Whats the equation to calculate the area under the curve of a normal distribution, given an upper and lower standard deviation?

Lets say I want to find out the area under the graph of normal distribution curve, between X1=standard deviation of -0.5 and X2 = standard deviation of 0.5. Is there a formula for this?

Case study: find the percentage chance of a stock remaining within +0.5 and -0.5 standard deviations within one trading day, given the daily implied volatility of that stock.

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This is simply the integral of the pdf from -0.5 to 0.5 (scaled to the SD of the distribution), also known as the cumulative distribution function or cdf. The cdf(x) function is indicated on the following wikipedia link: Normal Distribution.

The normal cdf(x) function computes the integral on [-Infinity, x], so to compute on your interval [x1,x2], is simply cdf(x2) - cdf(x1).

So if you meant [-.5, +.5] SDs,then would evaluate cdf(0.5*sd) - cdf(-0.5*sd).

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I should note that the above assumes the mean is 0. If not 0 then add the mean in the arguments for the cdf. –  Jonathan Shore May 24 '11 at 12:01
... and for anything finance related, the time series is never stationary, and this technique is only relevant to stationary time series. –  Contango May 25 '11 at 10:48

The source code for the CDF is listed in the book "The Complete Guide to Option Pricing Formulas - 2nd. Edition - Espan Gaarder Haug, Ph.D (McGraw-Hill, 2006, ISBN 9780071389976)".

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You can also find implementations in QuantLib and many other open source packages. –  Jonathan Shore May 24 '11 at 12:02
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