# Historical Volatility Calculator

The annualized volatility of stock prices can be calculated from historical data using several formulas. If you are inputing the annual volatility into the Black-Scholes formula, you use the standard deviation of the logarithmic returns since the Black-Scholes model assumes that stock prices are distributed log-normally. (And thus their returns are distributed normally.)

For other models or applications, you may use the mean absolute deviation of the daily returns. The mean absolute deviation is more robust than the standard deviation because extreme values are not squared.

To convert fluctuations in daily returns to an annual number, you multiply by the sqaure root of 252, which is the number of trading days in a year. The equations are explained below and built into the calculator on the left. To compute historical volatility with the calculator, simply enter the daily stock prices (not the daily returns) as a comma separated list starting with the oldest price.

### Standard Deviation of Logarithmic Returns (Black-Scholes)

To compute the Black-Scholes volatility from historical stock price data, you first compute the daily logarithmic returns. If X_{i}and X

_{i+1}are consecutive prices, then the i

^{th}logarithmic return is

L

_{i}= ln(X

_{i+1}/X

_{i})

If there are N + 1 stock prices, then there will be N returns. The standard deviation of these returns can be calculated by dividing by N (population standard deviation formula) or by N - 1 (sample standard deviation formula). Both are given in the calculator above.

To get the annualized volatility, multiply the standard deviation by sqrt(252) = 15.8745.

### Mean Absolute Deviation from the Mean

Another statistical measure of dispersion is the mean absolute deviation from the mean. If {p_{1}, p

_{2},..., p

_{k}} is a collection of

*k*data points and their arithmetic average is p, then the MAD is given by the equation

MAD = (1/k)Σ

_{j=1...k}|p

_{j}- p|

To compute the volatility of stock prices with this measure of dispersion, you first compute the daily returns R

_{i}with the formula

R

_{i}= (X

_{i+1}- X

_{i})/X

_{i}

Then you find the mean absolute deviation of these returns and annualize the number by multiplying by 15.8745.

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