# Understanding Historical Volatility

**Nick Pritzakis**

To truly be an effective options trader it's essential that you understand volatility. After all, having a sense of whether an option is "cheap" or "expensive" should help in your option strategy selection. For the most part, traders focus on two types of volatility, implied volatility and historical volatility. This article concentrates on the latter.

Actual volatility is the measure of the amount of randomness in the underlying stocks price at any point in time. However, actual volatility is instantaneous and cannot be measured. Because of this, traders associate volatility with specific time periods. For example, a trader may use daily, weekly or monthly periods in order to put volatility into context. This type of volatility is often referred to as historical volatility, statistical volatility and realized volatility (all three terms refer to the same thing).

In theory, asset price movements are normally distributed and the standard deviation of the distribution can be measured by historical data.

Source: http://en.wikipedia.org/wiki/Standard_deviation

The chart above depicts the normal distribution, also referred as the bell-curve. Standard deviation is a statistic that measures the amount of variability (randomness) around the mean (the highest point on the bell curve). In many ways, the standard deviation tells us how "risky" a stock is.

For example, let's say WPL has a 50% chance that it will return 50% in a year and a 50% chance it will return -10% in a year. The average return is 20%

Now, let's assume WBC has a 50% chance that it will return 15% in a year and a 50% chance it will return 25% in a year. The average return is 20%

As you can see, we have two different stocks with an expected return of 20%. However, WPL appears to be riskier. The standard deviation helps identify which stock is riskier.

There are a number of different methods used to measure the standard deviation. This article will focus on the most used method, the close-close method. We'll be using Tesla Motors in our example.

Tesla Motors Inc (NASDAQ:TSLA) Closing prices during October

Step One: Compute the one-day returns based on the closing prices. Take the natural logarithm of today's closing price and divide it by yesterday's closing price

ln ( 183.07/180.98) = 0.0115

ln (174.73/183.07) = -0.0466266

ln (168.78/ 174.73) = -0.034645

ln (172.93/168.78) = 0.0243

ln (178.7/172.93) = 0.0328

ln (174.73/183.07) = -0.0466266

ln (168.78/ 174.73) = -0.034645

ln (172.93/168.78) = 0.0243

ln (178.7/172.93) = 0.0328

Step Two: Sum up the one-day returns and divide it by the number of observations (5)

-0.012678106 / 5 = -0.002535621

Step Three: Compute the mean-squared differences, our mean is -0.002535621

.0115 - (-) 0.002535621 = 0.014017687 then multiply by 0.014017687 = 0.000196496

-0.0466266 (-) 0.002535621 = -0.044091047 then multiply it by -0.044091047 = 0.00194402

-0.034645 (-) 0.002535621 = -0.032110212 then multiply it by -0.032110212 = 0.001031066

0.0243 - (-) 0.002535621 = 0.026826418 then multiply it by 0.026826418 = 0.000719657

0.0328 - 0.002535621 = 0.035357155 then multiply it by 0.035357155 = 0.001250128

-0.0466266 (-) 0.002535621 = -0.044091047 then multiply it by -0.044091047 = 0.00194402

-0.034645 (-) 0.002535621 = -0.032110212 then multiply it by -0.032110212 = 0.001031066

0.0243 - (-) 0.002535621 = 0.026826418 then multiply it by 0.026826418 = 0.000719657

0.0328 - 0.002535621 = 0.035357155 then multiply it by 0.035357155 = 0.001250128

Step Four: Sum up our total from the data gathered in Step Three

0.005141367

Step Five: We square root the sum from Step Four and divide it by the number of observations minus one.

Square Root (0.005141367./4) = 3.58516

The daily volatility for this sample period is 3.585%

Step Six: As you may know, volatility is usually expressed in annualized terms. In order to convert our daily volatility into annualized volatility, we take our daily volatility and multiply it by the square root of the number of trading days in a year (252)

.03585 x the square root of 252 = .5691

Annualized the volatility for this sample period was .5691 or 56.91%

What we've done is calculate the 5-day historical volatility for Tesla Motors. Of course, this is simply done for illustrative purposes. In reality, traders use a larger sample size than 5 days because of sampling error.

For example, if a recent smoker decided to quit smoking for good and were on their fifth day of being smoke free, you wouldn't necessarily call them a non-smoker yet. However, if they were smoke free for a couple months it would be a lot more significant.

For the most part, traders will look at historical volatility in 20-day trading periods and beyond. A trading period less than 20 days includes too many sampling errors and noise. However, if your sample includes too much data then it might not be reflective of the current price behaviour of the stock.

Near term price action tends to be wilder (greater randomness) and contains sampling errors. This means that volatility has the potential to see extremely low or high readings in the short term, but smooth itself out in the longer term.

Historical volatility is not forward looking, it simply tells us about the past. However, the options implied volatility is forward looking.

Volatility is said to mean-reverting, periods of high volatility tend to revert back to the mean and vice versa during periods of low volatility. In respect to trading, selling options when premiums are relatively low could be dangerous. In addition, buying options when volatility is relatively high could be a losing proposition.

Tesla Motors Inc (NASDAQ:TSLA) Hodges Tompkins Volatility Cone displays mean reverting attributes

The above chart is a historical volatility cone of Tesla Motors, covering 10/15/10 to 10/11/13. The sample periods are 20, 40, 60 and 90 trading days. Based on the Hodges Tompkins volatility estimator (which is basically the close to close volatility adjusted for sampling bias) the current 20 trading day volatility is 51.5% By observing the volatility cone, we see that the highest 20-day volatility over the last 3 years was 127.4%, the lowest was 22.6%, the median is 52.7%, the upper range (75tth percent quartile) is 65.4% and the lower range (25th percent quartile) is 43.0%.

The volatility cone clearly shows how historical volatility is mean reverting; it displays how volatility smooths itself out as we gather more data (near term moves tend to show greater extremes, both high and low volatility).

If you compare the median to the present volatility (52.7% vs. 51.5%) you'll notice that we are pretty much in line with where it's been over the last 3 years. Remember, historical volatility data is not forward looking. There could be some fundamental shifts in the company, an event or news surrounding the company that could potentially drive volatility higher or lower. However, it does help us in putting the present volatility into context.

We would then use this information and compare it to the options market implied volatility. Ideally, you want to compare 20-trading day volatility to options that 30 days till expiration, 40-trading day volatility to options that have 60 days till expiration, so on and so forth. The best options to use in this comparison are the at the money options.

Ideally, you'd like to compare the spread between implied volatility and historical volatility to get a sense if options are cheap or expensive (IV/HV). Ideally, you'd look to sell volatility when historical volatility is relatively high and the spread between implied and realized is positive (IV>HV). On the contrast, you should look to be a premium buyer when historical volatility is relatively low and the spread between implied and realized is negative (IV<HV).

Remember, implied volatility is forward looking and it does take into account future events like earnings and other relevant information. With that said, there might be a reason why implied volatility is much more (or less) than historical volatility, which is always backwards looking.