# Are there volatility models dependent on returns?

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).

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?

• 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. – Brian B Jan 10 at 17:01

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.