Advances in Risk Management

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.