Gamma: the rate of change of delta

Gamma, the curvature Greek

Delta tells you how fast your option price moves. Gamma tells you how fast delta itself is changing. If delta is the speedometer, gamma is the gas pedal. A high-gamma option has a delta that swings hard — every $1 move in the stock kicks delta up or down by a lot. A low-gamma option has a stable delta that barely budges. Gamma is the convexity Greek. It's why options can deliver outsized moves once they start working.

Where is gamma highest? At-the-money options near expiry. Picture the delta curve from lesson one — that S-shape. Gamma is the slope of that slope. It's biggest where the S-curve is bending fastest, which is right at the strike. The closer to expiry, the sharper the bend gets, because the option is collapsing toward a kinked payoff (either ITM or OTM with no in-between). On the last day of an at-the-money option, gamma can spike dramatically.

Long option positions (calls or puts you bought) have positive gamma. As the underlying moves in your direction, your delta increases — you make money faster the further it runs. Short option positions (calls or puts you sold) have negative gamma. As the underlying moves against you, your delta gets worse — losses accelerate. That asymmetry is the trade. Sellers earn the time-decay premium in exchange for absorbing that gamma risk.

Deep ITM and deep OTM options have low gamma because their delta is already pinned (near 1 or near 0). There's not much room for delta to change. The action is at-the-money, where the option is most uncertain about which way it'll end up.

Quick example to anchor this. Imagine you bought a $100-strike call when the stock was at $97 with three days to expiry. Delta is around 0.35 and gamma is high — say 0.10. Stock rallies to $100. Delta jumps to about 0.55. Stock pushes to $103. Delta jumps to about 0.80. Each $1 move accelerated your delta because gamma was working in your favor. That's the convexity bonus long-option buyers chase. The flip side: if the stock had dropped from $97 to $94, delta would have collapsed quickly from 0.35 toward 0.10 — gamma working against you on the way down.

Recap: gamma = how fast delta is changing. Highest for ATM options near expiry. Positive for long options, negative for short options.