SVI implied vol model parameterization

Question

It would be nice if anybody could help me. I'm trying to understand the raw SVI parameterization of Gatheral. We have: $$\omega(k; \chi_R)=a+b \left\{\rho(k-m)+ \sqrt{(k-m)^2+ \sigma^2}\right\} $$ I'm not clear about what exactly is given to calculate this and how to determine the parameters. Can anybody explain me this, please? Thanks for your help.

Answer 1

In the SVI model, you are modeling the implied volatility $\omega$ (at a fixed time) as a function of the strike $k$, with other parameters $a, b, \rho, m, \sigma^2$ fixed.

The way you compute this is, given a bunch of quotes at the same maturity, run numerical optimization to calibrate the parameters, and once they are fixed, you have implied vol as a one-dimensional function of the strike $k$.

UPDATE

Here is a practical guide to do this.

Implement the functions

1. BSvol2price -- straight-forward implementation of the BS formula calculations

2. BSprice2vol -- rootfinder inversion of the calculation above, matching the vol to produce a given price

Now you have 2 options, both using multi-dimensional numerical optimization:

• fit in vol space - convert incoming price quotes to vols (using BSprice2vol), and fit the parameters to the resulting vols
• fit in price space -- fit the parameters by producing prices from vols and measuring errors in price