Gamma

Gamma measures how much delta changes when the underlying moves one point. If delta is speed, gamma is acceleration.

Gamma = change in delta / change in underlying

Why it matters

Delta tells you your current directional exposure; gamma tells you how fast that exposure grows or shrinks as price moves. High gamma means delta swings quickly — your position can flip from mildly directional to heavily directional with a small move.

A worked example

Call delta 0.50, gamma 0.05, stock at 100.
Stock rises 1 point to 101:
  new delta ~ 0.50 + 0.05 = 0.55
Stock rises another point to 102:
  new delta ~ 0.55 + 0.05 = 0.60

The position gets "longer" the more the stock rises — gains accelerate. The reverse is also true: against you, losses decelerate (delta shrinks), which is the favourable curvature option buyers pay for.

Where gamma is largest

Gamma peaks for at-the-money options and near expiry. A short-dated ATM option has explosive gamma: its delta can lurch from 0.50 to near 1 or near 0 in hours.

Honest risk for sellers

Option sellers are short gamma — their losses accelerate as the underlying moves against them, fastest right when they can least afford it (ATM, near expiry). This is why a quiet short-premium strategy can blow up violently on a single sharp move.