Question

Doug Bernard specializes in cross-rate arbitrage. He notices the following quotes:

1. Crédit Lyonnais: Yen / US dollar = JPY 111.0000 / USD

2. Barclays: US dollar / euro = USD 1.2200/ EUR

3. Crédit Agricole: Yen / EUR = JPY 137.0000 / EUR

Ignoring transaction costs, show how much Doug Bernard can make a triangular arbitrage profit by trading at these prices. Assume that he has $1 million available to conduct the arbitrage and calculate the arbitrage profit in both $ and percentage terms, showing clearly the steps of the arbitrage.

What yen/euro price will eliminate the triangular arbitrage opportunity?

Answer 1

The simple way to solve arbitrage profit problems is to start from the currency you have and then using exchange rate through different routes, arrive back at the same currency. We have two options with us -

1. Go from USD to JPY to EUR to USD

2. Go from USD to EUR to JPY to USD

We'll go both ways to find out any arbitrage opportunity -

1. Convert USD 1,000,000 to JPY = USD 1,000,000 x JPY 111 / USD = JPY 111,000,000

Then, convert JPY to EUR = JPY 111,000,000 / (JPY 137 / EUR) = EUR 810,218.978102

Finally go back to USD = EUR 810,218.978102 x USD 1.2200 / EUR = USD 988,467.153284

As you can see, we have less money than before so, their is no arbitrage possible going this route.

2. Convert USD to EUR = USD 1,000,000 / (USD 1.2200 / EUR) = EUR 819,672.131147

Then, convert EUR to JPY = EUR 819,672.131147 x JPY 137 / EUR = JPY 112,295,081.967

Finally, go back to USD = JPY 112,295,081.967 / (JPY 111 / USD) = USD 1,011,667.4051

Arbitrage Profit = USD 1,011,667.4051 - USD 1,000,000 = USD 11,667.4051

Arbitrage Profit (in %) = (USD 11,667.4051 / USD 1,000,000) x 100 = 1.16674051 % or 1.17%

Last part of finding the exchange rate - .

You need to have exact JPY 111,000,000 to arrive at USD 1,000,000 (see no arbitrage way). Also, you need to have EUR 819,672.131147 to arrive at USD 1,000,000. (See arbitrage way)

So, the exchange rate would be = JPY 111,000,000 / EUR 819,672.131147 = JPY 135.4200 / EUR