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

where
with
( 1
A(t,T )−B(t,T )r
P (t, T, r) =e
£ ¤
â
B (t, T ) =
r
1 − e
−â r(T −t)
, if β =0,
−δr (T −t)
1−e κ (r)
1 −κ(r) 2 e−δr(T −t) 2 , , and, (9)
⎧ R ³
´
T 1
t 2
⎪⎨
rB (τ,T) 2 − θ r (τ) B (τ,T) dτ
=ln P (0,T )
∂ ln P (0,t)
ln A (t, T ) =
P (0,t)
− B (t, T )
∂t
¡
− σ2 r
4â e
−â r
⎪⎩
− e ¢ −â rt 2 ¡ e
2â r t − 1 ¢ if β =0,
(10)
,
3
r
− R T
t
θ r (τ) B (τ,T) dτ, if β = 1 2 ,
with
δ x = p â 2 x +2σ 2 x and κ (x)
1/2 = âx
2 ± 1 2 δ x. (11)
Proof.
See, e.g., Hull & White (1990).
Using Equation (2) and Equations (6) - (8) we can calculate the price of a
defaultable zero-coupon bond in the Schmid and Zagst model:
Theorem 2 (Price of a defaultable zero-coupon bond) The price of a defaultable
zero-coupon bond at time t