I'm trying to construct a model which shows how much the closing price of a security ($P_t$) differs from the VWAP of that security on that day ($VWAP_t$). I'm calling this measure the "VWAP Premium": $$VWAP_{premium} = \frac{VWAP_t}{P_t}-1$$ By simply plotting this for the MSCI ACWI ETF, I see that it exhibits heteroskedasticity, but not necessarily any other trend:

Time Series of VWAP Premium for ACWI ETF

One thought I had was that intraday volatility ($\sigma_{ID}$) could help signal these larger absolute values of $VWAP_{premium}$, so I plotted the scatterplot and it does indeed look like that's the case:

Scatterplot of Intraday Volatility vs. VWAP Premium

Days with larger intraday volatilities also are more likely to have larger absolute VWAP Premiums, which is intuitive enough. My question is, how is this model fitted? It's not a GARCH model, since the heteroskedasticity is not dependent on prior volatility, but on another variable altogether. It seems to me like this would be something like:

$$VWAP_{premium} = (\sigma_{ID})(\sigma_{VWAP})$$

Is there a simple way to fit this model in Python?



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