# Triangular arbitrage -- which direction to go?

1 million USD investor is trying for an arbitrage profit and is given the following bid-ask quotes for the USD, GBP and EUR.

USD:EUR: 0.7000 - 0.7010

GBP:USD: 1.7000 - 1.7010

GBP:EUR: 1.2000 - 1.2010

There is an arbitrage opportunity here. The given USD:EUR is under quoted. I tried in the following way

USD --> EUR --> GBP --> USD? I got -ve value

USD --> GBP --> EUR --> USD? I got correct value.

How do we start with which direction to go? I am afraid I will loose time, if I hit the wrong direction.

1. Find out the theoretical price

2. Then Alway remember Buy Low and Sell High

3. I will discuss you a problem tomorrow

can you post the answer and the USD:EUR under quoted?

1. The quoted USD:EUR: 0.7000 - 0.7010

To find out if arbitrage opportunity exists, I calculated the same currency pair using GBP:USD and GBP:EUR and resulted in 0.7055 - 0.7065. So there is an arbitrage opportunity here.

1. The answer for the arbitrage profit is \$6372

USD –> EUR –> GBP –> USD? (0.7000/1.2010) x 1.7000 = \$0.9908, I have a loss here.

USD –> GBP –> EUR –> USD? (1.2000/1.7010) x (1/0.7010) = \$1.006372. I have a gain \$6372

your process is good in both ways.

you have to ask youserlf in term of the domestic currency USD ,which one will give you more in term of the other currency when your are looking at the misspriced quote :

you can get it directly 700 000Eur with the direct quote 0.7000 - 0.7010

or you can get 705 500Eur by strating up with GBP and doing the triangle 0.7055 - 0.7065

so you always want to get more. if the quote was over quote, then you would start with the one that pays more. which would be the direct quote and do the inverse triangle

I think this answers it. I never focused on the the inverse triange value 0.7055 - 0.7065, except to find out if arbitrage is possible or not. So I know upfont now, that I am getting 0.7055 EUR/USD using the inverse triangle. If the inverse quote is less (say 0.6055 - 65), then I should avoid this route and start with 0.7000 - 0.7010.

Does the question tell you that the USD:EUR is under quoted or are we supposed to figure out ourselves which pairing is mispriced?

You cannot figure out which pairing is mispriced. Any one of the three could be.

Like some have commented I only worked out if possible OR not. I would use the ladder approach, bid and ask column then place your FX pairs underneath each other - 3 in total, starting and ending pair should have the currency that’s in the nominal amount. Obviously you may need to take the inverse etc to make that happen.

borrowing in USD.

USD/NZD

USD/GBP

EUR/GBP

____

1 1

USD/ NZD

EUR/GBP

GBP/USD

if you are on the bid, move up the bid so multiply, or down the ask so divide down the FX pairs in each column. if < 1 then no arbitrage exists, if > 1 then it does.

Didnt realise you had the pairs:

B (Multiply) A (Divide)

1 1

USD:EUR: 0.7000 0.7010

EUR:GBP 0.8326 0.8333 *** FLIPPED

GBP:USD: 1.7000 1.7010

= 0.99 1.00641

1.00641 means we can make \$\$\$