# Arbitrage Free Cross Rate

                  Spot Rates


EUR/USD €1.0000 €1.0015
GBP/USD £2.0000 £2.0100
EUR/GBP €0.3985 €0.4000

The EUR/USD and GBP/USD rates imply that the arbitrage free cross rates for the EUR/GBP are:

bid = €1.000/₤2.0100 = €0.4975

I"m confused why you don’t inverse the GBP/USD. I was working it out and it looks like it gets the same answer:

EUR/USD €1.0000 €1.0015
GPB/USD £2.000 £2.0100

(1/2.01) = .4975 .4975 * €1.0000 = €.4975
(1/ £2.000) * €1.0015 = €.5008

Last question is how do you know which order to take the rates? Why do you do bid = €1.000/₤2.0100 = €0.4975 in the answer above and not £2.000 / €1.0015 ?

To answer my last question is it because the 3rd rate we have is EUR/GBP? Therefore EUR must be on the numerator and GBP must be on the denominator?

First, it would be nice if these examples used reasonable exchange rates. Where did you get this question?

For there to be no arbitrage opportunity:

• GBP → USD → EUR → GBP must give you no more GBP than you started with, and,
• EUR → USD → GBP → EUR must give you no more EUR than you started with

So, if you start with GBP 1.00, then the first gives you:

GBP\ 1.00\ × \left(\dfrac{USD\ 1.0}{GBP\ 2.0100}\right) × \left(\dfrac{EUR\ 1.0000}{USD\ 1.0}\right) = EUR\ 0.4975

Therefore, GBP/EUR cannot be any greater than \left(\dfrac{GBP\ 1.0}{EUR\ 0.4975}\right) = GBP/EUR\ 2.0100