What Is Speed?

Speed is the third derivative of option value with respect to the underlying price (partial3 V / partial S3). Equivalently, speed measures how gamma itself changes as spot moves. In the Black-Scholes model, speed equals -gamma / S × (d1 / (sigma sqrt(T)) + 1). Speed captures the convexity of convexity and becomes important in large-move regimes and short-tenor options.

What Is Speed in Options?

Speed tells you how much gamma changes per $1 move in the underlying. A position with gamma 0.04 and speed 0.0005 has gamma that increases to 0.045 if spot moves up $10 (ignoring the curvature of the speed function itself). Speed is the third-order analog of gamma, measuring the rate at which the rate-of-change-of-delta itself moves.

Two intuitions for speed. First, speed is the curvature of gamma - how the convexity itself bends. Second, speed is the structural cause of large-move asymmetry: the second-order (gamma-driven) P&L estimate from a $20 spot move is incomplete; speed adds the third-order correction that becomes material in tail moves.

Worked Example

SPY at $500, 7-day ATM call, IV 14%, rate 4%. Speed for this option:

The negative sign means gamma decreases as spot rises above $500 (because we are moving toward the wing where gamma falls off). For a $10 spot move (from 500 to 510), gamma changes by approximately -0.000291 × 10 = -0.00291, dropping from 0.041 to 0.038. The change is small per dollar move but compounds: across a $30 move, gamma drops by roughly 0.009 (about 22% of starting gamma), materially shifting the position's hedging requirements.

Why Speed Matters

Speed is a niche Greek for normal day-to-day trading but becomes essential in three operational contexts. First, gamma scalping. A trader running a gamma-positive position is "long gamma" and rebalances at a profit on every spot move; the realized profit per move scales with gamma, but speed controls how that gamma changes during the move itself. Failing to account for speed leads to underestimating gamma decay through the move.

Second, large move analysis. Stress testing of options portfolios uses gamma as a first-cut convexity estimator; speed is the next-order correction. For 5-sigma stress moves on short-tenor positions, the gamma estimate underestimates P&L because gamma itself moves substantially. Risk teams running fat-tail scenario analysis include speed in production.

Third, dealer convexity in 0DTE flow. Aggregate dealer speed describes how aggregate dealer gamma evolves with spot - a dealer book that is short gamma at the current strike may become even shorter gamma if spot moves toward another concentrated-strike level. This is a microstructural input to GEX dynamics that vanilla gamma analysis misses.

How Pricing Models Compute Speed

Speed Across Moneyness and Time

Speed is largest in absolute terms for short-tenor options near the strike, falling off for both long-tenor and deep-OTM/ITM options. The sign of speed is generally negative for ATM and slightly OTM options (gamma decreases as spot moves further from the strike), positive for deep OTM options (gamma increases as spot approaches the strike from far away).

The 1/T scaling of gamma compounds for speed: a 1-DTE ATM option has speed many times larger in magnitude than the same option at 30 DTE. This is the structural reason 0DTE risk analysis must include third-order Greeks: the convexity profile itself is changing rapidly as expiration approaches, and second-order analysis misses the dynamic.

Special Cases

Related Greeks

Speed is one of three third-order Greeks involving gamma. Zomma is gamma's sensitivity to vol (cross-derivative with sigma). Color is gamma's sensitivity to time (cross-derivative with t, "gamma decay"). Together, speed, zomma, and color describe the full third-order structure around gamma.

Related Concepts

Gamma · Zomma · Color · Dealer Gamma · Gamma Exposure · 0DTE Options · Black-Scholes · All 17 Greeks

References & Further Reading

View SPY gamma exposure profile →

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.