Stochastic Modeling in Economics and Finance Applied Optimization Volume 75 Series Editors: Panos M. Pardalos University of Florida, U.S.A. Donald Hearn University of Florida, U.S.A. Stochastic Modeling in Economics and Finance by Jan Hurt and Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University, Prague KLUWER ACADEMIC PUBLISHERS NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW eBookISBN: 0-306-48167-7 Print ISBN: 1-4020-0840-6 ©2003 Kluwer Academic Publishers NewYork, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: http://kluweronline.com and Kluwer's eBookstore at: http://ebooks.kluweronline.com To my husband Václav To Jarmila, Eva, and in memory of my parents Jan Hurt To my wife Iva v CONTENTS Preface xi Acknowledgments xiii Part I Fundamentals I.1 Money, Capital, and Securities 1 1.1 Money and Capital 1 1.2 Investment 1 1.3Interest 1 1.4 Cash Flows 2 1.5 Financial and Real Investment 2 1.6 Securities 3 1.7 Financial Market 12 1.8 Financial Institutions 12 1.9 Financial System 12 I.2Interest Rate 13 2.1 Simple and Compound Interest 13 2.2 Calendar Conventions 14 2.3 Determinants of the Interest Rate 15 2.4 Decomposition ofthe Interest Rate 16 2.5 Term Structure of Interest Rates 18 2.6ContinuousCompounding 19 I.3 Measures ofCash Flows 21 3.1 Present Value 21 3.2 Annuities 23 3.3FutureValue 24 3.4 Internal Rate of Return 26 3.5 Duration 29 3.6 Convexity 30 3.7 Comparison of Investment Projects 31 3.8 Yield Curves 36 I.4Return, Expected Return, and Risk 39 4.1 Return 39 4.2 RiskMeasurement 43 I.5 Valuation of Securities 48 5.1 Coupon Bonds 48 5.2Options 52 5.3 Forwards and Futures 63 I.6 Matching of Assets and Liabilities 64 6.1 Matching and Immunization 64 6.2 Dedicated Bond Portfolio 65 6.3 A Stochastic Model ofMatching 67 I.7 IndexNumbers and Inflation 68 7.1 Construction ofIndex Numbers 68 7.2 Stock Exchange Indicators 70 7.3Inflation 71 vii I.8 Basics ofUtility Theory 73 8.1 The Concept ofUtility 73 8.2 Utility Function 73 8.3 Characteristics ofUtility Functions 74 8.4 Some Particular Utility Functions 75 8.5 Risk Considerations 76 8.6 Certainty Equivalent 77 I.9MarkowitzMean-VariancePortfolio 79 9.1 Portfolio 80 9.2 Construction ofOptimal Portfolios and Separation Theorems 81 I.10 Capital Asset Pricing Model 92 10.1 Sharpe-Lintner Model 92 10.2 Security Market Line 93 10.3 Capital Market Line 95 I.11 Arbitrage Pricing Theory 96 11.1 Regression Model 96 11.2 Factor Model 97 I.12 Bibliographical Notes 101 Part II Discrete Time Stochastic Decision Models II.1 Introduction and Preliminaries 103 1.1 Problem of a Private Investor 104 1.2 Stochastic Dedicated Bond Portfolio 105 1.3 Mathematical Programs 106 II.2 Multistage Stochastic Programs 108 2.1 Basic Formulations 108 2.2 Scenario-Based Stochastic Linear Programs 112 2.3 Horizon and Stages 115 2.4 The Flower-Girl Problem 117 2.5 Comparisonwith Stochastic Dynamic Programming 119 II.3 Multiple Criteria 123 3.1 Theory 123 3.2 Selected Applications to Portfolio Optimization 127 3.3 Multi-Objective Optimization and Stochastic ProgrammingModels 131 II.4 Selected Applications in Finance and Economics 137 4.1 PortfolioRevision 137 4.2TheBONDSModel 139 4.3 Bank Asset and Liability Management – Model ALM 141 4.4 General Features ofMultiperiod Stochastic Programs in Finance 144 4.5 Production Planning 148 4.6 Capacity Expansion ofElectric Power Generation Systems – CEP 150 4.7 Unit Commitment and Economic Power Dispatch Problem 153 4.8 Melt Control: Charge Optimization 154 II.5 Approximation Via Scenarios 158 5.1 Introduction 158 5.2 Scenarios and their Generation 159 5.3 HowtoDrawInference aboutthe TrueProblem 164 viii 5.4 Scenario Trees for Multistage Stochastic Programs 169 II.6 Case Study: Bond Portfolio Management Problem 180 6.1 The Problem and the Input Data 180 6.2 The Model and the Structure ofthe Program 182 6.3 Generation of Scenarios 187 6.4 Selected Numerical Results 190 6.5 “What if” Analysis 192 6.6 Discussion 197 II.7 Incomplete Input Information 199 7.1 Sensitivity for the Black-Scholes Formula 199 7.2MarkowitzMean-VarianceModel 200 7.3 Incomplete Information about Liabilities 204 II.8 Numerical Techniques and Available Software (by Pavel Popela) 206 8.1 Motivation 206 8.2 Common OptimizationTechniques 208 8.3 Solution Techniques for Two-Stage Stochastic Programs 214 8.4 Solution Techniques for Multistage Stochastic Programs 218 8.5 Approximation Techniques 224 8.6 ModelManagement 226 II.9 Bibliographical Notes 228 Part III Stochastic Analysis and Diffusion Finance III.1 Martingales 231 1.1 Stochastic Processes 231 1.2 Brownian Motion and Martingales 238 1.3 Markov Times and Stopping Theorem 244 1.4 Local Martingales and Complete Filtrations 252 1.5 and Density Theorem 257 1.6Doob-MeyerDecomposition 263 1.7 Quadratic Variation ofLocal Martingales 269 1.8 Helps to Some Exercises 275 III.2 Stochastic Integration 277 2.1 StochasticIntegral 277 2.2 Stochastic Per Partes and Itô Formula 286 2.3 Exponential Martingales and Lévy Theorem 295 2.4 Girsanov Theorem 300 2.5 Integral and Brownian Representations 308 2.6 Helps to Some Exercises 316 III.3 Diffusion Financial Mathematics 319 3.1 Black-Scholes Calculus 319 3.2 Girsanov Calculus 333 3.3 Market Regulations and Option Pricing 350 3.4 Helps to Some Exercises 363 III. 4 Bibliographical Notes 366 References 369 Index 377 ix PREFACE The three authors of this book are my colleagues (moreover, one of them is my wife). I followed their work on the book from initial discussions about its con(cid:1) cept, through disputes over notation, terminology and technicalities, till bringing the manuscript to its present form. I am honored by having been asked to write the preface. The book consists of three Parts. Though they may seem disparate at first glance, they are purposively tied together. Many topics are discussed in all three Parts, always from a different point of view or on a different level. Part I presents basics of financial mathematics including some supporting topics, such as utility or index numbers. It is very concise, covering a surprisingly broad range of concepts and statements about them on not more than 100 pages. The mathematics of this Part is undemanding but precise within the limits of the chosen level. Being primarily an introductory text for a beginner, Part I will be useful to the enlightened reader as well, as a manual of notions and formulas used later on. The more extensive Part II deals with stochastic decision models. Multistage stochastic programming is the main methodology here. The scenario(cid:1)based approach is adopted with special attention to scenarios generation and via scenarios appro(cid:1) ximation. The output analysis is discussed, i.e. the question how to draw inference about the true problem from the approximating one. Numerous applications of the presented theory vary from portfolio optimal control to planning electric power ge(cid:1) neration systems or to managing technological processes. A case study on a bond investment problem is reported in detail. A survey of numerical techniques and available software is added. Mathematics of Part II is still of standard level but the application of the presented methods may be laborious. The final Part III requires from the reader higher mathematical education inclu(cid:1) ding measure(cid:1)theoretical probability theory. In fact, Part III is a brief textbook on stochastic analysis oriented to what is called diffusion financial mathematics. The apparatus built up in chapters on martingales and on stochastic integration leads to a precise formulation and to rigorous proving of many results talked about already in Part I. The author calls his proofs honest; indeed, he does not facilitate his task by unnecessarily simplifying assumptions or by skipping laborious algebra. The audience of the book may be diverse. Students in mathematics interested in applications to economics and finance may read with benefit all Parts I,II,III and then study deeper those topics they find most attractive. Students and researchers in economics and finance may learn from the book of using stochastic methods in their fields. Specialists in optimization methods or in numerical mathematics will get acquainted with important optimization problems in finance and economics and with their numerical solution, mainly through Part II of the book. The probabilistic Part III can be appreciated especially by professional mathematicians; otherwise, this Part will be a challenge to the reader to raise his/her mathematical culture. After all, a challenge is present in all three Parts of the book through numerous unsolved exercises and through suggestions for further reading given in bibliographical notes. I wish the book many readers with deep interest. xi ACKNOWLEDGMENTS This volume could not come into being without support of several institutions and a number of individuals. We wish to express our sincere gratitude to every one of them. First of all we thank to Ministry of Education of Czech Republic1, Grant Agency of Czech Republic2 and Directorate General III (Industry) of the European Com- mission3 who supported the scientific and applied projects listed below that sub- stantially influenced the contents and form of the text. We gratefully acknowledge the financial support from the companies NEWTON Investment Ltd and ALAX Ltd and appreciate the particularly helpful technical assistance provided by the Czech Statistical Office. The authors are very indebted to Pavel Popela from the Brno University of Technology who, using his extensive experience with the numerical solutions to the problems in the field of Stochastic Programing, wrote Chapter II.8. Horrand I. Gassmann from the Dalhousie University read very carefully this Chapter and offered some valuable proposals for improvements. We thank also Marida Bertocchi from the University of Bergamo whose effective cooperation within the project (3) influenced the presentation of results in Chapter II.6. We have to say many thanks, indeed, to our colleagues and friends and Josef Machek who agreed to read the text. They expended a great effort using their extensive knowledge both of Mathematics and English to make many invaluable suggestions, pressing for higher clarity and consistency of our presentation. Further, we are particularly grateful to Jarom(cid:1)r Antoch for his invaluable help in the process of technical preparation of the book. The authors are also indebted to their present and former PhD students at the Charles University of Prague: Alena Henclová and deserve credits for their efficient and swift technical assistance. Part III owes much to Petr Dostál, Daniel Hlubinka, and who, cruelly tried out as the first readers, have then become enthusiastic and respected critics. Finally, we thank our publisher KluwerAcademic Publishers and, above all, the senior editor John R. Martindale for publishing the book. J. Hurt, and 1 MSM 1132000008 Mathematical Methods in Stochastics 2 402/99/1136, 201/99/0264, 201/00/0770 3 INCO’95, HPC/Finance Project, no. 951139 xiii