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Adding a Smile to a Three-Factor FX Hybrid Model

Article Quant Q&A · Author: BrownianBread
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Summary

The document asks how practitioners might extend a long-dated foreign exchange option hybrid model that captures volatility level and skew but may not reproduce the full smile. The questioner wants to retain a three-factor setup, avoid stochastic volatility, and use PDE pricing, while considering local-volatility parameterizations such as SABR or SVI.

The response suggests using local volatility for FX together with stochastic interest rates, noting that PDE speed is not viewed as a constraint in that setup. It does not specify a calibration procedure, define the factors, or show how the proposed model fits market smiles. As a result, this is a brief direction for model design rather than a worked extension or a comparison of parameterizations; the trade-offs in calibration stability and smile dynamics remain open.

Key ideas

  • The question concerns fitting the volatility smile in a long-dated FX option hybrid model.
  • The proposed direction combines local volatility in FX with stochastic interest rates.
  • PDE pricing is presented as compatible with that approach when computation speed is manageable.
  • The response does not provide calibration details or establish how well the proposal fits the smile.

Tags

ForexOptionsVolatilityDerivatives pricing
Full text
# FX Hybrid Model with Smile


# FX Hybrid Model with Smile












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

## Answer by Lech (score 1)

https://quant.stackexchange.com/a/66505

Why not handle it via Local Vol FX with stochastic interest rates? If you wish to stick to PDEs then the speed is not an issue anyway, right?

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