The lecture says expected stock return does NOT enter the BSM formula. How is that possible — is it really true?

It really is true, and $\sigma$ does not secretly contain $\mu$. This is one of the most famous surprising results in finance and the strongest test of whether you have absorbed the replication argument.

The intuition (informal):

Consider two stocks:

• Stock A: expected return 5%/year, volatility 25%/year
• Stock B: expected return 25%/year, volatility 25%/year

Naively you might think a call on stock B is worth more — bigger expected upside. But under BSM, the two calls have identical prices (assuming same $S_0$, $K$, $r$, $T$).

Why? Because anyone who thinks B has higher expected return can simply lever long stock B directly instead of buying the call. The call's value comes from its asymmetric payoff (downside truncated to zero), not from being a leveraged play on drift. The market must price the call such that there is no free arbitrage between (a) buying the call and (b) replicating it with stock + bond. Replication does not care about drift, so neither does the call price.

The mathematical reason:

In the BSM PDE derivation, $\mu$ shows up only inside the drift of the option price $f$. When you form the delta-neutral hedge by choosing $\Delta = \partial f / \partial S$, the $\mu \cdot S \cdot \partial f / \partial S$ term in $df$ is exactly cancelled by the $-\mu \cdot S \cdot \Delta$ term in the dynamics of the hedged portfolio. The drift literally subtracts out.

Where IS the stock's growth in the formula?

This is the most common follow-up question. The drift in the formula is $r - q$ (the risk-free rate minus dividend yield), not $\mu$. Under the risk-neutral measure $Q$, the stock grows at $r - q$ on average, not at its real-world expected return. That is the substitution that makes the replication work.

Where IS volatility?

Volatility $\sigma$ enters in two places: as a multiplier on $\sqrt{T}$ in both $d_1$ and $d_2$, and as $\sigma^2$ inside $d_1$. Volatility is the only property of the stock's distribution that BSM cares about. Higher $\sigma$ → wider distribution of $S_T$ → bigger expected payoff on the asymmetric option → higher call/put price.

The takeaway:

BSM cares about the uncertainty ($\sigma$) and the risk-free rate ($r$), not about expected returns. This is one of the most elegant and useful results in finance, because expected returns are nearly impossible to estimate while volatility is empirically tractable.