Triangular arbitrage with explicit bid and ask

Question

You have live two-sided quotes: - EUR/USD: bid 1.2000, ask 1.2002 (dollars per euro) - GBP/USD: bid 1.5000, ask 1.5003 (dollars per pound) - EUR/GBP: bid 0.7980, ask 0.7982 (pounds per euro) Is there an executable arbitrage? Trace the trades on £1,000,000 and give the profit.

Accepted Answer

First the implied cross, roughly \frac{1.2001}{1.50015} \approx 0.8000 pounds per euro. The market cross is about 0.7981, so the euro is cheap in pound terms: buy euros with pounds, then unwind euro to dollars to pounds, always crossing the side that is against you. Trace the loop GBP to EUR to USD to GBP on £1{,}000{,}000: 1. Buy EUR with GBP. Buying euros, you pay the EUR/GBP ask 0.7982 pounds per euro: 	frac{1{,}000{,}000}{0.7982} \approx €1{,}252{,}819. 2. Sell EUR for USD. Selling euros, you receive the EUR/USD bid 1.2000 dollars per euro: 1{,}252{,}819 	imes 1.2000 \approx \1{,}503{,}…

Triangular arbitrage with explicit bid and ask

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