A Computational Analysis of the Black-Scholes Equation Open Access

Abstract

This paper explores the most decorated option pricing model in recent history of the financial industry: the Black-Scholes Equation. We will first study the framework of the Black-Scholes Equation in detail by introducing its object of evaluation, distinguished assumptions, and deduction of the Black-Scholes partial differential equation. Although Black and Scholes(1973) has proposed the famous Black-Scholes formula to evaluate the European option, the PDE form has proposed struggle in finding the exact analytical solution, thus giving rise to the enormous interest in the numerical approach. In the second part of this paper, we will introduce three primary numerical and simulation methods including Finite Element Method(FEM), Finite Difference Method(FDM) and Monte Carlo Simulation(MC). We will discuss extensively about each method and present its advantages and shortcomings. In general, FEM are better founded mathematically on extensive theoretical analysis. Nevertheless, FDM and MC can have some advantages, in particular in terms of the easiness of implementation. We will consider some of these aspects in the present paper.

Table of Contents

1. Introduction to the Black-Scholes Pricing Model

1.1 Preliminaries

1.2 The Black-Scholes Pricing

1.2.1 Brownian Motion

1.2.2 Ito's Lemma

1.2.3 Assumptions of the Black-Scholes Equation

1.3 Black-Scholes Equation

1.4 Black-Scholes Formula

2. Finite Difference Method

2.1 Basics of Finite Difference Method

2.2 Finite Difference Method in the Black-Scholes Model

2.3 Explicit Euler Method

2.4 Alternative FD Schemes

3. Finite Element Method

3.1 Basics of the Finite Element Method

3.2 Finite Element Method in the Black-Scholes Model

3.3 Advantages of FEM and Comparison with FDM

4. Numerical Results

4.1 A Test Case

4.1.1 Domain Truncation

4.1.2 Domain Transformation

4.2 Numerical Results and Error Analysis

5. Monte Carlo Methods in Option Pricing

5.1 Basics of Monte Carlo Simulation

5.2 An illustrative example of Monte Carlo method on European option

5.3 Monte Carlo Simulation Process and Methodology

5.4 Simulations and Results

5.5 Advantages and Shortcoming of Monte Carlo Method in Option Pricing

6. Conclusion

About this Honors Thesis

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School	Emory College
Department	Mathematics and Computer Science
Degree	BS
Submission	Honors Thesis
Language	English
Research Field	Mathematics Economics, Finance
Keyword	Finite Element Method Finite Difference Method Monte Carlo Method The Black-Scholes Equation Domain Transformation Domain Truncation
Committee Chair / Thesis Advisor	Veneziani, Alessandro, Emory University
Committee Members	Nagy, James, Emory University Chen, Kaiji, Emory University

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