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Ito Process and Geometric Brownian Motion

Geometric Brownian motion
dS=μS dt+σS dWdS = \mu S\,dt + \sigma S\,dW

Both terms scale with the price, so the process compounds and can never reach zero from above.

Geometric Brownian motion: drift and volatility both scale with the price, making moves multiplicative.

Every path here has the same drift and the same volatility. Nothing separates the winners from the losers except the draws, which is the uncomfortable half of modelling a price this way.

Ito's lemma

For a function f(S,t)f(S,t) of an Ito process, ordinary calculus is not enough:

df=(∂f∂t+μS∂f∂S+12σ2S2∂2f∂S2)dt+σS∂f∂S dWdf = \left(\frac{\partial f}{\partial t} + \mu S\frac{\partial f}{\partial S} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2}\right)dt + \sigma S\frac{\partial f}{\partial S}\,dW

Everything matches the chain rule except the third term in the bracket. That term exists because (dW)2=dt(dW)^2 = dt rather than being negligible: the squared increments of Brownian motion accumulate at a predictable rate.

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