Essential Mathematics for Market Risk Management.

By: Hubbert, SimonMaterial type: TextTextSeries: The Wiley Finance SerPublisher: Hoboken : John Wiley & Sons, Incorporated, 2011Copyright date: ©2012Edition: 1st edDescription: 1 online resource (354 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9781119953012Subject(s): Capital market -- Mathematical models | Risk management -- Mathematical modelsGenre/Form: Electronic books.Additional physical formats: Print version:: Essential Mathematics for Market Risk ManagementDDC classification: 658.15/50151 LOC classification: HD61 -- .H763 2012ebOnline resources: Click to View
Contents:
Intro -- Essential Mathematics for Market Risk Management -- Contents -- Preface -- 1 Introduction -- 1.1 Basic Challenges in Risk Management -- 1.2 Value at Risk -- 1.3 Further Challenges in Risk Management -- 2 Applied Linear Algebra for Risk Managers -- 2.1 Vectors and Matrices -- 2.2 Matrix Algebra in Practice -- 2.3 Eigenvectors and Eigenvalues -- 2.4 Positive Definite Matrices -- 3 Probability Theory for Risk Managers -- 3.1 Univariate Theory -- 3.1.1 Random variables -- 3.1.2 Expectation -- 3.1.3 Variance -- 3.2 Multivariate Theory -- 3.2.1 The joint distribution function -- 3.2.2 The joint and marginal density functions -- 3.2.3 The notion of independence -- 3.2.4 The notion of conditional dependence -- 3.2.5 Covariance and correlation -- 3.2.6 The mean vector and covariance matrix -- 3.2.7 Linear combinations of random variables -- 3.3 The Normal Distribution -- 4 Optimization Tools -- 4.1 Background Calculus -- 4.1.1 Single-variable functions -- 4.1.2 Multivariable functions -- 4.2 Optimizing Functions -- 4.2.1 Unconstrained quadratic functions -- 4.2.2 Constrained quadratic functions -- 4.3 Over-determined Linear Systems -- 4.4 Linear Regression -- 5 Portfolio Theory I -- 5.1 Measuring Returns -- 5.1.1 A comparison of the standard and log returns -- 5.2 Setting Up the Optimal Portfolio Problem -- 5.3 Solving the Optimal Portfolio Problem -- 6 Portfolio Theory II -- 6.1 The Two-Fund Investment Service -- 6.2 A Mathematical Investigation of the Optimal Frontier -- 6.2.1 The minimum variance portfolio -- 6.2.2 Covariance of frontier portfolios -- 6.2.3 Correlation with the minimum variance portfolio -- 6.2.4 The zero-covariance portfolio -- 6.3 A Geometrical Investigation of the Optimal Frontier -- 6.3.1 Equation of a tangent to an efficient portfolio -- 6.3.2 Locating the zero-covariance portfolio.
6.4 A Further Investigation of Covariance -- 6.5 The Optimal Portfolio Problem Revisited -- 7 The Capital Asset Pricing Model (CAPM) -- 7.1 Connecting the Portfolio Frontiers -- 7.2 The Tangent Portfolio -- 7.2.1 The market's supply of risky assets -- 7.3 The CAPM -- 7.4 Applications of CAPM -- 7.4.1 Decomposing risk -- 8 Risk Factor Modelling -- 8.1 General Factor Modelling -- 8.2 Theoretical Properties of the Factor Model -- 8.3 Models Based on Principal Component Analysis (PCA) -- 8.3.1 PCA in two dimensions -- 8.3.2 PCA in higher dimensions -- 9 The Value at Risk Concept -- 9.1 A Framework for Value at Risk -- 9.1.1 A motivating example -- 9.1.2 Defining value at risk -- 9.2 Investigating Value at Risk -- 9.2.1 The suitability of value at risk to capital allocation -- 9.3 Tail Value at Risk -- 9.4 Spectral Risk Measures -- 10 Value at Risk under a Normal Distribution -- 10.1 Calculation of Value at Risk -- 10.2 Calculation of Marginal Value at Risk -- 10.3 Calculation of Tail Value at Risk -- 10.4 Sub-additivity of Normal Value at Risk -- 11 Advanced Probability Theory for Risk Managers -- 11.1 Moments of a Random Variable -- 11.2 The Characteristic Function -- 11.2.1 Dealing with the sum of several random variables -- 11.2.2 Dealing with a scaling of a random variable -- 11.2.3 Normally distributed random variables -- 11.3 The Central Limit Theorem -- 11.4 The Moment-Generating Function -- 11.5 The Log-normal Distribution -- 12 A Survey of Useful Distribution Functions -- 12.1 The Gamma Distribution -- 12.2 The Chi-Squared Distribution -- 12.3 The Non-central Chi-Squared Distribution -- 12.4 The F-Distribution -- 12.5 The t-Distribution -- 13 A Crash Course on Financial Derivatives -- 13.1 The Black-Scholes Pricing Formula -- 13.1.1 A model for asset returns -- 13.1.2 A second-order approximation -- 13.1.3 The Black-Scholes formula.
13.2 Risk-Neutral Pricing -- 13.3 A Sensitivity Analysis -- 13.3.1 Asset price sensitivity: The delta and gamma measures -- 13.3.2 Time decay sensitivity: The theta measure -- 13.3.3 The remaining sensitivity measures -- 14 Non-linear Value at Risk -- 14.1 Linear Value at Risk Revisited -- 14.2 Approximations for Non-linear Portfolios -- 14.2.1 Delta approximation for the portfolio -- 14.2.2 Gamma approximation for the portfolio -- 14.3 Value at Risk for Derivative Portfolios -- 14.3.1 Multi-factor delta approximation -- 14.3.2 Single-factor gamma approximation -- 14.3.3 Multi-factor gamma approximation -- 15 Time Series Analysis -- 15.1 Stationary Processes -- 15.1.1 Purely random processes -- 15.1.2 White noise processes -- 15.1.3 Random walk processes -- 15.2 Moving Average Processes -- 15.3 Auto-regressive Processes -- 15.4 Auto-regressive Moving Average Processes -- 16 Maximum Likelihood Estimation -- 16.1 Sample Mean and Variance -- 16.2 On the Accuracy of Statistical Estimators -- 16.2.1 Sample mean example -- 16.2.2 Sample variance example -- 16.3 The Appeal of the Maximum Likelihood Method -- 17 The Delta Method for Statistical Estimates -- 17.1 Theoretical Framework -- 17.2 Sample Variance -- 17.3 Sample Skewness and Kurtosis -- 17.3.1 Analysis of skewness -- 17.3.2 Analysis of kurtosis -- 18 Hypothesis Testing -- 18.1 The Testing Framework -- 18.1.1 The null and alternative hypotheses -- 18.1.2 Hypotheses: simple vs compound -- 18.1.3 The acceptance and rejection regions -- 18.1.4 Potential errors -- 18.1.5 Controlling the testing errors/defining the acceptance region -- 18.2 Testing Simple Hypotheses -- 18.2.1 Testing the mean when the variance is known -- 18.3 The Test Statistic -- 18.3.1 Example: Testing the mean when the variance is unknown -- 18.3.2 The p-value of a test statistic -- 18.4 Testing Compound Hypotheses.
19 Statistical Properties of Financial Losses -- 19.1 Analysis of Sample Statistics -- 19.2 The Empirical Density and Q-Q Plots -- 19.3 The Auto-correlation Function -- 19.4 The Volatility Plot -- 19.5 The Stylized Facts -- 20 Modelling Volatility -- 20.1 The RiskMetrics Model -- 20.2 ARCH Models -- 20.2.1 The ARCH(1) volatility model -- 20.3 GARCH Models -- 20.3.1 The GARCH(1, 1) volatility model -- 20.3.2 The RiskMetrics model revisited -- 20.3.3 Summary -- 20.4 Exponential GARCH -- 21 Extreme Value Theory -- 21.1 The Mathematics of Extreme Events -- 21.1.1 A naive attempt -- 21.1.2 Example 1: Exponentially distributed losses -- 21.1.3 Example 2: Normally distributed losses -- 21.1.4 Example 3: Pareto distributed losses -- 21.1.5 Example 4: Uniformly distributed losses -- 21.1.6 Example 5: Cauchy distributed losses -- 21.1.7 The extreme value theorem -- 21.2 Domains of Attraction -- 21.2.1 The Fréchet domain of attraction -- 21.3 Extreme Value at Risk -- 21.4 Practical Issues -- 21.4.1 Parameter estimation -- 21.4.2 The choice of threshold -- 22 Simulation Models -- 22.1 Estimating the Quantile of a Distribution -- 22.1.1 Asymptotic behaviour -- 22.2 Historical Simulation -- 22.3 Monte Carlo Simulation -- 22.3.1 The Choleski algorithm -- 22.3.2 Generating random numbers -- 23 Alternative Approaches to VaR -- 23.1 The t-Distributed Assumption -- 23.2 Corrections to the Normal Assumption -- 24 Backtesting -- 24.1 Quantifying the Performance of VaR -- 24.2 Testing the Proportion of VaR Exceptions -- 24.3 Testing the Independence of VaR Exceptions -- References -- Index.
Summary: Everything you need to know in order to manage risk effectively within your organization You cannot afford to ignore the explosion in mathematical finance in your quest to remain competitive. This exciting branch of mathematics has very direct practical implications: when a new model is tested and implemented it can have an immediate impact on the financial environment. With risk management top of the agenda for many organizations, this book is essential reading for getting to grips with the mathematical story behind the subject of financial risk management. It will take you on a journey-from the early ideas of risk quantification up to today's sophisticated models and approaches to business risk management. To help you investigate the most up-to-date, pioneering developments in modern risk management, the book presents statistical theories and shows you how to put statistical tools into action to investigate areas such as the design of mathematical models for financial volatility or calculating the value at risk for an investment portfolio. Respected academic author Simon Hubbert is the youngest director of a financial engineering program in the U.K. He brings his industry experience to his practical approach to risk analysis Captures the essential mathematical tools needed to explore many common risk management problems Website with model simulations and source code enables you to put models of risk management into practice Plunges into the world of high-risk finance and examines the crucial relationship between the risk and the potential reward of holding a portfolio of risky financial assets This book is your one-stop-shop for effective risk management.
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Intro -- Essential Mathematics for Market Risk Management -- Contents -- Preface -- 1 Introduction -- 1.1 Basic Challenges in Risk Management -- 1.2 Value at Risk -- 1.3 Further Challenges in Risk Management -- 2 Applied Linear Algebra for Risk Managers -- 2.1 Vectors and Matrices -- 2.2 Matrix Algebra in Practice -- 2.3 Eigenvectors and Eigenvalues -- 2.4 Positive Definite Matrices -- 3 Probability Theory for Risk Managers -- 3.1 Univariate Theory -- 3.1.1 Random variables -- 3.1.2 Expectation -- 3.1.3 Variance -- 3.2 Multivariate Theory -- 3.2.1 The joint distribution function -- 3.2.2 The joint and marginal density functions -- 3.2.3 The notion of independence -- 3.2.4 The notion of conditional dependence -- 3.2.5 Covariance and correlation -- 3.2.6 The mean vector and covariance matrix -- 3.2.7 Linear combinations of random variables -- 3.3 The Normal Distribution -- 4 Optimization Tools -- 4.1 Background Calculus -- 4.1.1 Single-variable functions -- 4.1.2 Multivariable functions -- 4.2 Optimizing Functions -- 4.2.1 Unconstrained quadratic functions -- 4.2.2 Constrained quadratic functions -- 4.3 Over-determined Linear Systems -- 4.4 Linear Regression -- 5 Portfolio Theory I -- 5.1 Measuring Returns -- 5.1.1 A comparison of the standard and log returns -- 5.2 Setting Up the Optimal Portfolio Problem -- 5.3 Solving the Optimal Portfolio Problem -- 6 Portfolio Theory II -- 6.1 The Two-Fund Investment Service -- 6.2 A Mathematical Investigation of the Optimal Frontier -- 6.2.1 The minimum variance portfolio -- 6.2.2 Covariance of frontier portfolios -- 6.2.3 Correlation with the minimum variance portfolio -- 6.2.4 The zero-covariance portfolio -- 6.3 A Geometrical Investigation of the Optimal Frontier -- 6.3.1 Equation of a tangent to an efficient portfolio -- 6.3.2 Locating the zero-covariance portfolio.

6.4 A Further Investigation of Covariance -- 6.5 The Optimal Portfolio Problem Revisited -- 7 The Capital Asset Pricing Model (CAPM) -- 7.1 Connecting the Portfolio Frontiers -- 7.2 The Tangent Portfolio -- 7.2.1 The market's supply of risky assets -- 7.3 The CAPM -- 7.4 Applications of CAPM -- 7.4.1 Decomposing risk -- 8 Risk Factor Modelling -- 8.1 General Factor Modelling -- 8.2 Theoretical Properties of the Factor Model -- 8.3 Models Based on Principal Component Analysis (PCA) -- 8.3.1 PCA in two dimensions -- 8.3.2 PCA in higher dimensions -- 9 The Value at Risk Concept -- 9.1 A Framework for Value at Risk -- 9.1.1 A motivating example -- 9.1.2 Defining value at risk -- 9.2 Investigating Value at Risk -- 9.2.1 The suitability of value at risk to capital allocation -- 9.3 Tail Value at Risk -- 9.4 Spectral Risk Measures -- 10 Value at Risk under a Normal Distribution -- 10.1 Calculation of Value at Risk -- 10.2 Calculation of Marginal Value at Risk -- 10.3 Calculation of Tail Value at Risk -- 10.4 Sub-additivity of Normal Value at Risk -- 11 Advanced Probability Theory for Risk Managers -- 11.1 Moments of a Random Variable -- 11.2 The Characteristic Function -- 11.2.1 Dealing with the sum of several random variables -- 11.2.2 Dealing with a scaling of a random variable -- 11.2.3 Normally distributed random variables -- 11.3 The Central Limit Theorem -- 11.4 The Moment-Generating Function -- 11.5 The Log-normal Distribution -- 12 A Survey of Useful Distribution Functions -- 12.1 The Gamma Distribution -- 12.2 The Chi-Squared Distribution -- 12.3 The Non-central Chi-Squared Distribution -- 12.4 The F-Distribution -- 12.5 The t-Distribution -- 13 A Crash Course on Financial Derivatives -- 13.1 The Black-Scholes Pricing Formula -- 13.1.1 A model for asset returns -- 13.1.2 A second-order approximation -- 13.1.3 The Black-Scholes formula.

13.2 Risk-Neutral Pricing -- 13.3 A Sensitivity Analysis -- 13.3.1 Asset price sensitivity: The delta and gamma measures -- 13.3.2 Time decay sensitivity: The theta measure -- 13.3.3 The remaining sensitivity measures -- 14 Non-linear Value at Risk -- 14.1 Linear Value at Risk Revisited -- 14.2 Approximations for Non-linear Portfolios -- 14.2.1 Delta approximation for the portfolio -- 14.2.2 Gamma approximation for the portfolio -- 14.3 Value at Risk for Derivative Portfolios -- 14.3.1 Multi-factor delta approximation -- 14.3.2 Single-factor gamma approximation -- 14.3.3 Multi-factor gamma approximation -- 15 Time Series Analysis -- 15.1 Stationary Processes -- 15.1.1 Purely random processes -- 15.1.2 White noise processes -- 15.1.3 Random walk processes -- 15.2 Moving Average Processes -- 15.3 Auto-regressive Processes -- 15.4 Auto-regressive Moving Average Processes -- 16 Maximum Likelihood Estimation -- 16.1 Sample Mean and Variance -- 16.2 On the Accuracy of Statistical Estimators -- 16.2.1 Sample mean example -- 16.2.2 Sample variance example -- 16.3 The Appeal of the Maximum Likelihood Method -- 17 The Delta Method for Statistical Estimates -- 17.1 Theoretical Framework -- 17.2 Sample Variance -- 17.3 Sample Skewness and Kurtosis -- 17.3.1 Analysis of skewness -- 17.3.2 Analysis of kurtosis -- 18 Hypothesis Testing -- 18.1 The Testing Framework -- 18.1.1 The null and alternative hypotheses -- 18.1.2 Hypotheses: simple vs compound -- 18.1.3 The acceptance and rejection regions -- 18.1.4 Potential errors -- 18.1.5 Controlling the testing errors/defining the acceptance region -- 18.2 Testing Simple Hypotheses -- 18.2.1 Testing the mean when the variance is known -- 18.3 The Test Statistic -- 18.3.1 Example: Testing the mean when the variance is unknown -- 18.3.2 The p-value of a test statistic -- 18.4 Testing Compound Hypotheses.

19 Statistical Properties of Financial Losses -- 19.1 Analysis of Sample Statistics -- 19.2 The Empirical Density and Q-Q Plots -- 19.3 The Auto-correlation Function -- 19.4 The Volatility Plot -- 19.5 The Stylized Facts -- 20 Modelling Volatility -- 20.1 The RiskMetrics Model -- 20.2 ARCH Models -- 20.2.1 The ARCH(1) volatility model -- 20.3 GARCH Models -- 20.3.1 The GARCH(1, 1) volatility model -- 20.3.2 The RiskMetrics model revisited -- 20.3.3 Summary -- 20.4 Exponential GARCH -- 21 Extreme Value Theory -- 21.1 The Mathematics of Extreme Events -- 21.1.1 A naive attempt -- 21.1.2 Example 1: Exponentially distributed losses -- 21.1.3 Example 2: Normally distributed losses -- 21.1.4 Example 3: Pareto distributed losses -- 21.1.5 Example 4: Uniformly distributed losses -- 21.1.6 Example 5: Cauchy distributed losses -- 21.1.7 The extreme value theorem -- 21.2 Domains of Attraction -- 21.2.1 The Fréchet domain of attraction -- 21.3 Extreme Value at Risk -- 21.4 Practical Issues -- 21.4.1 Parameter estimation -- 21.4.2 The choice of threshold -- 22 Simulation Models -- 22.1 Estimating the Quantile of a Distribution -- 22.1.1 Asymptotic behaviour -- 22.2 Historical Simulation -- 22.3 Monte Carlo Simulation -- 22.3.1 The Choleski algorithm -- 22.3.2 Generating random numbers -- 23 Alternative Approaches to VaR -- 23.1 The t-Distributed Assumption -- 23.2 Corrections to the Normal Assumption -- 24 Backtesting -- 24.1 Quantifying the Performance of VaR -- 24.2 Testing the Proportion of VaR Exceptions -- 24.3 Testing the Independence of VaR Exceptions -- References -- Index.

Everything you need to know in order to manage risk effectively within your organization You cannot afford to ignore the explosion in mathematical finance in your quest to remain competitive. This exciting branch of mathematics has very direct practical implications: when a new model is tested and implemented it can have an immediate impact on the financial environment. With risk management top of the agenda for many organizations, this book is essential reading for getting to grips with the mathematical story behind the subject of financial risk management. It will take you on a journey-from the early ideas of risk quantification up to today's sophisticated models and approaches to business risk management. To help you investigate the most up-to-date, pioneering developments in modern risk management, the book presents statistical theories and shows you how to put statistical tools into action to investigate areas such as the design of mathematical models for financial volatility or calculating the value at risk for an investment portfolio. Respected academic author Simon Hubbert is the youngest director of a financial engineering program in the U.K. He brings his industry experience to his practical approach to risk analysis Captures the essential mathematical tools needed to explore many common risk management problems Website with model simulations and source code enables you to put models of risk management into practice Plunges into the world of high-risk finance and examines the crucial relationship between the risk and the potential reward of holding a portfolio of risky financial assets This book is your one-stop-shop for effective risk management.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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