Testing Butterfly Arbitrage from an Implied Volatility Smile

Summary

The note addresses why a W-shaped implied-volatility curve does not by itself determine whether options have butterfly arbitrage. The relevant no-arbitrage condition is convexity of the call price with respect to strike, not convexity of implied volatility as a plotted quantity. Because the call price depends on both strike and the strike-specific implied volatility, its total second derivative includes the direct strike curvature, mixed strike-volatility effects, volatility curvature, and a term involving the option’s volatility sensitivity.

The displayed chain-rule expression gives a way to test a volatility surface by mapping it into call prices and checking the resulting strike convexity. It does not provide a numerical evaluation for the cited event surface, nor does it give a standalone rule based only on the second derivative of implied variance. A brief additional answer points to positivity of local volatility and a volatility-space expression elsewhere, but the expression itself is not reproduced. Thus the main practical lesson is to assess price convexity after accounting for the full implied-volatility dependence.

Key ideas

• Butterfly arbitrage is tested through convexity of call prices across strike.
• A strike-dependent implied volatility makes the call’s second strike derivative a total derivative.
• The convexity condition includes strike, mixed strike-volatility, volatility curvature, and vega-related terms.
• A W-shaped volatility curve alone is insufficient to establish butterfly arbitrage.

Tags

# W-shaped Event Vol and Butterfly Arbitrage


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

## Answer by Kermittfrog (score 2, accepted)

https://quant.stackexchange.com/a/73447

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 by Misha Fomytskyi (score 0)

https://quant.stackexchange.com/a/76168

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.