## Jul 18, 2010

### Volatility and Expected Range

Despite what you read on many blogs, volatility and expected range are not the same, not even close. This fallacy is common among internet pundits who do not know any better, but to traders like me who have real money on the line such mistakes can be dangerous. This is not a post to correct some abstract mathematical technicality, or merely a semantic point. Rather I hope to shed some light on widespread mis-estimation of important risk metric.

For example, certain Mark Bail provides us with the following gem of ignorance here

"So, why is the VIX important? For one, it provides you with a reasonable projection of the expected range within which the S&P 500 is likely to trade within the next month. To use the current environment as an example, the S&P 500 closed on June 19 at 1240.14. The June 19 closing VIX reading of 17.83 suggests that options traders and investors anticipate that between now and July 19, the S&P 500 is likely to trade roughly within 1.49% range (17.83 divided by 12) of 1240.14 — or between 1221.71 and 1258.57."

First of, VIX is not the expected range. Second, VIX is not a measure of where the index is likely to trade. Third, none of these risk measures scale linearly with time, so dividing VIX by 12 months is rubbish. Fourth, simply from intuition, does is make sense that S&P will trade in a 1.49% range in a given month? To me as a trader, that number seems ridiculously low, even in low volatility market.

Ok, so what does VIX mean, and what does it imply? Technical definition of VIX (available from CBOE here) is the annualized square root of 30-day variance swap rate calculated from option prices. This is different from volatility swap rate, but for the purpose of this discussion I will use the terms interchangeably. The volatility that VIX references is not the volatility of prices, but of returns - another difference that I will ignore for today. We will treat VIX as simply market's expected annualized volatility.

To convert VIX from annual to monthly volatility, one needs to divide not by 12, but by square root of 12. Likewise, if you want to find daily volatility, don't divide by 252, but by √252 ≈ 16. So in Mr Bail's example above, expected monthly return volatility of S&P 17.83 / √12 = 5.14%, and monthly price volatility 5.14% * 1240.14 = \$63. Daily volatility is 1.12%, or \$13.92.

As I wrote here, expected range of a random walk is 2 * √(2/π) ≈ 1.6 times the volatility. Therefore expected range for S&P for the month is 1.6 * 5.14% = 8.21%, or \$101.85. Expected daily range is 1.79%, or \$22.22. Expected monthly high, and low come out to \$1291.06, and \$1189.21. Please note the magnitude of difference between Mark Bail's forecast and mine - his expected range for the month is smaller than my daily range!!!

Now let's compare these predictions using historical data... After I wrote the previous sentence I tried to figure out when the article was written. There is no date on the article, the date that is referenced in the paragraph above is "June 19", however I could not any such date on which S&P and VIX closed at the levels reported in the article. I really do not know if the author just made a mistake or simply confabulated the numbers.

What I did find is that VIX happened to close at 17.83 on May 30th, 2008, so for comparison purposes I will use that date, dealing only with percent forecasts, since S&P level on that day differs significantly from the one in the article.

During the calendar month from 5/30/2008 to 6/30/2008 S&P index had a daily return volatility of 20.44% (annualized), which is a bit higher than the VIX, but not radically so. That month S&P had an open of 1400.38, high 1404.46, low 1272, closing at 1280. The range (1404.46-1272)/1400.38 = 9.4%, is much higher than 1.49% reported in the article, and much closer to my calculation expected range of 8.21%. Make your own conclusions from here.

Traders, pay attention to numbers and formulas you use in your trading. Mistakes can costs you money!