Contemporary Mathematics Volume 351, 2004 Credit Barrier Models in a Discrete Framework Claudio Albanese and Oliver X. Chen ABSTRACT. We formulate credit barrier models in the framework of jump pro- cesses with absorption on a discrete lattice. The lattice model is formulated in terms of finite state Markov processes related to the Hahn family of hyper- geometric polynomials. The continuous limit we obtain as the lattice spacing goes to zero corresponds to the Jacobi process. The model is designed to relate real-world and risk neutral measures. 1. Introduction Credit barrier models are derivative pricing models for credit sensitive instru- ments. The underlying is a credit quality variable with the meaning of distance to default, a measure of an obligor's leverage relative to the volatility of its asset values. The first models in this class appeared in working papers and internal doc- uments [HLPQ99], [GHOl], [DJ02] and made their way to the open literature in articles by Hull and White [HWOl] and Avellaneda and Zhu [AZOl]. This first generation of credit barrier models involves estimations against a single spread curve. Real world estimates instead are applied to barrier models of the Merton type used for risk management applications as in the CreditMetrics TM technical document by Gupton et al. [GFB97]. A new class of credit barrier models was introduced by the authors in [ACCZ03] and [AC03] in an attempt to reconcile the real-world and the risk-neutral measure. In this new class, the estimation frame- work is extended to include a more comprehensive set of statistical data such as historical migration rates, default frequencies over several time horizons and aggre- gate spread curves across all ratings. Within the extended framework, one obtains metrics for relative liquidity spreads across credit ratings. One also obtains a new methodology to extrapolate implied migration rates in [AC03] we compare with 2000 Mathematics Subject Classification. Primary 91B28 Secondary 33C45, 39A70, 60J60. Key words and phrases. Credit risk models, discretization schemes, Jacobi process, Hahn process. The authors were supported in part by the Natural Sciences and Engineering Research Coun- cil of Canada. We thank Giuseppe Campolieti, Francesco Corielli, Alexei Kuznetsov, Stephan Lawi and Andrei Zavidonov for discussions. We thank the participants to the AMS-SIAM Conference in Mathematical Finance (Snowbird 2003), the Riskwaters Credit Risk Summit (London 2003), the GARP Credit Risk Summits (London and New York 2003), Risk Europe (Paris 2003) and Risk Italia (Milano 2003) for discussions. This research was conducted in part while the authors were at the National University of Singapore. @2004 American Mathematical Society
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