Spot the triangular FX arbitrage

Question

You see three spot FX rates (ignore bid-ask spreads for now):

EUR/USD $= 1.20$ (dollars per euro)
GBP/USD $= 1.50$ (dollars per pound)
EUR/GBP $= 0.75$ (pounds per euro)

Is there an arbitrage? If so, give the exact sequence of trades and the profit per $1.5M of capital.

Accepted Answer

The no-arbitrage (implied) cross rate is 	ext{EUR/GBP}{implied} = \frac{	ext{EUR/USD}}{	ext{GBP/USD}} = \frac{1.20}{1.50} = 0.80 	ext{ pounds per euro}. The market quotes 0.75, the euro is too cheap in pound terms. So: buy EUR with GBP at the quoted cross, and unwind through dollars. Starting with \1{,}500{,}000: 1. Buy GBP with USD at 1.50: receive £1{,}000{,}000. 2. Buy EUR with GBP at 0.75: receive €	frac{1{,}000{,}000}{0.75} = €1{,}333{,}333.33. 3. Sell EUR for USD at 1.20: receive €1{,}333{,}333.33 	imes 1.20 = \1{,}600{,}000. Riskless profit: \100{,}000, about 6.7\%, executed simu…

Spot the triangular FX arbitrage

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