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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!

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Please see below:

enter image description here

Copied from Quantitative Methods in Derivative Pricing by Tavella.

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