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Shifting an Implied Volatility Surface Without Creating Static Arbitrage

Article Quant Q&A · Author: JMC
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Summary

The document considers how to adjust a fitted vanilla-option implied volatility surface to represent a lower-volatility market or an abrupt volatility shock. It cautions that blindly adding a constant to implied volatility or multiplying it by a scale factor can both create static arbitrage. The relevant constraints include vertical call-spread relationships, which require call prices to behave consistently across strikes; the answer also refers to other static-arbitrage conditions without detailing their formulas in the provided text.

Two modeling routes are suggested. A practitioner can apply additive or multiplicative changes and then check whether they preserve arbitrage constraints, potentially using historical behavior for a particular underlying as supporting evidence. Alternatively, they can fit an arbitrage-free parameterization such as SSVI and adjust its parameters to represent changes in volatility level or term structure. That approach requires a more complex initial fit. No specific shock calibration or empirical validation is supplied.

Key ideas

  • An additive implied-volatility shift can introduce static arbitrage.
  • A multiplicative volatility scaling can also violate static-arbitrage conditions.
  • Adjusted surfaces should be checked against option-price constraints across strikes and maturities.
  • An arbitrage-free parameterization such as SSVI can provide a structured way to adjust the surface.
  • The SSVI approach adds complexity to fitting the original surface.

Tags

OptionsVolatilityArbitrageDerivatives pricing
Full text
# Does shifting/scaling the IV surface relatively/absolutely introduce arbitrage?


# Does shifting/scaling the IV surface relatively/absolutely introduce arbitrage?












I am fitting a volatility surface for vanilla call options. I do this by fitting low-degree polynomials (or cubic splines) along the strike dimension per maturity and then linearly interpolating implied variance along iso-moneyness lines. I would like to make a rough guess about how the surface might change if I assume that the volatility of the overall market as displayed by the VIX is going to decrease in the future.

Would it be a bad idea to assume that a given stock's IV surface would change the same way in terms of an absolute shift or a relative scaling of each option's IV?

As far as I understand, an absolute shift will introduce arbitrage, while relative scale of IV might not. What would be simple and not-so-bad approach to model the market shifting into a lower-volatility period or a sudden IV shock?

## Answer by Quantuple (score 6, accepted)

https://quant.stackexchange.com/a/60108

Long story short: yes both might introduce static arbitrage opportunities if performed blindly.

There are 3 types of static arbitrage to consider:



- Vertical arbitrage (or call spread arbitrage): call spreads should have a positive price



By writing the corresponding conditions under additive/multiplicative transformations of the original IV in (time to maturity, strike) axes:







So I guess you have two options:

- Use additive/multiplicative spreads but make sure they do not introduce arbitrage in the first place (depending on the market/underlying you're looking at, you might even show that historically this type of shift never introduced arbitrage, so you can safely "overlook" them).

- Use an arbitrage-free parametrisation of your IV, which you can easily "bump" to reflect level changes (or term-structure changes). An example would be SSVI. This however adds to your original problem the complexity of fitting such a parametrisation.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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