Using SVI model for IV surface

Question

I am using well-known paper of J. Gatheral & A. Jacquier Arbitrage-free SVI volatility surface to explore SVI model.

1. on the page 6 in the bottom is statet that

   The SVI-Jump-Wings (SVI-JW) parameterization of the implied variance v (rather than the implied total variance

I am confused as my understanding is that all 3 parametrisation (raw, neutural, JW) mentioned in article are describing total impled variance. Can someone please explain the quote provided above?

2. Main question I am clueless about is how SVI parametrisations defines volatility surface. Does one set of parameters defines only one slice of the volatility surface? Or one set of parameters defines the whole surface? If last-mentioned is true, then I do not understand how the volatility ATM term structure is managed: usually implied ATM volatility is not constant over the time, it is usualy increasing function over time when short-term volatility is low, and decreasing function when short-term volatility is high. if we take SVI JW we can see that with fixed set of parameters implied ATM volaitility is constant over the time.

3. When we fit SVI to market data, am I right that we fisrt calculate implied volatility (IV) from option prices, then convert IV to total implied variance = IV^2*t and then we try to find parameters of SVI that fits our obtained total implied variance data in the best way?

I have econimical background, hereat I am sorry if my questions are silly.

Answer 1

I use Gatheral's notations.

The SVI-Jump-Wings (SVI-JW) parameterization of the implied variance v (rather than the implied total variance

The raw and natural parametrizations describe the total implied variance for one slice (fixed tenor). The SVI-JW describes the implied variance for one slice (fixed tenor). The total implied variance slice for a fixed tenor $T$ is defined as $k \mapsto \sigma^2_{BS}(k,T)\cdot T$ (or $w(k,T)$) whereas the implied variance is defined as $k \mapsto \sigma^2_{BS}(k,T)$ (or $w(k, T)/T$).

Main question I am clueless about is how SVI parametrisations defines volatility surface. Does one set of parameters defines only one slice of the volatility surface? Or one set of parameters defines the whole surface?

SVI/SVI-JW are used to describe one slice (single tenor) at the time; Surface SVI (SSVI) is used to fit the whole surface (multiple tenors).

When we fit SVI to market data, am I right that we fisrt calculate implied volatility (IV) from option prices, then convert IV to total implied variance = IV^2*t and then we try to find parameters of SVI that fits our obtained total implied variance data in the best way?

You can also go the other way around and directly calibrate to the market quotes. To do that you need to convert the total implied variances to implied volatilities and then apply Black-Scholes. This is of course assuming that you are working with European options. This is what is suggested in Gatheral's paper. Other authors (see "Quasi-Explicit Calibration of Gatheral's SVI model") optimize over the total implied variance.