Saddlepoint approximation method for pricing
CDOs
Jingping Yang∗
Department of Financial Mathematics
Peking University, Beijing 100871
P. R. China
T. R. Hurd† Xuping Zhang‡
Dept. of Mathematics and Statistics
McMaster University, Hamilton ON L8S 4K1
Canada
September 17, 2006
Abstract
A critical issue in the credit risk industry is the accurate, efficient and
robust pricing of collateralized debt obligations (CDO) in a variety of math-
ematical models. These and many similar basket default products are very
complex, due to characteristics of the large number of individual firms upon
which they depend. Despite this complexity and because of their versatil-
ity, such products have become popular in the market. A central difficulty
which arises in most models of CDOs is the efficient computation of condi-
tional default loss distributions. Since exact computation is feasible only in
highly symmetric situations, it is necessary to have a variety of acceptable
approximation schemes. The present paper explores one general method, the
saddlepoint approximation, and shows that it offers an improvement when
compared with simpler methods.
Key words: large deviations; credit risk; basket credit derivative; collateralized
debt obligation; tranche function
∗Research supported by MITACS (Mathematics of Information Technology and Complex Sys-
tems) and NSFC10471008, e-mail: yangjp@math.pku.edu.cn
†Research supported by the Natural Sciences and Engineering Research Council of Canada and
MITACS, Canada, e-mail: hurdt@mcmaster.ca
‡e-mail: zhangx@math.mcmaster.ca
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1 Introduction
A collateralized debt obligation (CDO), or more generally any asset–backed security
(ABS), is a structured product based on an underlying portfolio of default risky
reference credits, such as corporate bonds, mortgages or loans. In essence, the
portfolio is sliced into separate securities called “tranches” ordered by seniority,
each of which receives its fair share of the revenue stream generated by the reference
credits. CDOs have become very popular in the market and it is now important to
price them accurately and efficiently, a problem which has been attracting more and
more attention from both practitioners and academic researchers. The problem is
intrinsically difficult regardless of the modeling approach adopted. The number of
credits or names in a typical CDO is moderately large: for example the highly traded
CDX index products are based on a portfolio of 125 credits. The credit structure
of the collateral pool is also complicated, comprising firms from different sectors
and with different credit ratings and in many cases the products are structured
with credit having different notional amounts. Finally, since the different CDO
tranches span the entire range of default probabilities, practical schemes must be
very robust with respect to underlying parameters. Computational schemes which
work in certain symmetric cases become infeasible in the general nonhomogeneous
setting. The goal of this paper is to investigate whether the method of saddlepoint
approximations, also called large deviation theory in probability, performs flexibly,
robustly and accurately enough to be a reliable and general method for pricing
CDOs.
The cash flow from a CDO is determined by the cumulative loss over time by
default of the underlying reference credits. Understanding this involves modeling
each credit’s default probability, its loss given default, and the correlations among
these quantities. There are a number of competing models which address this prob-
lem. [6] focused on modeling the correlation of default times by intensity-based
models; [3], [11] have continued along these lines. The normal copula approach,
pioneered by [15] and developed by [13], [1] and [10], is a simpler approach to mod-
eling multifirm default, and forms the basis for most practical CDO computations
because in important special cases it leads to tractable calibration and evaluation.
The basic ingredient of models such as these which controls default correlations is
the presence of one or more conditioning factors or common risk factors thought
of as “macro–environmental variables”. Conditional on knowing the values of these
latent factors, firms’ default times and loss amounts are assumed to be independent
random variables. Therefore, conditional on the risk factors, the cumulative loss
random variable is the sum of a large number of independent, but not identical
Bernoulli random variables.
Many approximation schemes focus on this conditional independence structure.
The normal proxy methods approximate the conditional loss by a normal random
variable with the same mean and variance. More generally, Edgeworth expansions
match the first n moments, and generate an asymptotic expansion for loss probabil-
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