Derivation of the Black-Scholes-Merton PDE › Boundary Conditions

The BSM PDE alone does not determine a unique solution. The equation describes how the derivative price must evolve, but it does not tell us where the price starts or ends. To price a specific derivative, we need boundary conditions that specify what happens at maturity and at extreme values of the stock price. These boundary conditions encode the contractual terms of the derivative, transforming an abstract PDE into a concrete pricing problem.

For a European call option with strike $K$ and maturity $T$:

$$\begin{aligned} V(S, T) &= \max(S - K, 0) && \text{(Terminal condition)} \\ V(0, t) &= 0 && \text{(Stock is worthless)} \\ V(S, t) &\approx S - Ke^{-r(T-t)} && \text{(Behaves like forward)} \end{aligned}$$

where:

• $V(S, t)$: value of the call option
• $S$: current stock price
• $K$: strike price
• $T$: maturity time
• $t$: current time
• $r$: risk-free interest rate

The terminal condition states that at maturity, the call option is worth $S - K$ if the stock price exceeds the strike, and zero otherwise. This is simply the contractual payoff of the option. The condition at $S = 0$ reflects that if the stock price falls to zero (and stays there, as geometric Brownian motion implies), the call option becomes worthless since there's no chance of the stock exceeding the strike. The condition for large $S$ captures the intuition that when the stock price is very high, the option is almost certain to be exercised, and its value approaches that of a forward contract with delivery price $K$.

For a European put option with strike $K$ and maturity $T$:

$$\begin{aligned} V(S, T) &= \max(K - S, 0) && \text{(Terminal condition)} \\ V(0, t) &= Ke^{-r(T-t)} && \text{(Worth discounted strike)} \\ V(S, t) &\to 0 && \text{(Put becomes worthless)} \end{aligned}$$

where:

• $V(S, t)$: value of the put option
• $S$: current stock price
• $K$: strike price
• $T$: maturity time
• $t$: current time
• $r$: risk-free interest rate

The terminal condition for the put is the mirror image of the call: the put pays off when the stock price is below the strike. When $S = 0$, the put is certain to pay $K$ at maturity, so its current value is the present value of that payment, which equals $Ke^{-r(T-t)}$. When $S$ is very large, the stock price is almost certain to remain above the strike, making the put worthless.

The combination of the PDE and these boundary conditions completely determines the option price. Together they form a well-posed mathematical problem: the PDE constrains how the solution must behave in the interior of the domain, while the boundary conditions pin down the solution at the edges. In the next chapter, we'll see how to solve this problem analytically to obtain the famous Black-Scholes formula.