The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 116 Friday, March 12, 2004 12:39 PM

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116 The Mathematics of Financial Modeling and Investment Management

Duration for continuously compounding constant interest rate in dis-

crete time:^8

dV 1 1 –

(

• Ni
  -------- ----= – ----[Ce


• i

• 2 Ce
  2 i


• ...+ NC + M)e ]
  diV V

We will now illustrate the chain rule of derivation by introducing

the concept of effective duration. In Chapter 2, we described the differ-

ent features of bonds. The bond valuation we presented earlier is for an

option-free bond. But when a bond has an embedded option, such as a

call option as discussed in Chapter 2, it is more complicated to value.

Similarly, the sensitivity of the value of a bond to changes in interest

rates is more complicated to assess when there is an embedded call

option. Intuitively, we know that the sensitivity of the value of a bond

with an embedded option would be sensitive to not only how changes in

interest rates affect the present value of the cash flows as shown above

for an option-free bond, but also how they would affect the value of the

embedded option.

We will use the following notation to assess the sensitivity of a call-

able bond’s value (i.e., a bond with an embedded call option) to a

change in interest rates. The value of an option-free bond can be decom-

posed as follows:

Vofb= Vcb + Vco

where

Vofb = value of an option-free bond

Vcb = value of a callable bond

Vco = value of a call option on the bond

The above equation says that an option-free bond’s value depends

on the sum of the value of a callable bond’s value and a call option on

that option-free bond. The equation can be rewritten as follows:

Vcb = Vofb – Vco

(^8) The duration for continuously compounding variable interest rate in discrete time is
dV
di

-^1
V
-^1
V

• Ce
  
  
  • ∫ 0 1 is()
    2 Ce
    
    
    • ∫ 0 2 is()
      ... NC M( + )e
      
      
      • ∫ 0 Nis()
        –= + + +
      
      
      
      
    
    
    
    
  
  
  
  

sd sd sd