Triangular Arbitrage

							D.       Triangular Arbitrage & Cross Exchange Rates

-        Cross exchange rates are exchange rates between two currencies (bilateral)

-        Daily cross rates are available between all major currencies

-        Given the exchange rate between two currencies and a third, you can calculate the
         exchange rate between those two currencies.

         Sa/c/Sb/c = Sa/b       Or     Sa/c = Sa/b x Sb/c

Or       1.0 = Sa/b x Sb/c x Sc/a      this is called the no-arbitrage condition (NOC)

-      If this condition is violated, then traders can make a risk-free profit by trading
currencies.

-        Trading currencies to exploit temporary violations of the no-arbitrage condition is
         called triangular arbitrage.

-        Triangular arbitrage will hold all currencies in equilibrium and enforce the NOC



Example:        Triangular Arbitrage & Spot Exchange Rates

Given:          SDm/$ =         Dm1.908/$              A/B

                S$/FFr =        $.1563/FFr             B/C


         What is the equilibrium rate of Dm per FFr?

         SDmFFr =       1.908 x .1563 = .2982 = Dm.2982/FFr             A/C



No-arbitrage condition:

Sa/b x Sb/c x Sc/a =    1.00           1.908 x .1563 x 1/.2982          =     1.0

Sa/b x Sb/c = Sa/c =    .2982          1.908 x 1.563 = .2982
Choosing the action to take in triangular arbitrage:

No-arbitrage condition:       Sa/b x Sb/c x Sc/a = 1.00

Arbitrage opportunity:        Sa/b x Sb/c x Sc/a > 1.00     Buy the numerator

                              Sa/b x Sb/c x Sc/a < 1.00     Buy the denominator



Example: Identify if an arbitrage opportunity exists and how to exploit it!

S$/Dm = $.524/DM              Price of Dm in U.S. dollars

SDm/C$ = Dm1.298/C$           Price of Canadian dollars in Dm

SC$/$ = C$1.30/$              Price of U.S. dollars in Canadian dollars


S$/Dm x SDm/C$ x SC$/$ = .524 x 1.298 x 1.30 = .8842        Checking the NOC



What action do you take?      Buy the denominator

Take $ and buy Dm:            $1/$.524 =      Dm1.9084

Take Dm and buy C$:           Dm1.9084/Dm1.298 = C$1.4703

Take C$ and buy $:            C$1.4703/C$1.30 = $1.131


You started with $1 and made an arbitrage profit of 13%!

Suppose that you started with:        SC$/$ = C$1.50/$

What would you do?