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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?

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    $\begingroup$ I was also researching the paper and trying to reproduce the results in Python. Understand the authors use TSPL (time-shifted power laws) kernels to avoid blow-up when the lag vanishes. They also claimed that K1 and K2 in Figure 3.1 are produced by the optimal parameters of alpha and delta in Table 3, however, I simply tried and cannot produce the similar alike K1 and K2. $\endgroup$
    – Steven Wang
    Oct 20, 2022 at 16:23

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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.

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