# Fitting Gatheral's SVI model

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.

I have the following variance plot:

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 $$\beta\notin [-1,1]$$

• Hi @ZRH, maybe it would help to have an example of the smile you are trying to fit. Could you provide $(K, \sigma(K))$ couples? – Quantuple Mar 18 at 8:24
• @Quantuple sure. Strikes were: 67.5,70,72.5,75,77.5,80,82.5,85,87.5,90,92.5, Vols in %: 22.14,20.99,19.61,19.16,18.57,17.73,17.11,16.60,16.20,15.73,15.37 – ZRH Mar 18 at 8:46
• Defining $k=log(K/80)$ as the log-forward moneyness for your problem, I find that raw SVI parameters $(a,b,\rho,m,\sigma)= (-0.0511,0.2613,-0.8375,-0.3383,0.4944)$ provide a good fit. My objective function is sum of squares type + a penalisation for absence of static arbitrage.I do work in a modified parameter space to account for constraints on the definition domain of the representation. Also, I determine the initial guess in the Jump-Wings SVI space (see section 3.3 in arxiv.org/pdf/1204.0646.pdf), since parameters bear a more straightforward interpretation in that space. – Quantuple Mar 19 at 13:57
• ok - that is way more advanced than my current approach. Thanks for the ref ! – ZRH Mar 19 at 16:02
• You are welcome. – Quantuple Mar 20 at 8:33