W-shaped Event Vol and Butterfly Arbitrage

Question

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 $\partial \sigma^2 / \partial K^2$.

Answer 1

Given the call option price $C$ as a function of strike $K$ and (strike-)implied volatility $\sigma(K)$, we have $C(K,\sigma(K))$. No-arbitrage requires the total derivative of the call option price w.r.t. the strike to be $\geq 0$, i.e.:

$$ \begin{align} \frac{\mathrm{d}^2C}{\mathrm{d}K^2}&\geq 0\\ \Rightarrow \quad\quad 0&\leq\frac{\partial^2C}{\partial K^2}+2\frac{\partial^2C}{\partial K\partial\sigma }\frac{\partial \sigma}{\partial K}+\frac{\partial^2C}{\partial\sigma^2 }\left(\frac{\partial \sigma}{\partial K}\right)^2+\frac{\partial C}{\partial\sigma }\frac{\partial ^2\sigma}{\partial K^2} \end{align} $$

Answer 2

Local volatility (LV) must be positive. The expression for LV in vol space can be found here on page 10.