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I am attempting to implement a local vol pricing model in finite difference for equity index options.

I have followed Gatheral's Lectures and fitted an SVI Model bringing me to the following local vol equation with VL being relatively straight forward to compute using a grid.

enter image description here

My problem currently is twofold

  1. y = log(K/FT) but I am using the normal (I think?) finite difference grid with log(K) transformation so the grids don't match up. What is typically done in practice? Do I have to interpolate w & y for each point in my finite difference grid?
  2. Cost of carry is not constant across tenors as is typically the case. I have not been able to find any reference for finite difference where r & q are not assumed to be constant. I am assuming this is important given that the vol surface is forward dependent? Do I simply use piece-wise constant forward rates in between tenors?

Can someone explain how this is usually implemented or point me towards a practical reference source?

***** update *****

After working on this a bit more. I took a different transformation

$$\ X=\ln(\frac{S}{S_0 e^{(r-q)T}}) $$

giving me

$$\ 0=rC+\frac{∂C}{∂T}-\frac{1}{2} σ^2 \frac{∂C}{∂X}+\frac{1}{2} σ^2 \frac{∂^2 C}{∂X^2} $$

which seems to be the right approach for 1. Both VL and finite difference grids are now consistent and in the forward space. I also noticed that the cost of carry term is gone. I am thinking this means that I no longer have to worry about about non constant r & q except for path dependent derivatives (in which case it should be dealt with in the boundary condition?)

I am hoping someone with more experience in the matter can validate this approach

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