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Why a VIX Hedging Regression Residual Can Drift

Article Quant Q&A · Author: Volwiz
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Summary

The document examines a VIX change hedged against S&P 500 returns using a zero-intercept linear regression. The resulting residual has very low correlation with the equity return series, yet shows apparent drift and a chart resembling the index. The author also considers whether a changing regression slope or volatility term-structure decay could explain the behavior.

The response explains that an OLS residual is uncorrelated with the regressor by construction, which does not imply mean reversion or absence of drift. A trend in index levels can make plots look similar even when return correlation is near zero. Omitting the intercept can leave the residual with a nonzero mean, and financial residuals may be autocorrelated. These points are statistical cautions, not proof of a tradable effect; the post does not supply tests of persistence, trading costs, or risk-adjusted performance.

Key ideas

  • An OLS residual’s zero sample correlation with its regressor does not imply mean reversion.
  • A zero-intercept regression can produce residuals with a nonzero average.
  • Similar-looking level plots can coexist with low correlation between returns.
  • Autocorrelation in financial residuals can contribute to apparent drift.

Tags

StatisticsVolatilityEquitiesRisk management
Full text
# VIX vs S&P: Drift in the hedging residual?


# VIX vs S&P: Drift in the hedging residual?












I am looking at the daily returns of the VIX index (dVIX ) and the daily returns of the S&P 500 (dS).

I am running a linear regression (using 0 intercept) and get a regression slope of -1.4, i.e.

dVIX = -1.4 * dS/S + epsilon

R^2 is around 70%.

Now I am looking at dVIX + 1.4 * dS/S. The way I think about this is that I hedge out the spot movements from the VIX. As expected, the correlation between this construct and dS/S is very low, around -2%.

Naively I had first expected that this would give me something that mean reverts, but actually I have some strong drifts in there:

Also, even though the correlation to spot is almost 0 it still "looks" similar to the plot of the S&P:

I first thought that maybe using a static regression slope over the full time period is not accurate, so I used a moving regression using exponentially weighted moving averages, but the result is very close to what I got above. Also the regression slope is very similar for different parts of the full time horizon, so that cannot be the reason.

Can anybody explain what is happening here?

I am aware that if I actually wanted to trade this, I would have to consider the decay of the vol curve, i.e. my VIX future would move down the vol term structure (if the vol term structure is downward sloping as it would be on average). I have a way of calculating the vol slope from market data (explaining all this would make this post too complicated and long), but even after considering this I still have quite a significant drift.

For background information (not necessary to read to understand my question), the reason why I was looking into this the following: For S&P 500 options the ATM volatility is strongly correlated to spot moves. So I was hoping to modify my delta hedge, so that it does not only remove the delta move, but also the spot exposure in the vega move, so that the option price would mainly be moved by theta and the higher order terms, i.e. gamma, volga, vanna etc.

## Answer by fes (score 1)

https://quant.stackexchange.com/a/55784

Here is an answer from a statistical angle:

- Your hedged VIX corresponds to a regression residual.

- If you estimate the regression by OLS, the residual will by construction be uncorrelated with stock returns.

- The two plots look similar because of a constant trend in stock prices that does not affect the correlation between returns and the VIX.

- If you do not include a constant in the regression, the residual (hedged VIX) will not be zero on average. This tends cause statistical issues and the coefficient estimates are "biased".

- In financial applications, it is very common to find autocorrelation in residuals, hence there can be "drifts" in the "hedged VIX"

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.