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Value-at-Risk
Theory and Practice
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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
Standard Normal Table
Standard Normal Table
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