Advancing Volatility Forecasting in Financial Indices: Integrating GARCH Models, Multifractal Indicators, and Deep Learning

Accurate volatility prediction is essential for effective investment strategies and risk awareness. Yet, the intricate and ever-changing characteristics of markets pose considerable challenges, motivating the use of hybrid frameworks by integrating heteroscedastic models, multifractal analysis, and deep learning techniques. While heteroscedastic models are simple and widely adopted, they often fail to reflect the inherent nonlinearities and multifractal properties of volatility. In contrast, LSTM, GRU, and Transformers, while capable of capturing complex structures, require well-chosen explanatory variables to deliver accurate forecasts. Accordingly, this study conducts a rigorous comparative investigation across the Dow Jones Islamic Market Index, the Dow Jones Global Index, and the S&P 500. We confirm the existence of multifractal scaling and evaluate the performance of deep learning models based on historical features against hybrid models integrating GARCH-type forecasts and multifractal indicators. Results demonstrate that integrating GARCH, EGARCH, and FIGARCH features significantly improves accuracy by embedding key stylized facts such as volatility clustering, asymmetry, and long memory, with statistical significance confirmed by the Diebold-Mariano test. Furthermore, findings indicate that while standalone multifractal features are insufficient, they serve as complementary inputs. Rather than proposing a single novel model, the contribution of this work lies in a systematic analysis of feature complementarity, demonstrating that guiding deep learning with econometric signals enhances predictive robustness across diverse market structures.