July 7th, 2017, 1:19 am
Well, it's your Monte Carlo and your FDM. The most benign choice to me is [$]\sigma(S,t)[$] continuous and bounded -- implying (among the choices you presented) piecewise linear S-interpolation with constant extrapolation for 0 < S < Smin and Smax < S < infinity. Here Smin and Smax are the min and max strikes of the data you posted. This will mean [$]\sigma(S,t)[$] is continuous and bounded at least to 6/2/2017. Then, make sure exactly the same continuous and bounded [$]\sigma(S,t)[$] is used for both the PDE and the Monte Carlo. (I suspect you aren't). With that, I believe both procedures (if decently implemented) will lead to the same option prices through 6/2/2017, which was your concern.
I would also begin to use piecewise linear interpolation in time starting with 6/2/2017 and later. Then, [$]\sigma(S,t)[$] will be continuous and bounded for your whole local vol surface. With that, again I believe both procedures will lead to the same option prices for all strikes, through the furthest expiration of your data.
Finally, if you want to consider t > tmax, where tmax is the furthest expiration of your data, I would just use constant extrapolation in time. Again this will assure [$]\sigma(S,t)[$] will be continuous and bounded -- now for [$](S,t) \in (0,\infty) \times (0,\infty)[$].