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Using SKEW and VIX to Choose a Return Distribution

Article Quant Q&A · Author: Jared
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Summary

The document asks whether the CBOE SKEW index can serve as the shape parameter of a skew-normal distribution alongside VIX. It explains the index conversion to a skewness measure and notes that the shown skew-normal parameter is constrained to a bounded range, whereas empirical market skewness may not be. The question also considers whether variance and skewness can fully specify a return distribution or parameterize a lognormal model.

The answer identifies the formula as a skew-normal density and reports a practical limitation: in the author's experience comparing it with S&P 500 data, its tails lacked enough kurtosis to match observed wings. This is a qualitative, personal observation rather than a systematic empirical study. The note does not provide a fitted alternative distribution or a complete procedure for combining SKEW and VIX, so it is best read as a warning about distribution choice and tail fit.

Key ideas

  • The displayed density is a skew-normal distribution with a bounded shape parameter.
  • The document relates the CBOE SKEW index to a skewness measure through a simple conversion.
  • The answer cautions that skew-normal tails may not have enough kurtosis to fit equity index wings.
  • Variance and skewness alone do not identify a specific distribution in the discussion.

Tags

OptionsVolatilityStatisticsUS markets
Full text
# SKEW Index as parameter in lognormal distribution


# SKEW Index as parameter in lognormal distribution












The CBOE publishes a SKEW index, which is `SKEW = 100 - 10*S`, so from the index itself we can get `S = (SKEW - 100)/10`.

I just want to do some preliminary analysis of distributions using `SKEW` and `VIX` together.

I have this python code from another SO question:

```
from scipy import linspace
from scipy import pi,sqrt,exp
from scipy.special import erf

def pdf(x):
    return 1/sqrt(2*pi) * exp(-x**2/2)

def cdf(x):
    return (1 + erf(x/sqrt(2))) / 2

# e = location
# w = scale

def skew(x,e=0,w=1,a=0):
    t = (x-e) / w
    return 2 / w * pdf(t) * cdf(a*t)
```

Can I get a Distribution using this skew parameter? The wiki page mentions the skew variable has to be in the range (-1,1).

Edit: I just needed to read the scipy.stats package more closesly -- it's well documented what shape, location, and scale are required for each distribution.

Edit 2: If SKEW is the 3rd statistical moment, VIX is the variance, what probability distribution can be completely specified by these two parameters? The lognormal is completely specified by variance and location. What are the alternatives? Can I parameterize the lognormal with these two distributions?

## Answer by onlyvix.blogspot.com (score 1, accepted)

https://quant.stackexchange.com/a/26055

The formula in your skew function is one of skew normal distribution. That distribution has a limit on skew parameter, while in the real world there is no such limit.

From personal experience, few years ago I tried doing exactly what you described in your question. After comparing skew normal distribution on SPX with the real world, I concluded that there is not enough kurtosis in skew normal distribution to match wings properly.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.