Option-chain structure

Convexity Arbitrage

Understand call and put convexity across strikes, weighted butterfly construction and static no-arbitrage constraints without confusing them with volatility trades.

Mechanics & mathematicsHypothetical examples

What Is It?

Option convexity across strike means the middle-strike price cannot exceed the appropriate interpolation of the outer-strike prices for matching European contracts. A violation can create a weighted butterfly purchased for a credit despite its nonnegative terminal payoff.

Strike convexity is a static no-arbitrage condition. It differs from gamma, which measures curvature with respect to the underlying price, and from trading a volatility forecast. A volatility strategy can lose money without violating any pricing identity.

How It Works

For a fixed terminal asset price, a vanilla call payoff is convex as a function of strike. Taking a positive discounted expectation preserves that property. Puts have the same strike convexity because their payoff is also pointwise convex.

Choose weights so the middle strike equals the weighted outer strikes. Buy that weighted combination of wings and sell one middle option. Convexity of the payoff makes this position nonnegative at expiry. Equal spacing gives half a lower wing plus half an upper wing minus one middle option, equivalent after doubling to the familiar 1:−2:1 butterfly.

A full option chain also needs nonnegative prices, strike monotonicity and bounded vertical-spread slopes. A convex curve alone can still violate absolute bounds or parity. Smooth interpolation should respect these constraints rather than create artificial signals between observed strikes.

Mathematical Formula

\lambda=\frac{K_3-K_2}{K_3-K_1},\qquad K_2=\lambda K_1+(1-\lambda)K_3

For K₁ < K₂ < K₃ the weight λ lies strictly between zero and one. All contracts must otherwise match.

C(K_2)\le\lambda C(K_1)+(1-\lambda)C(K_3)

A matching weighted butterfly must have a nonnegative price in the frictionless European model. The same inequality applies to puts.

-D(0,T)\le\frac{C(K_2)-C(K_1)}{K_2-K_1}\le0

A European call vertical is worth between zero and the discounted strike difference under deterministic discounting. Across increasing strikes, these secant slopes must also be nondecreasing for convexity.

Worked Example

Illustrative prices · not market data

Consider strikes 90, 100 and 120 with the same European expiry. Lower and upper call asks are 15 and 2, while the middle bid is 11.

The lower-wing weight is (120 − 100) / (120 − 90) = 2/3; the upper-wing weight is 1/3.
The executable weighted cost is (2/3) × 15 + (1/3) × 2 − 11 = −1/3.
Scale by three to buy two 90 calls, sell three 100 calls and buy one 120 call. The gross credit is 1 per underlying unit of the scaled package.
Assuming aggregate costs of 0.40 for that package, the net opening credit is 0.60. Its terminal payoff is nonnegative and peaks at 20 when the underlying is 100.

Integer scaling makes the hedge feasible in contract counts but also increases fees and required depth. A negative mid-price interpolation gap alone is insufficient.

Execution Considerations

Evaluate raw executable prices before fitting an implied-volatility surface. A visually curved volatility smile is not itself a violation: convexity concerns option price as a function of strike, not the appearance of implied volatility.

Use the correct deliverables and multipliers, enforce contract-count constraints and check all legs' quote times. This topic is educational and has no dedicated scanner. The related butterfly article links to the options interface and explains its connection requirements.

Risks & Limitations

Sparse strikes, interpolation error, stale bids, minimum sizes and margin can make an estimated convexity gap nontradeable. A weighted portfolio must retain the exact strike relationship after rounding contract counts.

Static European expiry arguments do not eliminate early-assignment exposure in American contracts or the risk of failed execution. Gamma scalping and volatility relative-value trades additionally depend on path, hedging costs and model assumptions.

Frequently Asked Questions

Is convexity the same as a volatility smile?

No. Convexity constrains prices across strikes. The implied-volatility smile is a transformation of those prices using a chosen pricing convention.

Why are unequal strike distances important?

The weights must reproduce the middle strike. Using equal counts on uneven gaps can change tail exposure and break the nonnegative-payoff argument.

Does a convex option surface have no arbitrage?

Convexity is necessary but insufficient. Check absolute bounds, vertical slopes, parity and appropriately conditioned maturity relationships too.

Platform Availability

This article explains a financial relationship. No dedicated scanner is claimed for this topic. Explore the related articles for existing tools and their stated limitations.

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.