Derivation of the Black-Scholes Formula

Step 1: Under the risk-neutral measure $\mathbb{Q}$, the stock price is $S_T = S_0 \exp((r - \sigma^2/2)T + \sigma B_T)$, where $B_T \sim \mathcal{N}(0, T)$.

Step 2: The call option price is $C = e^{-rT} \mathbb{E}_\mathbb{Q}[(S_T - K)^+]$.

Step 3: Write $S_T > K$ as $\sigma B_T > \log(K/S_0) - (r - \sigma^2/2)T$. Using the lognormal distribution and Gaussian integrals:

$C = S_0 \Phi(d_1) - K e^{-rT} \Phi(d_2),$

where $d_1, d_2$ are defined in terms of $\log(S_0/K)$, $r$, $\sigma$, and $T$.

Step 4: The formula can also be derived by solving the Black-Scholes PDE with boundary condition $(S - K)^+$.