Comparative Analysis of European Option Pricing

Using the Binomial Tree Model & Black-Scholes Model

This project explores and compares two foundational models for pricing European options:

• Binomial Tree Model (step-based, discrete-time)
• Black-Scholes-Merton Model (closed-form, continuous-time)

It includes model implementation, sensitivity analysis, convergence checks, and real-world validation using AAPL options data via yfinance.

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Features

Implemented:

• Binomial Tree pricing (with variable step size)
• Black-Scholes pricing (with Greeks)
• Sensitivity analysis for:
  • Volatility
  • Time to maturity
  • Strike price
  • Risk-free interest rate
• Convergence analysis of Binomial Tree → Black-Scholes
• Real market comparison using AAPL options (via yfinance)
• Outlier detection with volatility/time diagnostics
• Visualizations of pricing and model errors

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Insights

• Black-Scholes is computationally fast and accurate under typical conditions.
• Binomial Tree converges to Black-Scholes as steps increase.
• Both models can fail in edge cases: ultra-low/high IV, short expiry, illiquid strikes.
• Real data shows outliers which reveal practical model limits.

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Requirements

pip install numpy pandas matplotlib scipy yfinance datetime

This project assumes European-style options. For American options, the Binomial Tree model can be extended, but the Black-Scholes model is not applicable without modification.