Normal Distribution Equation

Link to Normal Distribution Explanation

Base equation for derivation of normal distribution function:
ΔS/S ~ φ(μ Δt, θ√Δt)

Calculation of mean return and standard deviation with lognormal distribution:
ln ST ~ φ[ lnS0 + (μ – θ2/2)T, θ√T ]

Calculation of standard deviation intervals:
e(μ-s*θ) < ln ST < e(μ+s*θ)

Where  μ is the mean of ln ST, s is the # of std dev from the mean, and θ is the std dev of ln ST.

Link to Example: Normal Distribution Calculations

Calculating Price Moves Using The Normal Curve

Volatility is proportional to the square root of time so to approximate volatility over a shorter period of time you divide the annual volatility by the square root of the number of trading periods in a year.  Assume a volatility of 20% and underlying trade price of 100.

Daily Volatility: About 256 trading days in the year.  20%/√256 = 1.25
Weekly Volatility: 52 weeks in a year.  20%/√52 = 2.75

The standard deviation is represented as a percentage i.e. 1.25% and 2.75% so to get the distance from the mean in stock price terms the calculated volatility is multiplied by the current price of the underlying.  If we assume the stock is trading at 100 then one standard deviation for daily and weekly volatility would be 1.25% * 100 = 1.25 and 2.75% * 100 = 2.75 respectively.  Standard deviations are additive so to get the second standard deviation you multiply by 2.

The normal distribution curve shows there is a 68.2% chance of the underlying being traded within one standard deviation (2 out of 3 days), a 95.4% chance within two standard deviations (19 out of 20 days), and a 0.3% chance of being traded above two standard deviations (1 out of 20 days).

Applying this to the normal distribution curve we then see that the calculation shows us the theoretical likelihood of a move over time.  In the following example we look at daily moves.

So during one week (theoretically) a stock with volatility of 20% and price of 100 will trade:
between 101.25 and 98.75 about 2 out 3 days (3.41/days a week)
between 102.50 and 97.50 about 19 out of 20 days (4.77/days a week)
and, above or below 102.50 and 97.50 about 1 out of 20 days (.215/days a week)