I’m currently working on a project to build a local volatility model out of implied volatility data and am struggling in the selection of an appropriate method to interpolate the volatility surface. I need to interpolate the discrete vendor data (in time and in strike) to create an arbitrage-free implied vol surface that I can then use to calculate local vol.

I am following Gatheral’s arbitrage-free SVI paper [Arbitrage-free SVI Volatility Surface] and there are three methods he discusses to construct an implied volatility surface.

• SVI with different parametrizations (raw, natural, jump-wing, Section 3)
• Surface SVI (SSVI) – (Section 4)
• Reduced SVI (jump-wing form, Section 5.1)

Reduced SVI Fitness:

The problem is that SVI gives an excellent fit but doesn’t guarantee that the result is arbitrage-free.  Reduced SVI and SSVI work the other way round - they guarantee arbitrage-free but the fit is not as good and can even be quite poor in places.

So here are the questions I have:

1. Is there any method that eliminate arbitrage for SVI but does not sacrifice too much quality of fit?
2. Are there methods other than SVI-related ones that can be applied for this project? Perhaps a functional form for the local vol surface directly

Current eSSVI fitness I get:

For the first question, there have been a number of improvements on SVI/SSVI that are much more flexible than SSVI and also come with easy-to-impose no-arb conditions. See below:

Paper 2 builds on Paper 1 and comes with a robust fitting procedure. One thing to note about Paper 2 is that they do not derive the Sufficient conditions for no calendar arb (only necessary). You'll have to derive it yourself using the theorems in Paper 1.

For your second question, there have been other non-SVI works on this front. Here is a recent paper that I have come across:

https://arxiv.org/pdf/2008.09454.pdf

• Thanks for the replies! I tried to implement the eSSVI model following the second paper you mentioned and the fitness is not at as good as described in the paper. I updated the question with the fitness result of reduced SVI model (also has three free parameters for each slice), the eSSVI result I get is comparable with this one and even worse for larger moneyness of middle maturities. Is this the kind of fitness I should expect? I’m concerned about some potential errors in my implementation. Nov 18, 2020 at 17:13
• Besides this, I also have a question about the interpolation along time dimension for SVI model. I follow the interpolation method introduced in Gatheral’ s paper (Section 5.3) where he basically does the interpolation with price . This method guarantees no static arbitrage, but it will cause a spike in the local volatility at beginning of each time interval. I’m wondering have you met similar problem by any chance? Or do you know any other available method for interpolating in time dimension for the SVI-type of models? Thanks a lot! Nov 18, 2020 at 17:13
• Adding the sufficient condition does help with the fitness, it looks much better now (I have uploaded another plot). Thank you so much! Nov 20, 2020 at 9:16
• Hm.. I am a little surprised that adding a condition improved the fit, as you are only constraining the calibration, but that's great to hear! If the fits aren't looking good, I would generally recommend different objective functions (I usually use sum of relative error ) or increasing the number of points in your rho-grid Nov 20, 2020 at 13:28

I agree that the above mentioned eSSVI extension is a very efficient and elegant method for calibration purposes. Arbitrage free slices and interpolations can easily be created by making use of the criteria in the papers. It is also described in great detail in the thesis "Extending the SSVI model with arbitrage-free conditions" (google it). It also describes the local volatility application.

From my experience the fit quality for equity index and single names is very good for maturities >=6m and for many names also in the shortest tenors. However, please be aware that no three parameter model (like the eSSVI) can provide a perfect fit if markets price second derivatives differently on the downside vs. the upside. Here you would need more sophisticated theoretical models with the drawback of probably non-existing arbitrage bounds on the parameters.

• Yes, I've also come across this, especially when fitting during the Pandemic period. One possibility I've wanted to try is to fit some type of double-exponential diffusion (Kou) model with time varying parameters. The discounted asset price is a martingale, so there should not be arbitrage in the prices it produces, hence the skews-of-best-fit can be viewed as an arbitrage-free parameterization of given arbitrageable data. Nov 17, 2020 at 23:34
• Thanks for your replies! I totally agree that the three-parameter model cannot give a fit as perfect as the five-parameter raw SVI model. I attached the fitness of reduced SVI model to my question and the eSSVI fitness I got is comparable or even a little bit worse than that. Do you think this is a reasonable fitness result for a three-parameter model? Nov 18, 2020 at 17:18