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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?

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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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