When VIX Is a Suitable Proxy for Market Variance
Summary
The document asks whether squared VIX, converted to decimal units, is a suitable proxy for instantaneous market variance in a continuous-time variance model. Since VIX reflects option-implied expected volatility over the next month, its square represents a smoothed forward-looking measure rather than a direct observation of today’s instantaneous variance. That smoothing may make it a poor fit when the goal is to model daily or intraday variance movements.
The response frames proxy choice around the model’s purpose. For applications focused on long-period dynamics, such as martingale pricing or maximum-likelihood estimation, a smoother and more stable VIX-based measure may be adequate, even if it misses short-lived variance fluctuations. A trader studying intraday options behavior may need a more responsive measure. The example notes that the fitted model’s long-run volatility is broadly consistent with observations, but that agreement alone does not establish proxy quality for every use. The central caveat is that suitability depends on the time horizon and intended conclusions.
Key ideas
- Squared VIX reflects option-implied expected variance over a forward period, not instantaneous variance alone.
- The forward averaging in VIX smooths short-term variance movements.
- A smoothed proxy may be adequate for models focused on longer-period dynamics.
- Intraday trading and short-horizon analysis may require a more responsive variance measure.
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Full text
# Can VIX be interpreted as a proxy for instantaneous volatility?
# Can VIX be interpreted as a proxy for instantaneous volatility?
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"?
## Answer by Brian B (score 7, accepted)
https://quant.stackexchange.com/a/2094
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.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.