Starting Currency and Profit in Triangular FX Arbitrage

Summary

The note considers whether the profit from a three-currency FX arbitrage depends on which currency starts the conversion cycle. It represents the cycle with exchange rates for EUR/USD, USD/JPY, and JPY/EUR, and identifies an arbitrage when their product differs from one.

Changing the starting currency reorders the same exchange-rate factors, so under the document’s simplified multiplicative setup, the cycle’s profit condition is unchanged. The answer is brief and gives no worked numerical example. It does not discuss bid-ask spreads, transaction costs, liquidity, or execution constraints, all of which can affect realized returns in actual FX markets.

Key ideas

• A triangular FX cycle is mispriced when the product of its three exchange rates differs from one.
• Changing the starting currency reorders the factors in the exchange-rate product.
• The note’s equivalence assumes a simplified setup and does not account for trading costs or execution.

Tags

# Does the starting currency matter in triangle arbitrage?


# Does the starting currency matter in triangle arbitrage?












Say I have an FX arbitrage opportunity by doing a transaction like: EUR -> USD -> YEN -> EUR

It seems to me like the (exact) same profit can be realized by starting the transaction in USD and doing: USD -> YEN -> EUR -> USD

Did I miss something?

## Answer by M. Jeunesse (score 3, accepted)

https://quant.stackexchange.com/a/26073

Let XXXYYY be the forex for currency pair XXX/YYY

I assume that when you say you have an arbitrage, you say that :

$$ \text{EURUSD} \times \text{USDYEN}\times\text{YENEUR}\neq 1$$

As you see, starting from another currency is just changing the order of the product, so you did not miss something.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.