Refined analysis of the no-butterfly-arbitrage domain for SSVI slices

Claude Martini and Arianna Mingone

Abstract

The no-butterfly-arbitrage domain of the Gatheral stochastic-volatility-inspired (SVI) five-parameter formula for the volatility smile has recently been described. It requires in general a numerical minimization of two functions together with a few root-finding procedures. We study here the case of the famous surface SVI (SSVI) model with three parameters, to which we apply the SVI results in order to provide the nobutterfly- arbitrage domain. As side results, we prove that, under simple requirements on parameters, SSVI slices always satisfy Fukasawa’s weak conditions of no arbitrage (ie, the corresponding Black–Scholes functions d₁ and d₂ are always decreasing), and we find a simple subdomain of no arbitrage for the SSVI model that we compare with the well-known subdomain of Gatheral and Jacquier. We simplify the obtained no-arbitrage domain into a parameterization that requires only one immediate numerical procedure, leading to an easy-to-implement calibration algorithm. Finally, we show that the long-term Heston SVI model is in fact an SSVI model, and we characterize the horizon beyond which it is arbitrage free.