Volatility Modeling with ARCH/GARCH

GARCH(1,1) conditional volatility estimation on S&P 500 returns

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Overview

This project fits a GARCH(1,1) model to S&P 500 daily log-returns using MLE via the arch library, producing time-varying conditional volatility estimates.

It serves two purposes:

1. Standalone analysis — demonstrating that volatility is not constant and that ARCH-family models significantly improve on the constant-variance assumption
2. Foundation for the Integrated Risk App — the MLE estimation methodology and GARCH parameter interpretation developed here directly inform the garch_mle.py module in Integrated-Risk-App, which upgrades the FHS (Filtered Historical Simulation) VaR model from fixed parameters to MLE-estimated ones

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Connection to Integrated-Risk-App

The Integrated-Risk-App risk engine uses a GARCH(1,1) volatility filter to implement Filtered Historical Simulation (FHS) VaR — a methodology that standardises historical returns by their conditional volatility before computing quantiles.

The original implementation used hardcoded parameters (α=0.05, β=0.94), which is a documented limitation. The garch_mle.py module in that repo closes this gap by implementing MLE estimation consistent with what is demonstrated here.

Relationship between the two repos:

This repo (Volatility-modeling)	Integrated-Risk-App
Fits GARCH(1,1) via arch library MLE	Implements GARCH MLE via scipy.optimize (no extra dependency)
Single-asset (S&P 500)	Multi-asset portfolio
Estimation and diagnostics focus	Risk measure application (VaR, ES, backtesting)
Jupyter notebook	Production Python package

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Key Findings

Volatility clustering is strongly present The GARCH(1,1) model captures high persistence in S&P 500 return volatility, with α+β close to 1 — consistent with the well-documented behaviour of equity return series. This means volatility shocks dissipate slowly, and periods of elevated volatility tend to follow one another.

MLE-estimated parameters materially differ from common fixed defaults The arch library MLE estimates produce α and β values that are data-specific. Using fixed defaults (e.g. α=0.05, β=0.94, commonly seen in textbooks and RiskMetrics) can introduce meaningful misspecification, particularly for assets with atypical volatility dynamics.

Conditional volatility provides better VaR inputs than unconditional volatility Rescaling historical returns by their GARCH-estimated conditional volatility before taking quantiles (Filtered Historical Simulation) produces VaR estimates that adapt to the current volatility regime — more conservative when vol is elevated, less so in calm periods. This directly addresses the main weakness of plain historical simulation.

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Project Structure

volatility-modeling/
├── notebooks/
│   └── arch_garch_model.ipynb   # Main analysis notebook
├── outputs/
│   ├── log_returns_plot.png
│   ├── summary_table.png
│   └── estimated_volatility.png
├── test_imports.py
├── requirements.txt
└── README.md

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Sample Outputs

Log Returns — S&P 500

GARCH(1,1) Model Summary

Estimated Conditional Volatility

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How to Run

git clone https://github.com/sensor-aae/Volatility-modeling.git
cd Volatility-modeling
 
python3 -m venv .venv
source .venv/bin/activate
 
pip install -r requirements.txt
jupyter notebook notebooks/arch_garch_model.ipynb

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Tech Stack

Library	Purpose
arch	GARCH model estimation (MLE)
yfinance	S&P 500 price data
pandas / numpy	Data handling
matplotlib	Visualisation

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Related Projects

• Integrated-Risk-App — Full market & credit risk engine (VaR, ES, backtesting, stress testing) that uses GARCH-estimated volatility for FHS VaR

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Author

Amanda Achiangia BSc Applied Mathematics (Financial Mathematics), York University LinkedIn | GitHub