CREDIT RISK: Lecture V
Lina ElJahel
2006
1
Reduced Form Models
Reduced form models are often advocated on the basis that they are easier
to implement than structural models.
In fact, one can always implement
structural model, by inferring hard to observe information on say capital
structure form market prices so this arguments is not very convincing.
A more significant argument in favor of reducedform or structural models
is that they may allow the user to link the default event to more variables
than firm value (using Cox processes) and this approach can be linked to the
entire set of tools usually used in term structure modelling. This means that
tricks used for pricing derivatives can be transferred to defaultable claims.
In ordinary term structure modelling our ignorance of what truly governs
the dynamics of interest rates is often hidden by our modelling of the short
rate for example. This idea is transferred to intensity based models where
most models use exogenously given specifications of the default intensity.
The mathematical structure of ordinary term structure models easily allows
however, for the short rate to depend on multidimensional state variable
processes and even to depend on observable state variable processes such as
yields and volatilities. The reduced form approach is surely moving in this
direction and many researchers are trying to isolate fundamental variables
to explain the default event. Once these have been convincingly identified
one can then incorporate them in an intensity model. In our presentation of
reduced form models we will focus on models using doubly stochastic Poisson
processes or Cox processes. This specification is more general than using a
constant hazard rate and also allows the intensity process to be linked to
other variables.
1
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The Cox Process Construction of a Single
Jump Time
Consider a zerocoupon issued by a risky firm at time 0. Assume that the
maturity of the bond is
T
and the default time
τ
of the issuing firm has an
intensity
λ
(
X
t
) under the riskneutral probability measure
Q
. Also, assume
there is a short rate process
r
(
X
t
) such that the zero coupon default free
bond can be computed as
p
(0
, T
) =
E
"
exp

Z
T
0
r
(
X
s
)
ds
!#
(1)
The price of the risky bond at time 0 is then, assuming
zero recovery
,
F
(0
, T
)
=
E
"
exp

Z
T
0
r
(
X
s
)
ds
!
1
{
τ>T
}
#
=
E
"
E
"
exp

Z
T
0
r
(
X
s
)
ds
!
1
{
τ>T
}

λ
(
t
)
∀
t
##
=
E
"
exp

Z
T
0
r
(
X
s
)
ds
!
E
h
1
{
τ>T
}

λ
(
t
)
∀
t
i
#
=
E
"
exp

Z
T
0
r
(
X
s
)
ds
!
exp

Z
T
0
λ
(
X
s
)
ds
!#
=
E
"
exp

Z
T
0
r
(
X
s
) +
λ
(
X
s
)
ds
!#
The short rate has been replaced by the intensityadjusted short rate
(
r
+
λ
)
This results can be easily modified to cover a contingent claim with a
promised payment
f
(
X
T
) and an actual payment
f
(
X
T
)1
τ>T
.
DuffieSingleton (1999) examine this issue and introduce the concept of
a discount rate higher than the risk free rate under the risk neutral measure
to take into consideration the risk of default.
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 Spring '05
 wqeww
 Probability theory, Default, Credit rating

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