies in the foreign exchange market.[1][2] A triangular arbitrage strategy involves three trades, exchanging the initial currency for a second, the second currency for a third, and the third currency for the initial. During the second trade, the arbitrageur locks in a zero-risk profit from the discrepancy that exists when the market cross exchange rate is not aligned with the implicit cross exchange rate.[3][4]
Cross exchange rate discrepancies
Triangular arbitrage opportunities may only exist when a bank's quoted exchange rate is not equal to the market's implicit cross exchange rate. The following equation represents the calculation of an implied cross exchange rate, the exchange rate one would expect in the market as implied from the ratio of two currencies other than the base currency.[5][6]
where
- is the implicit cross exchange rate for dollars in terms of currency a
- is the quoted market cross exchange rate for b in terms of currency a
- is the quoted market cross exchange rate for dollars in terms of currency b
- is merely the reciprocal exchange rate for currency b in dollar terms, in which case division is used in the calculation
If the market cross exchange rate quoted by a bank is equal to the implicit cross exchange rate as implied from the exchange rates of other currencies, then a no-arbitrage condition is sustained.[5] However, if an inequality exists between the market cross exchange rate, , and the implicit cross exchange rate, , then there exists an opportunity for arbitrage profits on the difference between the two exchange rates.[3]
Mechanics of triangular arbitrage
Some international banks serve as market makers between currencies by narrowing their bid-ask spread more than the bid-ask spread of the implicit cross exchange rate. However, the bid and ask prices of the implicit cross exchange rate naturally discipline market makers. When banks' quoted exchange rates are move out of alignment with cross exchange rates, any banks or traders who detect the discrepancy have an opportunity to earn arbitrage profits via a triangular arbitrage strategy.[4] To execute a triangular arbitrage trading strategy, a bank would calculate cross exchange rates and compare them with exchange rates quoted by other banks to identify a pricing discrepancy.
For example, Citibank detects that Deutsche Bank is quoting dollars at a bid price of 0.8171 €/$, and that Barclays is quoting pounds at a bid price of 1.4650 $/£ (Deutsche Bank and Barclays are in other words willing to buy those currencies at those prices). Citibank itself is quoting the same prices for these two exchange rates. A trader at Citibank then sees that Crédit Agricole is quoting pounds at an ask price of 1.1910 €/£ (in other words it is willing to sell pounds at that price). While the quoted market cross exchange rate is 1.1910 €/£, Citibank's trader realizes that the implicit cross exchange rate is 1.1971 €/£ (by calculating 1.4650 × 0.8171 = 1.1971, meaning that Crédit Agricole has narrowed its bid-ask spread to serve as a market maker between the euro and the pound. Although the market suggests the implicit cross exchange rate should be 1.1971 euros per pound, Crédit Agricole is selling pounds at a lower price of 1.1910 euros. Citibank's trader can hastily exercise triangular arbitrage by exchanging dollars for euros with Deutsche Bank, then exchanging euros for pounds with Crédit Agricole, and finally exchanging pounds for dollars with Barclays. The following steps illustrate the triangular arbitrage transaction.[4]
- Citibank sells $5,000,000 to Deutsche Bank for euros, receiving €4,085,500. ($5,000,000 × 0.8171 €/$ = €4,085,500)
- Citibank sells €4,085,500 to Crédit Agricole for pounds, receiving £3,430,311. (€4,085,500 ÷ 1.1910 €/£ = £3,430,311)
- Citibank sells £3,430,311 to Barclays for dollars, receiving $5,025,406. (£3,430,311 × 1.4650 $/£ = $5,025,406)
- Citibank ultimately earns an arbitrage profit of $25,406 on the $5,000,000 of capital it used to execute the strategy.
The reason for dividing the euro amount by the euro/pound exchange rate in this example is that the exchange rate is quoted in euro terms, as is the amount being traded. One could multiply the euro amount by the reciprocal pound/euro exchange rate and still calculate the ending amount of pounds.
Evidence for triangular arbitrage
Studies of high-frequency exchange rate data have found that mispricings do arise in the foreign exchange market so that triangular arbitrage appears possible.[7] However, most of these apparent arbitrage opportunities typically only exist for 1 or 2 seconds and potentially only yield very small profits (usually less than $100 USD on a $1 million USD trade).
There are variations in the number of triangular arbitrage opportunities that occur during different hours of the 24 hour foreign exchange trading day. Perhaps slightly counter-intuitively, more triangular arbitrage opportunities arise during hours when market liquidity is at its highest; for example, from about 8-10 AM GMT when both Asian and European foreign exchange traders are active, and from 2-4 PM GMT when both European and American traders are active. During more liquid periods, triangular arbitrage opportunities also tend to exist for shorter durations than during less liquid periods. The reason for these differences is that in liquid periods the bid-ask spread is narrower and prices move around at a higher frequency due to the large volume of trading. This results in more price misalignments and thus more potential arbitrages. The high trade frequency, however, also ensures that the mis-pricings are quickly traded away and thus that any arbitrage opportunities are short-lived.
In recent years, there has been a decrease in the number of triangular arbitrage opportunities and a reduction in the potential profit that can be realized from the opportunities that do appear. This can be explained by the increasingly wider use of electronic trading platforms and trading algorithms. These systems enable traders to execute trades faster, and to react more quickly to price changes, which gives rise to increased trading efficiency, fewer mis-pricings, and fewer triangular arbitrage opportunities.[7]
Although triangular arbitrage opportunities exist, this does not necessarily mean that a trading strategy that seeks to take advantage of these mis-pricings is profitable. The three constituent trades of a triangular arbitrage transaction can be submitted extremely fast using an electronic trading system, but there is still a delay from the time that the opportunity is identified, and the trades initiated, to the time that the trades arrive at the price source. Although this delay is typically only of the order of milliseconds, it is nonetheless significant. If the trader places each trade as a limit order that will only be filled at the arbitrage price, if one of the prices moves, due to trading activity or the removal of a price by the party posting it, the transaction will not be completed. If a trader does not complete an arbitrage transaction it will cost them the amount by which the price has moved from the arbitrage price to exit their position.
In the foreign exchange market there are many market participants competing for each arbitrage opportunity; for arbitrage to be profitable a trader would need to identify and execute each arbitrage opportunity faster than their competitors. These competitors are also likely to be continually striving to increase their execution speeds - leading to an electronic trading “arms race”. Given the resources needed to stay ahead in this race, it is extremely costly to maintain the fastest execution speeds, and thus to regularly beat other competitors to the arbitrage prices over a prolonged period of time. This, along with other transaction costs such as brokerage, the network connectivity required to access the market, and the cost of developing and supporting a sophisticated electronic trading system, is likely to severely restrict the profitability of triangular arbitrage. In practice, to profit from triangular arbitrage over prolonged periods, a trader would need to trade more quickly than other market participants to a degree that appears unfeasible.[7]
References
- ^ Carbaugh, Robert J. (2005). International Economics, 10th Edition. Mason, OH: Thomson South-Western. ISBN 978-0-32-452724-7.
- ^ Pilbeam, Keith (2006). International Finance, 3rd Edition. New York, NY: Palgrave Macmillan. ISBN 978-1-40-394837-3.
- ^ a b Madura, Jeff (2007). International Financial Management: Abridged 8th Edition. Mason, OH: Thomson South-Western. ISBN 0-324-36563-2.
- ^ a b c Eun, Cheol S.; Resnick, Bruce G. (2011). International Financial Management, 6th Edition. New York, NY: McGraw-Hill/Irwin. ISBN 978-0-07-803465-7.
- ^ a b Feenstra, Robert C.; Taylor, Alan M. (2008). International Macroeconomics. New York, NY: Worth Publishers. ISBN 978-1-4292-0691-4.
- ^ Levi, Maurice D. (2005). International Finance, 4th Edition. New York, NY: Routledge. ISBN 978-0-41-530900-4.
- ^ a b c Fenn, Daniel J.; Howison, Sam D.; McDonald, Mark; Williams, Stacy; Johnson, Neil F. (2009). "The Mirage of Triangular Arbitrage in the Spot Foreign Exchange Market". International Journal of Theoretical and Applied Finance. 12 (8): 1105–1123. Retrieved 2011-07-06.