Triangular Arbitrage

EUR/USD, USD/JPY, and EUR/JPY are quoted by different desks as if they were unrelated markets. They are not. Because dollars appear in the first two and euros and yen appear in all three, the rate you get converting euros to yen directly must match the rate you get converting euros to dollars and then dollars to yen. If the two disagree, a trader can walk a euro around the triangle — into dollars, into yen, back into euros — and come out with more euros than they started with, for no risk.

Why the rates are forced to agree

The no-arbitrage condition says the direct EUR/JPY quote must equal the rate implied by going through dollars:

$$(\text{EUR/JPY})_{implied} = (\text{EUR/USD}) \times (\text{USD/JPY})$$

In words: how many yen a euro buys, worked out the roundabout way through dollars, has to equal how many yen it buys directly. If it doesn't, the "long way round" and the "direct way" are two different prices for the same conversion, and anyone can buy at the cheap one and sell at the rich one.

Worked example

Quotes: EUR/USD = 1.0850, USD/JPY = 155.00, direct EUR/JPY = 168.80. The implied cross is $1.0850 \times 155.00 = 168.175$ — the direct quote of 168.80 is richer than that, so euros are worth more buying JPY directly than routing through dollars. Start with JPY 168,175,000 and take the synthetic route first, then sell the resulting euros directly:

1. JPY 168,175,000 → USD at 155.00: $168{,}175{,}000 / 155.00 = 1{,}085{,}000$ dollars.
2. USD 1,085,000 → EUR at 1.0850: $1{,}085{,}000 / 1.0850 = 1{,}000{,}000$ euros.
3. EUR 1,000,000 → JPY at the direct rate 168.80: $1{,}000{,}000 \times 168.80 = 168{,}800{,}000$ yen.

Started with JPY 168,175,000, ended with JPY 168,800,000 — a risk-free JPY 625,000 (roughly $4,000) for three simultaneous trades, before transaction costs.

What this means in practice

Human traders haven't captured this kind of edge in decades; it's the province of low-latency algorithms scanning all three legs and firing the instant a gap opens, closing it before a retail platform's quote even refreshes. What the concept still buys you is intuition: it's why cross-rates you never trade directly (say, AUD/CHF) still move exactly as if two liquid legs against the dollar were multiplied together, and it's a standard mental-math interview question — give two rates, ask for the consistent third.