When market volatility spikes or stalls, financial websites, bloggers, social media, newspapers and television commentators all refer to the VIX^{®}. Formally known as the CBOE Volatility Index, the VIX is a benchmark index designed specifically to track S&P 500 volatility. Most investors familiar with the VIX commonly refer to it as the “fear gauge,” because it has become a proxy for market volatility.

The VIX was created by the Chicago Board Options Exchange (CBOE), which bills itself as “the largest U.S. options exchange and creator of listed options.” The CBOE runs a for-profit business selling (among other things) investments to sophisticated investors. These include hedge funds, professional money managers and individuals that make investments seeking to profit from market volatility. To facilitate and encourage these investments, the CBOE developed the VIX, which tracks market volatility on a real-time basis.

While the math behind the calculation and the accompanying explanation takes up most of a 15-page white paper published by the CBOE, we’ll provide the highlights in an overview. As my statistics professor once said: “It’s not so important that you are able to complete the calculation. Rather, I want you to be familiar with the concept.” Keeping in mind that he was teaching statistics to a room full of people who were not math majors, let’s take a layman’s look at the calculations behind the VIX, courtesy of examples and information provided by the CBOE.

#### Volatility

### **A Look at the VIX for the Mildly Curious**

The CBOE provides the following formula as a general example of how the VIX is calculated:

$\sigma^2 = \frac{2}{T}\sum_i \frac{\Delta K_i}{K_i^2}e^{RT} Q(K_i) - \frac{1}{T}\left [\frac{F}{K_0} - 1\right]^2$

The calculations behind each part of the equation are rather complex for most people who don’t do math for a living. They are also far too complex to fully explain in a short article, so let’s put some numbers into the formula to make the math easier to follow:

$\begin{aligned} \sigma^2 =& \ \tfrac{21,600}{525,600} \times 0.066472 \times \left ( \tfrac{61,920 - 43,200}{61,920 - 21,600} \right ) +\\ \phantom{\sigma^2 =}& \ \tfrac{61,920}{525,600} \times 0.063667 \times \left ( \tfrac{43,200 - 21,600}{61,920 - 21,600} \right ) \times \tfrac{525,600}{43,200} \\ \sigma^2 =& \ 0.0643180321 \\ \sigma =& \ 0.253610 \\ \end{aligned}$

$VIX = 100 \times \sigma = 25.36$

### **Delving into the Details of the Volatility Index**

The VIX is calculated using a "formula to derive expected volatility by averaging the weighted prices of out-of-the-money puts and calls.” Using options that expire in 16 and 44 days, respectively, in the example below, and starting on the far left of the formula, the symbol on the left of “=” represents the number that results from the calculation of the square root of the sum of all the numbers that sit to the right multiplied by 100.

To get to that number:

- The first set of numbers to the right of the “=” represents time. This figure is determined by using the time to expiration in minutes of the nearest term option divided by 525,600, which represents the number of minutes in a 365-day year. Assuming the VIX calculation time is 8:30 a.m., the time to expiration in minutes for the 16-day option will be the number of minutes within 8:30 a.m. today and 8:30 a.m. on the settlement day. In other words, the time to expiration excludes midnight to 8:30 a.m. today and excludes 8:30 a.m. to midnight on the settlement day (full 24 hours excluded). The number of days we’ll be working with will technically be 15 (16 days minus 24 hours), so it's 15 days x 24 hours x 60 minutes = 21,600. Use the same method to get the time to expiration in minutes for the 44-day option to get 43 days x 24 hours x 60 minutes = 61,920 (Step 4).
- The result is multiplied by the volatility of the option, represented in the example by 0.066472.
- The result is then multiplied by the result of the difference between the number of minutes to expiration of the next term option (61,920) minus the number of minutes in 30 days (43,200). This result is divided by the difference of the number of minutes to expiration of the next term option (61,920) minus the number of minutes to expiration of the near term option (21,600). Just in case you’re wondering where 30 days came from, the VIX uses a weighted average of options with a constant maturity of 30 days to expiration.

- The result is added to the sum of the time calculation for the second option, which is 61,920 divided by the number of minutes in a 365-day year (526,600). Just as in the first calculation, the result is multiplied by the volatility of the option, represented in the example by 0.063667.
- Next we repeat the process covered in step 3, multiplying the result of step 4 by the difference of the number of minutes in 30 days (43,200), minus the number of minutes to expiration of the near-term options (21,600). We divide this result by the difference of the number of minutes to expiration of the next-term option (61,920) minus the number of minutes to expiration of the near-term options (21,600).
- The sum of all previous calculations is then multiplied by the result of the number of minutes in a 365-day year (526,600) divided by the number of minutes in 30 days (43,200).
- The square root of that number multiplied by 100 equals the VIX.

Clearly, the order of operations is critical in the calculation and, for most of us, calculating the VIX isn’t the way we would choose to spend a Saturday afternoon. And if we did, the exercise would certainly take up most of the day. Fortunately, you will never have to calculate the VIX because the CBOE does it for you. Thanks to the Internet, you can go online, type in the ticker VIX and get the number delivered to your screen in an instant.

### **Investing in Volatility**

Volatility is useful to investors, as it gives them a way to gauge the market environment. It also provides investment opportunities. Since volatility is often associated with negative stock market performance, volatility investments can be used to hedge risk. Of course, volatility can also mark rapidly rising markets. Whether the direction is up or down, volatility investments can also be used to speculate.

For example, on June 13, 2016, the VIX surged by more than 23%, closing at a high of 20.97, which represented its highest level in over three months. The spike in the VIX came about due to a global sell-off of U.S. equities.

As one might expect, investment vehicles used for this purpose can be rather complex. VIX options and futures provide popular vehicles through which sophisticated traders can place their hedges or implement their hunches. Professional investors use these on a routine basis.

Exchange-traded notes – a type of unsecured, unsubordinated debt security – can also be used. ETNs that track volatility include the iPath S&P 500 VIX Short-Term Futures (VXX) and the VelocityShares Daily Inverse VIX Short-Term (XIV).

Exchange-traded funds offer a somewhat more familiar vehicle for many investors. Volatility ETF options include the ProShares Ultra VIX Short-Term Futures (UVXY) and ProShares VIX Mid-Term Futures (VIXM).

There are pros and cons to each of these investment vehicles that should be thoroughly evaluated before making investment decisions.

### **The Bottom Line**

Regardless of purpose (hedging or speculation) or the specific investment vehicles chosen, investing in volatility is not something to jump into without taking some time to understand the market, the investment vehicles and the range of possible outcomes. Failing to do the proper preparation and taking a prudent approach to investing can have a more detrimental result on your personal bottom line than making a mathematical error in your VIX calculation.