FX · X4 · 7 min
Cross rates
Work out the exchange rate between two currencies from their dollar rates, including its bid and offer.
In the news: “The euro rose against sterling”: EUR/GBP, worked out from two dollar rates.
The essentials
- Cross rate
- A pair without the dollar, such as EUR/GBP or EUR/JPY. Built from two dollar pairs it is the synthetic rate: EUR/USD 1.0850 and GBP/USD 1.2650 give EUR/GBP 0.8577.
- Divide or multiply
- Make the dollar cancel. If it sits on the same side of both pairs, divide; if it sits on opposite sides, multiply: dollars per euro × yen per dollar = yen per euro, so 1.10 × 150 gives EUR/JPY 165.
- The cross’s bid and offer
- When dividing, pair bid with offer: a client selling euros for pounds sells euros at the EUR/USD bid and buys pounds at the GBP/USD offer. When multiplying, it is bid × bid and offer × offer.
- Why crosses cost more
- A synthetic cross crosses two spreads, so in percentage terms it is about as wide as both added together: 0.02% plus 0.03% gives about 0.05%.
- When the cross is off
- A cross quoted away from the synthetic rate can be traded against the two dollar pairs for a profit: 3 on €10m is £3,000. Such gaps close within moments, so they are rare.
Cross rate through the dollar
EUR/GBP = EUR/USD ÷ GBP/USD; EUR/JPY = EUR/USD × USD/JPY
- ÷
- the dollar is on the same side of both pairs; bid = bid ÷ offer
- ×
- the dollar is on opposite sides; bid = bid × bid
How it works
1. Going through the dollar
Most currencies trade mainly against the dollar. To change euros into pounds, you can sell euros for dollars, then buy pounds with the dollars. If €1 = $1.10 and £1 = $1.25, then €100 becomes $110, and $110 ÷ 1.25 = £88.
2. A cross rate
A pair without the dollar, such as EUR/GBP, is a . From the last step, €1 = £0.88, and 0.88 is just 1.10 ÷ 1.25: EUR/USD divided by GBP/USD. A cross built this way is called the synthetic rate.
3. When to multiply
For EUR/JPY, use EUR/USD and USD/JPY. €1 = $1.10 and $1 = ¥150, so €1 = 1.10 × 150 = ¥165. Here you multiply, because the dollars cancel: dollars per euro × yen per dollar = yen per euro.
4. The rule
Look at where the dollar sits. If it is on the same side of both pairs, divide: EUR/USD ÷ GBP/USD, or USD/JPY ÷ USD/CHF for CHF/JPY. If it is on opposite sides, multiply: EUR/USD × USD/JPY. The answer should have the cross’s first currency on top.
5. The cross’s bid
A client selling euros for pounds does two trades: sells euros for dollars at the EUR/USD bid, then buys pounds with those dollars at the GBP/USD offer. So the EUR/GBP bid = EUR/USD bid ÷ GBP/USD offer. When you multiply instead, it is bid × bid.
6. Why crosses cost more
The client pays a spread on both trades, so a synthetic cross is wider than either of its two pairs: in percentage terms, about the two spreads added together. From the quotes above, EUR/GBP is 0.8574/0.8578, about 4 pips wide.
7. When the cross is off
If a dealer bids for EUR/GBP above the synthetic rate, sell euros to that dealer for pounds and buy them back through the dollar for less. The profit is the gap times the amount: 3 pips on €10m is £3,000. Gaps like this close within moments, so they are rare.
Worked example
EUR/USD is 1.0848/50 and GBP/USD is 1.2648/52. A client asks for a two-way price in EUR/GBP.
- The dollar is the second currency in both pairs, so divide.
- Bid (client sells euros): EUR/USD bid ÷ GBP/USD offer = 1.0848 ÷ 1.2652 = 0.85741.
- Offer (client buys euros): EUR/USD offer ÷ GBP/USD bid = 1.0850 ÷ 1.2648 = 0.85784.
- Width: 0.85784 − 0.85741 = 0.00043, about 4.3 pips.
EUR/GBP 0.8574/0.8578: about 4 pips wide, carrying both pairs’ spreads.
Check yourself
7 quick questions on the explanation above.
Practice
New numbers every set. Misses and guesses go into your review deck.
In the interview
Answer out loud in under a minute, then compare.
How would you price EUR/JPY with only dollar pairs?
Why is the cross spread wider than either leg?
What would you do if a quoted cross was off the synthetic rate?
Remember
EUR/GBP = EUR/USD ÷ GBP/USD
Make the dollar cancel: divide when it sits on the same side of both pairs, multiply when it sits on opposite sides, and pair bid with offer when you divide.