Numerical Approximation of the Black-Scholes Equations: A Practical Experience Pubblico
Connor, Dylan Michael (2012)
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
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