black-scholes-model

Derivation and implementation of the Black-Scholes model using Stochastic Calculus and PDEs.

Black-Scholes Model: Probabilistic & PDE Approach

Overview

This project explores the mathematical foundations of the Black-Scholes option pricing model, establishing the duality between two major approaches in quantitative finance:

1. Stochastic Calculus: Using Martingales, Itô's Lemma, and Girsanov's Theorem to define the risk-neutral measure.
2. Partial Differential Equations (PDEs): Deriving the pricing equation via dynamic hedging and replication arguments.

The repository includes a comprehensive theoretical report and a Python implementation for numerical simulations.

Key Features

• Theoretical Proofs: Rigorous derivation of the Black-Scholes PDE and the risk-neutral valuation formula.
• Monte-Carlo Simulation: Generation of Geometric Brownian Motion (GBM) paths to visualize asset dynamics.
• Convergence Analysis: Numerical verification of the Law of Large Numbers applied to option pricing.
• The Greeks: Sensitivity analysis (Delta, Gamma, Theta, Vega, Rho) implemented from scratch.

Visualizations

Brownian Motion Simulation

Project Structure

├── report/
│   └── Black_Scholes_Report.pdf  # Full mathematical report (LaTeX)
├── src/
│   ├── brownian_simulation.py    # Standard Geometric Brownian Motion paths
│   ├── volatility_simulation.py  # Analysis of Time-Dependent Volatility
│   └── rate_simulation.py        # Analysis of Interest Rate impact
├── requirements.txt
└── README.md
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## 👥 Authors
* **Eliott Oster** * **Bernard Tao**
* **Périne Gabarret**

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*Developed as part of the GMM Department curriculum at INSA Toulouse.*