Gamma Explained: Why Delta Changes, Gamma Risk Near Expiry, and Dealer Gamma

Gamma is the rate at which an option’s delta changes when the underlying stock moves $1. If a call has a delta of 0.50 and a gamma of 0.10, a $1 rise in the stock should lift its delta to about 0.60, so the option gains more on each successive dollar of the move.

In short: delta tells you how much an option behaves like the stock right now; gamma tells you how quickly that changes. Gamma is highest for at-the-money options and rises sharply as expiration approaches. Option owners have positive gamma: their gains accelerate and their losses slow down as the stock moves. Option sellers have negative gamma: the reverse. Gamma and theta come as a pair: the positions with the most gamma also pay or collect the most time decay. “Dealer gamma” refers to the estimated net gamma of the market makers who hedge customer trades, which some analysts believe can dampen or amplify market moves.

From delta to gamma

Delta, explained in delta explained, measures how much an option’s price moves per $1 in the stock. But delta is not fixed. A call that is far out of the money has a delta near 0; if the stock rallies until the call is deep in the money, its delta approaches 1. Gamma measures the speed of that transition.

Gamma is always positive for an option you own (calls and puts alike) and negative for an option you have sold. It is quoted per share; for a 100-share contract, multiply by 100 to get the change in share-equivalent exposure per $1 move.

Gamma rises as expiry approaches

Hypothetical Maple Widgets Inc. trades at $50. Using the Black-Scholes model with 30% implied volatility, a 4% interest rate, and no dividends, here is the gamma of the $50 call at different times to expiry:

Time to expiry	Call price	Delta	Gamma
3 months (0.25 years)	$3.23	0.556	0.053
30 days	$1.80	0.532	0.093
7 days	$0.85	0.516	0.192
1 day	$0.32	0.506	0.508

With one day left, gamma is almost ten times its three-month level. And gamma is concentrated right at the strike. With one day left:

Stock price	Delta	Gamma
$49	0.102	0.231
$50	0.506	0.508
$51	0.899	0.221

A $1 move either way swings delta from about 0.5 to 0.1 or 0.9. Near expiry, an at-the-money option behaves less like a smooth bet and more like a switch.

Why delta alone gets big moves wrong

A Greek-based price estimate for a move of size m is: delta x m + one half x gamma x m squared. The gamma term grows with the square of the move, so it matters more as moves get larger and as gamma rises.

For a $1 rise in Maple Widgets from $50 to $51, with no time passing:

	Delta only	Delta plus gamma	Full model
30 days left	+$0.532	+$0.579	+$0.578
1 day left	+$0.506	+$0.760	+$0.728

With 30 days left, adding gamma makes the estimate almost exact. With one day left, even the gamma-adjusted estimate is off by 3 cents per share, because gamma itself changes so much over a $1 move that a single number can’t capture it.

Long gamma vs short gamma

A worked example

You buy one 30-day $50 call for $1.80 ($180) and, to remove your directional exposure, short 53 shares (roughly matching the call’s delta of 0.532 x 100). This is a delta-hedged long call, a way of isolating gamma.

If the stock instantly jumps $3 in either direction:

Up to $53: the call gains $1.98 per share ($198). The 53 short shares lose 53 x $3 = $159. Net: about +$39.
Down to $47: the call loses $1.18 per share ($118). The short shares gain $159. Net: about +$41.

You make money whichever way the stock moves. That is positive gamma. But each day you hold the call, theta costs about $0.031 per share, or $3.12 per contract.

The gamma-theta trade-off

How big a daily move pays for one day of theta? Setting one half x gamma x move squared equal to theta: one half x 0.0925 x move squared = 0.0312, which gives a move of about $0.82 a day.

Now compare that with what the 30% implied volatility predicts: $50 x 30% divided by the square root of 365 calendar days is about $0.79 a day. The two are nearly the same, and that is not a coincidence. In the model, an option is fairly priced when the gamma gains you expect to collect from moves roughly equal the theta you pay. Long-gamma holders profit if the stock moves more than implied volatility suggests; short-gamma sellers profit if it moves less. Neither side has a free edge built into the arithmetic. Time decay is covered in detail in theta and time decay.

Short gamma

The seller of that call has the mirror image: they collect $3.12 a day in theta, and lose about $39 to $41 on a sudden $3 move after hedging (more if unhedged). Short-gamma positions tend to produce frequent small gains and occasional large losses. Near expiry, when gamma spikes, those losses can arrive in minutes, a central risk in 0DTE options explained.

Gamma and the level of volatility

Lower implied volatility concentrates gamma more tightly around the strike. Here is a 7-day Maple Widgets call (stock at $50, 4% rate) at different volatility levels, for the $50 strike and for a $52 strike:

Implied volatility	Gamma, $50 strike	Gamma, $52 strike
20%	0.288	0.112
30%	0.192	0.128
45%	0.128	0.108

When volatility is low, an at-the-money option’s delta is very sensitive to small moves, while options a few dollars away have much less gamma. When volatility is high, gamma is spread across a wider range of strikes. Combined with the time effect above, this is why at-the-money options in quiet markets, close to expiry, carry the most concentrated gamma of all.

Gamma for puts

A call and a put with the same strike and expiry have the same gamma in the Black-Scholes model. Owning either gives positive gamma; selling either gives negative gamma. A long straddle (a call plus a put, see straddles and strangles explained) has twice the gamma of either leg, which is why it gains from large moves in either direction and pays twice the theta while waiting.

Gamma risk near expiration

For anyone short an at-the-money option in its final days:

A small move can flip the option from worthless to in the money, changing whether shares will be delivered at assignment.
Hedges need constant adjustment, because delta jumps between about 0.1 and 0.9 within a dollar or two.
If the stock closes very near the strike on expiration day, it may be unclear whether the option will be exercised. That is pin risk, covered in what happens at expiration.

Dealer gamma, at an explainer level

Market makers who sell options to customers usually hedge their delta by trading the underlying stock or futures, as described in how options markets work. Their hedging behaviour depends on the sign of their gamma:

If dealers are net long gamma (for example, customers have sold them options), delta hedging means selling as prices rise and buying as prices fall. That hedging flow leans against moves and may dampen volatility.
If dealers are net short gamma (customers have bought options from them), hedging means buying as prices rise and selling as prices fall. That flow pushes in the same direction as the move and may amplify it.

Commentary about “positive” or “negative” dealer gamma is common, especially around large index option expiries. Treat it carefully: dealer positions are not public. Published gamma estimates are models built on assumptions about who bought and who sold each option, and different providers can reach different conclusions from the same data. Hedging is also only one of many forces moving prices.

What can go wrong

Underestimating how fast exposure changes. A short option that is comfortably out of the money in the morning can be at the money by afternoon, with its delta, and the position’s risk, multiplying.
Hedging lag. Delta hedges are adjusted in steps. A gap between adjustments, such as an overnight move, realises the full gamma loss at once.
Confusing positive gamma with an edge. Long gamma wins on big moves but pays theta every day the moves don’t come.
Over-reading dealer gamma narratives. They are estimates, not observations.

Takeaway

Gamma measures how quickly an option’s delta, and therefore its risk, changes as the stock moves. It peaks for at-the-money options close to expiry, which is why those options can swing so violently. Owners pay theta for positive gamma; sellers collect theta and take on negative gamma. In the model, the two balance when the stock moves as much as implied volatility predicted.