Probability and Statistics for Finance

SVETLOZAR T. RACHEV, PhD, DSC, is Chair Professor at the University of Karlsruhe in the School of Economics and Business Engineering, and Professor Emeritus at the University of California, Santa Barbara, in the Department of Statistics and Applied Probability. He was cofounder of Bravo Risk Management Group, acquired by FinAnalytica, where he currently serves as Chief Scientist.MARKUS HÖCHSTÖTTER, PhD, is an Assistant Professor in the Department of Econometrics and Statistics, University of Karlsruhe.FRANK J. FABOZZI, PhD, CFA, CPA, is Professor in the Practice of Finance and Becton Fellow at the Yale School of Management and Editor of the Journal of Portfolio Management. He is an Affiliated Professor at the University of Karlsruhe's Institute of Statistics, Econometrics and Mathematical Finance, and is on the Advisory Council for the Department of Operations Research and Financial Engineering at Princeton University.SERGIO M. FOCARDI, PhD, is a Professor of Finance at EDHEC Business School and founding partner of the Paris-based consulting firm Intertek Group plc.

• Preface xvAbout the Authors xviiChapter 1 Introduction 1Probability vs. Statistics 4Overview of the Book 5Part One Descriptive Statistics 15Chapter 2 Basic Data Analysis 17Data Types 17Frequency Distributions 22Empirical Cumulative Frequency Distribution 27Data Classes 32Cumulative Frequency Distributions 41Concepts Explained in this Chapter 43Chapter 3 Measures of Location and Spread 45Parameters vs. Statistics 45Center and Location 46Variation 59Measures of the Linear Transformation 69Summary of Measures 71Concepts Explained in this Chapter 73Chapter 4 Graphical Representation of Data 75Pie Charts 75Bar Chart 78Stem and Leaf Diagram 81Frequency Histogram 82Ogive Diagrams 89Box Plot 91QQ Plot 96Concepts Explained in this Chapter 99Chapter 5 Multivariate Variables and Distributions 101Data Tables and Frequencies 101Class Data and Histograms 106Marginal Distributions 107Graphical Representation 110Conditional Distribution 113Conditional Parameters and Statistics 114Independence 117Covariance 120Correlation 123Contingency Coefficient 124Concepts Explained in this Chapter 126Chapter 6 Introduction to Regression Analysis 129The Role of Correlation 129Regression Model: Linear Functional Relationship Between Two Variables 131Distributional Assumptions of the Regression Model 133Estimating the Regression Model 134Goodness of Fit of the Model 138Linear Regression of Some Nonlinear Relationship 140Two Applications in Finance 142Concepts Explained in this Chapter 149Chapter 7 Introduction to Time Series Analysis 153What Is Time Series? 153Decomposition of Time Series 154Representation of Time Series with Difference Equations 159Application: The Price Process 159Concepts Explained in this Chapter 163Part Two Basic Probability Theory 165Chapter 8 Concepts of Probability Theory 167Historical Development of Alternative Approaches to Probability 167Set Operations and Preliminaries 170Probability Measure 177Random Variable 179Concepts Explained in this Chapter 185Chapter 9 Discrete Probability Distributions 187Discrete Law 187Bernoulli Distribution 192Binomial Distribution 195Hypergeometric Distribution 204Multinomial Distribution 211Poisson Distribution 216Discrete Uniform Distribution 219Concepts Explained in this Chapter 221Chapter 10 Continuous Probability Distributions 229Continuous Probability Distribution Described 229Distribution Function 230Density Function 232Continuous Random Variable 237Computing Probabilities from the Density Function 238Location Parameters 239Dispersion Parameters 239Concepts Explained in this Chapter 245Chapter 11 Continuous Probability Distributions with Appealing Statistical Properties 247Normal Distribution 247Chi-Square Distribution 254Student’s t-Distribution 256F-Distribution 260Exponential Distribution 262Rectangular Distribution 266Gamma Distribution 268Beta Distribution 269Log-Normal Distribution 271Concepts Explained in this Chapter 275Chapter 12 Continuous Probability Distributions Dealing with Extreme Events 277Generalized Extreme Value Distribution 277Generalized Pareto Distribution 281Normal Inverse Gaussian Distribution 283α-Stable Distribution 285Concepts Explained in this Chapter 292Chapter 13 Parameters of Location and Scale of Random Variables 295Parameters of Location 296Parameters of Scale 306Concepts Explained in this Chapter 321Appendix: Parameters for Various Distribution Functions 322Chapter 14 Joint Probability Distributions 325Higher Dimensional Random Variables 326Joint Probability Distribution 328Marginal Distributions 333Dependence 338Covariance and Correlation 341Selection of Multivariate Distributions 347Concepts Explained in this Chapter 358Chapter 15 Conditional Probability and Bayes’ Rule 361Conditional Probability 362Independent Events 365Multiplicative Rule of Probability 367Bayes’ Rule 372Conditional Parameters 374Concepts Explained in this Chapter 377Chapter 16 Copula and Dependence Measures 379Copula 380Alternative Dependence Measures 406Concepts Explained in this Chapter 412Part Three Inductive Statistics 413Chapter 17 Point Estimators 415Sample, Statistic, and Estimator 415Quality Criteria of Estimators 428Large Sample Criteria 435Maximum Likehood Estimator 446Exponential Family and Sufficiency 457Concepts Explained in this Chapter 461Chapter 18 Confidence Intervals 463Confidence Level and Confidence Interval 463Confidence Interval for the Mean of a Normal Random Variable 466Confidence Interval for the Mean of a Normal Random Variable with Unknown Variance 469Confidence Interval for the Variance of a Normal Random Variable 471Confidence Interval for the Variance of a Normal Random Variable with Unknown Mean 474Confidence Interval for the Parameter p of a Binomial Distribution 475Confidence Interval for the Parameter λ of an Exponential Distribution 477Concepts Explained in this Chapter 479Chapter 19 Hypothesis Testing 481Hypotheses 482Error Types 485Quality Criteria of a Test 490Examples 496Concepts Explained in this Chapter 518Part Four Multivariate Linear Regression Analysis 519Chapter 20 Estimates and Diagnostics for Multivariate Linear Regression Analysis 521The Multivariate Linear Regression Model 522Assumptions of the Multivariate Linear Regression Model 523Estimation of the Model Parameters 523Designing the Model 526Diagnostic Check and Model Significance 526Applications to Finance 531Concepts Explained in this Chapter 543Chapter 21 Designing and Building a Multivariate Linear Regression Model 545The Problem of Multicollinearity 545Incorporating Dummy Variables as Independent Variables 548Model Building Techniques 561Concepts Explained in this Chapter 565Chapter 22 Testing the Assumptions of the Multivariate Linear Regression Model 567Tests for Linearity 568Assumed Statistical Properties about the Error Term 570Tests for the Residuals Being Normally Distributed 570Tests for Constant Variance of the Error Term (Homoskedasticity) 573Absence of Autocorrelation of the Residuals 576Concepts Explained in this Chapter 581Appendix A Important Functions and Their Features 583Continuous Function 583Indicator Function 586Derivatives 587Monotonic Function 591Integral 592Some Functions 596Appendix B Fundamentals of Matrix Operations and Concepts 601The Notion of Vector and Matrix 601Matrix Multiplication 602Particular Matrices 603Positive Semidefinite Matrices 614Appendix C Binomial and Multinomial Coefficients 615Binomial Coefficient 615Multinomial Coefficient 622Appendix D Application of the Log-Normal Distribution to the Pricing of Call Options 625Call Options 625Deriving the Price of a European Call Option 626Illustration 631References 633Index 635