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As I understood, the SVI is widely used among practitioners. However, it is mentioned in many published papers (including ones written by Gatheral), that the SVI model does not fit well short-maturity options. For example, Fabien Le Floc'h provides a specific example with weekly options here. Basically my question is why?

I have a couple of thoughts with this regard, but I couldn't find any approval or comments, so, any references would be much appreciated:

1) When fitting SVI, people usually impose some arbitrage-free restrictions on the parameters of the model. Might it be the case that for short-maturity options some of these arbitrage conditions fail, but imposing it in the model leads to failure in a fit? If so, which arbitrage condition fails and why?

2) It's known that short maturity options are more sensitive to jumps, especially OTM options. It is also known that the Heston model converges to SVI when $T\to \infty$. Because the Heston model does not include jumps, I thought that maybe svi for short maturity shows a bad fit as volatility smile becomes highly sensitive to jumps. But, again, it is not clear to me why the presence of jumps weakens the svi fit then.

As I said any thoughts, comments and references would be very helpful, thanks!

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    $\begingroup$ Hi. I would say that the SVI intrinsic representation is ill-suited to capture high-curvatured IV, and this high-curvature would typically happen on the short term as short-term option prices are more event-driven than medium to long-dated option prices. An example would be bimodal distributions: consider a stock that may end up in one regime or another depending on some news => W-shaped smile. SVI is incapable of fitting that. $\endgroup$ – Quantuple Mar 13 '20 at 8:40
  • $\begingroup$ @Quantuple how would you model such w-shaped smiles? I've come across these for Amazon near earnings on shorter maturities. Thanks! $\endgroup$ – BrownianBread Mar 13 '20 at 14:36
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    $\begingroup$ Timothy Klassmen has papers on that subject and runs a firm that produces vol fitters that are cable of fitting such curves. voladynamics.com/marketEquityUS_AMZN.html $\endgroup$ – roz Mar 13 '20 at 17:15
  • $\begingroup$ @roz that's very interesting website, thanks! $\endgroup$ – Evgenii Mar 13 '20 at 18:53
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    $\begingroup$ @Evgenii high IV curvature translates to "spikes" in the risk neutral pdf which the SVI formulation cannot intrinsically handle. Le Floc'h has another paper where he mentions this (link.springer.com/article/10.1007/s10203-019-00238-x) but there is no proof attached $\endgroup$ – Quantuple Mar 13 '20 at 23:50
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The issue has much more to do with the SVI parameterization per se, and not with any arbitrage constraint. The fact that Heston as $T \to \infty$ becomes close to SVI is not very useful either to explain this. It is merely a nice way to make the SVI parameterization have some stochastic root. If you read Timothy Klassen papers, they suggest that the SVI parameterization largely predates the link with Heston.

What is the SVI parameterization? I found two useful ways to look at it:

  1. you want linear wings in variance. So you set one side to be just a line, and the other to be the square root of a quadratic, and you join the two by an addition.
  2. it is just a specific case of multiquadratic RBF interpolation where the number of nodes is 1 or 2 (depending on how you look at the linear part - RBF interpolation is expressed as a polynomial plus a linear combination of multiquatrics).

Why doesn't it fit well the specific case of short term options? This has to do with the curvature and its link with the two wings. As soon as you have one side slightly concave, and lots of points to fit there, SVI will break. I think one of the wings has to be flat then, which is visible in one of the graphs of the link provided.

How to improve it? There are many proposals I find sort of silly where they add parameters to SVI for the wings. If you want to stay within the same sort of parameterization, I believe this is where the second way of looking at SVI is more interesting: you could just use an RBF interpolation with more nodes.

Otherwise there are many other parameterizations which are more flexible than SVI that I discuss in my book.

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