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Is this true and how would you prove it

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  • $\begingroup$ Did you mean "monotonic"? $\endgroup$ Nov 10 '14 at 11:15
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The OTM binary is not increasing with volatility. Simply plot the price as a function of implied volatility. eg take

S = 0.99 K = 1 r=0 T =1

and $\sigma$ vary upwards in steps of 0.01 starting at 0. It peaks around 0.04.

Differentiating $d_2$ with respect to $\sigma$ makes this behaviour obvious.

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