The VIX Index

The VIX is the market's "fear gauge," but under the hood it is a precise object: the model-free implied volatility of the S&P 500 over the next 30 days, computed as the square root of a variance-swap fair strike. It is not a Black-Scholes implied vol of any single option; it is a $1/K^2$-weighted average across the whole option chain, the exact discretized version of variance replication. Understanding the VIX means understanding that formula and the futures term structure built on top of it.

Model-free implied variance

Recall from variance swaps that the fair variance is a $1/K^2$-weighted integral of out-of-the-money option prices, no volatility model required. The VIX is that integral, discretized over the S&P 500 option chain and annualized. The CBOE formula for the variance of a single expiry is

$\sigma^2 = \frac{2}{T}\sum_i \frac{\Delta K_i}{K_i^2}\,e^{rT}\,Q(K_i) \;-\; \frac{1}{T}\Big(\frac{F}{K_0}-1\Big)^2,$

where the sum runs over all listed OTM options, $Q(K_i)$ is the mid-price of the OTM option at strike $K_i$ (puts below the forward $F$, calls above), $\Delta K_i$ is the strike spacing, $K_0$ is the highest strike below $F$, and the final term is a small correction for the forward not sitting exactly on a strike. This is precisely the log-contract strip made discrete: each option enters with the $1/K_i^2$ variance weight.

From two expiries to 30 days

The VIX targets a constant 30-day horizon, but listed options rarely expire exactly 30 days out. CBOE computes $\sigma^2$ for the two expiries bracketing 30 days (the "near" and "next" term) and interpolates in total variance (variance is additive in time) to a 30-day figure, then annualizes and takes the square root:

$\text{VIX} = 100 \times \sqrt{\;\Big[T_1\sigma_1^2\,\tfrac{N_{T_2}-N_{30}}{N_{T_2}-N_{T_1}} + T_2\sigma_2^2\,\tfrac{N_{30}-N_{T_1}}{N_{T_2}-N_{T_1}}\Big]\times\frac{N_{365}}{N_{30}}}.$

The $\times 100$ makes it a percentage; a VIX of 20 means the market's risk-neutral expectation of S&P realized vol over the next 30 days is 20% annualized. Since October 2014 the VIX has used weekly options to bracket 30 days precisely.

What the VIX is, and is not

• It is risk-neutral, so like any implied vol it sits above the market's true (physical) forecast of realized vol by the variance risk premium. VIX minus subsequent realized vol is, on average, positive.
• It is not tradeable directly, you cannot hold "the VIX." Exposure comes through VIX futures and VIX options, whose values depend on the forward 30-day variance at the future date, not on today's spot VIX. This disconnect is the source of most VIX-product P&L.
• It is a variance average dressed as a vol number: because of the $1/K^2$ weighting, the wings (crash puts) contribute heavily, so the VIX spikes when tail protection is bid, hence "fear gauge."

Futures, term structure, and roll

VIX futures settle to the spot VIX at their expiry, so they price the market's expectation of future 30-day implied vol. The term structure of VIX futures is usually in contango (upward-sloping) in calm markets, distant futures above spot VIX, because vol is mean-reverting and the risk premium is positive. In stress it inverts to backwardation, spot spikes above the futures, which price a return to normal.

This shape drives the notorious behaviour of VIX ETPs. A long-vol product like VXX rolls from a cheaper near future into a richer next future every day; in persistent contango this negative roll yield bleeds value continuously, VXX loses to decay even when spot VIX is flat. The inverse (short-vol) products like the former XIV harvest that roll, until a vol spike (February 2018's "Volmageddon") wiped out XIV in a day when backwardation and a VIX doubling turned the carry trade lethal. The roll is the carry; the tail is the risk.

Worked example

The near-term S&P variance computes to $\sigma_1^2 = 0.040$ ($T_1 = 23$ days) and the next-term to $\sigma_2^2 = 0.045$ ($T_2 = 37$ days). Interpolating the 30-day variance and annualizing gives roughly $\sigma^2_{30} \approx 0.042$, so $\text{VIX} = 100\sqrt{0.042}\approx 20.5$. If one-month VIX futures trade at 22 while spot VIX is 20.5, the curve is in contango: a long VXX position rolling up that 1.5-point slope loses roll yield daily until either the curve flattens or a spike rewards the long.

What breaks in practice

• Wing truncation and the discreteness correction. The sum is truncated where option quotes give out (CBOE stops after two consecutive zero-bid strikes), biasing the index; in a fast crash the missing deep-OTM puts mean the VIX can understate true fair variance.
• Microstructure and stale quotes. The index is a weighted sum of mid-quotes; wide spreads and stale prices in the illiquid wings inject noise, and the opening/closing auctions have caused documented settlement-manipulation concerns.
• Spot-vs-future confusion. The single most common trader error is treating VIX futures or VXX as tracking spot VIX. They track forward variance and carry roll, VXX can fall 15% in a month the VIX is unchanged.
• Jumps and the 30-day cliff. The replication assumes the variance-swap idealization; jumps and the discrete 30-day interpolation introduce basis versus a true 30-day variance swap.

In interviews

Explain that the VIX is model-free implied variance, i.e. the square root of a 30-day variance-swap fair strike, a $1/K^2$-weighted strip of OTM S&P options, not a Black-Scholes IV of one option. Write or sketch the CBOE formula $\sigma^2 = \frac{2}{T}\sum\frac{\Delta K}{K^2}e^{rT}Q(K) - \frac1T(F/K_0-1)^2$ and note the two-expiry interpolation to 30 days. Be sharp on why it is a risk-neutral expectation (above physical, variance risk premium) and on the term structure: contango normally, backwardation in stress, and the negative roll yield that decays long-vol ETPs, the Volmageddon story is the canonical example. Distinguish spot VIX (untradeable) from VIX futures (forward variance).