What Is Gamma?
Gamma (Γ) is the second derivative of option value with respect to the underlying price (and equivalently the rate of change of delta). In the Black-Scholes model, gamma equals phi(d1) / (S sigma sqrt(T)), where phi() is the standard normal probability density and d1 uses the same definition as in the delta formula. Gamma is the structural source of an option position's convexity and the second-order Greek that drives the entire dealer-flow story in modern options markets.
What Is Gamma in Options?
Gamma tells you how much delta changes per $1 move in the underlying. A long-call position with gamma 0.04 sees its delta rise from 0.50 to 0.54 if spot moves up $1. Gamma is the curvature of the option-value function: it is what makes long options profit on big moves regardless of direction (because delta keeps moving in the favorable direction) and short options lose on big moves (because delta moves against the seller).
Three intuitions for gamma. First, gamma is the convexity of the position - the amount by which actual P&L on a move exceeds the linear delta estimate. Second, gamma is the rebalancing cost of a delta-hedged position - the more gamma, the more you must rebalance to stay delta-neutral, and each rebalance happens at a slightly different price than the previous one. Third, gamma is positive for long options (calls and puts both) and negative for short options. A delta-neutral long-volatility position is structurally long gamma; a delta-neutral short-volatility position is structurally short gamma.
Worked Example
SPY trading at $500, 30-day ATM call, IV 14%, rate 4%. Black-Scholes inputs:
d1 = [0 + (0.04 + 0.142/2)(30/365)] / (0.14 sqrt(30/365)) = 0.00409 / 0.04014 = 0.102phi(d1) = 0.397- Gamma = 0.397 / (500 × 0.14 × sqrt(30/365)) = 0.397 / (500 × 0.14 × 0.287) = 0.0198 per share
That gamma of 0.0198 means each $1 move in SPY shifts the call's delta by 0.0198 (about 2 delta cents). Across a 5% SPY move ($25), delta would shift roughly 0.5 - meaning a position that started at delta 0.50 ends at delta 1.0 if spot rallies $25 (or delta 0.0 if spot drops $25). This is the gamma profile that makes long-call positions self-reinforcing on the upside: the more spot rallies, the more delta you accumulate, the more spot moves matter.
Gamma Across Moneyness and Time
Gamma peaks at-the-money and falls off in both wings (deep ITM and deep OTM positions have low gamma because delta is already pinned near +/-1 or 0). ATM gamma scales roughly as 1/sqrt(T), so the peak gamma value rises as expiration approaches: a 7-DTE ATM call has roughly 2.1x the gamma of a 30-DTE ATM call (sqrt(30/7) = 2.07), and a 1-DTE ATM call has roughly 5.5x (sqrt(30) = 5.48). This is the structural reason short-tenor options are unstable - gamma rises near expiration, and delta becomes extremely spot-sensitive.
Volatility moves gamma in the opposite direction: higher IV produces lower (smaller) ATM gamma because the option-value curve becomes flatter (more diffuse) when vol is high. Gamma of a 1-month ATM call at 14% IV is roughly twice the gamma of the same option at 28% IV. This is the structural reason vol regimes matter for dealer hedging cost: in low-vol regimes, dealers running short-gamma books are exposed to outsized rebalancing P&L when realized vol exceeds implied.
How Pricing Models Compute Gamma
- Black-Scholes: closed-form analytical gamma. Same gamma applies to calls and puts (because both have the same N(d1) curvature, just shifted). With continuous dividend yield, gamma equals
exp(-qT) phi(d1) / (S sigma sqrt(T)). - Heston (stochastic volatility): gamma is computed by Fourier inversion of the second derivative of the Heston pricing formula. Heston gamma is generally lower than BS gamma at the same IV because Heston accounts for vol-of-vol smearing the price distribution. The gap between Heston and BS gamma grows with the magnitude of
nu(vol-of-vol) and the absolute value ofrho(spot-vol correlation). - SABR: gamma is computed via the Hagan-implied-vol approximation plus smile corrections. SABR gamma differs from BS gamma at the same ATM vol because the SABR smile has different curvature than the BS-flat assumption.
- Local volatility (Dupire): LV gamma is computed by finite difference on the LV PDE solution. Because LV calibrates exactly to today's surface, LV gamma matches BS-implied gamma at every traded strike. The trade-off is in forward dynamics: LV gamma at a future date may diverge sharply from what stochastic-vol models predict.
- Jump diffusion: diffusion-component gamma plus a jump-correction term that becomes large when the option is near a strike where jumps could push it ITM or OTM. Jump-induced gamma dominates short-tenor pricing.
- Monte Carlo: gamma is typically computed by pathwise differentiation (Malliavin or vibrato) or by finite-difference resimulation. For exotic options where closed forms do not exist, MC gamma is the standard production method.
- Binomial tree: gamma is the second-difference: gamma = (Vup,up - 2 Vmiddle + Vdown,down) / (Sup,up - Smiddle)2, computed at the central node two time steps in.
The Theta-Gamma Tradeoff
The fundamental identity that ties gamma to theta through the Black-Scholes PDE is: theta + 0.5 sigma2 S2 gamma = r V - r S delta. For an at-the-money delta-neutral position, this simplifies approximately to theta = -0.5 sigma2 S2 gamma + r V, meaning theta scales negatively with gamma: more gamma costs more theta, and vice versa. This is the "no free lunch" relationship - you cannot have positive convexity (long gamma) without paying for it with negative time decay (short theta), and you cannot collect time decay (short theta) without taking convexity risk (short gamma).
The tradeoff is operational. Long-gamma traders accept negative theta in exchange for profiting from realized-vol moves; short-gamma traders accept gamma risk in exchange for collecting positive theta. The breakeven is where realized vol exactly matches implied vol over the option's life. The volatility risk premium (IV systematically exceeding RV) is the empirical reason short-vol strategies have positive expected returns despite the asymmetric tail risk.
Dealer Gamma and Market Microstructure
The aggregate gamma sitting on dealer books is the most consequential gamma quantity in modern markets. Dealer gamma exposure (GEX) measures the second-order hedging-flow derivative: when dealers are net long gamma (typical when retail sells puts), dealer rebalancing flows damp price moves (dealers buy weakness, sell strength). When dealers are net short gamma (typical when retail buys calls heavily), dealer flows amplify moves (dealers buy strength, sell weakness), producing the gamma squeeze phenomenon.
The gamma-flip level (the spot price where dealer aggregate gamma crosses zero) is the most-watched microstructure number on the SPX. Above the flip, dealers are typically long gamma and price dynamics are mean-reverting. Below the flip, dealers are short gamma and price dynamics become trend-amplifying. Major regime transitions (Aug 2024, March 2020) feature the gamma-flip level breaking and dealer flows changing sign rapidly.
Gamma in 0DTE and Expiration-Day Trading
Gamma reaches its theoretical maximum at the strike on expiration day - this is "pin gamma" and it is mathematically infinite (delta jumps from 0 to 1 instantaneously across the strike). Practical 0DTE options exhibit gamma that increases as 1/sqrt(T) - meaning gamma at 30 minutes to expiration is roughly sqrt(30 days / 30 min) = 38x the gamma at 30 days. This is the structural reason 0DTE dealer flows are increasingly viewed as a microstructure factor: aggregate gamma on 0DTE options can rival aggregate gamma on the entire monthly expiration in absolute terms.
Special Cases
- Deep ITM/OTM: gamma approaches zero. Delta is already pinned, so it cannot move further with spot.
- ATM at long expiration: gamma is small (delta sigmoid is flat). Position behaves nearly linearly with spot.
- ATM at short expiration: gamma is large. Position behaves nonlinearly; small spot moves produce big delta swings.
- At expiration exactly on the strike: gamma is undefined (mathematically infinite). This is the "pin risk" zone.
Related Greeks
Gamma is the second derivative in the spot direction. Three third-order Greeks describe how gamma itself moves: speed (gamma's change with spot, the curvature of the curvature), zomma (gamma's change with vol), and color (gamma's change with time, also called gamma decay). Gamma's first-derivative siblings are delta and vega; the cross-Greek connecting them is vanna.
Related Concepts
Delta · Speed · Zomma · Color · Dealer Gamma · Gamma Exposure · Gamma Squeeze · 0DTE Options · All 17 Greeks
References & Further Reading
- Hull, J. (2018). Options, Futures, and Other Derivatives, 10th ed. Pearson. Chapter 19.
- Wilmott, P. (2006). Paul Wilmott on Quantitative Finance. Wiley. Standard quantitative-finance reference for higher-order Greek behavior.
- Sinclair, E. (2010). Option Trading: Pricing and Volatility Strategies and Techniques. Wiley. Chapters 9 and 11 cover hedging and volatility-trading practice.
View live SPY gamma exposure (GEX) by strike →
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.