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Volatility is the standard deviation of the change in value of a financial instrument with a specific time horizon. It is often used to quantify the risk of the instrument over that time period. Volatility is typically expressed in annualized terms, and it may either be an absolute number (100$ +- 5$) or a fraction of the initial value (100$ +- 5%).
For a financial instrument whose return follows a Gaussian random walk, or Wiener process, the volatility increases by the square-root of time as time increases. Conceptually, this is because there is an increasing probability that the instrument's price will be farther away from the initial price as time increases. Mathematically, this is a direct result of applying Itô's lemma to the random process.
Historical volatility is the volatility of a financial instrument based on historical returns. This phrase is used particularly when it is wished to distinguish between the actual volatility of an instrument in the past, and the current volatility implied by the market.
Defined
The annualized volatility <math>\sigma<math> is proportional to standard deviation <math>\sigma_{SD}<math> of the instrument's returns by the square-root of time period of the returns:
<math>\sigma = {\sigma_{SD}\over\sqrt{P}}<math>,
where <math>P<math> is time period in years of returns. The generalized volatility <math>\sigma_T<math> for time horizon <math>T<math> is expressed as:
<math>\sigma_T = \sigma \sqrt{T}<math>.
For example, if the daily returns of a stock have a standard deviation of 0.01 and there are 252 trading days in a year, then the time period of returns is 1/252 and annualized volatility is
<math>\sigma = {0.01 \over \sqrt{1/252}} = 0.1587<math>.
The monthly volatiliy (i.e., <math>T = 1/12<math> of a year) would be
<math>\sigma_{month} = 0.1587 \sqrt{1/12} = 0.0458<math>.
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See also
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