Calculating Normal Probability Between Two Standardized Bounds

Summary

The document explains how to find the probability mass under a normal distribution between a lower and upper bound. It uses the cumulative distribution function (CDF), which gives the accumulated probability from negative infinity up to a chosen value. Subtracting the CDF at the lower bound from the CDF at the upper bound yields the probability within the interval.

The motivating application is estimating the chance that a stock’s one-day move stays within half a standard deviation in either direction, using daily implied volatility. The answer states that the bounds must be expressed on the scale of the distribution’s standard deviation. This is a probability calculation, not evidence that actual stock returns follow a normal distribution. The excerpt does not discuss how implied volatility maps to a return distribution, nor does it address skew, jumps, or other departures from the normal assumption.

Key ideas

• The probability between two bounds is the CDF at the upper bound minus the CDF at the lower bound.
• The CDF gives cumulative probability from negative infinity through a specified value.
• Standardized bounds must be scaled by the distribution’s standard deviation when using unstandardized inputs.
• The stock-move example applies the calculation to a range around zero using implied volatility.
• The method relies on a normal distribution assumption and does not assess its fit to returns.

Tags

# Answer by Jonathan Shore (score 5, accepted)


# 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.

## Answer by Jonathan Shore (score 5, accepted)

https://quant.stackexchange.com/a/1221

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)`.

## Answer by Contango (score 0)

https://quant.stackexchange.com/a/1223

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)".

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.