LINEBURG

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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:

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

Boston

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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 вЂњnqqвЂќ 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

ownersвЂ™ 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 вЂњconstrвЂќ 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

OPTIMAL CONTROL MODELS IN FINANCE

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