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OPTIMAL CONTROL MODELS
IN FINANCE
Applied Optimization
Volume 95

Series Editors:

Panos M. Pardalos
University of Florida, U.S.A.

Donald W. Hearn
University of Florida, U.S.A.
OPTIMAL CONTROL MODELS
IN FINANCE
A New Computational Approach

by

PING CHEN
Victoria University, Melbourne, Australia


SARDAR M.N. ISLAM
Victoria University, Melbourne, Australia




Springer
0-387-23570-1
eBook ISBN:
0-387-23569-8
Print ISBN:



©2005 Springer Science + Business Media, Inc.


Print ©2005 Springer Science + Business Media, Inc.
Boston

All rights reserved


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Created in the United States of America


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Contents




ix
List of Figures
List of Tables xi
Preface xiii
Introduction xv

1. OPTIMAL CONTROL MODELS 1
2
1 An Optimal Control Model of Finance
4
2 (Karush) Kuhn-Tucker Condition
Pontryagin Theorem 6
3
4 Bang-Bang Control 7
7
5 Singular Arc
6 8
Indifference Principle
Different Approaches to Optimal Control Problems 10
7
8 Conclusion 20
2. THE STV APPROACH TO FINANCIAL OPTIMAL CONTROL
MODELS 21
1 Introduction 21
2 Piecewise-linear Transformation 21
Non-linear Time Scale Transformation
3 23
4 A Computer Software Package Used in this Study 25
5 An Optimal Control Problem When the Control can only Take
the Value 0 or 1 26
6 Approaches to Bang-Bang Optimal Control with a Cost of
Changing Control 27
7 An Investment Planning Model and Results 30
vi OPTIMAL CONTROL MODELS IN FINANCE

8 Financial Implications and Conclusion 36
3. A FINANCIAL OSCILLATOR MODEL 39
1 Introduction 39
2 Controlling a Damped Oscillator in a Financial Model 40
3 Oscillator Transformation of the Financial Model 41
4 Computational Algorithm: The Steps 44
5 Financial Control Pattern 47
6 Computing the Financial Model: Results and Analysis 47
7 Financial Investment Implications and Conclusion 89
4. AN OPTIMAL CORPORATE FINANCING MODEL 91
1 Introduction 91
2 Problem Description 91
3 Analytical Solution 94
4 Penalty Terms 98
Transformations for the Computer Software Package for the
5
Finance Model 99
6 Computational Algorithms for the Non-linear Optimal Control
Problem 101
Computing Results and Conclusion 104
7
8 Optimal Financing Implications 107
Conclusion
9 108
109
5. FURTHER COMPUTATIONAL EXPERIMENTS AND RESULTS
1 Introduction 109
2 Different Fitting Functions 109
The Financial Oscillator Model when the Control Takes Three
3
Values 120
4 Conclusion 139
6. CONCLUSION 141
145
Appendices
145
A CSTVA Program List
1 Program A: Investment Model in Chapter 2 145
2 Program B: Financial Oscillator Model in Chapter 3 149
Program C: Optimal Financing Model in Chapter 4 153
3
156
4 Program D: Three Value-Control Model in Chapter 5
Contents vii

161
B Some Computation Results
1 Results for Program A 161
2 Results for Program B 163
167
3 Results for Program C
4 Results for Program D 175
C Differential Equation Solver from the SCOM Package 181
183
D SCOM Package
E Format of Problem Optimization 189
191
F A Sample Test Problem
References 193
Index 199
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List of Figures




31
2.1 Plot of n=2, forcing function ut=1,0
Plot of n=4, forcing function ut=1,0,1,0 31
2.2
32
2.3 Plot of n=6, forcing function ut= 1,0,1,0,1,0
32
2.4 Plot of n=8, forcing function ut= 1,0,1,0,1,0,1,0
Plot of n=10, forcing function ut= 1,0,1,0,1,0,1,0,1,0 33
2.5
Plot of the values of the objective function to the num-
2.6
34
ber of the switching times
2.7 Plot of the cost function to the cost of switching control 35
87
3.1 Plot of integral F against 1/ns at ut=-2,2
87
3.2 Plot of integral F against 1/ns at ut=2,-2
3.3 Plot of cost function F against the number of large time
intervals nb 88
Plot of n=4, forcing function ut=1,0,1,0 110
5.1
112
5.2 Plot of n=10, forcing function ut= 1,0,1,0,1,0,1,0,1,0
5.3 Results of objective function at n=2,4,6,8,10 113
5.4 Plot of n=4, forcing function ut=1,0,1,0 116
117
5.5 Plot of n=8, forcing function ut=1,0,1,0,1,0,1,0
119
5.6 Plot of n=8, forcing function ut= 1,0,1,0,1,0,1,0
5.7 Plot of nb=9, ns=8, forcing function ut=-2,0,2,-2,0,2,-2,0,2 123
5.8 Relationship between two state functions during the
time period 1,0 123
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List of Tables




Objective functions with the number of the switching times 33
2.1
2.2 Costs of the switching control attached to the objective
34
function
3.1 Results of the objective function at control pattern -2,2, ... 48
Results of the objective function at control pattern 2,-2, ... 86
3.2
4.1 Computing results for solution case [1] 105
106
4.2 Computing results for solution case [2]
4.3 Computing results for solution case 2 with another map-
106
ping control
114
5.1 Results of objective function at n=2,4,6,8,10
118
5.2 Results of objective functions at n=2,6,10
120
5.3 Test results of the five methods
121
5.4 Results of financial oscillator model
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Preface




This book reports initial efforts in providing some useful extensions in fi-
nancial modeling; further work is necessary to complete the research agenda.
The demonstrated extensions in this book in the computation and modeling
of optimal control in finance have shown the need and potential for further
areas of study in financial modeling. Potentials are in both the mathematical
structure and computational aspects of dynamic optimization. There are needs
for more organized and coordinated computational approaches. These exten-
sions will make dynamic financial optimization models relatively more stable
for applications to academic and practical exercises in the areas of financial
optimization, forecasting, planning and optimal social choice.
This book will be useful to graduate students and academics in finance,
mathematical economics, operations research and computer science. Profes-
sional practitioners in the above areas will find the book interesting and infor-
mative.
The authors thank Professor B.D. Craven for providing extensive guidance
and assistance in undertaking this research. This work owes significantly to
him, which will be evident throughout the whole book. The differential equa-
tion solver “nqq” used in this book was first developed by Professor Craven.
Editorial assistance provided by Matthew Clarke, Margarita Kumnick and Tom
Lun is also highly appreciated. Ping Chen also wants to thank her parents for
their constant support and love during the past four years.

PING CHEN AND SARDAR M.N. ISLAM
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Introduction




Optimal control methods have significant applications in finance. This book
discusses the general applications of optimal control methods to several areas
in finance with a particular focus on the application of bang-bang control to
financial modeling.
During the past half-century, many optimization problems have arisen in
fields such as finance management, engineering, computer science, production,
industry, and economics. Often one needs to optimize (minimize or maximize)
certain objectives subject to some constraints. For example, a public utility
company must decide what proportion of its earnings to retain to the advantage
of its future earnings at the expense of gaining present dividends, and also
decide what new stock issues should be made. The objective of the utility is
to maximize the present value of share ownership, however, the retention of
retained earnings reduces current dividends and new stock issues can dilute
owners™ equity.
Some optimization problems involve optimal control, which are consider-
ably more complex and involve a dynamic system. There are very few real-
world optimal control problems that lend themselves to analytical solutions.
As a result, using numerical algorithms to solve the optimal control problems
becomes a common approach that has attracted attention of many researchers,
engineers and managers. There are many computational methods and theoreti-
cal results that can be used to solve very complex optimal control problems. So
computer software packages of certain optimal control problems are becoming
more and more popular in the era of a rapidly developing computer industry.
They rescue scientists from large calculations by hand.
Many real-world financial problems are too complex for analytical solu-
tions, so must be computed. This book studies a class of optimal financial
control problems where the control takes only two (or three) different discrete
values. The non-singular optimal control solution of linear-analytic systems in
finance with bounded control is commonly known as the bang-bang control.
The problem of finding the optimal control becomes one of finding the switch-
xvi OPTIMAL CONTROL MODELS IN FINANCE

ing times in the dynamic financial system. A cost of switching control is added
to usual models since there is a cost for switching from one financial instrument
to another. Computational algorithms based on the time scaled transformation
technique are developed for this kind of problems. A set of computer software
packages named CSTVA is generated for real-world financial decision-making
models.
The focus in this research is the development of computational algorithms
to solve a class of non-linear optimal control problems in finance (bang-bang
control) that arise in operations research. The Pontryagin theory [69, 1962] of
optimal control requires modification when a positive cost is associated with
each switching of the control. The modified theory, which was first introduced
by Blatt [2, 1976], will give the solutions of a large class of optimal control
problems that cannot be solved by standard optimal control theories. The the-
orem is introduced but not used to solve the problems in this book. However,
the cost of changing control, which is attached to the cost function, is used
here for reaching the optimal solution in control system. In optimization com-
putation, especially when calculating minimization of an integral, an improved
result can be obtained by using a greater number of time intervals.
In this research, a modified version of the Pontryagin Principle, in which
a positive cost is attached to each switching of the control, indicates that a
form of bang-bang control is optimal. Several computational algorithms were
developed for such financial control problems, where it is essential to com-
pute the switching times. In order to achieve the possibility of computation,
some transformations are included to convert control functions, state functions
and the integrals from their original mathematical forms to computable forms.
Mainly, the MATLAB “constr” optimization package was applied to construct
the general computer programs for different classes of optimal control prob-
lems. A simplified financial optimal control problem that only has one state
and one control is introduced first. The optimal control of such a problem
is bang-bang control, which switches between two values in successive time
intervals. A computer software package was developed for solving this partic-
ular problem, and accurate results were obtained. Also some transformations
are applied into the problem formalization. A financial oscillator problem is
then treated, which has two states and one control. The transformation of sub-
division of time interval technique is used to gain a more accurate gradient.
Different sequences of control are then studied. The computational algorithms
are applied to a non-linear optimal control problem of an optimal financing
model, which was original introduced by Davis and Elzinga [22, 1970]. In that
paper, Davis and Elzinga had an analytical solution for the model. In this book
a computer software package was developed for the same model, including set-
ting up all the parameters, calculating the results, and testing different initial
points of an iterative algorithm. During the examination of the algorithms, it
INTRODUCTION xvii

was found that sometimes a local minimum was reached instead of a global
optimum. The reasons for the algorithms leading to such a local minimum are
indicated, and as a result, a part of the algorithms are modified so as to obtain
the global optimum eventually.
The computing results were obtained, and are presented in graphical forms
for future analysis and improvement here. This work is also compared with
other contemporary research. The advantages and disadvantages of them are
analyzed. The STV approach provides an improved computational approach
by combing the time discretization method, the control step function method,
the time variable method, the consideration of transaction costs and by coding
the computational requirements in a widely used programming system MAT-
LAB. The computational experiments validated the STV approach in terms of
computational efficiency, and time, and the plausibility of results for financial
analysis.
The present book also provides a unique example of the feasibility of model-
ing and computation of the financial system based on bang-bang control meth-
ods. The computed results provide useful information about the dynamics of
the financial system, the impact of switching times, the role of transaction
costs, and the strategy for achieving a global optimum in a financial system.
One of the areas of applications of optimal control models is normative so-
cial choice for optimal financial decision making. The optimal control models
in this book have this application as well. These models specify the welfare
maximizing financial resource allocation in the economy subject to the under-
lying dynamic financial system.
Chapter 1 is an introduction to the optimal control problems in finance and
the classical optimal control theories, which have been successfully used for
years. Some relevant sources in this research field are also introduced and
discussed.
Chapter 2 discusses a particular case of optimal control problems and the
switching time variable (STV) algorithm. Some useful transformations intro-
duced in Section 2.2 are standard for the control problems. The piecewise-
linear transformation and the computational algorithms discussed in Section
2.6 are the main work in this book. A simple optimal aggregate investment
planning model is presented here. Accurate results were obtained in using
these computational algorithms, and are presented in Section 2.7. A part of
the computer software SCOM developed in Craven and Islam [18, 2001] and
Islam and Craven [38, 2002] is used here to solve the differential equation.
Chapter 3 presents a financial oscillator model (which is a different version
of the optimal aggregative investment planning model developed in Chapter 2)
whose state is a second-order differential equation. A new time-scaled trans-
formation is introduced in Section 3.3. The new transformation modifies the
old transformations that are used in Chapter 2. All the modifications are made
xviii OPTIMAL CONTROL MODELS IN FINANCE

to match the new time-scale division. The computational algorithms for this
problem and the computing results are also discussed. An extension of the con-
trol pattern is indicated. The new transformation and algorithms in this chapter
are the important parts in this research.
Chapter 4 contains an optimal financing model, which was first introduced
by Davis and Elzinger [22, 1970]. A computer software package for this
model is constructed in this book (for details see Appendix A.3 model 1_1.m -
model1_5.m). The computing result is compared with the analytical result and
another computing result obtained from using the SCOM package.
Chapter 5 reports computing results of the algorithms 2.1-2.3 and algorithms
3.1-3.4 in other cases of optimal control problems. After analyzing the results,
the computer packages in Appendix A.1 and Appendix A.2 (project1_1.m -
project1_4.m and project2_2.m - project2_4.m) have been improved.
Chapter 6 gives the conclusion of this research. Optimal control methods
have high potential applications to various areas in finance. The present study
has enhanced the state of the art for applying optimal control methods, es-
pecially the bang-bang control method, for financial modeling in a real life
context.
Chapter 1

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