How to calibrate a volatility surface using SVI

Question

I've read the following paper by Gatheral and Jacquier and have several question regarding the calibration of a volatility surface in a arbitrage free way and some theoretical aspects. Let me first introduce some notation. They define the log strike as

$$k:=\log{\frac{K}{F}}$$

where $F$ denotes the forward. Moreover, $\sigma_{BS}(k,t)$ denotes the implied Black-Scholes volatility with strike $k$ and maturity $t$ and

$$w(k,t):=\sigma_{BS}^2(k,t)t$$

the so called total implied variance. With $\theta_t:=\sigma_{BS}^2(0,t)t$ we denote the at the money implied total variance. What follows they present different parametrization of a single slice in the surface, i.e. not depending on $t$. My first question:

1. Question: Why do they authors use the total implied variance instead of the directly observable $\sigma_{BS}(k,t)$ for the parametrization? Is there any advantage / meaning of that? Naturally, I would fit a model to the implied volatility

There are three parametrizations of a single surface slide:

• raw SVI: For a parameter set $\xi_R:=\{a,b,\rho,m,\sigma\}$ the raw parametrization is given by: $$ w(k,\xi_R):=a+b\left(\rho(k-m)+\sqrt{(k-m)^2+\sigma^2}\right)$$
• natural SVI: For a parameter set $\xi_N:=\{\Delta,\mu,\rho,\omega,\zeta\}$ the natural parametrization is given by: $$ w(k,\xi_N):=\Delta+\frac{\omega}{2}\left(1+\zeta\rho(k-\mu)+\sqrt{(\zeta(k-\mu)+\rho)^2+(1-\rho^2)}\right)$$
• SVI Jump Wings (SVI_JW): For a given time to expiry $t >0$ and a parameter set $\xi_J:=\{v_t,\psi_t,p_t,c_t,\tilde{v_t}\}$ the SVI-JW parametrization is given in raw SVI parameters: $$\begin{align} v_t &= \frac{a+b\left(-\rho m+\sqrt{m^2+\sigma^2}\right)}{t}\\ \psi_t &=\frac{b}{2\sqrt{w_t}}\left(-\frac{m}{\sqrt{m^2+\sigma^2}}+\rho\right)\\ p_t &= \frac{b}{\sqrt{w_t}}(1-\rho)\\ c_t &= \frac{b}{\sqrt{w_t}}(1+\rho)\\ \tilde{v_t} &= \frac{1}{t}\left(a+b\sigma\sqrt{1-\rho^2}\right)\\ \end{align}$$ where $w_t:=v_tt$.

2. Question: Why is it an advantage of having a dependency on time to expiration $t$ in the SVI-JW parametrization? As far as I see, you still fit the model to a given slice in all of the above parametrization, that is: You fix time to expiry and fit the model to the observed quotes. So that you could also introduce a time to expiry parameter in the raw/natural SVI.

The authors introduce now a new parametrization for a complete surface, the SSVI.

• SSVI: For a smooth function $\phi$ (with some additional properties) the SSVI parameterization is given by: $$ w(k,\theta_t):=\frac{\theta_t}{2}\left(1+\rho\phi(\theta_t)k+\sqrt{(\phi(\theta_t)k+\rho)^2+(1-\rho^2)}\right)$$ a common choice is $\phi(\theta) = \frac{\eta}{\theta^\gamma(1+\theta)^{1-\gamma}}$

They are translations how to convert one parametrization to another.

3. Question: Is it correct that the SSVI tries to fit a whole surface not just a single slice at once?

My last question is more about the actual calibration. For the raw and natural parametrization you would try to find optimal parameters so that $$\sum_{i=1}^n(w(k_i,\xi_R)-w(k_i)_{market})^2$$ is minimized, where $w(k_i)_{market}$ are observed market quotes (calculated from $\sigma_{BS}$) for strike $k_1,\dots,k_n$ for a fixed time to expiry $t$.

Now for the SSVI, if its really about fitting the whole surface, what function are you minimizing?

$$\sum_{t_i}\left(\sum_{i=1}^n(w(k_i,\theta_{t_i})-w(k_i,t_i)_{market})^2\right)$$ where you also sum over the maturities?

4. Question: How does the minimization function for the SSVI look like? It seems that the authors are using still for a fixed time to expiry $t_i$ a slice parametrization and then compare it with previous / next slice, run additional calibration if needed to avoid calendar spread arbitrage. See page 21 "An example SVI calibration recipe".

Answer 1

I will just answer your first question as I do not know the details of SSVI.

Total variance is more intrinsic than volatility.

The BS formula can be rewritten in terms of 3 parameters: the log-strike (log-moneyness would be more accurate) $k$, the total variance $w$ and the discount factor. Volatility never appears without a $\sqrt{T}$. It is just there because we chose make it appear.

Most arbitrages are easier to derive and interpret in terms of $w$: For example calendar arbitrage is just $\partial_t w \ge 0$ and follows from the fact that variance is additive. Roger Lee's tail formula is also better expressed in terms of growth of $w$.

Note that this is in part due to the face that $w$ is scaleless when $\sigma$ is not which is also the reason why $\sigma$ is more intuitive to us. So it usually makes sense to use $w$ as an internal parameter even if the results are expressed in terms of $\sigma$.

Answer 2

(1) as AFK says, total remaining variance is somewhat more natural mathematically. Of course it is just a choice of coordinate, and mathematically you can do changes of coordinate so it is for aesthetic rather than hard mathematical reasons.

(2) time dependence in SVI-JW parameters is carefully chosen so that if the parameters are held constant across maturities, the vol surface stays approximately constant as a function of delta as maturity changes. This gives a way to extrapolate volatilities that is much more similar to how real markets look than e.g. holding raw SVI parameters fixed.

(3) yes, SSVI is trying to fit the near-the-money part of the whole vol surface.

(4) my understanding is for SSVI you fit exactly the at-the-money vol level and at-the-money skew at each maturity. That is an exact calibration to two guven quantities, rather than an optimisation.