DFC-LSTM: A Novel LSTM Architecture Integrating Dynamic Fractal Gating and Chaotic Activation for Value-at-Risk Forecasting

Accurate Value-at-Risk (VaR) forecasting is challenged by the non-stationary, fractal, and chaotic dynamics of financial markets. Standard deep learning models like LSTMs often rely on static internal mechanisms that fail to adapt to shifting market complexities. To address these limitations, we propose a novel architecture: the Dynamic Fractal–Chaotic LSTM (DFC-LSTM). This model incorporates two synergistic innovations: a multifractal-driven dynamic forget gate that utilizes the multifractal spectrum width ($\Delta \alpha$) to adaptively regulate memory retention, and a chaotic oscillator-based dynamic activation that replaces the standard tanh function with the peak response of a Lee Oscillator’s trajectory. We evaluate the DFC-LSTM for one-day-ahead 95% VaR forecasting on S&P 500 and AAPL stock data, comparing it against a suite of state-of-the-art benchmarks. The DFC-LSTM consistently demonstrates superior statistical calibration, passing coverage tests with significantly higher p-values—particularly on the volatile AAPL dataset, where several benchmarks fail—while maintaining competitive economic loss scores. These results validate that embedding the intrinsic dynamical principles of financial markets into neural architectures leads to more accurate and reliable risk forecasts.