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Testing Forecast Bias in the VIX Term Structure

Article Quant Q&A · Author: Thessa
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Summary

The document presents a question about applying a regression from Asensio (2013) to test forecast bias in the VIX term structure, motivated by the possibility that VIX futures are overpriced. It identifies the equation’s forecast errors and its horizon indices, but the author is unsure how to map them to VIX and VIX futures observations. The post also raises a related question about adapting a risk-neutral expectations formula attributed to Campa and Chang, including its tenor adjustment, to the bias regression.

No resolution or empirical results are provided: the document consists of the original question and its equations. It therefore serves as a prompt for clarifying the interpretation of the time indices and the relationship between option-based variance expectations and VIX futures data. Any application would need to consult the cited paper’s definitions and assumptions; the post does not establish that VIX futures are overpriced or give a tested procedure for evaluating that claim.

Key ideas

  • The post asks how the horizons in Asensio’s regression map to VIX and VIX futures observations.
  • The regression is presented as a way to examine forecast bias in the VIX term structure.
  • The author identifies forecast errors and horizon indices but is uncertain how to choose their values.
  • A related question concerns adapting a risk-neutral expectations formula to VIX futures.
  • The document supplies no answer, empirical evidence, or validated application procedure.

Tags

VolatilityFuturesStatisticsDerivatives pricing
Full text
# Forecast biasness of VIX term structure


# Forecast biasness of VIX term structure












I'm interested in the topic of VIX futures being overpriced, so I'm looking for different models to find evidence for it. Asensio 2013 uses a regression to evaluate the forecast biasness of the VIX term structure. $$ \left(\frac{1}{k}\right)\sum_{i=1}^{k-1}\left[(\sigma^{VIX}_{i,i\rightarrow i+j})^2-(\sigma^{VIX}_{0,0\rightarrow j})^2\right] = \alpha_0 + \beta_0 \left[(\sigma^{VIX}_{0,0\rightarrow k+j})^2-(\sigma^{VIX}_{0,0\rightarrow j})^2\right] + \sum_{i=1}^{k-1}u_i\,. $$ That's equation (14) in the paper, such that

- $u_i$ represent the expectational errors.

- $j$ represents the future date.

- $k$ represents the number of periods.

Now I am asking myself, due to confusion of these indices, which values should I plug into this equation, when I want to apply this to the VIX and VIX futures.

I would be very thankful if someone could explain this to me!

EDIT: First, thanks to Farahvartish for correcting my first version.

Second, Asensio uses in a previous step (equation 13) Campa and Chang's (1995) risk neutral formula to test expectations hypothesis:

$$ \sigma^2_{0,km} = (\frac{1}{k})E_0[\sum_{i=0}^{k-1}\sigma^2_{im,(i+1)m}](\frac{\theta_{km}}{\theta_m}) $$

- $\theta$ represents the concavity adjustment for a given tenor (assumed to be 1).

- $m$ represents the number of months until expiration.

- $k$ represents the number of periods.

Originally this formula is applied to options. I don't know how to use it for VIX futures and how to imply it to the formula of forecast biasness. Is it that: $$\sigma^2_{0,km} = (\sigma^{VIX}_{0,0\rightarrow j})^2$$

Again, thanks for your help!

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