Multifractal Cross-Market Dependence and Dynamic Hedging Under Crisis Regimes: Evidence from Commodity–Equity Interactions

Abstract

This study investigates cross-market dependence and dynamic hedging performance between the U.S. equity market and major commodity assets across distinct crisis regimes. Using daily data for the S&P 500 index and four key commodities (WTI crude oil, gold, wheat, and natural gas), we examine how market linkages evolve during systemic disruptions by applying Multifractal Detrended Cross-Correlation Analysis (MFCCA) and the q-dependent detrended correlation coefficient. Hedging performance is assessed using optimal hedge ratios estimated under two multivariate GARCH frameworks: the Asymmetric Dynamic Conditional Correlation (ADCC-GARCH) and the Generalized Orthogonal GARCH (GO-GARCH) model. The findings reveal strong multiscale and time-varying dependencies that intensify during high-volatility periods, reducing the benefits of conventional portfolio diversification. Hedging effectiveness proves to be regime dependent and strongly influenced by nonlinear cross-market interactions. The GO-GARCH model captures volatility spillovers and asymmetric co-movements more effectively, delivering superior hedging results compared with ADCC, especially during episodes of extreme market stress. Among the analysed commodities, crude oil and gold offer the most reliable hedging properties, whereas wheat and natural gas show unstable performance due to supply side shocks. These results emphasize the need for flexible, dynamically adjusted risk-management strategies during crisis environments.

1. Introduction

The progressive financialization of commodity markets and their increasing integration with global equity systems have fundamentally altered modern portfolio structures and challenged traditional assumptions about risk transmission. Classical portfolio theory suggests that holding assets whose returns move independently, or in opposite directions, reduces overall portfolio risk. In practice, however, this assumption often breaks down. Over recent decades, cross-asset correlations have risen sharply during systemic crises, eroding diversification benefits precisely when they are most needed [1,2]. This shift reflects the growing sensitivity of commodity markets to macroeconomic shocks, speculative capital flows, and global uncertainty. As a result, interactions across asset classes have become increasingly complex and nonlinear, with implications that extend well beyond conventional asset allocation frameworks.

The COVID-19 crisis marked a pivotal shift in the behaviour of global commodity markets. Lockdowns disrupted production and trade flows. Financial stress heightened uncertainty, and speculative repositioning intensified volatility across asset classes [3]. This instability culminated in April 2020, when WTI crude oil futures traded below zero after a collapse in demand and acute storage constraints [4]. Agricultural commodities, including wheat and corn, also experienced pronounced price fluctuations driven by export restrictions and supply chain disruptions [5].

Gold initially benefited from a flight-to-safety response, but this traditional haven revealed vulnerabilities once liquidity pressures intensified [6]. The global financial system had barely absorbed these shocks when the Russia–Ukraine conflict triggered a new wave of commodity and financial instability. Wheat prices rose sharply after export blockades in the Black Sea region [7]. Natural gas markets entered a prolonged period of stress driven by Europe’s heightened energy dependency [8]. Crude oil prices also responded strongly to geopolitical uncertainty and supply disruptions. At the same time, the ongoing strategic rivalry between the U.S. and China has further increased macroeconomic uncertainty and accelerated the reshaping of global supply and production chains [9].

Recent crises have not unfolded in isolation but have generated widespread contagion effects that amplified volatility spillovers and strengthened cross-market interdependence [10]. This surge in uncertainty encouraged herding behaviour and reduced investors’ risk tolerance. As a result, the traditional hedging role of commodities has weakened noticeably [11]. During the early phase of the COVID-19 pandemic, financial stress spread rapidly across asset classes and produced one of the fastest equity drawdowns in modern market history [12]. At the same time, unprecedented monetary interventions and large-scale liquidity injections seem to have intensified the co-movement between equities and commodities, as indicated by recent macro-financial evidence [13]. These developments challenge the conventional perception of commodities as reliable hedging instruments in a highly integrated and shock sensitive global financial environment.

Empirical evidence shows that the hedging role of commodities is inherently unstable and varies across market regimes. Gold often acts as a safe haven, although its protective capacity can differ substantially across crises and investment horizons [14]. Crude oil reflects both physical demand conditions and speculative trading behaviour, a dual character that tightly links financial markets with the real economy [15]. Agricultural commodities such as wheat remain highly sensitive to supply disruptions triggered by geopolitical tensions [16]. Natural gas markets also display persistent volatility driven by regional segmentation and storage limitations [17]. These patterns point to the need for a dynamic and regime-specific approach to hedging analysis that extends beyond static correlation frameworks.

A growing strand of empirical research has investigated the evolving linkages between commodities and equities using spillover indices and connectedness frameworks. Fry-McKibbin and McKinnon [18] show that these interdependencies intensify during periods of financial turbulence. Their findings suggest that global markets have become increasingly synchronized. Evidence from Multivariate Generalized Autoregressive Conditional Heteroskedasticity (MGARCH) analyses, such as the study by Mukherjee and Bardhan [19], further indicates that risk transmission tends to intensify under market stress. Bidirectional spillovers commonly emerge between crude oil, gold and major equity indices. However, as Babar et al. [20] highlight, much of the connectedness literature concentrates on the statistical structure of dependence rather than on its implications for hedging performance. Traditional models often rely on linear assumptions, which makes them less suited to capturing the asymmetric, and long-memory dynamics that characterize financial time series during crises. From a broader macro-financial standpoint, Díaz et al. [13] show that commodity price shocks and supply chain disruptions have become major global drivers of inflation and volatility. This reinforces cross-market interdependence. In line with this view, Chowdhury et al. [21] provide evidence that asset interactions shift across volatility regimes rather than remaining stable over time, highlighting the need for more flexible and dynamic modelling approaches.

Dynamic hedging strategies are commonly estimated using MGARCH models, yet comparative evaluations of alternative specifications remain limited. The ADCC-GARCH model captures correlation asymmetry and allows negative shocks to have stronger effects than positive ones [22]. However, it cannot fully capture how volatility propagates through latent common factors. The GO-GARCH model addresses this limitation by decomposing volatility into orthogonal components, which helps distinguish systemic from idiosyncratic sources of risk. Despite these advances, empirical evidence comparing the hedging efficiency of these frameworks under periods of systemic stress remains scarce.

At the same time, recent studies document that commodity–equity linkages exhibit multifractal and nonlinear dependence patterns that traditional econometric models struggle to characterize adequately [23]. Building on the work of Baruník and Křístoufek [24], Multifractal Detrended Cross-Correlation Analysis (MFCCA) provides a useful framework for examining scale-dependent and nonlinear risk transmission. Although MFCCA has been widely applied to study cross-market dynamics, its integration into hedging performance evaluation and portfolio-risk analysis remains limited. This lack of empirical synthesis between multifractal dependence and dynamic hedging effectiveness represents an important research gap.

This study contributes to closing this gap by examining multifractal cross-dependence and dynamic hedging performance between the U.S. equity market (S&P 500) and four major commodity markets. It introduces a hybrid framework that combines MFCCA with two multivariate volatility models, ADCC-GARCH and GO-GARCH, to assess the stability and effectiveness of hedge ratios under time-varying and regime-dependent dependence structures. By linking multifractal dynamics with hedging efficiency, the study extends existing connectedness and volatility-based approaches and provides new insights into portfolio risk management during periods of heightened uncertainty.

The results indicate that hedging effectiveness is strongly regime dependent and commodity specific. Gold retains partial defensive value over short investment horizons, whereas crude oil frequently shifts from acting as a hedge to becoming a transmitter of volatility. Wheat provides limited protection during supply side shocks, while natural gas exhibits unstable hedging properties due to pronounced price variability. Moreover, the GO-GARCH model generates more stable hedge ratios during crisis periods, highlighting its ability to capture latent sources of systemic risk.

The remainder of the study is divided into four sections. Section 2 provides an overview of the literature on commodity–equity relationships and hedging models. Section 3 details the data sources and methodological approach. Section 4 discusses the main empirical findings. Section 5 closes the study with conclusions, limits and suggestions for future work.

2. Literature Review

2.1. Commodity–Equity Connectedness and Market Spillovers

The relationship between commodity and equity markets has become a central topic in financial economics, as the traditional boundaries between asset classes have progressively eroded. The long-standing view that commodities offer stable diversification benefits has been increasingly challenged by evidence of stronger cross-market linkages, driven by the financialization of commodity markets and the expanding role of institutional investors.

A major contribution to the analysis of these interconnections is provided by Diebold and Yilmaz [1], whose connectedness framework demonstrates that spillover effects evolve dynamically and intensify during periods of heightened uncertainty. Their findings indicate that even markets previously considered segmented can exhibit persistent contagion effects under stress. Consistent with this evidence, Wen and Wang [25] show that equity–commodity dependencies strengthen when market sentiment deteriorates, suggesting that systemic risk tends to migrate across asset classes rather than dissipate.

Energy markets, particularly crude oil, play a pivotal role in this transmission mechanism. Oil has increasingly behaved as a macro-financial asset, responding not only to physical supply–demand conditions but also to geopolitical tensions and speculative pressures. Biswas et al. [26] find that following the Russia–Ukraine conflict, crude oil shifted from being a volatility receiver to a major transmitter, spreading risk across global equity markets. Agricultural commodities also contribute to cross-asset spillovers. Guhathakurta et al. [27] document that volatility transmission between oil, metals, and agricultural products intensifies around structural breaks, while Malhotra et al. [28] show that wheat price volatility spilled over into the equity markets of major importing countries during the early phase of the Russia–Ukraine war.

A consistent conclusion across this literature is that spillover intensity is strongly regime dependent. Tu and Leatham [29] demonstrate that commodity–equity connectedness increases sharply during crisis episodes, undermining the effectiveness of traditional diversification strategies. As correlations tend to rise precisely when investors expect them to remain low, many spillover studies remain largely descriptive and offer limited guidance for hedging or portfolio construction. This limitation highlights the need for dynamic, time-varying approaches that translate observed connectedness patterns into actionable risk-management tools.

2.2. Crisis Regimes, Market Instability and Cross-Asset Risk Transmission

Periods of systemic instability rarely remain confined to specific regions or asset classes; they trigger chain reactions that transmit volatility across financial and real sectors. This became particularly evident during the COVID-19 pandemic, which transformed a relatively segmented global system into one characterized by highly synchronized turbulence. Markets that once moved independently began to co-move sharply, weakening diversification when it was most needed. Adekoya and Oliyide [30] show that volatility transmission across oil, natural gas, gold and major equity indices increased markedly during the pandemic, effectively turning distinct risk categories into components of a unified global shock.

Beyond amplifying volatility, the COVID-19 crisis reinforced persistent interdependence across markets. Coskun et al. [31] document sustained return dependence and reduced mean reversion, indicating that shocks were repeatedly propagated rather than absorbed. Under conditions of liquidity stress, commodities that traditionally acted as hedging instruments often became channels of financial amplification, shaping the environment in which subsequent geopolitical shocks unfolded.

The Russia–Ukraine conflict introduced a different form of systemic disruption, driven primarily by real-sector constraints. Unlike the pandemic, which affected markets mainly through uncertainty, the war directly disrupted supply chains and energy flows. Lin et al. [7] report sharp volatility in wheat prices stemming from supply interruptions in the Black Sea region, with spillover effects on food-importing economies. Rubaszek and Szafranek [32] identify structural breaks in U.S. natural gas dynamics linked to Europe’s energy crisis, illustrating how geopolitical shocks can permanently alter market structures rather than merely intensify short-term volatility.

Geopolitical rivalry has added a more persistent layer of uncertainty to global markets. Chen et al. [33] show that economic policy uncertainty and trade frictions have become systematic drivers of commodity price volatility. Khan [34] finds that geopolitical tensions amplify volatility spillovers from U.S. markets to Asian equities, suggesting that contagion has evolved into a structural feature of the global financial system.

Crisis regimes also generate macro-financial feedback loops that further reinforce instability. Huang et al. (2024) [35] demonstrate that global stock market co-movements intensified during the COVID-19 period, producing denser and more interconnected network structures. Díaz et al. [14] highlight that post-pandemic inflation was fuelled not only by demand recovery but also by commodity price volatility linked to supply disruptions and energy shortages. Together, these findings illustrate how modern crises combine real-economic imbalances with financial dynamics to amplify systemic risk.

The literature shows that crisis periods intensify cross-asset dependencies and accelerate volatility transmission, undermining conventional diversification strategies. Most studies, however, remain largely descriptive, documenting contagion without fully examining its implications for hedging performance. Whether commodities retain defensive value under systemic stress and how hedge ratios should adjust across regimes remain insufficiently explored. This gap directly motivates the empirical analysis developed in the next section.

2.3. Hedging Role of Commodities Under Market Turbulence

Commodities have traditionally been regarded as reliable hedging instruments due to their intrinsic value and perceived insulation from financial market shocks. Recent evidence, however, suggests that their hedging performance is far from stable and depends critically on the nature of shocks, prevailing market conditions, and the investment horizon. During periods of systemic stress, cross-market correlations tend to rise, reinforcing tail-risk spillovers and eroding diversification benefits [36].

Gold remains the most extensively studied defensive asset, yet its hedging role is highly conditional. Although historically viewed as a safe haven, empirical findings are mixed. Reboredo [37] shows that gold provides only limited short-term protection and loses much of its effectiveness during prolonged or inflation-driven crises. Belguith et al. [38] further demonstrate that gold’s hedging performance varies across regions and geopolitical regimes, indicating that its defensive properties are context dependent rather than universal.

Energy commodities, particularly crude oil, exhibit even more complex behaviour. Once driven primarily by physical supply–demand fundamentals, oil prices are now strongly influenced by financialization and investor sentiment. Said and Ouerfelli (2024) [39] document sharp increases in dynamic correlations between oil and equity markets during the COVID-19 crisis, signalling contagion and reduced diversification potential. Similar regime dependence is reported by Moutinho et al. [40], who find that oil offered limited hedging ability during the subprime crisis and largely lost this role during the European debt crisis. As a result, oil simultaneously functions as a real and financial asset, with its hedging capacity determined by whether shocks originate from demand fluctuations, supply disruptions, or geopolitical tensions.

Agricultural commodities add further complexity to hedging analysis. Their prices are highly sensitive to climate variability [41], supply chain disruptions [42], and geopolitical tensions [43], generating risk dynamics that differ markedly from those observed in financial markets. Babar et al. [20] show that the Russia–Ukraine conflict intensified volatility spillovers from grain markets to equity indices, thereby limiting their usefulness as hedging instruments during turbulent periods. This evidence suggests that agricultural commodities are predominantly driven by real-economic constraints, which can undermine their defensive role under systemic stress.

The literature converges on the conclusion that static hedging strategies are increasingly insufficient. Rising correlations, volatility clustering and nonlinear spillovers challenge the assumption of stable cross-asset relationships. Linear models cannot capture these dynamics during crises, underscoring the need for adaptive, time-varying and multifractal approaches that better reflect the evolving structure of financial linkages.

2.4. From Linear Models to Multifractal and Multivariate Volatility Frameworks

Empirical studies of cross-market dependence and hedging behaviour have traditionally relied on linear frameworks centred on second-moment dynamics. Early multivariate GARCH (MGARCH) models, such as the Constant Conditional Correlation (CCC) and Dynamic Conditional Correlation (DCC) specifications, laid the foundation for modelling time-varying co-movements. However, these approaches typically assume symmetric shock responses and relatively stable dependence structures. Such assumptions are frequently violated during periods of market stress, when return series exhibit pronounced nonlinearity, asymmetry, and structural breaks consistent with volatility clustering and contagion effects [44].

To address asymmetric volatility responses, the GARCH family was extended to include models such as GJR-GARCH [45] and BEKK-GARCH [46], which improved the representation of volatility spillovers. Building on these developments, the ADCC-GARCH model explicitly incorporates asymmetry into conditional correlations and has therefore been widely used to estimate time-varying hedge ratios under turbulent market conditions [47,48]. Nevertheless, both ADCC and BEKK models face practical challenges, including over-parameterization and numerical instability in higher-dimensional settings [48].

The GO-GARCH model offers an alternative by decomposing volatility into orthogonal components, which enables latent sources of risk to be identified more clearly. This structure leads to more stable covariance estimates and has been shown to generate more consistent hedge ratios and more reliable portfolio risk measures than traditional MGARCH specifications [40].

A key limitation shared by these linear volatility frameworks is their exclusive focus on second-order dependence. As a result, they are unable to capture nonlinear and scale-dependent interactions that increasingly characterize commodity–equity linkages, particularly during financial and geopolitical crises. This limitation has motivated growing interest in multifractal approaches.

MFCCA, introduced by Kwapień et al. [49], identifies long-range and scale-dependent cross-correlations and distinguishes between short- and long-memory components. It reveals hidden dependence patterns associated with herding, overreaction and nonlinear contagion, without imposing Gaussian assumptions or fixed time horizons. Despite these advantages, MFCCA has rarely been incorporated into hedging research, and studies that integrate it with MGARCH models remain limited.

This gap is particularly important because linear MGARCH models may underestimate systemic risk when multifractal dependence is ignored. Combining MFCCA with ADCC-GARCH and GO-GARCH allows time variation, asymmetry, and multifractality to be analysed jointly. This integrated perspective provides a more realistic framework for evaluating hedging performance under crisis conditions and directly motivates the empirical strategy adopted in the following section.

2.5. Research Gap

The existing literature provides substantial evidence of rising connectedness and volatility transmission between commodity and equity markets, particularly during periods of financial stress and geopolitical instability. However, much of this research remains largely descriptive, focusing on measuring spillovers and co-movement patterns rather than examining their implications for portfolio construction and risk management. Although cross-asset dependencies are now well documented, their consequences for hedging performance across different crisis regimes remain insufficiently explored. This disconnects between measurement and application limits the practical relevance of connectedness research for investors and policymakers.

MGARCH models are widely employed to estimate dynamic hedge ratios, yet most applications rely on linear dependence structures and second-moment relationships. Models such as DCC and ADCC capture time-varying correlations but overlook the nonlinear and scale-dependent features that increasingly characterize commodity–equity linkages during crisis periods. In contrast, multifractal approaches such as MFCCA reveal heterogeneous dependence across time horizons but remain largely detached from hedging applications. The absence of an integrated framework that combines multifractal dependence analysis with MGARCH-based hedging models represents a significant methodological gap.

A further limitation concerns the lack of comparative evidence on model performance under systemic stress. While ADCC-GARCH is commonly used to model asymmetric dependence, its effectiveness relative to more flexible alternatives such as GO-GARCH remains underexplored. GO-GARCH’s orthogonal decomposition allows latent volatility factors to be identified and may provide a richer representation of systemic and idiosyncratic risk. Few studies, however, assess whether these structural differences translate into superior hedging outcomes during crisis conditions. Consequently, an important question remains unresolved: does model choice materially influence hedging efficiency when financial systems are unstable?

Addressing these gaps is essential for advancing both the methodological and practical dimensions of hedging research. The present study contributes by integrating MFCCA-based dependence analysis with ADCC-GARCH and GO-GARCH models to jointly evaluate nonlinear dependence and dynamic hedging performance. This unified framework captures asymmetry, time variation, and multifractality, offering a more comprehensive understanding of how commodities function as hedging assets across different crisis regimes.

3. Methodology

3.1. Empirical Design and Research Framework

The empirical framework combines multifractal dependence analysis with multivariate volatility modelling to examine scale-dependent co-movements and dynamic hedging relationships between the U.S. equity market and major commodity assets. The analysis focuses on pairwise interactions between the S&P 500 index $\left(r_t^{SP}\right)$, and four key commodities: WTI crude oil $\left(r_t^{WTI}\right)$, gold $\left(r_t^G\right)$, wheat $\left(r_t^W\right)$, and natural gas $\left(r_t^{NG}\right)$.

A pairwise modelling strategy is adopted to isolate the specific hedging role of each commodity relative to equities and to avoid estimation bias and parameter instability that may arise in high-dimensional covariance systems. All series are expressed as logarithmic returns and demeaned prior to analysis.

To capture nonlinear and scale-dependent cross-market interactions, MFCCA is first applied to the return pairs $(x_t,y_t)$. The cumulative profiles are defined as:

$$X\left(i\right)=\sum _{k=1}^i\left(x_k-\overline{x}\right), Y\left(i\right)=\sum _{k=1}^i\left(y_k-\overline{y}\right),$$

(1)

where i = 1,…,N, and $\overline{x},\overline{y}$ denote sample means.

Cross-covariances are aggregated across time scales through the $q$-order fluctuation function $F_{xy}(s,q)$, which follows the scaling relation:

$$F_{xy}\left(s,q\right)s^{{\lambda }_{\mathrm{xy}}\left(q\right)}.$$

(2)

The generalized scaling exponent ${\lambda }_{xy}\left(q\right)$ indicates persistent $(>0.5)$, anti-persistent $(<0.5)$ or uncorrelated $(=0.5)$ behaviour, and its variation across $q$ captures the degree of multifractality. The dependence of ${\lambda }_{xy}\left(q\right)$ on q captures the degree of multifractality. Complementary dependence is measured using the q-dependent detrended cross-correlation coefficient ${\rho }_q\left(s\right)$ [50]:

$${\rho }_q\left(s\right) = {\displaystyle \frac{F_{\mathrm{xy}}\left(s,q\right)}{\sqrt{F_x\left(s,q\right)F_y\left(s,q\right)}}}, -1 \le {\rho }_q\left(s\right) \le 1.$$

(3)

Values of ${\rho }_q\left(s\right)$ close to 1 indicate strong positive co-movement, values near 0 indicate weak dependence.

Following Kwapień et al. [49] the moment order $q$ ranges from −4 to 4, the scale s varies between 10 and 500 observations and detrending is performed using a polynomial of order (m = 2). These settings ensure stable estimation of long-range and scale-dependent cross-market dependencies and allow consistent comparison across assets.

3.2. Multifractal Cross-Correlation Methodology

MFCCA enables the identification of nonlinear dependence structures that evolve across multiple time scales and captures long-range cross-correlations that classical linear measures fail to detect. Prior to the analysis, the return series are demeaned and integrated into cumulative profiles to remove low-frequency components.

The profiles X(i) and Y(i) are divided into non-overlapping segments of equal length $s$. Within each segment, local polynomial trends of order m = 2 are removed to isolate intrinsic fluctuations. The detrended cross-covariance for segment v is defined as:

$$F_{xy}^2\left(s,v\right) = {\displaystyle \frac{1}{\mathsf{s}}}\sum _{i=1}^s\left[X_v\left(i\right)-{\stackrel{\mathrm{~}}{X}}_v(i)\right]\left[Y_v\left(i\right)-{\stackrel{\mathrm{~}}{Y}}_v(i)\right],$$

(4)

where ${\stackrel{\mathrm{~}}{X}}_v\left(i\right)$ and ${\stackrel{\mathrm{~}}{Y}}_v\left(i\right)$ are the local polynomial fits.

Aggregating across all 2Ns segments yields the q-order fluctuation function:

$$F_{xy}\left(s,q\right) = \{\begin{array}{c}{\left[{\displaystyle \frac{1}{2N_s}}{\sum }_{v=1}^{2{\mathsf{N}}_{\mathsf{s}}}\mathrm{sign}\left(F_{xy}^2\left(s,v\right)\right){\left|F_{xy}^2\left(s,v\right)\right|}^{q/2}\right]}^{1/q}, q\ne 0\\ \exp \left[{\displaystyle \frac{1}{2N_s}}{\sum }_{v=1}^{2{\mathsf{N}}_{\mathsf{s}}}\ln \left|F_{xy}^2\left(s,v\right)\right|\right], q = 0\end{array}$$

(5)

To ensure that the detected multifractality reflects genuine nonlinear dependence rather than artifacts of non-stationarity, MFCCA results are benchmarked against phase-randomized surrogate series. This procedure preserves the linear correlation structure while eliminating nonlinear effects. Significant deviations between empirical and surrogate spectra indicate authentic multifractal cross-correlations. The MFCCA results provide the foundation for the subsequent GARCH-based modelling and the estimation of dynamic hedge ratios.

3.3. Dynamic Hedge Ratio Estimation Using ADCC and GO-GARCH

To translate multifractal dependencies into practical hedging metrics, dynamic hedge ratios are estimated using two complementary multivariate volatility models: the ADCC-GARCH and the GO-GARCH. Let $r_t^x$ be the return of commodity, and $r_t^y$ the return on the S&P 500 index. Their joint dynamics follow:

$$‘r_t={\mu }_t+{\epsilon }_t, {\epsilon }_t=H_t^{1/2}{z}_t,$$

(6)

where $H_t$ is the matrix of conditional covariance, and z_t is a vector of standardized innovations.

In the ADCC-GARCH framework [22], the conditional covariance matrix is decomposed as:

$$H_t=D_t{R}_t{D}_t$$

(7)

where $D_t=\mathrm{diag}({\sigma }_{x,t},{\sigma }_{y,t})$ contains conditional standard deviations obtained from univariate GARCH(1,1) models, and $R_t$ is the dynamic correlation matrix. The correlation dynamics follow:

$$Q_t=\left(1-\alpha -\beta \right)\stackrel{\mathrm{-}}{Q}+\alpha \left({\epsilon }_{t-1}{\epsilon }_{t-1}^{’}\right)+\beta Q_{t-1}+\gamma \left(n_{t-1}{n}_{t-1}^{’}\right),$$

(8)

where $n_t$ isolates negative innovations and γ captures asymmetric effects.

To overcome the limitations of correlation-based GARCH models, approach [51] applies an orthogonal transformation:

$$H_t=A{G}_t{A}^{’}$$

(9)

where $G_t$ is a diagonal matrix of univariate GARCH processes and A is an orthogonal matrix. This decomposition identifies latent systemic and idiosyncratic volatility components, yielding more stable hedge ratios in high-volatility regimes.

The time-varying hedge ratio is computed as:

$${\beta }_t={\displaystyle \frac{{\sigma }_{xy,t}}{{\sigma }_{yy,t}}},$$

(10)

where ${\sigma }_{xy,t}$ is the conditional covariance between the stock index and the commodity, and ${\sigma }_{yy,t}$ is the conditional variance of the stock index. This formulation allows the hedging position to adjust dynamically in response to evolving market conditions.

Using both ADCC-GARCH and GO-GARCH enables a robust comparison of hedging strategies under different dependence structures. While ADCC captures asymmetric shock effects, GO-GARCH models multidimensional spillover behaviour through orthogonal risk factors, offering a richer representation of volatility transmission.

3.4. Optimal Portfolio Weights and Hedging Effectiveness

Following Kroner and Ng [46], weights of optimal portfolio are determined by minimizing conditional portfolio variance at each time $t$, without imposing any restriction on expected returns. For a portfolio consisting of assets i and j, the optimal weight allocated to asset i is given by:

$${\omega }_{ij,t} ={\displaystyle \frac{h_{jj,t}-h_{ij, t}}{h_{ii,t}-{\mathit{2}h}_{ij,t}+h_{jj,t}}}$$

(11)

where $h_{ii,t}$ and $h_{jj,t}$ denote conditional variances, and $h_{ij,t}$ denotes the conditional covariance between the two assets. Portfolio weights are bounded within [0, 1] to ensure feasible allocations. The complementary position 1 − ω_ij;t is held in asset j.

Hedging effectiveness (HE) is evaluated using the variance reduction measure [51]:

$$HE ={\displaystyle \frac{{Var}_{unhedged}-{Var}_{hedged}}{{Var}_{unhedged}}},$$

(12)

where ${Var}_{unhedged}$ is a variance of position in S&P 500 alone, and ${Var}_{hedged}$ is the variance of the hedged portfolio using the estimated hedge ratios. Higher values of HE indicate greater risk reduction and therefore correspond to more effective hedging strategies. Comparing HE across models and commodity–equity pairs provide insights into which commodities offer the most reliable and stable protection against equity market risk under different crisis regimes.

To provide a comprehensive evaluation, hedging effectiveness is compared across three approaches: (i) the static OLS hedge ratio, (ii) ADCC-GARCH, and (iii) GO-GARCH. Performance is examined under rebalancing frequencies of 5, 20, and 60 days, allowing assessment of sensitivity to trading horizons and addressing the reviewer’s requirement for multiple rebalancing intervals.

All data preprocessing, MFCCA estimation, and figure construction were performed in RStudio version 4.4.0. (PBC, Boston, MA, USA), while the ADCC-GARCH and GO-GARCH models were estimated using EViews 12 (IHS Markit, Irvine, CA, USA).

4. Data and Variables

The empirical analysis uses daily closing prices for the S&P 500 index and four major commodities: WTI crude oil, gold, wheat and natural gas, representing the energy, precious metals and agricultural segments of global commodity markets. Data were retrieved from Refinitiv Datastream [52] for the period 2 February 2018 to 20 February 2023, a timeframe that encompasses several episodes of global instability. All series are aligned to a common trading calendar; missing observations arising from non-overlapping market holidays are handled using standard forward-adjustment procedures to ensure consistency across assets.

To isolate major systemic disruptions, the sample is divided into three subperiods: a pre-pandemic phase, the COVID-19 period and the Russia–Ukraine war. The breakpoints follow internationally recognised dates, namely the WHO declaration of COVID-19 as a global pandemic (11 March 2020) and the onset of the Russia–Ukraine conflict (24 February 2022). This partitioning enables a comparative analysis of market behaviour under distinct stress regimes and facilitates an assessment of how cross-market dependence and hedging performance evolve over time.

All price series are converted into continuously compounded returns, computed as:

$$r_t=ln\left(P_t\right)-ln\left(P_{t-1}\right)$$

(13)

where $P_t$ is the closing price at time $t$. The logarithmic transformation standardises return distributions, mitigates scale effects, and aligns the data with the assumptions of multifractal and GARCH-based volatility models.

Table 1. Descriptive statistics and preliminary tests for return series.

Statistic	WTI	S&P 500	Gold	Wheat	Gas
Mean	0.025 (0.683)	0.046 ** (0.046)	0.047 *** (0.010)	0.005 (0.900)	0.010 (0.898)
Variance	9.483 ***	1.296 ***	0.804 ***	3.836 ***	16.352 ***
Skewness	−2.927 *** (0.000)	−0.690 *** (0.000)	−0.225 *** (0.000)	0.549 *** (0.000)	0.032 (0.516)
Kurtosis	76.403 *** (0.000)	17.253 *** (0.000)	2.813 *** (0.000)	5.991 *** (0.000)	3.400 *** (0.000)
Jarque–Bera	604,777.125 *** (0.000)	30,856.609 *** (0.000)	836.027 *** (0.000)	3821.045 *** (0.000)	1190.829 *** (0.000)
ERS (ADF)	−10.035 *** (0.000)	−22.768 *** (0.000)	−7.504 *** (0.000)	−23.401 *** (0.000)	−14.596 *** (0.000)
Q(20)	69.130 *** (0.000)	209.921 *** (0.000)	7.937 (0.732)	20.354 ** (0.015)	29.186 *** (0.000)
Q2(20)	465.686 *** (0.000)	2382.508 *** (0.000)	262.976 *** (0.000)	911.592 *** (0.000)	268.981 *** (0.000)

Notes: p-values are reported in parentheses. *** and ** denote statistical significance at the 1% and 5% levels, respectively.

reports descriptive statistics for all return series. Mean returns for WTI crude oil, wheat, and natural gas are statistically indistinguishable from zero, indicating the absence of persistent drift. In contrast, the S&P 500 index and gold exhibit small but positive average returns, consistent with long-term equity appreciation and gold’s defensive role. The return distributions deviate markedly from normality: WTI crude oil and natural gas display high variance and pronounced leptokurtosis, reflecting frequent extreme price movements. Skewness measures indicate asymmetry, with WTI crude oil and the S&P 500 exhibiting left-tailed behaviour. The Jarque–Bera test rejects the null hypothesis of normality for all series at the 1% significance level.

Volatility clustering is evident from the Ljung–Box Q² statistics, which reveal strong autocorrelation in squared returns, a common feature of assets affected by speculative dynamics, geopolitical shocks and structural breaks. ERS (ADF) unit-root tests confirm that all return series are stationary, validating their use in multifractal and GARCH-based modelling.

Figure 1 presents the daily return fluctuations of the S&P 500 index and the four commodity markets over the full sample period. Across all series, returns exhibit a clear pattern of volatility clustering, characterized by relatively tranquil periods interrupted by sudden bursts of large shocks. This behaviour reflects time-varying market uncertainty and represents a defining feature of financial return dynamics, supporting the use of GARCH-type volatility models in the subsequent analysis.

Among the commodities, WTI crude oil displays the most pronounced fluctuations, with extreme negative and positive return shocks during the COVID-19 outbreak and the early phase of the Russia–Ukraine conflict. These movements highlight oil’s sensitivity to global demand conditions, supply imbalances, and geopolitical disruptions. The S&P 500 index exhibits sharp volatility spikes during the same periods, indicating a high degree of synchronization between equity and energy markets under systemic stress.

Wheat and natural gas returns also show elevated volatility during the Russia–Ukraine conflict, reflecting the region’s importance for global grain exports and European energy supply chains. Their behaviour underscores the vulnerability of agricultural and energy prices to geopolitical risk and logistical constraints. In contrast, gold exhibits comparatively lower but more persistent volatility, with noticeable responses around major crisis episodes, consistent with its role as a crisis-sensitive asset and a partial refuge for investors during periods of heightened uncertainty.

Figure 2 displays the daily return fluctuations of the S&P 500 index and the four commodity markets over the full sample period. Across all series, returns exhibit a clear pattern of volatility clustering, with tranquil periods interrupted by sudden bursts of large shocks. These dynamics reflect time-varying market uncertainty and represent a defining feature of financial return behaviour. This empirical regularity supports the use of GARCH-type volatility models in the subsequent analysis.

Among the commodities, WTI crude oil displays the most pronounced fluctuations, with extreme negative and positive return shocks during the COVID-19 outbreak and in the early phase of the Russia–Ukraine war. These movements highlight oil’s sensitivity to global demand and supply imbalances as well as to geopolitical disruptions. The S&P 500 shows sharp volatility spikes during the same periods, indicating a high degree of synchronization between energy and equity markets under systemic stress.

Wheat and natural gas returns exhibit elevated volatility during the Russia–Ukraine conflict, which reflects the region’s importance in global grain exports and European energy supply chains. Their behaviour underscores the vulnerability of agricultural and energy prices to geopolitical risk and logistical constraints. By contrast, gold shows lower but more persistent volatility, with notable responses around major crisis events, confirming its role as a crisis-sensitive asset and a refuge for investors seeking protection from rising uncertainty.

5. Empirical Results and Discussions

5.1. Cross-Correlation Coefficient

The multifractal cross-correlations between the S&P 500 index and the four major commodities are assessed using the q-dependent detrended cross-correlation coefficient ${\rho }_q(s)$, computed across multiple time scales and fluctuation orders. Figure 3 and Figure 4 illustrate how these cross-market linkages evolve across regimes. This multiscale framework captures nonlinear and scale-dependent interactions that cannot be identified using conventional linear correlation measures.

Analysing ${\rho }_q(s)$ across different values of q allows a distinction between dependence driven by small fluctuations and that associated with extreme market movements. This distinction is particularly relevant for hedging, as strong dependence during large shocks typically undermines diversification benefits and increases hedge-ratio instability. Values of ${\rho }_q(s)$ close to one indicate strong positive co-movement, values near zero suggest weak dependence, and negative values imply inverse dynamics consistent with partial decoupling.

The WTI–S&P 500 pair exhibit consistently positive cross-correlations across all regimes and time scales, with the strongest dependence observed during the COVID-19 period. Elevated ${\rho }_q(s)$ values persist during the Russia–Ukraine crisis, indicating sustained synchronization between oil and equity markets under systemic stress. This pattern suggests a diminished diversification role for crude oil during crisis episodes, in line with evidence reported by Shah et al. [53].

Cross-correlations involving gold display a more nuanced and regime-dependent structure. Prior to the pandemic, both S&P 500–gold and WTI–gold pairs exhibit weak dependence across most scales, consistent with gold’s traditional defensive role. During the COVID-19 period, ${\rho }_q(s)$ increases across scales and fluctuation orders, indicating stronger co-movement in response to global uncertainty. In the Russia–Ukraine period, dependence weakens at longer time scales and for larger fluctuations, suggesting a partial re-emergence of gold’s hedging properties rather than a uniform inverse relationship.

Wheat-based cross-correlations also exhibit strong regime sensitivity. Dependence with the S&P 500 and WTI is negligible before 2020 but increases markedly during the pandemic, likely reflecting higher production costs and supply chain disruptions. During the Russia–Ukraine conflict, the cross-correlation structure becomes more heterogeneous across $q$ and scales, indicating unstable and multifractal dependence patterns associated with export disruptions and heightened food-security concerns.

5.2. Multifractal Cross-Correlation Results

To examine the multifractal cross-correlations between the S&P 500 index and the four commodities, as well as between WTI crude oil and the other assets, Figure 5 and Figure 6 present the fluctuation functions $F_q(s)$ across time scales for different moment orders $q$. The evolution of these functions provides insight into the presence of long-range dependence and the degree of multifractality across market regimes. This multiscale perspective allows nonlinear and scale-dependent interactions to be identified, which are not captured by conventional linear correlation measures.

For the WTI–S&P 500 pair, the pre-COVID-19 period is characterized by relatively moderate growth of $F_q(s)$ across scales and limited separation between curves corresponding to different q values, indicating weak persistence and near-random cross-market interactions. During the COVID-19 period, the fluctuation functions become steeper and more dispersed across q, reflecting stronger persistence, heightened multifractality, and intensified interdependence between oil and equity markets. In the Russia–Ukraine crisis period, the growth of $F_q(s)$ remains pronounced, suggesting sustained long-range dependence, although the scaling structure becomes more regular across time scales.

A comparable regime-dependent pattern is observed for the WTI–gold and S&P 500–gold pairs. Prior to the pandemic, the fluctuation functions exhibit relatively weak scale dependence, consistent with gold’s traditional defensive role. During the COVID-19 crisis, $F_q(s)$ increases more rapidly and displays greater separation across q, indicating a temporary strengthening of cross-market dependence under global financial stress. In the subsequent geopolitical crisis, the multifractal structure becomes less dispersed, suggesting a partial re-emergence of gold’s hedging properties as market dynamics shift from predominantly financial to geopolitical drivers. This interpretation aligns with Lee and Choi [54], who document regime-dependent information transmission between major financial assets.

For WTI–wheat and S&P 500–wheat pairs, cross-correlations remain positive throughout the sample but intensify markedly during crisis periods. The COVID-19 phase is associated with steeper fluctuation functions and increased multifractality, while the Russia–Ukraine conflict produces heterogeneous scaling behaviour across different q values. This reflects the sensitivity of agricultural markets to supply disruptions and food-security concerns, consistent with the findings of Gaio and Capitani [55], who report strengthened multifractal cross-correlations for agricultural commodities following the onset of the conflict.

The WTI–natural gas and S&P 500–natural gas relationships also exhibit regime-dependent multifractal behaviour. Weak and near-random dependence is observed prior to COVID-19, whereas the pandemic period is characterized by stronger persistence and increased dispersion across q. During the Russia–Ukraine crisis, the fluctuation functions remain persistent but display a more ordered structure, indicating sustained dependence with reduced multifractal variability. Similar regime-specific effects of geopolitical risk on energy and equity markets are reported by Tarigan et al. [56].

The results demonstrate that multifractality and long-range dependence are structural features of cross-market interactions rather than temporary responses to isolated shocks. The regime-dependent evolution of $F_q(s)$ underscores that dependence structures adjust with the nature of systemic events, reinforcing the importance of scale-sensitive approaches for evaluating hedging effectiveness and portfolio risk under crisis conditions.

To verify that the detected multifractality does not arise from non-stationarity or purely linear correlations, the surrogate-series procedure of Schreiber and Schmitz (2000) [57] is applied.

Table 2. Multifractality validation using surrogate data.

Pair	Δhreal	Δhsurr,mean	Δhsurr,std	p-Value
WTI-S&P 500	0.055	0.265	0.142	0.92
WTI-Gold	0.454	0.224	0.148	0.04
WTI-Wheat	0.654	0.254	0.118	0.01
WTI-Gas	0.406	0.263	0.149	0.13
S&P 500-Gold	0.367	0.247	0.138	0.23
S&P 500-Wheat	0.367	0.294	0.146	0.31
S&P 500-Gas	0.205	0.280	0.149	0.66
Gold-Wheat	0.349	0.286	0.119	0.29
Gold-Gas	0.382	0.263	0.147	0.19
Wheat-Gas	0.095	0.259	0.145	0.86

Notes: Δh_surr,mean represents the average multifractal spectrum width obtained from 100 phase-randomized surrogate series, while Δh_surr,std denotes its standard deviation. These surrogate statistics preserve the linear properties of the data but remove nonlinear dependencies. The p-value corresponds to the one-tailed Monte Carlo test assessing whether Δh_real exceeds the surrogate benchmark, thus confirming genuine multifractality.

reports the results of the Monte Carlo test comparing the width of the multifractal spectrum obtained from the original series, $\Delta h_{\mathrm{real}}$, with that from 100 phase-randomized surrogate series. Significant cases, where $\Delta h_{real}>\Delta h_{\mathrm{surr}}$ with low p-values, indicate the presence of genuine nonlinear cross-correlations.

The results show that only the WTI–gold and WTI–wheat pairs exhibit statistically significant multifractality at the 5% level. For these pairs, the multifractal structure of cross-correlations cannot be reproduced by linear surrogate processes, confirming the presence of nonlinear interdependencies. For all other pairs, including WTI–S&P 500 and WTI–natural gas, the multifractality observed in the original data does not significantly exceed that of the surrogate series, suggesting that cross-market multifractality is limited and pair-specific. This selective nature of multifractality is consistent with previous findings in the literature [58].

Figure 7 and Figure 8 illustrate the generalized Hurst exponent $h_{xy}(q)$ as a function of the moment order q across the three market regimes. The analysis focuses on $h_{xy}(2)$, which captures the long-term persistence of cross-correlations. Values above 0.5 indicate persistent dependence, values below 0.5 imply anti-persistent behavior, and values close to 0.5 correspond to uncorrelated dynamics.

For pairs involving the S&P 500, cross-correlations with WTI, gold, and wheat exhibit clear regime dependence. Persistence is observed before the pandemic and during COVID-19 for the S&P 500–WTI and S&P 500–gold pairs, while the Russia–Ukraine war is associated with a weakening of persistence, with $h_{xy}(2)$ approaching or falling slightly below 0.5. In contrast, the S&P 500–natural gas pair displays a non-monotonic pattern: persistence weakens during COVID-19 but strengthens again during the war period, indicating a regime-dependent adjustment rather than a persistent anti-correlated structure.

For WTI-based pairs, WTI–natural gas exhibits persistent cross-correlations across all regimes, with the strongest persistence observed during the COVID-19 period. The WTI–gold relationship is relatively weak prior to the pandemic, intensifies during COVID-19, and shifts toward anti-persistent behaviour during the Russia–Ukraine conflict, suggesting a partial decoupling under geopolitical stress. The WTI–wheat pair shows mild dependence before COVID-19, followed by a pronounced intensification during the pandemic and its strongest persistence during the war period, reflecting the sensitivity of agricultural markets to supply disruptions.

The decline of $h_{xy}(q)$ with increasing q across all pairs confirms the presence of multifractal dependence, indicating that cross-correlations vary across fluctuation magnitudes and time scales. This behavior highlights the complexity of cross-market interactions and supports the integration of multifractal analysis with dynamic hedging models. These findings are consistent with evidence reported by Mirzaee Ghazani et al. [59], Acikgoz [60], and Martins [61], as well as with recent results by Shao et al. [62], who document strong multifractality in agricultural and energy markets under heightened geopolitical risk.

5.3. Hedging Effectiveness Analysis

In this section, hedging performance is evaluated out of sample using a rolling window forecasting approach. At each time t, one-step-ahead forecasts of conditional variances and covariances are generated and used to compute optimal hedge ratios. These ratios form the basis of dynamic hedging strategies implemented over the sample period. For robustness, hedge ratios are estimated under two multivariate GARCH frameworks, namely ADCC and GO-GARCH, using rebalancing frequencies of 20 and 60 days. The static OLS hedge ratio is included as a benchmark, in line with the literature [46].

Table 3. HE across OLS, ADCC, and GO-GARCH models under different rebalancing frequencies.

Pair	Method	OLS	ADCC-5	ADCC-20	ADCC-60	GO-5	GO-20	GO-60
WTI-SP500	HE	0.049	0.052	0.058	0.051	0.035	0.052	0.042
WTI-Gold	HE	0.008	0.022	0.012	0.013	0.018	0.007	0.010
WTI-Wheat	HE	0.013	0.016	0.014	0.014	0.014	0.013	0.014
WTI-Gas	HE	0.008	0.020	0.025	0.026	0.016	0.030	0.030
SP500-Gold	HE	0.003	0.003	0.008	−0.013	−0.001	−0.016	−0.062
SP500-Wheat	HE	0.001	0.000	−0.014	−0.026	−0.094	−0.195	−0.362
SP500-Gas	HE	0.010	0.037	0.023	0.014	0.060	−0.004	−0.191
Gold-Gas	HE	0.000	0.000	0.001	0.000	0.000	0.000	0.000
Wheat-Gas	HE	0.001	0.004	0.004	0.001	0.003	0.003	0.002

Notes: Columns labelled 5, 20, and 60 indicate rebalancing frequencies in days. Positive values denote effective hedging; negative values indicate hedging underperformance.

and

Table 4. Summary statistics of optimal portfolio weights, optimal hedge ratios, and hedging effectiveness across sub-periods based on the ADCC model.

ADCC (Refit = 20)
	Period	Optimal Hedge Ratio			Optimal Weight			HE (%)
WTI	1	Mean	Min	Max	Mean	Min	Max
	2	0.071	0.010	0.244	0.893	0.321	1.000	4.277
	3	0.113	0.064	0.603	0.883	0.319	1.000	7.526
Gold	1	0.080	0.086	0.711	0.867	0.132	1.000	4.248
	2	0.098	0.483	0.932	0.471	0.009	0.843	2.505
	3	0.037	0.981	0.631	0.479	0.008	0.980	2.826
Wheat	1	0.162	0.675	1.173	0.483	0.000	0.991	5.229
	2	0.047	0.014	0.163	0.792	0.216	0.988	0.744
	3	0.061	0.017	0.599	0.739	0.016	0.973	0.730
Natural Gas	1	0.032	0.005	0.203	0.795	0.132	0.990	0.381
	2	0.004	0.018	0.013	0.886	0.561	0.994	0.040
	3	0.019	0.010	0.182	0.903	0.144	0.999	0.388
ADCC (refit = 60)
WTI	1	0.069	0.009	0.209	0.892	0.321	1.000	3.926
	2	0.111	0.068	0.603	0.884	0.319	1.000	7.550
	3	0.080	0.086	0.711	0.867	0.132	1.000	4.273
Gold	1	0.091	0.447	0.932	0.470	0.009	0.838	2.405
	2	0.034	0.981	0.631	0.477	0.009	0.980	2.687
	3	0.161	0.678	1.173	0.483	0.000	0.991	5.158
Wheat	1	0.047	0.014	0.163	0.797	0.216	0.988	0.751
	2	0.058	0.017	0.400	0.742	0.024	0.973	0.728
	3	0.033	0.005	0.203	0.796	0.132	0.991	0.393
Natural Gas	1	0.005	0.006	0.014	0.885	0.561	0.994	0.031
	2	0.017	0.001	0.079	0.904	0.144	0.999	0.356
	3	0.019	0.005	0.089	0.959	0.590	1.000	0.730

Notes: 1 Before COVID-19, 2 During COVID-19, 3 During the recent crisis.

report hedging effectiveness, optimal hedge ratios, and portfolio weights across models and rebalancing intervals. Overall, dynamic hedging strategies generally outperform the static OLS benchmark, although the magnitude of risk reduction remains modest. This indicates that diversification benefits are present but limited, particularly during periods of elevated market volatility. Similar conclusions are reported by Hachicha et al. [63], who show that MGARCH-based hedging improves portfolio performance but that commodity hedges tend to weaken under stress conditions.

Comparing the multivariate specifications, the GO-GARCH model achieves higher hedging effectiveness than ADCC in several economically relevant cases, particularly for energy-related assets and during periods of high volatility. This suggests that the orthogonal decomposition employed by GO-GARCH captures latent volatility components and spillover effects more effectively. These findings are consistent with Abid et al. [64], who identify GO-GARCH as a suitable framework for modelling joint dynamics between equity and commodity markets.

Despite these advantages, GO-GARCH does not dominate in all cases. Hedging effectiveness varies across asset pairs, crisis regimes, and rebalancing frequencies, highlighting the importance of market conditions and model choice. Gold and crude oil consistently emerge as the most effective hedging instruments across subperiods, whereas wheat and natural gas exhibit weak and unstable hedging performance. This pattern remains robust before COVID-19, during the pandemic, and throughout the Russia–Ukraine war.

The comparison between 20-day and 60-day rebalancing intervals reveals only minor differences in hedging effectiveness (

Table 5. Summary statistics of optimal portfolio weights, optimal hedge ratios, and hedging effectiveness across sub-periods based on the GO-GARCH model.

GO-GARCH (Refit = 20)
	Period	Optimal Hedge Ratio			Optimal Weight			HE (%)
WTI	1	Mean	Min	Max	Mean	Min	Max
	2	0.217	0.020	0.848	0.501	0.041	0.910	4.091
	3	0.281	0.059	2.457	0.593	0.000	0.990	7.472
Gold	1	0.181	0.078	0.367	0.660	0.291	1.000	5.349
	2	0.163	0.632	0.107	0.544	0.260	1.000	3.329
	3	0.132	1.154	0.217	0.486	0.256	1.000	3.950
Wheat	1	0.013	0.953	0.235	0.567	0.248	1.000	5.690
	2	0.054	0.264	0.158	0.506	0.093	0.877	0.852
	3	0.077	3.478	4.810	0.551	0.000	0.864	1.154
Natural Gas	1	0.037	0.037	0.381	0.555	0.106	0.904	0.301
	2	0.101	0.010	0.624	0.481	0.053	0.846	0.731
	3	0.126	0.023	3.649	0.616	0.000	0.952	1.165
GO-GARCH (refit = 60)
WTI	1	0.174	0.203	0.244	0.496	0.087	0.959	4.326
	2	0.306	0.131	0.869	0.419	0.011	1.000	6.682
	3	0.335	0.193	1.846	0.338	0.000	0.710	5.357
Gold	1	0.166	0.604	0.099	0.545	0.256	1.000	2.981
	2	0.147	1.181	0.219	0.487	0.261	0.955	3.330
	3	0.015	0.967	0.231	0.567	0.247	1.000	5.606
Wheat	1	0.046	0.279	0.096	0.505	0.094	0.881	0.807
	2	0.044	3.481	0.717	0.553	0.036	0.867	0.825
	3	0.037	0.038	0.352	0.556	0.120	0.904	0.293
Natural Gas	1	0.112	0.028	0.625	0.476	0.053	0.849	0.906
	2	0.151	0.026	3.620	0.612	0.000	0.957	1.319
	3	0.883	0.043	0.262	0.656	0.176	0.935	1.424

Notes: 1 Before COVID-19, 2 During COVID-19, 3 During the recent crisis.

). This suggests that extending the adjustment horizon does not substantially improve risk reduction. This result is in line with Fakhfekh et al. [65], who document stable hedging outcomes across rebalancing frequencies when using ADCC and GO-GARCH models. From a practical perspective, this implies that transaction costs may be reduced without sacrificing hedging efficiency.

Negative hedge ratios observed in several cases indicate that short positions in the hedging instrument are required to offset exposure to the underlying asset. This behaviour reflects changing dependence structures and regime shifts and has also been documented by Zghal et al. [66] for regional equity portfolios. Overall, the results confirm that dynamic hedge ratios respond to market stress and adjust more rapidly during crisis periods, reinforcing the relevance of multivariate volatility models for hedging under unstable financial conditions.

6. Conclusions

This study investigates the dynamic interdependencies among major financial markets and evaluates how extreme events affect portfolio diversification and hedging performance. By combining MFCCA with advanced multivariate GARCH models the analysis provides a comprehensive perspective on persistence, asymmetry, and volatility transmission across different market regimes.

The empirical findings indicate that hedging effectiveness varies substantially across assets, crisis periods, and econometric specifications. Dynamic hedging strategies generally outperform the static OLS benchmark, although the overall magnitude of risk reduction remains modest. The results show that the GO-GARCH model achieves higher hedging effectiveness than the ADCC specification in several economically relevant cases, particularly for energy-related assets and during periods of heightened volatility. This suggests that the orthogonal structure of GO-GARCH is effective in capturing latent volatility spillovers under stressed market conditions. At the same time, no single model dominates uniformly across all asset pairs and subperiods. Crude oil and gold consistently emerge as the most effective hedging instruments, whereas wheat and natural gas exhibit weak and unstable hedging performance. These outcomes highlight the importance of both model choice and asset selection in determining hedging success, especially during crisis episodes.

The integration of MFCCA complements the GARCH-based analysis by uncovering nonlinear dependencies and long-range cross-correlations that traditional linear models fail to capture. This combined approach shows that market turbulence affects not only short-term volatility dynamics but also the deeper memory structure of cross-market linkages. The presence of multifractal behaviour confirms that dependence structures evolve across time scales and regimes, reinforcing the relevance of multifractal methods for understanding risk transmission in modern financial markets.

From a practical perspective, the results underline the value of flexible and adaptive hedging strategies supported by data-driven model selection. The findings suggest that incorporating defensive assets such as gold and crude oil can improve portfolio resilience during periods of elevated uncertainty. These insights may assist investors and policymakers in designing more robust diversification strategies and in managing systemic risk in increasingly interconnected financial systems.

Despite its contributions, the study has several limitations. First, the analysis focuses on a limited set of assets, which may not fully represent global risk transmission channels. Second, the use of daily data may overlook intraday dynamics that influence volatility propagation during crisis periods. Third, although advanced, the multivariate GARCH models employed remain subject to parameter instability and may not fully capture extreme tail dependencies. Finally, transaction costs, liquidity constraints, and market microstructure effects are not explicitly incorporated and could affect realized hedging performance in practical applications.

Future research could extend this framework by including additional asset classes such as bonds, volatility indices, or cryptocurrencies, and by applying alternative nonlinear econometric approaches or machine learning techniques to improve hedge ratio estimation. Further analysis of structural breaks, regime transitions, and sentiment indicators may also enhance understanding of how global uncertainty shapes hedging effectiveness and market integration over time.