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Zomma Greek: How Implied Volatility Changes Gamma

For educational purposes only; not investment advice.

Direct answer

Section titled “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

Section titled “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

Section titled “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

Section titled “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

Section titled “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.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”