A Generic One-Factor Lévy Model for Pricing Synthetic CDOs

Abstract

The one-factor Gaussian model is well-known not to fit simultaneously the prices of the different tranches of a collateralized debt obligation (CDO), leading to the implied correlation smile. Recently, other one-factor models based on different distributions have been proposed. Moosbrucker used a one-factor Variance Gamma model, Kalemanova et al. and Guegan and Houdain worked with a NIG factor model and Baxter introduced the BVG model. These models bring more flexibility into the dependence structure and allow tail dependence. We unify these approaches, describe a generic one-factor Levy model and work out the large homogeneous portfolio (LHP) approximation. Then, we discuss several examples and calibrate a battery of models to market data. CDOS • Collateralized Credit Obligations (CDOs) are complex multivariate credit risk derivatives. • A CDO transfers the credit risk on a reference portfolio of assets in a tranched way. • The risk of loss on the reference portfolio is divided into tranches of increasing seniority: – The equity tranche is the first to be affected by losses in the event of one or more defaults in the portfolio. – If losses exceed the value of this tranche, they are absorbed by the mezzanine tranche(s). – Losses that have not been absorbed by the other tranches are sustained by the senior tranche and finally by the super-senior tranche. CDOS •When tranches are issued, they usually receive a rating by rating agencies. • The CDO issuer typically determines the size of the senior tranche so that it is AAA-rated. • Likewise, the CDO issuer generally designs the other tranches so that they achieve successively lower ratings. • The CDO investors take on exposure to a particular tranche, effectively selling credit protection to the CDO issuer, and in turn collecting premiums (spreads). •We are interested in pricing tranches of synthetic CDOs. • A synthetic CDO is a CDO backed by credit default swaps (CDSs) rather than bonds or loans, i.e. the reference portfolio is composed of CDSs. • Recall that a CDS offers protection against default of an underlying entity over some time horizon. CDOS • Take the example of the DJ iTraxx Europe index. • It consists of a portfolio composed of 125 actively traded names in terms of CDS volume, with an equal weighting given to each • Below, we give the standard synthetic CDO structure on the DJ iTraxx Europe index. Reference portfolio Tranche name K1 K2 Equity 0% 3% 125 Junior mezzanine 3% 6% CDS Senior mezzanine 6% 9% names Senior 9% 12% Super-senior 12% 22% Table 1: Standard synthetic CDO structure on the DJ iTraxx Europe index. PROBLEMS IN CDO MODELING • The problem is high dimensional : 125 dependent underlyers. • The behavior of the underlying firm’s values typically show all the stylized features like, jumps, stochastic volatility, ... • The tranching complicates mathematics. • You can write down a fancy model, but prices should be generated within a sec. to be practically useful. A GENERIC LEVY MODEL FOR PRICING CDOS •We are going to model a homogeneous portfolio of n obligors: each obligor – has the same weight in the portfolio – has the same recovery value R – has the same individual default probability term structure p(t), t ≥ 0, which is the probability an obligor will default before time t. • The inhomogeneous case is also possible but a bit more involved in notation and calculations. • Basic idea is to come up with a vector of n dependent random variables who indicate the value of the firms. A GENERIC LEVY MODEL FOR PRICING CDOS • The Gaussain one-factor model (Vasicek, Li) assumes the following dynamics: – Ai(T ) = √ ρ Y + √ 1− ρ ǫi, i = 1, . . . , n; – Y and ǫi, i = 1, . . . , n are i.i.d. standard normal with cdf Φ. • The ith obligor defaults at time T if the firm value Ai(T ) falls below some preset barrier Ki(T ) (extracted from CDS quotes see later): Ai(T ) ≤ Ki(T ) • This model is actual based on the Gaussian Copula with its known problems (cfr. correlation smile). • The underlying reason is the too-light tail behavior of the standard normal rv’s. (Note that a large number of joint defaults will be caused by a very negative common factor Y ). • Therefore we look for models where the distribution of the factors has more heavy tails. HOW TO GENERATE MULTIVARIATE FIRM VALUES •We want to generate standardized (zero mean, variance one) multivariate random vectors with a prescribed correlation. • Basic idea: correlate by letting Lévy processes run some time together and then let them free (independence) 0 0.2 0.4 0.6 0.8 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time Correlated outcomes ρ=0.3 0 0.2 0.4 0.6 0.8 1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Correlated outcomes ρ=0.8

Cite this paper

@inproceedings{Schoutens2006AGO, title={A Generic One-Factor Lévy Model for Pricing Synthetic CDOs}, author={Wim Schoutens and Sophie Ladoucette}, year={2006} }