Smiles in delta
刻画了满足无蝶式套利条件的隐含波动率微笑集合，利用Delta参数化得到一组参数表示，可用于校准弱无套利微笑，并证明该微笑与对称微笑可通过双射变换到执行价格空间。
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.