VIX Futures & Options Pricing Lab

Model mean-reverting term structures, roll yield carry, Black-76 option Greeks, positive volatility skew, and cumulative ETP drag.

Market Presets:

Typical bull market equity regime: Spot VIX trades well below the long-term mean. Upward-sloping curve creates negative roll carry for long volatility ETPs (VXX/UVXY decay).

Parameters

14.50
Current cash VIX index level.
19.50
Long-term equilibrium volatility attractor.
1.80
Annualized speed of return to mean attractor.
85.0
Implied volatility of the VIX futures contract.
4.50
Annualized short-term risk-free discount rate.
18.00
Exercise price of target VIX option.
30
Time until option and future expiration.
0.35
Upward smile steepness for OTM VIX calls.
Underlying Future
15.22
Expiration $F(T)$
VIX Call Price
$1.42
Black-76 with Skew
VIX Put Price
$3.98
Black-76 with Skew
Roll Yield
-22.4%
Contango
Spot Delta ($\Delta_{\text{Spot}}$)
0.245
$\Delta_{\text{Fut}} \times e^{-\kappa T}$
Futures Delta ($\Delta_{\text{Fut}}$)
0.312
Direct Futures Hedge
Vega (VVIX)
$0.048
Per 1% VVIX move
Front Month (M1) vs Spot: +0.72 pts M2 vs M1 Calendar Spread: +0.68 pts Daily Roll Decay: -0.089% / day
Skew-Adjusted VVIX: 88.5% Gamma ($\Gamma$): 0.0412 Theta ($\Theta$): -$0.024 / day
Demonstrates 120-day cumulative compounding drag of daily 1/21st front-to-second month rolling for constant term structure regimes.

VIX Futures Term Structure Grid

Contract Tenor Label Days to Expiration Theoretical Future Price Spread vs Spot VIX

Matrix 1: Spot VIX vs Reversion Speed on 1M Future ($F_1$)

Matrix 2: Strike vs Vol-of-Vol (VVIX) on 30-Day Call Price ($)

Institutional Theory: VIX Futures, Roll Yield & Options Mechanics

1. Absence of Cash-and-Carry Spot Parity

Unlike equity or commodity futures, the CBOE VIX index is a computed 30-day implied volatility statistic derived from S&P 500 option prices. Because spot VIX cannot be bought, stored, or borrowed in cash markets, standard no-arbitrage cost-of-carry parity does not apply:

$$F_t(T) \neq V_t \cdot e^{(r - q)T}$$

Instead, VIX futures trade as expectations of future realized 30-day volatility plus an embedded Volatility Risk Premium (VRP) demanded by institutional market makers providing short volatility liquidity.

2. Risk-Neutral Mean-Reversion Dynamics

In the framework of Grunbichler & Longstaff (1996) and Cox-Ingersoll-Ross (CIR), VIX follows an affine mean-reverting square-root diffusion:

$$dV_t = \kappa^*(\theta^* - V_t) dt + \sigma_v \sqrt{V_t} dW_t^\mathbb{Q}$$ $$F(T) = \theta^* + (V_0 - \theta^*) \cdot e^{-\kappa^* T}$$

where $\theta^*$ is the risk-neutral long-term mean attractor and $\kappa^*$ is the annualized reversion speed. When spot VIX is low, $F(T) > V_0$ (contango); during panics, $V_0 > F(T)$ (backwardation).

3. Contango, Backwardation & Roll Yield

Exchange-traded products (such as VXX, UVXY, or SVXY) maintain constant 30-day maturity exposure by daily rolling $\frac{1}{21}$ of their position from front-month ($M_1$) into second-month ($M_2$) futures:

$$\text{Daily Roll Yield} \approx \frac{M_1 - M_2}{M_1 \times 21}$$

In upward-sloping contango, $M_2 > M_1$, causing investors to sell low and buy high daily. This produces continuous negative carry exceeding 5%–10% per month during extended bull runs.

4. Positive Skew & Spot Delta Attenuation

VIX options price on the forward future $F(T)$ using Fischer Black's 1976 formula with Volatility of VIX (VVIX). Two unique features define the market:

$$\Delta_{\text{Spot}} = \frac{\partial C}{\partial V_0} = \frac{\partial C}{\partial F} \cdot \frac{\partial F}{\partial V_0} = \Delta_{\text{Fut}} \cdot e^{-\kappa^* T}$$

Because VIX mean-reverts, distant options have minimal sensitivity to spot spikes ($\Delta_{\text{Spot}} \ll \Delta_{\text{Fut}}$). Furthermore, crash protection demand forces out-of-the-money calls to trade at high implied volatilities, establishing an inverted positive skew curve.

Quantitative Mastery Quiz

1. Why does traditional cost-of-carry spot-futures parity fail for VIX futures?

2. What happens to a long VIX rolling ETP (e.g. VXX) when the term structure is in Contango?

3. How does implied volatility skew in VIX options differ fundamentally from equity index options?

4. Why is Spot VIX Delta ($\Delta_{\text{Spot}}$) strictly smaller than Futures Delta ($\Delta_{\text{Fut}}$)?

Frequently Asked Questions

Spot-futures parity ($F = S \cdot e^{(r - q)T}$) fails because the CBOE VIX index is not a physically investable or dividend-paying cash asset that can be bought and held in storage. Instead, VIX futures price the risk-neutral expected value of 30-day implied volatility at contract expiration, reflecting mean-reverting stochastic processes and a volatility risk premium.

In upward-sloping contango markets (normal ~80% of trading days), front-month futures trade at a discount to second-month futures. Rolling positions requires continuously selling cheaper near-term futures and buying more expensive deferred futures, generating structural negative roll carry that compounds over time.

VIX options expire into the cash settlement value of VIX futures at terminal maturity. Because the underlying is an uninvestable spot index governed by a futures contract rather than an underlying stock, Fischer Black's 1976 model on forward futures prices is the market standard.

In equity markets, downside crash risk causes deep OTM puts to trade at high implied volatility (negative skew). Conversely, VIX behaves as an asymmetric hedge: market crashes create massive upward volatility spikes. Consequently, out-of-the-money VIX calls command heavy risk premiums, producing an upward-sloping positive volatility skew.

Futures Delta ($dC/dF$) measures option sensitivity to the underlying futures price. Because the futures contract mean-reverts toward the long-term mean attractor, a one-point move in spot VIX translates to only $\exp(-\kappa^* T)$ in the futures price. Therefore, Spot Delta equals Futures Delta multiplied by $\exp(-\kappa^* T)$, decaying as tenor lengthens.

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