Third-order Greek · ∂³V/∂S²∂σ

Zomma: how gamma changes when implied volatility moves

Zomma, sometimes written DgammaDvol, is the change in gamma for a change in implied volatility. When vol rises, gamma spreads out: lower at the money, higher in the wings.

The formula

d₁ = [ ln(S/K) + (r + σ²/2)·T ] / (σ√T)
d₂ = d₁ − σ√T

Zomma = ∂Γ/∂σ = Γ·(d₁·d₂ − 1) / σ

Black-Scholes, European options, no dividends. S spot, K strike, T years to expiry, r the risk-free rate, σ implied volatility, N(·) the standard normal CDF and φ(·) its density.

At the money d₁·d₂ ≈ 0, so zomma ≈ −Γ/σ: more vol, less gamma. In the wings, where d₁·d₂ > 1, zomma is positive: more vol spreads gamma out to them.

Why a gamma map changes shape

The same open interest produces a sharp, peaky gamma profile when implied vol is low and a smoother, wider one when vol is high. Zomma is that effect, strike by strike. It's also why gamma exposure figures can change after a vol move with no new trades at all.

Do they predict anything?

We tested speed, color and zomma, plus four fourth-order exposures (∂⁴V/∂S⁴, ∂⁴V/∂S³∂t, ∂⁴V/∂S²∂σ², ∂⁴V/∂S³∂σ), computed from the dealer book every minute, against the next 30 minutes' realised volatility and move, on seven archived sessions across SPX, SPY, ES, NDX, QQQ, NQ and IWM. Trained on five days, checked on two unseen ones, controlling for the last 30 minutes' realised volatility, gamma and time of day: none added anything, and the best result on training reversed on the unseen days. Small sample, correlated symbols; we rerun it as the archive grows.

Not financial advice. This article explains how options and dealer hedging work. It is not a recommendation to buy or sell anything.