I'm trying to think about the right way to estimate the delta of a VX contract to the S&P 500. VX futures are on the VIX index, which is a basket of S&P 500 options. By extension, VX and ES (E-mini futures on S&P 500) returns have a strong contemporaneous relationship. Having a good estimate allows for construction of a portfolio where you isolate the return of the "pure volatility" move, stripping out the returns due to equities.

I have thought of some very rudimentary ways to estimate it:

1. Simple statistical relationship (VX returns / ES returns over some historical window)
2. Linear regression using expanding/rolling window (y = mu + beta(x) + error, y=ES returns, x=VX returns)

Both of these seem very naïve. There seems to be very little literature on this topic, are there any better estimators?

I'm trying to think about the right way to estimate the delta of a VX contract to the S&P 500. VX futures are on the VIX index, which is a basket of S&P 500 options. By extension, VX and ES (E-mini futures on S&P 500) returns have a strong contemporaneous relationship. Having a good estimate allows for construction of a portfolio where you isolate the return of the "pure volatility" move, stripping out the returns due to equities.

I have thought of some very rudimentary ways to estimate it:

1. Simple statistical relationship (VX returns / ES returns over some historical window)
2. Linear regression using expanding/rolling window (y = mu + beta(x) + error, y=VX returns, x=ES returns)

Both of these seem very naïve. There seems to be very little literature on this topic, are there any better estimators?