Itô's Lemma Applied in Finance

It would not be exaggerated to say that Itô's Lemma is one of the building blocks in the stochastic analysis. Itô's Lemma is essentially the chain rule for stochastic functions.  The lemma is an important part in valuing derivatives since a derivative is a function of the price of the underlying and time. Changes in a variables, for example change in stock price, involve a deterministic component which is a function of time and a stochastic component.

For instance:

x is the stock price at time t

dx is the change in x over the interval of time dt.

dz is the change in the random variable z over this interval of time is dz (stated briefly, dz is a Wiener process).

The change in stock price is:

dx=adt+bdz,

where a and b are functions of x. The variable x has a drift a and standard deviation b.

Under the Itô's Lemma the function F of x and t follows the process (we denote here as partial derivative; according to Wikipedia partial derivatice  of a function of several variables is its derivative with respect to one of those variables, with the pthers held constant, as opposed to the total derivative, in which all variables are allowed to vary)

:

Looking at perspective, applying the Itô's Lemma in valuing a forward contract on shares:

So we have:

This is the last post for 2015. I am wishing you an Ispiring New Year!