Volatility Innovations as Unexpected Changes in Forecasts
Summary
The document explains a volatility innovation as the part of future volatility that was not captured by the forecast made with currently available information. If realized volatility differs from its prediction, that forecast error is the innovation. In time-series terms, a well-formed forecast error is unpredictable from the information used to make the forecast; the excerpt also notes the connection between innovations and noise terms in stochastic models.
The discussion links this concept to a study’s claim that stocks more sensitive to aggregate volatility innovations have lower average returns. One explanation offered is that investors value assets that perform favorably when volatility is unexpectedly high, since such surprises complicate portfolio decisions and increase uncertainty. Greater demand could then be associated with lower expected returns. This is an interpretation rather than a demonstrated causal result in the excerpt. The answer also gives a GARCH-style example, but its notation is simplified, so readers should distinguish the return shock from the resulting forecast update in a full model.
Key ideas
- A volatility innovation is the unexpected component of future volatility relative to the available forecast.
- Forecast errors should be unpredictable using the information that formed the forecast.
- Sensitivity to volatility innovations measures how an asset relates to unexpected volatility changes.
- The proposed explanation for lower returns is that investors may value assets that hedge undesirable volatility surprises.
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Full text
# What is meant by innovations in volatility?
# What is meant by innovations in volatility?
I am currently reading about stocks with "high sensitivity to innovations in aggregate volatility". I am not a native speaker so this might be a trivial question, but I truly cannot find an answer anywhere on the internet. Ang, Xing and Zhang (2006) state that "stocks with high sensitivities to innovations in aggregate volatility have low average returns".
Currently I am reading it as follows: Stocks that are sensitive to market volatility developments/movements/changes have low average returns. Is this how I should interpret the sentence stated before? I have found a book online describing stochastic volatility, but it's all relatively new to me.
Many thanks!
## Answer by Freelunch (score 5)
https://quant.stackexchange.com/a/37882
A volatility innovation is the difference between our best prediction of future volatility and what is actually observed. Say that we predict the volatility at the next time step as $E_t[ \sigma_{t+1}] = \hat{\sigma}_{t+1}$, but instead observe $\sigma_{t+1} = \hat{\sigma}_{t+1} + \varepsilon_{t+1}$. Here $\varepsilon_{t+1}$ is the innovation, the unpredictable component of future volatility.
Unpredictability in the volatility is bad since it creates an uncertainty in the optimal portfolio decision, especially when the volatility is higher than expected. This means that assets which are highly correlated to the innovations $\varepsilon_{t+1}$ (that have high sensitivities) are attractive and will hence have lower average return as demand increase the price of those assets (by assumption).
## Answer by vanguard2k (score 2)
https://quant.stackexchange.com/a/37883
Without context, I am not 100% sure but this is my interpretation:
Just from wikipedia:
> In time series analysis (or forecasting) — as conducted in statistics, signal processing, and many other fields — the innovation is the difference between the observed value of a variable at time t and the optimal forecast of that value based on information available prior to time t. If the forecasting method is working correctly, successive innovations are uncorrelated with each other, i.e., constitute a white noise time series.
Or more in relation to your case:
If you look at a time series model for volatility, say for example a GARCH model:
$$ h_t = \omega + \alpha \epsilon_{t-1}^2 + \beta h_{t-1}$$
then $\epsilon_{t-1}$ would be the innovation. In the discrete time models, the Brownian term takes the role of the innovator. In a Heston model for example, you have one mean reverting term and one innovation term.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.