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The concept of multifractality offers a powerful formal tool to filter out a multitude of the most relevant characteristics of complex time series. The related studies thus far presented in the scientific literature typically limit themselves to evaluation of whether a time series is multifractal, and width of the resulting singularity spectrum is considered a measure of the degree of complexity involved. However, the character of the complexity of time series generated by the natural processes usually appears much more intricate than such a bare statement can reflect. As an example, based on the long-term records of the S&P500 and NASDAQ—the two world-leading stock market indices—the present study shows that they indeed develop the multifractal features, but these features evolve through a variety of shapes, most often strongly asymmetric, whose changes typically are correlated with the historically most significant events experienced by the world economy. Relating at the same time the index multifractal singularity spectra to those of the component stocks that form this index reflects the varying degree of correlations involved among the stocks.

Multifractality is a concept that is central to the science of complexity. The related multiscale approach [

At present, there exist two distinct, commonly accepted, and complementary computational methods that serve quantification of the multifractal characteristics of the time series. One of them—the wavelet transform modulus maxima (WTMM) [

One thus considers two time series

Since

Fractal cross-dependencies between the time series

The conventional MFDFA procedure of calculating the singularity spectra for single time series can be considered a special case of the above MFCCA procedure and corresponds to taking

A family of the fluctuation functions as defined by (

In the present study, two sets of data are used:

Daily prices of the S&P500 and NASDAQ indices covering the period January 03, 1950–December 29, 2016 (16,496 data points). The values of the NASDAQ before 1971 (official launching date of the index is February 05, 1971) were reconstructed from the historical data [

Daily prices of 9 stocks listed on the NYSE over the period from January 1, 1962, to July 07, 2017 (13,812 points). The analysed companies are GE (General Electric), AA (Alcoa), IBM (International Business Machines), KO (Coca-Cola), BA (Boeing), CAT (Caterpillar), DIS (Walt Disney), HPQ (Hewlett-Packard), and DD (DuPont). These in fact are the only stocks that participate in the Dow Jones Industrial Average (DJIA) over such a long period of time and thus also in the S&P500. They, however, represent a large spectrum of the economy sectors and may thus be considered as a reasonable representation for the larger American indices.

For each time series, the logarithmic returns are calculated according to the equation:

The MFDFA multifractal spectra

Main panels: multifractal spectra calculated for the S&P500 and NASDAQ returns (black dots) covering the period January 03, 1950–December 29, 2016. Average spectra obtained for the Fourier phase-randomized surrogates and for the randomly shuffled time series are denoted by blue squares and green triangles, respectively. Upper-left insets display cumulative distributions of return fluctuations, and lower-right insets display the fluctuation functions calculated for the original S&P500 and NASDAQ series.

Figure

(a) For the S&P500 from January 03, 1950, to December 29, 2016, the sequence of singularity spectra

The same as in Figure

In Figures

For the S&P500, the Hurst exponents

For the NASDAQ, the Hurst exponents

Two upper panels: daily returns for the S&P500 and for the NASDAQ over the period January 03, 1950–December 29, 2016. The bottom panel: the corresponding detrended variance for the S&P500 (red line) and for the NASDAQ (black line).

The window-probed multifractal spectra of Figures

Cross-correlation fluctuation functions between the S&P500 and NASDAQ calculated according to (

It is natural to expect that significant changes in time of the multifractal features of the two indices seen in the previous subsection reflect different market phases, and such phases vary in a degree of coupling among the component shares [

Daily prices of the S&P500, of the Dow Jones, and of the sum of 9 DJIA stocks listed on the NYSE over the period from January 1, 1962, to July 07, 2017 (13,812 points). The companies included are GE (General Electric), AA (Alcoa), IBM (International Business Machines), KO (Coca-Cola), BA (Boeing), CAT (Caterpillar), DIS (Walt Disney), HPQ (Hewlett-Packard), and DD (DuPont).

The results of calculations relating to the multifractal spectra

Projections onto the time

An especially interesting related case occurs in the period between October 1990 and April 1994 indicated in Figure

In this expression,

Changes of the magnitude of the largest eigenvalue

The blue line displays the time dependence of the largest eigenvalue

Quantification of the complex time series in terms of multifractality nowadays finds a multitude of applications in diverse areas. Thus far, however, majority of the related studies presented in the scientific literature limit themselves to a sole estimation of the singularity spectrum, and if found multifractal, it usually is treated as evidence of the hierarchical organization of such series, and the width of such a spectrum is considered a measure of the degree of complexity involved. While this indicates some kind of a cascade-like, hierarchical organization indeed, in realistic cases, such an organization is rarely uniform. The time series generated by natural processes may include many convoluted components with different hierarchy generators each, which results in asymmetry of the singularity spectra. Even more, contribution of such components may vary in time, and this thus may introduce further dynamical variability. Definitely, the financial markets constantly functioning in evolving external conditions represent a natural candidate to become a subject of such effects. This can be anticipated to apply almost straightforwardly to the stock market indices as they by construction constitute an average (typically weighted but not always) of the prices of selected stocks representing different economy sectors, thus not necessarily obeying the same multiscaling characteristics. The degree of correlations among such stocks is also known to depend on the global market phases. In the present paper, based on over half a century daily recordings of the S&P500 and NASDAQ, the two world-leading stock market indices, it is shown that they reveal the multiscaling features which expressed in terms of the multifractal spectrum evolve through a variety of shapes whose changes typically appear correlated with the historically most significant events experienced by the world economy. From a more general perspective, these results indicate that the form of the multifractal spectrum, and especially its departures from the model mathematical cases of the uniform cascades, contains richness of information that, if properly interpreted and potentially disentangled, may provide very valuable insight into the underlying dynamics which may be of crucial value for a more accurate modelling of the financial markets. Taking into consideration the effects exposed here may also be very helpful for market regulators and policymakers in stabilizing markets as well as for a flexible portfolio optimization.

Finally, the methodology introduced in Section

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This research was supported in part by PLGrid Infrastructure.