Simply stated, we can define volatility as a 1 standard deviation (StdDev) price change over the course of one trading year. The value is normally expressed as a percentage of the security price. If a stock trades at $100 with a volatility of 30%, a 1 standard deviation change in a 1 year time frame will raise the price to $130 or lower the price to $70. Using the normal distribution, we can assume a 68% chance that the final price will fall within this range. Some adjustment for interest rates is also necessary. If the risk-free rate of return during this time frame is 5%, a 1 standard deviation price change at year end would be $105 × 30% = $31.50. We would expect the stock to trade between $73.50 and $136.50 68% of the time. This view, however, is somewhat simplistic. A more precise definition of volatility is the standard deviation of the return provided over one year when the return is expressed using continuous compounding.

Returns are normally distributed, and future prices are lognormally distributed. The distinction is important. If prices were normally distributed, a $20 stock would have the same probability of rising to $50 as falling to –$10. Clearly this cannot happen. Suppose, however, that price changes were continuously compounded. Five 10% upward price changes would raise a $20 stock by $12.21 to $32.21. The corresponding downward price changes would reduce the price by $8.19 to $11.81. The continuously compounded upward change is 61%, and the corresponding downward change is only –41%. The distribution of final prices is skewed so that no price ever falls below zero. The continuous compounding of normally distributed price changes will cause the prices at maturity to be lognormally distributed.

Assuming continuous compounding, it can be shown that volatility is proportional to the square root of time. A 1 standard deviation change for any time frame is given by the following formula, where σ is the annual volatility and ∆t is the percentage of a year: σ√∆t

Returning to the example of a $100 stock with 30% annual volatility, we can calculate the size of a 1 standard deviation change in one week (1/52 year): .30 x√.0192 = .0416 x $100 = $4.16

Much debate has occurred over the length of a trading year and the number that should be used for volatility calculations. Although a calendar year has 365 days, there are actually only 252 trading days. We will use a 252-day trading year when calculating volatility based on daily price changes. This approach makes sense, because there are 252 daily closing prices each year, and historical volatility is based on close-to-close price changes. However, some of the Black-Scholes calculations are more granular in the sense that they use the number of minutes remaining until options expiration. These calculations specifically take into account all the remaining time, including weekends. The difference becomes significant as expiration approaches, because the final weekend of the cycle represents two of the seven remaining days. Simply stated, we can define volatility as a 1 standard deviation (StdDev) price change over the course of one trading year.

The volatility edge in options trading – Jeff Augen.
ISBN 0-13-235469-1