Cross-Rate Arbitrage Check Across Three Currency Pairs
A junior trader at a bank sees the following interbank quotes simultaneously: EUR/USD 1.0842/1.0848; USD/JPY 149.20/149.26; EUR/JPY 161.55/161.65. The trader believes a triangular arbitrage exists and wants to execute a USD 10,000,000 round trip. The desk's compliance manual requires that any apparent arbitrage be checked for stale quotes, transaction costs, and settlement timing before execution, and that the trader document the calculation. The trader's limit for cross-rate positions is USD 25,000,000 notional. The bank charges no explicit commission but the bid-ask spread is the cost. The trader must decide whether to execute and, if so, in which direction, and must record the implied cross rate versus the quoted cross rate.
1. Calculate the implied EUR/JPY bid and ask from the EUR/USD and USD/JPY quotes. Determine whether a triangular arbitrage exists and, if so, the direction and approximate profit on USD 10,000,000.
Read the worked answer
Implied EUR/JPY bid = EUR/USD bid × USD/JPY bid = 1.0842 × 149.20 = 161.7626. Implied EUR/JPY ask = 1.0848 × 149.26 = 161.9172. So the implied market is 161.76/161.92, while the quoted EUR/JPY is 161.55/161.65. The quoted EUR/JPY bid (161.55) is below the implied bid (161.76), and the quoted ask (161.65) is below the implied ask (161.92). This means EUR is cheap in JPY terms via the quoted cross relative to the synthetic route.
Arbitrage direction: sell USD for EUR at 1.0842 (get EUR 9,223,391), sell EUR for JPY at the quoted EUR/JPY bid 161.55 (get JPY 1,490,038,000 approx), then sell JPY for USD at USD/JPY ask 149.26 (get USD 9,982,840). That loses money. Reverse: start with JPY, buy EUR at 161.65, sell EUR for USD at 1.0848, buy JPY at 149.20. Let's check the correct profitable route: buy EUR with USD at ask 1.0848 (EUR 9,218,300), sell EUR for JPY at 161.55 (JPY 1,489,215,000), sell JPY for USD at 149.20 (USD 9,981,300). Still a loss. The apparent gap is inside the spread: the quoted cross (161.55/161.65) is actually below implied (161.76/161.92) on both sides, so selling EUR/JPY at 161.55 and buying the synthetic at 161.92 loses 0.37. No arbitrage exists. The apparent discrepancy is explained by the spread and possibly stale quotes. The trader should not execute and should document that implied 161.76/161.92 vs. quoted 161.55/161.65 shows no executable arbitrage because the quoted market is outside the synthetic on the wrong side for a profitable round trip.
2. The trader later learns one quote was 30 seconds stale. Explain how stale quotes and settlement timing can create phantom arbitrage and what controls the desk should apply.
Read the worked answer
Stale quotes create phantom arbitrage because the implied cross is computed from prices that no longer reflect the market. If EUR/USD moved after the USD/JPY quote was captured, the implied EUR/JPY is a historical artifact, not an executable rate. When the trader attempts the round trip, the counterparty will either reject the stale price or fill at the current market, eliminating the apparent profit. Settlement timing adds risk: if the three legs settle on different value dates or in different time zones, the trader faces Herstatt (settlement) risk and funding gaps, and a leg may fail, leaving an open position. Controls: (1) timestamp all quotes and reject quotes older than a defined threshold (e.g., 5 seconds for liquid pairs); (2) require executable, firm quotes from a single venue or verify with a second source; (3) compute arbitrage net of spreads and expected slippage, not gross; (4) confirm value dates match across legs or explicitly fund the mismatch; (5) pre-clear counterparty and settlement limits; (6) document the check in the trade blotter, including the reason for no-trade. The desk should also monitor for latency arbitrage and ensure the trader is not exploiting a client or stale internal price, which would raise ethical and market-conduct issues.