I'm currently reading Hull's book and I was wondering about an apparant inconsistency in my thoughts:

1. In the Black Scholes PDE, the only rate exist is risk free rate 'r'. The derivation from Ito Lemma is pretty straightforward (using the riskless portfolio assumption). The drift of underlying '\mu' is removed when we consider a riskless portfolio. My personal interpretation is that the option's value have little to do with the drift of underlying.

2. If we instead opt to use monte carlo method, we will have to simulate based on the GBM equation:

\(S_t = S_0 exp \large( \large( \mu -\frac{\sigma^2}{2} \right) t + \sigma W_t \right)\)

This however, does not assume \(\mu = r\) anywhere. In practice however, we usually take \(\mu = r\) to get obtain the result (which is equal to Black Scholes' formula).

My question is where did we assumed in case (2) that \(\mu = r\)? I'll be glad if anyone could point me to some reading about it.. Thanks.