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My understanding is you can calibrate with atm vol which should be close with either model then the otm vols are taken care of by the beta parameter from sabr


Yes, multivariate GARCH is what you should consider. Imho, you can look at a book, analysis of financial time series, by Ruey S. Tsay, in chapter 10, they discussed multivariate volatility models. You can google this book, download the pdf of second edition, hope it helps.

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