couping with copulas

couping with copulas - Contents 1 Coping with Copulas...

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Unformatted text preview: Contents 1 Coping with Copulas Thorsten Schmidt 1 Department of Mathematics, University of Leipzig Dec 2006 Forthcoming in Risk Books ”Copulas - From Theory to Applications in Finance” Contents 1 Introdcution 1 2 Copulas: first definitions and examples 3 2.1 Sklar’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Copula densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Conditional distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Bounds of copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Important copulas 7 3.1 Perfect dependence and independence . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Copulas derived from distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Copulas given explicitly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3.1 Archimedean copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3.2 Marshall-Olkin copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Measures of dependence 14 4.1 Linear correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Rank correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Tail dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5 Simulating from copulas 18 5.1 Gaussian and t-copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Archimedean copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.3 Marshall-Olkin copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6 Conclusion and a word of caution 20 1 Introdcution Copulas are tools for modelling dependence of several random variables. The term copula was first used in the work of Sklar (1959) and is derived from the latin word copulare , to connect or to join. The main purpose of copulas is to describe the interrelation of several random variables. The outline of this chapter is as follows: as a starting point we explore a small example trying to grasp an idea about the problem. The first section then gives precise definitions and fundamental relationships as well as first examples. The second section explores the most important examples of copulas. In the following section we describe measures of dependence 1 Dep of Mathematics, Johannisgasse 26, 04081 Leipzig, Germany. Email: [email protected] The author thanks S. L. Zhanyong for pointing out a typo in Formula (15). 1 Introdcution 2 as correlation and tail dependence and show how they can be applied to copulas. The fourth section shows how to simulate random variables (rvs) from the presented copulas. The final section resumes and gives a word of caution on the problems arising by the use of copulas....
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This note was uploaded on 03/14/2012 for the course IEOR 4602 taught by Professor Martin during the Spring '12 term at Columbia.

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couping with copulas - Contents 1 Coping with Copulas...

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