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When I look at the relationship between volatility and price, I see a clear negative correlation as shown in this figure (SPY and VIX prices today looking back 1 year). enter image description here

The common volatility models (GARCH, Heston, etc.) do not seem to exploit this correlation. I'm sure they exist, but I just haven't found them. Can anyone point me towards models that do?

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    $\begingroup$ The Heston model explicitly includes a stock-volatility correlation parameter $\rho$, which is of course strictly less than zero in nearly all practical use cases. $\endgroup$ – Brian B Jan 10 '20 at 17:01
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The Heston model can have that property. If you make the correlation negative between the Brownian motions in the $dS_{t}$ process and the $d\nu_{t}$ process you imply that price is negatively correlated with variance.

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  • $\begingroup$ This is key. I forgot (or didn't realize) the two noise sources in the Heston model are correlated. $\endgroup$ – EpicAdv Jan 11 '20 at 16:12
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Yeah this is often called Spot-Vol correlation and is well known. Most people take this into account. I think if you just google spot-vol correlation you will come up with many example/models.

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