# Is the market really Normal. Is Implied Volatility Historically Correct?

Ok. So as of 6/10/2014's market close the SPY was 195.6 and the VIX closed at a ridiculous recent low of 10.99. Now because the VIX (IV) is the implied volatility of 1 month contracts on the SPX and because this 1 month figure is quoted annually, we take the current value of the VIX and multiply it by sqrt(20/250) to convert it to a true monthly figure.

So given a VIX of 10.99, the SPX a 1stdv move to the upside and downside will be 195.6(+/-)10.99*sqrt(20/250)*195.6, ie, 201 and 189 on the SPY, 1 month from now. If it goes above 201 or below 189, the move is greater than a 1stdv move.

So I ran a historical test to see how close the SPX has abided by the VIX, 1 month prior. So I ran a historical test from 1/2/2001 to 6/10/2014, on daily data of the SPY and the VIX. Because I lose one of month of VIX data because due to the forward lag, we have a sample size of 3358 days.

Running through how many days we broke above:

1stdev  234 329 1074.56
2stdev  0   47  167.9
3stdv   0   9   10.074
4stdev  0   1


The 1st number is the days above the upper stdev band The 2nd number is the days below the lower stdev band The 3rd number is the total # of days beyond nth stdev by the normal distribution.

From the results it seems that the market is more normal that we actually though. But other empirical evidence has shown otherwise. From the results I am seeing, It seems to suggest equity prices are normally distributed. We don't see more fat tailed events more than the normal distribution suggests. Can someone shed light if I ran the analysis incorrectly??

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I see a lot of statements. What's your question? – Joshua Ulrich Jun 11 '14 at 21:29
...and how does 195+/-608 give you 233/177? Even if it did give you 233/177, don't you see a problem with your 1-month standard-deviation band being ~3x larger than the annual VIX? – Joshua Ulrich Jun 11 '14 at 21:50
This is sort of similar to this image from the wikipedia page for VIX: en.wikipedia.org/wiki/VIX#mediaviewer/File:Vix.png – User1996 Jun 11 '14 at 23:54
233 is 37 points above, 177 is 18 points or so below. will be good if you can clean and restate your results. Clearly it is not normal because it spends far more time in 1 std dev than normal would suggest only 1074.56 rather than 3358-234-329 days. – user12348 Jun 13 '14 at 23:00
Sorry corrected the Upper bands and lower band value. Typo on my end. – jessica Jun 14 '14 at 1:44

You need to use log of prices, because log of returns are normally distributed. So or where x is return- $$x=-\frac{1}{\tau} ln(\frac{S_{t+\tau}}{S_{t}})$$ The annualized standard deviation can be scaled as +/-$n\frac{\sigma}{\sqrt{\tau}}$ where n is your multiple. You can either ignore or estimate drift. or look at it another way, S refers to the index $$ln (S_t) = ln(S_{t-1}) + \left( \left(\mu - \frac{\sigma^2}{2} \right)dt + \sigma dW_t\right)$$