**Question**: Current exchange rates are 0.80 euros per dollar, 114.56 yen per dollar?

Assuming arbitrage exists what is the yen-euro exchange
rate?

**Thoughts on Triangular Arbitrage**: There are several nice explanations of triangular arbitrage on the web.

Let’s assume a person starts with dollars and wants
yen. There are two ways to get the
yen. First, he could convert directly
and get 114.56 yen at current dollars or he could convert the dollars to euros
and then convert the euros to yen.

Traders will insure that the yen-euro exchange rate is such
so that the cost of getting yen directly through dollars is the same as the
cost as converting dollars to euros and then converting the euros to yen. Deviations from the prices established under
this triangular relationship would result in profits. Traders find profit opportunities quickly so
the triangular arbitrage relationship tends to hold very closely.

**Calculation of the Yen-Euro rate**:

Note that when one divides a ratio that is denominated in
terms yen/dollar by a ratio that is denominated in terms of euros/dollar one
gets a ratio denominated in terms of yen/euro.

(yen/dollar)/(euro/dollar) = yen/euro

The yen/euro exchange rate should be 143.2 (114.56/0.80)

So a person starts with $1,000 and converts to yen. How many yen does she have?

Answer is $1,000 x 114.56 = 114560 yen.

So a person starts with $1,000 converts to Euros and then
coverts to yen. How many yen does she
have?

After converting the $1,000 to Euros we have 800 (0.80 x
1000) Euros. Convert the 800 Euros to
Yen and get 114560 yen (880 x 143.2.)

This is how arbitrage works.
It requires small transaction costs and quick computers.

My Fraction Tips blog has a lesson on exchange rate
math. See the link below.

Thanks for a very interesting blog.

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