Wavelet Entropy for Efficiency Assessment of Price, Return, and Volatility of Brent and WTI During Extreme Events

Abstract

This study analyzes the market efficiency of crude oil markets, namely Brent and West Texas Intermediate (WTI), during three different periods: pre-COVID-19, during the COVID-19 pandemic, and during the ongoing Russia–Ukraine military conflict. To evaluate the efficiency of crude oil markets, wavelet entropy is computed from price, return, and volatility series. Our empirical results show that WTI prices are predictable during the Russia–Ukraine military conflict, but Brent prices are difficult to predict during the same period. The prices of Brent and WTI were difficult to predict during the COVID-19 pandemic. Returns in Brent and WTI are more difficult to predict during the military conflict than they were during the pandemic. Finally, volatility in Brent and WTI carried more information during the pandemic compared to the military conflict. Also, volatility series for Brent and WTI are difficult to predict during the military conflict. These findings offer insightful information for investors, traders, and policy makers in relation to crude oil energy under various extreme market conditions.

1. Introduction

Crude oil is one of the most crucial strategic energy resources on the planet. In addition, fluctuations in crude prices can affect not only individual firms and specific industries, but also the overall economy, economic development, and economic stability worldwide. Furthermore, crude oil plays a central role in investment decisions like asset allocation and risk management. Therefore, understanding the price dynamics of crude oil is important for energy and investment management.

In this regard, according to the Efficient Markets Hypothesis (EMH) [1], in a competitive market with rational with profit-maximizing agents, asset prices integrate all available information. Therefore, in such efficient markets, prices are not predictable and riskless profit opportunities cannot be utilized. Hence, in empirical finance, the EMH is widely examined in various markets including stock markets [2,3,4], exchange markets [5,6,7], energy markets [8,9,10,11,12,13], and commodities [14,15,16,17,18,19,20]. However, the general approach to testing EMH is based on statistical methods under strong assumptions like linearity and normality [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In addition, there is a growing interest in adopting an econophysics-based approach, as it is robust and assumption-free. Indeed, econophysics-based approaches have been used to estimate fractals [21,22,23], chaos [24,25,26], and entropy [27,28,29,30] to test whether the markets are predictable. In this study, we aim to examine the EMH in crude oil markets by means of wavelet-based entropy.

The remainder of the paper is organized as follows: Section 2 reviews recent related studies. Section 3 describes the algorithm of the wavelet-based Shannon entropy used to measure complexity and irregularity in the data. Section 4 presents the data and provides the results. Finally, Section 5 discusses the study and concludes the paper.

2. Related Literature

In recent years, crude oil and commodity markets have faced serious challenges and disruptions triggered by the COVID-19 pandemic and the ongoing Russia–Ukraine conflict. Indeed, with respect to the COVID-19 pandemic, recent works show evidence of significant large correlations between oil and stock market returns during the COVID-19 pandemic, which restrict opportunities for portfolio diversification [31], as well as increased volatility transmission from crude oil to G20 equity markets [32]. In addition, dynamic dependencies and volatility spillovers between the Baltic dry index, iron ore price, and Brent crude oil price have been strongly reinforced [33]; volatility spillover from media coverage of crude oil mostly arises in the short term [34]; West Texas Light crude oil and Brent crude oil showed a strong negative financial bubble [35]; the impact of COVID-19 on oil surpasses that of the 2008 financial crisis, and the crude oil market shows altered patterns before and after COVID-19 [36]; the asymmetric volatility transmission between oil and sector stocks in both the US and Chinese economies was more noticeable during the pandemic [37]; the positive interaction between climate policy uncertainty and WTI was the largest during the pandemic [38]; the COVID-19 pandemic has diminished oil consumption, while the 2008 financial crisis increased oil consumption [39]; inflation significantly and positively affected crude oil [40]; multifractal in Brent and WTI strengthened during the pandemic [41]; causality shifted direction during the pandemic, with cryptocurrencies exerting influence on the gold and crude oil markets [42]; clustering structures in fossil energy markets were affected and similarity between them increased [43]; and the pandemic affected complexity in fossil energy markets including Brent and WTI [44].

In the context of the effect of the Russia–Ukraine military conflict on crude oil markets, it was found that fake news shows a highly dominant role in driving crude oil’s future price volatility [45]; the war has significantly altered the future leverage effect in crude oil markets (WTI, Brent, Oman) and reduced the volatility spillover effect between them [46]; there are increased linkages between oil and exchange rate markets from pre-war to higher levels, and war may cause market risk and drive spillover of opportunities [47]; the influence of COVID-19 on the stability of and relationships between energy and financial markets is larger compared to the influence of the war [48]; there was a significant rise in the spillover between fossil energy, electricity, and carbon markets during the COVID-19 pandemic and another during the Russia–Ukrainian conflict [49]; there has been a strong intensification in systemic risk in both the European gas and the oil markets [50]; there are strengthened dynamic associations between crude oil futures (WTI, Brent, Chinese oil futures) and commodity futures (agricultural, industrial metals, and precious metals) [51]; fossil energy markets have a bigger impact due to continuing adjustments in high- and low-volatility regimes compared to renewable energy markets [52]; and the uncertainty caused by the Russo-Ukrainian conflict pointedly increases the volatility linkages between energy and agricultural commodity markets [53].

The literature is abundant in examinations of the effects of the pandemic [41,42,43,44] and war [45,46,47,48,49,50,51,52,53] on crude oil markets. However, little attention has been given to the comparison of their effects on these markets [18,19]. In addition, to the best of our knowledge, no work has examined and compared the efficiency of these markets during extreme events. Therefore, the main purpose of the current study is to compare the effects of the COVID-19 pandemic and Russia–Ukraine military conflict on crude oil markets; namely, Brent and WTI. While the limited previous works focused on the effects of these two extreme events on the connectedness and spillovers between crude oil markets [48,49], we seek to examine their effect on the efficiency of crude oil markets from an econophysics perspective. Specifically, we analyze the predictability of prices, returns, and volatilities pre-extreme events (COVID-19 pandemic and Russia–Ukraine military conflict), during the COVID-19 pandemic, and during the Russia–Ukraine military conflict. In this regard, we seek to evaluate the predictability that refers to the intrinsic property of a time series in relation to its deterministic, stochastic, or chaotic behavior. In this study, the predictability of crude oil data (price, return, volatility) is evaluated by computing the wavelet entropy [54] used to represent the information and uncertainty in the data; hence, it is useful to quantify the intrinsic predictability of time series data. Indeed, entropy incorporates much more information than volatility, it does not depend on any distribution, it is independent of the mean for all types of distributions, and it also serves as a measure of dispersion [55,56]. As a result, entropy is a robust method for identifying and measuring the uncertainty associated with equity and commodity markets [57,58,59,60,61,62,63,64,65,66,67].

The conceptual framework follows. A sharp change in price, returns, and volatility information content is expected in commodity markets, particularly those under economic and geopolitical stress due to uncertainty. Entropy, as a measure of information irregularity and randomness, is a robust tool to quantify uncertainty without loss of information during periods of economic turmoil like the COVID-19 pandemic and the Russia–Ukraine military conflict, where extreme values in prices are observed due to sudden and large shifts in prices. In this regard, significant changes in market informational irregularity and efficiency are expected and can be captured by entropy. In other words, in the current work, efficiency in the crude oil market is assessed by means of information irregularity in prices, returns, and volatility. In this regard, the market is efficient if prices, returns, and volatility carry regular information. In contrast, it is inefficient if prices, returns, and volatility carry irregular information. In this study, wavelet-based entropy is adopted as a measure of information regularity to assess efficiency in crude oil markets.

In summary, the contributions of our study are as follows. First, we study predictability in the price, return, and volatility of crude oil markets. This topic is missing from the literature. Second, we enrich the limited comparative literature [48,49] on the effects of the COVID-19 pandemic and the Russia–Ukraine military conflict on crude oil markets. Third, we employ a robust technique, wavelet-based Shannon entropy, to measure complexity and irregularity in price, return, and volatility series. To the best of our knowledge, our study is the first to assess the efficiency of the prices, returns, and volatilities of the Brent and WTI markets based on wavelet entropy. This approach is robust and widely employed in science and engineering [68,69,70,71,72,73,74] and in the assessment of efficiency in equity and commodity markets [75,76,77,78,79,80]. Fourth, we shed light on the predictability of prices, returns, and volatility as they attract large amounts of attention from investors, traders, and scholars.

3. Methods

3.1. Wavelet Entropy

The wavelet entropy [54] of signal y is computed in three main steps.

(a)

The signal y is decomposed by applying the continuous wavelet transform given by the following:

$$W_{\psi }\left(a,b\right)={\displaystyle \frac{1}{\sqrt{a}}}{\int }_{-\infty }^{+\infty }y\left(t\right){\psi }^{*}\left({\displaystyle \frac{1}{ab}}\right)dt$$

(1)

(b)

Given the number of translation coefficients N at scale a_j, the wavelet coefficients $W_{\psi }\left(a,b\right)$ are normalized to obtain the wavelet energy density:

$$p_j={\displaystyle \frac{1}{N}}{\sum }_{k=1}^N{\left|W_{\psi }\left(a_j,b_k\right)\right|}^2$$

(2)

(c)

Finally, the wavelet entropy (WE) is then calculated as the Shannon entropy of the wavelet energy density at scale M, as follows:

$$WE=-{\sum }_{j=1}^M{p}_j{log}_2\left(p_j\right)$$

(3)

The higher the WE, the higher the complexity and irregularity in the signal y. In contrast, the signal y is regular and predictable when WE is low.

3.2. EGARCH Model

In this study, the WE is estimated from price, return, and volatility series ${\sigma }_t^2$. We calculated the daily returns (R_t) of prices (P_t) as the difference in natural logarithms, for instance, R_t = log (P_t/P_t−1). The volatility of returns is estimated by using the well-known exponential generalized autoregressive conditional heteroscedasticity (EGARCH(p,q)) model [81] under the assumption that the errors follow a t-distribution. The EGARCH model allows the capture of asymmetric behavior of returns R_t. For instance, the EGARCH provides an asymmetric specification to consider the leverage effects of return R_t on conditional variance ${\sigma }_t^2$. That means a large decline in return can have a bigger impact on volatility than a large return increase. The EGARCH(p,q) model can be represented as follows:

$$r_t={\sigma }_t{\epsilon }_t$$

(4)

$$log\left({\sigma }_t^2\right)=\omega +{\sum }_{j=1}^p{\beta }_{j}log\left({\sigma }_{t-j}^2\right)+{\sum }_{i=1}^q{\gamma }_j\left({\displaystyle \frac{r_{t-i}}{{\sigma }_{t-i}}}-\sqrt{{\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.}}\right)+{\sum }_{i=1}^q{\alpha }_i{\displaystyle \frac{r_{t-i}}{{\sigma }_{t-i}}}$$

(5)

The terms α and γ, respectively, capture the leverage effect on the market shock and the asymmetric shock by leverage effect. In addition, the use of log form in the EGARCH specification allows the parameters to be negative without the conditional variance becoming negative. In other words, no restrictions are imposed on the parameters of the EGARCH model. In this study, the errors ε_t are assumed to be identically and independently distributed under the Student-t distribution. In this work, the Akaike criterion information (AIC) [82] is used to determine the appropriate orders p and q. The AIC is expressed as follows:

$$AIC=2k-2log\left(LL\right)$$

(6)

Here, k is the number of estimated parameters in Equation (5) and LL is the log-likelihood function of the estimated Equation (5).

4. Results

We collected daily price data on Brent and West Texas Intermediate (WTI) for the period from 2 January 2018 to 30 September 2024 from the Federal Reserve Economic Data (FRED) database of the Federal Reserve Bank of St. Louis, USA. The full period is split into three subperiods: prior to the COVID-19 pandemic (2 January 2018 to 31 December 2019), during the COVID-19 pandemic (2 January 2020 to 31 December 2021), and during the Russia–Ukraine military conflict (3 January 2022 to 30 September 2024). The goal is to find how wavelet entropy varies across the calm, pandemic, and war periods. Figure 1 plots the price, return, and volatility series of Brent and WTI in the full period of study. The full-sample estimation results of the EGARCH model in Equation (5) are provided in

Table 1. EGARCH estimation results for Brent.

Parameter	Value	t-Statistic	p-Value
ω	−0.30952	−4.479	7.499 × 10−6
β	0.96644	129.19	0.0000
γ	0.19674	6.793	1.098 × 10−11
α	−0.085366	−4.9498	7.4271 × 10−7

The estimation results are from the full sample under a 5% statistical significance level.

for Brent and in

Table 2. EGARCH estimation results for WTI.

Parameter	Value	t-Statistic	p-Value
ω	−0.1959	−3.278	0.0010455
β	0.97888	152.65	0.0000
γ	0.20043	7.4821	7.3136 × 10−14
α	−0.059715	−3.6685	0.00024396

The estimation results are from the full sample under a 5% statistical significance level.

for WTI. As shown, all model parameters (ω, β, γ, α) are all highly significant.

The computed values of wavelet entropy WE are shown in

Table 3. Computed wavelet entropy (WE).

	Before	Pandemic	War
Price
Brent	0.2719	0.2418	0.8166
WTI	0.8116	0.3144	0.2878
Return
Brent	0.8319	0.5348	0.8625
WTI	0.8053	0.6256	0.8413
Volatility
Brent	0.1	0.2259	0.695
WTI	0.7669	0.4653	0.7003

The values across markets and periods.

. Accordingly, for Brent prices, the WE decreased during the COVID-19 pandemic (from 0.2719 to 0.2418) but increased during the Russia–Ukraine military conflict period (from 0.2418 to 0.8166). For WTI prices, the WE decreased during the COVID-19 pandemic (from 0.8116 to 0.3144) and during the Russia–Ukraine military conflict period (from 0.3144 to 0.2878). Therefore, Brent prices were more regular during the pandemic compared to WTI prices. However, they are highly irregular during the Russia–Ukraine military conflict compared to WTI.

For Brent returns, the WE decreased during the COVID-19 pandemic (from 0.8319 to 0.5348) but increased during the Russia–Ukraine military conflict period (from 0.5348 to 0.8625). For WTI returns, the WE decreased during the COVID-19 pandemic (from 0.8053 to 0.6256) and during the Russia–Ukraine military conflict period (from 0.6256 to 0.8413). As a result, Brent returns show more regular dynamics during the pandemic compared to WTI returns. Conversely, they are more irregular during the Russia-Ukraine military conflict compared to WTI.

For Brent volatility, the WE decreased during the COVID-19 pandemic (from 0.1000 to 0.2259) and increased during the Russia–Ukraine military conflict period (from 0.2259 to 0.6950). For WTI volatility, the WE decreased during the COVID-19 pandemic (from 0.7669 to 0.4653) and increased during the Russia–Ukraine military conflict period (from 0.4653 to 0.7003). Consequently, Brent volatility was less complex and irregular during the COVID-19 pandemic compared to the Russia–Ukraine military conflict period. However, Brent volatility shows similar irregularity to WTI during the Russia–Ukraine military conflict.

In summary, WTI prices are predictable during the Russia–Ukraine military conflict, but Brent prices are difficult to predict during this period. The prices of Brent and WTI were difficult to predict during the COVID-19 pandemic. Returns in Brent and WTI are more difficult to predict during the military conflict than they were during the pandemic. Finally, volatility in Brent and WTI carried more information during the pandemic compared to the military conflict. Also, volatility series for Brent and WTI are difficult to predict during the military conflict.

5. Discussion and Conclusions

The COVID-19 pandemic and the Russia–Ukraine military conflict have significantly impacted energy, commodity, and financial markets [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53], drawing significant attention from scholars who have analyzed these effects from various perspectives.

In this paper, by using wavelet entropy, we examined the predictability of the prices, returns, and volatilities of the Brent and WTI crude oil markets. Specifically, we evaluated the irregularity in these time series across three periods: calm, COVID-19 pandemic, and Russia–Ukraine military conflict. Accordingly, the main purpose was to study the effect of the COVID-19 pandemic and the Russia–Ukraine military conflict on the irregularity of prices, returns, and volatilities.

This paper’s contributions are multiple. First, we examine the effects of extreme events, like the pandemic and the war in eastern Europe, on the predictability of crude oil market data. Secondly, we consider the most active crude oil markets in the West, namely Brent and WTI. Third, to the best of our knowledge, this is the first paper to examine such issues in prices, returns, and volatilities. Fourth, to examine predictability, the study applies wavelet entropy, which is a well-known technique in science and engineering applications thanks to its robustness in analyzing data [68,69,70,71,72,73,74]. Fifth, it enriches the limited literature on the comparison between the effects of the COVID-19 pandemic and the Russia–Ukraine military conflict on crude oil markets [48,49].

The main findings follow. Compared to the pre-pandemic period, during the pandemic, irregularity slightly decreased in Brent prices and was strongly diminished in WTI prices. Also, irregularity strongly decreased in the returns of both markets. Finally, the irregularity in the volatility of Brent was significantly augmented and the irregularity in WTI volatility was strongly diminished. In addition, compared to the pandemic period, during the Russia–Ukraine military conflict, irregularity has increased in Brent prices and decreased in WTI prices. This could be explained by the fact that the military conflict is occurring on European soil and affects the European economy more than the US economy. In addition, returns in both markets have shown a strong increase in irregularity during the conflict. Furthermore, the irregularity in the volatility of both markets has greatly increased during the conflict. Overall, our findings are in accordance with the literature; indeed, both the pandemic and the Russia–Ukraine war have significantly affected energy and commodity markets.

In summary, compared to the COVID-19 pandemic, Brent price, Brent and WTI returns, and Brent and WTI volatilities have become more difficult to predict during the Russia–Ukraine military conflict. This could be explained by the fact that timely access to reliable information during periods of high geopolitical risk could be limited and the information is definitively inaccurate, restricted, or missing. The information collected during periods of serious geopolitical risk is highly irregular, as shown by the high values of wavelet entropy during the Russia–Ukraine war. In this regard, our findings are significant for portfolio and risk managers, traders, and economic policy makers. Indeed, the findings should be taken into consideration for better investment decision making related to the Brent and WTI markets. Certainly, economic agents should pay more serious attention to geopolitical risk to aid their decision making as prices, returns, and volatilities become strongly irregular in terms of information content. For instance, during periods of high geopolitical risk, like military conflicts, economic agents should reduce their investment in and purchase of oil-based energy resources, consider alternative energy resources that are supposed to be less affected by wars, diversify oil markets, and consider sources not located in or close to war zones.

For future work, a large set of fossil energy markets, like natural gas, propane, kerosene, gasoline, heating oil, and coal will be considered. In addition, machine and deep learning models will be implemented by taking into account wavelet entropy information to improve their prediction ability across calm, pandemic, and war periods.