Does a shorter or longer option carry more gamma

Question

Two vanilla options are identical except for their time to expiry. Which one has the bigger gamma?

Short answer

Near the money, the shorter-dated one. At the money $\Gamma \approx \frac{\phi(d_1)}{S\sigma\sqrt T}$ grows like $1/\sqrt T$ as expiry approaches, because delta must snap from near $0$ to near $1$ over a narrowing price range. Far from the money the ranking can flip: a short-dated deep in- or out-of-the-money option has almost no gamma, while a longer one still does.

Adapted from Crack, Heard on the Street, Q2.2, p.23; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Greeks and Numerical Sensitivities
Derivation — From Binomial Trees to Black-Scholes in the Continuous Limit
The Black-Scholes Formula Derived via Change of Measure

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