## Estimating Volatility from Historical Data

Volatility (θ) is the standard deviation of the return provided by the stock in 1 year using continuous compounding.  Stocks typically have a volatility between 15% and 60%.

The standard deviation of the % change in stock price in time T is:
θ√T

The standard deviation increases by the square root of time (T) i.e. 4 weeks is about 2* the standard deviation of 1 week.  The equation to estimate volatility is the following:
where, ui = ln(Si / Si-1)
s = √[(Σni=1 u2i) / (n-1) – (Σni=1 ui)2 / n(n – 1)]

## Geometric Mean Equation for Calculating Multi-Period Returns

When calculating the return of \$100 invested in a fund for 5 years of varying returns the geometric mean has to be used.  It is more appropriate than the arithmetic mean because it calculates the mean of proportional or exponential growth.  The arithmetic mean finds the average returns and annually compounds it while the geometric mean takes into consideration the nonlinear nature of rates of return and compounds them continuously.

In some jurisdictions, regulations require fund managers to report returns this way because it reflects the actual average return realized over a period of time.  The geometric mean is always less than the arithmetic mean so many fund managers mislead the public by simply taking the arithmetic mean of each year’s return.

We begin with the following function for calculation of distribution of the continuously compounded rate of return:
x ~  φ(μ – θ2/2, θ/√T)

This is equivalent to the following geometric mean equations shown below:
ni=0 ai)1/n = exp(1/n * Σni=0 ln ai) = E(x) = μ – θ2/2

The geometric mean = the log-average = E(x), the expected value of the rate of return

Link to Example: Rate of Return Calculations

## Normal Distribution Equation

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.