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Dynamic Copulas: Applications to finance and economics

by Daniel Totouom-Tangho of MINES ParisTech

November 6, 2007

Abstract: Credit derivatives are one of the fastest developing parts of market finance at present. Not only have the nominal amounts increased phenomenally, but new products are continually being created. After reviewing the evolution of these products since their inception in the early 90s, we demonstrate that this development has been market driven, with the mathematical models used for pricing lagging behind. As the market developed, the weak points of the models became apparent and improved models had to be developed. In October 2003 when the work on this thesis started, CDOs (Collateralised Debt Obligations) were becoming standard products. A new generation of products which we will refer to as third generation credit derivatives were starting to come on line: these include forward-starting CDS, forward-starting CDOs, options on CDOs and CPDOs. In contrast to early products, these derivatives require a dynamic model of the evolution of the "correlation" between the names over time, something which base correlation was not designed to do. The aim of this doctorate has been to develop a mathematical consistent framework for pricing these types of products.

We present the first formulation of the dynamic copula processes and then a new model for pricing CDO tranches based on a dynamic copula process with low tail dependence as in the Clayton copula. A formula for the asymptotic loss distribution, similar to the Vasicek formula, is derived. This simplifies the computations for the case of a fine grained portfolio.

We propose a model for the joint dynamics of credit ratings of several firms based on copulas and dynamic copulas. Namely, individual credit ratings are modelled by a continuous time Markov chain, while their joint dynamics is modelled using copulas.

A multifactor or Nested dynamic copula approach is developed within the new formulated dynamic copula processes, and a time changed levy process is used to introduce dependency on spreads dynamics. We show that the building block of time changed approach fail within the frameworks of dynamic copulas.

JEL Classification: G12, G13, C0, C00, C16, C30.

Keywords: Default risk, CDOs, Forward Starting CDOs, correlation smile, LÚvy Process, Non Gaussian Ornstein-Uhlenbeck process, Sub Prime, Archimedean copulas, multivariate stochastic processes, lower tail dependence.

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