Two-Dimensional Markovian Model for Dynamics of Aggregate Credit Loss

Abstract

We propose a new model for the dynamics of the aggregate credit portfolio loss. The model is Markovian in two dimensions with the state variables being the total accumulated loss L t and the stochastic default intensity λ t . The dynamics of the default intensity are governed by the equationThe function ρ depends both on time t and accumulated loss L t , providing sufficient freedom to calibrate the model to a generic distribution of loss. We develop a computationally efficient method for model calibration to the market of synthetic single tranche CDOs. The method is based on the Markovian projection technique which reduces the full model to a one-step Markov chain having the same marginal distributions of loss. We show that once the intensity function of the effective Markov chain consistent with the loss distribution implied by the tranches is found, the function ρ can be recovered with a very moderate computational effort. Because our model is Markovian and has low dimensionality, it offers a convenient framework for the pricing of dynamic credit instruments, such as options on indices and tranches, by backward induction. We calibrate the model to a set of recent market quotes on CDX index tranches and apply it to the pricing of tranche options.JEL Classification: C63, C65, G12, G13

Hedging default risks of CDOs in Markovian contagion models

“…Intensities depending upon the number of defaults are related to mean-field approaches (see Frey & Backhaus (2006) . This is discussed in van der Voort (2006), Arnsdorf & Halperin (2007) or Lopatin & Misirpashaev (2007). Later on, we provide a calibration procedure of such unconstrained intensities onto market inputs such as expected losses on CDO tranches.…”

“…Such ideas have been put in practice by Arnsdorf & Halperin (2007), de Koch & Kraft (2007), Herbertsson (2007), Lopatin & Misirpashaev (2007) and Herbertsson & Rootzén (2006) for the pricing of basket credit derivatives and also with respect to calibration issues.…”

An extension of Davis and Lo's contagion model

“…All along years, the contagion phenomena was modeled using Bernoulli random variables (Davis and Lo (2001)), copula functions (Schönbucher and Schubert (2001)), interacting particle systems (Giesecke and Weber (2004)), incomplete information models (Frey and Runggaldier (2010)) or Markov chains (see for example Schönbucher (2006), Graziano and Rogers (2009) or Kraft and Steffensen (2007)). As far as the risk management of synthetic CDO tranches is concerned, Markov chain contagion models have also been investigated by several papers such as Van der Voort (2006), Herbertsson and Rootzén (2006), Herbertsson (2007), Frey and Backhaus (2010), Frey and Backhaus (2008), De Koch, Kraft and Steffensen (2007), Epple, Morgan and Schloegl (2007), Lopatin and Misirpashaev (2007), Arnsdorf and Halperin (2008), Cont and Minca (2008), Cont, Deguest and Kan (2010) among others. The hedging issue for CDO tranches is also addressed by Laurent, Cousin and Fermanian (2010) and Cousin, Jeanblanc and Laurent (2010) in the class of Markovian contagion models.…”

Multiname and Multiscale Default Modeling

“…The main drawback of this approach is the fact that these are static models which do not take into account the time evolution of joint default risks. This has been recognized in the academic literature on dynamic models with recent developments in the multiname structural approach [10,13], reduced form models [2,16,15,4], and top-down models [6,14,18].…”