In a further extension of the Black Scholes, Hagen et al go directly to a new measure $\mathbb{Q}_T$. The justification here is that what we are really interested in (as traders, investors, brokers, economists as well as old age pensioners) is the forward price of $S$ i.e $S_T$ and not the spot price of $S$. Where $S(t)$ is headed is indeed what we speculate on every time we look at $S(t)$.

Numeraire $X = B^T$

We define