# Estimating delta of VX futures to S&P 500

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?

• I'm afraid I don't have a good answer but I would add that you need to consider the futures roll as it will affect both the futures return and the beta to the S&P 500. Jan 26 '21 at 21:57

A simple linear regression is probably not such a bad idea to estimate the "delta" of the VIX future wrt to SPX if you do not believe in / assume any particular model. It is more natural however to write $$x = a + \beta y$$ where $$y$$ is the SPX return. The "delta" in this case is correlation, which is not delta in the strict sense.