couping with copulas

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

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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).
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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. Finally, Table 1 at the very end of this chapter shows all the introduced copulas and their definitions. The important issue of fitting copulas to data is examined in the next chapter of this book, “The Estimation of Copulas: Theory and Practice”, by Charpentier, Fermanian and Scaillet. Let us start with an explanatory example: Consider two real-valued random variables X 1 and X 2 which shall give us two numbers out of { 1 , 2 , . . . , 6 } . These numbers are the outcome of a more or less simple experiment or procedure. Assume that X 1 is communicated to us and we may enter a bet on X 2 . The question is, how much information can be gained from the observation of X 1 , or formulated in a different way, what is the interrelation or dependence of these two random variables (rvs). The answer is easily found if the procedure is just that a dice is thrown twice and the outcome of the first throw is X 1 and the one of the second is X 2 . In this case the variables are independent: the knowledge of X 1 gives us no information about X 2 . The contrary example is when both numbers are equal, such that with X 1 we have full information on X 2 .
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