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Bakshi et al., (2006) Estimation of continuous-time models with an application to equity volatility dynamics (Table 2) estimate the following Cox-Ingersoll-Ross model for market variance, $\sigma^2_t$:

$\mathrm{d}\sigma^2_t = (\alpha_0 + \alpha_1\sigma^2_t)\mathrm{d}t + \sqrt{\beta_1}\sigma_t\mathrm{d}W_t$

To estimate their model they use $\left(\frac{VIX_t}{100}\right)^2$ as a proxy to $\sigma^2_t$, where $VIX_t$ is the daily VIX price.

But VIX measures expected volatility (in percentage terms) of the market over the next 30-day period (as implied by S&P index options). So $\left(\frac{VIX_t}{100}\right)^2$ is basically a moving average over future daily market variance. This extra MA structure makes it a poor proxy to the true instantaneous market variance $\sigma^2_t$---especially when trying to model daily market variance dynamics.

What am I missing? Have I misunderstood something? Or have I understood things correctly and using the $VIX_t$ proxy is considered a "good enough" approach?

NOTE: The authors do end up estimating $(\alpha_0, \alpha_1, \beta_1) = (0.3141, -8.0369, 0.1827)$, which implies a long-run market volatility of 0.20 in annualized terms. That more or less agrees with observations. So maybe "good enough"?

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    $\begingroup$ Hi brianjd, welcome to quant.SE and thanks for posting your question. $\endgroup$ – Tal Fishman Oct 3 '11 at 20:30
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    $\begingroup$ VIX index (or V2X for the Eurostoxx50) cannot be used to estimate the instantaneous volatility. It is rather used for long-period comparison purpose. For example let's consider an article of 2003 of Polson and Stroud. A forecast for the Heston instant volatility is built from stock historic data. Then turning to real data, they compare VIX and the historic instant vol estimator on a 10Y period. The two curves have almost the same behavior. However, on a 1Y comparison one observes an excessive smoothness of the VIX and the lack of interactivity of this index compared to the instant estimator.. $\endgroup$ – Beer4All Oct 4 '11 at 13:32
  • $\begingroup$ @Beer4All: Thx for pointing this out. Makes sense if VIX is thought of as a 30-day-MA over future daily volatility $\endgroup$ – lowndrul Oct 4 '11 at 22:22
  • $\begingroup$ Here's my 5 cents. VIX does not estimate Local Volatility, for that you can look at Ait-Sahalia + Jacod book named High-Frequency Financial Econometrics. In fact, if you estimate the local volatility (under ito semimartingale price movements) and compare it to the VIX, you get two series with very different behaviors, especially across important news announcements (like the FOMC). In fact, across FOMC announcements the average Local Volatility jumps up, but the average VIX does not change. $\endgroup$ – Guilherme Salomé Feb 15 '18 at 23:44
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    $\begingroup$ The link to the paper is broke. It would be better if you write out the title and the author of the paper. $\endgroup$ – Hans Mar 22 '18 at 0:13
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You have to ask yourself what the ultimate purpose of this parameterization is. In their case, they imply the "end-goal is martingale pricing or maximum-likelihood estimation", both of which are ultimately about capturing long-period dynamics rather than intraday or interday behavior.

For this reason, the fact that intraday variance may, ahem, vary around a smoother VIX estimate is not so material to any ultimate conclusions. And the resulting model has the advantage of working off a much better-behaved estimator than you get by actually computing live variances of underlying price series.

Someone trading options intraday might well avoid this parameterization, but for 1-week risk or exotics vauation I see it as sensible.

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  • $\begingroup$ Got it. Still not sure why they just wouldn't use end-of-month VIX to estimate their model if long-run dynamics are the interest. They'd at least avoid using a proxy time series with an embedded MA component. $\endgroup$ – lowndrul Oct 4 '11 at 22:30
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    $\begingroup$ What do you mean by "end-of-month VIX" Once a month? Then they will have 20x less observation points, and estimate will not be nearly as efficient. $\endgroup$ – onlyvix.blogspot.com Jul 21 '15 at 0:17

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