The Black-Scholes model essentially says that, if we assume some things (lognormal, constant variance, etc.) then the following the fair price of a
$$C = N(d_1) S_t - N(d_2) K e^{-rt}$$
However, another interpretation of the Black-Scholes models is that if we make the above assumptions, then we can merely pretend the drift of the underlying security is the risk-free interest rate and then interpret a derivative's expected payout as equal to its fair price.
Clearly, the lognormal assumption is required for the standard Black-Scholes formula. However, I want to know if the lognormal assumption is required for the "pretend riskless drift and then compute expected-value" perspective.
