Triangular Arbitrage
A riskless profit captured when three currencies’ cross rates are momentarily inconsistent.
Formula
Variables
Example
Rates misaligned by 0.2% → 0.2% riskless gain
How it works
Triangular arbitrage checks whether converting money through three FX markets brings you back with more than you started. The term inside the parentheses multiplies the exchange rates around the loop A → B → C → B; if markets are perfectly consistent, that product should equal 1 after normalizing for the direct A-to-B quote. Any value above 1 implies a profit opportunity, while a value below 1 means the round trip loses value and no arbitrage exists.
When to use it
Use this when you are comparing three related currency pairs and want to test whether their quoted rates are internally consistent. It is especially useful in FX trading, market-making, and exam problems involving cross rates or arbitrage detection.
Worked example, step by step
Suppose the quoted rates are: R_AB = 1.2500 (1 A buys 1.2500 B) R_AC = 0.9100 (1 A buys 0.9100 C) R_CB = 1.4200 (1 C buys 1.4200 B) Plug into the formula: Profit = ((1 / R_AB) × R_AC × R_CB) - 1 Profit = ((1 / 1.2500) × 0.9100 × 1.4200) - 1 Profit = (0.8000 × 0.9100 × 1.4200) - 1 Profit = 1.03376 - 1 Profit = 0.03376 So the triangular arbitrage profit is 0.03376, or 3.376% per cycle.
Common mistakes to avoid
- Using the wrong quote direction for one of the exchange rates
- Forgetting to subtract 1 at the end, which makes the result a growth factor instead of profit
- Ignoring transaction costs and bid-ask spreads, which can eliminate a theoretical arbitrage
Did You Know?
High-frequency trading firms close these gaps in milliseconds, so they rarely survive long enough to exploit by hand.
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More in Currency & Forex
View allCurrency Conversion
BasicConvert an amount from one currency to another using the quoted exchange rate.
Inverse Exchange Rate
BasicFlip a quoted rate to express it from the other currency’s point of view.
Currency % Change
BasicThe percentage move in an exchange rate between two points in time.
Cross Rate
IntermediateDerive the exchange rate between two currencies via a common third currency.
Bid-Ask Spread
IntermediateThe gap between the price to buy and to sell a currency — the dealer’s markup.
Spread Percentage
IntermediateThe bid-ask spread expressed as a percentage of the ask price.