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Why SPX and VIX Models Should Be Calibrated Consistently

Article Quant Q&A · Author: Sinbad The Sailor
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Summary

The document asks why a volatility model should calibrate the S&P 500 index and VIX together, and whether an underlying asset’s model should use implied or realized volatility. Its answer links the indices through VIX’s construction: VIX is derived from a weighted portfolio of European S&P 500 options, so their prices and implied volatility are related.

This relationship makes S&P 500 options a potential hedge for VIX exposure. Because those options may be more liquid than VIX products, consistent joint calibration can support hedging and help avoid pricing inconsistencies that could create arbitrage opportunities. The document offers this as an explanation rather than a formal arbitrage example or proof. It does not resolve the question about using realized volatility in an underlying model, and it gives no empirical test of the hedging claim.

Key ideas

  • VIX is derived from a weighted portfolio of S&P 500 options.
  • S&P 500 options can be used to hedge exposure to VIX-related products.
  • Joint calibration can keep related S&P 500 and VIX prices consistent.
  • The explanation motivates consistent calibration but does not provide a formal arbitrage proof.

Tags

EquitiesOptionsVolatilityDerivatives pricingArbitrage
Full text
# The Holy Grail of Volatility Modelling: The SPX & VIX - Why?


# The Holy Grail of Volatility Modelling: The SPX & VIX - Why?












I am currently researching a pre-print article by Julien Guyon & Jordan Lekeufack (2022): Volatility Is (Mostly) Path-Dependent.

Their model is quite impressive in both its simplicity, as well as its efficacy. However, I have one fundamental question in doing research about the joint calibration of the SPX and VIX.

The question is simply: why?

The continued answer to this question, which I've found in other articles, is a hand wavy response that there might be arbitrage otherwise. However, I have never seen such an example put forward.

The only sensible heuristic reasoning I've come up with is that:

- The SPX and VIX are intrinsically linked by definition. Regarding pricing, we would not want to price e.g. SPX put options and VIX call options too differently.

Lastly, another source of confusion is the fact that the model being put forward of the underlying uses VIX as its volatility parameter. Shouldn't the model of an underlying use realised volatility?

## Answer by Daneel Olivaw (score 4)

https://quant.stackexchange.com/a/73543

Recall that the VIX can be expressed as a weighted portfolio of European call and put options on the S&P500. Thus, there is a relationship between the VIX and the S&P implied volatility, so that you could use options on the S&P to hedge VIX exposure - the advantage being that S&P options are likely more liquid than products on the VIX, so I suspect many VIX products are actually hedged by using S&P options.

Thus it should be desirable to have a model where the S&P500 and the VIX indices are calibrated consistently to avoid arbitrages and to enable a trader to confidently hedge VIX exposures using S&P options.

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.