Volatility Forecasting (GARCH, HAR-RV, LSTM)

This project compares econometric and deep learning approaches for forecasting financial market volatility.

Implemented models:

• GARCH(1,1)
• HAR-RV
• LSTM (PyTorch, walk-forward retraining)

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Features

• Yahoo Finance data ingestion via yfinance
• Rolling realized volatility computation
• Out-of-sample walk-forward validation
• Diebold–Mariano statistical comparison
• CSV outputs for reproducibility

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Example Output

Plots and metrics are saved for each ticker (AAPL, MSFT).
Models are compared via RMSE, MAE, MAPE, R², and Diebold–Mariano tests.

Below are sample volatility forecasts showing realized volatility versus GARCH and LSTM predictions.

Apple (AAPL)

Microsoft (MSFT)

File Structure

VOL-FORECASTING/ │ ├── VolatilityForecasting.py # Main pipeline │ ├── vol_accuracy_comparison.csv # Model performance summary ├── dm_tests.csv # Diebold–Mariano test results │ ├── AAPL_vol_outputs.csv # AAPL volatility outputs ├── MSFT_vol_outputs.csv # MSFT volatility outputs │ ├── AAPL_garch_params.csv # AAPL GARCH(1,1) parameters ├── MSFT_garch_params.csv # MSFT GARCH(1,1) parameters │ ├── images/ │ ├── AAPL_vol_forecast.png # AAPL realized vs forecasted volatility │ └── MSFT_vol_forecast.png # MSFT realized vs forecasted volatility │ ├── .gitignore └── README.md

Requirements

• Python ≥ 3.9
• PyTorch
• ARCH
• scikit-learn
• yfinance
• matplotlib

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Methodology Overview

1. GARCH(1,1) Model

The conditional variance evolves as:

σₜ² = ω + αεₜ₋₁² + βσₜ₋₁²

where:

• σₜ² — conditional variance (volatility²)
• ω — long-run mean variance
• α — reaction to recent shocks
• β — persistence of past volatility

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2. HAR-RV Model

The Heterogeneous AutoRegressive model for realized volatility:

RVₜ = β₀ + β₁RVₜ₋₁ + β₂RV̄ₜ₋₅:ₜ₋₁ + β₃RV̄ₜ₋₂₂:ₜ₋₁ + εₜ

This model captures multi-horizon volatility components (daily, weekly, monthly).

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3. LSTM Model

A recurrent neural network using Long Short-Term Memory cells to capture nonlinear temporal dependencies in volatility:

hₜ = LSTM(RVₜ₋₁, RVₜ₋₂, …, RVₜ₋ₚ)

Hidden states are trained via Backpropagation Through Time (BPTT), and the model is retrained in a walk-forward fashion to simulate live forecasting conditions.

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4. Diebold–Mariano Test

Used to compare forecast accuracy between two models A and B:

DM = 𝑑̄ / √((2π f̂_d(0)) / T)

where:

• dₜ = L(e_A,t) − L(e_B,t)
• L is squared error loss

A negative DM statistic implies Model A is more accurate.

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Sample Results (2018–2025)

Ticker	Model	RMSE	MAE	MAPE (%)	R²
AAPL	GARCH(1,1) (in-sample)	0.0583	0.0438	17.88	0.81
AAPL	LSTM (WF)	0.0345	0.0238	8.79	0.94
AAPL	HAR-RV (WF)	0.0212	0.0113	4.12	0.98
MSFT	GARCH(1,1) (in-sample)	0.0538	0.0398	17.34	0.82
MSFT	LSTM (WF)	0.0344	0.0229	9.56	0.93
MSFT	HAR-RV (WF)	0.0188	0.0102	4.01	0.98

Key Takeaway:
Both LSTM and HAR-RV models achieve significant accuracy improvements over GARCH baselines — reducing RMSE by over 50% and explaining 93–98% of realized volatility variance.

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Diebold–Mariano Test Results (Forecast Accuracy Comparison)

Ticker	Comparison	DM Stat	p-value	Interpretation
AAPL	LSTM WF vs GARCH OOS	-9.54	< 0.001	LSTM significantly outperforms GARCH
AAPL	LSTM WF vs HAR-RV OOS	+12.13	< 0.001	HAR-RV significantly outperforms LSTM
AAPL	HAR-RV OOS vs GARCH OOS	-10.95	< 0.001	HAR-RV significantly outperforms GARCH
MSFT	LSTM WF vs GARCH OOS	-6.95	< 0.001	LSTM significantly outperforms GARCH
MSFT	LSTM WF vs HAR-RV OOS	+8.67	< 0.001	HAR-RV significantly outperforms LSTM
MSFT	HAR-RV OOS vs GARCH OOS	-7.95	< 0.001	HAR-RV significantly outperforms GARCH

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Future Work

• Extend models to include EGARCH and GJR-GARCH (asymmetric volatility)
• Combine HAR and LSTM architectures for hybrid modeling
• Apply framework across asset classes (FX, bonds, crypto)
• Integrate forecasts into option pricing and risk management workflows

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📜 License

MIT License © Kevin Wood