In the black-scholes model, the hedging portfolio is given (in some textbooks) by $$\Pi_t = V_t - \Delta S_t,$$ i.e., the portfolio consits of a long position in the option $V$ and $\Delta$ units of short positions in the stock $S_t$.
How does we get the change in the portfolio value, i.e., how does we get $$d\Pi_t = dV_t - \Delta dS_t$$ In some textbook they argue it is due to Ito's lemma. I know Ito's lemma, but I don't know how to apply it to get the above.
My Idea: In some textbooks they start with a (replicating) portfolio given by $$\Pi_t = \alpha S_t + \beta B_t,$$ where $B_t$ is the risk-free asset. Then, by assuming that the replicating portfolio is self-financing, we have $$d\Pi_t = \alpha dS_t + \beta dB_t,$$ i.e., the change in portfolio value is due to changes in market conditions and not to either infusion or extraction of cash.
Does we have $$d\Pi_t = dV_t - \Delta dS_t$$ because of the self-financing assumption? But why do some books say that we get it by Ito's lemma? How do we apply Ito's lemma to $$\Pi_t = V_t - \Delta S_t \qquad ?$$