In the Hull textbook's derivation of Black's model (Section 27.6), they apply equation (27.20), which is $f_0 = P(0,T)E_T(f_T)$, where $P(0,T)$ is the value of a zero coupon bond at time $0$ expiring at $T$, and $E_T$ is the expectation with respect to the forward risk neutral measure of the zero coupon bond.
They set $f_T=\max(S_T-K,0)$ to the call payoff, and go from there.
However, $f_0 = P(0,T),E_T(f_T)$ was derived in Section 27.3 assuming that $f_t$ satisfies $df = \mu f dt + \sigma f dz$.
Why is it valid to set $f_T=\max(S_T-K,0)$? That is, how do we show that $f_t$ satisfies $df = \mu f dt + \sigma f dz$?
Are all European derivatives $f_T$, not necessarily a call, also of this form?