Under what circumstances can implied volatility smile be concave (ATM implied volatility higher than OTM put and call)? I know that a slight concavity is not prohibited by no-arbitrage... What are some real-life examples for that?
You can see concavity in mean-reverting underlying assets where the option tenor is comparable to the characteristic reversion time of the asset. For a geometric brownian motion, all underlying prices are possible, so any mean reversion or other limitation on large changes that might occur in reality would ultimately appear as a skinny tail and negative curvature.
A good example of real-life negative curvature is options on VIX futures. I include a chart from Bloomberg data below, with the caveats that
- Bloomberg is terrible at implying vols for VIX, so one can only see this in their put vols, and
- Somehow I typed 3014 in the title
Properly implied VIX vols do have negative curvatures as seen here, and were especially pronounced in it during 2008 and 2009 when longer tenors were available. Typically they also have wrong-way skew, which somewhat obscures the negative curvature effects. There are some nice charts, properly done, on page 23 of this paper by Jim Gatheral.
Negative excess kurtosis leads to a concave vol smile.
By the way, no-arbitrage arguments are of theoretical nature: implied volatilities can exhibit no-arbitrate violations in the theoretical sense for extended periods given that such arbitrate cannot be traded due to other factors, such as liquidity, spreads, transaction related costs...not saying this happens often but it does at times.
In addition to the presented answers, I just wanted to mention that such a situation is described in Hull, page 419 (Chapter 19 Volatility Smiles, 19.8: "When a single large jump is anticipated"). This happens when probability distribution of returns is binomial. It can occur in a situation when market is expecting some announcement which will either significantly lift the asset price or drop it.