Grid points are placed over the range where the equation is to be solved, and each derivative is replaced with a difference between neighboring points1. The result is a system of equations in the grid values, or a formula that gives the values at the next time step from those at the previous one. There are three ways to step through time: the explicit method, which finds the next values from the previous step alone; the implicit method, which solves a system of equations; and the Crank-Nicolson method, which averages the two.
Consider a hypothetical example of a call with underlying price 100, strike price 100, one year to expiry, interest rate 1%, and volatility 20%. Solved with the Crank-Nicolson method on a log-price grid with spacing 0.02 and 200 time steps, it is worth 8.4235. That is 0.0098 below the Black-Scholes formula value of 8.4333, and the gap shrinks as the grid is refined.
A binomial tree is also a backward calculation on a grid. For a diffusion equation without discounting or drift, the explicit method has, under the stability condition, the same form as a calculation walking a trinomial tree backward1. The Monte Carlo method differs in approximating the price by an average over paths, rather than solving the equation on a grid.