SSVI parametrization motivation , SSVI implementation

Question

I've read the following paper of Gatheral and Jacquier https://arxiv.org/pdf/1204.0646.pdf about volatility surfaces. I'm thinking about the SSVI surface. Is there any motivation why they choose exactly this parameterization? Moreover I had a look at this https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/cfm-imperial-institute-of-quantitative-finance/events/distinguished-lectures/Gatheral-2nd-Lecture.pdf. On page 17 he writes that: This smile is completely defined given three observables. The ATM volatility and ATM skew are obvious choices for two of them. The most obvious choice for the third observable in equity markets would be the asymptotic slope for negative and in FX markets and interest rate markets, perhaps the ATM curvature of the smile might be more appropriate. Does these mean I need this data for calibrating SSVI? And then when Implementing he does this in a quite complicated way, it seems to me. Is there an easier method? And if I fix $\theta=\theta_t$ and plot $\omega(k,\theta)$ I should have a smile? Am I right? Because I only get a line. For $\phi$ I used the heston like function. Can maybe anybody tell me values for $\rho$ and $\lambda$ to get a curve, so I can see this is possible with this parameterization. Thanks in advance!

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Update Something with my update doesn't work. So I have given these data: k=[ -0.0193, -0.0070, 0.0040, 0.0150, 0.0260] implied volatility= [0.0366, 0.0331, 0.0320, 0.0329, 0.0344] t=0.25. So total implied variance=0.25*(implied volatility)^2. By interpolation I can find $\theta_t=3.0000e-04$ Now my program gives me $\rho=-0.9993, \lambda=3.8324$. If I plot this it looks like the file attached. Can anybody tell me where I made a mistake?

Answer 1

In short no you don't need to estimate the values for these quantities to calibrate the model. Instead what you want to do is to perform a least squares optimization on the model parameters against market volatilities, this gives you the calibrated surface to the market prices and if you want to output the model quantities you mentioned then you can.

So you want to minimise the error between

$$\sigma_{model} = \sqrt{w(k,\theta_t;t)/t}$$ and $$\sigma_{market}(k,t)$$ as

$$\sum_{k, t} |\sigma_{model}(k, \theta_t;t) - \sigma_{market}(k,t)|^2 < \epsilon$$

for some $\epsilon >0$. You will either have quotes from the market in terms of prices or volatilites, if you have prices then you must use a numerical method such as Newton-Raphson to obtain the market volatilities. During the optimisation procedure you will need to enforce the constraints mentioned in the paper.