Geopolitical risk (GPR), according to Caldara and Iacoviello [1], is defined as the risk associated with terror threats, war threats, nuclear threats and military build-ups between states or countries that disrupt the usual, peaceful conduct of international affairs. For instance, Russia’s recent invasion of Ukraine on 24 February 2022 has rattled human capital, physical infrastructure, financial markets, international peace and the security system [2]. It is producing a major humanitarian crisis and running amok on an already frail global economy, which was recently hit by the COVID-19 pandemic [3]. GPR has been on the rise in recent decades, with extreme events such as the US bombing of Libya (April 1986), invasion of Kuwait (August 1990), Iraq airstrikes (January 1993), Bosnian war (February 1994), 9/11 attacks (September 2001), Iraq war (March 2003), London bombing (July 2005), the global financial crisis (GFC 2007–2008), Paris attacks (November 2015), US–North Korea (2017–2018), US–Iran tensions (2020), and the current COVID-19 pandemic (2019) [1]. GPR is now ranked even higher than economic uncertainty [4] and is emphasized as a key driver of the state of the economy [5]. Carney [6] includes GPR with economic and policy uncertainty as the “uncertainty trinity” with major financial and economic impacts. “Uncertainty” and “risk” are different, as “uncertainty” cannot be quantified while “risk” is measurable using probabilities, either subjective or physical. Since 1921, economists have debated and modeled this issue. However, the macroeconomics literature muddles this distinction, and explicitly measures “uncertainty” by developing indices described in the text, which most likely also capture some parts of “risk” [7].

Asset prices reflect investors’ hopes and fears for the future and generate a tidal wave of activity. In an uncertain and risky environment, investors search for other safe heavens and withdraw their investments, thereby adversely affecting markets. GPR immediately affects the financial and commodity markets by increasing the risk aversion of investors, consumers and firms, which leads to lower consumption and investments, triggering economic slowdown and spilling over to other alternative markets. Furthermore, due to psychological fear, ambiguity, and the desire to avoid future loss, individual investors become reluctant to trade, which negatively affects financial markets [7]. Despite this, there is a lack of literature on the impact of GPR on financial markets because of the difficulty in measuring GPR. Some studies examined the impact of GPR on stock markets ^[8][9][10][8,9,10], cryptocurrencies ^[11][12][11,12], metals ^[13][14][13,14], energies ^[15][16][15,16], and oil markets ^{[4][17][18][19]}[4,17,18,19].

Despite the development of electric cars, renewable-energy production, and ambitious climate goals, the oil market is still regarded as the lifeblood of the world’s economic engine [20], meeting around two-thirds of the global energy demand [21]. For this reason, a large body of literature has investigated oil markets from various perspectives, such as the pricing formation ^[22][23][24][22,23,24], the relationship with the macroeconomy ^[25][26][27][25,26,27], interlinkages with stock markets ^[28][29][28,29], financialization ^[30][31][30,31], forecasting ^[32][33][32,33] or multifractal characteristics ^[34][35][34,35], among others. The crude-oil market is extraordinarily large and complicated. According to the Crude Oil Global Market Report 2020 (https://www.thebusinessresearchcompany.com/report/crude-oil-global-market-report#:~:text=Crude%20Oil%20Market%20Size,1.2%25%20during%20the%20forecast%20period (accessed on 19 April 2022)) and Natural Gas Global Market Report 2020 (https://www.thebusinessresearchcompany.com/report/natural-gas-global-market-report#:~:text=Natural%20Gas%20Market%20Size,7.7%25%20during%20the%20forecast%20period (accessed on 19 April 2022)), the crude-oil and natural-gas markets are expected to reach market values of about $1407.65 billion and $1031.55 billion by 2022, respectively. It is the geopolitical aspect which distinguishes crude-oil markets from other energies, commodities, and financial assets. Heating oil, on the other hand, commonly known as No. 2 fuel oil, accounts for around 25% of a barrel’s yield and had a market value of $163.3 billion in 2019 (https://www.verifiedmarketresearch.com/product/fuel-oil-market/ (accessed on 19 April 2022)).

The consumption of oil, natural gas and coal increases carbon emissions, which represents a barrier to sustainable economic development and contributes to the creation of new geopolitical conditions. For example, GPR can reduce carbon emissions by limiting economic growth and energy consumption. On other hand, it may deter innovation and clean energy and result in increased carbon emissions [36]. Therefore, oil-related issues should be widely investigated from a geopolitical perspective, as governments frequently regard crude oil as a political weapon [37]. Even though oil prices have recovered from historic lows in 2014 and 2015, recent volatility fueled by the Russians, Iran sanctions, United States (US) and China conflicts, and the recent shale revolution has left many concerned, and it shows no signs of easing. This means that GPR has an impact on economic aspects including oil price, production, resource mobility, demand and supply, extraction costs, exchange rates and other alternative investments. A large imbalance could occur if supply channels are blocked or demand collapses due to economic shutdowns triggered by unrest, as happened during the recent COVID-19 pandemic.

Other major energy markets such as natural gas are also prone to geopolitical risk. Russia is one of the world’s main producers of primary energy resources, with a particularly strong position in global gas markets [38]. It has the world’s largest gas reserves and is the world’s second-largest gas producer, trailing only the US, which recently overtook Russia due to the shale revolution [39]. Most Russian gas exports go to European and Commonwealth of Independent States (CIS) countries, while Asian exports are likely to grow significantly in the future (https://scholarship.rice.edu/bitstream/handle/1911/91291/CES-pub-GeoGasRussiax-022114.pdf?sequence=1 (accessed on 19 April 2022)). As a result, Russia wields huge influence over prices and geopolitical leverage as well as on the “rules of the game”. Along with oil, natural gas is also being used as a key geopolitical weapon. Therefore, the importance of the natural-gas industry for the current and future political and economic situation should not be overlooked. However, the relationship between geopolitical-risk indicators and natural gas has been largely ignored in previous studies. Natural gas is not only the energy sector’s backbone, but also one of the most important national and foreign policy tools, being greatly influenced by domestic economic and political events.

GPR’s connection with other commodity markets has been studied via numerous analytical methods. These include the fixed-effect regression model [40], random-effect regression model [41], quantile regression ^[42][43][42,43], linear and non-linear probabilistic models and feasible generalized least-square estimator ^[44][45][44,45], time–frequency-based wavelet analysis [46], decomposition and the STVAR model [47] or Bayesian graphical structural VAR [48], among others. However, multifractal aspects of GPR with energies as well as other financial markets have been largely ignored in these studies.

Since their popularization by the Polish-born mathematician Mandelbrot [49], the idea of fractals has roused curiosity and has now been applied in various fields to examine the self-similarity and Hausdorff dimension of an object [50]. Mandelbrot [51] used fractals to study the behavior of cotton prices and discovered that commodity prices follow self-similar complicated patterns rather than being random. Primarily, there are two types of fractal-based methodologies, i.e., mono-fractal and multifractality. The anti-persistent or persistent behavior, also known as long-memory features, were mainly studied by mono-fractals. However, scholars later found that financial markets have complex multi-scale properties, which present a challenge for mono-fractality. Multifractality, according to Mandelbrot [52], may quantify the complexity of financial time series better than mono-fractality and it has a wider use in empirical studies such as physics ^[53][54][53,54], chemistry ^[55][56][55,56] (10–11), biology ^[57][58][57,58], hydrology [59], environment [60], linguistics [61], physiology [62], psychology ^[63][64][63,64], behavioral sciences [65] economics [66] and even in music [67].

At the same time, several researchers recognize multifractality in energy markets as a stylized fact [68]. Multifractal dynamics, for example, give  us a new model with appealing stochastic qualities that can reproduce some stylized facts including volatility clustering, fat tails, multi-scaling, and long-term dependence [69]. However, the combinatorial character of older versions of multifractal models, as well as their non-stationarity due to the constraint to a finite interval, limit their practical application. The pioneer methodologies involved rescaled range analysis (R/S) [70] and detrended fluctuation analysis (DFA) [71] for mono-fractality. However, R/S is prone to causing the bias error because of its vulnerability to short-range dependence (Lo, [72]). Hence, DFA compared to R/S and other above-mentioned methodologies has the benefit of long-range correlation detection in non-stationary time series. Furthermore, it eliminates the spurious analysis of long-range correlations, which is a non-stationary artifact [73].

Later, an extension of the DFA, i.e., multifractal detrended fluctuation analysis (MFDFA) by Kantelhardt, et al. [74] was derived, and this has been employed to examine the multifractality of various financial time series such as crude oil [75], stock markets ^[76][77][78][76,77,78], cryptocurrencies [79], and even sin markets [80]. Meanwhile, based on the concept of the DFA, Podobnik, et al. [81] developed the detrended cross-correlation analysis (DCCA) to examine the long-range cross-correlations between two non-stationary time series, which has been applied to various analyses ^[82][83][82,83]. However, it is easy to obtain λ as a scaling exponent in the case of the DCCA, but it lacks complete interpretation and severely distorts or even spuriously amplifies multifractal cross-correlation measures. To overcome this issue, Zhou [84] proposed the multifractal detrended cross-correlation analysis (MF-DCCA) by combining the DCCA and MFDFA, which has lately become popular ^[85][86][87][85,86,87]. The MF-DCCA approach can detect and quantify subtle features of multifractal cross-correlations between two financial time series. Furthermore, the multifractal spectrum analysis in the MF-DCCA quantifies the multifractal intensity of cross-correlations and explains the time series’ internal complexity and local properties.

Understanding the relationship and the role of geopolitical risk in asset prices is important for investors, companies, and government policymakers, in order to incorporate the magnitude of geopolitical risk into the valuation of asset prices and risk insurance, as well as to support markets in effectively absorbing the impacts of such risks.  Our purpose is thus to use the robust econophysics-based MF-DCCA to investigate the multifractal aspects of the cross-correlation between geopolitical-risk indicators and four major energy markets, i.e., West Texas Intermediate (WTI), Brent, natural gas and heating oil, from 1 January 1985 to 30 August 2021. Our study is different from others in three major aspects. Firstly, to the best of our knowledge, this paper is the first to investigate multifractal features in terms of cross-correlations. Only Bouoiyour, Selmi, Hammoudeh and Wohar [5] examined the effect of geopolitical risk on the informational efficiency of the oil market by employing MFDFA. Secondly, the interactions among oil prices and geopolitical risk and uncertainties have been a focal point of research in academia and policy circles [88], and have largely ignored the other major energy markets such as natural gas and heating oil. This study includes natural-gas and heating-oil markets as the major variables. Thirdly, we use the large daily data sets of about 36 years to reach conclusive results for market participants and government policy makers.

2. GeopolLitical Riskerature Review

The pioneering literature on GPR was based on individual geopolitical events ^[88][89][90][90,91,92]. However, after the recent development of the novel GPR index, a new stream of literature has investigated its impact on stock markets, exchange rates, renewable energy markets and energy markets. For instance, Yang and Yang [9] employed the monthly GPR index for stock markets and found that GPR is significant enough to capture the long and medium-term trends of stock markets. Similarly, Yang, et al. ^[91][93] employed the GARCH-MIDAS and found that global and regional monthly GPR indices have a significant impact on Chinese stock markets. The monthly GPR index is also found to be the best predictor of Kuwaiti and Omani stock markets [45]. However, Das, et al. ^[92][94] and Kannadhasan and Das ^[93][95] found that GPR has a less negative impact than Economic Policy Uncertainty (EPU) on the stock markets of emerging countries. Likewise, GPR is a major long run driver for the exchange rate of ASEAN countries ^[94][96] as well as the exchange rates of the UK, Republic of Korea, Japan, China and Canada ^[95][97]. Yang, Wei, Li and He [15] employed delta conditional Value-at-Risk techniques and found that the geopolitical-risk spillover to renewable energy markets is much smaller than that from equity and oil markets. The major shortcoming in previous literature is that most of it focused either on individual geopolitical events or the monthly frequency of the GPR indices, where major geopolitical events could be missed ^[1][1]. Hence, this study fills this gap and employs the daily frequency of the GPR index to provide the multifractal dynamics for energy markets in a more robust way.  The literature on GPR in energy markets deals mostly with the crude-oil market rather than other energy markets. For instance, Antonakakis, et al. ^[96][98] employ the VAR-BEKK-GARCH model and find that GPR has a severe impact on the mean return and variability of oil markets compared to stock markets. Similarly, using nonlinear Granger causality tests and a DCC-MVGARCH model, Huang, et al. ^[97][99] find that the impact of GPR on the volatility of oil through the jump component is higher than its returns. While the correlation between volatility jumps and GPR seems to be positive, Mei, Ma, Liao and Wang ^[98][89] report that GPR is positively linked with the realized volatility of oil and can be used to predict the short-term volatility of oil futures. Liu, Ma, Tang and Zhang [17] proposed a new model, GARCH-MIDAS-GPR, which uses GPR and serious GPR to forecast the volatility of oil futures in order to gain higher economic gains. Despite a great number of empirical studies on the relationship between GPR and energy markets, the multifractal dimension of GPR with energies is mostly overlooked.  Therefore, this study employs MF-DCCA to uncover the inner dynamics of multifractality for GPR and major energy markets. In addition, for better policy making, it is important for policy makers, and market participants to have extensive knowledge about the different impact of geopolitical acts and threats on financial markets. However, only a few studies have examined this. For instance, Bouoiyour, Selmi, Hammoudeh and Wohar [5] separate the shocks and find that the impact of geopolitical threats on oil price dynamics is moderate or non-significant while the impact of geopolitical acts is stronger and positive. Geopolitical acts, as opposed to threats, help to forecast the long-term volatility of oil markets ^[98][89]. Likewise, Salisu, Pierdzioch and Gupta [4] report that geopolitical acts decrease tail risk at longer forecasting horizons in oil markets, while threats increase tail risk. Hence, geopolitical threats are the major predictors of oil markets’ tail risk. Society’s survival and advancement depends on energy, which can be seen as a key driver of global economic expansion ^[99][100][100,101]. Energy plays an important role in any country’s development ^[101][102]. There has been a rise in the amount of energy consumed as a percentage of world consumption. Energy price stability has become a major concern for many countries because of the importance of energy for economic growth. However, because of its scarcity, vital strategy, the geographical dispersion of supply and demand, and low price elasticity of demand, the price of energy is particularly susceptible to geopolitical risk [18]. Consequently, it is important to study the impact of geopolitical risk on the energy market. The current literature has focused primarily on the relationship between GPR and oil markets and ignored other energy markets such as natural gas and heating oil. ResWearchers could only find the study by Qin, et al. ^[102][103], who examined the relationship between GPR indices and energy markets of crude oil as well as natural gas and heating oil by using the quantile regression approach. These authors found no significant impact of GPR on natural gas, while the impact of GPR on heating oil and crude oil seems to be negatively significant. Geopolitical threats, compared to acts, have a statistically negative impact on the volatility of heating oil and natural gas in various quantiles. In a recent study, Aloui and Hamida ^[103][104] demonstrate the relevance of geopolitical risk in the oil-stock nexus in a time-frequency domain. The authors applied the wavelet coherence method to show that geopolitical risk weakens oil-stock connectedness in the short term and lowers the oil-stock magnitude and volatility correlation. Similarly, Bouri, et al. ^[104][105] used logistic regressions to conclude that Bitcoin jumps are dependent on jumps in the geopolitical-risk index. Other studies on the relationship of GPR focus on precious metals ^[13][14][13,14], gold ^[105][106], corporate cash holdings ^[106][107], corporate investments ^[107][108], financial constraints ^[108][109], insurance ^[109][110], merger and acquisition ^[110][111], natural resource rents ^[111][112], tourism ^[112][113] and others.