How VIX Interpolates Near- and Next-Term Implied Volatility

Summary

The document presents the VIX calculation's time-weighted interpolation of variance from near-term and next-term SPX options to a constant 30-day horizon. It gives the formula and defines the time inputs in minutes and years, including minutes to each option settlement and the minutes in a 30-day period and a 365-day year. The result is expressed as a volatility after combining the two maturity contributions and annualizing.

The text asks why this interpolation is appropriate and how it was derived, but supplies no answer, derivation, empirical evidence, or comparison with alternative methods. It therefore serves as a statement of the calculation and a conceptual question rather than a complete explanation of VIX construction. It also does not cover how each maturity's implied variance is obtained from option prices, so the formula alone is not enough to reproduce the full index calculation.

Key ideas

• The VIX calculation combines implied variance from near- and next-term options.
• The weights depend on each maturity's distance from the target 30-day horizon.
• The formula uses minutes to settlement and converts the interpolated result to annualized volatility.
• The document poses, but does not answer, the question of why the interpolation is suitable or how it was derived.
• It does not explain how implied variance for each option maturity is calculated.

Tags

# How does VIX interpolate implied volatilities?


# How does VIX interpolate implied volatilities?












In the CBOE VIX white paper (direct link to PDF), it is explained that once the implied volatility of the near and next-term options $\sigma_1^2$, $\sigma_2^2$ are found, the constant-maturity 30-day implied volatility is calculated as follows:

$$\sqrt{\left\{T_1\sigma_1^2\left[\frac{N_{T_2}-N_{30}}{N_{T_2}-N_{T_1}}\right]+T_2\sigma_2^2\left[\frac{N_{30}-N_{T_1}}{N_{T_2}-N_{T_1}}\right]\right\}\times\frac{N_{365}}{N_{30}}}$$

where $$\begin{align} N_{30} &= \text{minutes in $30$ days}\\ N_{365} &= \text{minutes in a $365$-day year}\\ N_{T_1} &= \text{minutes to settlement of near-term options}\\ N_{T_2} &= \text{minutes to settlement of next-term options}\\ T_1 &= \text{time to expiration (in years) for near-term options, i.e.} N_{T_1}/N_{365}\\ T_2 &= \text{time to expiration (in years) for next-term options, i.e.} N_{T_2}/N_{365}\\ \end{align}$$

Now, the formula makes somewhat of an intuitive sense, but I'd like to understand exactly why it's a good method to interpolate implied volatilities and how it was derived. I have searched extensively, both on Google and here, but I didn't find an explanation.

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