Frequency Diagram for the Implied Volatility Index, January 2, 1990 – October 10, 2008

According to a Reuters article published yesterday (October 10; see “Wall Street’s fear gauge hits record highs“),

“An index regarded as Wall Street’s fear gauge notched record highs again on Friday, reflecting unprecedented investor anxiety after the broad U.S. stock market fell in a turbulent session. The Chicago Board Options Exchange Volatility Index, or the VIX, surged 20.4 percent to an all-time high at 76.94, before closing at 69.95.”

I made reference to the VIX in my comments at the “Understanding the Financial Crisis” panel discussion on Thursday (see pp. 12-13 of “Moral Hazard and Financial Hazard”). I also reference VIX in my blog entry entitled “Volatility“. Besides being the worst 1 week period in recorded history for the major stock market indices, last week was also marked by a daily progression of new all-time highs for the VIX. Here are the closing values for each trading day last week:

10/6/2008 52.05
10/7/2008 53.68
10/8/2008 57.53
10/9/2008 63.92
10/10/2008 69.95

Since January 2, 1990, there have been a total of 4,733 daily observations for VIX. The addition of only 5 data points last week actually marginally increased the average VIX, the VIX standard deviation, as well as the positive skewness and “fat-tailedness” of the entire data series. Friday’s closing value of 69.95 is 7.75 standard deviations to the right of the average of 19.23. If this were a normal distribution (it’s definitely not), the probability of such an extreme value would be 0.00000000000047%.

Here’s a frequency diagram for VIX during the period January 2, 1990 – October 10, 2008. (note that if you click on the graphic and look closer, the last four bars from 52 out to 70 represent the 5 closing values for VIX last week). Visually, you can see that this is a positively skewed distribution of outcomes (skewness is 1.128), and it is also fat-tailed (kurtosis is 5.22). For the normal distribution, skewness is 0 (the normal distribution is not skewed), and kurtosis is 3 (the normal distribution is not fat-tailed either). The joint effect of positive skewness and fat tails is that this means that the odds of extreme values (particularly to the upside) are higher than they would be for a normally distributed random variable. Suffice it to say, these are extremely rare events now matter how you look at them.

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