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 $s, t$.