# Lévy Processes for Credit models

Lévy processes form a particular class of mathematical models ment to study randomness in a dynamic framework. They constitute a fundamental class, among which lies the Brownian motion, the Poisson process and Stable processes. They can be found in Risk Management, Queuing theory and Stochastic analysis applied to Finance.

In particular, such processes allow for a more faithful fit of Assets dynamic and a correction of democratized formula for pricing financial derivatives, where higher moments of distrubutions play a key part. Thanks to recent developments, a robust mathematical framework allows to handle a wide variety of analysis as well as the obtention of closed formula, notably thanks to tools borrowed from signal processing and Stochastic Calculus.

In the context of the subprime crisis, it is yet not sufficient to be able to account for jump uncertainty, the question of dependence should also be raised. Where in the case of the commonly used Gaussian process the dependence structure is fairly straightforward, and given thanks to a correlation matrix, it is not so simple when the marginal distributions get fancy. Since 1997, Copula are studied and characterized, and present a possible tool for the mentionned problem. Already appropriated by practionners, their main application is regarding the study of extreme events and tail distribution, in Risk Management, and the prricing of Credit derivatives (CDO) in Mathematical Finance.

This level of sophistication is admittedly required in today's volatile markets. An accurate formalization of these concepts would have significant impact in the quality of Portfolio and Risk Management, in the precision of pricing algorithms and could lead to opportunities in the field of Statistical Arbitrage.