Volatility Modeling with ARCH/GARCH

This project demonstrates how to model time-varying volatility in financial returns using ARCH and GARCH models. The dataset used is the S&P 500 index (^GSPC) from Yahoo Finance.

Project Structure

volatility-modeling/
├── .venv/                  # Python virtual environment
├── data/                  # Folder for storing downloaded data
├── notebooks/             # Jupyter notebooks
│   └── arch_garch_model.ipynb
├── outputs/               # Plots or generated model outputs
├── test_imports.py        # Test for library imports
├── requirements.txt       # Installed Python packages
└── README.md              # Project overview

How to Run

1. Clone the repo and navigate to the folder.
2. Create a virtual environment:

   python3 -m venv .venv
   source .venv/bin/activate
   

3. Install dependencies:

   pip install -r requirements.txt
   

4. Launch the notebook:

   jupyter notebook notebooks/arch_garch_model.ipynb
   

Key Tools

• Python
• arch for GARCH models
• yfinance for market data
• matplotlib for visualization

Output

• Estimated conditional volatility from a GARCH(1,1) model
• Comparison with raw return series

📈 Sample Outputs

Log Returns Time Series

Model Summary Table

GARCH(1,1) Estimated Volatility

Results & Key Findings

• Volatility Clustering Observed
  The GARCH(1,1) model captures strong volatility persistence in S&P 500 returns, with ( \alpha + \beta ) close to 1, consistent with well-documented financial time-series behavior.

• Improved Fit Over Constant Volatility
  Compared to a constant-variance baseline, the GARCH model produces time-varying volatility estimates that better align with periods of market stress and calm.

• Mean-Reverting Conditional Variance
  Conditional variance shows mean reversion after volatility spikes, reflecting realistic post-shock dynamics in equity markets.

• Practical Risk Insight
  Estimated volatility paths provide inputs for downstream risk measures such as Value-at-Risk (VaR), stress testing, and scenario analysis.

👤 Author

Amanda Achiangia
BSc Applied Mathematics (Financial Mathematics), York University
Aspiring Quantitative Finance Professional
LinkedIn | GitHub