Local volatility SVI parametrization

Question

In this paper Gatheral presents the following parametrization of the implied total variance $w(k,T) = \sigma_{BS}(k,T)^2T$ for each slice $k \mapsto w(k,T)$:

$$ w(k) = a + b\{\rho (k-m) + \sqrt{(k-m)^2 + \sigma^2} \}.$$

As far as I understand it, for each expiry $T$ one will have to calibrate a set of five parameters $\{a,b,m,\rho,\sigma\}$.

On the other hand I found the following article where in appendix A, a calibrated volatility surface is presented. But in their example there is an explicit dependence on $T$, so then they will have a "simple" specific expression for the whole volatility surface.

Another thing I've noticed when reading articles about various parametrizations that there seems to be some inconsistencies regarding implied total variance. Gatheral defines it as $\sigma_{BS}(k,T)^2T$ but I've seen in other articles people parametrizing on $\sigma_{BS}(k,T)^2$ or $\sigma_{BS}(k,T)$ instead.

Summarizing: My question is primarily if one has to calibrate SVI for each expiry slice or if it is possible to parametrizing the whole surface in a way such that the total number of parameter does not increase if more expiries are added.

Answer 1

Gatheral and Jacquier discuss this issue in section 4 of the paper. Instead of using the raw parameterization of the SVI, they use the natural parameterization of the total implied variance: $$ w(k) = \Delta + \frac{\omega}{2} \left\{ 1 + \zeta \rho (k - \mu) + \sqrt{(\zeta (k-\mu) + \rho)^2 + (1-\rho^2)} \right\} (\text{p. 61 of the published paper}) $$

In order to fit the entire surface of the total implied variance, they propose the following generalization. To ensure that the fit is free of arbitrage, they define the surface in terms of the log-moneyness and the at-the-money implied total variance $\theta_t := \sigma_{BS}^2(0,t)t$. The Surface SVI then has the form: $$ w(k,\theta_t) = \frac{\theta_t}{2} \left\{ 1 + \rho \phi(\theta_t) k + \sqrt{(\phi(\theta_t) k + \rho)^2 + (1-\rho^2)} \right\} (\text{p. 63 of the published paper}) $$ Where $\phi$ is a smooth function from $\mathbb{R}_{+}$ to $\mathbb{R}_{+}$ such that the limit $\lim_{t\rightarrow 0} \theta_t \phi(\theta_t)$ exists in $\mathbb{R}$. The parameters that you need to fit for the entire surface are therefore $\rho$ and whatever is needed to fit $\phi$. In practice, you need some interpolation to get $\theta_t$ because you almost never observe a log moneyness of exactly 0.

The function $\phi$ and the parameters have to satisfy certain restrictions for the parameterization to be free of arbitrage. The paper discusses these at length.

Heston and Jacquier propose two possible $\phi$ functions: $$\phi(\theta) = \frac{1}{\lambda \theta} \left( 1 - \frac{1-e^{-\lambda\theta}}{\lambda \theta} \right)$$ Which they call a Heston-like parameterization and the power law $$ \phi(\theta) = \eta \theta^{-\gamma}$$ A while back, I implemented the paper in MATLAB. In the end, I didn’t use the codes so they are not extensively tested. I have uploaded them to the file exchange. Maybe they are helpful to you: http://ch.mathworks.com/matlabcentral/fileexchange/49962-gatherals-and-jacquier-s-arbitrage-free-svi-volatility-surfaces

Answer 2

For short maturity SPX option chain, the analytic form of the V-shape volatility smile has been fully worked out in my latest paper on SSRN. You can take a look.