Triangular FX Arbitrage Requires Executable Bid and Offer Quotes

Summary

The document explains how exchange rates across three currencies are related in a triangular arbitrage calculation. Given any two rates, the third cross rate is implied by their relationship, so traders can compare a quoted rate with the rate implied by the other two. The example asks whether one of two candidate rates is the relevant cross rate, illustrating the confusion that can arise when rate conventions are unclear.

The response emphasizes that indicative levels alone cannot establish an arbitrage opportunity. Each dealer's bid and offer are needed to know the prices at which the currency trades can actually be executed. A difference between an implied rate and a quote may disappear once bid–ask spreads and transaction costs are considered; the stated example notes that a sufficiently large half-spread could consume the apparent gross profit. The document does not provide a complete execution sequence or account for other frictions such as latency, funding, or market impact, so its central lesson is about quote inputs and spread-aware evaluation.

Key ideas

• Any two exchange rates among three currencies imply a cross rate for the third pair.
• A theoretical discrepancy does not prove that a triangular arbitrage can be executed profitably.
• The bid and offer for each dealer quote are required to evaluate executable trades.
• Bid–ask spreads and transaction costs can eliminate apparent gross arbitrage profit.

Tags

# How to determine the cross rate in a triangular arbitrage


# How to determine the cross rate in a triangular arbitrage












I am very confused about what two currencies are to be chosen as the cross rate in a triangular arbitrage.

For example, when the bank quotes are

- ¥180/£

- $1.5/£

- ¥130/$

Does the cross rate have to be ¥120/$?

Couldn't it be $1.38/£?

Can I use any two currencies to calculate the profit?

## Answer by Ezy (score 1, accepted)

https://quant.stackexchange.com/a/43271

Any 2 rates imply the third.

Now what is missing here is that you did not provide actual quotes. A quote is the given of a bid and offer values. If you don’t know each dealer’s bid and offer there is no way to assert whether the levels offer an arbitrage opportunity or not. You could see an implied 1.38 level against a fair value of 1.5 however if the bid/offer half-spread is say 0.2 then this spread opportunity would not be a real one as the expected gross profit would be less than transaction costs.

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