Multiname and Multiscale Default Modeling

Abstract

Multiname default modeling is crucial in the context of pricing credit derivatives such as Collaterized Debt Obligations (CDOs). We consider here a simple reduced form approach for multiname defaults based on the Vasicek or Ornstein-Uhlenbeck model for the hazard rates of the underlying names. We analyze the impact of volatility time scales on the default distribution and CDO prices. We demonstrate how correlated fluctuations in the parameters of the name hazard rates affect the loss distribution and senior tranches of CDOs. The effect of stochastic parameter fluctuations is to change the shape of the loss distribution and cannot be captured by using averaged parameters in the original model. Our analysis assumes a separation of time scales and leads to a singular-regular perturbation problem [7,8]. This framework allows us to compute perturbation approximations that can be used for effective pricing of CDOs.

Monte Carlo Algorithms for Default Timing Problems

“…In these and many other cases, the probability of n hitting any finite level H in finite time is strictly positive. Models with this feature are ubiquitous in the literature (see Arnsdorf and Halperin 2008, Bayraktar and Yang 2009, Berndt et al 2005, Carr and Linetsky 2006, Cont and Jessen 2011, Ding et al 2009, Duffie and Garleanu 2001, Eckner 2009, Errais et al 2010, Feldhütter 2007, Fouque et al 2009, Jarrow et al 2005, Longstaff and Rajan 2008, Mortensen 2006, Pan and Singleton 2008, Papageorgiou and Sircar 2007, and many others).…”

Graphical Models for Correlated Defaults

“…The Gaussian copula models (Schönbucher 2003; Li 2000; Vasicek 1987) separate the modeling of the interdependence of random variables from the modeling of their marginal distributions. Dynamic models include Papageorgiou and Sircar (2007) and Fouque, Sircar, and Solna (2009), which put more of an emphasis on matching the marginals and default distribution of single names.…”

Utility valuation of multi-name credit derivatives and application to CDOs

“…A necessary and sufficient condition for positive definiteness is therefore (1 − ρ) N + ρ > 0, which leads to the first condition in (15).…”

“…The conditions (15) guarantee that the Brownian motions have a legitimate correlation structure (in other words that we have a positive definite covariance matrix), and are derived in Appendix A (see also [15]). Under this assumption, we have that…”