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Is there an equivalent BS volatility formula for the Heston model, something like Hagan's formula for the SABR model? Of course, such a formula will be an approximation as in Hagan's formula.

Under the Heston model, we can price European options with the inverse Fourier transform (or FFT) quite precisely. So it's possible to numerically invert the price to the BS volatility. Nevertheless, an analytic volatility formula (albeit approximation) will be still helpful for many occasions. For example, FFT method seems unstable for extreme inputs (deep out-of-the-money or short time-to-maturity). See this question.

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Answering the question in a way coming from the motivation at the back: There is not one universal approximation, but different ones depending on which asymptotics you want to look into. Just to state a few:

I am happy to admit that I have not worked on related ideas for several years there might be well newer and better results out, I hope others can fill in the gaps.

PS. While quite early in the asymptotics literature, I feel it is always appropriate to point to point to the 2010 Zeliade Heston White Paper by Jacquier and Martini, I think it really helps to understand the intricacies of Heston asymptotics besides any specific expansion, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1769744.

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  • $\begingroup$ thanks so much for the references! $\endgroup$
    – jChoi
    Jun 6 at 1:43

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