Triangle arbitrage
Encyclopedia
Triangular arbitrage is the act of exploiting an arbitrage
opportunity resulting from a pricing discrepancy among three different currencies
in the foreign exchange market
. 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.
'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.
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. 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.
s 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. 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.
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.
. A study examining exchange rate data provided by 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, 95% of them lasted for 5 seconds or less, and 60% lasted for 1 second or less. Further, most arbitrage opportunities were found to have small magnitudes, with 94% of JPY and CHF opportunities existing at a difference of 1 basis point
, which translates into a potential arbitrage profit of $100 USD per $1 million USD transacted.
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 during 10:00am and 1:00am UTC, contrasted with a greater number of opportunities and short average duration during 1:00pm and 4:00pm 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 during 12:00am and 10:00am UTC, for Europe during 7:00am and 5:00pm UTC, and for America during 1:00pm and 11:00pm UTC. The overall foreign exchange market is most liquid during 8:00am and 4:00pm UTC, and the least liquid during 10:00pm and 1:00am 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.
Researchers have shown a decrease in the incidence of triangular arbitrage opportunities from 2003 to 2005 for the Japanese yen and Swiss franc and have attributed the decrease to broader adoption of electronic trading platform
s and trading algorithms
during the same period. Such electronic systems have enabled traders to trade rapidly and react hastily to price changes. The speed gained from these technologies improved trading efficiency and the correction of mispricings, allowing for less incidence of triangular arbitrage opportunities.
systems allow the three constituent trades in a triangular arbitrage transaction to be submitted very rapidly. However, there exists a delay between the identification of such an opportunity, the initiation of trades, and the arrival of trades to the party quoting the mispricing. Even though such delays are only milliseconds in duration, they are deemed significant. For example, if a trader places each trade as a limit order to be filled only at the arbitrage price and a price moves due to market activity or new price is quoted by the third party, then the triangular transaction will not be completed. In such a case, the arbitrageur will face a cost to close out the position that is equal to the change in price that eliminated the arbitrage condition.
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 competitors. Competing arbitrageurs are expected to persist in striving to increase their execution speed of trades by engaging in what some researchers describe as an "electronic trading 'arms race'." The costs involved in keeping ahead in such a competition present difficulty in consistently beating other arbitrageurs over the long term. Other factors such as transaction costs, brokerage fees, network access fees, and sophisticated electronic trading platforms further challenge the feasibility of significant arbitrage profits over prolonged periods.
Arbitrage
In economics and finance, arbitrage is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices...
opportunity resulting from a pricing discrepancy among three different currencies
Currency
In economics, currency refers to a generally accepted medium of exchange. These are usually the coins and banknotes of a particular government, which comprise the physical aspects of a nation's money supply...
in the foreign exchange market
Foreign exchange market
The foreign exchange market is a global, worldwide decentralized financial market for trading currencies. Financial centers around the world function as anchors of trading between a wide range of different types of buyers and sellers around the clock, with the exception of weekends...
. 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
Exchange rate
In finance, an exchange rate between two currencies is the rate at which one currency will be exchanged for another. It is also regarded as the value of one country’s currency in terms of another currency...
is not aligned with the implicit cross exchange rate.
Cross exchange rate discrepancies
Triangular arbitrage opportunities may only exist when a bankBank
A bank is a financial institution that serves as a financial intermediary. The term "bank" may refer to one of several related types of entities:...
'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.
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 calculationIf 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. 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.Mechanics of triangular arbitrage
Some international banks serve as market makerMarket maker
A market maker is a company, or an individual, that quotes both a buy and a sell price in a financial instrument or commodity held in inventory, hoping to make a profit on the bid-offer spread, or turn. From a market microstructure theory standpoint, market makers are net sellers of an option to be...
s between currencies by narrowing their bid-ask spread
Bid-offer spread
The bid–offer spread for securities is the difference between the prices quoted for an immediate sale and an immediate purchase...
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. 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
Citibank
Citibank, a major international bank, is the consumer banking arm of financial services giant Citigroup. Citibank was founded in 1812 as the City Bank of New York, later First National City Bank of New York...
detects that Deutsche Bank
Deutsche Bank
Deutsche Bank AG is a global financial service company with its headquarters in Frankfurt, Germany. It employs more than 100,000 people in over 70 countries, and has a large presence in Europe, the Americas, Asia Pacific and the emerging markets...
is quoting dollars at a bid price
Bid price
A bid price is the highest price that a buyer is willing to pay for a good. It is usually referred to simply as the "bid."In bid and ask, the bid price stands in contrast to the ask price or "offer", and the difference between the two is called the bid/ask spread.An unsolicited bid or purchase...
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
Crédit Agricole
Crédit Agricole S.A. is the largest retail banking group in France, second largest in Europe and the eighth largest in the world by Tier 1 capital according to The Banker magazine. It is also part of the CAC 40 stock market index....
is quoting pounds at an ask price
Ask price
Ask price, also called offer price, offer, asking price, or simply ask, is the price a seller states she or he will accept for a good....
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.
- 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
Research examining high-frequency exchange rate data has found that mispricings do occur in the foreign exchange market such that executable triangular arbitrage opportunities appear possible. In observations of triangular arbitrage, the constituent exchange rates have exhibited strong correlationCorrelation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
. A study examining exchange rate data provided by HSBC Bank
HSBC Bank (Europe)
HSBC Bank plc is one of the four major clearing banks in the United Kingdom and is a wholly owned subsidiary of HSBC Holdings. The business ranges from the traditional High Street roles of personal finance and commercial banking, to private banking, consumer finance as well as corporate and...
for the Japanese yen
Japanese yen
The is the official currency of Japan. It is the third most traded currency in the foreign exchange market after the United States dollar and the euro. It is also widely used as a reserve currency after the U.S. dollar, the euro and the pound sterling...
(JPY) and the Swiss franc
Swiss franc
The franc is the currency and legal tender of Switzerland and Liechtenstein; it is also legal tender in the Italian exclave Campione d'Italia. Although not formally legal tender in the German exclave Büsingen , it is in wide daily use there...
(CHF) found that although a limited number of arbitrage opportunities appeared to exist for as many as 100 seconds, 95% of them lasted for 5 seconds or less, and 60% lasted for 1 second or less. Further, most arbitrage opportunities were found to have small magnitudes, with 94% of JPY and CHF opportunities existing at a difference of 1 basis point
Basis point
A basis point is a unit equal to 1/100 of a percentage point or one part per ten thousand...
, which translates into a potential arbitrage profit of $100 USD per $1 million USD transacted.
Tests for seasonality
Seasonality
In statistics, many time series exhibit cyclic variation known as seasonality, periodic variation, or periodic fluctuations. This variation can be either regular or semi regular....
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 during 10:00am and 1:00am UTC, contrasted with a greater number of opportunities and short average duration during 1:00pm and 4:00pm UTC. Such variations in incidence and duration of arbitrage opportunities can be explained by variations in market liquidity
Market liquidity
In business, economics or investment, market liquidity is an asset's ability to be sold without causing a significant movement in the price and with minimum loss of value...
during the trading day. For example, the foreign exchange market is found to be most liquid for Asia during 12:00am and 10:00am UTC, for Europe during 7:00am and 5:00pm UTC, and for America during 1:00pm and 11:00pm UTC. The overall foreign exchange market is most liquid during 8:00am and 4:00pm UTC, and the least liquid during 10:00pm and 1:00am 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.
Researchers have shown a decrease in the incidence of triangular arbitrage opportunities from 2003 to 2005 for the Japanese yen and Swiss franc and have attributed the decrease to broader adoption of electronic trading platform
Electronic trading platform
In finance, an Electronic trading platform is a computer system that can be used to place orders for financial products over a network with a financial intermediary. This includes products such as shares, bonds, currencies, commodities and derivatives with a financial intermediary, such as a...
s and trading algorithms
Algorithmic trading
In electronic financial markets, algorithmic trading or automated trading, also known as algo trading, black-box trading or robo trading, is the use of electronic platforms for entering trading orders with an algorithm deciding on aspects of the order such as the timing, price, or quantity of the...
during the same period. Such electronic systems have enabled traders to trade rapidly and react hastily to price changes. The speed gained from these technologies improved trading efficiency and the correction of mispricings, allowing for less incidence of triangular arbitrage opportunities.
Profitability
Mere existence of triangular arbitrage opportunities does not necessarily imply that a trading strategy seeking to exploit currency mispricings is necessarily profitable. Electronic tradingElectronic trading
Electronic trading, sometimes called etrading, is a method of trading securities , foreign exchange or financial derivatives electronically...
systems allow the three constituent trades in a triangular arbitrage transaction to be submitted very rapidly. However, there exists a delay between the identification of such an opportunity, the initiation of trades, and the arrival of trades to the party quoting the mispricing. Even though such delays are only milliseconds in duration, they are deemed significant. For example, if a trader places each trade as a limit order to be filled only at the arbitrage price and a price moves due to market activity or new price is quoted by the third party, then the triangular transaction will not be completed. In such a case, the arbitrageur will face a cost to close out the position that is equal to the change in price that eliminated the arbitrage condition.
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 competitors. Competing arbitrageurs are expected to persist in striving to increase their execution speed of trades by engaging in what some researchers describe as an "electronic trading 'arms race'." The costs involved in keeping ahead in such a competition present difficulty in consistently beating other arbitrageurs over the long term. Other factors such as transaction costs, brokerage fees, network access fees, and sophisticated electronic trading platforms further challenge the feasibility of significant arbitrage profits over prolonged periods.