Simple question that I was wondering about over during the weekend. I have done a little FEM during the last years and my university time and did not spend a lot of time with FDM. For a new job I have started brushing up my numerical skillset a bit again and read up on FDM a bit more in detail.
For derivatives pricing, is there actually a situation in which you explicitly choose FEM over FDM? I would choose FEM for a problem in which I
- either have complicated boundaries or
- am more more concerned with a weak solution of a PDE or I
- can be really smart about the mass- and stiffness matrix
Neither of those are necessarily concerns in QF, right? I mean I can also for high-dimensional problems pretty much always use at least some OS method and do dimensional splitting with an approach like Hundsdorfer-Verwer, with which I essentially have a sparse multiplication and a few tridiagonal matrix systems to solve - I therefore pretty much always land at (or close to) a $O(N)$ complexity, with a high constant maybe but that's already about as good as I would have got with MG methods and so on. That seems to be the current state of the art for anything from (low-dim) basket-options to american options to SV models or SI models and so on. Even if you check "slightly" nonlinear problems like UVM and lending rates, you have policy iterations with, again, stepwise FDM solvers.
I know that there are books about FEM in finance but what I am missing a bit is the reason - most publications go about it as "look, that also works". I wouldn't really care about higher complexity (after all, we don't need to program in Excel anymore...), just about superior solvers. So my question would be:
What type of problems favor FEM over FDM in quantitative finance? Is FDM here just the objectively better, faster and more versatile ansatz in this field?