Finite differences in options pricing
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...
