I was considering using Gatheral's formula for fitting option skew. In the specific (commodity) market that I am concerned with, the underlying is ca. at 50, and typically 5 integer strikes left and right of that would be observable, i.e. $k$ ranges from ca. -0.1 to +0.1. I really struggle to get a stable fit on this... I am wondering if this is down to the data just not really being enough for fitting this 5-parameter function (which i guess is the case), or is it just me not using well-chosen start values for the fit. I was hoping there would be seasoned practitioners to comment on this.
with above 2nd order polynomial fit, I get $p_t=-0.0015$, $c_t=-1.7146e-04$, $\psi_t=-8.1992e-04$ $b=-1.2818e-04$, $\rho=-0.7950$, and that means $\beta\notin [-1,1]$