# Zomma Greek: How Implied Volatility Changes Gamma
Understand Zomma as Gamma's sensitivity to implied volatility, derive its Black-Scholes form, normalize units, and use full repricing for event and surface risk.
Canonical: https://wiki.fcontext.com/options/zomma-greek/
Fact checked: 2026-07-22
> For educational purposes only; not investment advice.
## Direct answer
**Zomma** measures how an option's Gamma changes when implied volatility changes, holding the model's other inputs fixed:
`Zomma = ∂Gamma/∂σ = ∂³V/(∂S²∂σ)`
Gamma describes the local change in Delta for a small underlying-price move; Zomma describes how that local curvature is itself reshaped by an IV change. It is a higher-order, model-dependent sensitivity—not a separately traded cash flow, a directional forecast, or proof of what Gamma will be after a large market move.
## Formula, sign, and units
For a continuously compounded Black-Scholes-style European option with dividend yield `q`:
`Gamma = e^(−qT) φ(d₁) / (Sσ√T)`
`Zomma = Gamma × (d₁d₂ − 1) / σ`
where:
`d₁ = [ln(S/K) + (r − q + σ²/2)T] / (σ√T)`
`d₂ = d₁ − σ√T`
Calls and Puts with the same strike and expiry have the same Gamma and Zomma under this model. Near ATM, `d₁d₂` is often below `1`, so Zomma is commonly negative: raising IV spreads Delta's transition over a wider price range and lowers peak Gamma; lowering IV can concentrate Gamma near the strike. Far from ATM, the sign can differ.
Units must be stated. If Gamma is Delta change per `$1` underlying move and `σ` is decimal annualized IV, Zomma is Gamma change per `1.00` IV change. A platform quoting change per one volatility point (`1% = 0.01`) should display `Zomma_decimal / 100`. Some systems also scale Gamma, spot, multiplier, or percentage moves, so raw vendor numbers are not comparable without conventions.
Zomma is a partial derivative: spot, time, rates, dividends, and the rest of the surface are held fixed. Real markets usually move several inputs together. Spot-IV dependence, skew movement, elapsed time, jumps, and early-exercise features can dominate this isolated sensitivity.
## Local estimate versus full repricing
Suppose one option has:
- Gamma `0.040` per `$1` underlying move;
- Zomma `−0.060` per `1.00` decimal IV;
- standard multiplier `100`.
For a small IV increase from `40%` to `41%`, `Δσ = +0.01`:
`ΔGamma ≈ Zomma × Δσ = −0.060 × 0.01 = −0.0006`
Estimated new Gamma is `0.0394`. Per one standard contract, the Gamma-scaled Delta change for a `$1` move shifts from about `4.00` to `3.94` share-equivalents, under the stated local convention.
Now consider an event IV drop from `80%` to `35%`, so `Δσ = −0.45`. Linear extrapolation would give:
`ΔGamma ≈ −0.060 × (−0.45) = +0.027`
and an estimated Gamma of `0.067`. That is not a reliable event forecast: the shock is large, Zomma changes along the path, time passes, spot can gap, and skew can reprice. Recalculate the full option and portfolio under the post-event surface; use the linear number only to understand the initial direction and rough scale.
## Analysis and risk controls
- Record valuation model, exercise style, spot or forward convention, rates, dividends, time clock, IV source, and surface interpolation.
- State whether IV input is decimal, percentage, or volatility points and whether Gamma/Zomma include the contract multiplier.
- Verify Zomma with central differences: reprice Gamma at `σ + h` and `σ − h`, then compare `[Γ(σ+h) − Γ(σ−h)]/(2h)` across several `h` values.
- Recompute each strike and expiry; do not multiply one representative Zomma across a heterogeneous portfolio.
- Shock spot, time, IV level, skew, and term structure jointly, especially around events and near expiration.
- Use full repricing for large IV moves, price gaps, barriers, discrete dividends, early exercise, or sparse surface wings.
- Aggregate long/short sign, quantity, multiplier, and currency consistently; net Zomma can conceal offsetting bucket risk.
- Compare predicted and realized changes in Gamma, Delta, and P&L; unexplained residual is model, data, surface, or execution risk.
- Prioritize Delta, Gamma, Vega, liquidity, maximum loss, and settlement before relying on a third-order sensitivity.
- Read product specifications and OCC disclosures; Zomma does not alter contractual exercise, assignment, or margin obligations.
## Common misconceptions
- **“Zomma predicts implied volatility.”** It measures a conditional sensitivity; it does not forecast IV direction.
- **“Negative Zomma is bearish.”** The sign describes Gamma's response to IV, not the underlying's expected return.
- **“Zomma is the same as Vega.”** Vega changes option value with IV; Zomma changes Gamma with IV.
- **“The same-strike Call and Put have opposite Zomma.”** In the stated European model they share Gamma and Zomma.
- **“A one-point IV move means `1.00`.”** One volatility point is normally `0.01` in decimal-IV input.
- **“One vendor's number can be copied into another system.”** Scaling, model, surface, clock, and dividend assumptions differ.
- **“Linear Zomma explains an IV crush.”** A large event shock requires full repricing along a changed surface.
- **“More Greeks automatically improve risk control.”** A precise higher-order number cannot repair bad prices, liquidity, sizing, or expiration planning.
## Related topics
- [Gamma Risk](/options/gamma-risk/)
- [Vega Risk](/options/vega-risk/)
- [Ultima Greek](/options/ultima-greek/)
- [Color Greek](/options/color-greek/)
## Authoritative sources
- [The Pricing of Options and Corporate Liabilities](https://doi.org/10.1086/260062) — Fischer Black and Myron Scholes, *Journal of Political Economy* (1973)
- [Theory of Rational Option Pricing](https://doi.org/10.2307/3003143) — Robert C. Merton, *Bell Journal of Economics and Management Science* (1973)
- [Options: The Basics and the Greeks](https://www.finra.org/investors/insights/options-z-basics-greeks) — FINRA
- [Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document) — Options Clearing Corporation