The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 120 Friday, March 12, 2004 12:39 PM

120 The Mathematics of Financial Modeling and Investment Management

d^2 V d^2 [Ce –i + Ce –^2 i + ...+ (CM+ )e –Ni]

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

di^2 di^2

= 12 ⋅Ce –i + 22 ⋅Ce –^2 i + ...+ N^2 ⋅ (CM+ )e –Ni

where we make use of the rule

d

2

---------()ex = ex

dx

2

We can now write the following formulas for convexity:

Convexity for constant interest rates in discrete time:

dV

2

1 1 2 C () 3 () 2 C NN ( + 1 )(CM+ )

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

di

(^2) V
V( 1 + i)
(^2) ( 1 + i)
( 1 + i)^2 ( 1 + i)N
Convexity for variable interest rates in discrete time:
d^2 V (^112) C () 3 () 2 C NN ( + 1 )(CM+ )
-------------- = -------------------------+ ---------------------+ ...+ ----------------------------------------------
dx
(^2) V V
( 1 + i 1 )
3
( 1 + i 2 )
4
( 1 + iN)
N + 2
Convexity for continuously compounding constant interest rate in dis-
crete time:^11
d^2 V 1 1 – –Ni
-------------- = ----[Ce +

• i
  
  
  • 2
    2
    Ce
    2 i
    
    
    • ...+ N
      2
      (CM)e ]
      di^2 V V
    
    
    
    
  
  
  
  

(^11) The convexity for continuously compounding variable interest rate in discrete time
is

d^2 V 1 1 – ∫^1 is()sd –∫^2 is()sd is()sd

+ 2

2

Ce

0 0

---------- ----= ---- Ce +

0

+ ...+ N

2

(CM)e

• ∫N^

di^2 V V^