Numerical singular perturbation approaches based on spline approximation methods for solving

185 Pages·2011·1.75 MB·English

Numerical singular perturbation approaches based
on spline approximation methods for solving
problems in computational finance
Mohmed Hassan Mohmed Khabir
A Thesis submitted in partial fulfillment of the requirements for the degree of Doctor
of Philosophy in the Department of Mathematics and Applied Mathematics at the
Faculty of Natural Sciences, University of the Western Cape
Supervisor: Prof. Kailash C. Patidar
May 2011
KEYWORDS
Computational Finance
Options Pricing
Black-Scholes Equation
Standard Options
Nonstandard Options
Free Boundary Problems
Spline Approximation Theory
Singular Perturbation Techniques
Numerical Methods
Convergence Analysis
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ABSTRACT
Numerical singular perturbation approaches based on spline approximation
methods for solving problems in computational finance
by
Mohmed Hassan Mohmed Khabir
PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of
Natural Sciences, University of the Western Cape.
Options are a special type of derivative securities because their values are derived from
the value of some underlying security. Most options can be grouped into either of
the two categories: European options which can be exercised only on the expiration
date, and American options which can be exercised on or before the expiration date.
American options are much harder to deal with than European ones. The reason being
the optimal exercise policy of these options which led to free boundary problems. Ever
since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659],
the differential equation approach in pricing options has attracted many researchers.
Recently, numerical singular perturbation techniques have been used extensively for
solving many differential equation models of sciences and engineering. In this thesis,
we explore some of those methods which are based on spline approximations to solve
the option pricing problems. We show a systematic construction and analysis of these
methods to solve some European option problems and then extend the approach to
solve problems of pricing American options as well as some exotic options. Proposed
methods are analyzed for stability and convergence. Thorough numerical results are
presented and compared with those seen in the literature.
May 2011.
ii
DECLARATION
I declare that Numerical singular perturbation approaches based on spline approxima-
tion methods for solving problems in computational finance is my own work, that it
has not been submitted before for any degree or examination at any other university,
and that all sources I have used or quoted have been indicated and acknowledged by
complete references.
Mohmed Hassan Mohmed Khabir May 2011
Signed ..........................................
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ACKNOWLEDGEMENT
Prof. Kailash C. Patidar encouraged me to take up an academic career and gave me
the opportunity to study at the University of the Western Cape. During my PhD
studies, he provided me with continuous guidance and advice on the directions this
thesis should pursue. I very much appreciated his support and his great ideas which I
found to be invaluable.
I would like to thank my university: Sudan University of Science and Technology
for providing financial support for my PhD programme.
I acknowledge the senior students and staff of the Department of Mathematics and
Applied Mathematics at the University of the Western Cape.
Special thanks to my friends Abdelmgid, Hasim, Mushal, Ammar, Gasim, Eihab, Za-
kariya and Sara. They shared every bit of pressure, anxiety, and joy that I experienced.
I also take this opportunity to thank my extended family, my sisters (Geehan, Eman,
Wigdan)andtheirhusbands(Kamal,Magdi,Amgad),andmyyoungerbrothers(Aasim
and Yaseen) for their all time support.
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DEDICATION
I dedicate this work to two of the finest people who ever graced this world and who
naturally brought me this far, my parents Hassan Khabir and Hager Mubarak.
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Contents
Keywords i
Abstract ii
Declaration iii
Aknowledgement iv
Dedication v
List of Tables x
List of Figures xii
List of Publications xiii
1 General introduction 1
1.1 Option pricing: A brief overview . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The Black-Scholes Equation . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Literature review on numerical methods for pricing standard options . . 17
1.4 Literature review on numerical methods for pricing nonstandard options 26
1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Splines approximations: Basic theory and applications to solve differ-
ential equations 30
vi
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Interpolation by different splin es . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Basic properties of splines . .  . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Applications of splines approximation methods . . . . . . . . . . . . . . 52
3 Comparison of some numerical methods for option pricing problems 54
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Solving option pricing problem by method of lines . . . . . . . . . . . . 58
3.4 Solving option pricing problem by cubic splines . . . . . . . . . . . . . 62
3.5 Numerical simulations and results . . . . . . . . . . . . . . . . . . . . . 70
3.5.1 Numerical results using method of lines . . . . . . . . . . . . . . 70
3.5.2 Numerical results using cubic spline . . . . . . . . . . . . . . . . 74
3.6 Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 B-spline approximation method for pricing European options 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Construction of the numerical method . . . . . . . . . . . . . . . . . . 81
4.4 Analysis of the numerical method . . . . . . . . . . . . . . . . . . . . . 87
4.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.6 Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 B-spline approximation method for pricing American options 99
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3 Computation of the American put options and analysis of the numerical
method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3.1 Construction of the numerical method . . . . . . . . . . . . . . 106
5.3.2 Convergence analysis of the numerical method . . . . . . . . . . 112
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5.3.3 Numerical computation of the American put option . . . . . . . 122
5.4 Numerical results . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . 125
5.5 Summary and discussions . .  . . . . . . . . . . . . . . . . . . . . . . . 127
6 B-spline approximation method for pricing the barrier options 128
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.3 Construction of the numerical method . . . . . . . . . . . . . . . . . . 135
6.4 Analysis of the numerical method . . . . . . . . . . . . . . . . . . . . . 140
6.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.6 Summary and discussions . . . . . . . . . . . . . . . . . . . . . . . . . 150
7 Concluding remarks and scope for future research 151
Bibliography 154
viii
List of Tables
1.1.1List of some notations used in the thesis . . . . . . . . . . . . . . . . . 7
2.2.1Values of the quadratic B-splines φ (x) and its derivatives with knots at
i
different points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.2Values of the cubic B-splines φ (x) and its derivatives with knots at
i
different points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.3Values of the quintic B-splines φ (x) and its derivatives with knots at
i
different points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.4Values of the sextic B-splines φ (x) and its derivatives with knots at
i
different points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.5Values of the septic B-splines φ (x) and its derivatives with knots at
i
different points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.1Comparison between the exact solution, Quasi-RBF solution [67] and
solution obtained by method of lines along with MATLAB solver ode45
for a European put option for two different space step-sizes . . . . . . . 72
3.5.2Comparison between the exact solution, B-spline solution and solution
obtained by method of lines along with MATLAB solver ode45 for a
European put option for two different space step-sizes . . . . . . . . . . 73
3.5.3Comparisonbetweentheexactsolutionandsolutionobtainedbymethod
of lines along with different MATLAB solvers for the European put option. 73
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