Arbitrage Constraints in Joint SPX and VIX Volatility Calibration

Summary

The document raises questions about constructing implied volatility surfaces for SPX options and VIX options after jointly calibrating a model to market data. It asks whether the resulting surfaces need further checks for calendar and butterfly arbitrage, or whether a successful joint fit already ensures those properties. It also asks whether separately arbitrage-free surfaces can still imply an arbitrage between the two related markets.

No answer, calibration method, or empirical evidence is included in the excerpt. The questions distinguish fit quality from arbitrage consistency: matching observed prices is not itself a stated guarantee that interpolated surfaces satisfy static no-arbitrage constraints, nor does checking each surface alone establish consistency across assets. Resolving the issue would require details of the model, the surface construction and interpolation, and the relationships imposed between SPX and VIX prices. The document is useful as a statement of validation questions, but it provides no specific procedure or conclusion.

Key ideas

• A successful joint model calibration does not, by itself, answer whether constructed volatility surfaces satisfy calendar and butterfly constraints.
• The document asks whether each surface requires separate arbitrage checks after fitting.
• Arbitrage freedom of individual SPX and VIX surfaces does not establish cross-market consistency.
• The excerpt gives no calibration details, evidence, or resolution to these questions.

Tags

# Joint SPX and VIX calibration - volatility surfaces construction


# Joint SPX and VIX calibration - volatility surfaces construction












I am currently researching the joint calibration problem of SPX options and VIX options. A question that comes to mind is the construction of each assets respective volatility surface.

In the articles I've looked there is no mention of constructing a volatility surface after having achieved a successful joint calibration.

To my knowledge, after having calibrated a model to market data it has to be checked/modified not to allow for calendar and butterfly arbitrage.

However, is this not also the case when building a volatility surface after a joint fit? Is this something that is obvious, so it is never mentioned? And lastly, even if you create arbitrage free volatility surfaces for each asset after the joint calibration, can we be certain that there is no arbitrage between the SPX and VIX?

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