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Chen P., Islam S.M.N. Optimal Control Models in Finance A New Computational Approach (2005)Chen P., Islam S.M.N. 
Contents List of Figures ix List of Tables xi Preface xiii Introduction xv 1. OPTIMAL CONTROL MODELS 1 1 An Optimal Control Model of Finance 2 2 (Karush) Kuhn-TuckerCondition 4 3 Pontryagin Theorem 6 4 Bang-Bang Control 7 5 Singular Arc 7 6 Indifference Principle 8 7 Different Approaches to Optimal Control Problems 10 8 Conclusion 20 2. THE STV APPROACH TO FINANCIAL OPTIMAL CONTROL MODELS 21 1 Introduction 21 2 Piecewise-linear Transformation 21 3 Non-linear Time Scale Transformation 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 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 5 Transformations for the Computer Software Package for the Finance Model 99 6 Computational Algorithms for the Non-linear Optimal Control Problem 101 7 Computing Results and Conclusion 104 8 Optimal Financing Implications 107 9 Conclusion 108 5. FURTHER COMPUTATIONAL EXPERIMENTS AND RESULTS 109 1 Introduction 109 2 Different Fitting Functions 109 3 The Financial Oscillator Model when the Control Takes Three Values 120 4 Conclusion 139 6. CONCLUSION 141 Appendices 145 A CSTVA Program List 145 1 Program A: Investment Model in Chapter 2 145 2 Program B: Financial Oscillator Model in Chapter 3 149 3 Program C: Optimal Financing Model in Chapter 4 153 4 Program D: Three Value-Control Model in Chapter 5 156 B Some Computation Results 161 1 Results for Program A 161 2 Results for Program B 163 3 Results for Program C 167 4 Results for Program D 175 C Differential Equation Solver from the SCOM Package 181 D SCOM Package 183 E Format of Problem Optimization 189 F A Sample Test Problem 191 References 193 Index 199 Preface This book reports initial efforts in providing some useful extensions in financial 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 extensions 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. Professional practitioners in the above areas will find the book interesting and informative. 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 equation 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 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 considerably 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 theoretical 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 solutions, 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 switching 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 theorem 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 computation, 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 compute 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 problems. 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 particular 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 subdivision 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 setting up all the parameters, calculating the results, and testing different initial points of an iterative algorithm. During the examination of the algorithms, it 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 modeling and computation of the financial system based on bang-bang control methods. 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 social 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 underlying 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 introduced 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 transformation is introduced in Section 3.3. The new transformation modifies the old transformations that are used in Chapter 2. All the modifications are made to match the new time-scale division. The computational algorithms for this problem and the computing results are also discussed. An extension of the control 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, especially the bang-bang control method, for financial modeling in a real life context. |
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Optimal Control Models in Finance A New Computational Approach (2005)