Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.

https://iopscience.iop.org/article/10.1088/1361-6633/ab42fb/pdf

Multifractal analysis of financial markets: a review.

In this paper, we investigate the cross-correlations between Chinese A-share and B-share markets. Qualitatively, we find that the return series of Chinese A-share and B-share markets were overall significantly cross-correlated based on the analysis of a statistic. Quantitatively, employing the detrended cross-correlation analysis, we find that the cross-correlations were strongly multifractal in the short-term and weakly multifractal in the long-term. Moreover, the cross-correlations of small fluctuations were persistent and those of large fluctuations were anti-persistent in the short-term while cross-correlations of all kinds of fluctuations were persistent in the long-term. Using the method of rolling windows, we find that the cross-correlations were weaker and weaker over time, especially after the price-limited reform. We attribute the fact to the improvement of market efficiency. On the volatility series, our results show that the cross-correlations were much stronger than those between return series. Results from rolling windows show that the short-term cross-correlations between volatility series are still high now. We also provide some relevant discussions later.

Cross-correlations between Chinese A-share and B-share markets

Multifractal analysis of financial markets

An efficient market has been theoretically proven to be a key component for effective and efficient resource allocation in an economy. This paper incorporates econophysics with Efficient Market Hypothesis to undertake a comparative analysis of Islamic and developed countries’ markets by extending the understanding of their multifractal nature. By applying the Multifractal Detrended Fluctuation Analysis (MFDFA) we calculated the generalized Hurst exponents, multifractal scaling exponents and generalized multifractal dimensions for 22 broad market indices. The findings provide a deeper understanding of the markets in Islamic countries, where they have traces of highly efficient performance particularly in crisis periods. A key finding is the empirical evidence of the impact of the ‘stage of market development’ on the efficiency of the market. If Islamic countries aim to improve the efficiency of resource allocation, an important area to address is to focus, among others, on enhancing the stage of market development.

An analysis of stock market efficiency: Developed vs Islamic stock markets using MF-DFA

The miltifractal properties and scaling behaviour of the exchange rate variations of the Iranian rial against the US dollar from a daily perspective is numerically investigated. For this purpose the multifractal detrended fluctuation analysis (MF-DFA) is used. Through multifractal analysis, the scaling exponents, generalized Hurst exponents, generalized fractal dimensions and singularity spectrum are derived. Moreover, contribution of two major sources of multifractality, that is, fat-tailed probability distributions and nonlinear temporal correlations are studied.

A Multifractal Detrended Fluctuation Description of Iranian Rial-US Dollar Exchange Rate

Analyzing the Shanghai stock price index daily returns using MF-DFA method, it is found that there are two different types of sources for multifractality in time series, namely, fat-tailed probability distributions and non-linear temporal correlations. Based on that, a sliding window of 240 frequency data in 5 trading days was used to study stock price index fluctuation. It is found that when the stock price index fluctuates sharply, a strong variability is clearly characterized by the generalized Hurst exponents h ( q ) . Therefore, two measures, Δ h and σ , based on generalized Hurst exponents were proposed to compare financial risks before and after Price Limits and Reform of Non-tradable Shares. The empirical results verify the validity of the measures, and this has led to a better understanding of complex stock markets.

Measuring multifractality of stock price fluctuation using multifractal detrended fluctuation analysis

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9359631

Pandemic-driven financial contagion and investor behavior: Evidence from the COVID-19

An empirical study of stable distribution and long-range correlation in Brent crude oil market was presented. First, it is found that the empirical distribution of Brent crude oil returns can be fitted well by a stable distribution, which is significantly different from a normal distribution. Second, the detrended fluctuation analysis for the Brent crude oil returns shows that there are long-range correlation in returns. It implies that there are patterns or trends in returns that persist over time. Third, the detrended fluctuation analysis for the Brent crude oil returns shows that after the financial crisis 2008, the Brent crude oil market becomes more persistence. It implies that the financial crisis 2008 could increase the frequency and strength of the interdependence and correlations between the financial time series. All of these findings may be used to improve the current fractal theories.

Stable distribution and long-range correlation of Brent crude oil market

Taking Baidu Index as a proxy for abnormal investor attention (AIA), the long memory property in the AIA of Shanghai Stock Exchange (SSE) 50 Index component stocks was empirically investigated using detrended fluctuation analysis (DFA) method. The results show that abnormal investor attention is power-law correlated with Hurst exponents between 0.64 and 0.98. Furthermore, the cross-correlations between abnormal investor attention and trading volume, volatility respectively are studied using detrended cross-correlation analysis (DCCA) and the DCCA cross-correlation coefficient ( ρ D C C A ). The results suggest that there are positive correlations between AIA and trading volume, volatility respectively. In addition, the correlations for trading volume are in general higher than the ones for volatility. By carrying on rescaled range analysis (R/S) and rolling windows analysis, we find that the results mentioned above are effective and significant.

Long memory of abnormal investor attention and the cross-correlations between abnormal investor attention and trading volume, volatility respectively

This study synthesizes interactions among real link and informational risk transmission channels to construct a spatial weight matrix and considers the effects of specific factors on returns through such interactions. We propose a spatial pricing model with multiple risk transmission channels and specific factors. The empirical results of an application of the spatial pricing model to industry assets in the Chinese stock market show that (1) the interaction plays a significant role in explaining asset return. The multiple risk transmission channels based spatial weight matrix improves the performance of the spatial pricing model, and specific factors further improve the performance; (2) the feedback effects and indirect effects of specific factors indicate the importance of interaction and specific factors in the pricing model; and (3) the spatial pricing models are asymmetric in different regimes.

Xiu Jin

Papers

Measuring multifractality of stock price fluctuation using multifractal detrended fluctuation analysis

Pandemic-driven financial contagion and investor behavior: Evidence from the COVID-19

Stable distribution and long-range correlation of Brent crude oil market

Long memory of abnormal investor attention and the cross-correlations between abnormal investor attention and trading volume, volatility respectively

Spatial pricing with multiple risk transmission channels and specific factors