Finite Difference Methods for Option Pricing › Limitations and Impact

Finite difference methods transformed quantitative finance by providing a general-purpose framework for solving the pricing PDEs that arise from no-arbitrage arguments. Before these methods became practical, traders relied heavily on analytical formulas that exist only for simple payoffs, or on binomial trees that converge slowly. The development of stable, accurate finite difference schemes enabled rapid and reliable pricing of American options, barrier options, and other derivatives where analytical solutions don't exist.

However, finite difference methods face fundamental limitations. The most significant is the "curse of dimensionality": computational cost grows exponentially with the number of underlying assets. A one-dimensional grid with 100 points becomes a two-dimensional grid with 10,000 points, a three-dimensional grid with 1,000,000 points, and so on. Beyond two or three dimensions, Monte Carlo methods typically dominate despite their slower convergence rate.

Boundary conditions also require care. The assumption that $V(S_{\max}, t) \approx S_{\max} - Ke^{-r(T-t)}$ for deep in-the-money calls is an approximation that can introduce errors if $S_{\max}$ is not chosen sufficiently large. Similarly, for options with complex features like barriers or early exercise, so boundary conditions accurately reflect the true option behavior requires domain expertise.

Numerical stability, while guaranteed for implicit and Crank-Nicolson schemes, doesn't guarantee accuracy. Coarse grids can produce stable but inaccurate solutions. You must balance computational cost against required precision, often through convergence studies like those presented above.

Despite these limitations, finite difference methods remain essential tools in quantitative finance. They provide deterministic, reproducible prices without the statistical noise of Monte Carlo simulation. For low-dimensional problems, they are typically faster and more accurate than alternatives. And they offer a direct numerical solution to the fundamental pricing PDE, and gives intuition about how option values evolve across price and time.