Currency cross-rates and triangular arbitrage

What is a cross-rate?

A cross-rate is the exchange rate between two currencies that is worked out from each currency’s rate against a third, instead of being quoted directly. Most currencies trade mainly against the US dollar, so the rate between, say, the Canadian dollar and the yen is usually built from their two dollar rates.

Recall the quoting rule. In an A/B quote, the price currency is A (the numerator) and the base currency is B (the denominator): A/B is the number of units of A that buy one unit of B. Write each quote as a fraction and the cross-rate follows from ordinary cancelling.

How do you calculate a cross-rate?

Line the quotes up so the common currency C cancels. What is left is the pair you want.

$$\frac{A}{B} = \frac{A}{C} \times \frac{C}{B}$$

What if a quote points the wrong way?

Sometimes the quote you hold is the reciprocal of the one you need. Invert it first.

$$\frac{C}{B} = \frac{1}{B/C}$$

Inverting one quote and multiplying is the same as dividing by that quote. So when the common currency plays the same role in both quotes, you divide.

Two quotes that share one currencywrite the target pair A/B, then make the common currency cancel

• Common currency plays opposite roles Memorize base in one quote, price in the other: multiply
  • (A/C) × (C/B) = A/B USD is the price of one and the base of the other
• Common currency plays the same role Memorize base in both, or price in both: divide
  • Same base: (A/C) ÷ (B/C) both quotes are per one USD
  • Same price: (C/B) ÷ (C/A) both quotes are priced in USD
• A quote points the wrong way Use it invert it with 1 ÷ x; the result is the same as dividing

$$\frac{A}{B} = \frac{A/C}{B/C} = \frac{C/B}{C/A}$$

Exam tip Sanity check: write the units, cancel, and the leftover must be the pair you want. If it reads B/A, flip your answer or one of your steps.

What is triangular arbitrage?

Use it

Anyone can see a quote for the cross-rate and the two quotes it is built from. The three prices must agree. If they do not, there is a riskless profit called triangular arbitrage (“triangular” because it uses three currencies). The base currency of the cross has two prices: the implied cross, and the dealer’s quote. Buy it where it is cheaper and sell it where it is dearer.

Dealer cross-rate against the implied cross-ratecompare the two prices of the base currency

• Dealer quote above the implied cross Use it the base currency is dear at the dealer
  • Buy the base currency through the common currency at the implied cross, the cheap route
  • Sell it to the dealer at the higher quoted price
• Dealer quote below the implied cross Use it the base currency is cheap at the dealer
  • Buy the base currency from the dealer at the lower quoted price
  • Sell it through the common currency at the implied cross, the dear route
• Profit per unit of base currency Use it the gap between the two prices, in the price currency

Human traders and algorithms watch for these gaps, so in practice they last for moments. For exam purposes, assume quoted crosses agree with the quotes they are built from, unless a question hands you a gap.

How do you find the percentage change in a cross-rate?

Use it

Treat a cross-rate as any other quote. Compute it at both dates and apply the base-currency formula from the previous module: new ÷ old − 1. Do not change one leg and ignore the other.

$$\%\Delta B = \frac{X_1}{X_0} - 1$$

Go deeper Why dealers work out crosses themselves

Know the idea

The two legs of a cross (for example dollar–Canada and dollar–yen) are usually traded by different desks. A dealer asked for a Canada–yen price calculates it from the legs, or an electronic platform does. A client normally gets the finished cross-rate, not the two quotes.

Go deeper Bid–offer spreads on crosses

Know the idea

Real quotes come in pairs, a bid and an offer. A cross built from two two-sided quotes is a little wider than a single price, because each leg adds its spread. This lesson (like the exam) uses one price per pair.

Go deeper Ranking currencies from cross-rates

Know the idea

To rank currencies against each other, put each one in the same role against the common currency (for example all as the base against the dollar) and compare the percentage changes. A currency that rises against the dollar while another falls against it is stronger than that other currency, and the cross-rate between them shows it.