Numerical Approximation of the Black-Scholes Equations: A Practical Experience Open Access

Abstract

Abstract
Numerical Approximation of the Black-Scholes Equations: A Practical Experience
Black and Scholes equations for pricing of derivatives are an interesting and up-to-date topic of research, where both backgrounds in math and finance are fundamentals. In this work we aim at experiencing the mathematical approach and the numerical approximation of this differential problem. We wil assume these equations to be a reliable model and will work on their numerical approximation with a mixed finite elements(FE)/finite differences (FD) approach. More precisely, we wil use the mathematically well-established Finite Element Method (FEM) for the underlying price dependence, while time dependence wil be discretized with the Finite Difference Method (FDM). The ultimate purpose of the work is to earn sensitivity when using numerical tools in financial mathematics, with particular attention to the critical analysis of the results. This includes an extensive error analysis and appropriate visualization of the results. This wil allow us to comment on the practicality of numerical mathematics used to solve the Black-Scholes equation.

Table of Contents

Table of Contents
1) Introduction .................................................................................................................. 1

1.1 Purpose of the Present Work .......................................................................................... 1
1.2 Theoretical and Practical Application ............................................................................. 2
1.2.1 Software ................................................................................................................... 4
1.3 Error Analysis .................................................................................................................. 4
1.4 Data Visualization ........................................................................................................... 6
2) Understanding Black-Scholes ......................................................................................... 9
2.1 Basic Terminology ........................................................................................................... 9
2.2 The 1D and 2D Black-Scholes Equation ......................................................................... 11
2.3 Building the Model: The Lognormal Density ................................................................. 11
2.4 Building the Model: The Equation ................................................................................ 13
2.5 Building the Model: The Boundary and "Final" Conditions .......................................... 16
3) Understanding Black-Scholes ....................................................................................... 21
3.1 Overview ....................................................................................................................... 21
3.2 Strengths of FEM ........................................................................................................... 22
3.3 Weak Form Solution ...................................................................................................... 23
3.4 Discretization ................................................................................................................ 24
3.4.1 1D Discretization .................................................................................................... 25
4) One Dimensional Experimentation .............................................................................. 29
4.1 Understanding FEM1D .................................................................................................. 29
4.2 Error Analysis ................................................................................................................ 33
4.3 Other Test Cases ........................................................................................................... 37
4.3.1 Analyzing the Results ............................................................................................. 38

5) One Dimensional Experimentation .............................................................................. 43

5.1 Coding ........................................................................................................................... 43
5.2 1D Augmentation .......................................................................................................... 44
5.3 2D Analysis .................................................................................................................... 46
6) Conclusion ................................................................................................................... 51
6.1 Findings ......................................................................................................................... 51
6.2 Going Forward .............................................................................................................. 52

Table of Figures
1.1 Discretization Error Schematic ........................................................................................................ 5
1.2 Discretization Error Break Down ..................................................................................................... 6
2.1 Pay-off Diagram ........................................................................................................................... 10
2.2 Well Posed Conditions .................................................................................................................. 17
3.1 Vh Versus V Space ......................................................................................................................... 22
4.1 Fem1D Screen 1 ........................................................................................................................... 30
4.2 FEM1D Screen 2 ........................................................................................................................... 31
4.3 Numeric Solution, View 1 ............................................................................................................. 32
4.4 Numeric Solution, View 2 ............................................................................................................. 33
4.5 Analytic Solution .......................................................................................................................... 34
4.6 Difference of Analytic and Numeric Solutions ............................................................................... 35
4.7 Error Table ................................................................................................................................... 38
4.8 Domain of Interest ....................................................................................................................... 39
4.9 Best Case Error ............................................................................................................................. 42
5.1 FreeFem++ 1D Error: Value ........................................................................................................... 44
5.2 FreeFem++ 1D Error: Time Value .................................................................................................. 45
5.3 Coarse Mesh 1D, Beginning .......................................................................................................... 46
5.4 10x10 Mesh .................................................................................................................................. 47
5.5 Coarse Mesh 1D, End .................................................................................................................... 48
5.6 60x60 2D Image End, Value .......................................................................................................... 48
5.7 60x60 2D Image End, Time Value .................................................................................................. 49
5.8 20x20 Adapt Mesh, 2D Image End ................................................................................................ 49

About this Honors Thesis

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School	Emory College
Department	Mathematics and Computer Science
Degree	BBA
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	Finite Element Method Finite Difference Method Numerical Financial Mathematics Black-Scholes
Committee Chair / Thesis Advisor	Veneziani, Alessandro, Emory University
Committee Members	Green, Clifton, Emory University Hamilton, Emily, Emory University

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