Delta is not fixed. Hedge a call at 55 shares, let the stock rally, and the call now behaves like 65 shares: your hedge is short 10. Gamma is the Greek that tells you how fast that happens, and it decides whether a position makes or loses money when the stock actually moves.
Gamma is the change in delta
Gamma is the change in an option's delta for a $1 move in the stock. Say NVDA is at $200 and the $200 call has a delta of 0.50 and a gamma of 0.02. If NVDA rises to $201, the delta becomes about 0.52. If it falls to $199, about 0.48. On one contract, 0.02 of gamma means the hedge changes by 2 shares for every $1 the stock moves, and a $10 rally turns a 50-share hedge into a 70-share one.
The chain's Gamma column is quoted per $1 move in the stock, per share. A call and a put at the same strike share the same gamma, and it is positive for whoever owns the option. The seller is short gamma by the same amount.
Where gamma lives
Gamma is largest where delta is changing fastest, and that is at the money. Deep in the money, delta is already near 1 and has nowhere to go. Far out of the money, it is near 0 and stays there. Between them, at the strike closest to the stock, a $1 move makes the biggest difference to the odds.
Gamma also grows as expiration approaches. With six months to go, the delta curve is gentle and gamma is small at every strike. With two days to go, the at-the-money option is a coin flip that a $2 move decides, so its delta swings from 0.20 to 0.80 and gamma is enormous. Short-dated, at-the-money options are where gamma concentrates.
The same call with 60, 20 and 5 days left. Short-dated gamma is a tall, narrow spike; long-dated gamma is low and wide.
Read from right to left as the days pass. At-the-money gamma explodes in the last week; out-of-the-money gamma rises for a while, then collapses as the strike moves out of reach.
Long gamma wants movement
Own an option and hedge its delta, and something useful happens. The stock rises, your delta grows, so you are now long more than your hedge covers: you sell some shares at the higher price. The stock falls, your delta shrinks, so you are short: you buy some shares back lower. Every re-hedge sells high and buys low. That is why a long gamma position profits from movement in either direction, and the bigger the moves, the better. What it costs is time decay, which is the next lesson.
Short gamma is the mirror image. The seller who hedges is forced to buy after rallies and sell after drops. Small moves are fine and the decay collected pays for them. A large move is where the losses come from, and they grow with the size of the move.
The dealer side of gamma
The same rules apply to market makers, often called dealers, who take the other side of most customer trades and hedge what is left. Long gamma, their hedging sells rallies and buys dips and damps moves; short gamma, it chases the market and amplifies them. Module 7 covers how the Terminal estimates dealer gamma from open interest and the key levels that come out of it.
The maths: why gamma peaks where it does
Gamma is e^(-qT) n(d1) / (S v √T), where n is the bell-shaped normal curve. The bell is highest when d1 is near zero, which is close to the money, and falls away quickly either side: that is the peak in the first chart. The √T in the denominator is why gamma grows into expiration. At the money, n(d1) is about 0.4, so gamma is roughly 0.4 / (S v √T). A $100 stock at 30% volatility with 30 days left gives 0.4 divided by (100 × 0.30 × 0.287), about 0.046. Cut the time to a quarter, 7.5 days, and √T halves, so gamma doubles to about 0.093.
Traders often quote dollar gamma instead: the change in delta, in dollars of stock, for a 1% move. It is gamma times the stock price squared, divided by 100, and it lets you compare gamma on a $20 stock with gamma on a $700 one.
Your turn
Open the NVDA chain on the monthly expiration. Read the gamma at the money, then at the call whose delta is near 0.10, well out in the wing, and see how much smaller it is.