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

where
P d (t, T, r, w, u) =e Ad (t,T )−B d (t,T )r(t)−E d (t,T )w(t)−D d (t,T )u(t)
(26)
with
B d (t, T ) = Λ ³
r
1 − e −â r(T −t)´
,
â r
D d (t, T ) = Λ ³
u −âu(T
1 − e
−t)´
,
â u
E d (t, T ) = Λ µ
r 1 − e
−â w (T −t)
â r
A d (t, T ) =
Proof.
See appendix.
Z T
â w
+ e−â w(T −t) − e −â
r(T −t)
,
â w − â r
1 ¡ σ
2
t 2 u D d (l, T) 2 +2σ u ρ r,u σ r B d (l, T)D d (l, T)
+σ 2 wE d (l, T) 2 + σ 2 rB d (l, T) 2 +2σ r ρ r,w σ w B d (l, T)E d (l, T) ¢
−θ u D d (l, T) − θ w E d (l, T) − Λ 0 dl.
Similar to the extended model of Schmid and Zagst the non-defaultable short
rate and the short rate spread are negatively correlated (in all our parameter
estimations Λ r is always between 0 and 1).
Note that
ds(t) = (Λ r − 1)dr(t)+Λ u du(t)
= (Λ u θ u +(Λ r − 1)(w(t) − a r r(t)) − Λ u a u u(t)) dt + σ s dZ s (t)
= (θ s − b sw w(t)+b su u(t) − a s s(t)) dt + σ s dZ s (t),
where θ s = Λ 0 a r + Λ u θ u , b su = Λ u (a r − a u ), b sw = 1 − Λ r , a s = a r ,andZ s is
the Itô-process given by
⎛
⎞
σ s dZ s (t) = ⎝(Λ r − 1)σ r
q1 − ρ 2 ρ
r,w + Λ u σ ru
u q ⎠ dW r (t)
1 − ρ 2 r,w
+(Λ r − 1)σ r ρ r,w dW w
s
+Λ u σ u 1 − ρ2 ru
1 − ρ 2 dW u (t).
r,w
Therefore, there is a close relationship between the stochastic differential equation
for s in the model of Bakshi, Madan and Zhang and the one in the extended
model of Schmid and Zagst.
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