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

mean reversion level of 5.21%. Thesevaluesarequiteintuitiveastheaverage
observed GDP growth rate is around 1.37% and the average observed 3−month
rate is approximately 5.12%. For rating categories BBB1 and A2 the mean
reversion levels of u are 20 bp, 17 bp, and for s 74 bp, 60 bp, respectively.
These numbers are rather intuitive as the average 3−month spreads for rating
categories BBB1 and A2 approximately equal 76 bp and 59 bp, respectively,
i.e. decrease for ratings of higher quality.
5 The Model of Bakshi, Madan and Zhang
In the model of Bakshi et al. (2001a) uncertainty is modeled with a threedimensional
standard Brownian motion W = (W r ,W w ,W u ) 0 on the Þltered
probability space (Ω, F, F, P). The dynamics of the non-defaultable short rate
are described by a two-factor model
dr(t) = (w(t) − a r r(t)) dt + σ r
q1 − ρ 2 r,wdW r (t)+σ r ρ r,w dW w (t), (22)
dw(t) = (θ w − a w w(t)) dt + σ w dW w (t), (23)
where a r , a w , σ r ,andσ w , are positive constants, θ w ≥ 0 and |ρ r,w | < 1. The
mean reversion level of the short rate r is dependent on w. w is assumed to be
unobservable. The short rate spread is modeled according to
where u is given by
du(t) =(θ u − a u u(t)) dt + σ u
ds(t) =(Λ r − 1)dr(t)+Λ u du(t), (24)
ρ r,u
q
1 − ρ 2 r,w
dW r (t)+σ u
s
1 − ρ2 r,u
1 − ρ 2 dW u (t).
r,w
(25)
a u and σ u are positive constants, θ u ≥ 0, andρ 2 r,u < 1 − ρ 2 r,w. Therefore,
the short rate spread is driven by a factor describing the general state of the
economy and a Þrm speciÞc component. Bakshi et al. (2001b) use for u Þrm
speciÞc data like stock prices. Note that the system of stochastic differential
equations as given by Equations (22) - (25) has a unique strong solution for
each given initial value (r 0 ,w 0 ,s 0 ,u 0 ) 0 ∈ R 4 . If we now deÞne a progressively
measurable process γ(t) =(γ r (t), γ w (t), γ u (t)) 0 such that
γ w (t) = λ w σ w w(t),
γ r (t) = λ r σ r r(t) −
ρ r,w
γ u (t) = λ u σ u u(t) − r
1 −
q γ w (t),
1 − ρ 2 r,w
ρ r,u
√
1−ρ 2 r,w
ρ2 r,u
1−ρ 2 r,w
γ r (t) =λ u σ u u(t) −
ρ r,u
q
γ r (t)
1 − ρ 2 r,w − ρ 2 r,u
16