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 HOCHSTOTTER, 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. About the Authors. CHAPTER 1 Introduction. Probability Versus Statistics. Overview of the Book. PART ONE Descriptive Statistics. CHAPTER 2 Basic Data Analysis. Data Types. Frequency Distributions. Empirical Cumulative Frequency Distribution. Data Classes. Cumulative Frequency Distributions. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 3 Measures of Location and Spread. Parameters versus Statistics. Center and Location. Variation. Measures of the Linear Transformation. Summary of Measures. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 4 Graphical Representation of Data. Pie Charts. Bar Chart. Stem and Leaf Diagram. Frequency Histogram. Ogive Diagrams. Box Plot. QQ Plot. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 5 Multivariate Variables and Distributions. Data Tables and Frequencies. Class Data and Histograms. Marginal Distributions. Graphical Representation. Conditional Distribution. Conditional Parameters and Statistics. Independence. Covariance. Correlation. Contingency Coefficient. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 6 Introduction to Regression Analysis. The Role of Correlation. Regression Model: Linear Functional Relationship Between Two Variables. Distributional Assumptions of the Regression Model. Estimating the Regression Model. Goodness of Fit of the Model. Linear Regression of Some Non-Linear Relationship. Two Applications in Finance. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 7 Introduction to Time Series Analysis. What Is Time Series? Decomposition of Time Series. Representation of Time Series with Difference Equations. Application: The Price Process. Concepts Explained in this Chapter (In Order of Presentation). PART TWO Basic Probability Theory. CHAPTER 8 Concepts of Probability Theory. Historical Development of Alternative Approaches to Probability. Set Operations and Preliminaries. Probability Measure. Random Variable. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 9 Discrete Probability Distributions. Discrete Law. Bernoulli Distribution. Binomial Distribution. Hypergeometric Distribution. Multinomial Distribution. Poisson Distribution Discrete Uniform Distribution. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 10 Continuous Probability Distributions. Continuous Probability Distribution Described. Distribution Function. Density Function. Continuous Random Variable. Computing Probabilities from the Density Function. Location Parameters. Dispersion Parameters. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 11 Continuous Probability Distributions with Appealing Statistical Properties. Normal Distribution. Chi-Square Distribution. Student's t -Distribution. F -Distribution. Exponential Distribution. Rectangular Distribution. Gamma Distribution. Beta Distribution. Log-Normal Distribution. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 12 Continuous Probability Distributions Dealing with Extreme Events. Generalized Extreme Value Distribution. Generalized Pareto Distribution. Normal Inverse Gaussian Distribution. a-Stable Distribution. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 13 Parameters of Location and Scale of Random Variables. Parameters of Location. Parameters of Scale. Concepts Explained in this Chapter (In Order of Presentation). Appendix: Parameters for Various Distribution Functions. CHAPTER 14 Joint Probability Distributions. Higher Dimensional Random Variables. Joint Probability Distribution. Marginal Distributions. Dependence. Covariance and Correlation. Selection of Multivariate Distributions. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 15 Conditional Probability and Bayes' Rule. Conditional Probability. Independent Events. Multiplicative Rule of Probability. Bayes' Rule. Conditional Parameters. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 16 Copula and Dependence Measures. Copula. Alternative Dependence Measures. Concepts Explained in this Chapter (In Order of Presentation). PART THREE Inductive Statistics. CHAPTER 17 Point Estimators. Sample, Statistic, and Estimator. Quality Criteria of Estimators. Large Sample Criteria. Maximum Likehood Estimator. Exponential Family and Sufficiency. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 18 Confidence Intervals. Confidence Level and Confidence Interval. Confidence Interval for the Mean of a Normal Random Variable. Confidence Interval for the Mean of a Normal Random Variable with Unknown Variance. Confidence Interval for the Parameter p of a Binomial Distribution. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 19 Hypothesis Testing. Hypotheses. Error Types. Quality Criteria of a Test. Examples. Concepts Explained in this Chapter (In Order of Presentation). PART FOUR Multivariate Linear Regression Analysis. CHAPTER 20 Estimates and Diagnostics for Multivariate Linear Regression Analysis. The Multivariate Linear Regression Model. Assumptions of the Multivariate Linear Regression Model. Estimation of the Model Parameters. Designing the Model. Diagnostic Check and Model Significance. Applications to Finance. Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 21 Designing and Building a Multivariate Linear Regression Model. The Problem of Multicollinearity. Incorporating Dummy Variables as Independent Variables. Model Building Techniques 561 Concepts Explained in this Chapter (In Order of Presentation). CHAPTER 22 Testing the Assumptions of the Multivariate Linear Regression Model. Tests for Linearity. Assumed Statistical Properties About the Error Term. Tests for the Residuals Being Normally Distributed. Tests for Constant Variance of the Error Term (Homoskedasticity). Absence of Autocorrelation of the Residuals. Concepts Explained in this Chapter (In Order of Presentation). APPENDIX A Important Functions and Their Features. Continuous Function. Indicator Function. Derivatives. Monotonic Function. Integral. Some Functions. APPENDIX B Fundamentals of Matrix Operations and Concepts. The Notion of Vector and Matrix. Matrix Multiplication. Particular Matrices. Positive Semidefinite Matrices. APPENDIX C Binomial and Multinomial Coefficients. Binomial Coefficient. Multinomial Coefficient. APPENDIX D Application of the Log-Normal Distribution to the Pricing of Call Options. Call Options. Deriving the Price of a European Call Option. Illustration. REFERENCES. INDEX.