Building a cross through the dollar
Almost everything trades against the dollar, so a rate between two other currencies is built by going through it. Write the units and the multiply-or-divide takes care of itself; cross two spreads on the worse side each time and you have a two-way price.
- Build EUR/JPY and EUR/GBP from the dollar pairs, knowing when to multiply and when to divide
- Make a two-way cross from two two-way dollar quotes, and see why it is wider than either leg
- Recover EUR/USD from EUR/JPY and USD/JPY
- Trace a move in one dollar pair through to the cross and to a holding's value
The intuition
If one euro buys 1.0800 dollars and one dollar buys 150.00 yen, one euro buys 1.0800 × 150.00 = 162.00 yen. Write the units and it is obvious: dollars per euro times yen per dollar leaves yen per euro, because the dollars cancel. When the dollar is the quote currency in both pairs, as in EUR/USD and GBP/USD, it cancels by dividing instead: 1.0800 dollars per euro over 1.2500 dollars per pound is 0.8640 pounds per euro.
A two-way cross is built from real trades, and every leg goes through on the side that is worse for you. To buy a euro from a client and pay in pounds, you sell that euro for dollars at the EUR/USD bid and buy pounds with the dollars at the GBP/USD offer, so your EUR/GBP bid is EUR/USD bid ÷ GBP/USD offer. The offer is the mirror image. Two spreads stack, so the cross is wider than either dollar pair it is made from, until a cross gets enough liquidity of its own to be quoted directly.
Dollar on opposite sides: multiply (EUR/JPY = EUR/USD × USD/JPY). Dollar on the same side: divide (EUR/GBP = EUR/USD ÷ GBP/USD). Two-way: multiplying pairs bid with bid and offer with offer; dividing pairs your bid with the other leg's offer. Implied EUR/USD = EUR/JPY ÷ USD/JPY.
Why it works
- The conventions here: BASE/QUOTE is units of the quote currency per one unit of the base. EUR/USD and GBP/USD to four decimals (a pip is 0.0001); USD/JPY and EUR/JPY to two (a pip is 0.01). Crosses go through the dollar. Two-way crosses take the worse side of each leg. Where a rate is printed and you work from it, the answer is computed from the printed rate.
- Units decide the operation. (USD per EUR) × (JPY per USD) = JPY per EUR. (USD per EUR) ÷ (USD per GBP) = GBP per EUR. If the dollar would not cancel, you have the wrong operation.
- Why the worse side, each time. The cross is manufactured from two trades you actually do. Each is done at the price the market gives you, which is its bid when you sell and its offer when you buy. Both spreads are paid.
- Why the cross spread is bigger than the sum of the legs' pips. Multiplying EUR/USD by USD/JPY magnifies the EUR/USD spread by the size of USD/JPY: a 1-pip EUR/USD spread is worth 150 × 0.0001 = 1.5 yen pips at 150, before the USD/JPY spread is added.
- Sanity checks. EUR/GBP below 1 when a pound buys more dollars than a euro. EUR/JPY moves by the same percentage as USD/JPY when EUR/USD is fixed. An implied EUR/USD that differs from the screen by a fraction of a pip is rounding, not arbitrage.
- Run it backwards. EUR/USD = EUR/JPY ÷ USD/JPY. Triangular arbitrage exists only when the gap beats all three spreads, and machines close it in milliseconds.
| EUR/JPY mid: 1.0800 × 150.00 | 162.00 |
| EUR/GBP mid: 1.0800 ÷ 1.2500 | 0.8640 |
| EUR/GBP bid: EUR/USD bid ÷ GBP/USD offer = 1.0800 ÷ 1.2502 | 0.8639 |
| EUR/GBP offer: EUR/USD offer ÷ GBP/USD bid = 1.0802 ÷ 1.2500 | 0.8642, about 3 pips wide |
| EUR/JPY two-way: 1.0800 × 150.00 to 1.0802 × 150.02 | 162.00 / 162.05, about 5 yen pips wide |
| Implied EUR/USD from EUR/JPY 162.00 and USD/JPY 150.00 | 1.0800 |
The formulas
Dollar on opposite sides of the two quotes: multiply.
Dollar on the same side: divide.
Each leg on the side that is worse for you.
Sell euros on the bid, buy pounds on the offer.
The backwards reading.
Worked example
Build each side from real trades: sell euros on the EUR/USD bid and buy pounds on the GBP/USD offer for your bid; the mirror for your offer. The follow-up walks through the bid in trades.
See it move
Same three dollar pairs. Change each rate and each spread.
- Raise EUR/USD. Both crosses rise: a euro that buys more dollars buys more yen and more pounds.
- Raise GBP/USD. EUR/GBP falls and EUR/JPY does not move. Raise USD/JPY: EUR/JPY rises and EUR/GBP does not move.
- Widen the EUR/USD spread. Both cross spreads widen, because both crosses are built through it. Widen GBP/USD and only EUR/GBP widens; widen USD/JPY and only EUR/JPY does.
Run it backwards
EUR/JPY and USD/JPY are on the screen. What EUR/USD do they imply?
EUR/JPY = EUR/USD × USD/JPY, so EUR/USD = EUR/JPY ÷ USD/JPY. Yen per euro divided by yen per dollar leaves dollars per euro.
Compare it with the screen. A gap of a fraction of a pip is rounding of EUR/JPY to two decimals. A real triangular arbitrage would need a gap bigger than all three spreads, and machines close those in milliseconds.
Traps
Say it in the interview
“EUR/USD is 1.0800 and USD/JPY is 150. What is EUR/JPY, and how would you make a two-way price?”
Check yourself
5 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Write the units. Dollar on opposite sides: multiply. Same side: divide.
- A two-way cross takes the worse side of each leg, so both spreads stack.
- Implied EUR/USD = EUR/JPY ÷ USD/JPY; sub-pip gaps are rounding.
- With EUR/USD fixed, EUR/JPY moves by the same percentage as USD/JPY.