The purpose of this paper is to study the Stochastic Volatility Inspired model (SVI) as implied volatility model: we study the analytic part of the SVI with the arbitrage conditions, we establish the initial guess and the parameter's boundaries. Until recently it was not possible to find sufficient conditions that would guarantee an SVI model calibration arbitrage-free. The main contribution in this paper is that we provide two methods to resolve the arbitrage problem (butterfly and calendar spread): the first one is numerical using the Sequential Least-Squares Quadratic Programming (SLSQP) algorithm, and the second one is analytical by using sufficient conditions that guarantee an SVI arbitrage-free.Our method guarantee to get SVI calibration with butterfly and calendar spread arbitrage-free, We provide many numerical examples with arbitrage such as Vogt Axel example and we show how to fix them. The calibration method is tested on 23 equity indexes with 14 maturities each and we get 322 slices fits using the same initial guess and the SVI parameters boundaries for all indexes.This new calibration method is very important and it meets practical need: resolving this arbitrage problem will pave the way to the surface calibration and the transition from implied volatility to local volatility using Dupire's formula, therefore, it allows price different kind of pathdependent options such as barrier options, and American options. The SVI model could also be applied to price interest rate derivatives such as swaptions, interest rate caps, and floors.

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