Copulas. Mathematisches Seminar (Prof. Dr. D. Filipovic) Di Uhr in E

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1 Copulas Mathematisches Seminar (Prof. Dr. D. Filipovic) Di Uhr in E41 A Short Introduction The above picture shows a scatterplot (500 points) from a pair (X 1,X 2 ) of random variables each uniformly distributed on [0, 1]. Now, to understand what a copula is we first notice a very trivial fact: Let F be any 1-dimensional distribution function and let C be the distribution function of the uniform distribution on [0, 1]. Then we trivially have F(x) = C(F(x)) for all x R. This carries over to 2- and higher dimensional distribution functions F (with F 1,...,F n denoting the 1-dimensional marginal distribution functions) (1) F(x 1,...,x n ) = C(F 1 (x 1 ),...F n (x n )) for all (x 1,...,x n ) R n. Here C denotes a distribution function on [0, 1] n with uniform marginals. It is also called a copula. The above scatterplot thus is a copula scatterplot, namely of a member of the famous Marshall-Olkin family. We shall see in Vortrag 3 which situation this copula is applied to. IMPORTANT: The fundamental benefit of (1) is a breaking up of a multivariate distribution into two parts, namely the 1-dimensional marginal distributions F 1,...,F n the dependence structure (given by C) 1

2 And in nearly all situations... there can be a value in seperation the marginal-modelling and dependence-modelling issues and looking at each in more detail. The copula approach to multivariate models faciliates this approach and allows us to consider, for example, the issue of whether tail dependence appears in our data. (A. McNeil, R. Frey, P. Embrechts (2005), Quantitative Risk Management, p.229). Problems to be discussed in this seminar are e.g. does (1) really hold for all F, and is the copula C uniquely determined? what different kind of copulas are used for modelling and what properties do they have? how can popular measures of dependency (e.g. the above mentioned tail dependence) be expressed in the framework of copulas? where and how are copulas applied in practice (finance, insurance, risk management...)? REFERENCES The talks will mostly ground on the following book: Roger B. Nelsen (2006), An Introduction to Copulas, Springer-Verlag. Other references are [2] C. Savu, M. Trede (2006), Hierarchical Archimedean Copulas, Institute of Econometrics, University of Münster. [3] Chr. Bluhm, L. Overbeck, Chr. Wagner (2003), An Introduction to Credit Risk Modeling, Chapman & Hall. [4] R. Cont, P. Tankov (2004), Financial Modelling With Jump Processes, Chapman & Hall. [5] A. McNeil, R. Frey, P. Embrechts (2005), Quantitative Risk Management, Princeton Series in Finance. List of possible talks Proofs: Mostly elemenary; prerequisite is an upper-level undergraduate course in probability and mathematical statistics 1. Vortrag Definition and basic properties of copulas p analytic definition of Subcopulas and copulas (in 2 dimension) study of properties of copulas (monotonicity, continuity, differentiability...) examples (e.g. Frechet, Mardia, Cuadras, Gumbel family of copulas...) extending the definition from bivariate to multivariate copulas singular and absolutely continuous component of a copula 2

3 2. Vortrag Sklar s theorem or copulas as (two-)dimensional distribution functions defining any two-dimensional distribution function via copulas (Sklar s theorem) regaining the copulas in Sklar s theorem from a two-dimensional distribution function (inverse of Sklar s theorem) rewriting Sklar s theorem using the notion of random variables invariance principles of copulas w.r.t. random variables how to construct dependent random variables using copulas extending Sklar s theorem to the multivariate case (without proof) 3. Vortrag Methods of constructing copulas and interesting examples the Marshall-Olkin resp. generalized Cuadras-Auge family of copulas as specail cases of survival copulas copula transformation method and extreme value copulas copulas with prescribed horizontal or vertical sections or (singular) copulas with prescribed support (geometric methods) construction of multivariate copulas 4. Vortrag Archimedean copulas definition of Archimedean copulas Characterization theorem for 2-dim. Archimedean copulas via properties of the generator properties of Archimedean copulas and examples one-parameter families (Gumbel, Clayton families...) 5. Vortrag Measures of Dependence (I) (distributional version of) Kendall s tau Kendall s τ expressed through copulas examples 3

4 6. Vortrag Measures of Dependence (II) [opt.] (distributional version of) Spearman s rho Spearman s ρ and Kendall s τ as measures of concordance relationship between ρ and τ definition of tail dependence tail dependence expressed via copulas coefficient of tail dependence 7. Vortrag Colpulas describing Markov Processes 6.4 product of copulas 8. Vortrag Lévy Copulas [4] 5.5 definition of Markov processes Chapman-Kolmogorov equations characterizing the Chapman-Kolmogorov equations via products of copulas (Theorem of Darsow-Nguyen-Olsen) interseting alternative for construction and study of Markov processes tail integrals substituting distribution functions definition of positive Lévy copulas tail integrals and Lévy measures dependence of compound Poisson processes construction of positive Lévy copulas 9. Vortrag Application in Risk Management [3] [5] factor copula models loss distribution by means of copulas Bernoulli mixture model based on a uniform asset value (KMV) model Gaussian and t-copulas with normal margins (tail) properties of gaussian and t-copulas estimation of mean and quantile of a loss distribution with t-copulas in the dependence structure 4

5 10. Vortrag Fitting copulas to data [5] 5.5 estimation of the parameters of a copula family method of moments using rank correlation forming a pseudo-sample from the copula maximum likelihood estimation 11. Vortrag Hierarchical Archimedean copulas [2] multivariate Archimedean copulas Hierarchical Archimedean copulas density and random variable generation for Hierarchical Archimedean copulas working example from finance 5

6 Collection of the talks 1. Vortrag Definition and basic properties of copulas 2. Vortrag Sklar s theorem 3. Vortrag Methods of constructing copulas and interesting examples 4. Vortrag Archimedean copulas 5. Vortrag Measures of Dependence (I) 6. Vortrag Measures of Dependence (II) 7. Vortrag Colpulas describing Markov Processes 8. Vortrag Lévy Copulas 9. Vortrag Application in Risk Management 10. Vortrag Fitting copulas to data 11. Vortrag Hierarchical Archimedean copulas 6