# How are FDE's implemented when one wants one particular price?

Say I want to price a particular call option in the Black Scholes model using finite difference methods.

The value process of this option $$V(s, t)$$ satisfies a PDE. I can use finite difference methods to determine $$V(s, t)$$ for $$s, t$$ lying in some discrete grid.

Working backwards in time, I can get to $$V(s, 0)$$ for all $$s$$ in that grid. This is my price, for all initial spots, as long as those spots are on the grid.

But my original goal was to just price 1 call option, and I already know the initial spot. So really I only want $$V(s', 0)$$ for some fixed $$s'$$.

My question is, how should I then implement my FDM scheme?

For example, should I "manually" make sure that my $$s'$$ will lie on that grid, hence avoiding the need to interpolate from the grid?

What other things ought I do?

TLDR: Basically my question is what changes should I make to a FDM-scheme when I want one particular price $$V(s', 0)$$ and am not really interested in the general FUNCTION $$V(s, t)$$ for all kind of $$s, t$$.