The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 114 Friday, March 12, 2004 12:39 PM

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114 The Mathematics of Financial Modeling and Investment Management

Now suppose that interest rates are subject to a parallel shift. In other

words, let’s assume that the interest rate for period t is (it + x). If we

compute the first derivative with respect to x for x = 0, we obtain

 

dV i() = -------d C C C 

-------------------------------+ -------------------------------+ ...+ ---------------------------------

dx^1

x = 0 dx



( 1 + i 1 + x) ( 1 + i 2 + x)

2

( 1 + iN + x)

N



x = 0

= –[C( 1 + i –^3 (

1 )

– (^2) + 2 C( 1 + i
2 ) + ...+ NC + M)(^1 + iN)
– N – (^1) ]
In this case we cannot factorize any term as interest rates are different in
each period. Obviously, if interest rates are constant, the yield curve is a
straight line and a change in the interest rates can be thought of as a
parallel shift of the yield curve.
In the continuous-time case, assuming that interest rates are con-
stant, the dollar duration is^7
dV dCe [ –^1 i + Ce –^2 i + ...+ (CM+ )e –Ni]
-------- = ----------------------------------------------------------------------------------------------
di di
= – 1 Ce –^1 i – 2 Ce –^2 i – ...– NC ( + M)e –Ni
where we make use of the rule
(^7) When interest rates are deterministic but time-dependent, the derivative dV/di is
computed as follows. Assume that interest rates experience a parallel shift i(t) + x and
compute the derivative with respect to x evaluated at x = 0. To do this, we need to
compute the following derivative:
d –∫ 0 t[is()+ x]sd d –∫t 0 is()sd –∫ 0 txsd –∫t 0 is()sd
-------e = -------e e = e d
-------(e

• xt
  )
  dx dx dx


• ∫tis()sd
  = –te –xte^0

d –∫ 0 t[is()+ x]sd –xt –∫ 0 tis()sd –∫t 0 is()sd

-------e = –te e x = 0 = –te

dx

x = 0

Therefore, we can write the following:

dV –∫^1 is()sd –∫^2 is()sd –∫Nis()sd

= – Ce^0 – 2 Ce^0 – ...– NC ( + M)e^0

dx x = 0

For i = constant we find again the formula established above.