Third-order Greek · ∂³V/∂S³

Speed: how fast gamma changes as price moves

Speed is the change in gamma for a $1 move in the underlying: the third derivative of value with respect to price. It tells you whether your gamma is about to grow or shrink if price keeps going.

The formula

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

Speed = ∂Γ/∂S = −(Γ / S)·(1 + d₁ / (σ√T))

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.

Gamma peaks near the money, so speed is the slope of that peak: positive below it (gamma grows as price rises toward the strike) and negative above it. Like gamma, it is the same for calls and puts and grows sharply into expiry.

Reading a book's speed

For a single option speed is a curiosity. For a dealer book it answers a practical question: if price rallies 0.5%, does the hedging pressure under price get bigger or smaller? A long-gamma book with positive speed above spot gets more stabilising as price rises into it; one with negative speed is about to run out of support.

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.

ES absolute gamma with the gamma build window and gamma ladder in Zanshin Flow — How gamma at each ES strike is building or fading, beside the gamma map.

Questions

What is the speed Greek?

Speed is the rate of change of gamma with respect to the underlying price, the third derivative of option value with respect to price.

Is speed ever used in practice?

Mostly by hedgers of large gamma books, to judge how their gamma will change over a price move larger than one delta-hedging interval.

Does dealer speed exposure predict price?

Not in our tests: across seven archived sessions it added nothing to the next 30 minutes' volatility or move once recent volatility and gamma were known.

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