The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 118 Friday, March 12, 2004 12:39 PM

118 The Mathematics of Financial Modeling and Investment Management

Vco ()i = fV[ ofb ()i]

This tells us that the value of the call option on an option-free bond

depends on the value of the option-free bond and the value of the

option-free bond depends on the interest rate. Now let’s apply the chain

rule. We get

dVco () i df dVofb

------------------- = --------------- ---------------

di dVofb di

The first term on the right-hand side of the equation is the change in

the value of the call option for a change in the value of the option-free

bond. This is the delta of the call option, ∆co. Thus,

dVco () i dVofb

------------------- = –∆co---------------

di di

Substituting this equation into the equation for the duration and rear-

ranging terms we get

Vofb

Durcb = Durofb -----------( 1 – ∆ )co

Vcb

This equation tells us that the duration of the callable bond depends on

the following three quantities. The first quantity is the duration of the

corresponding option-free bond. The second quantity is the ratio of the

value of the option-free bond to the value of the callable bond. The dif-

ference between the value of an option-free bond and the value of a call-

able bond is equal to the value of the call option. The greater (smaller)

the value of the call option, the higher (lower) the ratio. Thus, we see

that the duration of the callable bond will depend on the value of the

call option. Basically, this ratio indicates the leverage effectively associ-

ated with the position. The third and final quantity is the delta of the

call option. The duration of the callable bond as given by the above

equation is called the option-adjusted duration or effective duration.

Application of the Second Derivative

We can now compute the second derivative of the bond value with

respect to interest rates. Assuming cash flows do not depend on interest