Grid methods for pricing equations

Describe the finite-difference method as used in derivative pricing.

It solves the pricing PDE numerically by replacing derivatives with differences on a grid of asset price and time. Starting from the payoff at expiry, the scheme marches backward to today, applying boundary conditions at extreme prices. Explicit, implicit and Crank-Nicolson schemes trade simplicity against stability and accuracy. It handles early exercise and barriers naturally and gives greeks from the grid, but becomes expensive beyond about three state variables.