  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.
