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

Quantitative Finance, Forthcoming

65 Pages Posted: 4 Oct 2017 Last revised: 17 Nov 2017

See all articles by Stephen H.T. Lihn

Stephen H.T. Lihn

Atom Investors LP; Novus Partners, Inc.

Date Written: November 15, 2017


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

Suggested Citation

Lihn, Stephen H.T., A Theory of Asset Return and Volatility Under Stable Law and Stable Lambda Distribution (November 15, 2017). Quantitative Finance, Forthcoming, Available at SSRN: https://ssrn.com/abstract=3046732

Stephen H.T. Lihn (Contact Author)

Atom Investors LP ( email )

3711 S Mopac Expressway
Austin, TX 78746
United States
917-603-4133 (Phone)

Novus Partners, Inc. ( email )

521 5th Ave
29th Floor
New York, NY 10175
United States
917-603-4133 (Phone)

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