Smiles in delta. (English) Zbl 1536.91322
Summary: M. Fukasawa [Math. Finance 22, No. 4, 753–762 (2012; Zbl 1272.91121)] introduced two necessary conditions for no butterfly arbitrage on a given implied volatility smile which require that the functions \(d_1\) and \(d_2\) 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.
MSC:
| 91G15 | Financial markets |
Citations:
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