Not investment advice. Educational reading. See Disclaimer.
L.7 · BEGINNER · 3 MIN
Gamma in Depth: The Curvature That Moves Delta
Every reading you took from the delta column has a quiet caveat attached: it was measured at today's stock price and it does not survive the next move. Gamma is the Greek that tells you how fast the reading changes. That makes it the least intuitive of the four and the most consequential, because it is the reason a buyer's gains come in slightly better than the straight-line estimate and the reason a writer's losses come in considerably worse. Nothing here requires calculus. It requires one idea held steadily: delta is a live number, and gamma is the rate at which it moves.
An at-the-money call has a delta of 0.50 and a gamma of 0.06. The stock falls $3. What is the call's delta now, before any other effect?
§ 04Gamma peaks at the money and near expiration
Gamma is not spread evenly across the board. It concentrates where the outcome is most undecided: at strikes near the money, and in contracts near expiration. A deep in-the-money call is already behaving almost exactly like stock, so there is little room for its delta to change — low gamma. A far out-of-the-money call is so unlikely to matter that a small move barely alters its odds — also low gamma. The at-the-money contract sits on the knife edge where a modest move flips the whole probability picture, and in the final days that edge becomes a cliff: a strike can travel from behaving like 20 shares to behaving like 80 shares in a single session. This is why the last week before expiration is a different instrument from the same strike a month earlier, even though the ticker and the strike have not changed.
§ 05Why the buyer beats the straight-line estimate
You are long a call with a delta of 0.50 and a gamma of 0.06, and the stock rises $4. A straight-line delta estimate says $2.00 per share. What does the position actually make?
§ 06Short gamma is the same curve, pointed the other way
Every dollar of convexity a buyer collects is a dollar a writer pays, and the writer's version arrives on a worse schedule. A short call that opened roughly neutral becomes progressively more short the stock as the stock rallies, so the position gets bigger precisely when it is losing. A short put becomes progressively more long the stock as the stock falls. In both cases the exposure grows in the direction of the pain, which is the mechanical reason a small adverse move can turn into a large loss without any new information arriving. This is what practitioners mean by gamma risk, and it is why writing options without a defined-risk structure sits at the top of every broker's approval ladder. Buyers experience gamma as a pleasant surprise; writers experience it as the thing that turned a routine week into a bad one.
§ 07What gamma does to a naked call writer
You wrote one naked call. Your position delta is −0.30, so you are short about 30 shares of equivalent exposure, and gamma is 0.05. The stock rises $4. What is your delta now?
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Sit with the ideas.
You are long one at-the-money call with a delta of 0.50 and a gamma of 0.06. The stock rises $4. What is the position's approximate gain per share, and why does it beat the simple delta estimate?
