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After implementing some FDM to price some option, there are gaps between our grid points that may be of interest. From reading around, it appears common to use bilinear interpolation to estimate these gap values. However, there are more accurate interpolation schemes available. Are more sophisticated forms of interpolation, like bicubic interpolation, commonly used in industry or wherever, and if so which ones? Or is bilinear, or even nearest-neighbour, generally sufficient?

I understand that when deciding on different numerical methods to tackle certain problems we must meet our required accuracy and computing budget. I'm mainly asking because when using implicit methods our grid can be courser, and while our solved grid points are more accurate and the method is unconditionally stable, we still lose some "information" when we need to estimate gap values.

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