Analysis of Financial Time Series, Third Edition

6.3 Ito's Lemma

In finance, when using continuous-time models, it is common to assume that the price of an asset is an Ito process. Therefore, to derive the price of a financial derivative, one needs to use Ito's calculus. In this section, we briefly review Ito's lemma by treating it as a natural extension of the differentiation in calculus. Ito's lemma is the basis of stochastic calculus.

6.3.1 Review of Differentiation

Let G(x) be a differentiable function of x. Using the Taylor expansion, we have

Taking the limit as Δx → 0 and ignoring the higher order terms of Δx, we have

When G is a function of x and y, we have

Taking the limit as Δx → 0 and Δy → 0, we have

6.3.2 Stochastic Differentiation

Turn next to the case in which G is a differentiable function of xt and t, and xt is an Ito process. The Taylor expansion becomes

(6.3)

A discretized version of the Ito process is

(6.4)

where, ...