Finite Difference Methods for Option Pricing › Boundary Conditions

Proper boundary conditions are essential for finite difference methods. They determine the option value at the edges of the computational grid, where the standard update formulas cannot be applied directly because they would require values outside the grid.

At expiration ($t = T$), the option value equals its payoff. This is the fundamental condition that initializes our backward-in-time computation:

$$\begin{aligned} V(S, T) &= \max(S - K, 0) && \text{(call)} \\ V(S, T) &= \max(K - S, 0) && \text{(put)} \end{aligned}$$

where:

• $V(S,T)$: option value at expiration
• $S$: stock price at expiration
• $K$: strike price

This condition initializes the grid before we begin the backward time-stepping process. The non-smooth nature of the payoff function at $S = K$ can introduce complications, as the second derivative is undefined at that point. You might sometimes smooth the payoff near the strike to improve numerical behavior.

When the underlying asset price is zero, it remains at zero forever (assuming geometric Brownian motion). This follows because geometric Brownian motion has the property that zero is an absorbing state: if $dS = \mu S dt + \sigma S dW$, then $S = 0$ implies $dS = 0$. Therefore:

• Call option:

  $$V(0, t) = 0$$

  where:

  • $V(0, t)$: option value at asset price zero

  (a call is worthless if the stock is worthless)

• Put option:

  $$V(0, t) = Ke^{-r(T-t)}$$

  where:

  • $V(0, t)$: option value at asset price zero
  • $K$: strike price
  • $r$: risk-free interest rate
  • $T-t$: time to expiration

  (a put pays $K$ at expiration with certainty)

At very high stock prices, we need to specify the option value at the upper edge of the grid. The exact value depends on the type of option:

• Call option:

  $$V(S_{\max}, t) \approx S_{\max} - Ke^{-r(T-t)}$$

  where:

  • $V(S_{\max}, t)$: option value at maximum grid price
  • $S_{\max}$: maximum asset price on the grid
  • $K$: strike price
  • $r$: risk-free interest rate
  • $T-t$: time to expiration

  (deep in-the-money, behaves like a forward)

• Put option:

  $$V(S_{\max}, t) \approx 0$$

  where:

  • $V(S_{\max}, t)$: option value at maximum grid price

  (deep out-of-the-money)

The boundary at $S_{\max}$ requires some care. If $S_{\max}$ is chosen too small, the boundary condition may distort the solution in the region of interest. A rule of thumb is to set $S_{\max} \geq 3K$ to $5K$ for typical volatilities.