No Arbitrage SVI
38 Pages Posted: 4 Jun 2020 Last revised: 25 May 2021
Date Written: May 25, 2021
We fully characterize the absence of Butterfly arbitrage in the SVI formula for implied total variance proposed by Gatheral in 2004. The main ingredient is an intermediary characterization of the necessary condition for no arbitrage obtained for any model by Fukasawa in 2012 that the inverse functions of the -d1 and -d2 of the Black-Scholes formula, viewed as functions of the log-forward moneyness, should be increasing. A natural rescaling of the SVI parameters and a meticulous analysis of the Durrleman condition allow then to obtain simple range conditions on the parameters. This leads to a straightforward implementation of a least-squares calibration algorithm on the no arbitrage domain, which yields an excellent fit on the market data we used for our tests, with the guarantee to yield smiles with no Butterfly arbitrage.
Keywords: Implied volatility, volatility smile, calibration, SVI, arbitrage-free parameterization
JEL Classification: G12, G13, C60, C63
Suggested Citation: Suggested Citation