Analytic Study and Numerical Method of the Skew Lognormal Cascade Distribution

Abstract

This working paper studies the "skew lognormal cascade distribution", which is proposed by the first time here, as the static solution of the simplified SIBM model (Stephen Lihn 2008, SSRN: 1149142). This distribution exhibits fat-tail, asymmetry tunable by a skew parameter, converges to the normal distribution, and has finite moments. These fine properties make it very useful in financial applications. The analytic formula of the raw moments and the cumulants are calculated for both the symmetric and skew forms. The implication to the multiscaling property is also studied for the symmetric distribution. The Taylor expansion on the distributions and their logarithms are carried out. A numeric method is carried out for the numerical computation of the probability density function. This method can be implemented via a computer algebra system and enable the numerical algorithm to produce high precision result. This distribution is implemented on http://www.skew-lognormal-cascade-distribution.org/ by the author. The author has tried to apply the distribution to the daily log returns of several financial time series, such as DJIA, WTI spot oil, XAU index, VIX index, 10-year Treasury, and several currencies. They all showed very good fit.

Keywords: lognormal cascade distribution, Taylor expansion, fat tail

JEL Classification: C61, C63