72 MANAGING INTEREST RATE RISK UNDER NON-PARALLEL CHANGES
A 1 (τ)=exp
(
−
σ^21
4 q 1
B^2 (τ)+s∗
(
B(τ)−τ
)
)
A 2 (τ)=exp
(
−
σ^22
4 q 2
C^2 (τ)+L∗
(
C(τ)−τ
)
)
(4.9)
B(τ)=
1 −e−q^1 τ
q 1
C(τ)=
1 −e−q^2 τ
q 2
with
q 1 =k 1 +bσ 1 ,s∗=ˆμ 1 −
1
2
σ 12
q^21
,μˆ 1 =
k 1 μ 1 −aσ 1
q 1
(4.10)
q 2 =k 2 +dσ 2 ,L∗=ˆμ 2 −
1
2
σ 22
q^22
,μˆ 2 =
k 2 μ 2 −cσ 2
q 2
4.3 GENERALIZED DURATION AND CONVEXITY
We will generalize the concepts of conventional duration and convexity
using the above two-factor model. Hence, we can measure the interest rate
risk with respect to both stochastic factors.
The price, at timet, of a default-free zero-coupon bond that pays $1 at
maturity,T=t+τ, is given by:
P(s,L,t,T)=P(t,T)=e−(T−t)Y(s,L,t,T) (4.11)
whereY(s,L,t,T)≡Y(s,L,τ) is the (continuously compounded) yield to
maturity of this bond.
Applying Itô’s lemma, using the closed-form expression (4.8) given
by Proposition 1 and the dynamics of the state variables (see (4.4)), the
instantaneous change in the bond price is given by:
dP(t,T)=μP(.)dt−(T−t)P(t,T)
[
∂Y(t,T)
∂s
σ 1 dw 1 +
∂lY(t,T)
∂L
σ 2 dw 2
]
(4.12)
with
μP(.)=Psk 1 (μ 1 −s)+PLk 2 (μ 2 −L)+Pt+
1
2
Pssσ 12 +
1
2
PLLσ 22 (4.13)
Next, we consider a coupon bond payingncouponsci at timesti,
i=1, 2,...,n. This bond has a nominal value equal to $1 and matures at
timeT=tn. LetP∗(s,L,t,T)≡P∗(s,L,τ) be the price, at timet, of this bond.