Advances in Risk Management

MANUEL MORENO 75

standard normal variable, and

h=

ln (P(t,Tb))−ln (KP(t,Tc))

σP ̃

−

1

2

σP ̃,

σP^2 ̃=Var(ln (P ̃)),P ̃=P(Tc,Tb) (4.23)

Computing the derivative of (4.22), applying the chain’s rule, we obtain that
the generalized duration of this option to spread changes^7 is given by:

∂C(.)

∂s

=[B(t,Tc)−B(t,Tb)]

[

P(t,Tb)

f(h+σP ̃)

σP ̃

−KP(t,Tc)

(

f(h)

σP ̃

−(h)

)]

−B(t,Tb)C(t,Tc;K,Tb) (4.24)

wheref(.) denotes the density function of a standard normal variable.
Similarly, the generalized duration of the call option to changes in the
long-term interest rate is given by:

∂C(.)

∂L

=[C(t,Tc)−C(t,Tb)]

[

P(t,Tb)

f(h+σP ̃)

σP ̃

−KP(t,Tc)

(

f(h)

σP ̃

−(h)

)]

−C(t,Tb)C(t,Tc;K,Tb) (4.25)

We will apply a relationship^8 that links the generalized durations of the
bond option with that of its underlying asset: this relationship establishes
that the generalized duration of an option is equal to the elasticity of this
option times the generalized duration of the bond where the elasticity of the
option is given by:

P(t,Tb)

C(t,Tc;K,Tb)

∂C(t,Tc;K,Tb)

∂P(t,Tb)

(4.26)

The elasticity of the option is the product of two terms, the leverage of the
option and the hedging ratio. Therefore, from equations (4.19) and (4.26), it
is deduced that the delta of the option is:

=

∂C(.)

∂s

1

B(t,Tb)

C(t,Tc;K,Tb)

P(t,Tb)

(4.27)

where∂C(.)/∂sis as given in (4.24).

4.5 A PROPOSAL OF A SOLUTION FOR THE LIMITATIONS

OF THE CONVENTIONAL DURATION

We consider two portfolios with the same generalized durations with respect
to the above factors. Both portfolios differ in yield and convexity. We will