Contents
PART I – OVERVIEW
Chapter 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
Chapter 1 – Value-at-Risk
1.1 Measures
1.2 Risk Measures
1.3 Market Risk
1.4 Value-at-Risk
Probabilistic Metrics of Market Risk (PMMRs)
1.5 Risk Limits
1.6 Other Applications of Value-at-Risk
Bank Regulatory Capital Requirements
1.7 Examples
Example: Australian Equities (Monte Carlo Transformation)
Example: Australian Equities (Linear Remapping)
Example: Australian Equities (Quadratic Transformation)
1.8 Value-ar-Risk Measures
1.9 History
Regulatory Value-at-Risk Measures
Proprietary Value-at-Risk Measures
1.10 Further Reading
PART II – ESSENTIAL MATHEMATICS
Chapter 2 – Mathematical Preliminaries
2.1 Motivation
2.2 Mathematical Notation
2.3 Gradient and Gradient-Hessian Approximations
2.4 Ordinary Interpolation
Example: Quadratic Interpolation
Ordinary Interpolation Methodology
2.5 Complex Numbers
2.6 Eigenvalues and Eigenvectors
2.7 Cholesky Factorization
Positive Definite Matrices
Matrix “Square Roots”
Cholesky Factorization
Computational Issues
Exercises
2.8 Minimizing a Quadratic Polynomial
2.9 Ordinary Least Squares
Ordinary Least Squares Methodology
2.10 Cubic Spline Interpolation
2.11 Finite Difference Approximations of Derivatives
2.12 Newton’s Method
Example: Univariate Newton’s Method
2.13 Change of Variables Formula
2.14 Numerical Integration: One Dimension
2.15 Numerical Integration: Multiple Dimensions
2.16 Further Reading
Chapter 3 – Probability
3.1 Motivation
3.2 Prerequisites
3.3 Parameters
Expectation of a Function of a Random Variable
Variance and Standard Deviation
3.4 Parameters of Random Vectors
Expectation of a Function of a Random Vector
3.5 Linear Polynomials of Random Vectors
3.6 Properties of Covariance Matrices
3.7 Principal Component Analysis
Definition of Principal Components
Choice of Weights With Principal Components
3.8 Bernoulli and Binomial Distributions
3.9 Uniform and Related Distributions
Multivariate Uniform Distribution
3.10 Normal and Related Distributions
3.11 Mixtures of Distributions
Parameters of Mixed Distributions
Mixed Joint-Normal Distributions
3.12 Moment-Generating Functions
3.13 Quadratic Polynomials of Joint-Normal Random Vectors
3.14 The Cornish-Fisher Expansion
Cornish-Fisher Expansion With Five Cumulants
3.15 Central Limit Theorem
3.16 The Inversion Theorem
3.17 Quantiles of Quadratic Polynomials of Joint-Normal Random Vectors
The CDF of a Quadratic Polynomial of a Joint-Normal Random Vector
Quantiles of a Quadratic Polynomial of a Joint-Normal Random Vector
3.18 Further Reading
Chapter 4 – Statistics and Time Series Analysis
4.1 Motivation
4.2 From Probability to Statistics
4.3 Estimation
4.4 Maximum Likelihood Estimators
ML Estimates of Scalar Parameters
ML Estimates of Non-Scalar Parameters
Example: Mixed Normal Distribution
4.5 Hypothesis Testing
4.6 Stochastic Processes
Conditional vs. Unconditional Distributions
Correlations, Autocorrelations and Cross Correlations
Stationarity Stochastic Processes
4.7 Testing for Autocorrelations
4.8 White Noise, Moving-Average and Autoregressive Processes
Autoregressive Moving-Average Processes
4.9 Garch Processes
Properties of CCC-GARCH and Orthogonal GARCH
4.10 Regime-Switching Processes
4.11 Further Reading
Chapter 5 – Monte Carlo Method
5.1 Motivation
5.2 The Monte Carlo Method
Example: Approximating a Standard Deviation
5.3 Realizations of Samples
5.4 Pseudorandom Numbers
Linear Congruential Generators
5.5 Testing Pseudorandom Number Generators
5.6 Implementing Pseudorandom Number Generators
5.7 Breaking the Curse of Dimensionality
Implications of Standard Error
5.8 Pseudorandom Variates
Joint-Normal Pseudorandom Vectors
Monte Carlo Simulation – Directly Modeling Relevant Random Vectors
5.9 Variance Reduction
Control Variates for Monte Carlo Estimators of Quantiles
5.10 Further Reading
PART III – VALUE-AT-RISK
Chapter 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
Constant-Maturity Futures Prices
6.7 Implied Volatilities
6.8 Further Reading
Chapter 7 – Covariance Matrix Construction
7.1 Motivation
7.2 Selecting Key Factors
Stationarity and Homoskedasticity
7.3 Current Practice
Uniformly Weighted Moving Average Estimates
Exponentially Weighted Moving Average Estimates
Covariance Matrices That are Not Positive Definite
7.4 Unconditional Leptokurtosis and Conditional Heteroskedasticity
An Experiment With Conditional Heteroskedasticity
Modeling Unconditional Leptokurtosis
Implications for Value-at-Risk Measures
7.5 Further Reading
Chapter 8 – Primary Portfolio Mappings
8.1 Motivation
8.2 Day Counts
Actual Days, Basis Days, and Trading Days
Example: Cash Valuation Discount Curve
Example: 2nd-Day Valuation Discount Curve
Example: Random Discount Curve
8.3 Primary Mappings
8.4 Example: Equities
8.5 Example: Forwards
8.6 Example: Options
8.7 Example: Physical Commodities
8.8 Further Reading
Chapter 9 – Portfolio Remappings
9.1 Motivation
Facilitating Another Remapping
9.2 Holdings Remappings
Example: Holdings Remappings of Fixed Cash Flows
Example: Holdings Remapping of Interest Rate Caps
9.3 Function Remappings
Linear Remappings with Gradient Approximations
Example: Gradient Approximations
Quadratic Remappings with Gradient-Hessian Approximations
Interpolation and The Method or Least Squares
Selecting Realizations for Interpolation of Least Squares
9.4 Variables Remappings
Principal-Component Remappings
Example: US Treasury Securities
9.5 Further Reading
Chapter 10 – Transformation Procedures
10.1 Motivation
10.2 Linear Transformation Procedures
10.3 Quadratic Transformation Procedures
Example Continued: Linear Polynomial Representation
Example Continued: Standard Deviation of Portfolio Value
Example Continued: Cornish-Fisher Expansion
Example Continued: Inverting the Characteristic Function
10.4 Monte Carlo Transformation Procedures
Empirical Analysis of Standard Error for Value-at-Risk
10.5 Variance Reduction
Stratified Sampling to Calculate Standard Deviation of Loss
Stratified Sampling to Calculate Value-at-Risk
Selective Valuation of Realizations
10.6 Further Reading
Chapter 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
PART IV – IMPLEMENTATION AND VALIDATION
Chapter 12 – Implementingvalue-at-risk
12.1 Motivation
12.2 Preliminaries
12.3 Purpose
12.4 Functional Requirements
12.5 Build vs. Buy
12.6 Implementation
Agile Software Development Methods
12.7 Further Reading
Chapter 13 – Model Risk, Testing and Validation
13.1 Motivation
13.2 Model Risk
Type A: Model Specification Risk
Type B: Model Implementation Risk
Type C: Model Application Risk
13.3 Managing Model Risk
Standard Assumptions and Modeling Procedures
13.4 Further Reading
Chapter 14 – Backtesting
14.1 Motivation
14.2 Backtesting
14.3 Backtesting with Coverage Tests
A Recommended Standard Coverage Test
The Basel Committee’s Traffic Light Coverage Test
14.4 Backtesting with Distribution Tests
Framework for Distribution Tests
A Recommended Standard Distribution Test
14.5 Backtesting with Independence Tests
Christoffersen’s (1998) Exceedence Independence Test
A Recommended Standard Loss-Quantile Independence Test
14.6 Example: Backtesting a One-Day 95% EUR Value-at-Risk Measure
Example: Applying Coverage Tests
Example: Applying Distribution Tests
Example: Applying Independence Tests
14.7 Backtesting Strategy
Backtesting as Hypothesis Testing
Designing a Backtesting Program
14.8 Further Reading
Back Matter
Endnotes
References
Appendix – Normal Table