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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 implied volatility|volatility implied by the market].

1 Defined

The annualized volatility is proportional to standard deviation of the instrument's returns by the square-root of time period of the returns:

,

where is time period in years of returns. The generalized volatility for time horizon is expressed as:

.

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

.

The monthly volatiliy (i.e., of a year) would be

.

2 External links

• The concept of volatility in theories of finance

3 See also

• Beta coefficient
• Derivative security
• Financial economics
• Implied volatility
• Risk
• Standard deviation