25-delta butterfly (smile curvature)
The 25-delta butterfly measures the curvature of a volatility smile: the average implied volatility of the 25-delta call and the 25-delta put minus the at-the-money implied volatility. Together with the 25-delta risk reversal, which measures the slope, it summarises the smile of one expiry in two numbers.
Senzoukria · Glossary · Updated September 2026
At a glance
- Formula
- BF25 = (σ25C + σ25P)/2 − σATM
- Companion measure
- Risk reversal RR25 = σ25C − σ25P (slope)
- Reading
- Higher BF = wings priced richer than the centre
Slope and curvature
A smile can be summarised by three points: the at-the-money implied volatility and the implied volatilities of the 25-delta put and call. The risk reversal, call minus put, measures the tilt of the line joining the two wings. The butterfly measures how far the midpoint of the wings sits above the centre: a flat smile has a butterfly of zero, a pronounced smile a positive one. Using deltas rather than fixed strikes makes the measures comparable across expiries and volatility levels.
Worked example
One expiry quotes 22% for the 25-delta put, 15% for the 25-delta call and 17% at the money. The butterfly is (22 + 15)/2 − 17 = 1.5 volatility points. The risk reversal is 15 − 22 = −7 points, a strong put skew. Knowing the at-the-money level, the risk reversal and the butterfly, the two wings can be recovered: put = 17 + 1.5 − (−7)/2 = 22, call = 17 + 1.5 + (−7)/2 = 15.
| Measure | Formula | Value |
|---|---|---|
| ATM implied volatility | σATM | 17% |
| 25-delta risk reversal | σ25C − σ25P | −7 points |
| 25-delta butterfly | (σ25C + σ25P)/2 − σATM | +1.5 points |
What curvature says
- A rising butterfly means both wings gained relative to the centre: the market pays more for large moves in either direction than for moderate ones.
- Skew and curvature can move independently: puts can richen while the butterfly stays flat, if calls cheapen at the same time.
- Conventions differ. Some markets quote a 'market' or 'broker' butterfly defined on a strangle price rather than on implied volatilities; the two can differ materially for steep smiles.
- Some tools quote skew as put minus call, the opposite sign of the risk reversal.
In Senzoukria
Senzoukria does not compute a 25-delta butterfly. Its GEX module shows a 25Δ skew defined as put implied volatility minus call implied volatility at 25 delta, positive for put skew, which is the risk reversal with its sign reversed, and an ATM IV. When the provider does not publish them, the app computes the skew from the first expiry carrying enough implied volatilities on both sides, using Black-Scholes deltas, and says so. A butterfly also needs the two 25-delta implied volatilities separately, which the screen summarises only as their difference.
Related
In the same section
- Butterfly spread
- 0DTE options
- 80% rule
- Absorption
- Acceptance and rejection
- Activation fee
- Adverse selection
- Aggregated vs order-by-order
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Frequently asked questions
- Why use 25 delta and not a fixed strike?
- Because a fixed percentage distance is far in the wing for a one-week option and near the money for a one-year option. A 25-delta point is at a comparable probability-weighted distance for every expiry, so butterflies and risk reversals from different maturities can be compared.
- Can the butterfly be negative?
- Rarely for liquid index options, since the wings usually carry more implied volatility than the centre. A negative value would mean the at-the-money volatility exceeds the wings' average, which can happen briefly around binary events or with poor data.