The fluctuations of returns in the stock market are highly irregular and nonlinear. The distribution function of returns deviates significantly from a normal distribution and is a distribution with long tails [1–5]. Events occurring in the tails of the return distribution function cannot be explained by random fluctuations, and the occurrence of such events is a unique characteristic of the stock market. In the stock market, large volatility occurs intermittently, much like an earthquake. Once a large volatility occurs, other large volatility events follow, creating clustered volatility [4–6]. This fat-tailed distribution of stock returns and the nonlinearity and correlation characteristics of the time series are called “stylized facts” [1–7].

Return time series in the stock market also exhibits statistical self-similarity and a multi-scale structure, too [1, 8–15]. To examine the scale structure hidden in the fluctuations of stock returns, a trend-removing analysis method has been proposed. The detrended fluctuation analysis (DFA) method, which removes trends from time series, has also been applied to various time series analyses [16]. Financial time series such as stock indices, cryptocurrency prices, and exchange rate exhibit multi-scale characteristics [11–14]. Multi-scale characteristics can be categorized into multifractal detrended fluctuation analysis (MF-DFA) and multifractal detrended moving average (MF-DMA), depending on the trend removal method [16, 17]. One method, utilizing the wavelet transformation, analyzes multi-scale time series without directly removing trends. The multifractal detrended cross-correlation analysis (MF-DCCA) method was introduced to identify multi-scale correlations between the stock return time series of two companies in the stock market [18].

Despite numerous studies demonstrating the existence of multiple scales in financial time series, the lack of multiscale structures in financial time series remains controversial [19]. The presence of multiple scales in surrogate time series or in randomly shuffled time series suggests that multifractal structures can arise from a variety of factors [13–16]. A recent report, based on a detailed analysis of large-scale transaction data, found that market influence on trading volume in the Tokyo Stock Exchange follows a universal square-root relationship, raising questions about the necessity of multifractal structures in financial time series. Another report found that when multifractal analysis was performed using intraday time series of several major stock indices, scaling exponents estimated using commonly used multifractal analysis techniques did not significantly differ between the raw data and the randomly shuffled series [20]. Further theoretical research is needed to understand the impact of various factors, such as finite-size samples of financial time series, thick-tailed distributions of returns, and long-term correlations in volatility, on multifractality. This paper briefly reviews various multi-scale analysis methods applied to financial time series. It introduces recent research exploring the principles of multi-fractal structure and outlines future work.

Fractal and multifractal analysis methods are well established for nonlinear dynamical time series thatexhibit chaotic behavior [21, 22]. To characterize the multiscale structure of a strange attractor, generalized dimensions can be computed in phase space using generalized box-counting methods. These generalized dimensions are obtained from the scale dependence of the generalized correlation sum of the time series. Equivalently, they can be expressed through a generalized partition function defined in terms of the probability measure assigned to each cell in the box-counting procedure.

For strange attractors arising in deterministic dynamical systems, the underlying phase space is well defined. In contrast, financial time series are typically one-dimensional observations for which the phase space is not explicitly given. To address this issue, an embedding space is constructed following the standard embedding approach developed in chaos theory, allowing the original time series to be projected into a higher-dimensional reconstructed phase space [21]. Within this reconstructed space, generalized dimensions can again be estimated using either generalized partition functions or generalized correlation sums.

The generalized dimensions can be further transformed into the spectrum of scaling exponents f α through a Legendre transformation [21]. Both the generalized dimensions and the f α spectrum can be readily related to the generalized Hurst exponent. For this reason, we first review partition function based dimensions and generalized dimensions and then discuss their transformation into the f α – α spectrum. Subsequently, we introduce the generalized Hurst exponent as a measure of the generalized scaling behavior of fluctuations in time series.

Understanding the scaling structure of financial time series cannot be achieved through simple fractal analysis alone; it requires an explicit characterization of multiscaling properties. The interactions among heterogeneous agents in financial markets give rise to the superposition of multiple fractal timeseries.

After reconstructing a one-dimensional financial series into a higher-dimensional embedding dimension, a q -th order partition function Z q R is derived for the probability p i of falling into a hypercube i of length R [21]. Then, the generalized dimension is defined as follows. Z q R = R τ q .

The partition function exponent τ q and the generalized dimension D q follow the relationship τ q = q − 1 D q [19].

In the stock market, the scaling and multi-scaling structures are found in stock returns and volatility. To understand the scaling structure of stock returns, we remove the trend from the normalized return time series and examine the pure fluctuation structure [12–18]. First, we create a cumulative time series by accumulating the normalized returns, and then we examine the multi-scaling structure of this cumulative time series. Widely used trend removal analysis methods are MF-DMA and MF-DFA. MF-DMA enhances the trend by subtracting the mean value of the cumulative time series over a certain range from the current point in time [12–22]. MF-DFA, on the other hand, fits the cumulative time series to a polynomial function over a fixed time window and then subtracts the trend fitting function from the time series to analyze pure fluctuations [16]. The q -th order fluctuation function F q s , which is weighted by the q -th power of the magnitude of the fluctuations, is multi-scaled over the window size s . F q s ∼ s H q .

Here, H q is the generalized Hurst exponent (GHE), and H q is an increasing function with respect to the power q if the time series is multi-scale. If the time series is monofractal, H q is constant and does not depend on q . The GHE H q is related to the scaling index α q and the singularity spectrum f α via Legendre transformation [16, 21] and the partition function exponent as τ q = q H q − 1 . f α = q α q − τ q and α q = d τ q d q = H q + q d H q d q .

Multifractality is a ubiquitous feature across financial markets, including stock indices, exchange rates, and cryptocurrencies [13–15]. The primary sources of multifractality are often attributed to the fat-tailed (non-Gaussian) distribution of returns and the non-linear temporal correlations (volatility clustering) within the time series [13–18]. The width of the f α spectrum, a measure of multifractal strength, often increases during periods of financial stress (e.g., the 2008 financial crisis, the COVID-19 pandemic) [13–15]. This suggests greater irregularity and heterogeneity of fluctuations during crises [23, 24].

Mensi et al. reported the presence of multifractality and long-memory correlations in the Shanghai and Hong Kong stock markets. In addition, they examined Bitcoin trading volume and Bitcoin-related time series and found that Bitcoin price returns did not significantly affect the efficiency of the Chinese stock markets [25]. Jayasankar applied the multifractal detrended fluctuation analysis (MF-DFA) method to time series of stablecoins and Bitcoin, specifically Tether (USDT) and USD Coin (USDC), and observed clear multifractal properties in their return series [26]. The study also reported the existence of long-term memory in both stablecoin and Bitcoin markets.

Li analyzed stock returns in the Nasdaq insurance sector during the market crash triggered by the outbreak of COVID-19 in March 2020 and identified pronounced volatility clustering [27]. Using the MF-DFA method, Li confirmed the multifractal nature of the return time series. Moreover, stronger multifractal characteristics were observed after the market collapse, indicating an increase in market inefficiency during periods of extreme stress.

For a truly multifractal signal, the f α spectrum is a convex-up curve to the Figure 1 illustrates f α versus α spectrum using the log-returns of the KOSPI index from 1 January 2004, to 31 December 2023 (data source: finance.yahoo.com). The KOSPI index clearly exhibits a multi-scaling phenomenon.

Methods like the wavelet transform modulus maxima (WTMM) and wavelet leader (WL) decompose the signal into time-localized components using a wavelet kernel [28–34]. They are particularly useful for detecting sharp transitions. When the time series behaviors the multi-scaling, the partition function Q q s based on the wavelet modulus maxima follows a power law [30] as Q q s ∝ s τ q , where s is the scaling size and τ q is the partition function exponent. When the time series shows the multi-scaling, τ q depends on the order q .

Wavelet transformations are extensively applied to the analysis of non-linear financial series, including stock indices, exchange rates, and cryptocurrencies [13, 15]. Drozdz et al. used the WTMM method to investigate the origins of multifractality in the German stock market [34]. Bai et al. applied discrete wavelet transformation to predict turning points in the stock market [35]. Tan et al. used the wavelet leader to detect stock market turning points in the US and Chinese markets [36]. Sarraj and Mabrouk introduced non-uniform wavelet analysis to characterize systemic risk during financial crises [37].

Saadaoui employed segmented multifractal analysis together with the maximal overlap discrete wavelet transform to evaluate the market efficiency of North African stock markets and reported the presence of asymmetric multifractality in the Maghreb stock indices [38]. Vogl and Kojic investigated the multifractal properties of time series associated with green cryptocurrencies and S&P 500 green bonds [39]. After filtering market trends using the maximal overlap discrete wavelet transform, they applied the multifractal detrended cross-correlation analysis (MF-DCCA) method. Their results revealed strong scaling behavior and pronounced multifractal cross-correlations between green cryptocurrency markets and green bond markets.

Multifractal detrended cross-correlation analysis (MF-DCCA) is a widely used method observing multi-scaling cross-correlation [13, 15, 28, 40–49]. MF-DCCA analyzes the joint scaling of the detrended cross-covariance function between two-time series, u i and v i where the index i indicates a particular time [13, 15, 40]. The generalized cross-covariance function F u v is defined as F u v ∼ s H u v q , where s is the size of the time segment and H u v q is the generalized cross-correlation Hurst exponent [13, 30]. If H u v q depends on the moment order q , the cross-correlation is multifractal [13, 30].

MF-DCCA has confirmed multifractal cross-correlations across numerous financial pairings, including stock indices [41, 42], sectoral stock markets [43, 44], commodity prices (e.g., crude oil) [45] and cryptocurrencies (e.g., Bitcoin) [13, 48]. The cross-correlation strength between assets often increases during high-volatility events or crises, indicating greater market contagion and synchronization (e.g., 2008 crisis, Russia-Ukraine conflict) [49].

Ling and Cao investigated the dynamics of asymmetric multifractal cross-correlations between cryptocurrency markets and gold as well as stock markets [46]. Their results showed that the strength of the correlation between Bitcoin and the stock markets of the G7 countries is higher than that between Bitcoin and the E7 markets. Kristjanpoller et al. examined asymmetric cross-correlation properties among blockchain exchange-traded funds (ETFs), cryptocurrencies, and the Nasdaq market using the MF-DCCA method [47]. They found that the cross-correlations between blockchain ETFs and the Nasdaq index exhibit stronger persistence during market downturns, whereas the correlations between blockchain ETFs and cryptocurrency time series display stronger persistence during periods of rising prices.

Various multifractal models have been proposed to explain the multi-scale fluctuation characteristics of financial time series. Among these, the multifractal model of asset returns (MMAR), first proposed by Mandelbrot et al, provides a theoretical framework that explains typical empirical facts observed in financial markets [50]. MMAR captures the heterogeneity of market trading activity and the resulting multi-scale structure of volatility by modeling financial returns as Brownian motion defined over multifractally distorted trading time frames [13, 15].

Günay analyzed stylized facts of stock returns for four emerging stock markets and compared predictability of the ARCH series and MMAR models by estimating key parameters from real-world data [51]. Meanwhile, Batten et al. applied a modified MMAR to 5-min return data of the EUR/USD foreign exchange market and evaluated its out-of-sample Value-at-Risk (VaR) forecasting performance [52]. They compared their approach with historical simulations and a GARCH model, confirming that the multifractal nature of EUR/USD returns has practical implications for risk management.

The Markov-switching multifractal (MSM) model was proposed to describe the multifractal volatility dynamics observed in financial markets [53–58]. The MSM model replaces the deterministic cascade hierarchy with a stochastic regime-switching mechanism governed by a Markov process [52]. The core assumption of the MSM model is that asset returns are determined by a latent volatility process comprised of multiple volatility components operating on different time scales. Each component evolves according to a Markov process with its own transition probability. This hierarchical structure naturally reproduces the volatility clustering and long-term dependencies observed in financial markets.

Calvet and Fischer applied MSM to capture the fat-tailed distribution of returns and multifractality of stock indices [54]. Liu et al estimated the generalized Hurst exponents by MSM model for daily returns of stock markets and simulated returns [55]. They observed apparent long memory of the volatility by the multifractal model.

The multifractal random walk (MRW), introduced by Bacry et al. presents a continuous-time probabilistic framework for describing the multifractal volatility structure observed in financial markets [13, 53, 56, 57]. MRW was modeled to overcome the limitations of discrete multiplicative cascade models or Markov switching-based formulations by directly constructing multifractality in a continuous-time setting while maintaining analytical tractability. In the MRW framework, the asset return process is modeled as Brownian motion governed by a stochastic multifractal volatility. This volatility is assumed to be a Gaussian process with a log-normal distribution, forming a long-term correlation structure over time, effectively reproducing the volatility clustering and long-term dependence observed in financial time series. The MRW model explains the multifractality of financial returns, such as the daily DJIA returns [58] and intraday S&P 500 future index and predicts the power-law tailed distribution of financial returns [59].

Research on stochastic process models with time-dependent Hurst exponents and their application to financial time series analysis is expanding. Ayache and Taqqu introduced a multifractal process with random exponent (MPRE) by replacing the Hurst exponent of fractional Brownian motion (fBM) with a stochastic process (Ayache 2005). Angelini and Bianchi reinterpreted the local irregularity of financial time series with time-varying Hurst–Hölder exponents [60–62]. They proposed the Fractional Stochastic Regularity Model (FSRM), a multifractional stochastic process with stochastically varying Hurst exponents H t . FSRM naturally generates multifractal characteristics. When the time-dependent Hurst exponent H t is given by a fractional Ornstein–Uhlenbeck process, the information content of the time series is quantified, allowing prediction of market trend continuation and mean reversion. FSRM was applied to predict future price returns of real stock indices [62]. Bianchi and Angelini proposed that financial risk is irregular, not volatility, and formalized the concept of “fair volatility” as the maximum level of risk acceptable in an efficient market [63]. These studies consistently suggest that inefficiencies and risks in financial markets should be understood from the perspective of local scale structures.

Agent-based market models provide a bottom-up framework for understanding financial markets as complex adaptive systems. While multifractal models phenomenologically attribute scaling properties, agent-based models aim to explain the mechanisms by which key empirical regularities, such as heavy-tailed returns, volatility clustering, and multifractality [64]. Lu et al. proposed an agent-based model that incorporates heterogeneous risk transmission among market participants [65]. In their framework, interactions between agents are time-dependent, and links between nodes are continuously formed and dissolved, thereby mimicking dynamic social interactions. The model considers two types of investors, namely, high-risk investors and low-risk retail investors, and assumes that trading decisions depend on the states of neighboring agents. This agent-based framework successfully reproduces several stylized facts of financial markets, including heavy-tailed return distributions, volatility clustering, and multifractal structures.

In these models, markets are comprised of heterogenous agents with different strategies, information sets, and time horizons. Commonly considered agent types include fundamental analysts, chartists, noise traders, and liquidity providers, each submitting buy or sell orders according to adaptive decision rules [66–69]. Prices are determined through market clearing rules or order-book mechanisms, and the collective effects of these micro-level interactions generate the nonlinear price movements and characteristic statistical properties observed at the macro level [68, 69].

Financial markets exhibit a variety of emergent phenomena. We introduce extensive research being conducted to observe and explain these emergent phenomena. This paper reviews methods for measuring multi-scale phenomena found in financial time series. Representative methods, such as multifractal fluctuation analysis which remove trends and the wavelet method, are introduced, along with research on their application to financial time series. MF-DCCA, a method for measuring multi-scale phenomena arising from the correlation between two or more time series, is introduced, and related research is reviewed.

We reviewed various models for explaining financial markets. The ABM model, which explains macro market characteristics based on the interactions of micro-level agents, offers important implications for understanding the interacting characteristics of economic actors that drive multi-scale phenomena.

Rather, these features may stem from finite-size effects, heavy-tailed distributions, nonstationary, or aggregation mechanisms in return and volatility time series. Recent empirical studies based on high-resolution and large-scale datasets demonstrate robust and universal scaling laws at the conditional mean level.

Future research should focus on clarifying the conditions for multifractality in financial time series and identifying its microscopic origins. This requires systematic comparisons with proxy data, rigorous statistical validation, and close integration with microstructural and agent-based models. From this perspective, multifractality should be viewed not as an established empirical fact, but rather as a working hypothesis whose validity largely depends on methodology, data quality, and observation scale. More fundamental macro- and micro-level models are needed to explain stylized facts and multi-scale phenomena in financial markets. In particular, understanding the impact of the collective interactions of micro-level actors on market fluctuations will contribute to understanding macro market characteristics.