A Theory of Asset Return and Volatility Under Stable Law and Stable Lambda Distribution

Abstract

We develop a new theory of asset return and volatility based on the stable law and Lihn's former work on the lambda distribution. The accomplishments are twofold: First, the newly discovered stable count distribution models the stationary distribution of VIX with high statistical precision. Secondly, the “stable lambda distribution” (SLD) models the SPX log-return distribution with accuracy to more than 4 stdev and/or 99.5% quantile. The foundation of the new theory is the Laplace transform of the one-sided stable distribution, which is used to decompose a lambda distribution into the product distribution of a stable count distribution and a Laplace distribution. The decomposition leads to a stable version of asset return SDE, which strictly defines what volatility and noise term should be. The stable count distribution is the conjugate prior of the one-sided stable distribution, whose moments are all finite. One-to-one mapping between a stable distribution and a lambda distribution is established, from which the stability index alpha can be understood as a measure of kurtosis. A novel microscopic level random walk concept is presented to explain the mechanism of financial asset returns, in which the absolute Lévy sum represents the elapsed time and a Laplace-style binomial random walk occurs in the log-price dimension. We find that, at quartic lambda aka lambda=4 (alpha=0.5) where the Lévy distribution and quartic lambda distribution reside, elegant closed form solutions are available at almost every level of our analysis. We use such analytical tractability to successfully generalize the lambda distribution. It combines the quartic stable count distribution with a new standardized leptokurtic stochastic process, called standardized Laplace process (SLP), which extends the Wiener process by convolutions of a Laplace distribution. These stable-law inspired processes and distributions produce highly accurate statistical results for SPX, VIX, and other financial assets.

Keywords: stable law, stable count distribution, time series modeling, asset return distribution, stable lambda distribution, standardized Laplace process, VIX, SPX

JEL Classification: G13, C46, C58