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https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2731359

http://janroman.dhis.org/finance/SABR/ZABR%20Andreasen.pdf

In the two articles listed above we see several ways to extend the original SABR (Hagan 2002) model and apply numerical schemes to solve it. Both articles mentions low/negative rates as the reason for why these models are useful.

How I interpret use of these methods: When the underlying product can be negative then it a good idea too apply these models rather than the original SABR.

My question: Is there any advantage in using these extended and more complicated models for options where the price of the underlying cannot be negative? For instance FX and equity options.

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You don't want to use the SABR (or an extension) to price equity options or FX options. The lag of mean-reversion in the model's volatility dynamics leads to explosive behavior and to a implied distribution that is absolutely not in line with empirics -- especially on longer time horizons.

To my knowledge people use it mostly for interest rate derivatives. This is stated by in the linked paper of Jesper Andreasen and Brian Huge as well. The SABR is quite simple since it only relies on 4 Paremeters and it has a quasi-closed form (approximation) formula for European options. So it is numerical quite tractable. One can extend the SABR to a full market model to price different kind of products, like caps, swaps and CMS spreads in one consistent setting. (often more than 50 contracts in total) Here the simplicity of the SABR really counts, because it leads to market models that can relatively easy and fast calibrated.

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