How finite difference methods price derivatives

Explain the idea behind finite difference methods for pricing derivatives.

Discretise the pricing PDE on a grid in stock price and time and step backward from the payoff. Derivatives are replaced by difference quotients, the terminal condition is the payoff and boundary conditions are set at the edges of the price range. Explicit schemes are simple but conditionally stable; implicit and Crank-Nicolson schemes solve a tridiagonal system per step and are unconditionally stable, Crank-Nicolson being second-order in time. American features are handled by taking the max with exercise value at each step.