Imagine there are two stocks: Stock A and Stock B. Both are trading at around $100 a share. Over the past month, a typical end-of-day net change in the price of Stock A has been up or down $5 to $7. During that same period, a typical daily move in Stock B has been something more like up or down $1 or $2. Stock A has tended to move more than Stock B as a percentage of its price, without regard to direction.

Therefore, Stock A is more volatile—in the common usage of the word—than Stock B. In the options vernacular, Stock A has a higher historical volatility than Stock B.Historical volatility (HV) is the annualized standard deviation of daily returns. Also called realized volatility, statistical volatility, or stock volatility,HV is a measure of how volatile the price movement of a security has been during a certain period of time.

But exactly how much higher is Stock A’s HV than Stock B’s?In order to objectively compare the volatilities of two stocks, historical volatility must be quantified. HV relates this volatility information in an objective numerical form. The volatility of a stock is expressed in terms of standard deviation.Standard Deviation: Although knowing the mathematical formula behind standard deviation is not entirely necessary, understanding the concept is essential. Standard deviation, sometimes represented by the Greek letter sigma (σ), is a mathematical calculation that measures the dispersion of data from a mean value.

In this case, the mean is the average stock price over a certain period of time.The farther from the mean the dispersion of occurrences (data) was during the period, the greater the standard deviation.Occurrences, in this context, are usually the closing prices of the stock.Some utilizers of volatility data may use other inputs (a weighted average of high, low, and closing prices, for example) in calculating standard deviation.Close-to-close price data are the most commonly used.

The number of occurrences, a function of the time period, used in calculating standard deviation may vary. Many online purveyors of this data use the closing prices from the last 30 consecutive trading days to calculate HV. Weekends and holidays are not factored into the equation since there is no trading, and therefore no volatility, when the market isn’t open. After each day, the oldest price is taken out of the calculation and replaced by the most recent closing price.

Using a shorter or longer period can yield different results and can be useful in studying a stock’s volatility.Knowing the number of days used in the calculation is crucial to understanding what the output represents. For example, if the last 5 trading days were extremely volatile, but the 25 days prior to that were comparatively calm, the 5-day standard deviation would be higher than the 30-day standard deviation.

Standard deviation is stated as a percentage move in the price of the asset. If a $100 stock has a standard deviation of 15 percent, a one-standarddeviation move in the stock would be either $85 or $115—a 15 percent move in either direction. Standard deviation is used for comparison purposes. A stock with a standard deviation of 15 percent has experienced bigger moves—has been more volatile—during the relevant time period than a stock with a standard deviation of 6 percent.

When the frequency of occurrences are graphed, the result is known as a distribution curve. There are many different shapes that a distribution curve can take, depending on the nature of the data being observed. In general,option-pricing models assume that stock prices adhere to a lognormal distribution.The shape of the distribution curve for stock prices has long been the topic of discussion among traders and academics alike.

Regardless of what the true shape of the curve is, the concept of standard deviation applies just the same. For the purpose of illustrating standard deviation, a normal distribution is used here.When the graph of data adheres to a normal distribution, the result is a symmetrical bell-shaped curve. Standard deviation can be shown on the bell curve to either side of the mean. Exhibit 3.1 represents a typical bell curve with standard deviation.

Large moves in a security are typically less frequent than small ones.Events that cause big changes in the price of a stock, like a company’s being acquired by another or discovering its chief financial officer cooking the books, are not a daily occurrence. Comparatively smaller price fluctuations that reflect less extreme changes in the value of the corporation are more typically seen day to day. Statistically, the most probable outcome for a price change is found around the midpoint of the curve. What constitutes a large move or a small move, however, is unique to each individual security.