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
- 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 = $.1563/FFr B/C
What is the equilibrium rate of Dm per FFr?
SDmFFr = 1.908 x .1563 = .2982 = Dm.2982/FFr A/C
Sa/b x Sb/c x Sc/a = 1.00 1.908 x .1563 x 1/.2982 = 1.0
Sa/b x Sb/c = Sa/c = .2982 1.908 x 1.563 = .2982
Choosing the action to take in triangular arbitrage:
No-arbitrage condition: Sa/b x Sb/c x Sc/a = 1.00
Arbitrage opportunity: Sa/b x Sb/c x Sc/a > 1.00 Buy the numerator
Sa/b x Sb/c x Sc/a < 1.00 Buy the denominator
Example: Identify if an arbitrage opportunity exists and how to exploit it!
S$/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?