# Volatility Defined

Volatility is a measure of price changes without regard to direction. With volatility, it is the percentage change that matters, not the absolute amount of change, the stock price, or the direction.This nondirectional nature of volatility can be difficult to grasp for traders who tend to think in terms of direction and in terms of good and bad. A trader with a bullish opinion, for example, views a price rise as good and a price decline as bad.

A trader with a bearish opinion thinks the opposite. Regardless of the size of the movement, a movement in the “right direction” is good. Years of trading with this mind-set can impede one’s full understanding of the nondirectional nature of volatility.A second complicating aspect of volatility is that one price change,in and of itself, is not important. Rather, only a series of price changes over several trading days, evaluated together, determines a stock’s volatility.One day’s price change is just one number; volatility describes a series of numbers.

Historic Volatility:Mathematicians look at a series of price changes over several days,weeks, or months and derive what is called the standard deviation of movement. Mathematically, for option traders, historic volatility is the annualized standard deviation of daily returns over a specific time period. A standard deviation is the average difference between each of the daily returns and the mean return over the period observed.

Do not let this definition intimidate you because the following discussion is conceptual, not mathematical.To make meaningful comparisons of volatility,the exact observation period must be specified. Daily closing prices typically are used, but daily opening prices or weekly closing prices or some other consistent method of observation also could be used.Comparing one specific price change with another seems like a simple process, but comparing two series of prices changes is moredifficult.

Stock 1 trades in a narrow range around \$100, whereas stock 2 falls fairly quickly below \$94, then rises to \$110, and then falls back to \$100.The question is: “Is stock 1 or stock 2 more volatile?” Take a moment to reflect on this question, and then compare your answer, which is based on your own subjective, visual evaluation with the technical answer that is presented below.The historic volatility of stock 1 is calculated from the information in Table.

The left column, “Day,” simply assigns a number to each closing price; in the real world, this number would be a date. The middle column, “Closing Price,” contains the 31 closing prices that are plotted in Figure 7-1. The right column, “Daily Return,” contains percentage changes in price from the previous day’s price.The daily return takes two steps to calculate. The closing price of the previous day is subtracted from the closing price of the current day, and then the difference is divided by the closing price of the previous day.

The daily return for day 1 of 1.80 percent, for example, is calculated as follows: The closing price on day 0 of 100 is subtracted from the closing price on day 1 of 101.80 to yield a difference of 1.80. This difference then is divided by the closing price on day 0 of 100. The result is 1.80, or 1.80 percent. There is no daily return for day 0 because this day marks the first price observation; the previous price is unknown.Using the data in the right column in Table 7-1, the standard deviation of these daily returns can be calculated.

A standard deviation is a measure of the spread of values in a set of data. In practice, the standard deviation of these daily numbers is converted to an annual standard deviation by multiplying it by the square root of the number of days in a year. This calculation produces 37.55 percent, which is shown at the bottom of the table. The calculation of a standard deviation is a standard spreadsheet function, so the mathematically inclined may easily do their own research. However, if you are not mathematically inclined, do not worry; Op-Eval Pro performs the many important volatility calculations.