Volatility Modeling using ARCH(1) & GARCH(1,1)

Overview

This project explores volatility estimation and forecasting in financial markets using Autoregressive Conditional Heteroskedasticity (ARCH) and Generalized ARCH (GARCH) models. It applies these models to Apple Inc. (AAPL) stock data to capture and analyze volatility clustering patterns and compare predicted volatility with realized market behavior.

Key Features

• Analyzed AAPL historical data from 2015 to 2025.
• Computed daily log returns and identified volatility clustering.
• Built and compared ARCH(1) and GARCH(1,1) models.
• Forecasted 5-day rolling volatility and compared it with realized volatility.
• Visualized conditional volatility patterns and rolling volatility comparisons.

Methodology

1. Collected historical stock price data.
2. Processed data to calculate daily returns.
3. Fitted ARCH(1) and GARCH(1,1) models to estimate conditional volatility.
4. Compared model forecasts with realized volatility for evaluation.
5. Visualized volatility clustering and trends for interpretability.

Tools & Libraries

• Python for data analysis and modeling
• numpy, pandas for data manipulation
• yfinance for financial data retrieval
• matplotlib for visualizations
• arch for volatility modeling

Results

Historical Stock Prices

Returns Series and Volatility Clustering

ARCH(1) Model Conditional Volatility

GARCH(1,1) Model Conditional Volatility

Predicted vs Realized Volatility (5-day Rolling)

Future Improvements

• Implement additional GARCH-family models such as EGARCH and GJR-GARCH.
• Extend analysis to multiple assets and asset classes.
• Incorporate model selection criteria like AIC and BIC for optimization.