By John Summa, CTA, PhD, Founder of OptionsNerd.com

Volatility is both an input to valuation models (statistical/historical) and an output (implied). Just why this is so will become clearer once the difference between both volatility types is understood. This tutorial segment will focus on historical volatility, which is also known as statistical volatility (SV). Historical volatility is a measure of the volatility of the underlying stock or futures contract. It is known volatility, because it is based on actual, recent price changes of the underlying.

Historical volatility can be thought of as the speed (rate of change) of the underlying stock price. Like a car speeding along at 75 mph (rate of change per hour), a stock or futures contract moves at a speed that is measured as a rate too, but a rate of change per year. The higher the historical volatility, the more movement the stock has experienced and, therefore, theoretically, the more it can move in the future, although this does not provide insight into either direction or trend. While there are different ways to calculate historical volatility (different parameter settings just like with any technical indicator) the basic idea underlying different calculations is fundamentally the same.

Historical volatility essentially is a way to tell how far the stock or future might move in the future based on how fast it has been moving in the recent past. Thinking in terms of a car traveling at 75 mph again, we know that in one year, this car will have traveled a distance of 657,000 miles (75 x 24 hours x 365 days = 657,000). But the catch here is that the rate of change of 75 mph may not stay the same, and it doesn't tell us much about the direction of car (it could be going back and forth, not just in one direction, meaning it could end up where it began). This is true for stocks or futures as well. But the calculation clearly depends on recent speeds, which means percentage price changes on a daily basis. If these speeds are increasing, historical volatility will generally be greater.

Calculating Historical Volatility
By walking through a calculation of historical volatility, the above description should become more tangible. Historical volatility is a quantifiable number based on past changes in the price of the stock or futures contract. It can be calculated simply by taking the past prices, in this example 10 days are used, and price changes (from close to close), and then taking an average of those price changes in percentage terms. Once we have an average percentage price change over 10 days, we can subtract the daily percentage price changes from this average change to derive deviations from the daily average change for the 10-day period.

Figure 1 contains an example using this method. The calculation involves use of a few concepts you might have encountered in a Statistics 101, such as mean, standard deviation and even square roots. The historical volatility calculation here uses the past 10 days of price changes. Typically a 10-, 20- or 30-day period of past price changes is analyzed

 Date IBM Close %Change 7/3/2007 106.58 - 7/5/2007 108.05 1.38 7/6/2007 109.03 0.91 7/9/2007 108.97 -0.06 7/10/2007 108.63 -0.31 7/11/2007 109.1 0.43 7/12/2007 109.28 0.16 7/13/2007 108.6 -0.62 7/16/2007 109.66 0.98 7/17/2008 110.77 1.01 7/18/2007 111.08 0.28
 Figure 1: IBM closing prices and daily percent price changes.

As you can see in Figure 1, the middle column contains daily closing prices for IBM between July 3 and July 18, 2007. The right-hand column contains the daily price changes (calculated by subtracting yesterday's price from today's closing price and dividing by yesterday's closing price). This is the raw material for computing historical volatility.

The data from Figure 1 is then used to compute the standard deviation of daily price changes, which can be done easily in an Excel spreadsheet using the STDEV function. Using the IBM price data for the calculation, Figure 2 presents current 10-day historical volatility for IBM - let's suppose it's 10.29%. The number is an annualized figure that is derived by multiplying the standard deviation calculation of .645 (computed from the daily percent price changes found in Figure 1) by 15.937 (the square root of 254 trading days in a year). The standard convention is to annualize historical volatility (and implied volatility); it's an easy task using the square root function in an Excel spreadsheet. (To learn more, read Microsoft Excel Features For The Financially Literate.)

 Standard Deviation Statistical Volatility 0.645 15.937 10.29%
 Figure 2: IBM statistical volatility using prices presented in Figure 1

One additional thing to keep in mind is that while we are talking about the speed of the underlying, it is interpreted as an expected 10.29% up or down potential move at current price changes over the next 254 trading days.

Let's now cook up some numbers to see how it changes historical volatility, which should facilitate a better feel for volatility. Figure 3 shows new hypothetical closing prices for IBM, with the resulting new standard deviation and historical volatility results in Figure 4.

 Date IBM Close %Change 7/3/2007 111.99 7/5/2007 110.66 -1.19 7/6/2007 107.6 -2.77 7/9/2007 109.28 1.56 7/10/2007 108.1 -1.08 7/11/2007 105.63 -2.28 7/12/2007 106.97 1.27 7/13/2007 108.03 0.99 7/16/2007 110.05 1.87 7/17/2008 107.58 -2.24 7/18/2007 104.01 -3.32
 Figure 3: IBM closing prices and daily percent price changes.

Figure 3 presents IBM prices that have larger percentage price changes most of the entire period (up and down). When you compute standard deviation based on the new daily price changes seen in Table 3 in the right-hand column, you get 1.97, up from .645. This simply reflects greater dispersion around the average price change (not shown here), which will therefore lead (when multiplied by 15.937 (square root of 254), to a new historical volatility of 29.75%.

IBM moved from 106.58 to 111.08 during the 10-day period, as seen in Figure 1, while the hypothetical data in Figure 3 has IBM moving down from 111.99 to 104.01. The size of daily price changes was altered to illustrate how that will impact historical volatility.

When a big cap stock is having a bearish cycle, the daily price swings typically will expand, generating higher historical volatility. This hypothetical case attempts to capture that phenomenon. We'll go back to this point toward the end of this tutorial on using volatility to predict market direction.

 Standard Deviation Statistical Volatility 1.97 15.937 31.40%
 Figure 4: IBM statistical volatility using prices presented in Figure 3.

There are a few issues worth mentioning here that go beyond the scope of this tutorial. A number of different methods are used to calculate historical volatility. Here close-to-close percentage changes in daily prices were used, but some have used high-minus-low prices, and even an average of high, low and close prices, in an attempt to capture some intraday information that is missing when close prices are used.

Another issue regards the calculation of historical volatility and trending versus trading range markets. It is possible for a strong trend to develop up or down without any change in the size of daily percentage price changes (let's say the magnitudes remain the same but more positive days develop). The average of daily price changes can become larger while historical volatility calculated using the popular method outlined becomes smaller. Likewise, it could be shown here, too, that if average daily price change declines in size, historical volatility can rise even when the market is trending stronger.

Conclusion
Now you've learned how to calculate statistical volatility (historical volatility) using a popular method. Ten days of daily percentage price changes were computed using closing prices from IBM, from which a standard deviation was calculated. The standard deviation was then used to derive historical volatility, which is nothing more than annualized standard deviation of daily percentage price changes. While 10 days was used in the example here, 20 and 30 days are also common time frames used to calculate historical volatility. We can now move on to discuss historical volatility's cousin, implied volatility, which is covered in the next section.

Options Volatility: Projected or Implied Volatility
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