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I am trying to get a grip of the current state of research regarding option pricing in the SABR model. Am I correct in that, so far, there is no known general formula for the option price in the SABR model? If there is, I would be very happy if someone could point out the any relevant reference.

I, of course, know about P. Hagans approximative formulas for the implied Black volatility. But as soon as the parameters are a bit more extreme, the approximation turns pretty bad.

I know that there are some formulas involving integrals for the zero correlation case and that this has been used to try to approximate the general case. But also in this case, some assumptions are made about the parameters.

Since the Heston model, by now, has very practical explicit integral formulas for its option prices for which there are very good numerical methods to calculate, I am a bit surprised that it does not seem to exist anything similar for the SABR model. Especially since those models on some deep level (as P.Hagan showed in a paper) are related.

The more practical problem I am trying to solve in a satisfactory way is the construction of a volatility surface based on the SABR model. By doing that a calibration to market option prices is needed.
If the Hagan approximate formulas are used, the calibration becomes pretty nonsensical at more extreme values, since we then calibrate to the Hagan approximation rather than the actual SABR model which would be quite different. If there are no good formulas for option pricing, how is this calibration typically done?

Edit: I found this reference regarding the calibration problem https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2467231
Seems good. Still interested in if there are any explicit formulas for the option prices though.
Also, after reading a bit in the article, I see they express the implied volatility as a series expansion in the time $T$-variable, with an error of $O(T^2)$. Why would this be a good idea for general $T$ when $T$ is not small? It seems we have no control of the error whatsoever then?

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