What Is Zomma?
Zomma is the third-order cross derivative of option value with respect to spot (twice) and volatility (partial3 V / partial S2 partial sigma). Equivalently, zomma measures how gamma itself changes when implied volatility moves. In Black-Scholes, zomma equals gamma × (d1 × d2 - 1) / sigma. Zomma captures gamma stability across volatility regimes.
What Is Zomma in Options?
Zomma quantifies how gamma changes when IV changes. A long ATM call with gamma 0.05 and zomma 0.002 sees its gamma change by 0.002 per 1-unit IV move (per 100 percentage-point change in IV). Per-1-vol-point scaling: 0.00002 per 1 vol point. The metric is small in absolute terms but matters for dynamic hedging during vol-regime shifts.
Two intuitions for zomma. First, zomma is the cross between gamma and vol - it tells you whether gamma is stable as vol moves. Second, zomma helps explain why ATM gamma falls when vol rises (a structural feature of the BS gamma formula): the zomma is part of that mechanism. Operationally, zomma matters most when running a delta-and-gamma-hedged book through a vol-regime change.
Worked Example
SPY at $500, 30-day OTM call (K=520), IV 14%, rate 4%. Computing zomma:
d1 = -0.875,d2 = -0.915,phi(d1) = 0.272- Gamma at K=520: phi(d1) / (S sigma sqrt(T)) = 0.272 / 20.07 = 0.0136 per share
- Zomma = gamma × (d1 × d2 - 1) / sigma = 0.0136 × (0.801 - 1) / 0.14 = 0.0136 × (-0.199) / 0.14 = -0.0193 per share, per unit-vol
- Per-1%-IV scaling: -0.000193
Operational reading: an IV move from 14% to 19% (5 vol points) reduces gamma by approximately 0.001 (from 0.0136 to 0.0126). For a hedge book sized at thousands of contracts, this gamma drift can require meaningful rebalancing during vol-regime shifts.
How Pricing Models Compute Zomma
- Black-Scholes: closed-form zomma. Same formula applies to calls and puts.
- Heston (stochastic volatility): zomma computed by Fourier inversion of the cross-second-derivative. Heston naturally produces vol-dependent gamma through stochastic volatility, so Heston zomma reflects the model's structural assumption.
- SABR: zomma is captured through the cross-curvature of the Hagan formula.
- Local volatility (Dupire): zomma computed by finite-difference on a bumped IV surface.
- Monte Carlo: zomma via pathwise differentiation. Standard production method for exotic options.
Why Zomma Matters
Zomma matters operationally in three places. First, vol-regime transitions. When SPX vol regime shifts (e.g., 14% to 28% during a stress event), the gamma profile of an aggregate dealer book changes meaningfully through zomma. Risk teams running stress scenarios that hold gamma constant underestimate the impact of regime shifts.
Second, vol event preparation. Pre-FOMC and pre-earnings windows see IV spike then fall (IV crush). The gamma profile at ATM during the spike is different from the post-crush profile - zomma describes the magnitude of the shift. Traders preparing for events explicitly hedge zomma.
Third, gamma-by-strike concentration analytics. Aggregate dealer-side zomma indicates whether the gamma-by-strike profile (used for GEX calculations) is stable across vol regimes. A book with negative aggregate zomma is more gamma-exposed in low-vol periods than in high-vol periods.
Special Cases
- ATM: zomma is small (d1×d2-1 approaches -1, but gamma is large; the product is moderate).
- Deep OTM/ITM: zomma falls off as gamma falls off.
- Short-tenor near-strike: zomma is large in absolute terms.
Related Greeks
Zomma is one of the three third-order Greeks involving gamma. Its siblings are speed (gamma's sensitivity to spot) and color (gamma's sensitivity to time). Together they describe the full third-order structure around gamma.
Related Concepts
Gamma · Speed · Color · Vega · Vomma · Volatility Skew · All 17 Greeks
References & Further Reading
- Hull, J. (2018). Options, Futures, and Other Derivatives, 10th ed. Pearson.
- Wilmott, P. (2006). Paul Wilmott on Quantitative Finance. Wiley.
This page is part of the 17 Greeks reference covering every options Greek with formula, intuition, worked example, and how each pricing model computes it. Browse the full documentation.