Black-Scholes vs. Monte Carlo (European Call Options)

This repository contains my MA 398 Senior Capstone (Fall 2025), which compares the Black–Scholes closed-form solution to a Monte Carlo (GBM) estimator for European call options under consistent assumptions, and demonstrates how Monte Carlo prices converge toward the Black–Scholes price as the number of simulated paths increases.

Contents

• SeniorCapstone_MonteCarlo_BlackScholes_Comparison.pdf
  Full paper: theory, assumptions, qualitative comparison, convergence discussion, and proof that GBM implies log-normal stock prices.
• BS_MS_Calculator.ipynb
  Reproducible notebook that pulls market inputs, computes implied volatility, and generates Black–Scholes vs. Monte Carlo prices (the pipeline used to produce the comparison table in the paper).

Reproduce the BS vs. MC Comparison Table

1. Clone the repo and install dependencies:

   python -m pip install numpy pandas scipy yfinance jupyter

2. Open the Notebook:

   jupyter notebook BS_MS_Calculator.ipynb

3. Run all cells. The notebook:

   • fetches the latest underlying price and options chain data via yfinance
   • estimates dividend yield
   • solves for implied volatility using the market option price
   • computes the Black–Scholes call price
   • runs a risk-neutral Monte Carlo GBM simulation and reports price + standard error

Notes / Assumptions

• Pricing is done under a risk-neutral measure with drift (𝑟 − 𝑞)
• The Monte Carlo estimator uses GBM with discretized steps (default ~252 trading days/year).
• Results may vary slightly run-to-run depending on the random seed, number of paths, and live market data at execution time.