I am looking into a new method of calculating sensitivities starting off with a proof of concept with Black Scholes PDE. Suppose I want to calculate Rho and take the derivative of the PDE (heresy!!) and end up with a new PDE wrt to the interest rate.
I wish to approximate the above PDE using Crank Nicolson. However I would like to know whether it is possible to create a central difference approximation of:
Any ideas would be great. Thanks!