How do you price a pair that nobody quotes?
Suppose you need EUR/JPY and your screen only shows EUR/USD and USD/JPY. You are not stuck. A rate between any two currencies can be built out of their rates against a third — and once you can do it, you also understand why the world's currency prices cannot contradict each other.
The unit-cancellation method
Forget rules about multiplying and dividing. Write every quote as a unit and let them cancel, exactly as in physics.
- EUR/USD = 1.0800 → 1.0800 USD per EUR
- USD/JPY = 150.00 → 150.00 JPY per USD
Multiply them:
(USD per EUR) × (JPY per USD) = JPY per EUR
1.0800 × 150.00 = 162.00
The USD cancels top and bottom, leaving exactly the unit EUR/JPY is measured in. EUR/JPY = 162.00.
Now a case where the shared currency sits in the same slot in both quotes:
- EUR/USD = 1.0800 → USD per EUR
- GBP/USD = 1.2500 → USD per GBP
Here the dollar is the quote currency in both, so multiplying would give nonsense units. Divide instead:
(USD per EUR) ÷ (USD per GBP) = GBP per EUR
1.0800 ÷ 1.2500 = 0.8640
EUR/GBP = 0.8640 — one euro costs 0.864 pounds. Sanity check it: the euro is worth less than the pound, so a number below 1 is right. Always run that check; it catches an inverted rate instantly.
Triangular consistency
Now the deeper point. Those three rates are not independent. Given any two, the third is determined.
If EUR/USD is 1.0800 and USD/JPY is 150.00, then EUR/JPY must be 162.00. Suppose a venue printed 162.40 instead. One euro would then be worth 162.40 yen directly, but only 162.00 yen via the dollar — the same euro, two different values, at the same instant. Convert euros to yen at 162.40, yen to dollars, dollars back to euros, and you end up with more euros than you started with, having taken no market risk at all.
That is triangular arbitrage, and its existence is why the discrepancy does not survive. In a market watched by automated systems measuring in microseconds, such gaps are closed faster than a human can read them. The practical consequence for you is not a trading opportunity — it is a structural guarantee:
Cross rates are derived, and the whole grid of world currency prices is internally consistent, because inconsistency is immediately profitable to eliminate.
The bid/ask complication
Real quotes are two-sided, and building a cross means taking the unfavourable side of both legs. Take:
- EUR/USD 1.0800 / 1.0802 (2 pips)
- USD/JPY 149.98 / 150.02 (4 pips)
The derived EUR/JPY:
- Bid = 1.0800 × 149.98 = 161.978
- Ask = 1.0802 × 150.02 = 162.052
Spread = 0.0732, which in yen pips (0.01) is 7.32 pips — the sum of the two leg spreads once each is expressed in the same units. Two spreads, one trade.
This is the arithmetic behind a fact the next unit states in words: a cross is wider than either of its legs. Not because anyone decided so, but because the price is manufactured from two quotes and inherits both costs.
In the data
Crosses are quoted in their own right, not computed from their legs on the fly. The table below is five pairs taken at the same moment, the two euro legs, their crosses and the pound: each has its own price, and the timestamps match, so any gap you find is not a clock difference.
On 29 September 2026 the two legs implied EUR/JPY 178.70 against a quoted 178.62, about 8 yen pips apart. Triangular arithmetic that ties on paper will not tie exactly on a screen of delayed quotes, and the gap is not an arbitrage.
Try it now
- From the table above, multiply EURUSD by USDJPY for the implied EUR/JPY and compare it with the quoted one. The timestamps match, so the few pips left over are three independent quotes rather than stale data. The three pairs over a year sit below.
- Repeat with EURUSD, GBPUSD and EURGBP from the same table, using division this time. Confirm your answer is below 1 before you check it.
- The two pairs below share the dollar. Derive AUD/JPY from them, then compare it with the quoted AUD/JPY in the third row. If your units cancel, your answer is right; if they do not, you have multiplied where you should have divided.