Forex Triangular Arbitrage Gap Calculator
Dislocations between three related currency pairs can create a fleeting profit opportunity called triangular arbitrage. This calculator compares a direct cross rate with the synthetic rate implied by two other pairs while deducting fees and slippage, so you can decide whether to execute a loop trade. It complements the forex pip value calculator and the margin call distance calculator by ensuring that any projected pip gains exceed the friction inherent in three simultaneous fills.
Triangular arbitrage in practice
Foreign exchange markets trade currency pairs that relate one currency to another. If you multiply the price of A/B by B/C, you obtain an implied price for A/C. When the actual quoted A/C price diverges from the implied price, you can simultaneously trade all three pairs to lock in a profit. Executing this strategy requires speed, access to competitive spreads, and awareness of transaction costs. AgentCalc already provides tools for sizing trades (through the pip value calculator) and for protecting positions (via the margin call distance calculator). The triangular arbitrage gap calculator completes the toolkit by checking whether the apparent mispricing survives after fees and slippage.
The calculator evaluates two loops. Loop 1 starts with currency A, sells it for B using the A/B bid, uses B to buy C via the B/C bid, and then converts C back to A using the A/C ask. Loop 2 mirrors the process in the opposite direction: start with A, buy C at the A/C bid, trade C for B at the C/B bid (equivalently 1 divided by the B/C ask), and finish by buying A with B at the A/B ask. Each leg multiplies the notional by the relevant rate after deducting transaction costs and slippage expressed in basis points. The final amount compared with the starting notional reveals the profit or loss.
The synthetic cross rate generated by the other two pairs is compared to the direct cross rate to quantify the raw mispricing. A positive gap does not guarantee profit because each leg erodes value through spreads and fees. The calculator therefore subtracts an “all-in cost factor” from each rate so you can test whether the mispricing is large enough to overcome friction. This is especially helpful for retail traders whose per-trade costs are higher than institutional desks.
Mathematics behind the gap
Suppose you quote pair A/B with bid price and ask price . Likewise, pair B/C has bid and ask , and pair A/C has bid and ask . Ignoring costs, the synthetic ask for A/C inferred from A/B and B/C is , while the synthetic bid equals . A triangular arbitrage opportunity exists when the actual bid/ask for A/C lies outside this synthetic interval. After deducting basis-point costs, the calculator computes the final capital after each loop via , where aggregates per-leg cost factors converted from basis points.
Costs in basis points are converted into multiplicative adjustments via . Each leg multiplies the traded amount by this factor to model commissions and slippage. Because costs accumulate, even a seemingly attractive mispricing may vanish after three deductions. The calculator surfaces both the net payoff and the breakeven mispricing—the percentage gap that would reduce your profit to zero.
Worked example
Imagine you monitor EUR/USD (A/B), USD/JPY (B/C), and EUR/JPY (A/C) quotes. EUR/USD trades at 1.06620/1.06626, USD/JPY at 149.52/149.54, and EUR/JPY at 159.35/159.39. You can start with €100,000. Transaction costs are 0.4 bps per leg with expected slippage of 0.2 bps per leg because of fast markets. Feeding these quotes into the calculator shows that Loop 1 (sell EUR for USD, USD for JPY, JPY for EUR) returns approximately €100,018 after costs, a profit of €18 or 1.8 pips. Loop 2 yields a slight loss because the actual cross is inside the synthetic interval for that direction. The tool additionally reports a breakeven gap of 0.00018 (1.8 pips); if the dislocation were any smaller, fees would erase the edge.
The output also includes the synthetic bid/ask interval for EUR/JPY and the percentage mismatch between the actual quote and the implied value. These diagnostics help confirm whether latency or stale quotes caused the opportunity. The net profit is displayed both in currency A and as a percentage of the starting notional, enabling comparisons across trade sizes. If you run the calculator multiple times per day, you can build a log of how often your broker quotes actionable mispricings versus merely theoretical ones.
Comparison of arbitrage conditions
Triangular arbitrage windows vary dramatically depending on spreads and volatility. The table contrasts three common scenarios using the same €100,000 notional. In each case the calculator deducts 0.6 bps in combined costs per leg.
| Market state | Raw Mispricing (pips) | Net Profit (€) | Execution comment |
|---|---|---|---|
| Calm Asian session | 0.7 | -4 | Spreads dominate; avoid trade. |
| Moderate London open | 1.8 | 12 | Viable if fills are simultaneous. |
| Volatile data release | 3.6 | 42 | Large edge but higher slippage risk. |
Logging these scenarios clarifies whether your infrastructure and cost structure support arbitrage. If most opportunities fall below the breakeven threshold, improving execution speed or negotiating tighter spreads might yield a bigger payoff than chasing more signals.
Interpreting the outputs
The calculator summarizes the direct versus synthetic cross rates, loop profits in both directions, and breakeven gap. You also see the all-in-cost percentage per leg so you can double-check commissions and market-impact assumptions. A warning appears if any calculated rate is non-positive, signaling inconsistent inputs. Because arbitrage requires near-simultaneous execution, the tool emphasizes percentage payoff so you can compare it with latency risk. A 0.02% edge can disappear during a single 100 ms delay.
Beyond immediate trading, the calculator aids in planning. Suppose you run a carry trade and want to hedge currency risk intraday. Evaluating triangular gaps reveals whether hedging is cheaper through direct crosses or synthetic paths. Similarly, algorithm designers can plug in historical spread data to stress-test whether their strategy would have cleared costs.
Staying cautious
Triangular arbitrage is low risk only when all three legs confirm and clear simultaneously. Retail brokers may reject one leg, leaving you with an unintended open position. Use the breakeven gap to set alarms that exceed your minimum fill probability. Consider automating the input feed from your trading platform to minimize manual entry errors. Always test with small notionals before scaling.
With disciplined monitoring, this calculator helps ensure that every arbitrage attempt is deliberate. Combined with AgentCalc’s existing forex suite, you can evaluate trade sizing, financing, margin, and execution quality within a single ecosystem.