The Black-Scholes Formula Derived via Change of Measure

The Black-Scholes formula $C = S\Phi(d_1) - Ke^{-rT}\Phi(d_2)$ is traditionally derived by solving a PDE (from the Feynman-Kac route). The measure-change route is shorter and more transparent: under the risk-neutral measure $\mathbb{Q}$ constructed by Girsanov's theorem, the discounted stock price $\tilde{S}_t = e^{-rt}S_t$ is a martingale, and the call price is just a Gaussian integral.

The punchline: the drift $\mu$ of GBM under $\mathbb{P}$ plays no role in the option price. The only parameters that matter are the current price $S_0$, the strike $K$, the risk-free rate $r$, the maturity $T$, and the volatility $\sigma$. This follows because changing from $\mathbb{P}$ (drift $\mu$) to $\mathbb{Q}$ (drift $r$) changes only the drift, leaving $\sigma$ unchanged.

Learning objectives. By the end of this note you should be able to: