Fixed Income and Interest Rate Derivative Analysis

148 FIXED INCOME AND INTEREST RATE DERIVATIVE ANALYSIS

11.2 Ito's lemma

Ito's lemma provides us with a description of the process followed by a

function

of a Weiner process. If f is a function of one variable with continuous

first and second derivatives ft and f" and

Xt

is a generalized Weiner process

dX = at dt + bt dW

then Ito's lemma states that the process

ft

defined by the value of the

function at

Xt

denoted

f(Xt)

follows the following stochastic differential

equation (sde):

)

ta t + 1

t.,,~2

dt + f'

J t ~'t tbt

dWt

This result is extremely useful as it provides a formula for the function