Triangular arbitrage

The triangular arbitrage is a variant of spatial arbitrage and describes a method for exploiting arbitrage opportunity arising from price differentials between three different currencies in the Forex market. In a triangular arbitrage strategy an output currency is exchanged for a second, the second is then exchanged for a third, which will be exchanged at the end in the original currency. During the second change, the arbitrageur uses the existing price difference, which results from a divergence between the exchange rate on the market and the implied exchange rate in order to generate a risk-free profit.

Exchange rate differences

Arbitrage opportunities only arise if the required by a bank exchange rate does not coincide with the implied exchange rate in the market. The following equation shows the calculation of the implied exchange rate ( exchange rate, which is derived from the ratio of two currencies with unequal base currency).

( Euros / pounds ) = ( euro / dollar ) × (dollar / pound )

Voices of the exchange rate on the market (first term ) and the implied exchange rate (second term ) match, then there is no arbitrage opportunity. Only when an imbalance there is a possibility to earn risk-free profits.

( Euros / pounds ) ≠ ( euro / dollar ) × (dollar / pound )

Mechanism of the triangular arbitrage

Voices of the exchange rate, which is required by a bank, and the implied exchange rate do not match, so can other banks or dealers use the existing divergence to in order to earn risk-free profits.

Example: A trader has given the following exchange rates:

1.1555 EUR / GBP or 0.86543 GBP / EUR

0.76388 EUR / USD or 1.3091 USD / EUR

1.5386 USD / GBP or 0.64994 GBP / USD

Step 1: The sum in pounds received by the dealer on the sale of 100.00 EUR is:

100.00 EUR × 0.86543 GBP / EUR = £ 86.543

Step 2: Sold the dealer then the £ 86.543 USD, he shall receive the sum of:

£ 86.543 × 1.5386 USD / GBP = 133.155 USD

Step 3: The dealer sold in the last step 133.155 USD against EUR and receive:

$ 133.155 × 0.76388 EUR / USD = 101.714 EUR

The resulting risk-free gain results from the difference:

101.714 EUR - 100.00 EUR = 1.714 EUR

The return on investment is:

1.714 % = ( 101.714 EUR / 100,00 EUR ) - 1

The equilibrium exchange rate is calculated by the cross rate between the USD and GBP:

1.3091 USD / EUR × 1.1555 EUR / GBP = 1.5126 USD / GBP

The given exchange rate ( $ 1.5386 / GBP), compared to the equilibrium exchange rate ( $ 1.5126 / GBP) to 1.714 % too high.

[( 1.5386 USD / GBP ) / ( 1.5126 USD / GBP )] - 1 = 1.174 %

The measurement of the change in the exchange rate can also be expressed by the following formula:

ΔUSD % = [( opening price - closing price ) / closing price ] × 100 = 1.174 %