Analytic Study and Numerical Method of The Skew Lognormal Cascade Distribution

  • Stephen H.-T. Lihn
  • Published 2009

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. 1 The Simplified SIBM Model 2 2 The Properties of the Symmetric Distribution 3 3 The Taylor Expansion of the Symmetric Distribution 5 4 The Numerical Method for The Symmetric Distribution 6 5 The Properties of the Skew Distribution 8 1 1 THE SIMPLIFIED SIBM MODEL 2 6 The Taylor Expansion of the Skew Distribution 10 7 The Numerical Method of The Skew Distribution 11 8 Appendix: Hermite Polynomials, Gamma Function, Etc. 13 Introduction. This short working paper studies the skew lognormal cascade distribution, which is the static solution of the simplified SIBM model (Lihn 2008, SSRN: 1149142). The skew distribution, which exhibits fat tail while maintains finite moments, can be a very useful distribution describing financial statistics, such as the stock market. The attempt is to work out the power expansion of this distribution so that the probability distribution function (pdf) can be computed numerically without using computationally expensive integrals. Even if this is not possible, the study should shed light on the inner structure of this distribution. Both the symmetric distribution and the skew distribution are studied. The analytic formula for the moments and cumulants are presented. The implication to the multiscaling property is studied for the symmetric distribution. Complicated algebraic results in this paper are carried out by the open source computer algebra system, Maxima. Numerical computation is prototyped on GNU Octave. 1 The Simplified SIBM Model In this section, we define the simplified SIBM model (Lihn 2008, SSRN: 1149142). We shall use a logarithmic model for continuous-time stock price processes. The stock price process X shall always be presented in its logarithm, log X (See 1.1 of Fernholz 2002), which is abbreviated as χ. Under the limit of τc → 0, the SIBM model can be reduced to the following stochastic equation for for the stock price process: dtχ(t) = Φ · eH [ dtW (t) + ( β · H + g) dt ] . (1) This expression could be very useful in finance since most stochastic equations in finance are written in the stock price processes, instead of the return processes. The parameters are defined as following: Φ is a global constant, β is ”the skewness parameter”, and g is the constant growth term. H is a time dependent normal process. However, to investigate the static return distribution of a fixed time lag T , we can assume H follows the higher order randomness hypothesis (HORN): H ∼ N (0, η2) . (2) In the computational order, averaging on H is applied last. With the help of the HORN, we can define the log-price change x = χ(T )−χ(0) and deduce the probability distribution function (pdf) p(x) by simply observing the functional form of the normal distribution: p(x) = ˆ ∞ −∞ dH 1 2π η σ(H) −H 2η2 e − (x−σ(H) (β·H+g))2 2σ(H)2 , (3) 2 THE PROPERTIES OF THE SYMMETRIC DISTRIBUTION 3 where σ(H) = Φ · eH. It is easy to verify that ́∞ −∞ p(x) dx = 1. This pdf is the same as the following distribution construct: Dη,β,g,Φ = {x = (a + β · b + g) · Φe, a ∈ N(0, 1), b ∈ N(0, η2)}. (4) With the substitution of h = H/η, Equation (3) becomes p1(x) = ˆ ∞ −∞ dh 1 2π σ(h) e− h2 2 e − (x−σ(h) (βηh+g))2 2σ(h)2 , (5) where σ(h) = Φ · e. When β = 0 and g = 0, we have the symmetric pdf of p2(x) = ˆ ∞ −∞ dh 1 2π σ(h) e− h2 2 e − x2 2σ(h)2 . (6) In this paper, we will focus on exploring the properties of p1(x) and p2(x). The complexity of p2(x) is less than p1(x), thus we shall work on p2(x) first. 2 The Properties of the Symmetric Distribution In this section, we attempt to work out the statistic properties for the symmetric case. First, by the substitution of z = h + η, we arrive at p2(x) = ˆ ∞ −∞ dz 1 2πΦ e η2 2 − z2 2 exp(− x 2 2Φ2 e−2(ηz−η 2)). (7) The characteristic function of p2(x) is Ψ2(t) = ́∞ −∞ dx e p2(x) and Ψ2(t) = ˆ ∞ −∞ dz 1 √ 2π e η2 2 − z2 2 eηz−η 2 exp(− t 2 e2(ηz−η 2)). (8) Applying Equation (58), we have

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Cite this paper

@inproceedings{Lihn2009AnalyticSA, title={Analytic Study and Numerical Method of The Skew Lognormal Cascade Distribution}, author={Stephen H.-T. Lihn}, year={2009} }