Gamma: Why the Payoff Bends
Gamma is the rate at which an option's delta changes as the underlying price moves. It is the quantity that makes an option's payoff curve rather than simply tilt, and it is largest at the strike price and in the final days before expiry, which is why short-dated positions behave so violently.
The curves on this page are shape-accurate illustrations drawn from the standard normal distribution rather than a full pricing model. They are correct about behaviour and are not quotes for any real contract.
How it works
Delta tells you how much an option moves per dollar. Gamma tells you how much that number itself changes as price moves.
If delta is speed, gamma is acceleration. A position with high gamma is one whose behaviour is changing rapidly, which is a very different thing from one that is simply moving.
That is what separates an option from shares. A share’s payoff is a straight line with a constant slope of one. An option’s bends, and gamma is the bend.
Where it lives
Gamma concentrates at the strike. Far above it, delta is already close to 1 and cannot change much. Far below, delta is near 0 and cannot change much either. Around the strike is where a small move flips the option’s character.
On the sweep computed for this site, gamma peaks near the strike and falls away on both sides. The shape is a hump, and its position is the strike rather than anything about the market.
Time concentrates it further. With a month left, a move of a few points barely settles anything. With a day left, the same move decides the outcome entirely — so delta swings between extremes, which is another way of saying gamma has become enormous.
That is why the last week of an option’s life is so violent, and why positions that behaved sensibly for a month stop doing so at the end.
In practice: the trade being made
Buying an option means owning gamma. Selling one means being short it. Long gamma is a position that gets better the further price moves, in either direction. Short gamma gets worse.
And that exposure is paid for in decay. Theta is the rent on gamma: a buyer pays it daily and a seller collects it. Nothing available gives you both, and any strategy claiming to is mispricing one of them.
In a flat market the seller wins that exchange. Nothing moves, the curvature is never needed, and the rent accrues to whoever sold it.
In a fast move the buyer is repaid in one go. Gamma means the position gains at an accelerating rate, which is why a single large move can pay for a long run of decayed premiums.
Managing it costs money. Keeping a position delta-neutral as gamma moves the delta means trading the underlying repeatedly, and each round trip costs 2% of a typical bar’s range on this site’s history.
What gamma is not
It is not a direction. Long gamma benefits from movement either way; it says nothing about which way.
It is not risk on its own. A small long-gamma position is barely noticeable. A large short-gamma one near expiry is the most dangerous ordinary position in retail options.
It is not constant. It changes with price, with time and with volatility, so the position you opened is not the position you hold a week later.
And it is not free to be long. Owning curvature costs decay every day, which is the trade rather than a side effect.
When it fails
A gap is the short-gamma nightmare. There is no opportunity to adjust, the position deteriorates at an accelerating rate, and it is already several points worse before anything can be done.
Which is why short-gamma strategies have the shape they do: a long run of small collected premiums and an occasional loss large enough to remove several of them. The credit spread and iron condor pages describe that shape in their own terms.
The second failure is selling short-dated options for the premium. They carry the most gamma, which is precisely why the premium looks generous relative to the days remaining.
A third is assuming a hedge holds. A delta-neutral position with significant gamma is neutral only at this price, and it stops being neutral the moment price moves.
A fourth is measuring risk at today’s price. A short option that looks comfortable now can have a completely different profile after a modest move, because gamma has changed what the position is.
And a fifth is holding a short option through expiry week out of habit. The premium remaining is smallest exactly when the gamma is largest, so the last few days offer the least reward for the most risk in the position’s entire life.
The practical rule that falls out of all this is short. If you are short options, the final week near the strike is where the exposure is worst and the remaining premium is smallest — closing early gives up little and removes a lot. If you are long, the same week is where the position is most capable of moving, which is the one argument for staying in it.
The original data
15 of the 24,971 videos measured for this site cover gamma, at a median of 18,398 views — the same supply as delta and a similar median, on a concept that is considerably harder and considerably more consequential.
The gamma curve on this page was computed for it, on the same 80-to-120 sweep used for delta, and it peaks near the 100 strike. The specific height depends on the stated assumptions; the hump shape and its position do not, and knowing where the hump sits is what tells you when a position is about to become someone else’s problem.
Related
Delta is the quantity gamma is the rate of change of. Theta is what being long gamma costs every day. And options expiry is when all of this reaches its maximum.
Gamma is the reason a position I had checked on Friday was unrecognisable on Monday. Nothing about my analysis had changed; the option had simply moved from the flat part of its curve to the steep part while I was not looking.
— Michael Whitman
This page is educational, not financial advice. Test every idea on your own charts before risking money.