Smiles in delta

AbstractFukasawa introduced in Fukasawa [The normalizing transformation of the implied volatility smile. Math. Finance, 2012, 22(4), 753–762] two necessary conditions for no butterfly arbitrage on a given implied volatility smile which require that the functions d1 and d2 of the Black–Scholes formula have to be decreasing. In this article, we characterize the set of smiles satisfying these conditions, using the parametrization of the smile in delta. We obtain a parametrization of the set of such smiles via one real number and three positive functions, which can be used by practitioners to calibrate a weak arbitrage-free smile. We also show that such smiles and their symmetric smiles can be transformed into smiles in the strike space by a bijection. Our result motivates the study of the challenging question of characterizing the subset of butterfly arbitrage-free smiles using the parametrization in delta.Keywords: Implied volatilityVolatility smileDeltaButterfly arbitrageJEL Classification: G13C60C63 AcknowledgmentsI sincerely thank Zeliade Systems and especially Claude Martini for giving the opportunity to work with them and daily broaden my knowledge. The results in this paper have been achieved thanks to the concrete need of client CCPs of a volatility calibration in the sigma space which can be converted in the strike space. I thank Stefano De Marco for the precise reading of the article, and for the improvements suggested. I thank Antoine Jacquier who pointed out crucial refinements, and Vladimir Lucic who shared his fundamental knowledge with enthusiasm.Disclosure statementNo potential conflict of interest was reported by the author(s).