Calculate a currency cross rate by cancelling the bridge currency, then choose the correct bid and ask sides for an executable two-leg conversion.
Cross exchange rate
A cross exchange rate, or cross rate, is an exchange rate calculated through a third currency. It lets a desk translate a foreign-currency exposure, check a direct quote, or compare a direct conversion with a route through another currency. The arithmetic is simple; getting the currency units and executable prices right is the important part.
Let the bridge currency cancel
Suppose two fictional foreign-exchange rates, observed at the same time for the same settlement date, are USD 1.50 per GBP and USD 1.20 per EUR. Treat these as reference prices, ignoring spreads and fees for now. To obtain euros per pound, divide:
1.20USD per EUR1.50USD per GBP=1.25EUR per GBP.
The dollar is the bridge: GBP 100 corresponds to USD 150, which corresponds to EUR 125. The same calculation can be written as a product by inverting the second rate:
(1.50GBPUSD)(1.201USDEUR)=1.25GBPEUR.
Pair names and units run in opposite orders. In currency-pair notation, GBP/EUR names a price in EUR per GBP: GBP is the base currency, and EUR is the quote currency. Reversing the pair gives EUR/GBP at 1/1.25=0.80GBP per EUR. Write the units before deciding whether to multiply or divide.
The bridge need not be USD. What matters is that the intermediate currency cancels and that the two observations describe compatible transactions. Calculating a cross rate also does not establish how a dealer actually executes a conversion: a direct market can trade the two endpoint currencies.
A reference cross is not a dealing price
A ratio of midpoints is useful for marking positions or checking a data feed. It does not promise a trade at that rate. With a bid-ask spread, follow what the customer buys and sells on each leg.
Consider this separate fictional quote sheet. Bid and ask are from the dealer's perspective: the dealer buys the named base currency at its bid and sells it at its ask.
Base currency being priced
Price units
Dealer bid
Dealer ask
GBP
USD per GBP
1.4900
1.5100
EUR
USD per EUR
1.1900
1.2100
Assume both quotes are available simultaneously for the required amounts and the same settlement date, with no fees, funding charges or other costs.
Selling pounds for euros: sell GBP to the dealer at its GBP bid, receiving USD; then buy EUR at the dealer's EUR ask. The synthetic bid for GBP is
bcross=aUSD per EURbUSD per GBP=1.211.49≈1.2314EUR per GBP.
Selling GBP 100 therefore receives USD 149 and then approximately EUR 123.1405.
Buying pounds with euros: sell enough EUR at the dealer's EUR bid to obtain the USD needed to buy GBP at its ask. The synthetic ask for GBP is
across=bUSD per EURaUSD per GBP=1.191.51≈1.2689EUR per GBP.
Buying GBP 100 requires USD 151, obtained by selling approximately EUR 126.8908. These are calculations before currency-amount rounding or fees. Dividing bid by bid would miss the fact that the customer buys one base currency while selling the other.
Two fictional execution routes: selling GBP 100 yields USD 149 then EUR 123.1405; buying GBP 100 requires EUR 126.8908 then USD 151. Synthetic bid 1.2314 and ask 1.2689 EUR per GBP surround the input-midpoint ratio 1.25.
Original synthetic cash-flow ledger. Both legs share a quote time and settlement date; sufficient quantity is assumed, with no fees. Displayed cross rates and euro amounts are rounded to four decimals.
The input midpoints are still USD 1.50 per GBP and USD 1.20 per EUR. Their ratio, 1.25 EUR per GBP, lies between the synthetic bid and ask. It is a reference calculation rather than either executable side of this quote sheet.
For more work with quotation units and dealer sides, the Finance Primer's FX lesson, available with a free signed-in account, develops bid-ask conversion, cross rates and FX forwards.
What a cross-rate discrepancy tells you
Compare prices for the same size, timestamp and settlement date. A direct cross midpoint differing from a synthetic midpoint is not enough to demonstrate arbitrage. The relevant comparison is a price at which you can buy against a price at which you can sell, after every leg's spread and other costs.
A direct ask below the synthetic bid, or a direct bid above the synthetic ask, can identify a potential circular trade. Establishing an executable profit also requires available quantity, access to each market and completion of all currency legs. Stale quotes, mismatched settlement dates and changes during execution can invalidate the apparent opportunity.
