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BJO06 (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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Hi brianjd, welcome to quant.SE and thanks for posting your question. – Tal Fishman Oct 3 '11 at 20:30
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.. – Beer4All Oct 4 '11 at 13:32
@Beer4All: Thx for pointing this out. Makes sense if VIX is thought of as a 30-day-MA over future daily volatility – lowndrul Oct 4 '11 at 22:22
up vote 4 down vote accepted

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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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. – lowndrul Oct 4 '11 at 22:30
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. – onlyvix.blogspot.com Jul 21 '15 at 0:17

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