Equivalent BS volatility formula under the Heston model?

Question

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.

Answer 1

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:

• For the extreme strike you have the results by Friz, Gerhold, Gulisashvili and myself, https://arxiv.org/abs/1001.3003, see also the book by Archil Gulisasvili, https://link.springer.com/book/10.1007/978-3-642-31214-4

• For the small time regime you have the results by Forde and Jacquier, https://nms.kcl.ac.uk/martin.forde/HestonSmallTime.pdf. That said, to the best of my knowledge Heston has all diffusion based models an issue with the short end of the vola surface. Besides adding jumps, there is a lot recent work on rough volatility models, also rough Heston, to address these issues. I am not very familiar with the recent literature on these topic.

• Finally, in the large time regime Gatheral and Jacquier have proved that the implied volatility converges to the SVI parametrization of the vol-surface, https://arxiv.org/abs/1002.3633

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.