Finite element solver for European Put Black-Scholes Equation

Numerical approximation of PDEs MATH-451 project, Spring 2025, EPFL

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Table of Contents

1. Motivation
2. Installation
3. Code structure
4. Report
5. License
6. Authors

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Motivation

This project aims to numerically solve the Black-Scholes PDE for European put options using finite element methods. By reformulating the equation in its variational form and applying time-stepping schemes like backward Euler and Crank-Nicolson, the project investigates stability, accuracy, and convergence. Both a constructed solution and the real model are used to validate the method. The work combines theory with implementation to explore a key problem in computational finance.

This project was assigned as part of the Numerical approximation of PDEs MATH-451 project at EPFL. The instructions of the exercises are given in europeanput.pdf.

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Installation

Ensure you have Python 3.10+ and a virtual environment activated. Next clone the repository onto your local machine:

>>> git clone https://github.com/MattiaBarbiere/FEM_for_Black_Scholes.git

To install the requirements, move into the code/ folder and run

>>> pip install -r requirements.txt

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Code structure

All the code is available in the code/ folder.

• fem_solver.py: Code for the 1D finite element solver.

• black_scholes_pde.py: Classes for analytical soluitons of the Black-Scholes Equation.

• main.py: Main file for solving and plotting the PDEs.

• quad.py: Class for the quadrature points needed for numerical integration.

• tests.py: Simple tests to check correct implementation of the P1 and P2 basis functions.

For more details, see code/README.md and docstrings in each file.

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Report

For the full report of the project, including some theoretical background and experimental analysis, visit Final_report_Mattia_Barbiere.

Result

For a visual example, below you find the analytical solution (blue) versus interpolated finite element solution (dashed red) for various methods at time $t = 5$. The PDE was solved with $h = 0.5$ and $\Delta t = 0.5$ using the Gauss-Legendre quadrature with 10 points. First and second row correspond to P1 and P2 finite elements respectively. First and second column correspond to backwards Euler and Crank-Nicolson’s method respectively. In legend the maximum errors is presented for all four plots.

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License

This project is licensed under the MIT License. See the LICENSE file for more details.

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Author

Developed by Mattia Barbiere as part of the Numerical approximation of PDEs MATH-451 course during the spring 2025 semester at EPFL.
GitHub: @MattiaBarbiere

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