Ito Processes
Ito processes are generalized Wiener processes in which the drift and volatility parameters can also be functions of S and t. That is:
Ito's lemma is a powerful result that allows us to find the price dynamic for a derivative of S. The intuition is that for any security, say, G, that we know to be a derivative of S, then Ito's lemma can be used to solve for the Brownian motion process of G. Ito's lemma says that G must satisfy the following relationship (A proof is given in Appendix 17.1):
Example 17.1
To illustrate, consider the following problem. Suppose a stock price follows a geometric Brownian motion, which is an Ito process 
. We want to find an expression describing the dynamics in the forward price F on a nondividend-paying stock, S with (T – t) time to delivery. That is, we want an expression for the security F given by the relationship 
if S follows an Ito process.
. We want to find an expression describing the dynamics in the forward price F on a nondividend-paying stock, S with (T – t) time to delivery. That is, we want an expression for the security F given by the relationship if S follows an Ito process.
First, we recognize that the discount rate d(0,T) = e–rT. We then compute the following partial derivatives according to Ito's lemma:
Making the necessary substitutions, we arrive at:
Finally, ...
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