In 1993, Cboe Global Markets, Incorporated (Cboe) introduced the Cboe Volatility Index (VIX Index). Originally designed to measure the market’s expectation of 30-day volatility implied by at-the-money S&P 100 Index (OEX Index) option prices, the VIX Index soon became the premier benchmark for U.S. stock market volatility.
Ten years later in 2003, Cboe collaborated with Goldman Sachs to update the VIX Index. The changes reflected a new way to measure expected volatility, a methodology that continues to be widely used by financial theorists, risk managers, and volatility traders alike. The new VIX Index is based on the S&P 500 Index, the core index for U.S. equities, and estimates expected volatility by aggregating the weighted prices of S&P 500 Index puts and calls (SPXTM options) over a wide range of strike prices.
On March 24, 2004, Cboe introduced the first exchange-traded VIX futures contract on its new, all-electronic Cboe Futures Exchange (CFE). Two years later in February 2006, VIX options were launched for trading on Cboe Options Exchange (C1).
The negative correlation of volatility to stock market returns is well-documented and suggests a diversification benefit to including volatility in an investment portfolio. VIX futures and options (collectively, “VIX derivatives”) are designed to isolate expected volatility exposure in a single, efficient package.
Cboe Volatility Index
Re: Cboe Volatility Index
The Cboe Volatility Index (VIX), often dubbed the market's "fear gauge," provides a real-time measure of the market's expectation of 30-day forward-looking volatility for the S&P 500 Index.
Unlike a simple average, the VIX is calculated using a methodology known as the Bracket Method, which selects and weights prices of a range of S&P 500 (SPX) index options. This method ensures that the VIX consistently reflects a 30-day implied volatility, even as time progresses and option expirations draw nearer.
Understanding the Bracket Method
The core principle of the Bracket Method is to always interpolate the 30-day volatility from two distinct option expiration series: a "near-term" series and a "next-term" series. These series are chosen based on specific criteria related to their time to expiration:
The "Roll" Mechanism: A Dynamic Selection
The critical aspect of the Bracket Method is its dynamic "roll" mechanism. As time passes, the days to expiration for the near-term option series decrease. When the near-term series reaches a point where it has exactly 23 days or fewer until expiration, it is no longer considered suitable for the "near-term" component. At this precise moment, a roll occurs:
Rolling: Near-term (23) is no longer > 23. The window shifts forward.
Note: The "Days to Expiration" refers to calendar days.
As illustrated, the "roll" is not a gradual shift but a distinct event triggered by the near-term option's proximity to the 23-day threshold. This mechanism is fundamental to the VIX's design, allowing it to consistently provide a robust and relevant measure of implied market volatility over a constant 30-day horizon.
Unlike a simple average, the VIX is calculated using a methodology known as the Bracket Method, which selects and weights prices of a range of S&P 500 (SPX) index options. This method ensures that the VIX consistently reflects a 30-day implied volatility, even as time progresses and option expirations draw nearer.
Understanding the Bracket Method
The core principle of the Bracket Method is to always interpolate the 30-day volatility from two distinct option expiration series: a "near-term" series and a "next-term" series. These series are chosen based on specific criteria related to their time to expiration:
- Near-Term Options: These are defined as the SPX options with the shortest time to expiration that are still more than 23 days away from the current date. Cboe's methodology specifies that only Friday-expiring SPX and SPXW options are considered for the standard VIX calculation, excluding daily (Monday-Thursday) SPXW expirations.
- Next-Term Options: This series consists of the SPX options that expire immediately after the selected near-term options.
The "Roll" Mechanism: A Dynamic Selection
The critical aspect of the Bracket Method is its dynamic "roll" mechanism. As time passes, the days to expiration for the near-term option series decrease. When the near-term series reaches a point where it has exactly 23 days or fewer until expiration, it is no longer considered suitable for the "near-term" component. At this precise moment, a roll occurs:
- The previously designated "next-term" option series becomes the new near-term series.
- A new, further-out option series (the one immediately following the new near-term) is then selected as the new next-term series.
| Current Day | Near-Term Expiry Candidate | Days to Expiry | Next-Term Expiry Candidate | Days to Expiry | VIX Selection | Rationale |
|---|---|---|---|---|---|---|
| Friday (Day 0) | Friday (+28 days) | 28 | Friday (+35 days) | 35 | 28 & 35 | Near-term (28) is > 23 days. |
| Saturday | Friday (+27 days) | 27 | Friday (+34 days) | 34 | 27 & 34 | Near-term (27) is > 23 days. |
| Sunday | Friday (+26 days) | 26 | Friday (+33 days) | 33 | 26 & 33 | Near-term (26) is > 23 days. |
| Monday | Friday (+25 days) | 25 | Friday (+32 days) | 32 | 25 & 32 | Near-term (25) is > 23 days. |
| Tuesday | Friday (+24 days) | 24 | Friday (+31 days) | 31 | 24 & 31 | Near-term (24) is > 23 days. |
| Wednesday | Friday (+23 days) | 23 | Friday (+30 days) | 30 | 30 & 37 | ROLL OCCURS. |
| Thursday | Friday (+22 days) | 22 | Friday (+29 days) | 29 | 29 & 36 | Window remains shifted. |
| Friday (Day 7) | Friday (+21 days) | 21 | Friday (+28 days) | 28 | 28 & 35 | Window remains shifted. |
Note: The "Days to Expiration" refers to calendar days.
As illustrated, the "roll" is not a gradual shift but a distinct event triggered by the near-term option's proximity to the 23-day threshold. This mechanism is fundamental to the VIX's design, allowing it to consistently provide a robust and relevant measure of implied market volatility over a constant 30-day horizon.
Re: Cboe Volatility Index
Let \(S_t\) be the price of the stock at time \(T\), \(0\le t \le T\). Only \(S_0\) is known, and \(S_t\) is modeled as a Geometric Brownian Motion. That is, \(dS = r S dt + \sigma S dz\), where \(r\) is the risk-free interest rate. For example, the forward price with maturity \(T\) is \(F_0 = \hat E[S_T] = S_0 e^{rT}\).
Let \(\sigma(t)\) be the variance of the stock price at time \(t\). For simplicity it is often assumed that it stays constant. In reality it also changes in time and is determined by the random variable \(S_t\). The realized average variance rate \(\bar V\) from \(0\) to \(T\) is defined as
\[ \bar V = \frac{1}{T} \int_0^T \sigma^2 dt.\]
From Ito's Lemma, we have \(\frac{1}{S}dS - d\ln S = \frac{1}{2}\sigma^2 dt.\) It follows that
\[ \bar V = \frac{2}{T} \int_0^T \frac{1}{S}dS - d\ln S = \frac{2}{T} \int_0^T \frac{1}{S}dS - \frac{2}{T} \ln \frac{S_T}{S_0}.\]
Taking the risk-neutral expectation \(\hat E\), we have:
\begin{align*}
\hat{E} [\bar V] &= \frac{2}{T} \int_0^T \hat E[\frac{1}{S}dS] - \frac{2}{T} \hat E[\ln \frac{S_T}{S_0}] \\
&=\frac{2}{T} rT - \frac{2}{T} \hat E[\ln \frac{S_T}{S_0}] = \frac{2}{T} \ln \frac{F_0}{S_0} - \frac{2}{T} \hat E[\ln \frac{S_T}{S_0}].
\end{align*}
Let \(\sigma(t)\) be the variance of the stock price at time \(t\). For simplicity it is often assumed that it stays constant. In reality it also changes in time and is determined by the random variable \(S_t\). The realized average variance rate \(\bar V\) from \(0\) to \(T\) is defined as
\[ \bar V = \frac{1}{T} \int_0^T \sigma^2 dt.\]
From Ito's Lemma, we have \(\frac{1}{S}dS - d\ln S = \frac{1}{2}\sigma^2 dt.\) It follows that
\[ \bar V = \frac{2}{T} \int_0^T \frac{1}{S}dS - d\ln S = \frac{2}{T} \int_0^T \frac{1}{S}dS - \frac{2}{T} \ln \frac{S_T}{S_0}.\]
Taking the risk-neutral expectation \(\hat E\), we have:
\begin{align*}
\hat{E} [\bar V] &= \frac{2}{T} \int_0^T \hat E[\frac{1}{S}dS] - \frac{2}{T} \hat E[\ln \frac{S_T}{S_0}] \\
&=\frac{2}{T} rT - \frac{2}{T} \hat E[\ln \frac{S_T}{S_0}] = \frac{2}{T} \ln \frac{F_0}{S_0} - \frac{2}{T} \hat E[\ln \frac{S_T}{S_0}].
\end{align*}
Re: Cboe Volatility Index
Now a trick is used to find the risk-free expectation \(\hat E[\ln\frac{S_T}{S_0}]\) using the prices of call options and put options with the same maturity \(T\): for any \(S_0>0\)
\begin{align}
\int_{0}^{S_0} \frac{1}{k^2} (k- S_T)^{+} dk + \int_{S_0}^{\infty} \frac{1}{k^2} (S_T - k)^{+} dk = \ln\frac{S_0}{S_T} + \frac{S_T}{S_0} - 1.
\end{align}
Note that \((k- S_T)^{+}\) is the payout at maturity \(T\) of a put option with strike price \(k\), and similarly \((S_T-k)^{+}\) for a call option. Recall that the current price of the put option is given by \(p(k, T)= e^{-rT} \hat{E}[(k- S_T)^{+}] \), and the current price of the call option is given by \(c(k, T)= e^{-rT} \hat{E}[(S_T - k)^{+}] \).
It follows that
\begin{align*}
\ln\frac{S_T}{S_0} &= \frac{S_T}{S_0} - 1 - \int_{0}^{S_0} \frac{1}{k^2} (k- S_T)^{+} dk - \int_{S_0}^{\infty} \frac{1}{k^2} (S_T - k)^{+} dk;\\
\hat{E}[\ln\frac{S_T}{S_0}] & = \frac{\hat E[S_T]}{S_0} - 1 - \int_{0}^{S_0} \frac{1}{k^2} \hat E[(k- S_T)^{+}] dk - \int_{S_0}^{\infty} \frac{1}{k^2} \hat E[(S_T - k)^{+}] dk \\
&= \frac{F_0}{S_0} - 1 - \int_{0}^{S_0} \frac{1}{k^2} e^{rT} p(k,T) dk - \int_{S_0}^{\infty} \frac{1}{k^2} e^{rT} c(k, T) dk.
\end{align*}
Therefore,
\begin{align*}
\hat{E} [\bar V] &= \frac{2}{T} \ln \frac{F_0}{S_0} - \frac{2}{T} \hat E[\ln \frac{S_T}{S_0}]\\
&= \frac{2}{T} \ln \frac{F_0}{S_0} - \frac{2}{T} (\frac{F_0}{S_0} - 1) + \frac{2}{T}\int_{0}^{S_0} \frac{1}{k^2} e^{rT} p(k,T) dk + \frac{2}{T}\int_{S_0}^{\infty} \frac{1}{k^2} e^{rT} c(k, T) dk
\end{align*}
Recall that \(\ln(1+x) \approx x - \frac{1}{2}x^2\). It follows that
\begin{align*}
\hat{E} [\bar V] &\approx -\frac{1}{T}(\frac{F_0}{S_0} - 1)^2 +\frac{2}{T} \sum_{k_i \le S_0}\frac{1}{k_i^2} e^{rT} p(k_i,T) \Delta k_i + \frac{2}{T}\sum_{k_i \ge S_0}^{\infty} \frac{1}{k_i^2} e^{rT} c(k_i, T) \Delta k_i.
\end{align*}
\begin{align}
\int_{0}^{S_0} \frac{1}{k^2} (k- S_T)^{+} dk + \int_{S_0}^{\infty} \frac{1}{k^2} (S_T - k)^{+} dk = \ln\frac{S_0}{S_T} + \frac{S_T}{S_0} - 1.
\end{align}
Note that \((k- S_T)^{+}\) is the payout at maturity \(T\) of a put option with strike price \(k\), and similarly \((S_T-k)^{+}\) for a call option. Recall that the current price of the put option is given by \(p(k, T)= e^{-rT} \hat{E}[(k- S_T)^{+}] \), and the current price of the call option is given by \(c(k, T)= e^{-rT} \hat{E}[(S_T - k)^{+}] \).
It follows that
\begin{align*}
\ln\frac{S_T}{S_0} &= \frac{S_T}{S_0} - 1 - \int_{0}^{S_0} \frac{1}{k^2} (k- S_T)^{+} dk - \int_{S_0}^{\infty} \frac{1}{k^2} (S_T - k)^{+} dk;\\
\hat{E}[\ln\frac{S_T}{S_0}] & = \frac{\hat E[S_T]}{S_0} - 1 - \int_{0}^{S_0} \frac{1}{k^2} \hat E[(k- S_T)^{+}] dk - \int_{S_0}^{\infty} \frac{1}{k^2} \hat E[(S_T - k)^{+}] dk \\
&= \frac{F_0}{S_0} - 1 - \int_{0}^{S_0} \frac{1}{k^2} e^{rT} p(k,T) dk - \int_{S_0}^{\infty} \frac{1}{k^2} e^{rT} c(k, T) dk.
\end{align*}
Therefore,
\begin{align*}
\hat{E} [\bar V] &= \frac{2}{T} \ln \frac{F_0}{S_0} - \frac{2}{T} \hat E[\ln \frac{S_T}{S_0}]\\
&= \frac{2}{T} \ln \frac{F_0}{S_0} - \frac{2}{T} (\frac{F_0}{S_0} - 1) + \frac{2}{T}\int_{0}^{S_0} \frac{1}{k^2} e^{rT} p(k,T) dk + \frac{2}{T}\int_{S_0}^{\infty} \frac{1}{k^2} e^{rT} c(k, T) dk
\end{align*}
Recall that \(\ln(1+x) \approx x - \frac{1}{2}x^2\). It follows that
\begin{align*}
\hat{E} [\bar V] &\approx -\frac{1}{T}(\frac{F_0}{S_0} - 1)^2 +\frac{2}{T} \sum_{k_i \le S_0}\frac{1}{k_i^2} e^{rT} p(k_i,T) \Delta k_i + \frac{2}{T}\sum_{k_i \ge S_0}^{\infty} \frac{1}{k_i^2} e^{rT} c(k_i, T) \Delta k_i.
\end{align*}
