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Introductory Mathematics and Statistics for Islamic Finance, + Website

Book Description

A unique primer on quantitative methods as applied to Islamic finance

Introductory Mathematics and Statistics for Islamic Finance + Website is a comprehensive guide to quantitative methods, specifically as applied within the realm of Islamic finance. With applications based on research, the book provides readers with the working knowledge of math and statistics required to understand Islamic finance theory and practice. The numerous worked examples give students with various backgrounds a uniform set of common tools for studying Islamic finance.

The in-depth study of finance requires a strong foundation in quantitative methods. Without a good grasp of math, probability, and statistics, published theoretical and applied works in Islamic finance remain out of reach. Unlike a typical math text, this book guides students through only the methods that directly apply to Islamic finance, without wasting time on irrelevant techniques. Each chapter contains a detailed explanation of the topic at hand, followed by an example based on real situations encountered in Islamic finance. Topics include:

  • Algebra and matrices

  • Calculus and differential equations

  • Probability theory

  • Statistics

  • Written by leading experts on the subject, the book serves as a useful primer on the analysis methods and techniques students will encounter in published research, as well as day-to-day operations in finance. Anyone aspiring to be successful in Islamic finance needs these skills, and Introductory Mathematics and Statistics for Islamic Finance + Website is a clear, concise, and highly relevant guide.

    Table of Contents

    1. Cover
    2. Title Page
    3. Copyright
    4. Dedication
    5. Preface
    6. Acknowledgments
    7. About the Authors
    8. Part One: Mathematics
      1. Chapter 1: Elementary Mathematics
        1. Basic Mathematical Objects
        2. Variables, Monomials, Binomials, and Polynomials
        3. Equations
        4. Equations of Higher Order
        5. Sequences
        6. Series
        7. Applications of Series to Present Value of Assets
        8. Summary
      2. Chapter 2: Functions and Models
        1. Definition of a Function
        2. Functions and Models in Economics
        3. Functions and Models in Finance
        4. Multivariate Functions in Economics and Finance
        5. Summary
      3. Chapter 3: Differentiation and Integration of Functions
        1. Differentiation
        2. Differentiation Rules
        3. Maximum and Minimum of a Function
        4. Mean Value Theorem
        5. Polynomial Approximations of a Function: Taylor's Expansion
        6. Integration
        7. Applications in Finance: Duration and Convexity of a Sukuk
        8. Summary
      4. Chapter 4: Partial Derivatives
        1. Definition and Computation of Partial Derivatives
        2. Total Differential of a Function with Many Variables
        3. Directional Derivatives
        4. Gradients
        5. Tangent Planes and Normal Lines
        6. Extrema of Functions of Several Variables
        7. Extremal Problems with Constraints
        8. Summary
      5. Chapter 5: Logarithm, Exponential, and Trigonometric Functions
        1. Logarithm Functions
        2. The Exponential Function
        3. Power Series of Logarithmic and Exponential Functions
        4. General Exponential and Logarithmic Functions
        5. Some Applications of Logarithm and Exponential Functions in Finance
        6. Integration by Parts
        7. Trigonometric Functions
        8. Summary
      6. Chapter 6: Linear Algebra
        1. Vectors
        2. Matrices
        3. Square Matrices
        4. The Rank of a Matrix
        5. Determinant of a Square Matrix
        6. Homogenous Systems of Equations
        7. Inverse and Generalized Inverse Matrices
        8. Eigenvalues and Eigenvectors
        9. Stability of a Linear System
        10. Applications in Econometrics
        11. Summary
      7. Chapter 7: Differential Equations
        1. Examples of Differential Equations
        2. Solution Methods for the Differential Equation
        3. First-Order Linear Differential Equations
        4. Second-Order Linear Differential Equations
        5. Linear Differential Equation Systems
        6. Phase Diagrams and Stability Analysis
        7. Summary
      8. Chapter 8: Difference Equations
        1. Definition of a Difference Equation
        2. First-Order Linear Difference Equations
        3. Second-Order Linear Difference Equations
        4. System of Linear Difference Equations
        5. Equilibrium and Stability
        6. Summary
      9. Chapter 9: Optimization Theory
        1. The Mathematical Programming Problem
        2. Unconstrained Optimization
        3. Constrained Optimization
        4. The General Classical Program
        5. Summary
      10. Chapter 10: Linear Programming
        1. Formulation of the LP
        2. The Analytical Approach to Solving an LP: The Simplex Method
        3. The Dual Problem of the LP
        4. The Lagrangian Approach: Existence, Duality, and Complementary Slackness Theorems
        5. Economic Theory and Duality
        6. Summary
    9. Part Two: Statistics
      1. Chapter 11: Introduction to Probability Theory: Axioms and Distributions
        1. The Empirical Background: The Sample Space and Events
        2. Definition of Probability
        3. Random Variable
        4. Techniques of Counting: Combinatorial Analysis
        5. Conditional Probability and Independence
        6. Probability Distribution of a Finite Random Variable
        7. Moments of a Probability Distribution
        8. Joint Distribution of Random Variables
        9. Chebyshev's Inequality and the Law of Large Numbers
        10. Summary
      2. Chapter 12: Probability Distributions and Moment Generating Functions
        1. Examples of Probability Distributions
        2. Empirical Distributions
        3. Moment Generating Function (MGF)
        4. Summary
      3. Chapter 13: Sampling and Hypothesis Testing Theory
        1. Sampling Distributions
        2. Estimation of Parameters
        3. Confidence-Interval Estimates of Population Parameters
        4. Hypothesis Testing
        5. Tests Involving Sample Differences
        6. Small Sampling Theory
        7. Summary
      4. Chapter 14: Regression Analysis
        1. Curve Fitting
        2. Linear Regression Analysis
        3. The Probability Distribution of the Estimated Regression Coefficients â and b
        4. Hypothesis Testing of â and b
        5. Diagnostic Test of the Regression Results
        6. Prediction
        7. Multiple Correlation
        8. Summary
      5. Chapter 15: Time Series Analysis
        1. Component Movements of a Time Series
        2. Stationary Time Series
        3. Characterizing Time Series: The Autocorrelation Function
        4. Linear Time Series Models
        5. Moving Average (MA) Linear Models
        6. Autoregressive (AR) Linear Models
        7. Mixed Autoregressive-Moving Average (Arma) Linear Models
        8. The Partial Autocorrelation Function
        9. Forecasting Based on Time Series
        10. Summary
      6. Chapter 16: Nonstationary Time Series and Unit-Root Testing
        1. The Random Walk
        2. Decomposition of a Nonstationary Time Series
        3. Forecasting a Random Walk
        4. Meaning and Implications of Nonstationary Processes
        5. Dickey-Fuller Unit-Root Tests
        6. The Augmented Dickey-Fuller Test (ADF)
        7. Summary
      7. Chapter 17: Vector Autoregressive Analysis (VAR)
        1. Formulation of the VAR
        2. Forecasting with VAR
        3. The Impulse Response Function
        4. Variance Decomposition
        5. Summary
      8. Chapter 18: Co-Integration: Theory and Applications
        1. Spurious Regression
        2. Stationarity and Long-Run Equilibrium
        3. Co-Integration
        4. Test for Co-Integration
        5. Co-Integration and Common Trends
        6. Co-Integrated VARs
        7. Representation of a Co-Integrated VAR
        8. Summary
      9. Chapter 19: Modeling Volatility: ARCH-GARCH Models
        1. Motivation for Arch Models
        2. Formalization of the Arch Model
        3. Properties of the Arch Model
        4. The Generalized Arch (Garch) Model
        5. Arch-Garch in Mean
        6. Testing for the Arch Effects
        7. Summary
      10. Chapter 20: Asset Pricing under Uncertainty
        1. Modeling Risk and Return
        2. Uncertainty and Efficient Capital Markets: Random Walk and Martingale
        3. Market Efficiency and Arbitrage-Free Pricing
        4. Basic Principles of Derivatives Pricing
        5. State Prices
        6. Martingale Distribution and Risk-Neutral Probabilities
        7. Martingale and Complete Markets
        8. Summary
      11. Chapter 21: The Consumption-Based Pricing Model
        1. Intertemporal Optimization and Implication to Asset Pricing
        2. Asset-Specific Pricing And Correction For Risk
        3. Relationship Between Expected Return and Beta
        4. The Mean Variance (mv) Frontier
        5. Risk-Neutral Pricing Implied by the General Pricing Formula pt = Et(mt+1xt+1)
        6. Consumption-Based Contingent Discount Factors
        7. Summary
      12. Chapter 22: Brownian Motion, Risk-Neutral Processes, and the Black-Scholes Model
        1. Brownian Motion
        2. Dynamics of the Stock Price: The Diffusion Process
        3. Approximation of a Geometric Brownian Motion by A Binomial Tree
        4. Ito's Lemma
        5. Discrete Approximations
        6. Arbitrage Pricing: Black-Scholes Model
        7. The Market Price of Risk
        8. Risk-Neutral Pricing
        9. Summary
    10. References
    11. Index
    12. End User License Agreement