In the Black Scholes model, the normalized price process is a martingale under the martingale measure $Q$.
What I don't understand is, why is this limited to the Black Scholes model only? Does this not hold for the general model where $$dS(t) = S(t)\alpha(t,S(t))dt + S(t)\sigma(s,S(t))dW(t)?$$
At least I was asked to prove this property, and I did not need to assume that $\alpha$ and $\sigma$ were constants. So was my proof wrong?
The way I proved it was that I first found out $d\Pi(t)$ using the neutral risk evaluation formula, and then I used the multi-dimensional Ito formula on the ratio of the price process over the riskfree asset. The dt part just cancelled out, and so I was left with only dW(t), and then it's a martingale.
In fact, here in Bjork's text. He too states the martingale property in the generalized section with the model as stated above, and then shortly after he has a subsection where he focuses strictly on the standard Black Scholes model. This merely adds to my confusion: if the result holds generally, why does he claim in the theorem that it only holds in the Black Scholes model? And why does he not put this theorem inside subsection 7.5 where he limits himself to precisely the Black Scholes model??