[scott00]

This method will work but will require a large grid and consequently be quite slow. And order of magnitude or two faster than this is possible if you are clever.

[quanto]

what's a faster method than a Crank Nicolson implementation discussed in the post?

[fakesson]

"High Performance American Option Pricing" by Leif Andersen et al is many orders of magnitude faster than any finite difference method or other PDE / tree method. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2547027

Given the exercise boundary, the American Option Price can be written exactly as a one-dimensional integral. That is the key insight to this superior method.

[scott00]

Crank-Nicolson is probably the least objectionable part of the method, but I prefer ADE.

There are two numerically painful parts of the problem: the advection term and the oscillation inducing terminal condition (because it has a discontiuous derivative). I like to deal with advection by transforming the equation to an advection free equation. I'm under NDA on the best solution to the oscillatory terminal condition so I can't give that one away unfortunately.

[logicchains]

You can train a small NN on a bunch of synthetic data and the inference will be pretty fast.

[gituliar]

Indeed, Andersen-Lake method is much faster, but is limited to vanilla problems.

The finite-difference covers wider range of problems, including stochastic volatility models, like SABR or Heston.