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Probability and Statistics for Finance

Book Description

A comprehensive look at how probability and statistics is applied to the investment process

Finance has become increasingly more quantitative, drawing on techniques in probability and statistics that many finance practitioners have not had exposure to before. In order to keep up, you need a firm understanding of this discipline.

Probability and Statistics for Finance addresses this issue by showing you how to apply quantitative methods to portfolios, and in all matter of your practices, in a clear, concise manner. Informative and accessible, this guide starts off with the basics and builds to an intermediate level of mastery.

  • Outlines an array of topics in probability and statistics and how to apply them in the world of finance

  • Includes detailed discussions of descriptive statistics, basic probability theory, inductive statistics, and multivariate analysis

  • Offers real-world illustrations of the issues addressed throughout the text

The authors cover a wide range of topics in this book, which can be used by all finance professionals as well as students aspiring to enter the field of finance.

Table of Contents

  1. Copyright
  2. Preface
  3. About the Authors
  4. 1. Introduction
    1. 1.1. PROBABILITY VS. STATISTICS
    2. 1.2. OVERVIEW OF THE BOOK
      1. 1.2.1. Part One: Descriptive Statistics
      2. 1.2.2. Part Two: Basic Probability Theory
      3. 1.2.3. Part Three: Inductive Statistics
      4. 1.2.4. Part Four: Multivariate Linear Regression
      5. 1.2.5. Appendixes
  5. I. Descriptive Statistics
    1. 2. Basic Data Analysis
      1. 2.1. DATA TYPES
        1. 2.1.1. How To Obain Data
        2. 2.1.2. The Information Contained in the Data
        3. 2.1.3. Data Levels and Scale
        4. 2.1.4. Cross-Sectional Data and Time Series
      2. 2.2. FREQUENCY DISTRIBUTIONS
        1. 2.2.1. Sorting and Counting Data
        2. 2.2.2. Formal Presentation of Frequency
      3. 2.3. EMPIRICAL CUMULATIVE FREQUENCY DISTRIBUTION
        1. 2.3.1. Accumulating Frequencies
        2. 2.3.2. Formal Presentation of Cumulative Frequency Distributions
      4. 2.4. DATA CLASSES
        1. 2.4.1. Reasons for Classifying
        2. 2.4.2. Formal Procedure for Classifying
        3. 2.4.3. Example of Classifying Procedures
      5. 2.5. CUMULATIVE FREQUENCY DISTRIBUTIONS
      6. 2.6. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    2. 3. Measures of Location and Spread
      1. 3.1. Parameters vs. Statistics
      2. 3.2. Center and Location
        1. 3.2.1. Mean
        2. 3.2.2. Median
        3. 3.2.3. Mode
        4. 3.2.4. Weighted Mean
        5. 3.2.5. Quantiles
      3. 3.3. Variation
        1. 3.3.1. Range
        2. 3.3.2. Interquartile Range
        3. 3.3.3. Absolute Deviation
        4. 3.3.4. Variance and Standard Deviation
        5. 3.3.5. Skewness
        6. 3.3.6. Data Levels and Measures of Variation
        7. 3.3.7. Empirical Rule
        8. 3.3.8. Coefficient of Variation and Standardization
      4. 3.4. Measures of the Linear Transformation
      5. 3.5. SUMMARY Of Measures
      6. 3.6. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    3. 4. Graphical Representation of Data
      1. 4.1. PIE CHARTS
      2. 4.2. BAR CHART
      3. 4.3. STEM AND LEAF DIAGRAM
      4. 4.4. FREQUENCY HISTOGRAM
      5. 4.5. OGIVE DIAGRAMS
      6. 4.6. BOX PLOT
      7. 4.7. QQ PLOT
      8. 4.8. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    4. 5. Multivariate Variables and Distributions
      1. 5.1. DATA TABLES AND FREQUENCIES
      2. 5.2. CLASS DATA AND HISTOGRAMS
      3. 5.3. MARGINAL DISTRIBUTIONS
      4. 5.4. GRAPHICAL REPRESENTATION
      5. 5.5. CONDITIONAL DISTRIBUTION
      6. 5.6. CONDITIONAL PARAMETERS AND STATISTICS
      7. 5.7. INDEPENDENCE
      8. 5.8. COVARIANCE
      9. 5.9. CORRELATION
      10. 5.10. CONTINGENCY COEFFICIENT
      11. 5.11. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    5. 6. Introduction to Regression Analysis
      1. 6.1. THE ROLE OF CORRELATION
        1. 6.1.1. Stock Return Example
        2. 6.1.2. Correlation in Finance
      2. 6.2. REGRESSION MODEL: LINEAR FUNCTIONAL RELATIONSHIP BETWEEN TWO VARIABLES
      3. 6.3. DISTRIBUTIONAL ASSUMPTIONS OF THE REGRESSION MODEL
      4. 6.4. ESTIMATING THE REGRESSION MODEL
        1. 6.4.1. Application to Stock Returns
      5. 6.5. Goodness of Fit of The Model
        1. 6.5.1. Relationship between Coefficient of Determination and Correlation Coefficient
      6. 6.6. LINEAR REGRESSION OF SOME NONLINEAR RELATIONSHIP
        1. 6.6.1. Linear Regression of Exponential Data
      7. 6.7. TWO APPLICATIONS IN FINANCE
        1. 6.7.1. Characteristic Line
        2. 6.7.2. Application to Hedging
      8. 6.8. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    6. 7. Introduction to Time Series Analysis
      1. 7.1. WHAT IS TIME SERIES?
      2. 7.2. DECOMPOSITION OF TIME SERIES
        1. 7.2.1. Application to S&P 500 Index Returns
      3. 7.3. REPRESENTATION OF TIME SERIES WITH DIFFERENCE EQUATIONS
      4. 7.4. APPLICATION: THE PRICE PROCESS
        1. 7.4.1. Random Walk
          1. 7.4.1.1. Application to S&P 500 Index Returns
        2. 7.4.2. Error Correction
      5. 7.5. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
  6. II. Basic Probability Theory
    1. 8. Concepts of Probability Theory
      1. 8.1. HISTORICAL Development of ALTERNATIVE Approaches to PROBABILITY
        1. 8.1.1. Probability as Relative Frequencies
        2. 8.1.2. Axiomatic System
      2. 8.2. SET OPERATIONS AND PRELIMINARIES
        1. 8.2.1. Set Operations
        2. 8.2.2. Right-Continuous and Nondecreasing Functions
        3. 8.2.3. Outcome, Space, and Events
        4. 8.2.4. The Measurable Space
      3. 8.3. PROBABILITY MEASURE
      4. 8.4. RANDOM VARIABLE
        1. 8.4.1. Random Variables on a Countable Space
        2. 8.4.2. Random Variables on an Uncountable Space
      5. 8.5. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    2. 9. Discrete Probability Distributions
      1. 9.1. DISCRETE LAW
        1. 9.1.1. Random Variable on the Countable Space
        2. 9.1.2. Mean and Variance
          1. 9.1.2.1. Mean
          2. 9.1.2.2. Variance
          3. 9.1.2.3. Standard Deviation
      2. 9.2. BERNOULLI DISTRIBUTION
      3. 9.3. BINOMIAL DISTRIBUTION
        1. 9.3.1. Application to the Binomial Stock Price Model
        2. 9.3.2. Application to the Binomial Interest Rate Model
        3. 9.3.3. Application to the Binomial Default Distribution Model
      4. 9.4. HYPERGEOMETRIC DISTRIBUTION
        1. 9.4.1. Application
      5. 9.5. MULTINOMIAL DISTRIBUTION
        1. 9.5.1. Multinomial Stock Price Model
      6. 9.6. POISSON DISTRIBUTION
        1. 9.6.1. Application to Credit Risk Modeling for a Bond Portfolio
      7. 9.7. DISCRETE UNIFORM DISTRIBUTION
        1. 9.7.1. Application to the Multinomial Stock Price Model
      8. 9.8. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    3. 10. Continuous Probability Distributions
      1. 10.1. CONTINUOUS PROBABILITY DISTRIBUTION DESCRIBED
      2. 10.2. Distribution Function
      3. 10.3. Density Function
        1. 10.3.1. Requirements on the Density Function
      4. 10.4. Continuous Random Variable
      5. 10.5. Computing Probabilities from the Density Function
      6. 10.6. Location Parameters
      7. 10.7. Dispersion Parameters
        1. 10.7.1. Moments of Higher Order
          1. 10.7.1.1. Variance
          2. 10.7.1.2. Standard Deviation
          3. 10.7.1.3. Skewness
      8. 10.8. Concepts Explained in this Chapter (In Order of Presentation)
    4. 11. Continuous Probability Distributions with Appealing Statistical Properties
      1. 11.1. NORMAL DISTRIBUTION
        1. 11.1.1. Properties of the Normal Distribution
        2. 11.1.2. Applications to Stock Returns
          1. 11.1.2.1. Applying Properties 1 and 2 to Stock Returns
          2. 11.1.2.2. Using the Normal Distribution to Approximate the Binomial Distribution
          3. 11.1.2.3. Normal Distribution for Logarithmic Returns
      2. 11.2. CHI-SQUARE DISTRIBUTION
        1. 11.2.1. Application to Modeling Short-Term Interest Rates
      3. 11.3. STUDENT'S t-DISTRIBUTION
        1. 11.3.1. Application to Stock Returns
      4. 11.4. F-DISTRIBUTION
      5. 11.5. EXPONENTIAL DISTRIBUTION
        1. 11.5.1. Applications in Finance
      6. 11.6. RECTANGULAR DISTRIBUTION
      7. 11.7. GAMMA DISTRIBUTION
        1. 11.7.1. Erlang Distribution
      8. 11.8. BETA DISTRIBUTION
      9. 11.9. LOG-NORMAL DISTRIBUTION
        1. 11.9.1. Application to Modeling Asset Returns
      10. 11.10. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    5. 12. Continuous Probability Distributions Dealing with Extreme Events
      1. 12.1. GENERALIZED EXTREME VALUE DISTRIBUTION
      2. 12.2. GENERALIZED PARETO Distribution
      3. 12.3. NORMAL INVERSE GAUSSIAN Distribution
        1. 12.3.1. Normal Distribution versus Normal Inverse Gaussian Distribution
      4. 12.4. α-STABLE DISTRIBUTION
      5. 12.5. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    6. 13. Parameters of Location and Scale of Random Variables
      1. 13.1. Parameters of location
        1. 13.1.1. Quantiles
          1. 13.1.1.1. Computation of Quantiles for Various Distributions
          2. 13.1.1.2. Value-at-Risk
          3. 13.1.1.3. Computing the VaR of Various Distributions
        2. 13.1.2. Mode
        3. 13.1.3. Mean (First Moment)
          1. 13.1.3.1. Mean of the Binomial Distribution
          2. 13.1.3.2. Mean of the Poisson Distribution
          3. 13.1.3.3. Mean of the Exponential Distribution
          4. 13.1.3.4. Mean of the Normal Distribution
      2. 13.2. Parameters of scale
        1. 13.2.1. Moments of Higher Order
          1. 13.2.1.1. Second Moment of the Poisson Distribution
          2. 13.2.1.2. Second Moment of the Log-Normal Distribution
          3. 13.2.1.3. Nonexistence of the Second Moment of the α-Stable Distribution
        2. 13.2.2. Variance
          1. 13.2.2.1. Variance of the Poisson Distribution
          2. 13.2.2.2. Variance of the Exponential Distribution
          3. 13.2.2.3. Variance of the Normal Distribution and α-Stable Distribution
        3. 13.2.3. Standard Deviation
        4. 13.2.4. Skewness
          1. 13.2.4.1. Skewness of Normal and Exponential Distributions
          2. 13.2.4.2. Skewness of GE Daily Returns
        5. 13.2.5. Kurtosis
        6. 13.2.6. Kurtosis of the GE Daily Returns
      3. 13.3. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
      4. 13.4. APPENDIX: PARAMETERS FOR VARIOUS DISTRIBUTION FUNCTIONS
    7. 14. Joint Probability Distributions
      1. 14.1. Higher dimensional random variables
        1. 14.1.1. Discrete Case
        2. 14.1.2. Continuous Case
      2. 14.2. Joint probability distribution
        1. 14.2.1. Discrete Case
        2. 14.2.2. Continuous Case
          1. 14.2.2.1. Contour Lines
      3. 14.3. Marginal distributions
        1. 14.3.1. Discrete Case
        2. 14.3.2. Continuous Case
      4. 14.4. Dependence
        1. 14.4.1. Discrete Case
        2. 14.4.2. Continuous Case
      5. 14.5. Covariance and correlation
        1. 14.5.1. Discrete Case
        2. 14.5.2. Continuous Case
        3. 14.5.3. Aspects of the Covariance and Covariance Matrix
          1. 14.5.3.1. Covariance of Transformed Random Variables
          2. 14.5.3.2. Covariance Matrix
        4. 14.5.4. Correlation
        5. 14.5.5. Criticism of the Correlation and Covariance as a Measure of Joint Randomness
      6. 14.6. Selection of multivariate distributions
        1. 14.6.1. Multivariate Normal Distribution
          1. 14.6.1.1. Properties of Multivariate Normal Distribution
          2. 14.6.1.2. Density Function of a General Multivariate Normal Distribution
          3. 14.6.1.3. Application to Portfolio Selection
        2. 14.6.2. Multivariate t-Distribution
        3. 14.6.3. Elliptical Distributions
          1. 14.6.3.1. Properties of the Elliptical Class
      7. 14.7. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    8. 15. Conditional Probability and Bayes' Rule
      1. 15.1. Conditional Probability
        1. 15.1.1. Formula for Conditional Probability
          1. 15.1.1.1. Illustration: Computing the Conditional Probability
      2. 15.2. Independent Events
      3. 15.3. Multiplicative Rule of Probability
        1. 15.3.1. Illustration: The Multiplicative Rule of Probability
        2. 15.3.2. Multiplicative Rule of Probability for Independent Events
        3. 15.3.3. Law of Total Probability
        4. 15.3.4. Illustration: The Law of Total Probability
        5. 15.3.5. The Law of Total Probability for More than Two Events
      4. 15.4. Bayes' Rule
        1. 15.4.1. Illustration: Application of Bayes' Rule
      5. 15.5. Conditional Parameters
        1. 15.5.1. Conditional Expectation
        2. 15.5.2. Conditional Variance
        3. 15.5.3. Expected Tail Loss
      6. 15.6. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    9. 16. Copula and Dependence Measures
      1. 16.1. Copula
        1. 16.1.1. Construction of the Copula
        2. 16.1.2. Specifications of the Copula
        3. 16.1.3. Properties of the Copula
          1. 16.1.3.1. Copula Density
          2. 16.1.3.2. Increments of the Copula
          3. 16.1.3.3. Bounds of the Copula
          4. 16.1.3.4. Invariance under Strictly Monotonically Increasing Transformations
        4. 16.1.4. Simulation of Financial Returns Using the Copula
        5. 16.1.5. The Copula for Two Dimensions
          1. 16.1.5.1. Gaussian Copula (d = 2)
          2. 16.1.5.2. t-Copula (d = 2)
          3. 16.1.5.3. Gumbel Copula and Clayton Copula
        6. 16.1.6. Simulation with the Gaussian Copula (d = 2)
          1. 16.1.6.1. Simulating GE and IBM Data Through Estimated Copulae
      2. 16.2. Alternative dependence measures
        1. 16.2.1. Rank Correlation Measures
          1. 16.2.1.1. Spearman's Rho
        2. 16.2.2. Tail Dependence
          1. 16.2.2.1. Tail Dependence of a Gaussian Copula
          2. 16.2.2.2. Tail Dependence of t-Copula
          3. 16.2.2.3. Spearman's Rho and Tail Dependence of GE and IBM Data
      3. 16.3. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
  7. III. Inductive Statistics
    1. 17. Point Estimators
      1. 17.1. SAMPLE, STATISTIC, AND ESTIMATOR
        1. 17.1.1. Sample
        2. 17.1.2. Sampling Techniques
        3. 17.1.3. Illustrations of Drawing with Replacement
        4. 17.1.4. Statistic
        5. 17.1.5. Estimator
        6. 17.1.6. Estimator for the Mean
        7. 17.1.7. Linear Estimators
        8. 17.1.8. Estimating the Parameter p of the Bernoulli Distribution
        9. 17.1.9. Estimating the Parameter λ of Poisson Distribution
        10. 17.1.10. Linear Estimator with Lags
      2. 17.2. QUALITY CRITERIA OF ESTIMATORS
        1. 17.2.1. Bias
        2. 17.2.2. Bias of the Sample Mean
        3. 17.2.3. Bias of the Sample Variance
        4. 17.2.4. Mean-Square Error
        5. 17.2.5. Mean-Square Error of the Sample Mean
        6. 17.2.6. Mean-Square Error of the Variance Estimator
      3. 17.3. LARGE SAMPLE CRITERIA
        1. 17.3.1. Consistency
          1. 17.3.1.1. Convergence in Probability
        2. 17.3.2. Consistency of the Sample Mean of Normally Distributed Data
        3. 17.3.3. Consistency of the Variance Estimator
        4. 17.3.4. Unbiased Efficiency
        5. 17.3.5. Efficiency of the Sample Mean
        6. 17.3.6. Efficiency of the Bias Corrected Sample Variance
        7. 17.3.7. Linear Unbiased Estimators
      4. 17.4. MAXIMUM LIKEHOOD ESTIMATOR
        1. 17.4.1. MLE of the Parameter λ of the Poisson Distribution
        2. 17.4.2. MLE of the Parameter λ of the Exponential Distribution
        3. 17.4.3. MLE of the Parameter Components of the Normal Distribution
        4. 17.4.4. Cramér-Rao Lower Bound
        5. 17.4.5. Cramér-Rao Bound of the MLE of Parameter λ of the Exponential Distribution
        6. 17.4.6. Cramér-Rao Bounds of the MLE of the Parameters of the Normal Distribution
      5. 17.5. Exponential family and sufficiency
        1. 17.5.1. Exponential Family
        2. 17.5.2. Exponential Family of the Poisson Distribution
        3. 17.5.3. Exponential Family of the Exponential Distribution
        4. 17.5.4. Exponential Family of the Normal Distribution
        5. 17.5.5. Sufficiency
        6. 17.5.6. Sufficient Statistic for the Parameter λ of the Poisson Distribution
      6. 17.6. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    2. 18. Confidence Intervals
      1. 18.1. Confidence LEVEL AND Confidence interval
        1. 18.1.1. Definition of a Confidence Level
        2. 18.1.2. Definition and Interpretation of a Confidence Interval
      2. 18.2. Confidence Interval for the Mean of a Normal Random Variable
      3. 18.3. Confidence Interval for the Mean of a Normal Random Variable with Unknown Variance
      4. 18.4. Confidence Interval for the Variance of a Normal Random Variable
      5. 18.5. Confidence Interval for the Variance of a Normal Random Variable with Unknown Mean
      6. 18.6. Confidence Interval for the Parameter p of a Binomial Distribution
      7. 18.7. Confidence Interval for the Parameter λ of an Exponential Distribution
      8. 18.8. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    3. 19. Hypothesis Testing
      1. 19.1. Hypotheses
        1. 19.1.1. Setting Up the Hypotheses
        2. 19.1.2. Decision Rule
          1. 19.1.2.1. Acceptance and Rejection Region
      2. 19.2. Error Types
        1. 19.2.1. Type I and Type II Error
        2. 19.2.2. Test Size
        3. 19.2.3. The p-Value
      3. 19.3. Quality criteria of a test
        1. 19.3.1. Power of a Test
          1. 19.3.1.1. Uniformly Most Powerful Test
        2. 19.3.2. Unbiased Test
        3. 19.3.3. Consistent Test
      4. 19.4. Examples
        1. 19.4.1. Simple Test for Parameter λ of the Poisson Distribution
        2. 19.4.2. One-Tailed Test for Parameter λ of Exponential Distribution
        3. 19.4.3. One-Tailed Test for μ of the Normal Distribution When σ2 Is Known
        4. 19.4.4. One-Tailed Test for σ2 of the Normal Distribution When μ Is Known
        5. 19.4.5. Two-Tailed Test for the Parameter μ of the Normal Distribution When σ2 Is Known
        6. 19.4.6. Equal Tails Test for the Parameter σ2 of the Normal Distribution When μ Is Known
        7. 19.4.7. Test for Equality of Means
        8. 19.4.8. Two-Tailed Kolmogorov-Smirnov Test for Equality of Distribution
        9. 19.4.9. Likelihood Ratio Test
      5. 19.5. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
  8. IV. Multivariate Linear Regression Analysis
    1. 20. Estimates and Diagnostics for Multivariate Linear Regression Analysis
      1. 20.1. THE MULTIVARIATE LINEAR REGRESSION MODEL
      2. 20.2. ASSUMPTIONS OF THE Multivariate LINEAR REGRESSION MODEL
      3. 20.3. ESTIMATION OF THE MODEL PARAMETERS
      4. 20.4. DESIGNING THE MODEL
      5. 20.5. DIAGNOSTIC CHECK AND MODEL SIGNIFICANCE
        1. 20.5.1. Testing for the Significance of the Model
        2. 20.5.2. Testing for the Significance of the Independent Variables
        3. 20.5.3. The F-Test for Inclusion of Additional Variables
      6. 20.6. APPLICATIONS TO FINANCE
        1. 20.6.1. Estimation of Empirical Duration
        2. 20.6.2. Predicting the 10-Year Treasury Yield
      7. 20.7. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    2. 21. Designing and Building a Multivariate Linear Regression Model
      1. 21.1. THE PROBLEM OF MULTICOLLINEARITY
        1. 21.1.1. Procedures for Mitigating Multicollinearity
      2. 21.2. INCORPORATING DUMMY VARIABLES AS INDEPENDENT VARIABLES
        1. 21.2.1. Application to Testing the Mutual Fund Characteristic Lines in Different Market Environments
        2. 21.2.2. Application to Predicting High-Yield Corporate Bond Spreads
      3. 21.3. MODEL BUILDING TECHNIQUES
        1. 21.3.1. Stepwise Inclusion Method
        2. 21.3.2. Stepwise Exclusion Method
        3. 21.3.3. Standard Stepwise Regression Method
      4. 21.4. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
    3. 22. Testing the Assumptions of the Multivariate Linear Regression Model
      1. 22.1. TESTS FOR LINEARITY
      2. 22.2. ASSUMED STATISTICAL PROPERTIES ABOUT THE ERROR TERM
      3. 22.3. TESTS FOR THE RESIDUALS BEING NORMALLY DISTRIBUTED
        1. 22.3.1. Chi-Square Statistic
        2. 22.3.2. Jarque-Bera Test Statistic
        3. 22.3.3. Analysis of Standardized Residuals
      4. 22.4. TESTS FOR CONSTANT VARIANCE OF THE ERROR TERM (HOMOSKEDASTICITY)
        1. 22.4.1. Modeling to Account for Heteroskedasticity
          1. 22.4.1.1. Weighted Least Squares Estimation Technique
          2. 22.4.1.2. Advanced Modeling to Account for Heteroskedasticity
      5. 22.5. ABSENCE OF AUTOCORRELATION OF THE RESIDUALS
        1. 22.5.1. Detecting Autocorrelation
        2. 22.5.2. Modeling in the Presence of Autocorrelation
        3. 22.5.3. Autoregressive Moving Average Models
      6. 22.6. CONCEPTS EXPLAINED IN THIS CHAPTER (IN ORDER OF PRESENTATION)
  9. A. Important Functions and Their Features
    1. A.1. CONTINUOUS FUNCTION
      1. A.1.1. General Idea
      2. A.1.2. Formal Derivation
    2. A.2. INDICATOR FUNCTION
    3. A.3. DERIVATIVES
      1. A.3.1. Construction of the Derivative
    4. A.4. MONOTONIC FUNCTION
    5. A.5. INTEGRAL
      1. A.5.1. Approximation of the Area through Rectangles
        1. A.5.1.1. Integral as the Limiting Area
      2. A.5.2. Relationship Between Integral and Derivative
    6. A.6. SOME FUNCTIONS
      1. A.6.1. Factorial
      2. A.6.2. Gamma Function
      3. A.6.3. Beta Function
      4. A.6.4. Bessel Function of the Third Kind
      5. A.6.5. Characteristic Function
  10. B. Fundamentals of Matrix Operations and Concepts
    1. B.1. THE NOTION OF VECTOR AND MATRIX
    2. B.2. MATRIX MULTIPLICATION
    3. B.3. PARTICULAR MATRICES
      1. B.3.1. Determinant of a Matrix
      2. B.3.2. Eigenvalues and Eigenvectors
    4. B.4. POSITIVE SEMIDEFINITE MATRICES
  11. C. Binomial and Multinomial Coefficients
    1. C.1. Binomial Coefficient
      1. C.1.1. Derivation of the Binomial Coefficient
        1. C.1.1.1. Special Case n = 3
        2. C.1.1.2. Special Case n = 4
        3. C.1.1.3. General Case
    2. C.2. Multinomial Coefficient
  12. D. Application of the Log-Normal Distribution to the Pricing of Call Options
    1. D.1. Call Options
    2. D.2. Deriving the Price of a European Call Option
    3. D.3. Illustration
  13. References