This book provides the most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management. Whether you are a financial risk analyst, actuary, regulator or student of quantitative finance, Quantitative Risk Management gives you the practical tools you need to solve real-world problems. Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book's methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives. Fully revised and expanded to reflect developments in the field since the financial crisis Features shorter chapters to facilitate teaching and learning Provides enhanced coverage of Solvency II and insurance risk management and extended treatment of credit risk, including counterparty credit risk and CDO pricing Includes a new chapter on market risk and new material on risk measures and risk aggregation

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Quantitative Risk Management: Concepts, Techniques, and Tools

Building on the work of Bedford, Cooke and Joe, we show how multivariate data, which exhibit complex patterns of dependence in the tails, can be modelled using a cascade of pair-copulae, acting on two variables at a time. We use the pair-copula decomposition of a general multivariate distribution and propose a method for performing inference. The model construction is hierarchical in nature, the various levels corresponding to the incorporation of more variables in the conditioning sets, using pair-copulae as simple building blocks. Pair-copula decomposed models also represent a very flexible way to construct higher-dimensional copulae. We apply the methodology to a financial data set. Our approach represents the first step towards the development of an unsupervised algorithm that explores the space of possible pair-copula models, that also can be applied to huge data sets automatically.

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Pair-copula constructions of multiple dependence

Many proposals have been made recently for goodness-of-fit testing of copula models. After reviewing them briefly, the authors concentrate on "blanket tests", i.e.,Â those whose implementation requires neither an arbitrary categorization of the data nor any strategic choice of smoothing parameter, weight function, kernel, window, etc. The authors present a critical review of these procedures and suggest new ones. They describe and interpret the results of a large Monte Carlo experiment designed to assess the effect of the sample size and the strength of dependence on the level and power of the blanket tests for various combinations of copula models under the null hypothesis and the alternative. To circumvent problems in the determination of the limiting distribution of the test statistics under composite null hypotheses, they recommend the use of a double parametric bootstrap procedure, whose implementation is detailed. They conclude with a number of practical recommendations.

Omnibus Goodness-of-Fit Tests for Copulas: A Review and a Power Study

Many proposals have been made recently for goodness-of-fit testing of copula models. After reviewing them briefly, the authors concentrate on “blanket tests”, i.e., those whose implementation requires neither an arbitrary categorization of the data nor any strategic choice of smoothing parameter, weight function, kernel, window, etc. The authors present a critical review of these procedures and suggest new ones. They describe and interpret the results of a large Monte Carlo experiment designed to assess the effect of the sample size and the strength of dependence on the level and power of the blanket tests for various combinations of copula models under the null hypothesis and the alternative. To circumvent problems in the determination of the limiting distribution of the test statistics under composite null hypotheses, they recommend the use of a double parametric bootstrap procedure, whose implementation is detailed. They conclude with a number of practical recommendations.

Goodness-of-fit tests for copulas: A review and a power study

Summary The t copula and its properties are described with a focus on issues related to the dependence of extreme values. The Gaussian mixture representation of a multivariate t distribution is used as a starting point to construct two new copulas, the skewed t copula and the grouped t copula, which allow more heterogeneity in the modelling of dependent observations. Extreme value considerations are used to derive two further new copulas: the t extreme value copula is the limiting copula of componentwise maxima of t distributed random vectors; the t lower tail copula is the limiting copula of bivariate observations from a t distribution that are conditioned to lie below some joint threshold that is progressively lowered. Both these copulas may be approximated for practical purposes by simpler, better-known copulas, these being the Gumbel and Clayton copulas respectively.

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The t copula and related copulas

Stylized facts for univariate high-frequency data in finance are well known. They include scaling behaviour, volatility clustering, heavy tails and seasonalities. The multivariate problem, however, has scarcely been addressed up to now. In this paper, bivariate series of high-frequency FX spot data for major FX markets are investigated. First, as an indispensable prerequisite for further analysis, the problem of simultaneous deseasonalization of high-frequency data is addressed. In the following sections we analyse in detail the dependence structure as a function of the timescale. Particular emphasis is put on the tail behaviour, which is investigated by means of copulas.

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Dependence structures for multivariate high-frequency data in finance

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The stylized facts of univariate high-frequency data in flnance are well known; see Dacorogna et al. (2001). In this paper, we concentrate on multivariate high-frequency data, and analyse the dependence structure, through the notion of copula. More in particular, we analyse the conditional copula for two dimensional high-frequency data. We flnd evidence of non-constancy over time of the conditional copula and investigate this fact at six difierent time aggregation frequencies. Further, we investigate the existence of change-points in the conditional copula. The data analysed are highfrequency foreign exchange spot rates for US Dollar quoted against German Mark and quoted against Japanese Yen, previously analysed in a non-dynamic setting in Breymann et al. (2003).

Dynamic copula models for multivariate high-frequency data in flnance

Modeling exchange rate dependence dynamics at different time horizons

Over the recent years, both in finance and insurance, the modelling of dependence beyond linear correlation has become a key area of research. The notion of copula has been used with success in order to model these more general dependence concepts. We will discuss changes in dependence structures by using change-point techniques for specific parametric copula families. Besides some basic theory, some applied examples will be presented.

Alexandra Dias

Papers

Dependence structures for multivariate high-frequency data in finance

Dependence structures for multivariate high-frequency data in finance

Dynamic copula models for multivariate high-frequency data in flnance

Modeling exchange rate dependence dynamics at different time horizons

Change-point analysis for dependence structures in finance and insurance