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Value-at-Risk

Theory and Practice

  • Cover
    • Title Page
    • Copyright
    • About the Author
    • Acknowledgements
    • Contents
  • 0 Preface
    • 0.1 What We’re About
    • 0.2 Voldemort and the Second Edition
    • 0.3 How To Read This Book
    • 0.4 Notation
  • 1 Value-at-Risk
    • 1.1 Measures
    • 1.2 Risk Measures
    • 1.3 Market Risk
    • 1.4 Value-at-Risk
    • 1.5 Risk Limits
    • 1.6 Other Applications of Value-at-Risk
    • 1.7 Examples
    • 1.8 Value-at-Risk Measures
    • 1.9 History of Value-at-Risk
    • 1.10 Further Reading
  • 2 Mathematical Preliminaries
    • 2.1 Motivation
    • 2.2 Mathematical Notation
    • 2.3 Gradient & Gradient-Hessian Approx.
    • 2.4 Ordinary Interpolation
    • 2.5 Complex Numbers
    • 2.6 Eigenvalues and Eigenvectors
    • 2.7 Cholesky Factorization
    • 2.8 Minimizing a Quadratic Polynomial
    • 2.9 Ordinary Least Squares
    • 2.10 Cubic Spline Interpolation
    • 2.11 Finite Difference Approximations
    • 2.12 Newton’s Method
    • 2.13 Change of Variables Formula
    • 2.14 Numerical Integration: One Dim.
    • 2.15 Numerical Integration: Multi Dim.
    • 2.16 Further Reading
  • 3 Probability
    • 3.1 Motivation
    • 3.2 Prerequisites
    • 3.3 Parameters
    • 3.4 Parameters of Random Vectors
    • 3.5 Linear Polynomials of Random Vectors
    • 3.6 Properties of Covariance Matrices
    • 3.7 Principal Component Analysis
    • 3.8 Bernoulli and Binomial Distributions
    • 3.9 Uniform and Related Distributions
    • 3.10 Normal and Related Distributions
    • 3.11 Mixtures of Distributions
    • 3.12 Moment-Generating Functions
    • 3.13 Quadratic Polynomials of Joint-Normal Random Vectors
    • 3.14 The Cornish-Fisher Expansion
    • 3.15 Central Limit Theorem
    • 3.16 The Inversion Theorem
    • 3.17 Quantiles of Quadratic Polynomials of Joint-Normal Random Vectors
    • 3.18 Further Reading
  • 4 Statistics and Time Series
    • 4.1 Motivation
    • 4.2 From Probability to Statistics
    • 4.3 Estimation
    • 4.4 Maximum Likelihood Estimators
    • 4.5 Hypothesis Testing
    • 4.6 Stochastic Processes
    • 4.7 Testing for Autocorrelations
    • 4.8 White Noise, Moving-Average and Autoregressive Processes
    • 4.9 GARCH Processes
    • 4.10 Regime-Switching Processes
    • 4.11 Further Reading
  • 5 Monte Carlo Method
    • 5.1 Motivation
    • 5.2 The Monte Carlo Method
    • 5.3 Realizations of Samples
    • 5.4 Pseudorandom Numbers
    • 5.5 Testing Pseudorandom Number Generators
    • 5.6 Implementing Pseudorandom Number Generators
    • 5.7 Breaking the Curse of Dimensionality
    • 5.8 Pseudorandom Variates
    • 5.9 Variance Reduction
    • 5.10 Further Reading
  • 6 Historical Market Data
    • 6.1 Motivation
    • 6.2 Forms of Historical Market Data
    • 6.3 Nonsynchronous Data
    • 6.4 Data Errors
    • 6.5 Data Biases
    • 6.6 Futures Prices
    • 6.7 Implied Volatilities
    • 6.8 Further Reading
  • 7 Inference
    • 7.1 Motivation
    • 7.2 Selecting Key Factors
    • 7.3 Current Practice
    • 7.4 Unconditional Leptokurtosis and Conditional Heteroskedasticity
    • 7.5 Further Reading
  • 8 Primary Portfolio Mappings
    • 8.1 Motivation
    • 8.2 Day Counts
    • 8.3 Primary Mappings
    • 8.4 Example: Equities
    • 8.5 Example: Forwards
    • 8.6 Example: Options
    • 8.7 Physical Commodities
    • 8.8 Further Reading
  • 9 Portfolio Remappings
    • 9.1 Motivation
    • 9.2 Holdings Remappings
    • 9.3 Function Remappings
    • 9.4 Variables Remappings
    • 9.5 Further Reading
  • 10 Transformation Procedures
    • 10.1 Motivation
    • 10.2 Linear Transformation Procedures
    • 10.3 Quadratic Transformation Procedures
    • 10.4 Monte Carlo Transformation Procedures
    • 10.5 Variance Reduction
    • 10.6 Further Reading
  • 11 Historical Simulation
    • 11.1 Motivation
    • 11.2 Generating Realizations Directly From Historical Market Data
    • 11.3 Calculating Value-at-Risk With Historical Simulation
    • 11.4 Origins of Historical Simulation
    • 11.5 Flawed Arguments for Historical Simulation
    • 11.6 Shortcomings of Historical Simulation
    • 11.7 Further Reading
  • 12 Implementing Value-at-Risk
    • 12.1 Motivation
    • 12.2 Preliminaries
    • 12.3 Purpose
    • 12.4 Functional Requirements
    • 12.5 Build vs. Buy
    • 12.6 Implementation
    • 12.7 Further Reading
  • 13 Model Risk, Testing and Validation
    • 13.1 Motivation
    • 13.2 Model Risk
    • 13.3 Managing Model Risk
    • 13.4 Further Reading
  • 14 Backtesting
    • 14.1 Motivation
    • 14.2 Backtesting
    • 14.3 Backtesting With Coverage Tests
    • 14.4 Backtesting With Distribution Tests
    • 14.5 Backtesting With Independence Tests
    • 14.6 Example: Backtesting a One-Day 95% EUR Value-at-Risk Measure
    • 14.7 Backtesting Strategy
    • 14.8 Further Reading
  • Back Matter
    • Endnotes
    • References
    • Standard Normal Table

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Standard Normal Table

Standard Normal Table

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