## Wednesday, November 5, 2014

### Triangular Arbitrage

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.

#### 1 comment:

1. Thanks for a very interesting blog.

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