Option-chain structure

Butterfly Arbitrage

Derive the nonnegative butterfly payoff, test executable strike-price convexity and distinguish a mispriced butterfly from a speculative butterfly trade.

Mechanics & mathematicsHypothetical examples

What Is It?

Butterfly arbitrage tests whether a portfolio with a nonnegative terminal payoff can be bought for a credit. For equally spaced European strikes, a long call butterfly buys one lower-strike call, sells two middle-strike calls and buys one upper-strike call. All contracts must have the same underlying, expiry, multiplier and settlement.

An ordinary butterfly spread usually costs a debit and expresses a view about terminal price or volatility. Its tent-shaped payoff does not make it an arbitrage. The special violation is an executable negative purchase cost after fees, together with the valid nonnegative-payoff construction.

How It Works

For strikes 95, 100 and 105, the payoff is zero below 95, rises to 5 at 100, falls back to zero at 105 and stays zero above. It is never negative at European expiry. A negative price would pay the trader to acquire this terminal claim in the frictionless model.

This is a discrete convexity test across strike. The 1:−2:1 weighting requires equal spacing. Unequally spaced strikes need weights that put the middle strike at the weighted average of the outer strikes; blindly keeping the same contract counts can leave a directional tail payoff.

Mathematical Formula

B=C(K_1)-2C(K_2)+C(K_3)\ge0,\quad K_2=\frac{K_1+K_3}{2}

Same-expiry European call values are convex in strike. The same butterfly inequality holds for matching European puts. It follows from the pointwise convexity of their payoffs.

X_T=(S_T-K_1)^+-2(S_T-K_2)^++(S_T-K_3)^+\ge0

For equal strike spacing the payoff peaks at K₂ with height K₂ − K₁ and vanishes outside the two wings.

B_{\mathrm{exec}}=C_{\mathrm{ask}}(K_1)-2C_{\mathrm{bid}}(K_2)+C_{\mathrm{ask}}(K_3)

An executable credit requires this cost plus fees and other implementation costs to be negative. A middle-strike sale needs enough bid depth for twice the wing quantity.

Worked Example

Illustrative prices · not market data

Assume European call asks at strikes 95 and 105 are 8.00 and 2.90. At strike 100, two units can be sold at a 5.50 bid.

Buy the wings and sell two bodies: 8.00 − 2 × 5.50 + 2.90 = −0.10. The gross opening credit is 0.10 per underlying unit.
With total opening and anticipated settlement costs of 0.06 measured today, the modeled net credit is 0.04.
At settlement prices 90, 97, 100, 103 and 110, the respective butterfly payoffs are 0, 2, 5, 2 and 0.
A hypothetical 100-unit contract multiplier makes the initial net credit 4.00 for a 1:−2:1 spread, while the maximum terminal payoff is 500.

The positive terminal payoff is not a fixed additional profit: it depends on the settlement price. The theoretical arbitrage argument comes from a positive opening net credit combined with no negative terminal payoff.

Execution Considerations

Use a package limit order where supported, or plan for incomplete fills. Count commissions on four option units even though there are only three distinct strikes. Include exchange exercise and settlement charges, and scale book depth for the two short middle contracts.

Check expiry matching and exercise style before applying the static payoff. For uneven strikes, use the weighted construction in the convexity guide and scale fractional weights to feasible integer contracts; minimum trade sizes can defeat the theoretical spread.

Risks & Limitations

A quote can be crossed or stale rather than fillable. Selling two middle contracts can consume deeper, worse bids. Partial fills leave delta, gamma and volatility exposure.

American assignment can create stock and cash-flow obligations before the offsetting wings exercise. Physical-settlement pin risk, fees and margin pressure can invalidate a frictionless nonnegative-payoff interpretation in practice.

Frequently Asked Questions

Is buying a butterfly for a small debit arbitrage?

No. The entire debit can be lost if the options finish outside the wings. Arbitrage requires a valid violation with executable prices after costs.

Can I use the same 1:−2:1 counts on uneven strikes?

Not as a general convexity test. Use strike-distance weights and verify the complete payoff.

Does the test require a volatility forecast?

The European terminal-payoff inequality does not. Execution and interim risk still require realistic market assumptions.

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References & Further Reading

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

NYU Stern, Aswath Damodaran: Options Arbitrage

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.