By Enrico Marcantoni

The writer makes a speciality of a mode to cost Collateralized Debt responsibilities (CDO) tranches. the unique technique is built through Castagna, Mercurio and Mosconi in 2012. The Thesis presents an extension of the unique paintings through generalizing the Gaussian dependence by way of Copula features. particularly the version is rewritten for the explicit case of the Clayton copula. the strategy is utilized to cost the tranches of a CDX. via evaluating the tranches costs, it really is attainable to note that the Clayton process results in smaller fairness and mezzanine tranches. The senior and great senior tranches degrees are better while the dependence is modeled via a Clayton copula.

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**Additional info for Collateralized Debt Obligations: A Moment Matching Pricing Technique based on Copula Functions**

**Example text**

This copula is given by: 37 where and refers to the joint distribution function of . The Gaussian copula does not have a closed form but can be expressed as an integral over the density of . Consider the 2-dimenional copula where is the correlation between the two random variable for | | . Then : ∫ ∫ ⁄ { } Product copula can be seen as a particular case of the Gaussian copulas. A Gaussian copula with correlation matrix , where is a identity matrix of order I. Gaussian copula make easy draw random sample from it.

It will present the general setup which the most common models share and the derivation of the models themselves. This overview contains the models, which will be in the centre of the empirical part of the next chapters. 1 The Bernoulli Model Assume the existence of a portfolio constituted by counterparty and denote the loss of the - obligor, with . Let ) be a vector of random variables, whose marginal distributions are Bernoulli. A two-state of world is assumed, where the obligors default with probability , when , and survive with probability , when .

The default leg (DL) is the sum of all the expected losses of a tranche and the premium leg (PL) is the sum of the expected premia the protection seller receives. Assuming a constant annual spread for a given tranche we can write the two present value legs as follow: ∑( ∑ where ) ( ) is the discounting factor for each maturity and is the number of maturities. The fair spread is obtained setting a premium by equaling the present values of the two legs. That is: ∑ ( ∑ Being ) ( ) constant by the previous assumption and fixed in advance, the only unknown variable is the expected loss of the tranches.