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Previously, Stock price simulation , we saw that stock prices can be described by geometric brownian motion (GBM).
In order to solve a partial differential equation GBM, we need to appreciate Ito’s lemma.
Let us consider G is a function of two variables, x and t.
The change in G for a small changes in variables can be solved appliying partial derivatives by differentiating with respect to one varible at a time, keeping the other variable constant:

Ito's lemma

Taylor series are used in the formula above to get a more accurate estimate.
Suppose we have a variable x that follows the Ito process: Δx = aΔt + b*e*sqrt(Δt), where “e” is a random number drawn from a normal distribution with mean = 0 and standard deviation = 1.
Now, we take this Ito process to the power 2 and see that power(Δx,2) = power(b*e,2)*Δt, where all Δt with power higher than 1 are ignored because the value is too small.
Also, the expected value of a random number is too small and can be ignored as well.
Thus, power(Δx,2) = power(b,2)*Δt. In the Taylor series we take approximate value of G:

Taylor series

Plugging into the equation for x in ito’s lemma, we obtain:

Taylor serie

or it can be rewritten as:


This is called Ito’s lemma.
Now, consider the stock price to be lognormally distributed, G=ln(S), x=S, a= μS and b=σS(according to GBM formula in Stock price simulation ) and taking into account that:


We obtain:

Ito process

That can be simply rewritten:

Ito stock movement

And it gives a stock movement formula:

Stock movement

Additional iformation is here.