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according to the BS formula, $\gamma = \frac{N'(d_1)}{S_0\sigma\sqrt{T}}$, gamma will decrease when volatility increase.

How does it intuitively make sense? rather than from the formula.

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  • $\begingroup$ $d_1$ is a function of the volatility. So even if you restrict yourself to the BS model, ignoring smile dynamics, it is not true that gamma will always decrease when when volatility increases. I would be glad to read a practitioner's opinion about sensibility of gamma wrt vol though. $\endgroup$ – AFK Jan 30 '15 at 22:03
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For ATMish options, as vol goes higher, the option looks even more ATM. That is, at higher vol, the difference between a 99% strike option and a 100% strike option is less pronounced than if vol were low; hence your deltas won't change as fast as spot moves and thus, less gamma.

For far OTM options, its the opposite. They would have very little delta at low vol, and remain dead even as you start to move a little closer to them, so have little gamma. However as vol picks up, its like the options are less OTM, so start to have some more gamma.

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