# W-shaped Event Vol and Butterfly Arbitrage

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

• Hint: $C(K) = C^{BS}(K,IV(K))$. Differentiate both sides twice wrt to $K$ and then find the equivalent condition on IVs for no butterfly arbitrage.
– user34971
Oct 8, 2022 at 10:52
• @Frido, Thanks. I found the answer in this paper: arxiv.org/pdf/1204.0646.pdf Oct 8, 2022 at 14:26
• Yes, so in that paper equation 2.1 is I think what you're looking for, i.e $g(k) \geq 0$.
– user34971
Oct 8, 2022 at 16:28

## 2 Answers

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}

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