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

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.