Discrete Tenor Models for Credit Risky Portfolios Driven by Time-Inhomogeneous Lévy Processes

Abstract

The goal of this paper is to specify dynamic term structure models with discrete tenor structure for credit portfolios in a top-down setting driven by time-inhomogeneous Lévy processes. We provide a new framework, conditions for absence of arbitrage, explicit examples, an affine setup which includes contagion, and pricing formulas for single tranche collateralized debt obligations (STCDOs) and options on STCDOs. A calibration to iTraxx data with an extended Kalman filter shows an excellent fit over the full observation period. The calibration is done on a set of CDO tranche spreads ranging across six tranches and three maturities. Introduction.Contrary to the single-obligor credit risk models, portfolio credit risk models consider a pool of credits consisting of different obligors and the adequate quantification of risk for the whole portfolio becomes a challenge. A good model for portfolio credit risk should incorporate two components: default risk, which includes, in particular, the dependence structure in the portfolio (also termed default correlation), and spread risk, which represents the risk related to changes of interest rates and changes in the credit quality of the obligors.The main application of such a portfolio model, which we discuss in section 8, is the valuation of tranches of collateralized debt obligations (CDOs) and related derivatives. We would like to emphasize that variants of this model can be used for the valuation of other asset-backed securities. Currently, due to the sovereign credit crisis that has affected Europe, the issuance of so-called European safe bonds (ESBs) is discussed, where the underlying portfolio would consist of sovereign bonds of EU member states with fixed weights set by a strict rule which is proportional to GDP. Our model is easily adapted for pricing of such and other similar asset-backed securities whatever the precise specification of these instruments would be.

A Credit-Risk Valuation under the Variance-Gamma Asset Return

“…The area of use covers both the stock (see Bäurer 2015;Carr and Wu 2010) and interest rate (Eberlein et al 2013;Grbac 2009;Madan 2014) frameworks. Let us notice that the hazard rate theory is an extension of the idea of default time modeling by the exponential distribution, see, for example, the book by Bielecki and Rutkowski (2002).…”