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

Abstract: We discuss modelling of SPX and DAX index option prices using the Shifted Log-Normal (SLN) model, (also known as Displaced Diffusion), and the SABR model. We found out that for SPX options, an example of strongly skewed option prices, SLN can produce a quite accurate fit. Moreover, for both types of index options, 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 skew parameter without involving directly the 3rd moment of the related probability distribution. Eventually, we can follow with a procedure in which the skew is calculated using the SLN model and further smile effects are added as a next iteration/perturbation. Furthermore, we point out that the SLN trajectories are exact solutions of the SABR model for rho = +/-1.