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For a stock trading at $27, $28 strike, 0% interest, 15% annual vol, and one day until expiration there is about a 1 in 17000 chance of it being exercised?

$d_2 = \frac{1}{.15\sqrt{1/252}}\left[\ln\left(\frac{27}{28}\right) + \left(0 - \frac{.15^2}{2}\right)(1/252)\right]$

$1/(0.5 (1 + Erf[(Log[27/28] - (1/252) (0.15^2/2))/(0.15 *2^{1/2} Sqrt[1/252])]))$

that seem way too small but it's the answer I got

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up vote 9 down vote accepted

With $15\%$ annual volatility we have $15\%/\sqrt{252}\approx0.94\%$ daily volatility. To go from $27$ to $28$ is a $1/27\approx 3.7\%$ move which is $3.7/0.94\approx 3.9$ standard deviations. For a normal distribution this is about $0.005\%$ probability which is in line with your result.

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Nice answer, practical, concise and to the point. +1 for this!!! – Matt Wolf Jul 15 '13 at 6:34

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