Abstract: AbstractIn this article, we show how to calibrate the widely used SVI parameterization of the implied volatility smile in such a way as to guarantee the absence of static arbitrage. In particular, we exhibit a large class of arbitrage-free SVI volatility surfaces with a simple closed-form representation. We demonstrate the high quality of typical SVI fits with a numerical example using recent SPX options data.Keywords: Volatility smile fittingVolatility surfacesArbitrage pricingArbitrage relationshipFinancial engineeringFinancial mathematicsJEL Classification: C60C63G12G13G AcknowledgmentsThe first author is very grateful to his former colleagues at Bank of America Merrill Lynch for their work on SVI and its implementation, in particular Chrif Youssfi and Peter Friz. We also thank Richard Holowczak of the Subotnick Financial Services Centre at Baruch College for supplying the SPX options data, Andrew Chang of the Baruch MFE programme for helping with the data analysis, and Julien Guyon and the participants of Global Derivatives, Barcelona 2012, for their feedback and comments. We are very grateful to the anonymous referees for their helpful comments and suggestions, and in particular to one of the referees who led us to tighten our results and correct an error in one proof.NotesThe condition is equivalent to , i.e. to the convexity of the smile.Explicit expressions for these coefficients can be found in the R-code posted on http://mfe.baruch.cuny.edu/jgatheral/.