Gamma: Why Delta Refuses to Stay Still

Delta tells you how much an option moves per dollar of stock. The problem is that delta will not hold still. The moment the stock moves, delta is a different number, and gamma is how fast that happens.

This is the greek that turns a quiet short put into a stock position overnight, and it is the single best explanation for why experienced premium sellers close positions before expiration week rather than squeezing out the last few dollars.

Delta of the delta

If delta is the speed of the option, gamma is its acceleration. Formally it is the rate of change of delta with respect to the stock price.

A put with delta −0.30 and gamma 0.04 becomes a −0.34 delta put if the stock falls a dollar, and roughly −0.38 if it falls another. Your directional exposure grew by 8 shares per contract while you did nothing at all.

Gamma is always positive for anyone who is long an option and negative for anyone short one, regardless of whether it is a put or a call. That asymmetry is the entire subject: buyers own acceleration, sellers owe it.

Gamma peaks at the money, and explodes near expiry

Move the expiration slider down in the widget above. At 90 days the gamma curve is a broad, low hill, no strike is very certain, so delta drifts gently. At 3 days it is a narrow spike centred on the strike.

The reason is the same one behind the extrinsic value hump. Near expiration, delta stops being a smooth curve and starts becoming a switch: in the money is heading for 1.00, out of the money is heading for 0.00, and the transition between them compresses into a smaller and smaller price range. Gamma measures the steepness of that transition, so it necessarily blows up as the range narrows.

Read the bottom row. On the final day, a 2% move in the stock adds 37 shares of exposure per contract. You did not choose that; the calendar chose it for you.

Theta and gamma are the same trade

Here is the part that ties this track together. The strikes and expirations that pay the most theta are exactly the ones carrying the most gamma. It is not a coincidence or a coding error in the model. Both greeks derive from the same curvature in the option’s price.

Selling a 3-day at-the-money option gives you spectacular daily decay and a position that reprices violently on every tick. Selling a 45-day option 16 delta out gives you modest decay and a position you can leave alone. There is no configuration that pays high theta with low gamma, and any strategy claiming otherwise has simply hidden the gamma somewhere.

What sellers actually do about it

Close before expiration week. The common practice of exiting around 21 days to expiration is a gamma decision, not a theta one. You surrender the steepest part of the decay curve specifically to avoid the steepest part of the gamma curve. See managing winners and losers for the arithmetic.

Size for the delta you might have, not the delta you opened with. A 0.16 delta put can be a 0.45 delta put after a bad week. If the position would be too large at 0.45, it was too large at 0.16.

Prefer more time when uncertain. Longer-dated contracts have lower gamma, which buys you the ability to be wrong slowly rather than suddenly.

What can go wrong

Judging risk by opening delta. It is a snapshot with a short shelf life, and gamma is the expiry date on it.

Holding short options through expiration day for the last $15. The remaining premium is small and the gamma is enormous. This is the worst risk-per-dollar in premium selling.

Selling very short-dated options because the annualised return looks huge. The number is real and so is the gamma paying for it. See weeklies against monthlies.