I’ve read the fantastic paper by Piterbarg on long dated FX options here: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=685084. One of the limitations of the paper is that the model mainly captures level and skew of the volatility surface but cannot always fit the smile.
What extensions can be made to make this model fit the smile from a practitioners perspective? Are there any other parametrisations of the local volatility that might be useful such as SABR/SVI? I do not want stochastic volatility dynamics as I want PDE pricing, so I’d like to stick to 3 factors. Thanks