Options Volatility Modeling

Comparing Black-Scholes and SABR Models on Real Market Data

Project Overview

Quick Summary: See PROJECT_SUMMARY.pdf for a concise visual overview of the project, methods, and key findings.

This project explores the limitations of the Black-Scholes options pricing model and demonstrates how the SABR (Stochastic Alpha Beta Rho) stochastic volatility model better captures real market behavior.

Using real SPY (S&P 500 ETF) options data, we:

• Extract implied volatilities to visualize the volatility smile
• Implement and calibrate the SABR model to real market data
• Compare Black-Scholes vs SABR quantitatively using RMSE

Key Results

• Black-Scholes RMSE: 11.03 vol points
• SABR RMSE: 1.02 vol points
• SABR is 90.78% more accurate than Black-Scholes

SABR captures the volatility skew that Black-Scholes completely misses, demonstrating why stochastic volatility models are essential in practice.

Repository Structure

options-volatility-modeling/
├── notebooks/
│   └── main.ipynb              # Main analysis notebook
├── notes/    
│   └── concepts.md             # Key concepts and learning notes
├── data/                       # Raw options data (not tracked by git)
├── references/    
│   └── papers.md               # Papers and reading materials
└── requirements.txt            # Python dependencies

Models Covered

Black-Scholes Model

• Assumes constant volatility across all strikes
• Produces a flat implied volatility line
• Cannot capture the volatility smile/skew

SABR Model (Hagan et al. 2002)

• Stochastic volatility model
• Parameters: Alpha (vol level), Beta (fixed=0.5), Rho (skew), Nu (curvature)
• Calibrated to real market data using L-BFGS-B optimization
• Captures the volatility skew seen in real equity markets

Data

• Underlying: SPY (S&P 500 ETF)
• Expiry: April 17, 2026
• Data collected: March 18, 2026
• Source: Yahoo Finance via yfinance

Getting Started

Prerequisites

• Python 3.8 or higher
• Git

Installation

1. Clone the repository

git clone https://github.com/MayuriKawale/options-volatility-modeling.git
cd options-volatility-modeling

2. Create and activate a virtual environment

   Mac/Linux:

python -m venv venv
source venv/bin/activate

Windows (Command Prompt):

python -m venv venv
venv\Scripts\activate.bat

3. Install required packages

pip install numpy pandas scipy matplotlib yfinance jupyterlab
#pip install -r requirements.txt

4. Launch Jupyter Notebook

jupyter lab

5. Open notebooks/main.ipynb and run the cells in order

Learning Resources

See notes/concepts.md for detailed explanations of all concepts covered including Black-Scholes, implied volatility, the volatility smile, SABR model parameters, calibration, and model comparison.

See references/papers.md for all papers and resources used.

References

• Hagan, P., Kumar, D., Lesniewski, A., & Woodward, D. (2002). Managing Smile Risk. Wilmott Magazine, 84-108.
• Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.

Author

Mayuri Kawale

For Erdos Institute Quant Finance Boot Camp, Spring 2026