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I need to test strike (butterfly) arbitrage on a discrete implied volatility grid.

I know that the traditional procedure for continuous case is (for a given maturity T):

  1. See the Dupire formula in terms of implied volatility (and not in terms of option prices). You can find it here.
  2. Check if the denominator of Dupire formula is positive for every K

The Dupire denominator from step 2 contains derivatives, thus it relies on the fact that the implied volatility surface is continuous.

Now, back to my discrete grid of implied volatility. How can I transform step 2 (with derivatives), in a condition adapted for discrete grid?

Joanna
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  • Not sure if you'd be satisfied with smoothing the IV surface in an arbitrage-free way. If you are, then this may help: https://core.ac.uk/reader/6978470 –  Apr 17 '22 at 09:00
  • Absence of arbitrage must hold, irrespective of a discrete or continuous vol surface. For a discrete surface, no-arbitrage dictates positive second (weighted) differences, AFAIR – Kermittfrog Apr 17 '22 at 09:18

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