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
http://dx.doi.org/10.1090/conm/351/06388
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