Foreign Exchange Triangular Arbitrage Detection

Scenario

In the foreign exchange market, currency quotes are disseminated via continuous bid/ask order books. Consider three major currency pairs: USD/EUR, EUR/GBP, and GBP/USD.

Your Task

1. Formulate the exact mathematical condition for triangular arbitrage accounting for the bid-ask spread.
2. Trace the cycle of currency exchanges starting from USD $\to$ EUR $\to$ GBP $\to$ USD.
3. Given the market quotes:
   • EUR/USD: Bid $1.0800$ / Ask $1.0805$
   • EUR/GBP: Bid $0.8500$ / Ask $0.8505$
   • GBP/USD: Bid $1.2720$ / Ask $1.2725$ Determine if a profitable arbitrage loop exists, and compute the profit on a capital base of \1,000,000$ USD.

Constraints

• Buy at Ask price, sell at Bid price.
• Instantaneous execution with zero market impact.

Examples A trader converting 1 USD to EUR at $\frac{1}{\text{Ask}_{EUR/USD}}$, EUR to GBP at $\text{Bid}_{EUR/GBP}$, and GBP to USD at $\text{Bid}_{GBP/USD}$ makes profit if product of rates exceeds 1.0.