Gamma: how fast delta changes
Lesson 9 · about 9 min
Delta tells you how an option moves for the first dollar. Gamma tells you how that changes for the second dollar. It is the Greek that makes long options accelerate in your favour and short options accelerate against you, and it is why the same position feels harmless one week and dangerous the next.
The definition
Gamma is the change in delta for a $1 change in the stock.
XYZ at $50, the $50 call has delta 0.52 and gamma 0.08.
- XYZ rises to $51: delta becomes about 0.52 + 0.08 = 0.60.
- XYZ rises to $52: delta becomes about 0.68.
- XYZ falls to $49: delta becomes about 0.44.
Long options (calls or puts) always have positive gamma. Short options always have negative gamma. Stock has zero gamma; its delta is always exactly 1.00.
Why gamma is the buyer's friend
Track the $50 call as XYZ climbs, using delta plus gamma to update at each step:
| XYZ | Delta at start of step | Gain on this $1 (approx) | Cumulative gain |
|---|---|---|---|
| 50→51 | 0.52 | $0.56 | $0.56 |
| 51→52 | 0.60 | $0.64 | $1.20 |
| 52→53 | 0.68 | $0.72 | $1.92 |
| 53→54 | 0.76 | $0.80 | $2.72 |
(Each step's gain uses the average delta across the step, so 0.52 + 0.04 = 0.56 and so on.)
Each dollar of stock move earns more than the last. Meanwhile on the way down:
| XYZ | Delta at start of step | Loss on this $1 (approx) | Cumulative loss |
|---|---|---|---|
| 50→49 | 0.52 | $0.48 | $0.48 |
| 49→48 | 0.44 | $0.40 | $0.88 |
| 48→47 | 0.36 | $0.32 | $1.20 |
| 47→46 | 0.28 | $0.24 | $1.44 |
Each dollar of adverse move costs less than the last. A $4 rally earned $2.72; a $4 sell-off cost $1.44. That asymmetry is what you are paying for with extrinsic value, and gamma is the name for it.
For a short option, flip every sign: gains shrink as the stock moves in your favour and losses grow as it moves against you.
Where gamma lives
Gamma is highest at the money and falls off in both directions. It is also much higher with little time left, because near expiration an ATM option has to decide quickly whether it is a share of stock or nothing.
| Days to expiry | Gamma of ATM $50 option | Gamma of $56 call | Gamma of $44 call |
|---|---|---|---|
| 90 | 0.045 | 0.030 | 0.025 |
| 30 | 0.080 | 0.035 | 0.020 |
| 7 | 0.160 | 0.020 | 0.005 |
| 1 | 0.420 | 0.002 | 0.000 |
With one day left, the ATM option's delta swings 0.42 for each $1 move. That means a $1 move in either direction takes an ATM option from behaving like 50 shares to behaving like 92 shares or 8 shares. For a buyer, this is the "lottery ticket" acceleration of expiration day. For a seller, it is the reason a short ATM option in the last days can go from almost worthless to a full stock-sized loss on a small move.
Gamma and theta are the same bill
Positive gamma is not free. The option that gains more on rallies than it loses on sell-offs must lose value when the stock does nothing, or nobody would sell it. That steady loss is theta, the next lesson. Long gamma pays theta; short gamma collects theta. Every option position is somewhere on that trade, and the two Greeks scale together: the more gamma you own, the more theta you pay per day.
Key idea: Gamma is why long options gain faster than they lose and short options lose faster than they gain. It concentrates at the money and explodes into expiration, and it is paid for, day by day, in theta.
Practical uses
Judging a short position's danger. A short 30-day ATM option with gamma 0.08 is manageable; the same strike with two days left and gamma 0.30 is a position where a $2 move can double the loss. Many premium sellers close or roll short options with about a week to go for exactly this reason, forgoing the last bit of decay to avoid the worst of the gamma.
Understanding a long option's behaviour. If you own an OTM call with 60 days left, its gamma is small; the stock has to move a fair way before the call starts behaving like stock. If the stock rallies to your strike with a week left, gamma is now large and every tick matters, in both directions. Plans should account for the fact that the position you hold in week eight is not the position you bought in week one.
Netting. Like delta, gamma nets across a position. A vertical spread (Module 6) has a long and a short option, so its gamma is small and changes sign depending on where the stock sits. That is the reason spreads are calmer than single options.
A gamma worked example, short side
You sold a $50 put for $1.85 with XYZ at $50 and 30 days left. Delta −0.48, gamma 0.08. XYZ drops $3 in a day to $47.
- Rough delta path: −0.48 at $50, −0.56 at $49, −0.64 at $48, −0.72 at $47.
- Rough loss: 0.52 + 0.60 + 0.68 = $1.80 per share, so the put is now worth about $3.65 and you are down $180 on a $185 credit.
- Your short position is now behaving like 72 short shares, not 48, and the next $1 down costs more than the last.
The premium you collected was nearly wiped out by a single 6% move, and the position is more dangerous now than when you opened it.
Try it: Pick an ATM option with about 30 days left and note its delta and gamma. Estimate the delta after a $2 rise and a $2 fall. Then look at the same strike expiring this week and repeat. Write a sentence about how differently the two would feel to hold overnight.
Recap
- Gamma is the change in delta per $1 move; long options have positive gamma, short options negative.
- Positive gamma makes gains accelerate and losses decelerate; negative gamma does the reverse.
- Gamma peaks at the money and rises sharply as expiration approaches.
- Long gamma is paid for with theta; short gamma is compensated by theta. They are two sides of one price.
- A short option near expiration and near the money is the highest-gamma, highest-danger position in basic options trading.
See it drawn
Original diagrams for the ideas on this page. Illustrative, not real market data.