Call gamma as expiry approaches

Question

Describe how the gamma of a vanilla call evolves as time to expiry shrinks.

Short answer

Near the money it rises sharply; away from the money it decays towards zero. Gamma is $\tfrac{\varphi(d_1)}{S\sigma\sqrt\tau}$ . For an at-the-money call it grows like $1/\sqrt\tau$ , blowing up at expiry, while for strikes far from spot $\varphi(d_1)$ vanishes faster, so gamma concentrates into a spike at the strike.

Adapted from Joshi, Denson & Downes, Quant Job Interview Questions and Answers, Q2.44, p.20; 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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