Finite Difference Methods for Option Pricing › Summary

This chapter showed how to use finite difference methods to solve the Black-Scholes PDE. These methods are a powerful alternative to binomial trees and Monte Carlo simulation.

The key concepts and takeaways are:

• Grid discretization turns the continuous PDE into a system of algebraic equations by approximating derivatives with finite differences on a mesh of price and time points

• The explicit scheme is conceptually simple, directly calculating option values at the current time step from values at the next step, but suffers from severe stability constraints requiring very small time steps

• The implicit scheme evaluates spatial derivatives at the current time step, leading to a tridiagonal linear system that must be solved at each step, but gains unconditional stability allowing much larger time steps

• The Crank-Nicolson scheme averages explicit and implicit approaches, achieving second-order accuracy in time while maintaining unconditional stability, making it the preferred choice for most applications

• Boundary conditions at $S = 0$, $S = S_{\max}$, and at expiration must be specified carefully to ensure accurate solutions; for calls, $V(0,t) = 0$ and $V(S_{\max}, t) \approx S_{\max} - Ke^{-r(T-t)}$

• American options require checking the early exercise constraint after each time step, setting $V_j^n = \max(V_j^n, \text{payoff}(S_j))$ to ensure the option value never falls below intrinsic value

• Convergence properties differ across methods: explicit and implicit schemes have first-order time accuracy while Crank-Nicolson achieves second-order accuracy

• Method selection depends on the problem: finite difference methods excel for low-dimensional problems with early exercise or barriers, while Monte Carlo dominates for high-dimensional or path-dependent options