Market risk, quantitative analysis, market risk measurement, probability, statistics, estimation, stochastic processes and stochastic calculus, random vectors, joint and marginal distributions, moments and characteristic function, mean, variance, correlation, Cholesky factorization, estimators of covariance, maximum likelihood estimators, multivariate Normal distribution, Normal mixture distributions, Normal variance mixture, Normal mean-variance mixture, generalized Hyperbolic distributions, expectation maximization, spherical distributions, elliptical distributions, estimating dispersion, M-estimators, dimension reduction techniques, factor models, regression analysis, principal component analysis
From the Authors - Financial risk models, whether for market or credit risks, are inherently multivariate. The value change of a portfolio of traded instruments over a fixed time horizon depends on a random vector of risk-factor changes or returns. The loss incurred by a credit portfolio depends on a random vector of losses for the individual counterparties in the portfolio. In this chapter we consider some models for random vectors that are particularly useful for financial data.We do this from a static, distributional point of view without considering time series aspects, which are introduced later in Chapter 4.
3.1 Basics of Multivariate Modelling
3.1.1 Random Vectors and Their Distributions
3.1.2 Standard Estimators of Covariance and Correlation
3.1.3 The Multivariate Normal Distribution
3.1.4 Testing Normality and Multivariate Normality
3.2 Normal Mixture Distributions
3.2.1 Normal Variance Mixtures
3.2.2 Normal Mean-Variance Mixtures
3.2.3 Generalized Hyperbolic Distributions
3.2.4 Fitting Generalized Hyperbolic Distributions to Data
3.2.5 Empirical Examples
3.3 Spherical and Elliptical Distributions
3.3.1 Spherical Distributions
3.3.2 Elliptical Distributions
3.3.3 Properties of Elliptical Distributions
3.3.4 Estimating Dispersion and Correlation
3.3.5 Testing for Elliptical Symmetry
3.4 Dimension Reduction Techniques
3.4.1 Factor Models
3.4.2 Statistical Calibration Strategies
3.4.3 Regression Analysis of Factor Models
3.4.4 Principal Component Analysis
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*** From the publisher ***
The implementation of sound quantitative risk models is a vital concern for all financial institutions, and this trend has accelerated in recent years with regulatory processes such as Basel II. This book provides a comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management and equips readers--whether financial risk analysts, actuaries, regulators, or students of quantitative finance--with practical tools to solve real-world problems. The authors cover methods for market, credit, and operational risk modelling; place standard industry approaches on a more formal footing; and describe recent developments that go beyond, and address main deficiencies of, current practice.
The book`s methodology draws on diverse quantitative disciplines, from mathematical finance through statistics and econometrics to actuarial mathematics. Main concepts discussed include loss distributions, risk measures, and risk aggregation and allocation principles. A main theme is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. The techniques required derive from multivariate statistical analysis, financial time series modelling, copulas, and extreme value theory. A more technical chapter addresses credit derivatives. Based on courses taught to masters students and professionals, this book is a unique and fundamental reference that is set to become a standard in the field.