## Feb 28, 2011

### CBOE SKEW Index, Part 3

Read the first part of SKEW post here, part 2 here.

There is a very interesting table on page 8 of SKEW index white paper: Estimated Risk-Adjusted Probabilities of S&P 500 Log Returns Two and Three Standard Deviations below the Mean. I think the author meant risk-neutral, not risk-adjusted, but regardless I reproduce it below:

 SKEW -2 Std. Dev -3 Std. Dev. 100 2.30% 0.15% 105 3.65% 0.45% 110 5.00% 0.74% 115 6.35% 1.04% 120 7.70% 1.33% 125 9.05% 1.63% 130 10.40% 1.92% 135 11.75% 2.22% 140 13.10% 2.51% 145 14.45% 2.81%

Since SKEW index is "adjusted" as SKEW Index = 100 - 10 * implied skew, SKEW Index value of 100 is skew of 0, SKEW Index value of 105 is skew of -0.5, etc.

The table is based on Gram-Charlier expansion of normal distribution, for various skew levels with zero excess kurtosis, however the paper that is referenced provides the formula for probability distribution function, and not cumulative density function (see Wikipedia):

PDF(z) = n(z)*[1+skew/6*(z3-3z)+kurtosis/12*(z4 − 6z2 + 3)],

where n(z) is the normal probability distribution function.

Since kurtosis = 0, the expression becomes F(z) = n(z)*[1+skew/6*(z3-3z)]

The formula for CDF is obtained by integration

CDF(z) = N(z) - skew/6*n(z)*(z2-1),

where N(z) is the normal cumulative density function. In excel this formula would read:

NORM.DIST(Z,0,1,TRUE) - SKEW/6*NORM.DIST(Z,0,1,FALSE)*(Z*Z-1),

where Z and SKEW are variables, Z is the Z-score e.g. -2 for negative 2 standard deviations move, and SKEW is implied skew, calculated from SKEW index. The results are not exactly the same, but sufficiently close to the ones in CBOE paper. I suspect the difference is due to weird rounding on CBOE's part. I do realize it may sound arrogant for me to assume CBOE's error rather than mine, however CBOE's numbers don't quite match standard results in the case of normal distribution (where skew is zero): e.g. for 2 stds the number should be 2.27501 to the 5 significant digits, where CBOE rounds it to 2.30, and for 3 stds the number should be 0.13499 to the 5 significant digits, where CBOE rounds it to 0.15, so I confidently stand by my numbers.

However there is a different problem that I would really like to point out - the Gram-Charlier approximation is a rather weak approximation. In some cases it produces rather strange numbers, for example in the case of Z=-1 (down 1 standard deviation) probabilities become independent of the skew parameter, which obviously makes no sense.

 SKEW -0.5 Std. Dev -1 Std. Dev. -1.5 Std. Dev. 100 30.85% 15.87% 6.68% 105 28.65% 15.87% 8.03% 110 26.45% 15.87% 9.38% 115 24.25% 15.87% 10.73% 120 22.05% 15.87% 12.08% 125 19.85% 15.87% 13.43% 130 17.65% 15.87% 14.78% 135 15.45% 15.87% 16.12% 140 13.25% 15.87% 17.47% 145 11.05% 15.87% 18.82%

So, if you want to calculate risk-neutral probabilities now you have the same formula CBOE used in the white paper, however know that the formula is only an approximation, and clearly does not work in some cases. I hope in the near future I'll blog about better formula, as well as other uses for SKEW index.

1. Anonymous11/23/2011

thanks for this analysis of the cboe's SKEW index. I too have been working on backing out estimated implied probabilities of S&P500 returns using the SKEW index. And I see the problems with the references provided by CBOE that you noted. I also dont get the same results to those in their table. Additionally, using your excel formula verbatim I get vastly different results than yours. I cannot figure what I am missing with it - for example for a skew of 100, and Z of -0.5, you show 30.85% which is exactly the first term normdist(-.5,0,1,true). Plugging values In the second term: -100/6*normdist(-.5,0,1,false)*(-.5*-.5-1) gives a value of 4.4. So when combining these comes up with odd result. Curious, have you done any more work on this or have a working solution? If so I'd love to see what you came up with if you wanted to email me at blatchle at bc dot edu.

2. In the excel formula SKEW means statistical skew (third moment), not skew index.

skew index = 100 - 10 * statistical skew
statistical skew = ( 100 - skew index ) / 10

Try that and see if you get the same numbers.

3. Anonymous11/23/2011

of course that worked! Apologies, I was unclear on your original post where you used "implied" SKEW derived from cboe's published skew! Thanks!