By Enrico Marcantoni
The writer makes a speciality of a mode to cost Collateralized Debt responsibilities (CDO) tranches. the unique process is constructed by way of Castagna, Mercurio and Mosconi in 2012. The Thesis presents an extension of the unique paintings by way of generalizing the Gaussian dependence when it comes to Copula capabilities. particularly the version is rewritten for the categorical case of the Clayton copula. the strategy is utilized to cost the tranches of a CDX. via evaluating the tranches costs, it's attainable to note that the Clayton method results in smaller fairness and mezzanine tranches. The senior and great senior tranches degrees are larger while the dependence is modeled via a Clayton copula.
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The writer specializes in a mode to cost Collateralized Debt responsibilities (CDO) tranches. the unique strategy is built through Castagna, Mercurio and Mosconi in 2012. The Thesis offers an extension of the unique paintings via generalizing the Gaussian dependence when it comes to Copula services. specifically the version is rewritten for the categorical case of the Clayton copula.
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Additional resources for Collateralized Debt Obligations: A Moment Matching Pricing Technique based on Copula Functions
V. Y are reported as well. Name Gumbel Clayton ( Frank ) ( ) ( -stable, ⁄ (no closed form is known) Gamma( ⁄ ⁄ Logarithmic series on ) , with ⁄ Table 1: Summary of some generators, their inverse and Laplace transforms for Gumbel, Clayton and Frank copulas 15 15 Marshall and Olkin (1988). 3 Tail dependence Definition A bivariate copula has a upper tail dependence with parameter if: C has a lower tail dependence with parameter if: The coefficients of upper and lower tail dependence are measures depending only on the copula of a pair of random variables and and with continuous marginals .
For all (rectangle inequality): for all with we have ∑∑ where and ∑ for all . The first property is a necessary requisite for any multivariate distribution functions and the second is the equivalent mathematic way to require uniform marginal distribution. The third property ensures that for any random vector distribution function C, with is non-negative. The following theorem shows the importance of copulas, stating that all multivariate distribution functions contain copulas, as well that copulas can be used to construct multivariate distribution functions, starting from the marginal distributions.
The portfolio loss can be written as the sum of the single obligor's losses: ∑ where is the indicator function of the default of the The stochastic variable a where ∑ - obligor. , following Gordy (2003), is assumed to be distributed as where the distinctive parameters have to be chosen such that: , is the mean of the 's distribution. According to Gordy (2003), Castagna et al. rewrite the standard deviation the skewness , which are functions of and in terms of and . That is: √ 47 E. 1007/978-3-658-04846-4_5, © Springer Fachmedien Wiesbaden 2014 √ √ The default is modeled through a structural approach based on the Merton's model (1974) where the obligor's asset dynamics are modeled on one or more common factors, thus obtaining a dependency of the obligor.
Collateralized Debt Obligations: A Moment Matching Pricing Technique based on Copula Functions by Enrico Marcantoni