Bachelors Thesis: Stochastic Differential Equations in Finance

Project Overview

This repository contains the work of my Bachelor's thesis, which explores the use of stochastic differential equations (SDEs) in financial mathematics. The central focus of this study is the derivation and application of the Black-Scholes formula for option pricing, along with its theoretical foundations and practical implementation.

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Topics Covered

• Algebraic Structures and Probability Spaces
• Conditional Probability, Expectation, and Convergence
• Stochastic Processes, Martingales, and Brownian Motion
• Itô Integral and Stochastic Differential Equations
• Derivation and Mathematical Justification of the Black-Scholes Formula
• Solution via Heat Equation and Fourier Transform
• Computational Implementation and Market Data Application
• Volatility and Interest Rate Estimation from Real Data
• Computational Implementation and Market Data Application
• Comparison of Theoretical Prices vs. Real Option Prices
• Translation of the Thesis using Physics Informed Neural Networks (PINN's) (in progress)

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Methodology

1. Mathematical Foundation:
   Developed a rigorous understanding of stochastic processes and Itô calculus to model asset price dynamics as Geometric Brownian Motion (GBM).

2. Black-Scholes Derivation:
   Derived the Black-Scholes Partial Differential Equation (PDE) for European call and put options.

3. Analytical Solution:
   Transformed the Black-Scholes PDE into the heat equation and solved it using the Fourier transform.

4. Empirical Application:
   Used real market data from Yahoo Finance to price options and compare theoretical vs. actual outcomes.

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Real-World Case Study

A case study was conducted using historical stock prices from Yahoo Finance to validate the Black-Scholes model. Results were analyzed to evaluate the model's performance in real-world financial markets.

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Ongoing Work

Currently, I am extending this work using PINN to translate and enhance the original thesis.

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Files

• Original_Thesis — My original thesis written in Spanish and the source code (no AI)
• PINN-BlackScholes — Contains the code of using PINN in solving the Black Scholes partial differential equation
• solution.pdf- Contains the english translation of the solution of the PDE

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Future Work

• Full translation of the thesis into English
• Implementation of RL agents for option pricing
• Comparison between analytical Black-Scholes results and RL-based results

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Contact

Ricardo Alonso Manjarrez Retes
Graduate Student, Computer Science
Email: ramr99@nmsu.edu

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License

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