Empirical Evaluation of Hybrid Defaultable Bond Pricing ... - risklab

2,5
2
1,5
1
0,5
1 Year
3 Years
5 Years
7 Years
10 Years
0
01.10.1993
16.09.1994
01.09.1995
16.08.1996
01.08.1997
17.07.1998
02.07.1999
16.06.2000
01.06.2001
17.05.2002
02.05.2003
Figure 3: 1- To 10-Year Credit Spreads of US Industrials BBB1 in %. Time
Period: 1993 — 2003. Source: Bloomberg.
3 The Model of Schmid and Zagst
In the following section we give a short overview of the Schmid and Zagst
three-factor defaultable term structure model. As a typical hybrid model it
combines elements of structural and reduced-form models. The underlying nondefaultable
short rate is assumed to either follow a mean reverting Hull-White
process or a mean-reverting square root process with time-dependent mean reversion
level. Therefore, the dynamics of the non-defaultable short rate are
given by the following stochastic differential equation (SDE):
dr (t) =[θ r (t) − a r r (t)] dt + σ r r (t) β dW r (t) , 0 ≤ t ≤ T ∗ , (3)
where a r , σ r > 0 are positive constants, β =0or 1 2 , and θ r is a non—negative
valued deterministic function. This speciÞcation implies that the current rate
r (t) is pulled towards θr(t)
a r
with a speed of adjustment a r ,andifβ = 1 2 the
instantaneous variance of the change in the rate is proportional to its level.
One of the factors that determine the credit spread is the so called uncertainty
index which can be understood as an aggregation of all information on the
quality of the Þrm currently available: The greater the value of the uncertainty
process the lower the quality of the Þrm. The uncertainty or signaling process
is assumed to follow a mean reverting square root process. The development of
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