What is Triangular arbitrage?

Triangular arbitrage

Triangular arbitrage (sometimes called triangle arbitrage) refers to taking advantage of a state of imbalance between three foreign exchange markets: a combination of matching deals are struck that exploit the imbalance, the profit being the difference between the market prices.

Triangular arbitrage offers a risk-free profit (in theory), so opportunities for triangular arbitrage usually disappear quickly, as many people are looking for them, or they simply never occur as everybody knows the pricing relation.

Example

Consider the three foreign exchange rates among the Canadian dollar, the U.S. dollar, and the Australian dollar. Triangular arbitrage will produce a profit whenever the following relation does not hold:

CD$/US$ * AU$/CD$ = AU$/US$.

For example if you can trade at these exchange rates

• the Canadian dollar (CD$) against the US dollar (US$) is CD$1.13/US$1.00 (1 US$ gets you CD$1.13)

• the Australian dollar (AU$) against the US dollar (US$) is AU$1.33/US$1.00 (1 US$ gets you AU$1.33)

• the Australian dollar (AU$) against the Canadian Dollar (CD$) is AU$1.18/CD$1.00 (1 CD$ gets you AU$1.18)

1.13 * 1.18 = 1.3334 > 1.3300, thus mispricing has occurred.

To take advantage of the mispricing, starting with US$10,000 to trade:

• 1st buy Canadian dollars with US dollars: US$10,000 * (CD$1.13/US$1) = CD$11,300

• 2nd buy Australian dollars with Canadian dollars: CD$11,300 * (AU$1.18/CD$1.00) = AU$13,334

• 3rd buy US dollars with Australian dollars: AU$13,334 / (AU$1.33/US$1.0000) = US$10,025

• Net risk free profit: US$25.00

A profit maximizing trader presented with these prices will trade up to the maximum size possible, or equivalently do the trade as many times as possible, until one of the traders on the other side of one of the deals changes his price. In practice currencies are quoted with a bid-ask spread, so a trader should be careful that he is actually buying at the quoted ask price, and selling at the quoted bid price. Other transaction costs, such as commissions often prevent the trade from being profitable. Traders who attempt to make a profit this way typically use sophisticated software programs to automate the process.

Traders may also use this triangular relation to take a synthetic position if, for some reason, they can not otherwise trade a specific currency pair.^[1] For example, if a trader wanted to buy Australian dollars with US dollars, he could instead first buy Canadian dollars with US dollars, then buy the Australian dollars with the Canadian dollars. Using the prices in the above example, he would

• 1st buy Canadian dollars with US dollars: US$10,000 * (CD$1.13/US$1) = CD$11,300

• 2nd buy Australian dollars with Canadian dollars: CD$11,300 * (AU$1.18/CD$1.00) = AU$13,334

for an effective price of AU$1.3334/US$1.00.

In practice, if two of the markets exist, the third will also be available to trade and any apparent benefit of using the synthetic purchase or sale will be overcome by increased transaction costs.

Do triangular arbitrage opportunities actually exist?

Studies of high-frequency exchange rate data have found that mis-pricings do arise in the foreign exchange market so that triangular arbitrage appears possible^[2]. Most of these apparent arbitrage opportunities, however, typically only exist for 1 or 2 seconds and potentially only yield very small profits (usually less than US$100 on a US$1 million 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.

Is triangular arbitrage profitable?

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.