I came across the Vola Dynamics page about the W-shaped vol before an event: https://voladynamics.com/marketEquityUS_AMZN.html
I'm a bit confused by "this term does not have any butterfly arbitrage". I thought butterfly arbitrage suggests that price against strike is convex, i.e., $\partial^2 C/\partial K^2 > 0$. But the W-shape around the forward is clearly not convex.
I guess it may be because the y-axis is not price but vol, but then I thought roughly as vol is higher the price is higher too.
Any formula to check the butterfly arbitrage in the vol space? I mean, with some rule to check including for example $\partial \sigma^2 / \partial K^2$.