# Triangular Arbitrage by nuhman10

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```									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?

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