Why gamma explodes for an at-the-money option at expiry

Question

You are short one 100-strike call that expires tomorrow. The stock is sitting at exactly \100.00. Why is this the single most dangerous place for a short-gamma position, and what is "pin risk"?

Accepted Answer

Gamma blows up at the money as time runs out. For an at-the-money option, gamma is approximately \Gamma \approx \frac{1}{S\,\sigma\sqrt{2\pi T}}, which grows without bound as T 	o 0. Intuitively, right at expiry the call's delta is a step function: it is 0 if the stock is a hair below 100 and 1 if a hair above. Delta flips from 0 to 1 over pennies, and gamma (the rate of that flip) is enormous. For you, short that call, this means your hedge is impossible to keep up with. A \0.30 move can swing your delta from near 0 to near 1, forcing you to buy or sell a huge slug of stock, and if the stoc…

Why gamma explodes for an at-the-money option at expiry

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