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Why Butterfly Constraints Also Rule Out Vertical Spread Arbitrage

Article Quant Q&A · Author: alexfrag
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Summary

The document asks why static arbitrage checks for an eSSVI volatility parametrization focus on butterfly and calendar spreads, and whether vertical spreads require a separate condition. Its accepted answer says the relevant limit for a vertical bull spread is that its price must be positive.

That positivity follows from the butterfly condition: the vertical spread can be represented as a sum of butterflies positioned above the lower strike, and the butterfly constraint requires those prices to be positive. The brief exchange therefore explains why a separate vertical spread test is not needed in the stated framework. It provides no derivation or broader treatment of all arbitrage conditions, so the claim should be read within the assumptions and setup of the eSSVI question.

Key ideas

  • The answer identifies positive price as the key constraint for a vertical bull spread.
  • A vertical spread can be represented as a sum of butterflies above its lower strike.
  • Butterfly price positivity therefore implies the stated vertical spread condition.
  • The discussion is brief and does not derive the full eSSVI arbitrage conditions.

Tags

OptionsVolatilityDerivatives pricingArbitrage
Full text
# Volatility surface static arbitrage and SVI


# Volatility surface static arbitrage and SVI












I have a very simple question in regards to vol parametrization using SVI and static arbitrage definition.

Why we consider only butterfly spreads(convexity of option prices with respect to strikes) and calendar spread arbitrage to deduct the necessary and sufficient conditions for our eSSVI parametrisation to be arbitrage free?

What about vertical spreads(a.k.a bull spread arbitrage). Why Gatheral in his paper never mentions this kind or arbitrage. Shouldn't in theory have a negative skew for any pair of strikes?

## Answer by dm63 (score 1, accepted)

https://quant.stackexchange.com/a/83903

The only arbitrage limit on a vertical bull spread is that it must have a positive price. But this is automatically true if the butterfly spread condition is satisfied. (Because the bull spread can be expressed as a sum of butterflies above the lower strike, each of which has a positive price due to the butterfly convexity)

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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