Explicit no arbitrage domain for sub-SVIs via reparametrization

TL;DR

This paper derives explicit no arbitrage domains for certain three-parameter SVI models, simplifying the process by eliminating the need for numerical minimization and root finding.

Contribution

It provides explicit no arbitrage domains for specific sub-SVI models, avoiding complex numerical procedures used in previous approaches.

Findings

Explicit no arbitrage domain for symmetric SVI

Explicit no arbitrage domain for Vanishing Upward/Downward SVI

Explicit no arbitrage domain for SSVI

Abstract

The no Butterfly arbitrage domain of Gatheral SVI 5-parameters formula for the volatility smile has been recently described. It requires in general a numerical minimization of 2 functions altogether with a few root finding procedures. We study here the case of some sub-SVIs (all with 3 parameters): the Symmetric SVI, the Vanishing Upward/Downward SVI, and SSVI, for which we provide an explicit domain, with no numerical procedure required.