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This paper investigates the impact of jumps in forecasting co-volatility in the presence of leverage effects for daily crude oil and gold futures. We use a modified version of the jump-robust covariance estimator of Koike (2016), such that the estimated matrix is positive definite. Using this approach, we can disentangle the estimates of the integrated co-volatility matrix and jump variations from the quadratic covariation matrix. Empirical results show that more than 80% of the co-volatility of the two futures contains jump variations and that they have significant impacts on future co-volatility but that the impact is negligible in forecasting weekly and monthly horizons.

We show that typical behaviors of market participants at the high frequency scale generate leverage effect and rough volatility. To do so, we build a simple microscopic model for the price of an asset based on Hawkes processes. We encode in this model some of the main features of market microstructure in the context of high frequency trading: high degree of endogeneity of market, no-arbitrage property, buying/selling asymmetry and presence of metaorders. We prove that when the first three of these stylized facts are considered within the framework of our microscopic model, it behaves in the long run as a Heston stochastic volatility model, where a leverage effect is generated. Adding the last property enables us to obtain a rough Heston model in the limit, exhibiting both leverage effect and rough volatility. Hence we show that at least part of the foundations of leverage effect and rough volatility can be found in the microstructure of the asset.

We first propose a reduced-form model in discrete time for S&P 500 volatility showing that the forecasting performance can be significantly improved by introducing a persistent leverage effect with a long-range dependence similar to that of volatility itself. We also find a strongly significant positive impact of lagged jumps on volatility, which however is absorbed more quickly. We then estimate continuous-time stochastic volatility models that are able to reproduce the statistical features captured by the discrete-time model. We show that a single-factor model driven by a fractional Brownian motion is unable to reproduce the volatility dynamics observed in the data, while a multifactor Markovian model fully replicates the persistence of both volatility and leverage effect. The impact of jumps can be associated with a common jump component in price and volatility. This article has online supplementary materials.

The Black's leverage effect hypothesis postulates that a negative stock return innovation increases the financial leverage of a firm since the value of equity decreases at a given level of debt, which, in turn, creates a higher equity return volatility in the future. The paper is aimed at investigating the authenticity of the Black's leverage effect hypothesis and the relationship between negative stock returns and the financial leverage of the UK oil and gas stocks from 2004 to 2015. For each stock, exponential generalised autoregressive conditional heteroscedasticity model was estimated using Fama-French-Carhart 4-factor asset pricing model to extract the difference between the effects of negative and positive stock return innovations, regarded as leverage effect. The leverage effect parameter was further regressed on the financial leverage ratios of the book value of long-term debt to total assets, interest expenses to total assets and long-term debt to market value of equity to examine whether variation in the leverage parameter was as a result of variation in the firm's financial leverage. The findings of the study show that Fama-French-Carhart four risk factors of market, size effect, value and momentum were significant in the stock returns of most of the oil and gas companies. The mixed results in the significance level of the factors were attributed to the differences in individual firm characteristics. An evidence of leverage effect was also found in all the oil and gas stock returns but no evidence to suggest it was derived from the changes in the financial leverage of the companies. The implication of these findings for financial managers in the oil and gas industry was that while asset pricing frameworks such as CAPM and its extensions are relevant in determining oil stock returns, the level of gearing is irrelevant, albeit it has been recognised as one of the determinants of the firm's level of risk.

We introduce the realized exponential GARCH model that can use multiple realized volatility measures for the modeling of a return series. The model specifies the dynamic properties of both returns and realized measures, and is characterized by a flexible modeling of the dependence between returns and volatility. We apply the model to 27 stocks and an exchange traded fund that tracks the S&P 500 index and find specifications with multiple realized measures that dominate those that rely on a single realized measure. The empirical analysis suggests some convenient simplifications and highlights the advantages of the new specification.

This study aims to explore the impact of coronavirus pandemic-related variables and non-pharmaceutical interventions on fluctuations in the Malaysian stock market during the period from January 7, 2020, to March 31, 2021. By employing GARCH-M (1,1) family models (GARCH-M, EGARCH-M, and PGARCH-M), the study seeks to understand the intricate dynamics of market volatility amidst the pandemic and associated interventions. The findings suggest that while past market volatility and conditional variance continue to influence current market fluctuations, their effects have diminished over time during the study period. Additionally, the EGARCH-M (1,1) model reveals a leverage effect, indicating increased market volatility following negative news compared to positive news. Interestingly, the EGARCH-M (1,1) model emerges as the optimal choice for accurately capturing data dynamics. Conversely, the PGARCH-M (1,1) model does not exhibit a statistically significant leverage effect. These insights contribute to a better understanding of market behavior during crises, informing future research and risk management strategies. AcknowledgmentThe authors are grateful to the Middle East University, Amman, Jordan, for the full financial support granted to this research paper.

Customers are the most important assets of most companies, such that customer equity has been used as a proxy for shareholder value. However, linking customer metrics to shareholder value without considering debt and nonoperating assets ignores their effects on relative changes in customer equity and leads to biased estimates. In developing a new theoretical framework for customer-based valuation, grounded in valuation theory, this article links the value of all customers to shareholder value and introduces a new leverage effect that can translate percentage changes in customer equity into shareholder value. The average leverage effect in more than 2000 companies across ten years is 1.55, which indicates that a 10% increase in customer equity is amplified to a 15.5% increase in shareholder value. This research also compares the influence of customer and financial metrics on shareholder value. The findings challenge previous notions about the dominant effect of the retention rate and underline the importance of predicting the number of future acquired customers for a company.

This study explores whether digitally transformed firms outperform non-digital counterparts in terms of stock return volatility asymmetry and leverage behaviour, focusing on firm-specific characteristics such as complexity and maturity within UAE firms. By examining the lagged effects of negative news on these behaviours, the study captures differences in market responses over time. Utilizing a firm-level Exponential Generalized Autoregressive Conditional Heteroskedasticity (E-GARCH) model and stock returns data from 2012 to 2024, the research analyses 661 firms (368 digitalised and 293 non-digitalised), categorised further by firm complexity and maturity. Leverage effects are evaluated across multiple lags, with significant findings emerging particularly at lag intervals four and five, indicating the relevance of historical return patterns and supporting theories of market underreaction and delayed information processing. To ensure robustness, Variance Decomposition Analysis (VDA) and Impulse Response Analysis (IRA) were applied, reinforcing the credibility of the results. Digitally transformed firms demonstrated strategic advantages through the use of digital technologies to enhance efficiency, expand market reach and innovate more rapidly, thereby positioning themselves competitively. The impact of macroeconomic variables, including GDP growth, inflation, and oil price fluctuations, was consistent with existing literature. These findings offer practical insights for investors seeking to optimise portfolio strategies by understanding leverage gaps and time-lagged market reactions between digital and non-digital firms. Moreover, policymakers may apply these insights to design more informed macroeconomic and industrial strategies. The study contributes to the growing literature on digital transformation by linking volatility dynamics with firm maturity and complexity in the UAE context.

This study investigates the volatility dynamics of the Bahrain All Share Index (BAX) between 2010 and 2025, a period marked by COVID-19 and regional geopolitical shocks. Using ARMA (1,1) to model returns and four GARCH-family models (ARCH, GARCH, EGARCH, GJR-GARCH) to capture volatility, we provide new evidence from a bank-based frontier market that has received limited empirical attention. The results reveal that returns are stationary and exhibit volatility clustering. Among the competing models, EGARCH (1,1) provides the best fit—exhibiting the lowest AIC and SIC values and the highest log-likelihood—revealing a significant leverage effect whereby negative shocks generate stronger volatility than positive shocks. This asymmetric volatility pattern contradicts earlier findings for Bahrain but aligns with theoretical expectations for bank-based financial systems. The findings carry implications for investors in terms of portfolio risk management, derivative pricing, and asset allocation. They also have important implications for regulators and policymakers, suggesting that counter-cyclical buffers and interest rate adjustments could be applied to stabilize the market in anticipation of negative shocks. These insights enrich the scarce literature on volatility in small frontier markets and contribute to a more nuanced understanding of the volatility dynamics in the MENA region.

This paper investigates the efficacy of various heterogeneous autoregressive models (HAR) in forecasting volatility across the U.S. financial markets. We address potential data measurement errors and leverage a comprehensive dataset of 22 years of tick-by-tick data encompassing three major stock indices: the S&P500, the Dow Jones Industrial Average (DJI), and the Nasdaq. Our analysis reveals several key findings: (1) Long-term (monthly) realized volatility (RV) has a stronger influence on future volatility compared to short-term (daily and weekly) RV. This aligns with the Heterogeneous Market Hypothesis, suggesting all market participants prioritize long-term volatility due to its impact on market direction. (2) Daily jumps have a short-term negative impact on future volatility, while aggregated monthly jumps have a positive effect due to their influence on market direction. The transient nature of jumps implies that the persistence of volatility stems from its continuous component. (3) The leverage effect is present and persists for up to 1 week. Models incorporating this effect demonstrate significantly better performance. (4) Across all models, forecast accuracy peaks at the 1-week horizon. More general models offer superior predictive power for short-term forecasts. For longer horizons, while there is no statistically significant difference among models, the loss function shows a slight improvement for more general models. (5) All models are able to confirm the theoretical link between expected return and volatility by identifying a positive correlation between return and risk in the data.