Special Issue “Volatility Modeling in Financial Market”

Dear Readers,

In this editorial, we provide a short overview of the Special Issue “Volatility Modelling in Financial Market” and summarise the main findings and contributions of the articles published within it. This Special Issue aims to bring together studies that advance the understanding of volatility, dependence, risk transmission, and forecasting in financial markets. We sincerely hope that this overview encourages readers to explore the full-length papers and engage further with the research questions addressed in this collection.

Volatility modelling remains one of the central areas of financial econometrics and risk management. Since volatility is not directly observable, researchers and practitioners rely on statistical, econometric, and computational methods to estimate and forecast it. In recent years, the field has evolved from classical ARCH- and GARCH-type models to more flexible approaches that incorporate realised measures, asymmetric dynamics, multivariate dependence structures, machine learning, soft computing, and hybrid frameworks. At the same time, the practical importance of volatility modelling has increased due to geopolitical shocks, changing market microstructure, the expansion of derivative markets, the rise of cryptocurrencies, and the growing use of artificial intelligence in financial decision-making.

Now that this Special Issue has been closed, we can state that it presents a broad and diverse set of contributions to volatility modelling and financial risk analysis. The published papers address different asset classes, including equities, exchange rates, corporate bonds, cryptocurrencies, derivatives, and credit portfolios. They also employ a wide range of methods, including GARCH-family models, DCC-GARCH, EGARCH, asymmetric GARCH specifications, realised-measure approaches, multiplicative error models, adaptive neuro-fuzzy inference systems, normalising flows, machine learning, and deep learning. Taken together, these studies show that volatility modelling is no longer limited to a single methodological tradition but is increasingly characterised by the integration of econometric interpretability with computational flexibility.

Several papers in this Special Issue contribute directly to the development and assessment of volatility forecasting methods. In “Historical Perspectives in Volatility Forecasting Methods with Machine Learning”, Qiu et al. provide a comprehensive overview of the evolution of volatility forecasting, moving from implied volatility and GARCH-type models to recurrent neural networks, LSTMs, transformers, and other state-of-the-art learning-based approaches. Their results indicate that machine learning models can substantially improve forecasting accuracy, but they also underline important limitations related to interpretability, data requirements, computational cost, and robustness. This paper therefore offers a valuable bridge between classical financial econometrics and modern artificial intelligence methods.

Another methodological contribution is provided by Kirby in “Using Daily Stock Returns to Estimate the Unconditional and Conditional Variances of Lower-Frequency Stock Returns”. This study shows that daily returns can be used to construct realised measures that are unbiased estimators of the unconditional and conditional variances of lower-frequency returns under relatively mild assumptions. The empirical analysis, based on S&P 500 data, suggests that multiplicative error models using these realised measures can outperform standard GARCH forecasts for weekly and monthly returns. This contribution is especially relevant when intraday data are unavailable or when long historical samples are required.

The paper “Normalising Flow Enhanced GARCH Models: A Two-Stage Framework for Flexible Innovation Modelling in Financial Time Series” by Hassan et al. proposes a hybrid NF-GARCH framework that preserves the interpretability of classical GARCH variance dynamics while replacing restrictive parametric innovation distributions with learned densities generated by normalising flows. The study shows that this approach can improve forecast accuracy, particularly for skewed-t GARCH baselines, while also providing more flexible modelling of heavy tails and asymmetric residual behaviour. At the same time, the authors point to computational costs and model-specific instability as important considerations for future research.

The Special Issue also includes papers focused on specific financial markets and instruments. Jongadsayakul’s paper “Foreign Exchange Futures Trading and Spot Market Volatility in Thailand” examines whether the introduction of EUR/USD and USD/JPY futures on the Thailand Futures Exchange stabilises or destabilises the underlying spot market. Using EGARCH and VAR models, the study finds that the introduction of FX futures reduces spot volatility, increases the speed at which new information is incorporated into spot prices, and decreases the persistence of volatility shocks. The results also show that unexpected trading volume has a destabilising effect, while unexpected open interest has a stabilising effect, with the latter dominating overall.

In “Estimating Corporate Bond Market Volatility Using Asymmetric GARCH Models”, Hadad et al. analyse the Israeli corporate bond market, which is characterised by high transparency and significant retail participation. The authors show that negative shocks have a stronger impact on volatility than positive shocks, confirming the importance of asymmetry and investor sentiment in corporate bond markets. Their findings indicate that the GJR-GARCH model with a Student’s t-distribution best captures the volatility dynamics of the Tel-Bond 20 and Tel-Bond 60 indices. This contribution is particularly important because volatility behaviour in corporate bond markets remains less extensively studied than volatility in equity markets.

Czech and Wielechowski, in “Do Uncertainty and Action Shocks Affect G7 Stock Market Synchronisation? DCC-GARCH Evidence from the 2024 U.S. Election and the Reciprocal Tariffs Announcement”, investigate how different types of U.S.-centred shocks affect conditional correlations between the U.S. equity market and other G7 markets. By distinguishing between an uncertainty shock represented by the 2024 U.S. presidential election and an action shock represented by the 2025 reciprocal tariff announcement, the study shows that different shocks produce different patterns of cross-market synchronisation. Election-related uncertainty is mainly associated with lower correlations for European markets, while the tariff-related action shock increases conditional correlations across all analysed U.S.–G7 pairs. This paper contributes to the literature on international spillovers, portfolio diversification, and state-dependent market co-movement.

The Special Issue also contains contributions that extend the discussion of volatility and risk modelling towards cryptocurrencies and credit risk. Orozco-Castañeda et al., in “Evaluating Volatility Using an ANFIS Model for Financial Time Series Prediction”, develop an ARIMA-ANFIS model for BTCUSD price prediction and risk assessment and compare it with an ARIMA-GARCH benchmark. Their findings suggest that ANFIS and GARCH capture different aspects of the data generation process. ANFIS may perform well under more stable conditions but can underestimate volatility during turbulent periods, whereas GARCH provides wider confidence intervals and stronger protection against high-volatility episodes. This study highlights the trade-off between flexibility and risk coverage in cryptocurrency volatility modelling.

Chang et al., in “Credit Risk Prediction Using Machine Learning and Deep Learning: A Study on Credit Card Customers”, broaden the risk management perspective of the Special Issue by examining machine learning and deep learning techniques for credit default prediction. The study compares several models, including neural networks, logistic regression, AdaBoost, XGBoost, and LightGBM, and finds that XGBoost achieves the strongest predictive performance. Although this contribution does not focus on market volatility in the narrow sense, it complements the Special Issue by showing how machine learning tools can improve financial risk classification, lending decisions, and customer risk segmentation.

This Special Issue addresses several important gaps in the current literature and shows that classical GARCH-family models remain highly relevant, especially when interpretability, diagnostic testing, and practical risk management are central concerns. Moreover, it demonstrates that hybrid and machine learning approaches can improve predictive accuracy, but only when their limitations are carefully managed, including overfitting, computational complexity, data requirements, and reduced transparency. Additionally, the Special Issue highlights the need to model not only volatility levels but also asymmetry, tail behaviour, market synchronisation, innovation distributions, and the interaction between derivatives and spot markets. Finally, the collection confirms that volatility and risk dynamics are strongly context-dependent, varying across asset classes, market structures, investor compositions, and shock types.

In summary, the Special Issue “Volatility Modelling in Financial Market” presents a collection of studies that jointly contribute to the theoretical, methodological, and applied development of volatility and financial risk modelling. The published articles show that modern volatility research requires both rigorous econometric modelling and openness to new computational tools. By bringing together studies on GARCH models, realised measures, dynamic correlations, machine learning, soft computing, normalising flows, derivatives, corporate bonds, cryptocurrencies, equities, and credit risk, this Special Issue contributes to a deeper understanding of how financial risk is measured, transmitted, forecasted, and managed in increasingly complex markets.