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Zomma Greek: How Implied Volatility Changes Gamma

Define Zomma precisely, normalize its sign and units, verify it by finite difference, and use full repricing for large volatility or surface moves.

Updated

Educational only; not individualized investment, legal, or tax advice. Options involve risk and may result in loss.

Direct answer

Zomma is the partial derivative of an option’s Gamma with respect to a chosen implied-volatility input, with the pricing model’s other inputs held fixed:

Zomma = ∂Gamma/∂σ = ∂³V/(∂S²∂σ)

Gamma is the local change in Delta for a small underlying-price move; Zomma is the local change in that Gamma for a small IV move. It is a higher-order, model-dependent sensitivity, not a separately traded cash flow, an IV forecast, or a promise of what Gamma will be after a large market move.

Scope: the formulas below describe a European-style Black–Scholes model with S > 0, K > 0, σ > 0, and T > 0, continuous compounding and dividend yield, and a constant volatility input. They do not by themselves cover American exercise, discrete dividends, jumps, barriers, or a moving volatility surface. The example is not a quote or account-level calculation. FINRA and OCC materials address U.S. exchange-traded options; products, market rules, approvals, margin, taxes, and legal availability vary by account, jurisdiction, and date. No live market or customer data are used. Check current official documents and qualified advisers.

Formula, sign, and units

For that model, with continuously compounded 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: higher IV spreads Delta’s transition over a wider price range and lowers peak Gamma. Far from ATM, the sign can differ. This is a model result, not a universal empirical rule.

Units must be explicit. 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 display quoting change per one volatility point (1% = 0.01) reports Zomma_decimal / 100. Vendors may also scale Gamma, spot moves, contract multipliers, or position quantities, so raw numbers are not comparable without conventions.

Zomma is a partial derivative: spot, time, rates, dividends, and unbumped surface nodes are held fixed. In markets these inputs can move together. Spot-IV dependence, skew and term-structure changes, elapsed time, jumps, and early-exercise effects can dominate the 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. For 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 gives:

ΔGamma ≈ −0.060 × (−0.45) = +0.027

and estimated Gamma 0.067. That is not a reliable event forecast: Zomma changes along the path, time passes, spot can gap, and the surface can reprice. Recalculate every option and the portfolio under a documented post-event surface; use the linear result only for initial direction and rough scale.

Analysis and risk controls

  • Record valuation timestamp, model, exercise style, spot or forward convention, rates, dividends, time clock, IV source, surface construction, and data quality.
  • State whether IV is decimal, percent, or volatility points and whether Gamma and Zomma include multiplier, quantity, position sign, or currency conversion.
  • Verify Zomma with central differences: [Γ(σ+h) − Γ(σ−h)]/(2h). Require h > 0 and σ−h > 0; compare several stable h values and avoid bumps across model kinks or invalid surface nodes.
  • Recompute each strike and expiry; do not apply one representative Zomma to 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, sparse wings, or results that are unstable across bump sizes.
  • Aggregate long/short sign, quantity, multiplier, and currency consistently; a small net Zomma can conceal offsetting bucket risk.
  • Compare predicted and realized changes in Gamma, Delta, and P&L; residuals may reflect model, data, surface, liquidity, or execution risk.
  • Prioritize Delta, Gamma, Vega, liquidity, maximum loss, settlement, assignment, and margin before relying on a third-order sensitivity.
  • Read current product specifications and OCC disclosures; Zomma does not change contractual exercise, assignment, settlement, or funding 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.
  • “Same-strike Calls and Puts 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 0.01 when IV input is decimal.
  • “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 on a changed surface.
  • “More Greeks automatically improve control.” A precise higher-order number cannot repair poor inputs, liquidity, sizing, or expiration planning.

Authoritative sources

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