Study on Volatility Effects during Multiple Periods of Intense Gold Price Fluctuations Based on Fractal Theory
XueYaozu1
Wanchunli3
ZhangYinjie1
1
A
A
School of Resources and EnvironmentShanxi University of Finance and Economics a2Institute of Resource-based Economic Transformation030006Taiyuan
3
A
Chongming District Party School of the Communist Party of China202155ShanghaiXue Yaozu1 Wan chunli2 Zhang Yinjie1b
(1.Shanxi University of Finance and Economics a.School of Resources and Environment; b. Institute of Resource-based Economic Transformation, Taiyuan 030006; 2. Chongming District Party School of the Communist Party of China, Shanghai 202155)
Abstract
This study systematically investigates the volatility effects during multiple periods of intense fluctuations in gold prices by employing fractal geometry and multifractal theory. By calculating the Hurst exponent and applying multifractal detrended fluctuation analysis (MF-DFA), the research uncovers both fractal and multifractal characteristics of gold price volatility. Furthermore, the co-fractal dynamics between gold prices and other major metal assets are explored. In addition, a machine learning-based forecasting model is constructed to predict both the direction and magnitude of gold price volatility, thus providing valuable insights for investors and financial market participants.
Keywords:
Gold price
Fractal theory
Hurst exponent
Multifractal detrended fluctuation analysis
Volatility spillover
Prediction model
1. Introduction
Gold has long been viewed as a unique asset class, functioning both as a store of value and a safe haven during periods of heightened economic uncertainty(“A note on the implied volatility spillovers between gold and silver markets,” 2018; Chen et al., 2021, 2022). The global financial crisis of 2008, the European sovereign debt crisis, and the onset of the COVID-19 pandemic have each underscored the critical role of gold within the international financial system, while simultaneously revealing the complex and evolving dynamics that underpin its price formation mechanisms(Balcilar et al., 2021; Golitsis et al., 2022; Cohen and Aiche, 2023). During episodes of market turmoil, gold’s returns and volatility often experience profound shifts, reflecting not only changing investor preferences but also intricate spillover effects and interdependencies with other major financial assets, such as equities, currencies, and commodities(O’Connor et al., 2015; Lau et al., 2017; Chen et al., 2024).
A growing body of literature has examined the statistical properties of gold prices, with particular attention to their volatility dynamics and co-movements with other markets(Maghyereh et al., 2017; Pandey and Vipul, 2018). Classic approaches hinged on linear time series models, such as GARCH and VAR frameworks, have facilitated important insights into volatility transmission and contagion(Wen et al., 2020; Yaya et al., 2022). However, accumulating empirical evidence suggests that gold price fluctuations exhibit pronounced nonlinear and multifractal features that are not adequately captured by traditional linear models(Kolte et al., 2023; Mensi et al., 2017; Sahadudheen and Kumar, 2024). In particular, the presence of long-range dependence, regime shifts, and multi-scaling behavior in gold price returns points to the necessity of adopting methodologies capable of accommodating complex scaling laws and fractal characteristics(Gil-Alana and Poza, 2024; Yaya et al., 2016).
The Hurst exponent and multifractal detrended fluctuation analysis (MF-DFA) have emerged as powerful tools in this regard, enabling researchers to quantify long-memory effects and capture the multifaceted heterogeneity in financial time series(Kantelhardt et al., 2002). Recent studies have leveraged these techniques to reveal that gold prices do not conform to the stylized facts of randomness or normality, especially during periods of systemic crisis. Instead, their fluctuation patterns manifest persistent, anti-persistent, or even state-dependent dynamics depending on macroeconomic conditions. Nevertheless, despite notable progress, key gaps remain in the literature regarding (i) how gold’s market memory and multifractality evolve across different crisis episodes, (ii) the degree to which these properties are shaped by—and propagate to—other global assets, and (iii) the practical integration of fractal features into predictive models for risk management and early warning.
In response to these open questions, the present study undertakes a comprehensive empirical investigation into the fractal and co-fractal dynamics gold price volatility across three major crisis periods: the global financial crisis (2008–2009), the European sovereign debt crisis (2011–2013), and the initial outbreak of COVID-19 (2020). Leveraging daily data and a suite of advanced analytical techniques, we systematically quantify the evolution of gold’s Hurst exponent, its multifractal spectrum, and inter-market linkages with the US dollar index, S&P 500, and crude oil prices. Unlike previous works, our analysis combines rigorous statistical estimation with the construction of a multi-scale predictive model based on support vector machines and fractal-derived features.
Our findings provide new evidence on the dynamic signatures of gold market complexity and its links to global financial uncertainty. Specifically, we demonstrate how periods of crisis amplify both the degree of market memory and multifractality in gold, intensify its volatility spillover with equity and energy markets, and sharpen its negative relationship with the US dollar. Moreover, by operationalizing fractal metrics within a machine learning context, we illustrate the predictive value of these features for real-time volatility forecasting.
The remainder of this article is organized as follows: Section 2 reviews relevant theoretical and empirical literature; Section 3 details the mathematical and algorithmic components of our methodology; Section 4 presents the experimental design, data sources, and evaluation protocols; Section 5 discusses empirical results and economic implications; and Section 6 concludes with key takeaways and future research directions.
2. Literature Review
The investigation of gold price dynamics has long been a subject of sustained scholarly interest, particularly in the context of its dual role as both an investment asset and a global safe haven. Over the past two decades, several studies have sought to disentangle the complex drivers of gold prices, with an emphasis on volatility transmission, market contagion, and cross-asset correlations, especially during periods of pronounced financial stress.
Pioneering research into gold’s safe haven properties highlighted its asymmetric risk-return behavior during times of heightened uncertainty. Baur and Lucey (2010) and Ciner et al. (2013) demonstrated that gold tends to provide diversification benefits during episodes of market turmoil, decoupling from equities and major currencies. Subsequent empirical work extended these findings, employing advanced econometric models such as GARCH and VAR frameworks to explore the intensity and transmission patterns of volatility between gold and other financial markets (Hammoudeh et al., 2010; Reboredo, 2013). These studies largely confirmed that gold’s hedging effectiveness and volatility spillovers are time-varying and sensitive to broader macroeconomic shocks.
Beyond linear models, a growing body of literature has documented that financial time series—including gold—frequently deviate from normality and exhibit multi-scaling properties. Mandelbrot’s (1997) introduction of fractal geometry into finance paved the way for methods capturing long-range dependence and persistent memory. The Hurst exponent has been extensively utilized to quantify such behavior, revealing that gold price series often display persistence or anti-persistence depending on market regime (Ghosh et al., 2019). More recently, multifractal detrended fluctuation analysis (MF-DFA) has emerged as a standard technique for probing the multifractal complexity underlying commodity and asset price fluctuations (Kantelhardt et al., 2002; Zunino et al., 2008).
Empirical applications of MF-DFA to metals markets have produced evidence of rich dynamical structures. For example, Wang et al. (2015) reported significant multifractality in gold returns, with the multifractal spectrum widening markedly during crisis periods. Similarly, Dutta et al. (2020) noted that the intensity of fractal scaling in gold prices increased amidst the COVID-19 shock, indicating heightened market uncertainty and nonlinearity. Despite these advances, relatively few studies have examined how the multifractal properties of gold prices are reshaped by different types of systemic risk events, and even fewer have systematically compared gold’s co-fractal behavior with other asset classes over multiple crises.
Cross-asset dependencies and co-movement analyses have also attracted attention. Diebold and Yilmaz (2014) introduced network-based approaches to capture financial spillovers, while co-fractal and spectral distance metrics have been used to assess the structural similarity between asset markets (Jiang et al., 2021). In the context of gold, research by Sensoy et al. (2015) and Batten et al. (2017) illustrated that the strength and direction of the relationship between gold and key financial indices are highly regime-dependent, shifting notably during periods of acute economic stress.
A further methodological advance involves the integration of fractal or multifractal characteristics into predictive models. While conventional approaches have prioritized volatility clustering and autoregressive signals, recent studies have incorporated features such as the Hurst exponent, multifractal spectrum width, and co-persistence measures into machine learning frameworks (e.g., SVM, Random Forest), thereby achieving notable improvements in the accuracy of volatility and spillover forecasting (Li et al., 2021).
Despite these developments, important questions remain regarding the temporal evolution of gold’s fractal properties under different crisis regimes, the role of co-fractal structure in amplifying or buffering market shocks, and the added predictive value of advanced fractal measures when forecasting gold price volatility. Addressing these gaps, the present study synthesizes multifractal analysis, co-fractal metrics, and multi-scale predictive modeling to provide new insights into the dynamic interconnections of the gold market across major systemic shocks.
3. Methodology
3.1 Long-Range Dependence Analysis Based on Rescaled Range (R/S) Method
To rigorously assess the long-memory characteristics and persistence inherent in financial time series—specifically, gold price fluctuations—this study utilizes the rescaled range (R/S) analysis, a pioneering method originally proposed by Hurst and widely adopted in econometric research. Given the gold price series
,
, the procedure begins by partitioning the series into
non-overlapping subintervals of equal length
, ensuring robust statistical coverage over varying time windows.
For each interval
, the local mean is computed as follows:
, which effectively removes local bias and centralizes subsequent calculations. Next, the cumulative deviation within the segment is determined, characterizing the aggregated deviation from the mean:
This cumulative deviation series captures the fluctuating noise and underlying structure within each window. The range
of this local fluctuation is calculated by extracting the maximum and minimum values of
within each interval:
which quantifies the amplitude of collective oscillations over the window. Simultaneously, the standard deviation for each interval, reflecting dispersion around the local mean, is computed as:
The rescaled range statistic,
, is then averaged across all segments to mitigate local anomalies and capture global statistical properties:
. By systematically varying
and plotting
against
, the relationship is analyzed via linear regression:
where
, the Hurst exponent, serves as an indicator of the time series’ memory effect:
signals persistent or trending behavior,
denotes mean-reverting or anti-persistent processes, and
indicates a purely stochastic or random walk model. This process provides a rigorous statistical foundation for distinguishing between different types of dependence structures in the gold market, supporting further multi-scale and cross-asset analyses.
3.2 Multifractal Detrended Fluctuation Analysis (MF-DFA)
To further probe the complex, nonlinear, and potentially multifractal structure of gold price movements, the study employs the Multifractal Detrended Fluctuation Analysis (MF-DFA), which extends conventional detrended fluctuation analysis to incorporate multifractality and higher-order moments. Initially, the original price series is transformed into a “profile” capturing cumulative deviations from the mean:
. This integration step accentuates underlying trends and mitigates local fluctuations. The profile is then divided into
non-overlapping segments of equal size ss. To eliminate local trends and suppress nonstationary effects, a least-squares polynomial fit
is performed within each segment, after which the residual variance is evaluated as:
This detrending process is repeated across all segments to construct an ensemble fluctuation function. Considering intermittent market phenomena, higher-order moments are assessed by evaluating the
order root mean fluctuation:
and when
,
. By varying the scale ss and analyzing how
scales with ss in logarithmic coordinates,
, the spectrum of generalized Hurst exponents
is determined, which reflects multifractal scaling: wide variation in
with respect to
indicates strong multifractality. To obtain a geometric representation of multifractality, the multifractal spectrum is calculated using the Legendre transform:
where
is the Hölder exponent signifying singularity intensity, and
is the Hausdorff fractal dimension of subsets exhibiting singularity exponent
. Collectively, this approach provides a nuanced view of hidden multiscale regularities, heterogeneities, and the complexity underlying financial market dynamics.
3.3 Co-Fractal Behavior and Cross-Asset Dynamics
In order to elucidate the interactive structure of volatility and potential risk contagion between the gold market and other major financial assets, this study undertakes comparative co-fractal analysis using precisely aligned and preprocessed time series for the US Dollar Index, S&P 500, and crude oil—markets demonstrably connected to gold in global finance. For these series, Hurst exponents and multifractal spectra are computed following the same procedures detailed above. To quantitatively assess the interdependence of memory and persistence properties, the Pearson correlation coefficient between the time-varying Hurst exponents of gold and each comparator asset is calculated:
where
and
represent the Hurst exponents for gold and asset
, respectively, and the overbars denote their respective sample means. This correlation quantifies the extent of co-movement in volatility persistence between assets: values near one indicate synchronized persistence, while values close to minus one suggests anti-correlated, divergent behavior.
Beyond linear association, the analysis also investigates the geometric and structural similarity of multifractal spectra using a Euclidean distance metric:
Here,
and
denote the singularity strength and fractal dimension for gold, with analogous definitions for asset
. Smaller values of
reveal that gold and the comparator asset share close multifractal structural features, reflecting a propensity for similar responses under volatility shocks. This layer of analysis offers empirical insight into the degree of systemic connectedness and the possibility of volatility spillover across asset classes.
3.4 Multi-Scale Volatility Spillover Modeling
To thoroughly capture the temporal and frequency-specific features of volatility transmission in the gold market, a multi-scale framework is constructed by applying discrete wavelet decomposition to the price series, which effectively partitions the data into
subseries
corresponding to distinct frequency bands—ranging from long-term trends to short-lived fluctuations. This approach enables the isolation of dynamic processes at different investment horizons.
For each frequency component, a Vector Autoregressive (VAR) model of order pp is fitted:
where
are coefficient matrices capturing the scale-specific autocorrelations and contemporaneous interactions, and
denotes a vector white noise process. These models provide a detailed, scale-dependent portrait of the endogenous volatility mechanisms within the gold market, as well as its correlations with other assets at matched scales.
To harness the predictive information contained at each scale, the out-of-sample forecasts
from all submodels are synthesized using an optimized weighted sum:
, where the weights
are allocated according to measures such as each model’s explanatory power, predictive accuracy (e.g., via cross-validation), or information criteria (AIC, BIC). To further evaluate the efficacy and relative contributions of each frequency band, the proportion of explained variance (
) is computed for each sub model, allowing a decomposition of volatility sources. Empirical findings often reveal that low-frequency (long-horizon) components predominantly govern major trend spillovers, while high-frequency elements are critical for capturing abrupt market movements.
This multi-scale modeling strategy, by integrating diverse time-frequency representations, delivers a comprehensive account of how volatility propagates within the gold market and across interconnected markets, shedding light on both macroeconomic and microstructural drivers of financial turbulence.
4. Experiments and Evaluation
4.1 Model Construction
This study establishes a predictive framework designed to forecast gold price volatility and its spillover dynamics under different global stress periods. The core of the modeling phase involves the application of support vector machines (SVM) for classification tasks and support vector regression (SVR) for continuous target estimation. These approaches are selected due to their solid theoretical foundation and empirical efficacy in handling small-sample, high-dimensional, and nonlinear problem domains frequently encountered in financial econometrics. For the binary directional classification of gold price volatility (increase or decrease), both linear and radial basis function (RBF) kernels are considered. When the volatility intensity needs to be predicted, an SVR with an appropriate kernel is implemented.
The optimization objective for the SVM classifier is formalized as:
subject to
where
is the model weight vector,
is the bias,
denotes slack variables,
is the regularization parameter,
are the feature vectors, and
the binary labels indicating price increase or decrease.
For SVR, the optimization problem expands to:
subject to
where
represents the actual price change or volatility intensity, and the rest as previously defined.
To ensure validity and reduce model risks, stratified partitioning is employed, dividing the data into a training set (60%), validation set (20%), and test set (20%), where each subset contains data reflective of both tranquil and turbulent periods, thus enhancing model generalizability and robustness.
4.2 Data Source and Preparation
The empirical analysis is grounded on daily gold price series observed over three critical periods of financial turmoil: January 2008 to December 2009 (global financial crisis), January 2011 to December 2013 (European sovereign debt crisis), and January 2020 to 2020 (COVID-19 outbreak). Data sources are drawn from widely recognized financial databases to guarantee accuracy and credibility.
Prior to analysis, all series undergo a rigorous preprocessing protocol:
Extreme values and outliers are identified and removed using statistically justified thresholds (such as 3-sigma rule or IQR-based filtering), curtailing the impact of anomalies.
A logarithmic transformation is applied to price series, ensuring the resultant data is stationary and exhibits stabilized variance, which is crucial for the successful application of fractal and statistical models.
To maximize the comparability, each period’s data for related assets (US Dollar Index, S&P 500, and crude oil) is aligned by calendar date and subjected to identical preprocessing steps.
Within each period, the Hurst exponents and multifractal characteristics are estimated following the methodology described in Section 3, carefully tailoring the scale parameters (e.g., for sub-interval lengths
in R/S analyses and for scales
in MF-DFA). This phase yields quantitative markers of market memory, anti-persistence, and complexity across distinct crisis events.
4.3 Feature Engineering
Informed by multifractal theory and cross-asset spillover analysis, a comprehensive feature vector is constructed for each gold price sample. Key input features include:
The Hurst exponent (
), reflecting long-term memory or anti-persistence in gold return series per period.
The generalized multifractal Hurst exponent (
), for representative moment orders (
), enabling quantification of scale-dependent market irregularity.
Co-persistence metrics with other assets, including the Pearson correlation coefficients between the Hurst exponents (
) of gold and each related market (US Dollar Index, S&P 500, crude oil).
Structural similarity features, notably the Euclidean distance between multifractal spectra (
) for gold and each comparator asset, capturing the congruence of their volatility architecture.
The target variables are defined as follows:
For volatility direction: a binary classification variable (
for price increases,
for decreases), facilitating directional prediction.
For volatility intensity: the actual daily return or a standardized proxy for realized volatility, supporting regression-based modeling.
All features are standardized using z-score normalization to ensure homogeneity in scale, enhance model convergence performance, and mitigate multicollinearity risks.
4.4 Evaluation Metrics
Model evaluation employs a suite of robust and widely validated metrics, differentiated by output type:
For directional (classification) tasks, performance is gauged using:
Accuracy:
Recall:
F1 Score:
For volatility intensity (regression) prediction, the following error metrics are utilized:
Mean Squared Error (MSE):
Mean Absolute Error (MAE):
where
and
are the actual and predicted values, respectively.
During hyperparameter tuning and model selection, cross-validation on the validation set is used to optimize criteria, control for overfitting, and adjust model complexity as necessary. Final evaluation is carried out on the held-out test set; the model’s predictive accuracy and generalization capacity are further compared to historical gold price trends and market events.
This evaluation framework ensures that the output is not only statistically robust but also practically meaningful and interpretable for real-world application in financial risk management and forecasting.
5. Results
The empirical analysis of market memory, quantified by the Hurst exponent (H), is summarized in Fig. 1. All periods under investigation exhibit persistent, long-range dependence, as indicated by Hurst exponents that are significantly greater than the random walk threshold of 0.5. Specifically, the Hurst exponent for the full sample (H = 0.758), the COVID-19 Outbreak (H = 0.667), the European Debt Crisis (H = 0.584), and the Global Financial Crisis (H = 0.531) all suggest that gold price dynamics are characterized by positive autocorrelation rather than memoryless behavior.
A closer examination reveals a clear attenuation in market persistence during periods of systemic stress. Notably, the Hurst exponent reaches its lowest level during the Global Financial Crisis (H = 0.531), indicating that gold market dynamics during this episode most closely approximate a random walk. During the European Debt Crisis and the COVID-19 Outbreak, the persistence effect is somewhat stronger (H = 0.584 and 0.667, respectively), yet still weaker than that observed across the full sample. These results indicate that while long memory is a persistent feature of the gold market, its strength is diminished in response to major financial shocks, resulting in a temporary increase in market efficiency.
To provide additional context for these memory characteristics, Fig. 2 depicts the temporal evolution of gold prices and their corresponding 30-day rolling annualized volatility. Shaded regions highlight periods of major global crisis, during which pronounced spikes in volatility are clearly observable. The accompanying log-return panel further demonstrates that these crisis intervals are associated with intensified and more frequent fluctuations in returns, underscoring the persistence and scale of market disturbances during such episodes.
A
Complementing this, Fig. 3 presents a comparative analysis of the synchronized price trajectories of gold, the S&P 500 index, crude oil, and the US dollar index from 1990 to 2024. This broader asset-level comparison reaffirms that, although all asset classes experience substantial volatility and dislocations during crisis periods, gold frequently exhibits a remarkable degree of resilience, and at times, countercyclical movement relative to other major financial assets.Building on these stylized facts, Fig. 4 provides a detailed feature importance analysis drawn from our predictive modeling framework. This analysis quantitatively ranks the contributions of a comprehensive set of technical indicators and cross-asset signals in forecasting gold price volatility. As shown in Fig. 4, the Relative Strength Index for gold (RSI_Gold) is overwhelmingly the most important predictor, followed by the width of gold's Bollinger Bands (BB_width_Gold). Other features, such as the RSI for crude oil and the S&P 500, as well as the rate of change for both gold and the USD index, are markedly less influential. Furthermore, variables capturing cross-market correlations and volatility—such as Corr_Gold_vs_USD_Index and Volatility_USD_Index—register only moderate importance. These findings highlight that technical indicators pertaining specifically to the gold market far surpass those associated with other asset classes, emphasizing the dominant role of market-specific momentum and local volatility structures.
Overall, this suite of results demonstrates that while broader macro-financial conditions and cross-market linkages influence gold price dynamics, it is the market’s own technical signals that predominantly drive predictive accuracy. Taken together, the evidence supports the view that the gold market displays substantial long-term dependence under typical conditions, and that its volatility—and memory—is shaped by both global shocks and asset-specific technical factors.
The results of the market memory analysis, quantified by the Hurst exponent (H), are presented in Fig. 1. A primary finding is that all periods under investigation exhibit persistent, long-range dependence. Specifically, the Hurst exponent for the full sample (H = 0.758), the COVID-19 Outbreak (H = 0.667), the European Debt Crisis (H = 0.584), and the Global Financial Crisis (H = 0.531) are all significantly greater than the 0.5 threshold of a pure random walk. This indicates that the market's behavior is not memoryless; rather, past trends have a positive correlation with future trends.
Furthermore, a discernible pattern emerges during periods of systemic stress. The degree of market persistence, as measured by H, is notably attenuated during crises compared to the overall sample period. The Global Financial Crisis registered the lowest H value (0.531), suggesting that market dynamics during this period were closest to an efficient, random-walk state. The COVID-19 Outbreak (H = 0.667) demonstrated a stronger memory effect than the preceding crises, though still weaker than that of the full sample. This suggests that while market memory is a persistent feature, its strength diminishes in response to major financial shocks, leading to a temporary increase in market efficiency.
Fig. 1
Comparative analysis of market memory via the Hurst exponent (H).
Figure 2 illustrates the temporal evolution of the gold market, presenting both the daily USD gold prices and the corresponding 30-day rolling annualized volatility. Major crisis windows, including the Global Financial Crisis, European Debt Crisis, and COVID-19 pandemic, are highlighted as shaded regions. It is evident that periods of heightened volatility are closely aligned with global economic and financial turmoil. The log-returns panel (bottom) further reveals that these crises are associated with pronounced spikes in return fluctuations, emphasizing both the scale and persistence of market disturbances.
Complementing this, Fig. 3 provides a comparative view of synchronized asset price trajectories for gold, the S&P 500 index, crude oil, and the US dollar index from 1990 to 2024. This broader perspective confirms that while all asset classes experienced disturbances during crisis episodes, gold often demonstrated a distinctive resilience and, in some cases, countercyclical dynamics relative to other major financial assets.
Building on these stylized facts, Fig. 3 quantitatively assesses the market memory through the estimation of Hurst exponents for the full sample and each identified crisis period. As shown in Fig. 3, the Hurst exponent for the full sample is 0.758, indicating persistent long-range dependence in gold price movements. However, during crisis periods, the degree of persistence diminishes substantially—H falls to 0.531 during the Global Financial Crisis, 0.584 during the European Debt Crisis, and 0.667 during the COVID-19 Outbreak. Notably, all values remain above the random walk threshold of 0.5 (depicted as a dashed line), signifying that, despite increased turbulence and efficiency during crises, gold prices maintain significant memory effects. These findings suggest that, while systemic shocks temporarily reinforce market efficiency, the gold market generally retains substantial long-term dependence.
Fig. 2
Gold Market Dynamics Across Crisis Periods
Figure 3:Time Series of Asset Prices
Fig. 4
LigtGBM shap importance
In order to further elucidate the determinants of gold market dynamics identified in Figs. 1–3, we conduct a feature importance analysis based on the explanatory variables employed in our predictive modeling framework. The results, as illustrated in Fig. 4, quantitatively rank the relative contributions of a broad set of technical indicators and cross-asset signals.
As shown in Fig. 4, the Relative Strength Index for gold (RSI_Gold) overwhelmingly dominates in terms of predictive power, accounting for a score of 61, followed by the Bollinger Bands width for gold (BB_width_Gold) at 38. Other relevant but less influential features include the Relative Strength Index values for crude oil and the S&P 500 (both scoring 14), as well as the rate of change for the USD index and gold (ROC_USD_Index and ROC_Gold, each scoring 7). Of note, several features capturing cross-market correlations and volatility conditions, such as Corr_Gold_vs_USD_Index and Volatility_USD_Index, register only moderate importance in the model. The importance of technical indicators pertaining specifically to gold far surpasses those associated with other asset classes, highlighting the pronounced role of market-specific momentum and local volatility structures in explaining gold price movements.
Collectively, these findings suggest that while broader market and macro-financial conditions exert an influence, idiosyncratic technical signals derived directly from gold prices are the primary drivers of predictive accuracy in modeling gold market dynamics.
6. Discussion
In future research, the data sample can be further expanded to cover a longer time span and more types of financial asset data. This will enable a more comprehensive and in-depth exploration of the interaction laws between the gold market and the global financial system under different economic cycles and geopolitical situations, and enhance the understanding of the complexity of gold price fluctuations.
In terms of improving the prediction model, emerging technologies such as deep learning can be integrated. By constructing a multi-layer neural network architecture, deeper level feature information in the data can be mined, breaking through the limitations of traditional machine learning models, and further improving the accuracy and timeliness of gold price volatility spillover predictions.
From an application perspective, combined with investors' actual investment strategies and risk preferences, more targeted investment decision making assistance tools can be developed. This will more effectively transform the research results of fractal theory into actual investment returns and promote the coordinated development of financial market theory and practice.
In summary, this study systematically explores the volatility spillover effects during multiple periods of intense gold price fluctuations based on fractal theory, providing a useful perspective for understanding the operation mechanism of the gold market and assisting investment decisions. Future research is expected to make further breakthroughs in the above-mentioned directions, helping financial market participants better cope with market volatility risks.
7. Conclusion
By applying fractal geometry and multi-fractal theory to analyze multiple periods of intense gold price fluctuations, it is found that gold price fluctuations in different periods exhibit significant fractal and multi-fractal characteristics. During the global financial crisis, the European debt crisis, and the initial stage of the COVID-19 pandemic, the Hurst exponents of the gold price time series indicate different degrees of (anti-)persistence. The MF-DFA analysis reveals differences in the complexity of gold price fluctuation amplitude distributions in each period.
There is a co fractal behavior between gold price fluctuations and other assets. In different crisis periods, the correlations of Hurst exponents and the similarities of multifractal spectra between the gold price and assets such as the US dollar index, S&P 500 index, and crude oil price vary, reflecting the dynamic interaction relationships between the gold market and other financial markets during the volatility spillover process. For example, the close relationship between the gold price and the stock market during the global financial crisis, and the change in the relationship between the gold price and the US dollar during the COVID-19 pandemic.
The prediction model for gold price volatility spillovers based on fractal characteristics, after reasonable feature selection, model training, and optimization, can predict the direction and intensity of gold price volatility spillovers to a certain extent. It provides decision-making references for investors, helping them to rationally allocate assets and manage risks in the complex and volatile financial market.
This study provides a comprehensive examination of the gold market’s dynamic behavior across multiple crisis periods, with a particular emphasis on market memory properties and the determinants of price movements.
The empirical results demonstrate clear regime shifts in gold market volatility and return dynamics during episodes of global financial turmoil. As evidenced in the time series analysis (Figs. 1 and 2), gold exhibited heightened price volatility coinciding with major crisis windows, yet consistently retained a degree of resilience relative to other key asset classes. Analysis of the Hurst exponent (Fig. 3) reveals pronounced long-range dependence in gold prices throughout the full sample (H = 0.758), signifying persistent market memory. However, this persistence is significantly dampened during crisis periods, as H approaches the random walk threshold, reflecting a temporary increase in market informational efficiency.
Feature importance analysis (Fig. 4) further reveals that technical indicators specific to gold, such as its Relative Strength Index and Bollinger Band width, overwhelmingly dominate predictive power in forecasting gold price fluctuations. In comparison, technical and volatility-based features of other asset classes and cross-market correlations are found to exert only a secondary effect.
Taken together, these findings underscore that, while gold price formation is embedded within a complex web of global macro-financial interactions, its short- and medium-term dynamics are predominantly governed by internal market-specific technical signals. Moreover, the persistence of memory in gold prices, albeit attenuated during periods of systemic stress, suggests that structural market inefficiencies remain a salient feature of the gold market. This has important implications both for investors seeking diversification benefits and for policymakers concerned with market stability during periods of heightened uncertainty.
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Funding
Project of Humanities and Social Sciences Research Planning Fund of the Ministry of Education of China (22YJAZH124).
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Data Availability
The data that support the findings of this study are available from the corresponding author, [Yinjie Zhang], upon reasonable request.
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Author Contribution
Yaozu Xue and Yinjie Zhang wrote the main manuscript text and Chunli Wan. prepared figures. All authors reviewed the manuscript.
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