Assume that the underlying $S$ is some index, hence the risk-return $\mu=0$, where $S$ meets $$d S = \sigma S d W_t.$$
Let $V$ denote the price of the corresponding call option. To construct the related BS formula, I construct a portfolio $\Pi=V-\Delta S$, after setting a correct value of $\Delta$, I want the portfolio to be risk-free. That is $$d \Pi = r\Pi d t= r(V-\Delta S) d t,$$ where $r$ is risk-free rate.
hence by the Ito formula, I can get the BS-equation.
However, someone told me that the identity $d \Pi = r(V-\Delta S) d t$ should be $$d \Pi = (r*V-\mu*\Delta S) d t,$$ and then get another equation.
Since in my opinion, in the risk-neutral world, $\mu$ turns to be $r$ after applying the Girsonov transformation, and making the portfolio to be risk-free is under risk-neutral world. I agree with the first identity.
So my question is which one is correct? If is the latter one, what's meaning of risk-neutral pricing?
Thank you very much!
Added 2016/10/26 10:39AM(+8)
Thanks for @MJ73550. I am sure the first one is right now.
However, if we distinguish the funding and lending rate for unsecurities (denote as $r_F$) and stock collateral(denote as $r_R$). Then maybe the identity $d \Pi = r(V-\Delta S) d t$ should be $$d \Pi = (r_F*V-r_R*\Delta S) d t,$$
Is this equation right?