The Theory of Higher Order Lognormal Cascade Distribution and the Origin of Fat Tails in the Fluctuations of Stock Market
32 Pages Posted: 7 Jun 2009
Date Written: June 5, 2009
This working paper presents the general theory of the higher order "skew lognormal cascade distribution" as a mathematical extension of the previously proposed skew lognormal cascade distribution. In particular, the second order distribution is studied in details, which incorporates the fat tails into the volatility (aka the volatility of volatility). We show that the second order distribution can handle very heavy tails and high kurtosis in the high frequency financial time series. It accurately fits the daily log-returns of Dow in 80 years, whose kurtosis is 26. The framework of the higher order lognormal cascade distributions also provides a new way to study the capital distribution (aka firm size distribution), the market index, and the market entropy of the stock market. Such study in the context of stochastic portfolio theory reveals that the origin of the fat tails in the fluctuations of the market index is from the lognormal cascade structure of the capital distribution in the market. We show from a simple stochastic model that the contraction and expansion of the underlying capital distribution is the fundamental driving force of the bull-bear market cycles and the market volatility in the past 20 years. A stochastic equation is derived to establish the relation between the market index and the capital distribution, which is the lognormal cascade equation in our theory. This shows that the fluctuations of the market index are a natural mathematical consequence of the stochastic calculus on the market portfolio in which weights are exponentially distributed. Therefore, we conclude that the phenomena of fat tails should exist everywhere in our financial system.
Keywords: lognormal cascade, fat tails, heavy tails, capital distribution, time series, stochastic portfolio theory
JEL Classification: C22, D30, E32, E37
Suggested Citation: Suggested Citation