Black-Scholes Partial Differential Equation

Theorem8.1Black-Scholes PDE

The price $V(t, S)$ of a European derivative with payoff $g(S_T)$ satisfies:

$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V = 0$

with terminal condition $V(T, S) = g(S)$ .

Derivation: By Feynman-Kac, this PDE corresponds to the risk-neutral expectation $V(t, S) = e^{-r(T-t)} \mathbb{E}_\mathbb{Q}[g(S_T) \mid S_t = S]$ , yielding the Black-Scholes formula for calls/puts.