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gateway/knowledge/trading/strategies/fx/triangular-arbitrage.md

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+---
+description: "FX triangular arbitrage that exploits momentary mispricing across three currency pairs by executing a circular chain of exchanges to lock in a riskless profit."
+tags: [fx, arbitrage, triangular, market-microstructure]
+---
+
+# FX Triangular Arbitrage
+
+**Section**: 8.5 | **Asset Class**: FX | **Type**: Arbitrage
+
+## Overview
+Triangular arbitrage involves three currencies (A, B, C) and three currency pairs. If the cross-rate implied by trading A→B→C→A differs from the direct rate, a riskless profit exists. Such mispricings are extremely short-lived and require fast market data feeds and execution systems.
+
+## Construction / Mechanics
+Given three currencies A, B, C and their bid/ask prices, there are two chains:
+1. A → B → C → A
+2. A → C → B → A (equivalent to swapping B and C in chain 1)
+
+Focus on chain 1. The relevant rates are:
+- Bid(A→B): rate at which A is exchanged into B
+- Bid(B→C): rate at which B is exchanged into C
+- Ask(C→A): rate at which C is exchanged back into A (cost = 1/Ask(C→A))
+
+Note: Bid(B→A) = 1/Ask(B→A) and Ask(A→B) = 1/Bid(A→B).
+
+The overall round-trip exchange rate for chain 1 is:
+
+```
+R(A→B→C→A) = Bid(A→B) × Bid(B→C) × (1 / Ask(C→A))         (453)
+```
+
+**Profit condition:** If R(A→B→C→A) > 1, the trader profits by executing all three legs simultaneously. Starting with 1 unit of A, the trader ends with R > 1 units of A.
+
+## Return Profile
+The P&L per unit of A is R - 1 when R > 1. Profits are typically very small per trade (fractions of a pip) but can be accumulated at high frequency. The strategy is notionally riskless if all three legs are executed simultaneously; execution lag introduces market risk.
+
+## Key Parameters / Signals
+| Parameter | Description |
+|-----------|-------------|
+| R(A→B→C→A) | Round-trip rate; > 1 signals a profitable arbitrage |
+| Bid(A→B), Bid(B→C) | Bid rates for the two intermediate legs |
+| Ask(C→A) | Ask rate for the closing leg |
+| Execution speed | Critical: mispricings disappear in milliseconds |
+
+## Variations
+- **Multi-currency arbitrage**: extend to more than 3 currency pairs (N-currency chains); computational complexity increases but can uncover deeper mispricings.
+- **Chain 2** (A→C→B→A): identical logic with B and C swapped; both chains should be monitored simultaneously.
+
+## Notes
+- Opportunities are ephemeral; the strategy is effectively a latency arbitrage and is dominated by high-frequency traders with co-located infrastructure.
+- Bid-ask spreads are the primary cost; even small spreads can eliminate the theoretical profit, so only very tight markets are viable.
+- Unlike statistical arbitrage, this is a near-deterministic arbitrage: no model risk, but significant execution/technology risk.
+- Applicable to crypto exchanges where cross-exchange and cross-pair mispricings can be larger and more persistent than in professional FX markets.