The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 119 Friday, March 12, 2004 12:39 PM

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Principles of Calculus 119

rates, this second derivative is called dollar convexity. Dollar convexity

divided by the bond’s value is called convexity. In the discrete-time fixed

interest rate case, the computation of convexity is based on the second

derivatives of the generic summand:

 

d

2

-------- -----------------^1 = -----d d^1  d^1

----- ----------------- = -----–t-------------------------

t

di

2

( 1 + i) didi ( 1 + i)

t 



di ( 1 + i)t +^1



d 1  1

= –t ----- ------------------------- = t( 1 + t)-------------------------

di ( 1 + i)t +^1 ( 1 + i)t +^2

Therefore, dollar convexity assumes the following expression:

d

2   

() d

2

Vi C C CM+

----------------- = -------- ------------------+ ------------------+ ...+ --------------------

di^2 di^2 ( 1 + i)^1 ( 1 + i)^2 ( 1 + i)N

d^21 d^21

= C-------- ------------------ + ...+ (CM+ )-------- --------------------

di^2 ( 1 + i)   1 di^2 ( 1 + i)N

= [ 2 C( 1 + i)–^3 + 23 ⋅ C( 1 + i)–^4 + ...

+ NN ( + 1 )(CM+ )( 1 + i) –(N +^2 )]

Using the same reasoning as before, in the variable interest rate case,

dollar convexity assumes the following expression:

d

2

Vi() = [ 2 C( 1 + i 3 – 4

1 )

– (^3) + 2 ⋅⋅C( 1 + i
2 ) + ...
dx
2

x = 0

• NN ( + 1 )(CM+ )( 1 + iN)– N –^2 ]
  This scheme changes slightly in the continuous-time case, where,
  assuming that interest rates are constant, the expression for convexity is^10
  (^10) For variable interest rates this expression becomes
  dV 0 is()sd 0 is()^ sd^0 is()sd
  = 12 Ce +

• ∫^1
  
  
  • 22 Ce
    
    
    • ∫^2
      
      
      • ...+ N^2 (CM)e
        
        
        • ∫N
        
        
        
        
      
      
      
      
    
    
    
    
  
  
  
  

dx x = 0