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I'm looking to replace the FX vol interpolation scheme at my firm, and was wondering what the industry standard was.

We used to do vanna-volga, but it only takes 3 points (25dp, atm, 25dc), and so doesn't fit well on 10dp, 10dc. We could do a vanna-volga on (10dp, atm, 10dc), and somewhat mix the 25d and 10d smiles together, but that doesn't sound great.

I'm assuming there is some kind of industry standard to interpolate a 5-point smile (10dp, 25dp, atm, 25dc and 10dc) properly. Probably some kind of curve fitting in delta space? Any idea?

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Take a look at the following paper by Wystup:

https://mathfinance.com/wp-content/uploads/2017/06/CPQF_Arbeits20_neu2.pdf

For a more modern take, look at the book by Ian Clark on FX Options. There are several interpolation schemes you can use to consistently reprice both 10 and 25 delta quotes such that the surface is consistent with both BF and RR.

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For traders' "back of the envelope" calculations, it's a cubic spline in delta space. Quants will do something more sophisticated, however.

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Here's a new volatility interpolation method by Rolloos that performs well for equity options, and so should perform even better for FX options (less negative correlation between index and volatility):

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3265046

What I like most about his paper/method is that not only does it give an interpolation method, but also gives you an excellent/practically exact approximation of the variance swap strike, and a good to very good approximation of the volatility swap strike. All this using only 3 pillar options! So can be used in illiquid markets as well.

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