Frontmatter Page i Monday, March 8, 2004 10:06 AM The Mathematics of Financial Modeling and Investment Management SERGIO M. FOCARDI FRANK J. FABOZZI John Wiley & Sons, Inc. Frontmatter Page ii Monday, March 8, 2004 10:06 AM 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. 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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 Frontmatter Page iii Monday, March 8, 2004 10:06 AM Contents Preface xiv Acknowledgments xvi About the Authors xviii Commonly Used Symbols xix Abbreviations and Acronyms xx CHAPTER 1 From Art to Engineering in Finance 1 Investment Management Process 2 Step 1: Setting Investment Objectives 2 Step 2: Establishing an Investment Policy 2 Step 3: Selecting a Portfolio Strategy 6 Step 4: Selecting the Specific Assets 7 Step 5: Measuring and Evaluating Performance 9 Financial Engineering in Historical Perspective 10 The Role of Information Technology 11 Industry’s Evaluation of Modeling Tools 13 Integrating Qualitative and Quantitative Information 15 Principles for Engineering a Suite of Models 17 Summary 18 CHAPTER 2 Overview of Financial Markets, Financial Assets, and Market Participants 21 Financial Assets 21 Financial Markets 25 Classification of Financial Markets 25 Economic Functions of Financial Markets 26 Secondary Markets 27 Overview of Market Participants 34 Role of Financial Intermediaries 35 Institutional Investors 37 Insurance Companies 41 Pension Funds 41 Investment Companies 42 Depository Institutions 43 Endowments and Foundations 45 Common Stock 45 iii Frontmatter Page iv Monday, March 8, 2004 10:06 AM iv Contents Trading Locations 45 Stock Market Indicators 46 Trading Arrangements 48 Bonds 51 Maturity 51 Par Value 52 Coupon Rate 52 Provisions for Paying off Bonds 55 Options Granted to Bondholders 56 Futures and Forward Contracts 57 Futures versus Forward Contracts 58 Risk and Return Characteristics of Futures Contracts 59 Pricing of Futures Contracts 59 The Role of Futures in Financial Markets 63 Options 64 Risk-Return for Options 66 The Option Price 66 Swaps 69 Caps and Floors 70 Summary 71 CHAPTER 3 Milestones in Financial Modeling and Investment Management 75 The Precursors: Pareto, Walras, and the Lausanne School 76 Price Diffusion: Bachelier 78 The Ruin Problem in Insurance: Lundberg 80 The Principles of Investment: Markowitz 81 Understanding Value: Modigliani and Miller 83 Modigliani-Miller Irrelevance Theorems and the Absence of Arbitrage 84 Efficient Markets: Fama and Samuelson 85 Capital Asset Pricing Model: Sharpe, Lintner, and Mossin 86 The Multifactor CAPM: Merton 87 Arbitrage Pricing Theory: Ross 88 Arbitrage, Hedging, and Option Theory: Black, Scholes, and Merton 89 Summary 90 CHAPTER 4 Principles of Calculus 91 Sets and Set Operations 93 Proper Subsets 93 Empty Sets 95 Union of Sets 95 Intersection of Sets 95 Elementary Properties of Sets 96 Distances and Quantities 96 n-tuples 97 Distance 98 Frontmatter Page v Monday, March 8, 2004 10:06 AM v Contents Density of Points 99 Functions 100 Variables 101 Limits 102 Continuity 103 Total Variation 105 Differentiation 106 Commonly Used Rules for Computing Derivatives 107 Higher Order Derivatives 111 Application to Bond Analysis 112 Taylor Series Expansion 121 Application to Bond Analysis 122 Integration 127 Riemann Integrals 127 Properties of Riemann Integrals 129 Lebesque-Stieltjes Integrals 130 Indefinite and Improper Integrals 131 The Fundamental Theorem of Calculus 132 Integral Transforms 134 Laplace Transform 134 Fourier Transforms 137 Calculus in More than One Variable 138 Summary 139 CHAPTER 5 Matrix Algebra 141 Vectors and Matrices Defined 141 Vectors 141 Matrices 144 Square Matrices 145 Diagonals and Antidiagonals 145 Identity Matrix 146 Diagonal Matrix 146 Upper and Lower Triangular Matrix 148 Determinants 148 Systems of Linear Equations 149 Linear Independence and Rank 151 Hankel Matrix 152 Vector and Matrix Operations 153 Vector Operations 153 Matrix Operations 156 Eigenvalues and Eigenvectors 160 Diagonalization and Similarity 161 Singular Value Decomposition 162 Summary 163 CHAPTER 6 Concepts of Probability 165 Representing Uncertainty with Mathematics 165 Probability in a Nutshell 167 Frontmatter Page vi Monday, March 8, 2004 10:06 AM vi Contents Outcomes and Events 169 Probability 170 Measure 171 Random Variables 172 Integrals 172 Distributions and Distribution Functions 174 Random Vectors 175 Stochastic Processes 178 Probabilistic Representation of Financial Markets 180 Information Structures 181 Filtration 182 Conditional Probability and Conditional Expectation 184 Moments and Correlation 186 Copula Functions 188 Sequences of Random Variables 189 Independent and Identically Distributed Sequences 191 Sum of Variables 191 Gaussian Variables 194 The Regression Function 197 Linear Regression 197 Summary 199 CHAPTER 7 Optimization 201 Maxima and Minima 202 Lagrange Multipliers 204 Numerical Algorithms 206 Linear Programming 206 Quadratic Programming 211 Calculus of Variations and Optimal Control Theory 212 Stochastic Programming 214 Summary 216 CHAPTER 8 Stochastic Integrals 217 The Intuition Behind Stochastic Integrals 219 Brownian Motion Defined 225 Properties of Brownian Motion 230 Stochastic Integrals Defined 232 Some Properties of Itô Stochastic Integrals 236 Summary 237 CHAPTER 9 Differential Equations and Difference Equations 239 Differential Equations Defined 240 Ordinary Differential Equations 240 Order and Degree of an ODE 241 Solution to an ODE 241 Systems of Ordinary Differential Equations 243 Frontmatter Page vii Monday, March 8, 2004 10:06 AM vii Contents Closed-Form Solutions of Ordinary Differential Equations 246 Linear Differential Equation 247 Numerical Solutions of Ordinary Differential Equations 249 The Finite Difference Method 249 Nonlinear Dynamics and Chaos 256 Fractals 258 Partial Differential Equations 259 Diffusion Equation 259 Solution of the Diffusion Equation 261 Numerical Solution of PDEs 263 Summary 265 CHAPTER 10 Stochastic Differential Equations 267 The Intuition Behind Stochastic Differential Equations 268 Itô Processes 271 The 1-Dimensional Itô Formula 272 Stochastic Differential Equations 274 Generalization to Several Dimensions 276 Solution of Stochastic Differential Equations 278 The Arithmetic Brownian Motion 280 The Ornstein-Uhlenbeck Process 280 The Geometric Brownian Motion 281 Summary 282 CHAPTER 11 Financial Econometrics: Time Series Concepts, Representations, and Models 283 Concepts of Time Series 284 Stylized Facts of Financial Time Series 286 Infinite Moving-Average and Autoregressive Representation of Time Series 288 Univariate Stationary Series 288 The Lag Operator L 289 Stationary Univariate Moving Average 292 Multivariate Stationary Series 293 Nonstationary Series 295 ARMA Representations 297 Stationary Univariate ARMA Models 297 Nonstationary Univariate ARMA Models 300 Stationary Multivariate ARMA Models 301 Nonstationary Multivariate ARMA Models 304 Markov Coefficients and ARMA Models 304 Hankel Matrices and ARMA Models 305 State-Space Representation 305 Equivalence of State-Space and ARMA Representations 308 Integrated Series and Trends 309 Summary 313 Frontmatter Page viii Monday, March 8, 2004 10:06 AM viii Contents CHAPTER 12 Financial Econometrics: Model Selection, Estimation, and Testing 315 Model Selection 315 Learning and Model Complexity 317 Maximum Likelihood Estimate 319 Linear Models of Financial Time Series 324 Random Walk Models 324 Correlation 327 Random Matrices 329 Multifactor Models 332 CAPM 334 Asset Pricing Theory (APT) Models 335 PCA and Factor Models 335 Vector Autoregressive Models 338 Cointegration 339 State-Space Modeling and Cointegration 342 Empirical Evidence of Cointegration in Equity Prices 343 Nonstationary Models of Financial Time Series 345 The ARCH/GARCH Family of Models 346 Markov Switching Models 347 Summary 349 CHAPTER 13 Fat Tails, Scaling, and Stable Laws 351 Scaling, Stable Laws, and Fat Tails 352 Fat Tails 352 The Class L of Fat-Tailed Distributions 353 The Law of Large Numbers and the Central Limit Theorem 358 Stable Distributions 360 Extreme Value Theory for IID Processes 362 Maxima 362 Max-Stable Distributions 368 Generalized Extreme Value Distributions 368 Order Statistics 369 Point Process of Exceedances or Peaks over Threshold 371 Estimation 373 Eliminating the Assumption of IID Sequences 378 Heavy-Tailed ARMA Processes 381 ARCH/GARCH Processes 382 Subordinated Processes 383 Markov Switching Models 384 Estimation 384 Scaling and Self-Similarity 385 Evidence of Fat Tails in Financial Variables 388 On the Applicability of Extreme Value Theory in Finance 391 Summary 392 Frontmatter Page ix Monday, March 8, 2004 10:06 AM ix Contents CHAPTER 14 Arbitrage Pricing: Finite-State Models 393 The Arbitrage Principle 393 Arbitrage Pricing in a One-Period Setting 395 State Prices 397 Risk-Neutral Probabilities 398 Complete Markets 399 Arbitrage Pricing in a Multiperiod Finite-State Setting 402 Propagation of Information 402 Trading Strategies 403 State-Price Deflator 404 Pricing Relationships 405 Equivalent Martingale Measures 414 Risk-Neutral Probabilities 416 Path Dependence and Markov Models 423 The Binomial Model 423 Risk-Neutral Probabilities for the Binomial Model 426 Valuation of European Simple Derivatives 427 Valuation of American Options 429 Arbitrage Pricing in a Discrete-Time, Continuous-State Setting 430 APT Models 435 Testing APT 436 Summary 439 CHAPTER 15 Arbitrage Pricing: Continuous-State, Continuous-Time Models 441 The Arbitrage Principle in Continuous Time 441 Trading Strategies and Trading Gains 443 Arbitrage Pricing in Continuous-State, Continuous-Time 445 Option Pricing 447 Stock Price Processes 447 Hedging 448 The Black-Scholes Option Pricing Formula 449 Generalizing the Pricing of European Options 452 State-Price Deflators 454 Equivalent Martingale Measures 457 Equivalent Martingale Measures and Girsanov’s Theorem 459 The Diffusion Invariance Principle 461 Application of Girsanov’s Theorem to Black-Scholes Option Pricing Formula 462 Equivalent Martingale Measures and Complete Markets 463 Equivalent Martingale Measures and State Prices 464 Arbitrage Pricing with a Payoff Rate 466 Implications of the Absence of Arbitrage 467 Working with Equivalent Martingale Measures 468 Summary 468