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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 $s, t$.

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Answer is very straightforward: always center your grid on the current spot value and make sure that it covers sufficiently many standard deviations on each side to cover enough of the terminal distribution at maturity. In particular it should cover the forward at maturity.

NB:

  1. By definition when you use a FDM you will get back the grid of all values of the function at the present time. This differs from say Monte-Carlo types of methods where all the random paths would start at the spot value.
  2. Also having the values of V(S,0) for various S allows to calculate delta/gamma "on the grid". Be careful though that this may not be the total delta if say you have a volatility model that depends on spot itself.
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