Advances in Risk Management

74 MANAGING INTEREST RATE RISK UNDER NON-PARALLEL CHANGES

Hence,B(τ) andC(τ) indicate the sensitivity of a zero-coupon bond to
changes in the spread and in the interest rates level, respectively. Then,
after assessing the behavior of the bond portfolio in the presence of both
changes, these measures can be an adequate tool for portfolio management.
Investors who want to immunize a portfolio against these changes must
equate its generalized durations to those of the asset to be replicated.
Convexity is a measure that can complement the estimation (obtained
from duration) of the change in the bond price when interest rates change
in a large amount. Similarly to duration, we generalize this measure:

Definition (generalized convexity) The generalized convexitiesδsand

δLof a bond that paysncouponsciat timesti,i=1, 2,...,nwith respect

to the factorssandLare given by the expressions:

δs=

1

P∗(t,T)

∑n

i= 1

ci(ti−t)P(t,ti)

∂^2 Y(t,ti)

∂s^2

(4.20)

δL=

1

P∗(t,T)

∑n

i= 1

ci(ti−t)P(t,ti)

∂^2 Y(t,ti)

∂L^2

whereP(t,ti) is the price, at timet, of a zero-coupon bond that matures at

timeti(see Proposition 1).

For a zero-coupon bond, we have:

δs=B^2 (t,T)=B^2 (τ)

δ (4.21)

L=C

(^2) (t,T)=C (^2) (τ)

4.4 HEDGING RATIOS

An alternative technique to duration for managing the interest rate risk may
be performed with bond options. Since duration measures the sensitivity
of a present value to changes in interest rates, it can be applied not only
to bonds but it can be extended to options. Thus, it is possible to define
measures of the sensitivity of different interest rate derivatives to several
factors and, then, to construct the corresponding hedging strategy.
We consider a European call option on a zero-coupon bond. LetKbe its
strike price. If this option is exercised at expiration,Tc, the call holder pays
Kand receives a discount bond that matures at timeTb>Tc.
The price at timet,C(s,L,t,Tc;K,Tb), of this option (see Moreno, 2003)
is given by:

C(s,L,t,Tc;K,Tb)=P(t,Tb)(h+σP ̃)−KP(t,Tc)(h) (4.22)

whereP(t,Ti) is the price, at timet, of a zero-coupon bond that matures
at timeTi(see Proposition 1),(.) denotes the distribution function of a