ARCH–LSTM Structural Equivalence with Hybrid Student-t Likelihood Loss for Financial Volatility Forecasting

Accurate conditional volatility forecasting is essential for risk management, asset pricing, and portfolio optimization. Despite the widespread use of GARCH family models and the proliferation of deep learning architectures, a methodological gap persists: neural networks are rarely trained under criteria statistically coherent with the distributional properties of financial returns. This paper makes two contributions. First, we establish formal structural equivalences between classical heteroscedastic models and neural architectures, showing that ARCH(p) is equivalent to a single-layer linear MLP and GARCH(1,1) to a constrained LSTM, with an explicit parameter correspondence. Second, we propose LSTM-SSE-t-Student, a parsimonious LSTM trained with a hybrid loss that combines the sum of squared errors with the Student-t negative log-likelihood, penalizing errors in the tails of the return distribution. The model is evaluated on six daily series spanning three asset classes—Bitcoin, Ethereum, Gold, Oil, the DJIA, and the S&P 500—across diverse regimes, including the COVID-19 period, against a broad set of econometric and deep learning benchmarks. Relative to GARCH(1,1), it significantly improves point forecast accuracy and probabilistic calibration over naive, short-memory, and regime-switching specifications, while matching the strongest GARCH family and deep learning competitors; a sensitivity analysis shows that the likelihood term lowers the QLIKE loss for most series, with a market-dependent optimal weighting. Statistical significance is assessed via Diebold–Mariano tests with a correction for multiple comparisons, and Value-at-Risk and Expected Shortfall backtests confirm adequate tail calibration for the equity and cryptocurrency series. Interpretability is preserved through the GARCH-consistent structure, whose learned gate dynamics are stable across random seeds.