Stochastic Volatility Modelling with LSTM Networks: A Hybrid Approach for S&P 500 Index Volatility Forecasting

Paper: arXiv 2512.12250

Authors: Anna Perekhodko, Robert Ślepaczuk

Abstract

Accurate volatility forecasting is essential in banking, investment, and risk management, because expectations about future market movements directly influence current decisions. This study proposes a hybrid modelling framework that integrates a Stochastic Volatility model with a Long Short Term Memory neural network. The SV model improves statistical precision and captures latent volatility dynamics, especially in response to unforeseen events, while the LSTM network enhances the model’s ability to detect complex nonlinear patterns in financial time series. The forecasting is conducted using daily data from the S and P 500 index, covering the period from January 1 1998 to December 31 2024. A rolling window approach is employed to train the model and generate one step ahead volatility forecasts. The performance of the hybrid SV-LSTM model is evaluated through both statistical testing and investment simulations. The results show that the hybrid approach outperforms both the standalone SV and LSTM models and contributes to the development of volatility modelling techniques, providing a foundation for improving risk assessment and strategic investment planning in the context of the S and P 500.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 8.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced statistical methodology (Stochastic Volatility modeling, statistical tests like Diebold-Mariano) combined with complex deep learning (LSTM networks), while demonstrating high empirical rigor through a backtest-ready design using long-term S&P 500 data, a rolling window approach, and providing open-source code on GitHub.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Forecast S&P 500 Volatility"] --> B["Data Input:<br>Jan 1998 - Dec 2024 Daily Data"]
  B --> C{"Hybrid Methodology"}
  C --> D["Stochastic Volatility SV<br>Captures Latent Dynamics"]
  C --> E["LSTM Neural Network<br>Identifies Nonlinear Patterns"]
  D & E --> F["Rolling Window Training<br>One-Step Ahead Forecasts"]
  F --> G{"Evaluation"}
  G --> H["Statistical Testing<br>Superior Accuracy"]
  G --> I["Investment Simulation<br>Enhanced Risk Management"]

Cited References (23)

1. Deep Learning for Financial Time Series Forecasting (2019) — link
2. The Pricing of Options and Corporate Liabilities (1973) — link
3. Generalized Autoregressive Conditional Heteroskedasticity (1986) — link
4. The Detection and Estimation of Long Memory in Stochastic Volatility (1998) — link
5. Realized Volatility Forecasting with Neural Networks (2020) — link
6. Comparing Predictive Accuracy (2002) — link
7. Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation (1982) — link
8. GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics (2001) — link

2 of 23 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.