Higher-order finite difference scheme for the Black-Scholes equation

[Brian Park]

f''(x) = [(-1/12)f(x-2h) + (4/3)f(x-h) - (5/2)f(x) + (4/3)f(x+h) - (1/12)f(x+2h)]*(1/h^2) + O(h^4)

f''(x) = [(5/6)f(x-h) - (5/4)f(x) - (1/3)f(x+h) + (7/6)f(x+2h) - (1/2)f(x+3h) + (1/12)f(x+4h)]*(1/h^2) + O(h^4)

Using these formulas, create a finite-difference scheme for the Black-Scholes equation with error O(delta_x^4)+O(delta_t^2). This scheme should be similar to the Crank-Nicolson scheme (unconditionally stable).

• Solve the Black-Scholes equation and compute the price of call option.
• Check that the error is of the order O(delta_s^4) + O(delta_t^2).
• Compare efficiency with Crank-Nicolson and explicit scheme.

I was given this project in an undergraduate course and need to implement the scheme in Matlab. Sorry for making a mess with the formulas (I don't know how to use LaTex). Could someone guide me to an article or a book that deals with this subject? Thank you.

 

[Daniel Duffy]

These schemes you write lead to banded matrices, so complex at the expence of higher-order accuracy.


Maybe compact FDM?

http://deepblue.lib.umich.edu/bitstream/2027.42/56138/1/20200_ftp.pdf