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The Mathematics of Financial Modeling and Investment Management SERGIO M. FOCARDI FRANK J. FABOZZI John Wiley & Sons, Inc. SMF To Dominique, Leila, Guillaume, and Richard FJF To my beautiful wife Donna and my children, Francesco, Patricia, and Karly Copyright © 2004 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-750-4470, or on the web at www.copyright.com. 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ISBN: 0-471-46599-2 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Contents Preface Acknowledgments About the Authors Commonly Used Symbols Abbreviations and Acronyms CHAPTER 1 From Art to Engineering in Finance Investment Management Process Step 1: Setting Investment Objectives Step 2: Establishing an Investment Policy Step 3: Selecting a Portfolio Strategy Step 4: Selecting the Specific Assets Step 5: Measuring and Evaluating Performance Financial Engineering in Historical Perspective The Role of Information Technology Industry’s Evaluation of Modeling Tools Integrating Qualitative and Quantitative Information Principles for Engineering a Suite of Models Summary CHAPTER 2 Overview of Financial Markets, Financial Assets, and Market Participants Financial Assets Financial Markets Classification of Financial Markets Economic Functions of Financial Markets Secondary Markets Overview of Market Participants Role of Financial Intermediaries Institutional Investors Insurance Companies Pension Funds Investment Companies Depository Institutions Endowments and Foundations Common Stock xiv xvi xviii xix xx 1 2 2 2 6 7 9 10 11 13 15 17 18 21 21 25 25 26 27 34 35 37 41 41 42 43 45 45 iii iv Contents Trading Locations Stock Market Indicators Trading Arrangements Bonds Maturity Par Value Coupon Rate Provisions for Paying off Bonds Options Granted to Bondholders Futures and Forward Contracts Futures versus Forward Contracts Risk and Return Characteristics of Futures Contracts Pricing of Futures Contracts The Role of Futures in Financial Markets Options Risk-Return for Options The Option Price Swaps Caps and Floors Summary CHAPTER 3 Milestones in Financial Modeling and Investment Management The Precursors: Pareto, Walras, and the Lausanne School Price Diffusion: Bachelier The Ruin Problem in Insurance: Lundberg The Principles of Investment: Markowitz Understanding Value: Modigliani and Miller Modigliani-Miller Irrelevance Theorems and the Absence of Arbitrage Efficient Markets: Fama and Samuelson Capital Asset Pricing Model: Sharpe, Lintner, and Mossin The Multifactor CAPM: Merton Arbitrage Pricing Theory: Ross Arbitrage, Hedging, and Option Theory: Black, Scholes, and Merton Summary CHAPTER 4 Principles of Calculus Sets and Set Operations Proper Subsets Empty Sets Union of Sets Intersection of Sets Elementary Properties of Sets Distances and Quantities n-tuples Distance 45 46 48 51 51 52 52 55 56 57 58 59 59 63 64 66 66 69 70 71 75 76 78 80 81 83 84 85 86 87 88 89 90 91 93 93 95 95 95 96 96 97 98 Contents Density of Points Functions Variables Limits Continuity Total Variation Differentiation Commonly Used Rules for Computing Derivatives Higher Order Derivatives Application to Bond Analysis Taylor Series Expansion Application to Bond Analysis Integration Riemann Integrals Properties of Riemann Integrals Lebesque-Stieltjes Integrals Indefinite and Improper Integrals The Fundamental Theorem of Calculus Integral Transforms Laplace Transform Fourier Transforms Calculus in More than One Variable Summary CHAPTER 5 Matrix Algebra Vectors and Matrices Defined Vectors Matrices Square Matrices Diagonals and Antidiagonals Identity Matrix Diagonal Matrix Upper and Lower Triangular Matrix Determinants Systems of Linear Equations Linear Independence and Rank Hankel Matrix Vector and Matrix Operations Vector Operations Matrix Operations Eigenvalues and Eigenvectors Diagonalization and Similarity Singular Value Decomposition Summary CHAPTER 6 Concepts of Probability Representing Uncertainty with Mathematics Probability in a Nutshell v 99 100 101 102 103 105 106 107 111 112 121 122 127 127 129 130 131 132 134 134 137 138 139 141 141 141 144 145 145 146 146 148 148 149 151 152 153 153 156 160 161 162 163 165 165 167 vi Contents Outcomes and Events Probability Measure Random Variables Integrals Distributions and Distribution Functions Random Vectors Stochastic Processes Probabilistic Representation of Financial Markets Information Structures Filtration Conditional Probability and Conditional Expectation Moments and Correlation Copula Functions Sequences of Random Variables Independent and Identically Distributed Sequences Sum of Variables Gaussian Variables The Regression Function Linear Regression Summary CHAPTER 7 Optimization Maxima and Minima Lagrange Multipliers Numerical Algorithms Linear Programming Quadratic Programming Calculus of Variations and Optimal Control Theory Stochastic Programming Summary CHAPTER 8 Stochastic Integrals The Intuition Behind Stochastic Integrals Brownian Motion Defined Properties of Brownian Motion Stochastic Integrals Defined Some Properties of Itô Stochastic Integrals Summary CHAPTER 9 Differential Equations and Difference Equations Differential Equations Defined Ordinary Differential Equations Order and Degree of an ODE Solution to an ODE Systems of Ordinary Differential Equations 169 170 171 172 172 174 175 178 180 181 182 184 186 188 189 191 191 194 197 197 199 201 202 204 206 206 211 212 214 216 217 219 225 230 232 236 237 239 240 240 241 241 243 Contents Closed-Form Solutions of Ordinary Differential Equations Linear Differential Equation Numerical Solutions of Ordinary Differential Equations The Finite Difference Method Nonlinear Dynamics and Chaos Fractals Partial Differential Equations Diffusion Equation Solution of the Diffusion Equation Numerical Solution of PDEs Summary CHAPTER 10 Stochastic Differential Equations The Intuition Behind Stochastic Differential Equations Itô Processes The 1-Dimensional Itô Formula Stochastic Differential Equations Generalization to Several Dimensions Solution of Stochastic Differential Equations The Arithmetic Brownian Motion The Ornstein-Uhlenbeck Process The Geometric Brownian Motion Summary CHAPTER 11 Financial Econometrics: Time Series Concepts, Representations, and Models Concepts of Time Series Stylized Facts of Financial Time Series Infinite Moving-Average and Autoregressive Representation of Time Series Univariate Stationary Series The Lag Operator L Stationary Univariate Moving Average Multivariate Stationary Series Nonstationary Series ARMA Representations Stationary Univariate ARMA Models Nonstationary Univariate ARMA Models Stationary Multivariate ARMA Models Nonstationary Multivariate ARMA Models Markov Coefficients and ARMA Models Hankel Matrices and ARMA Models State-Space Representation Equivalence of State-Space and ARMA Representations Integrated Series and Trends Summary vii 246 247 249 249 256 258 259 259 261 263 265 267 268 271 272 274 276 278 280 280 281 282 283 284 286 288 288 289 292 293 295 297 297 300 301 304 304 305 305 308 309 313 viii Contents CHAPTER 12 Financial Econometrics: Model Selection, Estimation, and Testing Model Selection Learning and Model Complexity Maximum Likelihood Estimate Linear Models of Financial Time Series Random Walk Models Correlation Random Matrices Multifactor Models CAPM Asset Pricing Theory (APT) Models PCA and Factor Models Vector Autoregressive Models Cointegration State-Space Modeling and Cointegration Empirical Evidence of Cointegration in Equity Prices Nonstationary Models of Financial Time Series The ARCH/GARCH Family of Models Markov Switching Models Summary CHAPTER 13 Fat Tails, Scaling, and Stable Laws Scaling, Stable Laws, and Fat Tails Fat Tails The Class L of Fat-Tailed Distributions The Law of Large Numbers and the Central Limit Theorem Stable Distributions Extreme Value Theory for IID Processes Maxima Max-Stable Distributions Generalized Extreme Value Distributions Order Statistics Point Process of Exceedances or Peaks over Threshold Estimation Eliminating the Assumption of IID Sequences Heavy-Tailed ARMA Processes ARCH/GARCH Processes Subordinated Processes Markov Switching Models Estimation Scaling and Self-Similarity Evidence of Fat Tails in Financial Variables On the Applicability of Extreme Value Theory in Finance Summary 315 315 317 319 324 324 327 329 332 334 335 335 338 339 342 343 345 346 347 349 351 352 352 353 358 360 362 362 368 368 369 371 373 378 381 382 383 384 384 385 388 391 392 Contents CHAPTER 14 Arbitrage Pricing: Finite-State Models The Arbitrage Principle Arbitrage Pricing in a One-Period Setting State Prices Risk-Neutral Probabilities Complete Markets Arbitrage Pricing in a Multiperiod Finite-State Setting Propagation of Information Trading Strategies State-Price Deflator Pricing Relationships Equivalent Martingale Measures Risk-Neutral Probabilities Path Dependence and Markov Models The Binomial Model Risk-Neutral Probabilities for the Binomial Model Valuation of European Simple Derivatives Valuation of American Options Arbitrage Pricing in a Discrete-Time, Continuous-State Setting APT Models Testing APT Summary CHAPTER 15 Arbitrage Pricing: Continuous-State, Continuous-Time Models The Arbitrage Principle in Continuous Time Trading Strategies and Trading Gains Arbitrage Pricing in Continuous-State, Continuous-Time Option Pricing Stock Price Processes Hedging The Black-Scholes Option Pricing Formula Generalizing the Pricing of European Options State-Price Deflators Equivalent Martingale Measures Equivalent Martingale Measures and Girsanov’s Theorem The Diffusion Invariance Principle Application of Girsanov’s Theorem to Black-Scholes Option Pricing Formula Equivalent Martingale Measures and Complete Markets Equivalent Martingale Measures and State Prices Arbitrage Pricing with a Payoff Rate Implications of the Absence of Arbitrage Working with Equivalent Martingale Measures Summary ix 393 393 395 397 398 399 402 402 403 404 405 414 416 423 423 426 427 429 430 435 436 439 441 441 443 445 447 447 448 449 452 454 457 459 461 462 463 464 466 467 468 468 x Contents CHAPTER 16 Portfolio Selection Using Mean-Variance Analysis Diversification as a Central Theme in Finance Markowitz’s Mean-Variance Analysis Capital Market Line Deriving the Capital Market Line What is Portfolio M? Risk Premium in the CML The CML and the Optimal Portfolio Utility Functions and Indifference Curves Selection of the Optimal Portfolio Extension of the Markowitz Mean-Variance Model to Inequality Constraints A Second Look at Portfolio Choice The Return Forecast The Utility Function Optimizers A Global Probabilistic Framework for Portfolio Selection Relaxing the Assumption of Normality Multiperiod Stochastic Optimization Application to the Asset Allocation Decision The Inputs Portfolio Selection: An Example Inclusion of More Asset Classes Extensions of the Basic Asset Allocation Model Summary CHAPTER 17 Capital Asset Pricing Model CAPM Assumptions Systematic and Nonsystematic Risk Security Market Line Estimating the Characteristic Line Testing The CAPM Deriving the Empirical Analogue of the CML Empricial Implications General Findings of Empirical Tests of the CAPM A Critique of Tests of the CAPM Merton and Black Modifications of the CAPM CAPM and Random Matrices The Conditional CAPM Beta, Beta Everywhere The Role of the CAPM in Investment Management Applications Summary CHAPTER 18 Multifactor Models and Common Trends for Common Stocks Multifactor Models Determination of Factors 471 472 474 477 478 481 482 482 482 484 485 487 487 488 490 490 491 492 494 495 500 503 507 509 511 512 513 516 518 518 518 519 520 520 521 522 523 524 525 526 529 530 532 Contents Dynamic Market Models of Returns Estimation of State-Space Models Dynamic Models for Prices Estimation and Testing of Cointegrated Systems Cointegration and Financial Time Series Nonlinear Dynamic Models for Prices and Returns Summary CHAPTER 19 Equity Portfolio Management Integrating the Equity Portfolio Management Process Active versus Passive Portfolio Management Tracking Error Backward-Looking versus Forward-Looking Tracking Error The Impact of Portfolio Size, Benchmark Volatility, and Portfolio Beta on Tracking Error Equity Style Management Types of Equity Styles Style Classification Systems Passive Strategies Constructing an Indexed Portfolio Index Tracking and Cointegration Active Investing Top-Down Approaches to Active Investing Bottom-Up Approaches to Active Investing Fundamental Law of Active Management Strategies Based on Technical Analysis Nonlinear Dynamic Models and Chaos Technical Analysis and Statistical Nonlinear Pattern Recognition Market-Neutral Strategies and Statistical Arbitrage Application of Multifactor Risk Models Risk Decomposition Portfolio Construction and Risk Control Assessing the Exposure of a Portfolio Risk Control Against a Stock Market Index Tilting a Portfolio Summary CHAPTER 20 Term Structure Modeling and Valuation of Bonds and Bond Options Basic Principles of Valuation of Debt Instruments Yield-to-Maturity Measure Premium Par Yield Reinvestment of Cash Flow and Yield The Term Structure of the Interest Rates and the Yield Curve Limitations of Using the Yield to Value a Bond Valuing a Bond as a Package of Cash Flows Obtaining Spot Rates from the Treasury Yield Curve Using Spot Rates to the Arbitrage-Free Value of a Bond xi 537 538 538 543 544 546 549 551 551 552 553 555 556 560 560 562 564 564 565 566 566 567 568 571 573 574 575 577 577 582 583 587 587 589 593 594 596 598 598 599 602 603 603 606 xii Contents The Discount Function Forward Rates Swap Curve Classical Economic Theories About the Determinants of the Shape of the Term Structure Expectations Theories Market Segmentation Theory Bond Valuation Formulas in Continuous Time The Term Structure of Interest Rates in Continuous Time Spot Rates: Continuous Case Forward Rates: Continuous Case Relationships for Bond and Option Valuation The Feynman-Kac Formula Multifactor Term Structure Model Arbitrage-Free Models versus Equilibrium Models Examples of One-Factor Term Structure Models Two-Factor Models Pricing of Interest-Rate Derivatives The Heath-Jarrow-Morton Model of the Term Structure The Brace-Gatarek-Musiela Model Discretization of Itô Processes Summary CHAPTER 21 Bond Portfolio Management Management versus a Bond Market Index Tracking Error and Bond Portfolio Strategies Risk Factors and Portfolio Management Strategies Determinants of Tracking Error Illustration of the Multifactor Risk Model Liability-Funding Strategies Cash Flow Matching Portfolio Immunization Scenario Optimization Stochastic Programming Summary CHAPTER 22 Credit Risk Modeling and Credit Default Swaps Credit Default Swaps Single-Name Credit Default Swaps Basket Default Swaps Legal Documentation Credit Risk Modeling: Structural Models The Black-Scholes-Merton Model Geske Compound Option Model Barrier Structural Models Advantages and Drawbacks of Structural Models Credit Risk Modeling: Reduced Form Models 606 607 608 612 613 618 618 623 624 625 626 627 632 634 635 638 638 640 643 644 646 649 649 651 652 654 654 661 664 667 672 673 677 679 679 680 681 683 683 685 690 694 696 696 Contents The Poisson Process The Jarrow-Turnbull Model Transition Matrix The Duffie-Singleton Model General Observations on Reduced Form Models Pricing Single-Name Credit Default Swaps General Framework Survival Probability and Forward Default Probability: A Recap Credit Default Swap Value No Need For Stochastic Hazard Rate or Interest Rate Delivery Option in Default Swaps Default Swaps with Counterparty Risk Valuing Basket Default Swaps The Pricing Model How to Model Correlated Default Processes Summary CHAPTER 23 Risk Management Market Completeness The Mathematics of Market Completeness The Economics of Market Completeness Why Manage Risk? Risk Models Market Risk Credit Risk Operational Risk Risk Measures Risk Management in Asset and Portfolio Management Factors Driving Risk Management Risk Measurement in Practice Getting Down to the Lowest Level Regulatory Implications of Risk Measurement Summary INDEX xiii 697 698 703 706 710 710 711 712 713 716 716 717 718 718 722 734 737 738 739 742 744 745 745 746 746 747 751 752 752 753 754 755 757 Preface Since the pioneering work of Harry Markowitz in the 1950s, sophisticated statistical and mathematical techniques have increasingly made their way into finance and investment management. One might question whether all this mathematics is justified, given the present state of economics as a science. However, a number of laws of economics and finance theory with a bearing on investment management can be considered empirically well established and scientifically sound. This knowledge can be expressed only in the language of statistics and mathematics. As a result, practitioners must now be familiar with a vast body of statistical and mathematical techniques. Different areas of finance call for different mathematics. Investment management is primarily concerned with understanding hard facts about financial processes. Ultimately the performance of investment management is linked to an understanding of risk and return. This implies the ability to extract information from time series that are highly noisy and appear nearly random. Mathematical models must be simple, but with a deep economic meaning. In other areas, the complexity of instruments is the key driver behind the growing use of sophisticated mathematics in finance. There is the need to understand how relatively simple assumptions on the probabilistic behavior of basic quantities translate into the potentially very complex probabilistic behavior of financial products. Derivatives are the typical example. This book is designed to be a working tool for the investment management practitioner, student, and researcher. We cover the process of financial decision-making and its economic foundations. We present financial models and theories, including CAPM, APT, factor models, models of the term structure of interest rates, and optimization methodologies. Special emphasis is put on the new mathematical tools that allow a deeper understanding of financial econometrics and financial economics. For example, tools for estimating and representing the tails of the distributions, the analysis of correlation phenomena, and dimensionality reduction through factor analysis and cointegration are recent advances in financial economics that we discuss in depth. xiv xv Preface Special emphasis has been put on describing concepts and mathematical techniques, leaving aside lengthy demonstrations, which, while the substance of mathematics, are of limited interest to the practitioner and student of financial economics. From the practitioner’s point of view, what is important is to have a firm grasp of the concepts and techniques, which will allow one to interpret the results of simulations and analyses that are now an integral part of finance. There is no prerequisite mathematical knowledge for reading this book: all mathematical concepts used in the book are explained, starting from ordinary calculus and matrix algebra. It is, however, a demanding book given the breadth and depth of concepts covered. Mathematical concepts are in bolded type when they appear for the first time in the book, economic and finance concepts are italicized when they appear for the first time. In writing this book, special attention was given to bridging the gap between the intuition of the practitioner and academic mathematical analysis. Often there are simple compelling reasons for adopting sophisticated concepts and techniques that are obscured by mathematical details; whenever possible, we tried to give the reader an understanding of the reasoning behind these concepts. The book has many examples of how quantitative analysis is used in practice. These examples help the reader appreciate the connection between quantitative analysis and financial decision-making. A distinctive feature of this book is the integration of notions deeply rooted in the practice of investment management with methods based on finance theory and statistical analysis. Sergio M. Focardi Frank J. Fabozzi Acknowledgments We are grateful to Professor Ren-Raw Chen of Rutgers University for coauthoring Chapter 22 (“Credit Risk Modeling and Credit Default Swaps”). The application of mean-variance analysis to asset allocation in Chapter 16 is from the coauthored work of Frank Fabozzi with Harry Markowitz and Francis Gupta. The discussion of tracking error and risk decomposition in Chapter 18 draws from the coauthored work of Frank Fabozzi with Frank Jones and Raman Vardharaj. In writing a book that covers a wide range of technical topics in mathematics and finance, we were fortunate enough to receive assistance from the following individuals: ■ Caroline Jonas of The Intertek Group read and commented on most chapters in the book. ■ Dr. Petter Kolm of Goldman Sachs Asset Management reviewed Chap- ters 4, 6, 7, 9, and 20. ■ Dr. Bernd Hanke of Goldman Sachs Asset Management reviewed Chapters 14, 15, and 16. ■ Dr. Lisa Goldberg of Barra reviewed Chapter 13. ■ Professor Martijn Cremers of Yale University reviewed the first draft of the financial econometrics material. ■ Hafize Gaye Erkan, a Post-General Ph.D. Candidate in the Department ■ ■ ■ ■ ■ xvi of Operations Research and Financial Engineering at Princeton University, reviewed the chapters on stochastic calculus (Chapters 8 and 10). Professor Antti Petajisto of Yale University reviewed Chapter 14. Dr. Christopher Maloney of Citigroup reviewed Chapter 5. Dr. Marco Raberto of the University of Genoa reviewed Chapter 13 and provided helpful support for the preparation of illustrations. Dr. Mehmet Gokcedag of the Istanbul Bilgi University reviewed Chapter 22 and provided helpful comments on the organization and structure of the book. Professor Silvano Cincotti of the University of Genoa provided insightful comments on a range of topics. Acknowledgments xvii ■ Dr. Lev Dynkin and members of the Fixed Income Research Group at Lehman Brothers reviewed Chapter 21. ■ Dr. Srichander Ramaswamy of the Bank for International Settlement prepared the illustration in Chapter 13 to show the importance of fattailed processes in credit risk management based on his book Managing Credit Risk in Corporate Bond Portfolios: A Practitioner’s Guide. ■ Hemant Bhangale of Morgan Stanley reviewed Chapter 23. Finally, Megan Orem typeset the book and provided editorial assistance. We appreciate her patience and understanding in working through several revisions of the chapters and several reorganizations of the table of contents. About the Authors Sergio Focardi is a founding partner of the Paris-based consulting firm The Intertek Group. Sergio lectures at CINEF (Center for Interdisciplinary Research in Economics and Finance) at the University of Genoa and is a member of the Editorial Board of the Journal of Portfolio Management. He has published numerous articles on econophysics and coauthored two books, Modeling the Markets: New Theories and Techniques and Risk Management: Framework, Methods and Practice. His research interests include modeling the interaction between multiple heterogeneous agents and the econometrics of large equity portfolios based on cointegration and dynamic factor analysis. Sergio holds a degree in Electronic Engineering from the University of Genoa and a postgraduate degree in Communications from the Galileo Ferraris Electrotechnical Institute (Turin). Frank J. Fabozzi, Ph.D., CFA, CPA is the Frederick Frank Adjunct Professor of Finance in the School of Management at Yale University. Prior to joining the Yale faculty, he was a Visiting Professor of Finance in the Sloan School of Management at MIT. Frank is a Fellow of the International Center for Finance at Yale University, the editor of the Journal of Portfolio Management, a member of Princeton University’s Advisory Council for the Department of Operations Research and Financial Engineering, and a trustee of the BlackRock complex of closed-end funds and Guardian Life sponsored open-end mutual funds. He has authored several books in investment management and in 2002 was inducted into the Fixed Income Analysts Society’s Hall of Fame. Frank earned a doctorate in economics from the City University of New York in 1972. xviii Commonly Used Symbols A(L) β ∆ εt ⋅ + T adj |A| B ℑ Rα ∪ ∩ ∈ ∉ → ∑ polynomial in the lag operator L k-vector [β1...βk]′ difference operator error, usually white noise vector scalar product x ⋅ y also written xy sum of vector or matrices A + B transpose of a vector or matrix AT adjoint of a matrix determinant of a matrix Borel σ-algebra Filtration regularly varying functions of index α union of sets intersection of sets belongs to does not belong to tends to summation with implicit range N ∑ summation over range shown ∏ product with implicit range i=1 N ∏ product over range shown Φ(x) Ω E[X] E[X|Z] cdf of the standardized normal sample space expectation conditional expectation i=1 xix