Finite-size effect and the components of multifractality in financial volatility

TL;DR

This paper investigates the finite-size effects in multifractal analysis of financial volatility, decomposes multifractality into PDF and nonlinearity components, and demonstrates a method to identify their contributions using Dow Jones data.

Contribution

It introduces a method to separate finite-size effects from true multifractality and decomposes multifractality into PDF and nonlinearity components in financial time series.

Findings

Finite-size effects significantly influence multifractality detection.

Effective multifractality arises from nonlinearity and PDF impacts.

Nonlinearity is necessary for true multifractality in financial data.

Abstract

Many financial variables are found to exhibit multifractal nature, which is usually attributed to the influence of temporal correlations and fat-tailedness in the probability distribution (PDF). Based on the partition function approach of multifractal analysis, we show that there is a marked finite-size effect in the detection of multifractality, and the effective multifractality is the apparent multifractality after removing the finite-size effect. We find that the effective multifractality can be further decomposed into two components, the PDF component and the nonlinearity component. Referring to the normal distribution, we can determine the PDF component by comparing the effective multifractality of the original time series and the surrogate data that have a normal distribution and keep the same linear and nonlinear correlations as the original data. We demonstrate our method by taking the daily volatility data of Dow Jones Industrial Average from 26 May 1896 to 27 April 2007 as an example. Extensive numerical experiments show that a time series exhibits effective multifractality only if it possesses nonlinearity and the PDF has impact on the effective multifractality only when the time series possesses nonlinearity. Our method can also be applied to judge the presence of multifractality and determine its components of multifractal time series in other complex systems.