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Interpolating Five-Point FX Volatility Smiles

Article Quant Q&A · Author: d--b
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Summary

The document discusses how to interpolate an FX volatility smile quoted at five delta points: 10-delta and 25-delta puts, at-the-money, and 25-delta and 10-delta calls. The question arises because basic vanna-volga uses three points and may not represent both the 10-delta and 25-delta wings consistently. One response recommends consulting established FX volatility references for methods that can reprice the 10- and 25-delta quotes while maintaining consistency with butterfly and risk-reversal quotes.

For quick trader calculations, a cubic spline in delta space is suggested; the reply notes that quantitative implementations may use more sophisticated techniques. Another answer cites an interpolation method developed for equity options and argues it may also suit FX, with potential approximations for variance- and volatility-swap strikes using a small number of pillar options. These are pointers rather than a tested comparison: the exchange gives no calibration details, validation results, or single agreed industry standard. Any choice therefore needs to be checked for smile fit and consistency with the desk’s quoting conventions.

Key ideas

  • A five-point FX smile includes 10-delta and 25-delta put and call quotes plus the at-the-money quote.
  • Three-point vanna-volga may not represent both 10-delta and 25-delta wings consistently.
  • A cubic spline in delta space is suggested for approximate trader calculations.
  • More sophisticated methods can aim to reprice both delta tenors consistently with butterfly and risk-reversal quotes.
  • The discussion provides candidate references and methods, but establishes no universal industry standard.

Tags

ForexOptionsVolatilityDerivatives pricing
Full text
# Industry standard for interpolating FX volatilities


# Industry standard for interpolating FX volatilities












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?

## Answer by BrownianBread (score 4)

https://quant.stackexchange.com/a/42068

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.

## Answer by cosplay-raven (score 1)

https://quant.stackexchange.com/a/41591

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

## Answer by user34971 (score 0)

https://quant.stackexchange.com/a/43356

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.