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We discuss modelling of SPX and DAX index option prices using the ShiftedLog-Normal (SLN) model, (also known as Displaced Diffusion), and the SABRmodel. We found out that for SPX options, an example of strongly skewed optionprices, SLN can produce a quite accurate fit. Moreover, for both types of indexoptions, the SLN model is giving a good fit of near-at-the-forward strikes.Such a near-at-the-money fit allows us to calculate precisely the skewparameter without involving directly the 3rd moment of the related probabilitydistribution. Eventually, we can follow with a procedure in which the skew iscalculated using the SLN model and further smile effects are added as a nextiteration/perturbation. Furthermore, we point out that the SLN trajectories areexact solutions of the SABR model for rho = +/-1.

Modelling the skew and smile of SPX and DAX index options using the Shifted Log-Normal and SABR stochastic models