Scaling VIX Implied Volatility for Realized Variance Comparisons

Summary

The document describes a comparison between intraday VIX-style implied volatility and realized variance computed from one-minute prices aggregated into hourly intervals. The implied measure is calculated from option prices for two maturities around 30 days, then interpolated to a 30-day horizon using the CBOE formula. Realized variance is calculated as the sum of squared log returns over the interval.

The author finds that the two series have very different numerical magnitudes and asks how to put them on compatible scales before regression. The document does not provide an answer or report a comparison result. A key limitation is that variance and volatility are distinct quantities, and comparisons also require matching the horizon and annualization convention. The stated calculation frequencies alone do not resolve those choices, so the intended regression would need consistent definitions before its output could be interpreted as a forecast test.

Key ideas

• The implied measure is derived from option prices and interpolated to a 30-day maturity.
• The realized measure sums squared log returns from intraday price observations.
• Variance and volatility use different scales, so the quantities need consistent definitions before comparison.
• A meaningful regression also requires aligned horizons and annualization conventions.
• The document poses the scaling question but supplies no resolution or test results.

Tags

# VIX and Realised Volatility Scaling


# VIX and Realised Volatility Scaling












I have calculated the VIX implied volatility according to the CBOE Whitepaper: \begin{equation*} \sigma^2 = \frac{2}{T} \left(\sum_i \frac{\Delta K_i}{K_i^2} Q(K_i) e^{rT} \right) - \frac{1}{T} \left( \frac{F_0}{K_0} - 1 \right)^2 \end{equation*} on an intraday basis (1-Minute) for two maturities around 30 days and interpolated the 30-Day Implied Volatility from there. I also have calculated the realised variance according to \begin{equation} RV = \sum_{t=0}^T [\ln(P_{t}) - \ln(P_{t-1})]^2 \end{equation} which I calculated for 1-hour intervals out of minutely recorded prices.

Theoretically the VIX impleid volatility is a forecast for the realised volaitlity in 30 days and one could check this via a simple linear regression. However, I can see from my data immideately that my RV results are in the range of $1\cdot10^{-5}$ and my VIX results in the range of $1\cdot 10^{-2}$. Most likely this is due to the VIX using another scale than the RV (note that I did not annualise my VIX data).

Can anyone point out ot me how to properly scale the VIX or RV so I can continue with the comparison?

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