SKEW Index as parameter in lognormal distribution

Question

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 1

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.