Butterfly Arbitrage Constraints for Implied Volatility Slices

Summary

The document presents a condition used to assess whether an implied volatility slice is free of butterfly arbitrage. It defines a function of log-moneyness using total implied variance and its first and second derivatives. The stated criterion requires that this function remain nonnegative across the domain, alongside a limiting condition on the right-wing behavior of the call-price expression. The question asks whether a trading strategy can expose arbitrage when that limiting condition fails.

The document explains that the limit condition is associated with call prices tending to zero at extreme positive moneyness, and suggests that this may be needed for a valid probability density. However, it does not provide an answer to the arbitrage-strategy question, a construction of trades, or evidence from market data. The criterion is presented as a mathematical test attributed to prior research, so practical use requires careful attention to assumptions, definitions, and whether the volatility slice satisfies the conditions needed for the pricing model.

Key ideas

• The stated butterfly-arbitrage test uses a function built from implied variance and its first two derivatives.
• The function must be nonnegative throughout the log-moneyness domain under the stated criterion.
• The document also requires a right-tail condition connected to call prices vanishing at extreme moneyness.
• It raises, but does not resolve, how a failed tail condition could be turned into an arbitrage trade.

Tags

# Butterfly Arbitrage condition


# Butterfly Arbitrage condition












I hope anybody can help me. According to Gatheral and Jacquier (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2033323) no Butterfly Arbitrage can be expressed like this:

Define the function $\displaystyle g(k):= \left( 1- \frac{k \omega'(k)}{2 \omega(k)}\right)^2- \frac{\omega'(k)}{4} \left(\frac{1}{\omega(k)}+\frac{1}{4} \right) + \frac{\omega''(k)}{2}$

A slice is free of Butterfly Arbitrage $\Leftrightarrow$ \ the function $g(k) \geq 0 \ \forall k \in \mathbb{R}$ and $ \lim \limits_{k \rightarrow \infty} d_+(k)=-\infty$. The second condition here is equivalent to call prices converging to 0 as $k \to \infty$ (if I saw this right this is needed to have a density). Now I'm asking is there an arbitrage strategy to "see" the arbitrage if this point isn't fullfilled?

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