-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathnanahtesisongoing.tex
2736 lines (2470 loc) · 107 KB
/
nanahtesisongoing.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
%\documentclass[a4paper,10pt]{article}
\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{longtable}
\usepackage{wrapfig}
%\usepackage{color}
%\usepackage{amsthm}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{listings}
\usepackage[usenames,dvipsnames]{color}
\usepackage{fancyhdr}
\usepackage{fullpage}
%\usepackage[top=tlength, bottom=blength, left=llength, right=rlength]{geometry}
%\usepackage{geometry}
%\usepackage[a4paper]{geometry}
\definecolor{MyDarkGreen}{rgb}{0.0,0.4,0.0}
%\usepackage[pdftex]{graphicx}
%\usepackage{mathtools}
\usepackage{listings}
\lstset{language=C++}
%===========================================================================================
% Numbering Equations
%\numberwithin{equation}{section} %sets equation numbers <chapter>.<section>.<index>
\numberwithin{equation}{subsection} %sets equation numbers <chapter>.<section>.<subsection>.<index>
%\numberwithin{equation}{subsubsection} %sets equation numbers <chapter>.<section>.<subsection>.<subsubsection>.<index>
%===================================================================================================================
% Basic Commands for Page settings,Chapters, Appendices, Sections, etc..
%====================================================================================================================
%\setlength{\oddsidemargin}{.5in} \setlength{\topmargin}{0in}
%\setlength{\headheight}{.2in} \setlength{\headsep}{.2in}
%\setlength{\textwidth = 6.0in} \setlength{\textheight = 8.3in}
\setlength{\oddsidemargin}{0.6in} \setlength{\topmargin}{-0.3in} %adjust side margins and top margins
\setlength{\headheight}{.2in} \setlength{\headsep}{.2in}
\setlength{\textwidth = 6.0in} \setlength{\textheight = 8.3in}
\def\thebiblio#1{\list
{[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth
\advance\leftmargin\labelsep
\usecounter{enumi}}
\def\newblock{\hskip .11em plus .33em minus .07em}
\sloppy\clubpenalty4000\widowpenalty4000
\sfcode`\.=1000\relax}
\let\endthebiblio=\endlist
\newcommand{\sect}[1]{% Basic settings for general chapters
\cleardoublepage
\clearpage
\newpage
\begin{center}
\addtocounter{section} {1}
\setcounter{subsection} {0}
\section* {\normalsize \bf{CHAPTER \thesection \\ #1}}
\addcontentsline{toc}{section}{CHAPTER
\protect\numberline{\thesection : } #1\dotfill}
\end{center}
\thispagestyle{myheadings} }
\newcommand{\appen}[1]{% Basic settings for appendix chapters
\cleardoublepage
\clearpage
\newpage
\begin{center}
\addtocounter{section} {1}
\renewcommand{\thesection}{\Alph{section}}
\setcounter{subsection} {0}
\setcounter{table}{0}
\section* {\normalsize \bf{APPENDIX \thesection \\ #1}}
\addcontentsline{toc}{section}{APPENDIX
\protect\numberline{\thesection : } #1\dotfill}
\end{center}
\renewcommand{\thesubsection}{\Alph{section}.\arabic{subsection}}
\renewcommand{\thesubsubsection}{\Alph{section}.\arabic{subsection}.\arabic{subsubsection}}
\renewcommand{\theequation}{\Alph{section}.\arabic{equation}}
\renewcommand{\thetable}{\Alph{section}.\arabic{table}}
\thispagestyle{myheadings} }
%===================================================================================================================
% Basic Commands for Theorem, Lemma,Proposition,etc.....
\renewcommand{\baselinestretch}{2}
\renewcommand{\arraystretch}{.5}
\newcommand{\qed}{\hfill$\Box$}
\newtheorem{fact}{Theorem}[section]
\newtheorem{claim}{Claim}
\newtheorem{theorem}[fact]{Theorem}
\newtheorem{word}[fact]{Definition}
\newtheorem{prop}[fact]{Proposition}
\newtheorem{ob}[fact]{Observation}
\newtheorem{Corollary}[fact]{Corollary}
\newtheorem{corollary}[fact]{Corollary}
\newtheorem{lemma}[fact]{Lemma}
\newtheorem{Guess}[fact]{Conjecture}
\newtheorem{conj}[fact]{Conjecture}
\def\theotheorem{A\arabic{theorem}}
\newtheorem{mydef}{Definition}
%\newtheorem{theorem}{Theorem}[section]
%\newtheorem{lemma}[theorem]{Lemma}
%\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}[thm]
%\newtheorem{corollary}[theorem]{Corollary}
%\newtheorem{cor}[theorem]{Corollary}
\newenvironment{proof}[1][Proof]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{definition}[1][Definition]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{example}[1][Example]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{remark}[1][Remark]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
%================================================================================
%%%%%%%%%%%commands for problem
%================================================================================
\makeatletter
\newenvironment{problem}{\@startsection
{section}
{1}
{-.2em}
{-3.5ex plus -1ex minus -.2ex}
{2.3ex plus .2ex}
{\pagebreak[3]%forces pagebreak when space is small; use \eject for better results
\large\bf\noindent{Problem }
}
}
{%\vspace{1ex}\begin{center} \rule{0.3\linewidth}{.3pt}\end{center}}
\begin{center}\large\bf \ldots\ldots\ldots\end{center}}
\makeatother
%
%Fancy-header package to modify header/page numbering
\pagestyle{fancy}
%\addtolength{\headwidth}{\marginparsep} %these change header-rule width
%\addtolength{\headwidth}{\marginparwidth}
\lhead{Problem \thesection} \chead{} \rhead{\thepage}
\lfoot{\small\scshape course name} \cfoot{} \rfoot{\footnotesize
PS\#}
\renewcommand{\headrulewidth}{0.3pt}
\renewcommand{\footrulewidth}{.3pt}
%\setlength\voffset{-0.25in} \setlength\textheight{648pt} \maketitle
%\maketitle
%\thispagestyle{empty}
%===========================================================================
% commands for displaying codes in appendix
%============================================================================
\lstloadlanguages{Matlab}%
\lstset{language=Matlab, % Use MATLAB
frame=single, % Single frame around code
basicstyle=\small\ttfamily, % Use small true type font
keywordstyle=[1]\color{Blue}\bf, % MATLAB functions bold and blue
keywordstyle=[2]\color{Purple}, % MATLAB function arguments purple
keywordstyle=[3]\color{Blue}\underbar, % User functions underlined and blue
identifierstyle=, % Nothing special about identifiers
% Comments small dark green courier
commentstyle=\usefont{T1}{pcr}{m}{sl}\color{MyDarkGreen}\small,
stringstyle=\color{Purple}, % Strings are purple
showstringspaces=false, % Don't put marks in string spaces
tabsize=5, % 5 spaces per tab
%
%%% Put standard MATLAB functions not included in the default
%%% language here
morekeywords={xlim,ylim,var,alpha,factorial,poissrnd,normpdf,normcdf},
%
%%% Put MATLAB function parameters here
morekeywords=[2]{on, off, interp},
%
%%% Put user defined functions here
morekeywords=[3]{FindESS, homework_example},
%
morecomment=[l][\color{Blue}]{...}, % Line continuation (...) like blue comment
numbers=left, % Line numbers on left
firstnumber=1, % Line numbers start with line 1
numberstyle=\tiny\color{Blue}, % Line numbers are blue
stepnumber=5 % Line numbers go in steps of 5
}
% Includes a MATLAB script.
% The first parameter is the label, which also is the name of the script
% without the .m.
% The second parameter is the optional caption.
\newcommand{\matlabscript}[2]
{\begin{itemize}\item[]\lstinputlisting[caption=#2,label=#1]{#1.m}\end{itemize}}
\begin{document}
\pagestyle{empty}
\pagenumbering{roman}
%================================================ Approval Page ===================================================
\newpage
%\nocounter
\pagestyle{plain}
\thispagestyle{empty}% takes out page numbering
\begin{center}
{\bf APPROVAL} \\ [.05in]
{\bf This is to certify that the Graduate Committee of }\\
Nana Akwasi Abayie Boateng \\ [-.1in] met on the \\ [-.1in] 14th \
day of \ June, 2012.
\\ [.25in]
\end{center}
\baselineskip=20 pt
The committee read and examined his/her thesis,
supervised his/her defense of it in an oral examination, and decided
to recommend that his/her study should be submitted to the Graduate
Council, in partial fulfillment of the requirements for the degree
of Master of Science in Mathematics.
\vspace{.3in}
\noindent
\makebox[3.4in][l]{}\makebox[2.0in][l]{\rule{2.5in}{.01in}}\\[-.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. A.Q.M.Khaliq}\\[-.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{Chair, Graduate Committee } \\[.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. Zachariah Sinkala} \\[.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. Yuri Melnikov} \\[.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. James Hart} \\[-.1in]
\makebox[3.4in][l]{}\makebox[2.0in][l]{ Graduate Coordinator,} \\[-.1in]
\makebox[3.4in][l]{}\makebox[2.0in][l]{ Department of Mathematical Sciences} \\[.1in]
\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
%\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. Don Nelson} \\[-.1in]
%\makebox[3.4in][l]{}\makebox[2.0in][l]{ Chair,} \\[-.1in]
%\makebox[3.4in][l]{}\makebox[2.0in][l]{ Department of Mathematical Sciences} \\[.1in]
%\makebox[3.4in][l]{Signed on behalf of } \makebox[2.0in][l] {\rule{2.5in}{.01in}}\\[-.1in]
\makebox[3.4in][l]{the Graduate Council}\makebox[2.0in][l]{\it Dr. Michael Allen} \\[-.1in]
\makebox[3.4in][l]{}\makebox[2.0in][l]{ Dean,} \\[-.1in]
\makebox[3.4in][l]{}\makebox[2.0in][l]{ School of Graduate Studies}
\newpage
%================================================ Title Page ======================================================
\begin{center}
\thispagestyle{empty} % takes out page numbering
{MESHFREE METHODS FOR THE THE BLACK-SCHOLES PARTIAL DIFFERENTIAL EQUATION \\ [.07in]} \rm
\rule{1.25in}{.01in}\\[.0 in]
\vspace{.6in}
A Thesis \\ [.06 in]
Presented to the Faculty of the Department of Mathematical Sciences \\[.06in]
Middle Tennessee State University \\ [.06in]
\rule{1.25in}{.01in}\\
\vspace{.6in}
In Partial Fulfillment \\[.06 in]
of the Requirements for the Degree \\ [.06 in]
Master of Science in Mathematical Sciences \\ [.06 in]
\rule{1.25in}{.01in}\\
\vspace{.6in}
by \\ [.06in]
{Nana Akwasi Abayie Boateng} \\[.06in]
{August 2012}
\end{center}
%%================================================ Approval Page ===================================================
%\newpage
%
%\pagestyle{plain}
%
%\begin{center}
%
%{\bf APPROVAL} \\ [.05in]
%{\bf This is to certify that the Graduate Committee of }\\
%Nana Akwasi Abayie Boateng \\ [-.1in] met on the \\ [-.1in] 14th \
%day of \ June, 2012.
%\\ [.25in]
%\end{center}
%
%\baselineskip=20 pt
%
% The committee read and examined his/her thesis,
%supervised his/her defense of it in an oral examination, and decided
%to recommend that his/her study should be submitted to the Graduate
%Council, in partial fulfillment of the requirements for the degree
%of Master of Science in Mathematics.
%
%\vspace{.3in}
%
%\noindent
%\makebox[3.4in][l]{}\makebox[2.0in][l]{\rule{2.5in}{.01in}}\\[-.1in]
%\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. A.Q.M.Khaliq}\\[-.1in]
%\makebox[3.4in][l]{} \makebox[2.0in][l]{Chair, Graduate Committee } \\[.1in]
%\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
%\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. Zachariah Sinkala} \\[.1in]
%\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
%\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. Yuri Melnikov} \\[.1in]
%\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
%\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. James Hart} \\[-.1in]
%\makebox[3.4in][l]{}\makebox[2.0in][l]{ Graduate Coordinator,} \\[-.1in]
%\makebox[3.4in][l]{}\makebox[2.0in][l]{ Department of Mathematical Sciences} \\[.1in]
%\makebox[3.4in][l]{} \makebox[2.0in][l]{\rule{2.5in}{.01in}} \\[-.1in]
%%\makebox[3.4in][l]{} \makebox[2.0in][l]{\it Dr. Don Nelson} \\[-.1in]
%%\makebox[3.4in][l]{}\makebox[2.0in][l]{ Chair,} \\[-.1in]
%%\makebox[3.4in][l]{}\makebox[2.0in][l]{ Department of Mathematical Sciences} \\[.1in]
%%\makebox[3.4in][l]{Signed on behalf of } \makebox[2.0in][l] {\rule{2.5in}{.01in}}\\[-.1in]
%\makebox[3.4in][l]{the Graduate Council}\makebox[2.0in][l]{\it Dr. Michael Allen} \\[-.1in]
%\makebox[3.4in][l]{}\makebox[2.0in][l]{ Dean,} \\[-.1in]
%\makebox[3.4in][l]{}\makebox[2.0in][l]{ School of Graduate Studies}
%================================================ Abstract Page =================================================
\newpage
\begin{center}
{\bf ABSTRACT}\\
\end{center}
\baselineskip=24pt
Meshfree radial basis functions (RBF) is an interpolation technique
for constructing an unknown function from scattered data. We apply
the RBF method in evaluating the price of standard American
options. The analytical solution of the European option exists and
can be obtained by the Black-Scholes formula. There is no exact
solution of the American option problem due to the existence of an
early exercise constraint which leads to a free boundary condition.
We evaluate the American Option by adding a small continuous
nonlinear penalty term to the Black-Scholes model to remove the free
boundary condition. The application of RBFs leads to a system
ordinary differential equations which are solved by a time
integration scheme known as the $\theta$-method. The option price is
approximated with RBF with unknown parameters at each time step. We
compare the accuracy, efficiency and computational cost of three
RBFs Gaussian, Multiquadric and the Inverse-multiquadric. Finally a
comparison is made between the three RBFs and the solution obtained
by finite difference approximations.
%================================================ Copyright Page =================================================
\newpage
\baselineskip=24 pt
\begin{center}
\ \ \
\vspace{3.in}
Copyright \copyright\ 2012, Nana Akwasi Abayie Boateng
\end{center}
%================================================ Dedication Page =================================================
\newpage
\begin{center}
{ \bf DEDICATION } \\ [.15in]
\end{center}
This thesis is dedicated to my parents. I am forever grateful to
them for their unconditional love,prayer,material and emotional
support throughout my whole life. I pray the good Lord reward them
for all their sacrifices in my life.
%================================================ Acknowledgments Page ==============================================
\newpage
\begin{center}
{ \bf ACKNOWLEDGMENTS} \\ [.15in]
\end{center}
I am thankful to my Lord Jesus Christ for granting me the strength
to make this work a possibility. I cannot forget about the enormous
help I have received from my advisor Dr. Khaliq throughout the
period of writing this thesis. I thank him so much for all his
advice,suggestions and directions.
I am particularly grateful to the graduate coordinator Dr. Hart for
all his concern and guidance for the time I have spent here as a student in
the Mathematics department.
I also would like to say a special thank you to my thesis
committee members Dr.Sinkala and Dr. Melnikov for their role in
making this whole work a success.
%================================================ Table of Content =================================================
\newpage
\tableofcontents
%%================================================ Chapter 1 ==============================================================
%{INTRODUCTION}
%%================================================ Chapter 2 ==============================================================
%{\uppercase{Option Pricing}}
%%--
%%================================================ Chapter 3 ==============================================================
%\uppercase{Finite Difference Methods}\\
%%-------------------------------------------------------------------------------------------------------------------------
%%================================================ Chapter 4 ==============================================================
%\uppercase{RBF-Meshfree Methods}\\
%%-------------------------------------------------------------------------------------------------------------------------
%
%%================================================ Chapter 5 ==============================================================
%\uppercase{Discretization And Algorithms}\\
%%-------------------------------------------------------------------------------------------------------------------------
%%================================================ Chapter 7 ==============================================================
%\uppercase{ Numerical Methods and Stability Analysis }\\
%%================================================ List of Tables ===================================================
%%================================================ Chapter 6 ==============================================================
%\uppercase{Numerical Experiments And Results}\\
%%-------------------------------------------------------------------------------------------------------------------------
\newpage
\addcontentsline{toc}{section}{\rm LIST OF TABLES}
\listoftables
%================================================ List of Figures ====================================================
\newpage
\addcontentsline{toc}{section}{\rm LIST OF FIGURES}
\listoffigures
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\la{\langle}
\def\ra{\rangle}
\def\dist{{\rm dist}}
\def\X{{\bf X}}
\def\C{{\bf C}}
\def\D{{\bf D}}
\def\I{{\bf I}}
\def\J{{\bf J}}
\def\x{{\bf x}}
\def\y{{\bf y}}
\def\z{{\bf z}}
\def\W{{\bf W}}
\def\g{{\bf g}}
\def\e{{\bf e}}
\def\b{{\bf b}}
\def\u{{\bf u}}
\def\Beta{{\bf \beta}}
\def\pen{{\rm pen}}
\def\argmin{{\rm argmin}}
\def\diag{{\rm diag}}
\def\sgn{{\rm sgn}}
\def\supp{{\rm\rm supp}}
\vspace*{1cm}
%============================================= Appendix Separation Page ===============================================
\newcommand{\Appendixpage}{
\setcounter{section}{0}
\renewcommand{\baselinestretch}{1}\small\normalsize
\thispagestyle{myheadings}
\addcontentsline{toc}{section}{APPENDICES\dotfill}
\mbox{}
\vfil
\begin{center}%
APPENDICES
\vfil
\end{center}%
\renewcommand{\baselinestretch}{1.66} \small\normalsize%
\cleardoublepage
}
%================================================ Chapter 1 ==============================================================
\sect{INTRODUCTION}
\pagestyle{myheadings} \markboth{ } { }
\pagenumbering{arabic}
%-------------------------------------------------------------------------------------------------------------------------
An option is a financial contract which gives the holder of the
option the right to purchase or sell a prescribed asset at a
prescribed time in the future known as the expiry date at a
prescribed amount which is the exercise or strike price.
In 1973,Fisher Black and Myron Scholes showed that the option value
of the European call option can be modeled by a lognormal diffusion
partial differential equation.There are two categories of options
namely, standard options(European and American options) and non
standard options.\\
Hon and Mao\cite{Hon} introduced a numerical scheme in which by
applying global radial basis function as a spatial approximation
for the numerical solution of the value of the option
and it's derivative in the Black-Scholes equation. From their
numerical results,they showed that the use of RBFs does not require
the generation of a rectangular grid and also the computational
domain is composed of scattered data points.\\
Khaliq \textit{et al}\cite{KK} investigated meshfree RBF approximation to
options with non-smooth payouts. By taking advantage of parallel
architecture, they developed a strongly stable time stepping fourth
order method which was a linear combination of four Backward
Euler-like solver on four concurrent processors.
Khaliq \textit{et al}\cite{ADS06} considered a penalty method
approach to solving American options. They observed that by
introducing a carefully chosen continuous penalty term to the
Black-Scholes equation, the free and moving boundary condition can
be removed and allow the problem to be solved on a fixed
domain. They introduced a linearly implicit scheme with superior
accuracy and stability by solving the nonlinear term explicitly.\\
The remaining chapters of this thesis are organized as follows.
We introduce option pricing and discuss two standard options,
the European and American options in chapter 2. In chapter 3 we
implement the $\theta$-method $(\theta=0)$, the backward Euler
method to evaluate the price of the option using finite difference approximation. The theory and development of
RBF Meshfree methods is also presented here. In chapter 4, we
elaborate on the discretization , algorithms of the RBFs methods and
the stability analysis of the numerical scheme. Finally all
numerical results from the experiments are presented in chapter 5
and interpreted in chapter 6.
%\subsection{Sample Section} This is a sample section.
%
%The study of algebraic structures using its associate graphs is a
%very exciting field which generates many fascinating results,
%conjectures and questions \cite{Abdollahi2006}. There are various
%ways to associate graphs to algebraic objects such as groups and
%rings. For instance, the prime graph defined in\cite{Williams1981},
%the conjugacy class graph defined in\cite{Bertram}, the
%non-commuting graph defined in\cite{Abdollahi2006}, and the nonzero
%divisor graph defined in\cite{DavidAnderson}
%\subsubsection{Sample Sub Section}
%This is a sample equation.
%\begin{align}
% \frac{\partial}{{\partial}t}\int\int\limits_{system}
% (V_{A_{r}} + V_{A_{s}})dxdy = 0 \label{eq: equation1}\\
% \frac{\partial}{{\partial}t}\int\int\limits_{system} V_B dxdy = 0
% \label{eq:equation2}
%\end{align}
%\begin{theorem}{\rm(\cite{Wang2008})}\label{thm1}
%Let $G$ be a group such that $\nabla{(G)} \cong\nabla(A_{n})$, where
%$n\ge 5$. If $n=5,6$ or at least one of $n,n-1,n-2$ is prime then
%$G\cong A_{n}$.
%\end{theorem}
%================================================ Chapter 2 ==============================================================
\sect{\uppercase{Option Pricing}}
% \numberwithin{Option Pricing}
%-------------------------------------------------------------------------------------------------------------------------
%\subsection{What is an Option?}
An option is a financial contract which gives the holder of the
option the right to purchase or sell a prescribed asset at a
prescribed time in the future known as the expiry date at a
prescribed amount which the exercise or strike price\cite{Wilmot}.
The most common kinds of prescribed assets which are traded on
financial markets are stocks, bonds, currency and commodities. An
option is a derivative product because it is traded on an underlying
asset. The holder of a call option makes profit if the price of the
underlying asset rises on the market whereas the holder of a put
option does so when the price of the underlying asset falls on the
financial market. The two primary uses of option are for hedging and
speculation\cite{Wilmot}.
There are numerous kinds of options which are
traded on financial markets. Vanilla options are options which do
not possess any special features or characteristics. Examples are
the European and American options. Exotic options possess special
features. Examples include Asian options,Barrier options,Basket
options. In this thesis, we consider the pricing of American option
using a penalty method approach for Black-Scholes partial
differential equation.
%\subsubsection{The European Options}
\subsection{The European Option}
%\numberwithin{equation}{The European Options}
The European option is an option
which can only be exercised at its maturity time. The exact or
analytical formula for estimating a fair price for the European
options exist. In 1973 Fisher Black and Myron Scholes by making a
set of explicit assumptions including the risk-neutrality of the
underlying asset price showed that the value of the European call
option satisfies a backward -in-time lognormal partial differential
equation of diffusion type. This has come to be known as the
Black-Scholes equation\cite{BS73}.
Let the $V(S,t)$ be the price of an option which is function of both
asset price and time. This option satisfies the following
Black-Scholes equation.
\begin{equation} \label{BSS}
\frac{\partial P}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
P}{\partial S^2} +rS\frac{\partial P}{\partial S}-rP=0, \quad \ 0
\le t < T.
\end{equation}
where $r$ is the risk-free interest interest rate, $\sigma$ is the
volatility of the asset price, $S$ is the asset price. The Final
condition is given by\cite{Wilmot}.
% \begin{eqnarray}\label{fcfc}
%\[V(S,t) = \left\{
%\begin{array}{l l}
% max\{E-X,0\} & \quad \mbox{ for a put option}\\
%max\{S-E,0\}& \quad \mbox{for a call option}\\
%\end{array} \right. \]
%\end{eqnarray}
%$$
\begin{equation}\label{fcfc}
V(S,t) = \left\{ \begin{array}{rl}
max\{E-S,0\}&\mbox{ for a put option} \\
max\{S-E,0\}&\mbox{for a call option }
\end{array} \right.
\end{equation}
%$$
where E is the strike price.\\
The Boundary condition of the European call option is given as
follows:
\begin{equation}\label{c1}
C(S,t)\thicksim S \quad
as S\rightarrow\infty ,\quad C(0,t)=0.
\end{equation}
where $C(S,t)$ is the value of the European call option satisfying
$(\ref{BSS})$. The Boundary condition at time $t$ of the European
put option is given as follows:
\begin{equation}\label{FC}
P(S,t)\rightarrow 0 \quad as \quad S\rightarrow\infty ,\quad
P(0,t)=E\exp^{-{\int_{t}^{T}}\tau(\tau)d\tau}
%E\exp^{\int^{T}_{t}\tau(\tau)d\tau}}.
\end{equation}
where $P(S,t)$ is the value of the European put option
satisfying equation $(\ref{BSS})$, for a time dependent interest rate.\\
Equation $(\ref{BSS})$ can be transformed exponentially by making the
substitution $S=e^y$ to
\begin{equation}
\frac{\partial U}{\partial
t}+\frac{1}{2}\sigma^2\frac{\partial^2U}{\partial
y^2}+(r-\frac{1}{2}\sigma^2\frac{\partial U}{\partial y})-rU=0
\end{equation}
\[U(y,T) = \left\{
\begin{array}{l l}\label{INC}
max\{E-e^y,0\} & \quad \mbox{ for a put option}\\
max\{e^y-E,0\}& \quad \mbox{for a call option}\\ \end{array} \right. \]
The analytical solution of the Black- Scholes partial differential
equation $(\ref{BSS})$ with corresponding final and initial
conditions $(\ref{fcfc})$ and $(\ref{FC})$ with a constant
volatility and interest rate
for the European call option is given as\cite{Wilmot}
\begin{equation}
C(S,t)=SN(d_1)-E\exp^{-r(T-t)}N(d_2)
\end{equation}
where $N(.)$ is the cumulative distribution function for the
standardized normal random variable given by
\begin{equation}
N(x)=\frac{1}{\sqrt{2\pi}}\int^{x}_{-\infty}\exp^{-\frac{1}{2}y^2dy}
\end{equation}
The corresponding analytical solution of the European put option is
given by
\begin{equation}
P(S,t)=E\exp^{-r(T-t)}N(-d_2)-SN(-d_1)
\end{equation}
where\\
\begin{equation*}
d_1=\frac{\log(\frac{S}{E}+(r+\frac{1}{2}\sigma^2)(T-t))}{\sigma\sqrt{T-t}}
\end{equation*}
\begin{equation*}
d_2=\frac{\log(\frac{S}{E}+(r-\frac{1}{2}\sigma^2)(T-t))}{\sigma\sqrt{T-t}}
\end{equation*}
%\begin{figure}[h]
%\begin{centering}
%\vskip -0.5in
%\includegraphics*[height=3in]{calloption.png}\
%\caption{The pay-off of the European Call Option }
%\includegraphics*[height=3in]{putoption2.png}\
%\caption{The pay-off of the European Put Option } \vskip -0.5in
%\end{centering}
%\end{figure}
%\subsubsection{The American Option}
\subsection{The American Option}
The American option can be exercised at any time prior to expiry.
The American option is complicated because at at each time $t$ not
only is one interested in the value of the option but also for each
asset price $S$,whether it should be exercised or not. This creates
a free boundary problem\cite{Wilmot}. At each time $t$ there is a
particular value of $S$ which lies in the boundary between two
regions: one where early exercise is optimal to the other where one
should hold on to the option. The optimal exercise price $S_{f}(t)$
which in general depends on time is not known \textit{priori} unlike
the case of European options. The American option valuation can be
uniquely specified by a set of constraints among which are that the
option value must be greater than or equal to the payoff function,
the option value must be a continuous function of $S$, replacing the
Black-Scholes equation by an inequality and lastly making the
derivative of the option with respect to the asset price(option
delta) continuous.
The value $V(S,t)$ of the American option satisfies the following
inequality
\begin{equation}
\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
V}{\partial S^2} +rS\frac{\partial V}{\partial S}-rV\leq 0
\end{equation}
The Final condition is at expiry time $T$ given by\cite{Wilmot}
\[V(S,t) = \left\{
\begin{array}{l l}\label{FNC}
max\{E-S,0\} & \quad \mbox{ for a put option}\\
max\{S-E,0\}& \quad \mbox{for a call option}\\ \end{array} \right. \]
where E is the strike price.\\
In the region $0\leq S\leq S_{f}(t)$ where early exercise is optimal, the value of the
American put option satisfies the following inequality
\begin{equation}
\frac{\partial P}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
P}{\partial S^2} +rS\frac{\partial P}{\partial S}-rP< 0
\end{equation}
and
\begin{equation}
P=E-S
\end{equation}
In the other region, $S_{f}(t)< S<\infty$, early exercise is not
optimal and the value of the American put option satisfies the
Black-Scholes equation
\begin{equation}
\frac{\partial P}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
P}{\partial S^2} +rS\frac{\partial P}{\partial S}-rP= 0
\end{equation}
and
\begin{equation}
P>E-S
\end{equation}
The boundary condition at $S_{f}(t)= S$ are that $P$ and its
slope(delta) are continuous.
\begin{equation}\label{pp}
P(S_{f}(t),t)=max(E-S_{f}(t),0)
\end{equation}
\begin{equation}\label{dv}
\frac{\partial P}{\partial S}(S_{f}(t),t)=-1
\end{equation}
The boundary condition$( \ref{pp})$ determines the option value at the
free boundary, whereas $(\ref{dv})$known as the \textit{smooth pasting condition} determines the location of the
free boundary and simultaneously maximizes the benefit to the
holder whiles avoiding arbitrage.
The value $C(S,t)$ of the American Call option satisfies the
corresponding equality in the holding region
$0\leq S\leq S_{f}(t)$
\begin{equation}
\frac{\partial C}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
C}{\partial S^2} +rS\frac{\partial C}{\partial S}-rC=0
\end{equation}
in the other region where early exercise is optimal $S_{f}(t)<
S<\infty$,the value $C(S,t)$ of the American call option satisfies
\begin{equation}
\frac{\partial C}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
C}{\partial S^2} +rS\frac{\partial C}{\partial S}-rC<0
\end{equation}
and
\begin{equation}
P=E-S
\end{equation}
\subsubsection{The Penalty Method}
%An elegant transformation of this moving boundary
%problem to one on a fixed domain was suggested in \cite{ZFV98} and
%later refined in \cite{NST02}.
We introduce a penalty term into the Black-Scholes equation to
obtain a parabolic nonlinear partial differential
equation\cite{BOA08}. The introduction of the penalty term changes
the problem from that of a constrained optimization problem to that
of a series of unconstrained optimization problem. The solution of
the unconstrained optimization problems converges to the original
constrained optimization problem.\cite{BOA08}
\begin{equation}\label{BSpenalty}
\frac{\partial P_\epsilon}{\partial
t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 P_\epsilon}{\partial S^2}
+rS\frac{\partial P_\epsilon}{\partial S}-rP_\epsilon+\frac{\epsilon
C }{P_\epsilon + \epsilon - q(S)}=0, \quad 0 \leq S \leq S_\infty, \
0 \leq t < T.
\end{equation}
The addition of a penalty term allows the problem to be solved on a
fixed domain.Here $0 < \epsilon \ll1$ is a small regularization
parameter, $C \geq rE$ is a positive constant, and $q(S) = E - S$ is
the barrier function. The value $S_\infty$ is a (relatively very
large) price for which the option is worthless. The following are
terminal and boundary conditions which accompany the nonlinear
partial differential equation\cite{BOA08}.
\begin{eqnarray}
P_\epsilon(S,T) & = & \max (E-S,0)\\
P_\epsilon(0,t) & = & E,\\
P_\epsilon(S_\infty,t) & = & 0.
\end{eqnarray}
%\begin{theorem}{\rm(\cite{Wang2008})}\label{thm1}
%Let $G$ be a group such that $\nabla{(G)} \cong\nabla(A_{n})$, where
%$n\ge 5$. If $n=5,6$ or at least one of $n,n-1,n-2$ is prime then
%$G\cong A_{n}$.
%\end{theorem}
%================================================ Chapter 3 ==============================================================
\sect{\uppercase{Finite Difference and RBF Approximations}}
%\numberwithin{fd1}
The method of Finite Difference Approximation which is based on Taylor series expansion
of functions near the point of interest is be used to discretize the
Black-Scholes partial differential equation\cite{Wilmot}.
\subsection{Finite Difference Approximations}
\begin{equation}\label{fd1}
\frac{\partial P}{\partial t}+\frac{1}{2}\sigma^2
S^2\frac{\partial^2 P}{\partial S ^2}+rS\frac{\partial P}{\partial
S}-rP +\frac{{\epsilon}C}{P+\epsilon-q(s)}=0
\end{equation}
The spatial derivatives are approximated by central differencing
with spatial step size $h$=$\frac{S_{f}-S_{0}}{N}$ and time step
size $k$=$\frac{T_{f}-T_{0}}{M}$ as
follows. We apply the $\theta$-method$(\theta=0)$ \cite{BOA08} by performing a
Backward-Euler approximation on
$\frac{\partial P}{\partial t}$ and treat the penalty term $ Q$
explicitly.The discretization of the derivatives of the asset
price $S$, is given as follows:
\begin{equation*}
\frac{\partial P^2}{\partial
S^2}=\frac{P_{m+1,n}-2P_{m,n}+P_{m-1,n}}{h^2},
\end{equation*}
\vspace{5mm}
\begin{equation*}
\frac{\partial P}{\partial S}=\frac{P_{m+1,n}-P_{m-1,n}}{2h}.
\end{equation*}
% Using the above approximations ,the Black-Scholes partial differential
% equation is then applied to mesh points $(ih,jk)$ ,$i=1,2...N-1$, at
% the time level $t=jk$,$j=1,2,...,M$.At each $i$ we have
Using the approximations above , the Black-Scholes partial differential
equation is then discretized at mesh points $(mh,nk)$ ,$m=1,2...N$,at
the time level $t=nk$,$n=1,2,...,M$.At each $n$ we obtain
\begin{equation}\label{fdd1}
\frac{P_{m,n}-P_{m,n-1}}{k}=\frac{1}{2}\sigma^2S^2\left[\frac{P_{m+1,n}-2P_{m,n}+P_{m-1,n}}{h^2}\right]+rs\left[\frac{P_{m+1,n}-P_{m-1,n}}{2h}\right]-rP_{m,n}+\frac{{\epsilon}C}{P+\epsilon-q(s)},
\end{equation}
\begin{equation}\label{fdd2}
P_{m,n}-P_{m,n-1}=\frac{1}{2}\frac{\sigma^2S^2k}{h^2}\left[
P_{m+1,n}-2P_{m,n}+P_{m-1,n}\right]+\frac{krS}{2h}\left[
P_{m+1,n}-P_{m-1,n}\right]-rkP_{m,n}+\frac{{\epsilon}Ck}{P+\epsilon-q(s)}.
\end{equation}
Let $\beta=\frac{1}{2}\frac{\sigma^2S^2k}{h^2}$ , $\alpha=\frac{krS}{2h}$
and $Q=\frac{{\epsilon}Ck}{P+\epsilon-q(s)}$\\
By substituting $\alpha$ and $\beta$ in equation $(\ref{fdd2})$, we
obtain,
\begin{equation*}
P_{m,n}-P_{m,n-1}=\beta P_{m+1,n}-2\beta P_{m,n}+\beta
P_{m-1,n}+\alpha P_{m+1,n}-\alpha P_{m-1,n}-rkp_{m,n} +Q
\end{equation*}
\begin{equation*}
P_{m,n}+2\beta P_{m,n}+rkP_{m,n}-\beta P_{m+1,n}-\alpha
P_{m-1,n}-\beta P_{m-1,n}=P_{m,n-1}+Q
\end{equation*}
\begin{equation}\label{fdd3}
(1+2\beta
+rk)P_{m,n}-(\alpha+\beta)P_{m+1,n}+(\alpha-\beta)P_{m-1,n}=P_{m,n-1}+Q
\end{equation}
This leads to the following tridiagonal system
\[A = \left( \begin{array}{cccc}
1+2\beta +rk& -(\alpha+\beta) & ...& 0\\
\alpha-\beta & 1+2\beta +rk & & \\
& \ddots & \ddots & \\
& & & \\
%0 &...&\alpha-\beta & 1+2\beta +rk &-(\alpha+\beta)\\
0&......& \alpha-\beta&1+2\beta+rk -(\alpha+\beta) \\
0 &...&\alpha-\beta & 1+2\beta
+rk \end{array} \right) .\] The boundary condition vector
$\textbf{b}_n$ resulting from writing $(\ref{fdd3})$ as a
tridiagonal matrix system is given by
\begin{equation*}
\textbf{b}_n= \left[
(\alpha-\beta)P_{1},0,.......0,-(\alpha+\beta)P_{N,}\right]^{T}
\end{equation*}
\begin{equation*}
AV_{m,n} + b_{n}=V_{m,n-1}+Q
\end{equation*}
The eigenvalues of the matrix A can be shown to be
%\begin{equation*}
% \begin{array}
%
% \lambda_{s}&=&1+2\beta+rk+2\sqrt{-(\alpha+\beta)(\alpha-\beta)}\cos\frac{S\pi}{N+1}
%
% \end{array}
% \end{equation*}
\begin{equation*}
\lambda_{s} =
1+2\beta+rk+2\sqrt{-(\alpha+\beta)(\alpha-\beta)}\cos\left(\frac{S\pi}{N+1}\right)\quad
S = 1,\cdots ,N
\end{equation*}
%\begin{equation*}
%\begin{array}{lcl} \lambda_{s} & = &1+2\beta+rk+2\sqrt{-(\alpha+\beta)(\alpha-\beta)}\cos(\frac{S\pi}{N+1})\\ S & = & 1,\cdots ,N \end{array}
%\end{equation*}
\begin{equation*}
\lambda_s=1+2\beta+rk+2\sqrt{\beta^2-\alpha^2}\cos\left(\frac{S\pi}{N+1}\right)
\end{equation*}
If $0<\alpha \leq \beta$ then the eigenvalues are real numbers
satisfying
\begin{equation*}
-(1+2\beta+rk+2\sqrt{\beta^2-\alpha^2})\leq \lambda_s \leq
1+2\beta+rk+2\sqrt{\beta^2-\alpha^2}
\end{equation*}
If $\beta \leq \alpha$,then the eigenvalues are complex numbers
satisfying
\begin{equation*}
\begin{array}{cccllllc}
\lambda_s&=&1+2\beta+rk+2\sqrt{-(-\beta^2-\alpha^2)}\\
&=&1+2\beta+rk+2\sqrt{\alpha^2-\beta^2}i\\
&=&1+2\beta+rk+2i\gamma_j\\
\end{array}
\end{equation*}
where
\begin{equation*}
-\sqrt{\alpha^2-\beta^2}\leq \gamma_j\leq\sqrt{\alpha^2-\beta^2}.
\end{equation*}
The eigenvalues of \textbf{A} are essential in determining the
stability of the numerical scheme above. The system is stable if
$max|\lambda_s | \leq 1$ for $S=1,...,N$.
%\subsection{Time Stepping Scheme}
%We Consider the following nonlinear parabolic
%initial-boundary value problem:
%\begin{eqnarray}\label{tms1}
%u_t + A u &=& F(t,u) \qquad \textrm {in} \; \Omega,\qquad t \in \left( 0,T \,\, \right], \label{Eq1}\\
%u &=& v \qquad \qquad \; \, \textrm {on} \;\partial \Omega,\quad t \in \left( 0,T \,\, \right], \nonumber \\
%u(\cdot,0) &=& u_0 \qquad \qquad \textrm {in} \;\Omega, \nonumber
%\end{eqnarray}
%where $\omega$ is a bounded domain in $\mathbb{R}^{d}$ which also
%has Lipschitz continuous boundary,we let $A$ represents a uniformly
%elliptic operator, and $F$ is a sufficiently smooth, nonlinear
%reaction term. We represent a differential operator as follows:
%\begin{eqnarray}
%A := -\sum_{j,k=1}^d \frac{\partial}{\partial x_{j}} \left (
%a_{j,k}(x)\frac{\partial}{\partial x_{k}} \right )+\sum_{j=1}^d
%b_{j}(x)\frac{\partial}{\partial x_{j}}+b_{0}(x),
%\end{eqnarray}
%where the coefficients $a_{j,k}$ and $b_{j}$ are $C^{\infty}$ (or
%sufficiently smooth) functions on $\overline{\Omega}$,
%$a_{j,k}=a_{k,j}$, $b_{0} \ge 0$, and for some $c_0 > 0$
%%\begin{eqnarray*}
% \sum_{j,k=1}^{d}a_{j,k}(\cdot)\xi_j\xi_k \geq
%c_0 \mid \xi \mid^2$,\hspace{5mm}on $\={\Omega},\hspace{5mm}for all
%\xi \epsilon\Re^d
%\end{eqnarray*}
%\begin{eqnarray}
%\sum_{j,k=1}^d a_{j,k}(\cdot) \xi_{j} \xi_{k} \geq c_{0}
%|\xi|^2,\quad \rm on \; \overline{\Omega}, \quad \rm{for\; all} \;
%\xi \in \mathbb{R}^{d}.
%\end{eqnarray}
%
%We chose $A$ and $F$ based on an abstract formulation for
%convenience to facilitate the development of the numerical scheme
%and its analysis. The initial value problem \ref{tms1} is reset to
%be posed in a Hilbert space ${X}$, as follows. Consider now $A$ to
%be a linear, self-adjoint, positive definite, closed operator with a
%compact inverse, defined on a dense domain $D(A) \subset {X}$. The
%operator $A$ could represent any of $\{A_h\}_{0< h \le h_0}$,
%obtained through spatial discretization, and ${X}$ We assume the
%resolvent set $\rho(A)$ satisfies, for some $\alpha \in
%(0,\frac{\pi}{2} )$, $\rho(A) \supset \overline{\Sigma}_{\alpha},
%\,\, $ where $\Sigma_{\alpha}:= { z \in := \alpha < |\arg(z)| \leq
%\pi,z \neq 0}$. Also, assume there exists $M \geq 1$ such that
%\begin{equation}
%\|(zI-A)^{-1}\| \geq M |z|^{-1}, \: z \in \Sigma_\alpha.
%%\label{assumption1}
%\end{equation}
%It follows that $-A$ is the infinitesimal generator of an analytic
%semigroup $\{e^{-tA}\}_{t\ge 0}$ which is the solution operator for
%\ref{tms1}, and $|e^{-tA}| \leq C$ for ${t \ge 0}$. Also, we assume
%that $F(t,u(t))$ is Lipschitz on $[0,T] \times X$, i.e. it satisfies