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.