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

TL;DR

This paper compares the Shifted Log-Normal and SABR models for accurately capturing the skew and smile in SPX and DAX index options, highlighting the SLN model's effectiveness near at-the-money strikes.

Contribution

It demonstrates that the SLN model provides a precise fit for index options and can be integrated with SABR for enhanced modeling of skew and smile effects.

Findings

SLN fits strongly skewed SPX options well

SLN accurately models near-the-money strikes for both indices

SLN trajectories are exact solutions of SABR for rho = ±1

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.