Coursework – Binomial Trees & Black–Scholes Model

This repository contains the code and reports for a two-part coursework in Stochastic Processes – The Fundamentals (Vrije Universiteit Amsterdam). The project connects empirical S&P 500 data, return modelling, and option pricing via binomial trees and the Black–Scholes framework.

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Repository structure

• spf-assignment-1-script.ipynb
  Modelling S&P 500 returns and index dynamics, and pricing simple derivatives.
• spf-assignment-2-script.ipynb
  Numerical option pricing and dynamic hedging under Black–Scholes.
• SP500.csv
  Historical monthly S&P 500 index levels used throughout.
• spf-1-report.pdf, spf-2-report.pdf
  Full write-ups of the theory, methodology, and results.

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Part I – Return modelling & index dynamics

Using historical monthly S&P 500 data:

• Estimate mean and volatility of simple net returns and test whether the mean differs from zero using a t-test. :contentReference[oaicite:0]{index=0}
• Study estimation risk: how many years of data are needed for a tight confidence interval on the mean, and how parameter uncertainty affects optimal risky investment for a risk-averse investor.
• Compare two modelling choices:
  • Linear model for simple net returns
  • Geometric Brownian Motion (GBM) for index levels
• Use GBM to:
  • Forecast the expected index level over multi-year horizons
  • Simulate index paths and analyse the (lognormal) distribution of future levels
  • Price digital options (puts/calls) and contrast real-world vs risk-neutral probabilities, including put–call parity.

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Part II – Option pricing & dynamic hedging

Working with a 3-month European option on the S&P 500:

• Numerical pricing methods

  • Binomial trees calibrated to empirical variance and expected return; step-size rescaling and convergence to Black–Scholes.
  • Monte Carlo pricing under GBM using Euler discretisation on prices and on log-prices; analysis of sampling error vs discretisation bias.
  • Crank–Nicolson finite differences to solve the Black–Scholes PDE and benchmark accuracy against the closed-form formula.
  • Extension to a gap call option, highlighting how payoff structure affects value.

• Dynamic delta hedging

  • Simulate GBM paths and replicate a short call with discrete rebalancing (monthly, weekly, daily, and intraday).
  • Study how hedging frequency affects P&L bias, variance, and tail risk, and how mis-specifying the drift impacts hedging performance.
  • Analyse gamma exposure: relation between average gamma along a path and hedging P&L, and discuss ways to mitigate gamma risk (higher frequency, delta–gamma hedging, scaling position).

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How to run

1. Create a Python 3 environment with:
   • numpy, pandas, scipy, matplotlib, jupyter
2. Open the notebooks:
   • spf-assignment-1-script.ipynb
   • spf-assignment-2-script.ipynb
3. Run all cells to reproduce the simulations, figures, and numbers reported in the PDFs.

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This project is meant as a compact bridge from historical return estimation to risk-neutral pricing and dynamic hedging, showing how the binomial model, Monte Carlo, PDE methods, and Black–Scholes all fit into one coherent framework.