Complexity

Volume 2018, Article ID 7015721, 13 pages

https://doi.org/10.1155/2018/7015721

## Dynamical Variety of Shapes in Financial Multifractality

^{1}Complex Systems Theory Department, Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Kraków, Poland^{2}Faculty of Physics, Mathematics and Computer Science, Cracow University of Technology, ul. Warszawska 24, 31-155 Kraków, Poland^{3}Faculty of Mathematics and Natural Sciences, University of Rzeszów, ul. Pigonia 1, 35-310 Rzeszów, Poland

Correspondence should be addressed to Stanisław Drożdż; lp.ude.jfi@zdzord.walsinats

Received 16 March 2018; Revised 5 July 2018; Accepted 17 July 2018; Published 16 September 2018

Academic Editor: Lingzhong Guo

Copyright © 2018 Stanisław Drożdż et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

Multifractality is a concept that is central to the science of complexity. The related multiscale approach [1–3] aims at bridging the wide range of time and length scales that are inherent in a number of complex natural phenomena, and as such, it pervades essentially all scientific disciplines [4]. By now, it finds applications in essentially all areas of the scientific activity, including physics [5, 6], biology [7–9], chemistry [10, 11], geophysics [12, 13], hydrology [14], atmospheric physics [15], quantitative linguistics [16, 17], behavioural sciences [18], cognitive structures [19], music [20, 21], songbird rhythms [22], physiology [23, 25], human behaviour [24, 26, 27], social psychology [28], and even ecological sciences [29], but especially frequently in economic and in financial contexts [30–41] as stimulated by practical aspects and by needs to develop models of the financial dynamics based on multifractality [31, 42–45] such that they help in making predictions. Indeed, the multifractal analyses of the financial time series have provided so far most of the quantitative evidence for the factors that induce the genuine multifractality, such as the temporal long-range nonlinear correlations and, only when such correlations are present, the fat tails in the distribution of fluctuations [46]. In order to unambiguously identify action of such factors and to suppress potential spurious multifractality, the time series under study have to be, however, sufficiently long [47]. In addition, the realistic time series, as generated by the natural phenomena, even if of multifractal character, are typically more involved in composition than the model mathematical uniform multifractals, and they may contain several components of different multifractality characteristics. In such frequent cases, the global hierarchical organization of the series gets distorted, and the multifractal spectrum becomes asymmetric, either left- or right-sided, as recently demonstrated in ref. [48]. Detecting such effects may provide even more valuable information about the mechanism that governs dynamics of a particular time series than just a bare statement that it is multifractal. Such effects of asymmetry are, for instance, already found to constitute a very helpful formal tool in identifying a specific organization of complex networks [49]. Furthermore, directions of the relevant distortions may vary in time parallel to changes of weight of the constituent components in a series. The most straightforward candidate to experience this kind of impact is the stock market index which, by construction, is already a sum, most often weighted, of prices of the constituent companies, and those companies themselves may react differently for the same external news depending of the sector they belong to. It is primarily for this reason that below, the world largest stock market indices are studied. Of course another, more specific, market-oriented reason for this study is to broaden our historical perspective on evolution of the stock market multiscale characteristics over periods comprising the global crashes or transitions due to the technological revolution in trading.

#### 2. Multifractal Formalism

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) [3]—makes use of the wavelet expansion of the time series under consideration, and the other one—the multifractal detrended fluctuation analysis (MFDFA) [50]—is based on inspecting the scaling properties of the varying order moments of fluctuations evaluated after an appropriate trend removal. While the former of those techniques allows a better visualization of the underlying patterns in the time series, the latter one often appears more accurate and more stable numerically, and it will therefore be used here. Furthermore, at present, there exists a consistent generalization of MFDFA such that it even allows to properly identify and quantify the multifractal aspects of cross-correlations between two time series [51–53]. This novel method, termed multifractal cross-correlation analysis (MFCCA), consists of several steps that at the beginning are common to all the methods based on detrending.

One thus considers two time series and , where . The signal profile is then calculated for each of them: where denotes averaging over the entire time series. Next, both these signal profiles are split up into () disjoint segments of length starting both from the beginning and the end of the profile, and in each , the assumed trend is estimated by fitting a polynomial of order ( for and for ). In typical cases, an optimal choice corresponds to [54]. This trend is subtracted from the series, and the detrended cross-covariance within each segment is calculated:

Since can assume both positive and negative values, the th-order covariance function is defined by the following equation: where denotes the sign of . The parameter in (3) can take any real number except zero. However, for , the logarithmic version of this equation can be employed [50]:

Fractal cross-dependencies between the time series and manifest themselves in the scaling relations: or for , where is the corresponding scaling exponent whose range of dependence on quantifies the degree of the complexity involved. Scaling with the -dependent exponents reflects a richer, multifractal character of correlations in the time series as compared to the monofractal case when is -independent.

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 and as identical. Equation (3) then reduces to and to a corresponding counterpart of (4) for . The signatures of multifractality (monofractality) are then reflected, analogously to (5), by where denotes the generalized Hurst exponent. The singularity spectrum (also referred to as multifractal spectrum) is then calculated from the following relations: where denotes the Hölder exponent characterizing the singularity strength and reflects the fractal dimension of support of the set of data points whose Hölder exponent equals . In the case of multifractals, the shape of the singularity spectrum typically resembles an inverted parabola and the degree of their complexity is straightforwardly quantified by the width of : where and correspond to the opposite ends of the values as projected out by different -moments (see (6)). For monofractal signals, the spectrum converges to a single point, though in practice, this often turns out to be a subtle matter [47]. Another important feature of the multifractal spectrum is its asymmetry (skewness), which can be quantified by the asymmetry coefficient [48]: where and , and for , the spectrum assumes maximum. The positive value of reflects the left-sided asymmetry of ; that is, its left arm is stretched with respect to the right one and, thus, more developed multifractality on the level of large fluctuations in the time series. Negative , on the other hand, reflects the right-sided asymmetry of the spectrum and indicates temporal organization of the smaller fluctuations as the main source of multifractality.

A family of the fluctuation functions as defined by (3) can also be used to define a -dependent detrended cross-correlation (*q*DCCA) [55] coefficient
which allows to quantify the degree of cross-correlations between two time series and after detrending and at varying time scales . Furthermore, by varying the parameter , one is able to identify the range of detrended fluctuation amplitudes that are correlated the most in the two signals under study [55]. This filtering ability of constitutes an important advantage as cross-correlations among time series typically are not uniformly distributed over their fluctuations of different magnitude [56].

#### 3. Data Specification

In the present study, two sets of data are used: (i)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 [57].(ii)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: where denotes the stock price or index value and stands for time interval (). All time series are normalized to have unit variance and zero mean.

#### 4. Results

##### 4.1. S&P500 and NASDAQ

The MFDFA multifractal spectra for the S&P500 and NASDAQ indices are shown in Figure 1. For both these indices, the fluctuation functions reveal a convincing power-law behaviour over almost two decades, which is shown in the corresponding lower-right insets; thus, is determined unambiguously. The parameters are taken within the interval , which is common in financial applications because it allows to safely avoid the danger of divergent moments when the fluctuation functions are computed. Cumulative distributions of the return fluctuations for the two indices considered here are shown in the corresponding upper-left panels of Figure 1 and can be seen not to develop thicker tails than the inverse cubic power law [58, 59], and there is thus no danger of the divergent moments. The width of the resulting spectra for the S&P500 and 0*.*32 for the NASDAQ, correspondingly. The significance of this result is also tested against the two null hypotheses of calculated from (i) the series obtained from the original ones by a random shuffling, thus destroying all the temporal correlations (green triangles), and (ii) Fourier-phase randomized counterparts of the original series which destroys the nonlinear correlations (blue squares). Clearly, the spectra in these two tests get shrank to a form characteristic to monofractals. An additional form of surrogates tested here are time series with the Gaussianized pdfs. In the latter case, the original pdf is replaced by a Gaussian distribution while the amplitude ranks of fluctuations remain preserved. The resulting multifractal spectra appear only slightly narrower than the original ones, and therefore, they are not shown in Figure 1. All these tests thus provide a convincing evidence for quite a rich multifractality of the original time series and, moreover, corroborate the fact that this multifractality is, as expected [47], due to the nonlinear temporal correlations. The obtained multifractal spectra are at the same time visibly left-sided asymmetric [48]. The asymmetry coefficient for the S&P500 and 0*.*31 for the NASDAQ. The left side of is determined by the positive values which filter out larger events, and the opposite applies to the right side of this spectrum. In the present context, this thus means that it is the dynamics of the large returns which develop more pronounced multifractal organization than those of the small returns.