Crypto markets

Triangular Arbitrage

Learn how triangular arbitrage works across three cryptocurrency pairs, with formulas, worked examples, trading fees and execution risks.

Mechanics & mathematicsHypothetical examples

What Is It?

Triangular arbitrage converts a starting asset through two other assets and back again, using three trading pairs. A USDT → BTC → ETH → USDT loop tests whether the direct and indirect exchange rates agree at prices that can actually be traded.

Temporary inconsistencies can reflect asynchronous order-book updates, uneven liquidity and order flow across markets. Mid-price inconsistencies are easier to find than executable ones because each conversion crosses the relevant bid or ask. Even a positive executable snapshot can disappear before all legs fill.

How It Works

For a BASE/QUOTE pair, the displayed price is quote currency per unit of base. Selling base earns the bid; buying base costs the ask. Thus buying BTC through BTC/USDT uses the reciprocal of its ask. Buying ETH through ETH/BTC also uses the reciprocal ask. Selling ETH through ETH/USDT uses its bid.

In a frictionless model a conversion cycle multiplies to one. In real order books, bid–ask spreads can make both directions unprofitable: the executable reverse cycle is not generally the reciprocal of the forward one. Evaluate each direction separately and size all three legs to the least liquid book.

Mathematical Formula

M_{\mathrm{gross}}=\frac{1}{a_{\mathrm{BTC/USDT}}}\,\frac{1}{a_{\mathrm{ETH/BTC}}}\,b_{\mathrm{ETH/USDT}}

Here a is the ask and b is the bid. Units cancel: BTC/USDT × ETH/BTC × USDT/ETH is dimensionless. Prices must represent the actual filled quantity, not only the best displayed unit.

Q_{\mathrm{end}}=Q_0 M_{\mathrm{gross}}\prod_{i=1}^{3}(1-f_i),\qquad R_{\mathrm{net}}=\frac{Q_{\mathrm{end}}}{Q_0}-1

This simplified fee model assumes each fee is proportional and deducted from the received asset. Fees paid in a separate token, rebates, rounding and minimum charges require a cash-flow calculation instead.

Worked Example

Illustrative prices · not market data

Assume sufficient depth at a BTC/USDT ask of 60,000 USDT per BTC, an ETH/BTC ask of 0.05 BTC per ETH, and an ETH/USDT bid of 3,020 USDT per ETH. Begin with 6,000 USDT.

Buy 0.1 BTC, then buy 2 ETH, then sell for 6,040 USDT before fees. The gross profit is 40 USDT, or 0.6667%.
With a hypothetical 0.1% received-asset taker fee on each leg: 6,040 × 0.999³ = 6,021.8981 USDT.
The fee-adjusted profit is 21.8981 USDT, or about 0.3650%. If additional slippage and rounding cost 15 USDT in total, the remaining profit is about 6.8981 USDT.

These are illustrative fills, not observed prices. A 25 USDT execution loss would instead turn the fee-adjusted edge into a loss. The displayed multiplier does not guarantee the final balance.

Execution Considerations

Walk each book to estimate volume-weighted execution rates. Apply tick-size, lot-size, minimum-notional and precision rules after each conversion; dust cannot necessarily be used by the next leg. Verify whether fees reduce the available base or quote balance.

Taker orders trade immediately against liquidity but pay the applicable taker schedule. Maker orders may reduce fees, yet introduce queue position, non-fill and adverse-selection risk. Three separate exchange orders are generally not an atomic transaction. Define how to unwind a partial loop, account for rate limits and timestamp the books consistently.

Risks & Limitations

Latency can change the second or third price after the first trade commits. Thin depth, competing orders, rejected quantities and connection loss can leave an unhedged BTC or ETH balance.

A single-exchange cycle avoids transfers during the loop but retains exchange custody and operational risk. Cross-exchange variants add transfer delays, withdrawal fees, inventory fragmentation and counterparty exposure. Never infer achievable size from a top-of-book percentage alone.

Frequently Asked Questions

Can I multiply the three quoted prices directly?

Only if every quote is already expressed as output units per input unit for its leg. In this example two markets require reciprocal asks, while the final sale uses a bid.

Does a negative forward edge imply a profitable reverse loop?

No. Bid–ask spreads and fees can make both loops lose money. Compute the reverse using its own executable sides.

Why can the scanner show profit but the trade lose?

The books can move, fees can differ from the assumed tier, or actual size can consume worse levels. A signal and a completed three-leg fill are different observations.

Explore the Existing Tool

Dashboard available · connection required

The triangular dashboard requires a connected arbitrage data service. Check its connection indicator before interpreting results. A displayed signal does not establish that all three trades can be executed.

Open Triangular Arbitrage Scanner

References & Further Reading

Background sources for the concepts above. Apply the assumptions stated in this guide to each formula.

Binance Spot API: Trading Filters

Educational material only; not investment advice or an offer to trade. Examples are hypothetical. Trading involves execution, financing and capital-loss risks; no return is guaranteed.