This is a project in Financial Engineering and Risk Management course Describe Copulas with graphs..

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This is a project in Financial Engineering and Risk Management course Describe Copulas with graphs Copula as an innovative tool Copula theory and Copula Function Credit Default Swaps Collateralized Debt Obligations Working of Collateralized Debt Obligations with examples Role of Collateralized Debt Obligations Subprime mortgage crisis
Refer attached file for complete answer Copulas The history of copulas may be said to begin with Frechet. A copula function is a function that links or marries univariate marginals to their full multivariate distribution. For m uniform random variables, U 1, U 2, Um, the joint distribution function C , defined as C (u 1 , u 2 ,, um, ?) =Pr[ U 1= u 1 ,U 2= u 2 , ,Um = um ] can also be called a copula function . Copula functions can be used to link marginal distributions with a joint distribution. For given univariate marginal distribution functions F 1 (x 1 ) , F 2 (x 2 ) , , Fm(xm) , the function C(F 1 (x 1 ), F 2 (x 2 ), , Fm(xm)) = F(x 1 , x 2 ,, xm), which is defined using a copula function C , results in a multivariate distribution function with univariate marginal distributions as specified F 1 (x 1 ) , F 2 (x 2 ) ,, Fm(xm) . Copula Theory Consider a N-dimensional vector of random variables X=(X1,.., X n ) with a fully general multivariate distribution represented by its pdf X~ fX. In the multivariate case the pdf fX is defined in such a way that, for any set of potential joint valuesX?R N for (X1, , Xn) the following identity holds P(X1, , Xn)? X = X (x1,, xn)dx 1 , dx N From the joint distribution f x we can in principle extract all the N marginal distributions X n ~fx n , where n= 1, , N , by computing the marginal pdfs asfollows. F xn (x n ) = fx(x 1 ,..x n )dx 1 . Dx n-1 dx n+1 dx N Then we can compute the marginal cdfs Fx n . Finally, we can

d each cdf Fx n , which is a function, with the respective entry of the vector X, namely the random variable X n . The outcome of this operation are the grades, which we know have a uniform distribution on the unit interval U n = F Xn (X n )~U [0,1] marginal X1 grade U1 copula U=(U1,U2) joint X=(X1,X2) marginal X2 The grades of the distribution f X can be interpreted as a sort of non-linear z-score, which forces all the entries X n to have a uniform distribution on the unit interval [0, 1]. By feeding each random variable X n into its own cdf, all the information contained in each marginal distribution fx n is swept away, and what is left is the pure joint information amongst the X n s, i.e. the copula f u . The copula is the information missing from the individual marginals to complete the joint distribution ” joint = copula + marginals ” From the definition of copula FU (u)= PU1= u1,,UN=u N = PFx 1 (X 1 ) = u 1 ,,F XN (X N ) = u N = PX 1 = F -1 X1 (u 1 ),, X N =F -1 XN (u N ) =F x (F -1 X1 (u 1 ),, F -1 XN (u N ) ) Sklars theorem provides the pdf of the copula from the joint pdf and the marginal pdfs. This allows us to use maximum likelihood to fit copulas to empirical data. Sklars theorem provides the theoretical foundation for the application of copulas. Sklars theorem states that a multivariate cumulative distribution function of a random vector with marginals can be written as where is a copula. The theorem also states that, given , the copula is unique on , which is the cartesian product of the ranges of the marginal cdfs. This implies that the copula is unique if the marginals are continuous. The converse is also true: given a copula and margins then defines a d-dimensional cumulative distribution function. We now derive another useful result. If we feed the grades into arbitrary inverse cdfsF -1 Yn , we obtain new transformed random variables with a givenjoint distribution, which we denote by f Y Y 1 =F -1 Y1 (F X1 (X1)) ( ) ~ f Y Y N =F -1 YN (F XN (X N )) The joint distribution f Y has marginals whose cdfs are F Yn Furthermore, the copula of Y is the same as the copula of X, because from the definition of Y and the definition of the copula we obtain F Y1 (Y 1 )=F X1 (X 1 ) ( ) ~ f U F YN (Y N )=F XN (X N ) Therefore, we derive that the copula of an arbitrary random variable X=(X1, ., X N ) does not change when we transform each X n into a new variable Y n =g n (X n ) by means of functionsg n (x)=F -1 Yn (Fx n (Fx n (x)), where FY n are arbitrary cdfs. It is easy to verify that such functions g n are a very broad class, namely all the increasing transformations, also known as co-monotonic transformations. Copulas are used to describe the dependence between random variables. The different types of copulas are Archimedean copulas, Copulas related to elliptical distributions and Extreme value copulas. One of the most powerful applications of copulas concerns the Risk Management. According to Eric Bouye et al. (2000), copulas are a viable alternative to Gaussian assumptions and joint distribution modelling, which is not normally possible in fat-tailed financial time series. Their value in statistics is that they provide a way of understanding how marginal distributions of single risks are coupled together to form joint distributions of groups of risks; that is, they provide a way of understanding the idea of statistical dependence. There are two principal ways of using the copula idea. We can extract copulas from wellknown multivariate distribution functions. We can also create new multivariate distribution functions by joining arbitrary marginal distributions together with copulas. The first application of the Gaussian Copula for CDOs was Lis paper (2000), “On Default Correlation: A Copula Function Approach.”

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