Triangular Arbitrage Calculator

How this triangular arbitrage calculator works

Triangular consistency links three pairwise exchange rates across three currencies. Version 1.0.0 requires all three unordered pair connections exactly once. Starting from the selected currency, it constructs the two possible closed paths: start → currency A → currency B → start, and start → currency B → currency A → start.

Every leg applies the quote side required by the conversion direction. Moving from an entered pair’s base into its quote multiplies by the bid. Moving from its quote into its base divides by the ask. The audit table preserves the pair, side, amount before conversion, amount before fee, fee in that leg’s output currency, and resulting amount.

The optional fee is a simplified proportional deduction from every leg output. Fees across different output currencies are not added into one misleading total. The page shows each loop before and after that fee so you can see why a small gross quote inconsistency may disappear once even simplified transaction costs are introduced.

A careful interpretation workflow

Start by checking units, quote direction and observation alignment. Relative arithmetic is reproducible only when every input describes the intended instrument in the intended unit. Keep the original source and timestamp beside any result you save.

• Confirm that the three pairs cover exactly the three possible connections among the selected currencies.
• Use bid and ask, never midpoint rates, when auditing a direction that is intended to represent exchange.
• Check that the starting currency appears in the triangle and that the starting amount matches the scale you want to inspect.
• Treat the fee as one simplified proportional assumption; add venue-specific commission, spread, financing and withdrawal costs elsewhere.
• Inspect both directions because quote orientation alone does not reveal which closed path produces the higher entered result.
• Do not automate or place orders from this page; collect synchronized venue data and implement explicit risk controls in a separate tested system.

Assumptions and limits

• Exactly three finite, positive bid/ask quotes forming one complete three-currency graph are required.
• Each unordered pair may appear once; reversing the way a pair is typed is allowed but duplicating it is rejected.
• The proportional fee can range from zero through 10% and is deducted from every leg’s output amount.
• Spreads are represented by the entered bid and ask; slippage, minimum commission, rebates, tiers and financing are excluded.
• The page has no timestamps, order book, venue identity, executable size, latency or sequence-risk model.
• It assumes arithmetic completion of all three legs and cannot represent rejects, partial fills, capital locks or settlement constraints.
• A positive gross or net residual is not labelled an opportunity, recommendation, expected profit or achievable return.

Prepare comparable inputs before calculating

Record each source, venue, instrument label, observation time, price side, currency and unit. Bid and ask should refer to the same pair orientation and a comparable executable size if you are studying real quote relationships. Gold and silver prices should use the same currency, one troy ounce and as close to the same observation time as your purpose requires.

Do not silently replace bid and ask with a midpoint. A midpoint can be useful for descriptive research, but it removes the spread and cannot reproduce directional exchange. Likewise, do not mix a spot per-ounce metal price with a whole futures-contract notional. Normalize first, calculate second, and preserve the conversion notes.

The calculator validates finite positive decimals, bid/ask ordering, pair connectivity and the declared structural rules. It cannot detect stale browser input, delayed feeds, a mislabeled symbol, venue-specific quantity tiers, daylight-saving mismatches or whether two values came from different sessions. Those are data-governance checks that belong beside the arithmetic.

For any real trade decision, verify current broker or venue specifications, effective spreads, commission, minimum size, margin, financing, slippage, order type and jurisdictional availability. An auditable calculator can remove arithmetic ambiguity, but it cannot transform manual observations into a safe or suitable transaction.

Sources and methodology

• CFA Institute — Currency Exchange Rates — Explains triangular arbitrage relationships and bid/ask quote mechanics.
• CFA Institute — Exchange Rate Calculations — Documents cross-rate and bid/ask calculations used on each path.
• Bank for International Settlements — FX intermediation working paper — Shows a triangular-arbitrage example and why transaction costs matter.

The operational contract is Entered Cross-Market Quotes version 1.0.0. Independent fixture oracles test bid/ask direction, loop traversal, fee placement, ratio normalization, inverse relationships, input scaling, equality boundaries, validation and production isolation. External sources define the broader financial relationships; the disclosed versioned rules on this page govern the output.