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Balancing SVI Fit Quality with No-Arbitrage Constraints

Article Quant Q&A · Author: Dovie Chu
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Summary

The discussion addresses how to fit an implied volatility surface for local volatility modeling when standard SVI can fit market observations well but may permit arbitrage, while reduced SVI and SSVI impose arbitrage controls at a potential cost to fit quality. It points to more flexible SVI extensions, including eSSVI, whose no-arbitrage criteria support fitting and interpolation. One cited extension is described as having a robust calibration procedure, though its treatment of calendar arbitrage provides necessary rather than sufficient conditions, so additional derivation is needed.

A response reports good eSSVI fit quality for equity indices and single names at maturities of six months or longer, and often at shorter maturities too. The exchange also mentions non-SVI approaches, including work on constructing local volatility more directly. These are practitioner observations and paper references rather than a systematic comparison. A key limitation is that a three-parameter model may fail to capture different second-derivative behavior on the downside and upside; richer models may improve flexibility but lack simple arbitrage bounds on their parameters.

Key ideas

  • Standard SVI can fit implied volatility data closely without guaranteeing an arbitrage-free surface.
  • Reduced SVI and SSVI provide arbitrage controls, but may fit market observations less closely.
  • Flexible SVI extensions such as eSSVI offer additional calibration and no-arbitrage tools.
  • Some proposed calendar-arbitrage conditions are only necessary, so further analysis may be required.
  • A three-parameter model may not capture asymmetric curvature across downside and upside strikes.

Tags

OptionsVolatilityDerivatives pricingArbitrageStatistics
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# Question About SVI and SSVI Tradeoff between Fitness and No-Arbitrage


# Question About SVI and SSVI Tradeoff between Fitness and No-Arbitrage












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:

- Is there any method that eliminate arbitrage for SVI but does not sacrifice too much quality of fit?

- 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:

## Answer by d_797 (score 4)

https://quant.stackexchange.com/a/59410

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:

- https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2971502

- https://arxiv.org/pdf/1804.04924.pdf

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

## Answer by SI7 (score 4)

https://quant.stackexchange.com/a/59446

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.