Arbitrage Free Cross Rate

CFA Level II Economics
#1 
                  Spot Rates
                    Bid Price	  Ask Price

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

ask = €1.0015/₤2.0000 = €0.5008

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 ?

#2

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?

#3

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

If you start with EUR 1.00, then the second gives you:

EUR\ 1.00\ × \left(\dfrac{USD\ 1.0}{EUR\ 1.0015}\right) × \left(\dfrac{GBP\ 2.0000}{USD\ 1.0}\right) = GBP\ 1.9970

Therefore, GBP/EUR cannot be any less than GBP/EUR\ 1.9970.

There’s your bid/ask: GBP/EUR\ 1.9970/2.0100.

For the record, as of today:

  • EUR/USD is approximately 1.1900
  • GBP/USD is approximately 1.3700
  • EUR/GBP is approximately 0.8700