SVQ · No-arbitrage
Butterfly arbitrage
Also called: convexity arbitrage, negative butterfly, strike arbitrage
Butterfly arbitrage exists when option prices are not convex in strike, so a butterfly spread would cost less than zero to own.
Definition
Butterfly arbitrage is a violation of convexity of option prices in strike. If call prices for one expiry are not convex as a function of strike, a long butterfly spread (buy one low strike, sell two middle, buy one high) would have a negative cost while its payoff can never be negative.
Equivalently, it means the implied risk-neutral density is negative somewhere, which no consistent pricing model allows.
Intuition
A butterfly spread pays off only if the underlying finishes near the middle strike, and never loses money at expiry. Its price must therefore be non-negative, and in the limit of tight strikes it is proportional to the probability density of finishing at that strike.
In practice, apparent butterfly arbitrage in listed markets usually comes from stale or one-sided quotes, wide spreads or a poorly constrained surface fit, rather than a genuine opportunity. That is why it is mainly a consistency check on data and models.
How it's calculated
For equally spaced strikes K − h, K, K + h with call prices C:
C(K − h) − 2C(K) + C(K + h) ≥ 0
In the continuous limit this is ∂²C / ∂K² ≥ 0, and by Breeden–Litzenberger the discounted density is q(K) = erT ∂²C/∂K², so convexity and a non-negative density are the same condition.
On a fitted implied volatility smile, the check is done in total-variance space with the Durrleman condition, which tests whether the density implied by the fit stays non-negative at every log-moneyness.
Example: calls at 95, 100 and 105 priced 7.00, 4.60 and 2.10 give a butterfly cost of 7.00 − 9.20 + 2.10 = −0.10, a convexity violation worth checking against the underlying quotes before anything else.
How to read it
A flagged butterfly violation is first a question about the data: are the quotes two-sided, recent and liquid? If they are not, the violation is likely noise. If the fitted surface shows it but the raw quotes do not, the fit is at fault.
Persistent violations on liquid strikes would be notable, but transaction costs, the bid-ask spread and early-exercise features of American options can all explain small apparent breaches.
Common mistakes
- Treating every flagged violation as a trade. After costs and with model marks rather than tradeable quotes, most disappear.
- Testing mid prices only; the condition should hold for the prices you could actually trade at.
- Ignoring early exercise on American-style options, which changes the bounds slightly.
- Fixing a fit by deleting strikes until the violation disappears instead of constraining the model.
In Saqrid
SVQ's Arb Scanner flags butterfly violations, along with calendar, vertical and Durrleman density violations, as model-consistency checks and says whether each looks real or like a fitting artifact.
Open SVQ.
Go deeper
Saqrid Institute lesson: No-Arbitrage in Practice: The Arb Scanner. Lesson text is open to read without an account.