  Fukasawa introduced in [Fukasawa, Math Financ, 2012] two necessary conditions
for no butterfly arbitrage which require that the $d_1$ and $d_2$ functions 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
via one real number and three positive functions. 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.
