Smiles in delta

Fukasawa 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.