Boundary Conditions for a Call

Consider a European call option $C(t, S)$ with strike $K$ and maturity $T$ on a stock following a geometric Brownian motion:

dS_t = \mu S_t \, dt + \sigma S_t \, dW_t

The option price satisfies the Black-Scholes PDE:

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

Question: Derive the boundary conditions satisfied by $C(t, S)$ at $S = 0$ and as $S \to +\infty$ .

Answer