Boundary Conditions In Option PDEs

The Black-Scholes equation describes how an option's value changes with the stock price and time, but the equation alone has infinitely many solutions — like knowing a ball obeys gravity without knowing where it started or where the ground is. To pin down the one solution that actually represents a specific option, you need boundary conditions: extra facts about the option's value at the edges of the problem, at expiry and at extreme stock prices.

Think of a wall at each end of a hallway: the PDE tells you how value flows in between, but you still need to know what's happening at the two walls to know exactly where everything settles. For a European call with strike $K$ at expiry $T$, the "wall at the end of time" is the payoff itself: $V(S,T) = \max(S - K, 0)$ — an option one second before expiry is worth exactly its intrinsic value, no more, no less. The "walls" in stock price are $V(0,t) = 0$ (a worthless stock makes a call worthless, since it can never regain value under geometric Brownian motion) and $V(S,t) \to S - Ke^{-r(T-t)}$ as $S \to \infty$ (a call on an enormous stock price is worth almost exactly the stock minus the discounted strike, since the chance of finishing below the strike becomes negligible).

A put has mirrored conditions: $V(S,T) = \max(K - S, 0)$ at expiry, $V(0,t) = Ke^{-r(T-t)}$ (certain exercise if the stock is worthless), and $V(S,t) \to 0$ as $S \to \infty$. Numerical solvers — finite difference grids used when there's no closed form, such as for American options — literally need these values plugged in at the edges of the grid at every time step, or the computed prices drift into nonsense near the boundaries.