Smiles in delta

TL;DR

This paper characterizes the set of implied volatility smiles in delta space that satisfy no butterfly arbitrage conditions, providing a parametrization and a transformation into strike space, advancing understanding of arbitrage-free models.

Contribution

It offers a new parametrization of arbitrage-free smiles in delta space and establishes a bijection to strike space, aiding the study of arbitrage conditions.

Findings

Parametrization of arbitrage-free smiles using one real number and three positive functions.

Transformation of symmetric smiles into strike space via a bijection.

Insight into the subset of butterfly arbitrage-free implied volatility smiles.

Abstract

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.