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Calculate Lévy Base Correlation

By Katholieke Universiteit Leuven, Dobránszky P, Péter Dobránszky, Finalyse Sa, Fortis Merchant Banking and Ku Leuven


This paper presents different numerical quadratures to calculate Lévy base correlation that was specified in details in Garcia et al (2007). The different numerical quadratures described here are alternatives to the Gauss-Laguerre quadrature proposed by Garcia et al (2007) in case when the shifted gamma model is used. One might observe, that deepening in the recent credit crises, the increasing risk averseness caused higher base correlations implied to the synthetic CDO tranche quotes available on the market. However, as the base correlations were climbing up, some market practitioners noticed that the earlier used Gauss-Laguerre quadrature may fail to calculate the unconditional joint default probabilities precisely enough. This remark gave me the intention to test different techniques on the integration problem. In this paper, at first I review the integration problem related to the calculation of Lévy base correlation. I show how the semi-infinite-part integration can be altered to finite-part integration. In the following section I present different numerical quadratures to solve either the finitepart or the infinite-part integration problem. At the end, I compare these quadratures by precision, robustness and performance. 2 Finite vs. Infinite As described in Garcia et al (2007), the Lévy base correlation model, just as the well-known Gaussian base correlation model is a one-factor model to price synthetic CDO tranches. In order to derive the unconditional joint default distribution, and thus to price the tranche, one has to numerically integrate the conditional joint default distribution function depending on the common market factor. Following the concept of Garcia et al (2007), the unconditional joint default distribution is found by integrating over the common market factor P l t = ∫ P l t X f X d X, (1

Year: 2010
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