Cross Rates and Arbitrage
Building cross rates
A cross rate is an exchange rate between two currencies derived from their respective rates against a common third currency, usually the US dollar. To compute a cross rate, multiply or divide the two dollar-based quotes so the common currency cancels. Candidates must be able to construct a cross rate in either direction and to check the result against the direct quote in the market for consistency.
Triangular arbitrage
Triangular arbitrage exploits a discrepancy between a quoted cross rate and the rate implied by two other quotes. The trader buys the underpriced currency and sells the overpriced one through a sequence of three trades, locking in a riskless profit before prices adjust. In practice, spreads, transaction costs, and execution speed eliminate most opportunities quickly. Candidates should be able to identify which leg of a triangle is mispriced.
Traps in questions
Common traps: multiplying when the common currency requires division (or vice versa); forgetting to apply bid/offer spreads on each leg, which can turn an apparent arbitrage into a loss; and assuming arbitrage is riskless when settlement timing or credit risk introduces exposure. Also watch for questions that confuse a cross rate with a synthetic forward.
Rule of thumb
Write the two quotes with the common currency in opposite positions (one as base, one as quote), then multiply; if the common currency is in the same position in both, divide instead.
Worked example
USD/CHF 0.9000 and USD/JPY 150.00. The implied CHF/JPY cross is 150.00 / 0.9000 = 166.67. If the market quotes CHF/JPY at 167.50, a trader can buy CHF with USD, sell CHF for JPY, and sell JPY for USD, capturing the difference before spreads and costs.