The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 117 Friday, March 12, 2004 12:39 PM

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Principles of Calculus 117

That is, the value of a callable bond is found by subtracting the value of

the call option from the value of the option-free bond. Both components

on the right side of the valuation equation depend on the interest rate i.

Using linearity to compute the first derivative of the valuation equation

with respect to i and dividing both sides of the equation by the callable

bond’s value gives

dVcb 1 dVofb 1 dVco 1

------------- ---------= --------------- ---------– ------------- ---------

di Vcb di Vcb di Vcb

Multiplying the numerator and denominator of the right-hand side

by the value of the option-free bond and rearranging terms gives

dVcb 1 dVofb 1 Vofb dVco 1 Vofb

------------- --------- = --------------- ----------- -----------– ------------- ----------- -----------

di Vcb di Vofb Vcb di Vofb Vcb

The above equation is the sensitivity of a callable bond’s value to

changes in interest rates. That is, it is the duration of a callable bond,

which we denote by DurCB.^9 The component given by

dVofb 1

di Vofb

is the duration of an option-free bond’s value to changes in interest

rates, which we denote by Durofb. Thus, we can have

Vofb dVco 1 Vofb

Durcb = Durofb -----------– ------------- ----------- -----------

Vcb di Vofb Vcb

Now let’s look at the derivative, which is the second term in the

above equation. The change in the value of an option when the price of

the underlying changes is called the option’s delta. In the case of an

option on a bond, as explained above, changes in interest rates change

the value of a bond. In turn, the change in the value of the bond changes

the value of the embedded option. Here is where we see a function of a

function and the need to apply the chain rule. That is,

(^9) Actually, it is equal to –Dur
cb, but because we will be omitting the negative sign for
the durations on the right-hand side, this will not affect our derivation.