Estimating the Equity Sensitivity of VIX Futures

Summary

The document considers how to estimate the relationship between VIX futures and S&P 500 returns, with the goal of separating an equity-driven component from a volatility-related component. It proposes two basic empirical approaches: a ratio of VX to ES returns over a historical window, or a regression of VX returns on ES returns using a rolling or expanding sample. The response considers linear regression a reasonable model-free estimate when no particular stochastic-volatility model is assumed, and suggests expressing the regression with the S&P return as the explanatory variable.

It cautions that this estimated relationship is not a derivative delta in the strict sense. Under a fractional stochastic-volatility view, the formal delta of a VIX future with respect to the underlying equity index is zero because the future is a volatility derivative. The discussion provides no data, estimation window, performance evidence, or implementation details. Its estimates should therefore be understood as statistical co-movement measures, not structural hedge ratios guaranteed to isolate pure volatility returns.

Key ideas

• A return ratio or regression can estimate the historical co-movement of VIX futures and S&P returns.
• A regression can serve as a model-free estimate when no specific stochastic-volatility model is assumed.
• The response distinguishes statistical sensitivity from delta in the strict derivatives-pricing sense.
• Under the stated stochastic-volatility view, the formal equity delta of a VIX future is zero.
• The document supplies no empirical validation or guidance on selecting a sample window.

Tags

# Estimating delta of VX futures to S&P 500


# 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:

- Simple statistical relationship (VX returns / ES returns over some historical window)

- 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?

## Answer by user34971 (score 1)

https://quant.stackexchange.com/a/60722

It depends. If you believe in (fractional) stochastic volatility then the delta, in the strict sense of the word, is zero, since the VIX future is a volatility derivative.

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.

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