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The correlation between the index returns (e.g SPX) and its changes in option-impled volatility (e.g. VIX), is strong, stable and negative (the implied volatility feedback effect). To me at least, it follows intuitively that index-options with lower strikes should price higher in standard (Black) implied volatility; expecting to see the vola-skew. This goes for indices and index-options but not for the corresponding relationships for individual stocks.

My first question: isn't it settled then that the SPX/VIX correlation is there to make intertemporal risk/return pricing of the underlying index versus its expected risk closer to buy-and-hold optimality? Or to put it the other way, isn't the leverage story ruled out by the weak correlations and skews for returns and implied volatility innovations for specific stock return components?

My second question: is there ways to read out, infer, calibrate with, directly model, the VIX/SPX correlation in standard (Heston?) or novel (index-) option pricing models?

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On the first question, yes, it is settled, VIX-beta is a risk thing rather than a balance-sheet leverage story according to some papers at least.

On the second, no, you need to look at option prices to match option pricing. However, Heston correlation between the volatility and return processes has a negative sign, just like it has between time-series of VIX differences (implied volatility changes) and SPX (index) returns.

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    $\begingroup$ If you're assuming a SV model describes the SPX then the short-dated covariance can actually be approximated without knowing the exact form of skew generating process. Things get trickier if you assume a LSV process for example: it's always possible to assume the correlation between the stochastic vol and the spot price is zero and still match option prices due to the local vol component. $\endgroup$
    – Frido
    Commented Nov 28 at 13:50
  • $\begingroup$ Thanks, yes, it is SV I am thinking about. $\endgroup$
    – Mats Lind
    Commented Nov 29 at 8:16

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