Finite difference schemes are very much similar to trinomial tree options pricing, where each node is dependent on three other nodes with an up movement, a down movement, and a flat movement. The motivation behind the finite differencing is the application of the Black-Scholes Partial Differential Equation (PDE) framework (involving functions and their partial derivatives) whose price is a function of , with as the risk-free rate, as the time to maturity, and as the volatility of the underlying security:

The finite difference technique tends to converge faster than lattices and approximates complex exotic options very well.

To solve a PDE by finite differences working backward in time, a discrete-time grid of size by is set up to reflect asset prices over a course of time, such that and take on the following values at each point on the grid:

It follows that by grid notation, . is a suitably large asset price that cannot be reached by the maturity...