🧠 Quantitative Volatility Modeling Using ARCH, GARCH & EWMA This repository contains a detailed implementation of volatility forecasting techniques used in quantitative finance. The models used are:

ARCH (Autoregressive Conditional Heteroskedasticity)

GARCH (Generalized ARCH)

EWMA (Exponentially Weighted Moving Average)

You’ll also find real vs. predicted volatility comparison, full Python code (in Jupyter Notebook), and step-by-step implementation.

📁 Files Included File Name Description volatilitymodeling.ipynb Main Jupyter Notebook implementing ARCH, GARCH, and EWMA models on JPMorgan stock data

📌 Table of Contents

• Project Objective
• Data Source
• ARCH Model
• GARCH Model
• EWMA Model
• Realized vs Predicted Volatility
• Libraries Used
• How to Run
• Conclusion

📊 Project Objective

To model and forecast the volatility of JPMorgan Chase (JPM) stock returns using time-series econometric techniques, and to compare predicted volatility with actual realized market volatility.

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🔹 Data Source

Two data sources are used for price information:

• Yahoo Finance via the yfinance Python API

Date Range: 2015-01-01 to 2025-01-01 Stock Ticker: JPM (JPMorgan Chase)

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🔄 ARCH Model

✅ What is ARCH?

The ARCH (1) model assumes that today's volatility depends on the squared return from the previous day.

⚖️ Formula:

$\sigma^2_t = \omega + \alpha_1 r_{t-1}^2$

Where:

• $\omega$ is the long-term base level of variance
• $\alpha_1$ is the coefficient on yesterday's squared return

⚙️ Steps:

1. Calculate daily returns
2. Fit ARCH(1) using arch_model from the arch package
3. Forecast 5-day ahead variance
4. Convert to volatility (sqrt of variance)
5. Compare with 5-day realized volatility

📊 Example Result:

Predicted Volatility (5-day): ~1.56%
Realized Volatility (5-day): ~1.69%

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🔄 GARCH Model

✅ What is GARCH?

The GARCH(1,1) model extends ARCH by including past variance in addition to past squared returns. It captures volatility clustering better than ARCH.

⚖️ Formula:

$\sigma^2_t = \omega + \alpha_1 r_{t-1}^2 + \beta_1 \sigma^2_{t-1}$

Where:

• $\omega$: base variance
• $\alpha_1$: impact of last return shock
• $\beta_1$: impact of past variance

⚙️ Steps:

1. Calculate daily returns
2. Fit GARCH(1,1) using arch_model
3. Forecast 5-day volatility horizon
4. Compare with actual volatility of that period

📊 Example Result:

Predicted Volatility (5-day): ~1.62%
Realized Volatility (5-day): ~1.69%

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🔄 EWMA Model

✅ What is EWMA?

EWMA estimates volatility by assigning exponentially decreasing weights to past squared returns. It's widely used due to its simplicity and responsiveness to recent data.

⚖️ Formula:

$\sigma^2_t = \lambda \cdot \sigma^2_{t-1} + (1 - \lambda) \cdot r^2_{t-1}$

Where:

• $\lambda$: decay factor (commonly 0.94 for daily data)
• $r^2_{t-1}$: squared return from previous day
• $\sigma^2_{t-1}$: previous day’s variance

⚙️ Steps:

1. Set lambda value (e.g. 0.94)
2. Calculate variance recursively from return series
3. Take square root to get volatility
4. Forecast is only for 1 day ahead

📊 Example Result:

Predicted Volatility (2025-01-02): 0.14
Realized Return (absolute): 0.12

Note: Since it's a 1-day forecast, we compare to absolute return instead of standard deviation.

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👀 Realized vs Predicted Volatility

• Predicted Volatility: Output of ARCH/GARCH/EWMA model

• Realized Volatility:

  • For ARCH/GARCH: std deviation of next 5 daily returns (annualized)
  • For EWMA: abs(return on next day)

This comparison helps evaluate model accuracy in practical terms.

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📚 Libraries Used

• pandas
• numpy
• yfinance
• arch
• matplotlib
• requests

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🔧 How to Run

1. Install required libraries:

pip install yfinance arch pandas numpy matplotlib

2. Open volatilitymodeling.ipynb in Jupyter or VS Code
3. Run all cells

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🚀 Conclusion

This project demonstrates how to:

• Apply volatility models on real-world stock data
• Compare predicted vs actual market movements
• Understand the dynamics of conditional heteroskedasticity in finance

Useful for:

• Quantitative finance enthusiasts
• Students of econometrics
• Risk analysts and traders

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Feel free to fork, star, or contribute!