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From GARCH to Neural Network for Volatility Forecast
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Shows GARCH(1,1), GJR-GARCH and FI-GARCH have NN (RNN/CNN) counterparts under a shared likelihood loss, and proposes GARCH-LSTM, which plugs the GARCH NN kernel into an LSTM for next-day volatility and VaR forecasting.
Problem
Volatility forecasting has evolved in separate streams: econometric GARCH-family models are interpretable but limited, while neural networks are flexible but opaque, overfit noisy financial data, and ignore volatility stylized facts.
Approach
Establish an equivalence between GARCH-family models and NN structures (RNN for GARCH(1,1)/GJR-GARCH, CNN for FI-GARCH) sharing inputs, parameters, loss and training, then embed the GARCH NN kernel as a component inside an LSTM (GARCH-LSTM) so stylized facts are built into the architecture, trained with a Student-t likelihood loss.
Key results
- GARCH-LSTM achieves the best MAE and MSE in most dataset and horizon settings, e.g. 0.205 MAE / 0.070 MSE at 1-day horizon on S&P 500 versus 0.636 / 0.546 for Autoformer.
- The authors report about 3% MAE and 10% MSE improvement on average over the second-best approach on S&P 500, NASDAQ and Gold.
- NN counterparts of GARCH(1,1), GJR-GARCH and FI-GARCH produce forecast MAE/MSE close to the stochastic originals on the five datasets.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Integrate GJR-GARCH as the GARCH-LSTM kernel rather than use standalone stochastic models | Volatility forecasting · S&P 500, DJI, NASDAQ, EUR-USD, and Gold (1D) | MAE: S&P 500: 0.205 vs 0.205; DJI: 0.170 vs 0.175; NASDAQ: 0.288 vs 0.296; EUR-USD: 0.118 vs 0.114; Gold: 0.244 vs 0.274 | Mixed |
| Integrate GJR-GARCH as the GARCH-LSTM kernel rather than use standalone stochastic models | Volatility forecasting · S&P 500, DJI, NASDAQ, EUR-USD, and Gold (1D) | MSE: S&P 500: 0.070 vs 0.074; DJI: 0.056 vs 0.056; NASDAQ: 0.144 vs 0.157; EUR-USD: 0.023 vs 0.023; Gold: 0.100 vs 0.123 | Mixed |
| Replace MSE loss with maximum-likelihood N-loss (ablation) | RNN and LSTM volatility models · Multiple datasets | MAE | Helped |
| Replace MSE loss with maximum-likelihood N-loss (ablation) | RNN and LSTM volatility models · Multiple datasets | MSE | Helped |
| Use Student's t T-loss instead of MSE loss (ablation) | RNN and LSTM volatility models · Multiple datasets | MAE | Helped |
| Use Student's t T-loss instead of MSE loss (ablation) | RNN and LSTM volatility models · Multiple datasets | MSE | Helped |
| Set Student's t degrees of freedom to v=5 rather than v=3 (ablation) | RNN and LSTM volatility models · Multiple datasets | MAE | Mixed |
| Set Student's t degrees of freedom to v=5 rather than v=3 (ablation) | RNN and LSTM volatility models · Multiple datasets | MSE | Mixed |
Surprising findings
- Transformer-based models (Autoformer, Informer, Reformer, Transformer) perform poorly on volatility forecasting, often worse than the simple GARCH family models.
- On the 1-day horizon for DJI and S&P 500, vanilla GARCH(1,1) matches or slightly beats GARCH-LSTM in some cases.
Limitations
- Only three GARCH variants (GARCH(1,1), GJR-GARCH, FI-GARCH) are shown to have NN counterparts; the equivalence for other family members is not established.
- Equivalence is only approximate in practice: NN counterparts are trained with Adam rather than SLSQP, and parameter estimates differ noticeably from the stochastic models.
- Evaluation uses only five daily series with a single 1-day VaR case study; the paper notes VaR violation counts favor models forecasting large volatility.
Experimental setup
Datasets: S&P 500 DJI NASDAQ EUR-USD Gold
Benchmarks: S&P 500 DJI NASDAQ EUR-USD Gold
Metrics: MAE MSE VaR violation rate
Baselines: GARCH(1,1) GJR-GARCH FI-GARCH Autoformer Informer Reformer Transformer
Builds on: GARCH GJR-GARCH FI-GARCH LSTM
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