Probability and Statistics for Finance

Preface xv

About the Authors xvii

CHAPTER 1 Introduction 1

Probability vs. Statistics 4

Overview of the Book 5

Part One Descriptive Statistics 15

Chapter 2 Basic Data Analysis 17

Data Types 17

Frequency Distributions 22

Empirical Cumulative Frequency Distribution 27

Data Classes 32

Cumulative Frequency Distributions 41

Concepts Explained in this Chapter 43

Chapter 3 Measures of Location and Spread 45

Parameters vs. Statistics 45

Center and Location 46

Variation 59

Measures of the Linear Transformation 69

Summary of Measures 71

Concepts Explained in this Chapter 73

Chapter 4 Graphical Representation of Data 75

Pie Charts 75

Bar Chart 78

Stem and Leaf Diagram 81

Frequency Histogram 82

Ogive Diagrams 89

Box Plot 91

QQ Plot 96

Concepts Explained in this Chapter 99

CHAPTER 5 Multivariate Variables and Distributions 101

Data Tables and Frequencies 101

Class Data and Histograms 106

Marginal Distributions 107

Graphical Representation 110

Conditional Distribution 113

Conditional Parameters and Statistics 114

Independence 117

Covariance 120

Correlation 123

Contingency Coefficient 124

Concepts Explained in this Chapter 126

CHAPTER 6 Introduction to Regression Analysis 129

The Role of Correlation 129

Regression Model: Linear Functional Relationship Between Two Variables 131

Distributional Assumptions of the Regression Model 133

Estimating the Regression Model 134

Goodness of Fit of the Model 138

Linear Regression of Some Nonlinear Relationship 140

Two Applications in Finance 142

Concepts Explained in this Chapter 149

CHAPTER 7 Introduction to Time Series Analysis 153

What Is Time Series? 153

Decomposition of Time Series 154

Representation of Time Series with Difference Equations 159

Application: The Price Process 159

Concepts Explained in this Chapter 163

Part Two Basic Probability Theory 165

CHAPTER 8 Concepts of Probability Theory 167

Historical Development of Alternative Approaches to Probability 167

Set Operations and Preliminaries 170

Probability Measure 177

Random Variable 179

Concepts Explained in this Chapter 185

Chapter 9 Discrete Probability Distributions 187

Discrete Law 187

Bernoulli Distribution 192

Binomial Distribution 195

Hypergeometric Distribution 204

Multinomial Distribution 211

Poisson Distribution 216

Discrete Uniform Distribution 219

Concepts Explained in this Chapter 221

CHAPTER 10 Continuous Probability Distributions 229

Continuous Probability Distribution Described 229

Distribution Function 230

Density Function 232

Continuous Random Variable 237

Computing Probabilities from the Density Function 238

Location Parameters 239

Dispersion Parameters 239

Concepts Explained in this Chapter 245

CHAPTER 11 Continuous Probability Distributions with Appealing Statistical Properties 247

Normal Distribution 247

Chi-Square Distribution 254

Student’s t-Distribution 256

F-Distribution 260

Exponential Distribution 262

Rectangular Distribution 266

Gamma Distribution 268

Beta Distribution 269

Log-Normal Distribution 271

Concepts Explained in this Chapter 275

CHAPTER 12 Continuous Probability Distributions Dealing with Extreme Events 277

Generalized Extreme Value Distribution 277

Generalized Pareto Distribution 281

Normal Inverse Gaussian Distribution 283

α-Stable Distribution 285

Concepts Explained in this Chapter 292

CHAPTER 13 Parameters of Location and Scale of Random Variables 295

Parameters of Location 296

Parameters of Scale 306

Concepts Explained in this Chapter 321

Appendix: Parameters for Various Distribution Functions 322

Chapter 14 Joint Probability Distributions 325

Higher Dimensional Random Variables 326

Joint Probability Distribution 328

Marginal Distributions 333

Dependence 338

Covariance and Correlation 341

Selection of Multivariate Distributions 347

Concepts Explained in this Chapter 358

Chapter 15 Conditional Probability and Bayes’ Rule 361

Conditional Probability 362

Independent Events 365

Multiplicative Rule of Probability 367

Bayes’ Rule 372

Conditional Parameters 374

Concepts Explained in this Chapter 377

CHAPTER 16 Copula and Dependence Measures 379

Copula 380

Alternative Dependence Measures 406

Concepts Explained in this Chapter 412

Part Three Inductive Statistics 413

Chapter 17 Point Estimators 415

Sample, Statistic, and Estimator 415

Quality Criteria of Estimators 428

Large Sample Criteria 435

Maximum Likehood Estimator 446

Exponential Family and Sufficiency 457

Concepts Explained in this Chapter 461

Chapter 18 Confidence Intervals 463

Confidence Level and Confidence Interval 463

Confidence Interval for the Mean of a Normal Random Variable 466

Confidence Interval for the Mean of a Normal Random Variable with Unknown Variance 469

Confidence Interval for the Variance of a Normal Random Variable 471

Confidence Interval for the Variance of a Normal Random Variable with Unknown Mean 474

Confidence Interval for the Parameter p of a Binomial Distribution 475

Confidence Interval for the Parameter λ of an Exponential Distribution 477

Concepts Explained in this Chapter 479

Chapter 19 Hypothesis Testing 481

Hypotheses 482

Error Types 485

Quality Criteria of a Test 490

Examples 496

Concepts Explained in this Chapter 518

Part Four Multivariate Linear Regression Analysis 519

CHAPTER 20 Estimates and Diagnostics for Multivariate Linear Regression Analysis 521

The Multivariate Linear Regression Model 522

Assumptions of the Multivariate Linear Regression Model 523

Estimation of the Model Parameters 523

Designing the Model 526

Diagnostic Check and Model Significance 526

Applications to Finance 531

Concepts Explained in this Chapter 543

CHAPTER 21 Designing and Building a Multivariate Linear Regression Model 545

The Problem of Multicollinearity 545

Incorporating Dummy Variables as Independent Variables 548

Model Building Techniques 561

Concepts Explained in this Chapter 565

CHAPTER 22 Testing the Assumptions of the Multivariate Linear Regression Model 567

Tests for Linearity 568

Assumed Statistical Properties about the Error Term 570

Tests for the Residuals Being Normally Distributed 570

Tests for Constant Variance of the Error Term (Homoskedasticity) 573

Absence of Autocorrelation of the Residuals 576

Concepts Explained in this Chapter 581

Appendix A Important Functions and Their Features 583

Continuous Function 583

Indicator Function 586

Derivatives 587

Monotonic Function 591

Integral 592

Some Functions 596

Appendix B Fundamentals of Matrix Operations and Concepts 601

The Notion of Vector and Matrix 601

Matrix Multiplication 602

Particular Matrices 603

Positive Semidefinite Matrices 614

APPENDIX C Binomial and Multinomial Coefficients 615

Binomial Coefficient 615

Multinomial Coefficient 622

APPENDIX D Application of the Log-Normal Distribution to the Pricing of Call Options 625

Call Options 625

Deriving the Price of a European Call Option 626

Illustration 631

References 633

Index 635