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Detecting Triangular Arbitrage from Three Currency Quotes

Article Quant Q&A · Author: user4933
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Summary

The document explains how to test three exchange quotes for a triangular arbitrage opportunity. It identifies the currencies in each quote, derives the implied RUB-per-TRY cross rate by dividing the RUB/USD rate by the TRY/USD rate, and compares that result with the bank’s direct RUB/TRY quote. The example then traces a $1 million investment through USD to TRY, TRY to RUB, and RUB back to USD, showing a small gain from completing the loop.

The calculation illustrates that quote direction matters: follow the units through each conversion, or test both directions around the currency triangle. The example’s stated profit assumes the quoted rates can be traded as given. It does not account for bid-ask spreads, transaction costs, taxes, execution timing, or whether the available market depth supports the trade, so the apparent discrepancy alone does not establish a realizable arbitrage.

Key ideas

  • Derive the implied cross rate from the two quotes that share a currency, then compare it with the direct quote.
  • Track the currency units at each conversion to keep the trade direction clear.
  • Test both directions around the currency triangle when the profitable route is uncertain.
  • The example’s apparent gain excludes transaction costs, taxes, and execution constraints.

Tags

ForexArbitrage
Full text
# Simple three-pair triangulation question


# Simple three-pair triangulation question












I have a question I came across whilst self-studying and I need to use cross-currency triangulation. I am not too sure how to apply the cross-rate formula, and was hoping someone could show me how to apply it here.

> As a market trader you are looking for an arbitrage opportunity. Using the following cross rates given by the banks you are working with, determine whether there is a triangular arbitrage opportunity for the US$ 1 million that you would like to invest. Bank A quotes: RUB 64.42 /USD Bank B quotes: TRY 4.48 /USD Bank C quotes: RUB 14.39 /TRY

Thank you in advance for any help provided.

## Answer by AKdemy (score 0, accepted)

https://quant.stackexchange.com/a/68069

RUBUSD - how many RUB per USD TRYUSD - how many TRY per USD RUBTRY - how many RUB per TRY

RUBUSD/TRYUSD = RUBTRY 64.42/4.48 = 14.3795 which is less RUB per TRY compared to the direct RUBTRY quote.

Use your USD to get TRY - gives 4.48M Use TRY to get RUB - gives 64.4672M Use RUB to buy USD - gives ~ 1.000.732,692 USD

If you are unsure about the direction, you can simply try going around the circle in either direction. One way you will end up with a profit, the other with a loss.

There will be numerous online resources for questions like this if you spend some time with google. Searching cross triangle arbitrage should show Investopedia's Triangular Arbitrage Definition - Investopedia as the first result. This page also has a working example with EUR / GBP and USD:

EUR/USD = 0.8631, EUR/GBP = 1.4600 and USD/GBP = 1.6939. With these exchange rates there is an arbitrage opportunity:

```
Sell dollars for euros: $1 million x 0.8631 = €863,100
Sell euros for pounds: €863,100 ÷ 1.4600 = £591,164.40
Sell pounds for dollars: £591,164.40 x 1.6939 = $1,001,373
Subtract the initial investment from the final amount: $1,001,373 – $1,000,000 = $1,373
From these transactions, you would receive an arbitrage profit of $1,373 (assuming no transaction costs or taxes).
```

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