Triangular arbitrage: Difference between revisions

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{{Short description|Forex arbitrage across three currencies}}

'''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 [[currency|currencies]] in the [[foreign exchange market]].<ref name="Carbaugh 2005">{{Cite book | title = International Economics, 10th Edition | author = Carbaugh, Robert J. | year = 2005 | publisher = Thomson South-Western | ___location = Mason, OH | isbn = 978-0-32324-45272452724-7}}</ref><ref name="Pilbeam 2006">{{Cite book | title = International Finance, 3rd Edition | author = Pilbeam, Keith | year = 2006 | publisher = Palgrave Macmillan | ___location = New York, NY | isbn = 978-1-404039-3948374837-3}}</ref><ref name="Aiba et al. 2002">{{Cite journal | title = Triangular arbitrage as an interaction among foreign exchange rates | journal = Physica A: Statistical Mechanics and Its Applications | volume = 310 | issue = 4 | year = 2002 | pages = 467–479 | author = Aiba, Yukihiro | author2 = Hatano, Naomichi | author3 = Takayasu, Hideki | author4 = Marumo, Kouhei | author5 = Shimizu, Tokiko | doi=10.1016/S0378-4371(02)00799-9| arxiv = cond-mat/0202391 | bibcode = 2002PhyA..310..467A }}</ref> 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 [[exchange rate|cross exchange rate]] is not aligned with the implicit cross exchange rate.<ref name="Madura 2007">{{Cite book | title = International Financial Management: Abridged 8th Edition | author = Madura, Jeff | year = 2007 | publisher = Thomson South-Western | ___location = Mason, OH | isbn = 978-0-324-36563-24}}</ref><ref name="Eun & Resnick 2011">{{Cite book | title = International Financial Management, 6th Edition | author = Eun, Cheol S. | author2 = Resnick, Bruce G. | year = 2011 | publisher = McGraw-Hill/Irwin | ___location = New York, NY | isbn = 978-0-07-803465-7}}</ref> A profitable trade is only possible if there exist market imperfections. Profitable triangular arbitrage is very rarely possible because when such opportunities arise, traders execute trades that take advantage of the imperfections and prices adjust up or down until the opportunity disappears.<ref name="Ozyasar 2013">{{Cite news | title = Strategy for FOREX Triangulation | author = Ozyasar, Hunkar | date = 2013 | publisher = The Nest | url = http://budgeting.thenest.com/strategy-forex-triangulation-21257.html | accessdate = 2014-06-15}}</ref>

==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 implicit 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.<ref name="Feenstra & Taylor 2008">{{Cite book | title = International Macroeconomics | author = Feenstra, Robert C. | author2 = Taylor, Alan M. | year = 2008 | publisher = Worth Publishers | ___location = New York, NY | isbn = 978-1-4292-0691-4 | url-access = registration | url = https://archive.org/details/internationaleco0000feen }}</ref><ref Thename="Levi following2005">{{Cite equationbook represents| thetitle calculation= ofInternational anFinance, implied4th crossEdition exchange| rate,author the= exchangeLevi, rateMaurice oneD. would| expectyear in= the2005 market| aspublisher implied= fromRoutledge the| ratio___location of= twoNew currenciesYork, otherNY than| theisbn base= currency.978-0-415-30900-4}}</ref>

:<math>S_{a/\$} = S_{a/b} S_{b/\$}</math>

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:<math>S_{a/b}</math> is the quoted market cross exchange rate for b in terms of currency ''a''

:<math>S_{b/\$}</math> is the quoted market cross exchange rate for dollars in terms of currency ''b''

:<math>S_{\$/b}</math> ''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.<ref name="Feenstra & Taylor 2008" /> However, if an inequality exists between the market cross exchange rate, <math>S_{a/\$}</math>, and the implicit cross exchange rate, <math>S_{a/b} S_{b/\$}</math>, then there exists an opportunity for arbitrage profits on the difference between the two exchange rates.<ref name="Madura 2007" />

==Mechanics of triangular arbitrage==

[[File:Triangular-arbitrage.svg|thumb|right|A visual representation of a realistic triangular arbitrage scenario, using sample bid and ask prices quoted by international banks]]

Some international banks serve as [[market maker]]s between currencies by narrowing their [[bid-offer spread|bid-askbid–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.<ref name="Eun & Resnick 2011" /> 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€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€1.1910 €/£ (in other words it is willing to sell pounds at that price). While the quoted market cross exchange rate is 1€1.1910 €/£, Citibank's trader realizes that the implicit cross exchange rate is 1€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.<ref name="Eun & Resnick 2011" />

# Citibank sells $5,000,000 to Deutsche Bank for euros, receiving €4,085,500. ($5,000,000 × 0€0.8171 €/$ = €4,085,500)

# Citibank sells €4,085,500 to Crédit Agricole for pounds, receiving £3,430,311. (€4,085,500 ÷ 1€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==

StudiesResearch ofexamining high-frequency [[exchange rate]] data havehas found that mis-pricings[[mispricing]]s do ariseoccur in the [[foreign exchange market]] sosuch that executable triangular arbitrage appearsopportunities appear possible.<ref name="Fenn et al. 2009">{{Cite journal | title = The Mirage of Triangular Arbitrage in the Spot Foreign Exchange Market | journal = International Journal of Theoretical and Applied Finance | volume = 12 | issue = 8 | dateyear = 2009 | pages = 1105-11231105–1123 | author = Fenn, Daniel J. | author2 = Howison, Sam D. | author3 = McDonald, Mark | author4 = Williams, Stacy | author5 = Johnson, Neil F. | urldoi = http://www.worldscinet10.com/ijtaf/12/12081142/S0219024909005609.html | accessdatearxiv = 2011-07-060812.0913 }}</ref> MostIn observations of thesetriangular apparentarbitrage, the constituent exchange rates have exhibited strong [[correlation]].<ref name="Aiba et al. 2002" /> A study examining exchange rate data provided by [[HSBC Bank (Europe)|HSBC Bank]] for the [[Japanese yen]] (JPY) and the [[Swiss franc]] (CHF) found that although a limited number of arbitrage opportunities appeared to exist for as many as 100 seconds, however95% of them lasted for 5 seconds or less, typicallyand only60% existlasted for 1 second or 2less. secondsFurther, andmost potentiallyarbitrage onlyopportunities yieldwere veryfound to have small profitsmagnitudes, (usuallywith less94% thanof JPY and CHF opportunities existing at a difference of 1 [[basis point]], which translates into a potential arbitrage profit of US$100 on aper US$1 million trade)transacted.<ref name="Fenn et al. 2009" />

Tests for [[seasonality]] in the amount and duration of triangular arbitrage opportunities have shown that incidence of arbitrage opportunities and mean duration is consistent from day to day. However, significant variations have been identified during different times of day. Transactions involving the JPY and CHF have demonstrated a smaller number of opportunities and long average duration around 01:00 and 10:00 [[UTC]], contrasted with a greater number of opportunities and short average duration around 13:00 and 16:00 UTC. Such variations in incidence and duration of arbitrage opportunities can be explained by variations in [[market liquidity]] during the [[trading day]]. For example, the foreign exchange market is found to be most liquid for Asia around 00:00 and 10:00 UTC, for Europe around 07:00 and 17:00 UTC, and for America around 13:00 and 23:00 UTC. The overall foreign exchange market is most liquid around 08:00 and 16:00 UTC, and the least liquid around 22:00 and 01:00 UTC. The periods of highest liquidity correspond with the periods of greatest incidence of opportunities for triangular arbitrage. This correspondence is substantiated by the observation of narrower bid-ask spreads during periods of high liquidity, resulting in a greater potential for mispricings and therefore arbitrage opportunities. However, market forces are driven to correct for mispricings due to a high frequency of trades that will trade away fleeting arbitrage opportunities.<ref name="Fenn et al. 2009" />

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.

InResearchers recenthave years, there has beenshown a decrease in the numberincidence of triangular arbitrage opportunities andfrom a2003 reductionto in2005 the potential profit that can be realized fromfor the opportunitiesJapanese thatyen doand appear.Swiss Thisfranc canand behave explained byattributed the increasinglydecrease widerto usebroader adoption of [[electronic trading platformsplatform]]s and [[algorithmic trading|trading algorithms]] during the same period. TheseSuch electronic systems enablehave enabled traders to execute trades faster,trade and to react more quicklyrapidly to price changes,. whichThe givesspeed risegained tofrom increasedthese technologies improved trading efficiency, fewerand mis-pricingsthe correction of mispricings, andallowing fewerfor less incidence of triangular arbitrage opportunities.<ref name="Fenn et al. 2009" />

==Profitability==

AlthoughMere existence of triangular arbitrage opportunities exist, this does not necessarily meanimply that a trading strategy that seeksseeking to takeexploit advantagecurrency ofmispricings theseis mis-pricings isconsistently profitable. The[[Electronic trading]] systems allow the three constituent trades ofin a triangular arbitrage transaction canto be submitted extremelyvery fastrapidly. using an electronic trading systemHowever, but there is stillexists a delay frombetween the timeidentification thatof thesuch an opportunity, isthe identifiedinitiation of trades, and the tradesarrival of initiated,trades to the timeparty thatquoting the trades arrive at the price sourcemispricing. AlthoughEven thisthough delaysuch isdelays typicallyare only ofmilliseconds thein order of millisecondsduration, itthey isare nonethelessdeemed significant. IfFor theexample, if a trader places each trade as a [[limit order]] that will onlyto be filled only at the arbitrage price, ifand onea of the pricesprice moves, due to tradingmarket activity or thenew removalprice ofis a pricequoted by the third party, posting it,then the triangular transaction will not be completed. IfIn such a tradercase, doesthe notarbitrageur completewill anface arbitragea transactioncost itto willclose cost themout the amountposition bythat whichis equal to the pricechange hasin movedprice that fromeliminated the arbitrage pricecondition.<ref toname="Fenn exitet their positional. 2009" />

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. TheseCompeting competitorsarbitrageurs are also likelyexpected to bepersist continuallyin striving to increase their execution speedsspeed -of leadingtrades toby anengaging electronicin tradingwhat “armssome race”.researchers Givendescribe theas resourcesan needed"electronic totrading stay'arms aheadrace'."<ref inname="Fenn thiset race,al. it2009" is/> extremelyThe costlycosts toinvolved maintainin thekeeping fastestahead executionin speeds,such anda thuscompetition topresent regularlydifficulty beatin otherconsistently competitorsbeating toother the arbitrage pricesarbitrageurs over athe prolonged period oflong timeterm. This,Other alongfactors withsuch otheras [[transaction costs]], such[[brokerage]] as brokeragefees, the network connectivity required to access the marketfees, and the cost of developing and supporting a sophisticated electronic trading system,platforms isfurther likely to severely restrictchallenge the profitabilityfeasibility of triangularsignificant arbitrage. In practice, to profit from triangular arbitrageprofits over prolonged periods, a trader would need to trade more quickly than other market participants to a degree that appears unfeasible.<ref name="Fenn et al. 2009" />

==See also==

* [[Covered interest arbitrage]]

* [[Uncovered interest arbitrage]]

*[[Arbitrage]]

==References==

{{Reflist}}

==SeeExternal alsolinks==

* [https://play.google.com/store/apps/details?id=wellbeing.currencyarbitrage Currency triangular arbitrage calculator] on Android

*[[Covered interest arbitrage]]

*[[Uncovered interest arbitrage]]

[[Category: Financial economicsArbitrage]]

[[Category: Financial terminologyeconomics]]

[[Category: Foreign exchange market]]

[[Category:International finance]]