CBOE VIX Index Formula & Replicating Portfolio Lab

Compute the official CBOE Volatility Index (VIX) from discrete options chains. Analyze $1/K^2$ log-contract replication, zero-bid truncation, and 30-day calendar roll interpolation.

Market Surface Presets:
30-Day VIX Level

14.25

Annualized %
ATM Implied Vol

13.50%

Moneyness K = F
Skew Premium

+0.75 pts

VIX − ATM Vol
Forward Index (F)

$5,005.20

Near-Term Term
Active Options

23 of 33

Excl. Zero-Bids
Roll Weights (w₁ / w₂)

14.3% / 85.7%

30-Day Maturity
Index & Surface Inputs
$
%
%
%
% / Log(K/F)
Controls how steeply OTM put implied volatility climbs.
%
Curvature of out-of-the-money wings.
Official CBOE Formula

Single-Term Variance ($\sigma_j^2$):

σ² = (2/T) Σ [ΔK / K² · e^(RT) · Q(K)] − (1/T) [F/K₀ − 1]²

30-Day Roll Interpolation:

VIX = 100 × √{ [T₁ σ₁² w₁ + T₂ σ₂² w₂] × (365 / 30) }
Near-Term Options Chain Execution Audit (T₁ = 24d)
Filtered by Zero-Bid Cutoff
Strike ($K) Moneyness (K/F) Type Bid ($) Ask ($) Mid Q(K) ΔK ($) Contribution (×10⁻⁴) CBOE Status
Sensitivity: Spot Price vs. Put Skew on VIX Level
Demonstrates how downside market crashes simultaneously elevate VIX via spot drop and steeper put skew.
Sensitivity: Smile Curvature vs. ATM Vol on VIX Spread (VIX − ATM)
Quantifies the Volatility Risk Premium (VRP) and wing convexity effect lifting VIX above at-the-money volatility.

Quantitative Derivatives Guide: The CBOE VIX Architecture

1. Model-Independent Variance & The Log Contract

In 1993, the original VIX was introduced based on at-the-money implied volatilities of S&P 100 options using the Black-Scholes formula. In September 2003, CBOE radically redesigned the VIX in collaboration with Goldman Sachs to become a model-independent measure derived from S&P 500 (SPX) index options.

The foundational breakthrough stems from continuous-time finance (Neuberger 1994; Demeterfi, Derman, Kamal & Zou 1999). By Ito's Lemma, holding a continuous portfolio of out-of-the-money calls and puts weighted by $1/K^2$ synthesizes the payout of a log contract $\ln(S_T / F)$, which eliminates all directional delta risk and leaves pure realized variance:

E^Q[∫₀ᵀ σₜ² dt] = 2 e^(RT) [ ∫₀^F (F - K)/K² P(K) dK + ∫_F^∞ (K - F)/K² C(K) dK ]

The CBOE formula discretizes this continuous integral into finite strike intervals $\Delta K_i$, creating a static replicating portfolio that trades directly in liquid options markets.

2. The Anatomy of Zero-Bid Truncation & The Forward Term

Real-world options markets have discrete strikes and liquidity limits. Two essential adjustments ensure robustness:

  • Put-Call Parity Forward ($F$): Rather than relying on underlying index cash quotes (which can suffer from non-synchronous stock trading), the forward price $F$ is extracted directly from the option chain at the strike $K$ with the minimum absolute difference $|C_{\text{mid}} - P_{\text{mid}}|$ via $F = K + e^{RT}(C - P)$.
  • Zero-Bid Cutoff: Scanning starts at the at-the-money strike $K_0$ and proceeds outward. Once two consecutive strikes have a bid of $\$0.00$, scanning stops immediately. This guards against stale, wide quotes in illiquid deep OTM wings from distorting the integral.
  • Forward Adjustment: The term $-\frac{1}{T}\left(\frac{F}{K_0} - 1\right)^2$ subtracts the slight difference between the theoretical forward $F$ and the discrete strike $K_0$ immediately below it.
CBOE VIX Mastery Quiz

1. Why does the CBOE VIX formula weight out-of-the-money options by $1/K^2$?

2. What triggers the CBOE zero-bid truncation cutoff?

3. Why is the calculated VIX typically higher than the at-the-money (ATM) implied volatility?

4. How does the CBOE ensure the VIX represents an exact 30-day constant maturity?

Frequently Asked Questions

The CBOE Volatility Index (VIX) is a benchmark index measuring 30-day expected annualized volatility of the S&P 500 index. Unlike historical realized volatility, it is calculated directly from market prices of out-of-the-money SPX call and put options across two consecutive expirations (near-term and next-term) using a model-independent discrete integral formula.

The 1/K^2 weighting arises from the mathematical synthesis of a variance swap's log contract ln(S_T/F). Neuberger (1994) and Demeterfi et al. (1999) proved that an asset's continuous variance can be replicated by a dynamic delta hedge combined with a static portfolio of options whose weights are proportional to 1/K^2. This weighting exactly neutralizes the contract's nonlinear spot exposure.

The CBOE zero-bid rule dictates that options scanning starts at the at-the-money strike K_0 and moves outwards into puts (downward) and calls (upward). Once two consecutive strikes have a bid price of zero ($0.00), scanning terminates immediately and all further out-of-the-money strikes are excluded. This prevents illiquid, stale, or wide quotes in the deep wings from distorting the calculated index.

Because standard options do not expire exactly 30 days into the future on every trading day, the CBOE calculates the single-term variance for both a near-term expiration (T_1, between 23 and 30 days) and a next-term expiration (T_2, between 31 and 37 days). It then performs minute-level linear interpolation between the two variances to fix the synthetic maturity at exactly 30 calendar days (43,200 minutes).

The VIX formula is mathematically identical to the fair strike of an annualized 30-day variance swap before convexity adjustment. Because holding a portfolio of options weighted by 1/K^2 replicates the log payoff, the square of the VIX represents the annualized risk-neutral expectation of total integrated variance E^Q[int_0^T sigma_t^2 dt] / T.
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