Why Hull–White Two-Factor Models Need Extensions for Volatility Skew

Summary

The document asks whether a two-factor Hull–White short-rate model, also called G2++, can represent swaption volatility smiles or skews, and whether fitting it to out-of-the-money swaptions would help. The answer points to local- and stochastic-volatility extensions of Hull–White developed in the 1990s, identifying the Cheyette model as a one-factor extension and Multi-Factor Cheyette as its multifactor counterpart.

It recommends Andersen and Piterbarg’s interest-rate modeling volumes for a fuller account and mentions a 2005 risk paper describing practical implementation and calibration with stochastic volatility. The text does not explain the model equations, calibration procedure, or the consequences of fitting the unextended Hull–White model to out-of-the-money swaptions. It therefore offers references and model names rather than a worked comparison or empirical evidence.

Key ideas

• Local- and stochastic-volatility extensions of Hull–White were developed to model features such as volatility skew.
• The Cheyette model is identified as a one-factor extension, with Multi-Factor Cheyette as its multifactor form.
• The response points readers to reference material and a practical stochastic-volatility implementation.
• The document does not explain calibration mechanics or demonstrate that fitting Hull–White to out-of-the-money swaptions captures skew.

Tags

# Why Hull White 2 Factor model can't capture vol skew?


# Why Hull White 2 Factor model can't capture vol skew?












Is there a way to stay with the short rate model (like HW2F or G2++) but extend it to capture vol term structure (vol smile or skew). What happens if I calibrate HW2F to OTM swaptions? (I don't want to use calibration to caps as they almost always ignore correlation). Thanks!

## Answer by Antoine Savine (score 4)

https://quant.stackexchange.com/a/43382

Local and/or stochastic vol extensions of HW (incl. multi-factor) were produced around the mid 1990s, more or less independently in a number of research papers, the most notable being Cheyette (1992) and Ritchken-Sankarasubramanian (1995). Quants generally call the one-factor extension "Cheyette Model" and the multi-factor version "Multi-Factor Cheyette" or MFC.

You may see the whole story in Andersen and Piterbarg 3-volume "bible" on interest rate models. Andreasen published a risk paper in 2005, with a practical implementation and calibration with stochastic volatility. I also have some professional training notes here: https://www.slideshare.net/AntoineSavine/introduction-to-interest-rate-models-117627629. I hope this helps.

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