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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!

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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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