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

for real constants λ r , λ w , λ u , by applying Girsanov’s theorem we can show that
cW (t) =W (t)+
Z t
0
γ(l)dl
is a standard Brownian motion under the measure Q. Then the Q-dynamics of
r, w, s, and u are given by
dr(t) = (w(t) − â r r(t)) dt + σ r
q1 − ρ 2 r,wdcW r (t)+σ r ρ r,w dcW w (t),
dw(t) = (θ w − â w w(t)) dt + σ w dcW w (t),
ds(t) = (Λ r − 1)dr(t)+Λ u du(t),
s
ρ r,u
du(t) = (θ u − â u u(t)) dt + σ u q dcW r (t)+σ u 1 − ρ2 r,u
1 − ρ 2 1 − ρ 2 dcW u (t),
r,w
r,w
q
r
where â r = a r +λ r σ 2 r 1 − ρ 2 r,w, â w = a w +λ w σ 2 w,andâ u = a u +λ u σ 2 u 1 − ρ2 r,u
1−ρ
. 2 r,w
Using Equation (1) and Equations (17) and (18) we can calculate the price of a
non-defaultable zero-coupon bond in the model of Bakshi, Madan and Zhang:
Theorem 5 (Price of a non-defaultable zero-coupon bond) The time t
price of a non-defaultable zero-coupon bond with maturity T is given by
where
with
P (t, T )=E Q h e − R T
t r(l)dl¯¯¯ Ft
i
= P (t, T, r(t),w(t)),
A(t,T )−B(t,T )r(t)−E(t,T )w(t)
P (t, T, r, w) =e
B(t, T ) = 1 â r
³
1 − e −â r(T −t)´
,
E(t, T ) = 1 µ 1 − e
−â w(T −t)
+ e−âw(T −t)
−âr(T −t)
− e
,
â r â w
â w − â r
Z T
1
A(t, T ) =
t 2 σ2 rB(l, T) 2 + 1 2 σ2 wE(l, T) 2 + σ r ρ r,w σ w B(l, T)E(l, T)
−θ w E (l, T) dl.
Proof.
See Bakshi et al. (2001b).
Theorem 6 (Price of a defaultable zero-coupon bond) The price of a defaultable
zero-coupon bond at time t