The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 106 Friday, March 12, 2004 12:39 PM

106 The Mathematics of Financial Modeling and Investment Management

function itself remains bounded. For example, the function that assumes

the value 1 on rational numbers and 0 elsewhere is of infinite variation

in any interval, though the function itself is finite.

Continuous functions might also exhibit infinite variation. The follow-

ing function is continuous but with infinite variation in the interval [0,1]:

0 for x =^0

 

fx()=  π --- x

xsinfor 0 ≤<^1



 x

DIFFERENTIATION

Given a function y = f(x) defined on the open interval (a,b), consider its

increments around a generic point x consequent to an increment h of the

variable x ∈(a,b)

∆y = f(x + h) – f(x)

Consider now the ratio ∆y/h between the increments of the depen-

dent variable y and the independent variable x. Called the difference

quotient, this quantity measures the average rate of change of y in some

interval around x. For instance, if y is the price of a security and t is

time, the difference quotient

yt ( + h)– yt()

∆y= ------------------------------------

h

represents the average price change per unit time over the interval

[t,t+h]. The ratio ∆y/h is a function of h. We can therefore consider its

limit when h tends to zero.

If the limit

fx ( + h)– fx()

f ′()x = lim ------------------------------------

h → (^0) h
exists, we say that the function f is differentiable at x and that its deriv-
ative is f ′,also written as
df dy
-------or -------
dx dx