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Smiles in delta

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  • Arianna Mingone

Abstract

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 $ d_1 $ d1 and $ d_2 $ 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.

Suggested Citation

  • Arianna Mingone, 2023. "Smiles in delta," Quantitative Finance, Taylor & Francis Journals, vol. 23(12), pages 1713-1728, October.
    • Handle: RePEc:taf:quantf:v:23:y:2023:i:12:p:1713-1728
    DOI: 10.1080/14697688.2023.2258932
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