Gamma: Why Delta Refuses to Stay Still

Delta tells you how much an option moves per dollar of stock. The catch is that delta does not hold still. As soon as 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. It is also the best explanation for why experienced sellers close positions before expiration week instead of squeezing out the last few dollars.

The delta of the delta

If delta is the option's speed, gamma is its acceleration. Formally, it is the rate at which delta changes as the stock price changes.

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

Gamma is positive for anyone long an option and negative for anyone short one, whether it is a put or a call. That is the whole subject in one line: buyers own the acceleration, sellers owe it.

Gamma peaks at the money and spikes near expiry

Drag the expiration slider down in the widget. At 90 days the gamma curve is a broad, low hill: no strike is settled, so delta drifts gently. At 3 days it is a narrow spike sitting on the strike.

The reason is the same one behind the extrinsic value hump. Near expiration, delta stops being a smooth curve and starts behaving like a switch. In the money heads for 1.00, out of the money heads for 0.00, and the transition between them squeezes into a narrower and narrower price range. Gamma measures how steep that transition is, so it has to blow up as the range shrinks.

Look at the bottom row. On the last 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 what ties this course together. The strikes and expirations that pay the most theta are the ones carrying the most gamma. That is not a coincidence or a bug in the model. Both greeks come out of the same curvature in the option's price.

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

What sellers do about it

Close before expiration week. Exiting around 21 days out is a gamma decision, not a theta one. You give up the steepest part of the decay curve to avoid the steepest part of the gamma curve. The managing winners and losers lesson does the arithmetic.

Size for the delta you might have, not the one 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 big at 0.45, it was too big at 0.16.

Take more time when you are unsure. Longer-dated contracts have lower gamma, which lets you be wrong slowly rather than all at once.

What can go wrong

Judging risk by the delta you opened with. 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 premium left is small and the gamma is enormous. Pound for pound it is the worst risk 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.