Collateralized Debt Obligations: A Moment Matching Pricing by Enrico Marcantoni

By Enrico Marcantoni

The writer makes a speciality of a style to cost Collateralized Debt tasks (CDO) tranches. the unique procedure is constructed by means of Castagna, Mercurio and Mosconi in 2012. The Thesis presents an extension of the unique paintings by means of generalizing the Gaussian dependence when it comes to Copula features. specifically the version is rewritten for the explicit case of the Clayton copula. the strategy is utilized to cost the tranches of a CDX. through evaluating the tranches costs, it truly is attainable to note that the Clayton strategy ends up in smaller fairness and mezzanine tranches. The senior and large senior tranches degrees are larger whilst the dependence is modeled through a Clayton copula.

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The main idea is that in a given point of time, a firm defaults if its asset value is smaller than a certain threshold. Consider, as usual the existence of counterparties, each of them characterized, at time , by an asset value Each company is supposed have a critical threshold , such that the firms default in the period ⌈ ⌉ if and only if at the end of this period, so when , . Applying to this specific environment the previous Bernoulli Mixture Model, we can define each loss statistic as a Bernoulli distribution with parameter 1 and , that is: ( [ ]) is the only random variable which determines the default event and in both models, it is assumed to be a process driven by underlying factors reflecting country and sector events.

In particular, lower tail dependence means that when tends to zero like the probability mass , and not like the area of the square the corner (0,0) of the square . That is, in there must be a strong singularity of the copula's density. Upper tail dependence means the same but in the corner (1,1) of the square . Roughly speaking, upper tail dependence means that there is a tendency of assume extremes values when to assume extremes values as well. Being copula functions used to model credit default dependence, tail dependence represent a huge problem when the random variable are financial losses.

Both model assume that the asset value log-returns are normally distributed, that is: Given that the driving variable threshold is replaced by the asset log-return is replaced by the corresponding , the . 9) we follow the same elaboration of the Bernoulli mixture model framework. 1) we specify the joint where, is a cumulative multivariate normal distribution with correlation matrix . is the asset correlation of the asset log-return. 4 √ ] One-Factor Model The one-factor model relies on both Moody's KMV and RiskMetrics context.

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