Why the CBOE Dispersion Index Can Exceed Index Variance

# CBOE dispersion index formula


# CBOE dispersion index formula












I came across the CBOE white paper Cboe S&P 500 Dispersion Index Methodology. The formula in Subsection Index Construction/Outline of the Dispersion Index Methodology on page 4 that defines the dispersion index DSPX uses a sum $\sum_{i=1}^L w_i\hat\sigma_{i,30}^2$ where $w_i$ is the market capitalization weight of constituent $i$. That is puzzling since the variance of a portfolio should be quadratic in $w_i$. What am I missing?

## Answer by Hans (score 1)

https://quant.stackexchange.com/a/77246

That is a puzzling and interesting construction because it is an overestimation. In the following I will explain why it is an overestimation and put forth a rationale for making this choice as well as a suggestion of a twin product.

Note: In the following, we are going to use the formulation setup of Avellane's PPT lecture note Dispersion Trading. I am using his setup and notations only but not his content. The propositions below are all my own. I am going to switch the market capitalization weight notation from $w_i$ to $p_i$ to respect Avellane's lecture note as he uses $w_i$ as the number of shares.

Throughout this answer, we have $$p_i,\sigma_i\ge0\, \forall i,\; \sum_ip_i=1.$$ For index $I$ \begin{equation} \frac{dI}I=\sum_ip_i\frac{dS_i}{S_i},\quad p_i\ge0\, \forall i,\; \sum_ip_i=1. \tag1 \end{equation} He defines dispersion $D$ as \begin{align} D^2 &:= \sum_i p_i\bigg(\frac{dS_i}{S_i}-\frac{dI}I\bigg)^2 \tag2 \\ &=\sum_i p_i\Big(\frac{dS_i}{S_i}\Big)^2-\Big(\frac{dI}I\Big)^2. \label{eq:D^2-} \end{align} where the second equation is arrived at simply by expanding Equation $(2)$ and substituting in Equation $(1)$. Assume $dS_i=\sigma_iS_idB_i$ where $dB_i$ is a random variable where $\mathbf E[dB_i^2]=dt$ and $\mathbf E[dB_i,dB_j]=\rho_{i,j}dt, |\rho_{i,j}|\le1$, $\forall i,j$. Substituting these into Equation $(1)$, we have \begin{equation}\label{eq:dI^2expans} \frac1{dt}\mathbf E\Big(\frac{dI}I\Big)^2=\sum_i(p_i\sigma_i)^2+2\sum_{i<j}\rho_{i,j}p_ip_j\sigma_i\sigma_j \end{equation} Taking expectation on $D^2$ together with the above expectation, we obtain \begin{equation}\label{eq:D2expans} \frac{\mathbf E[D^2]}{dt}=\sum_i p_i\sigma_i^2-\Big(\sum_i(p_i\sigma_i)^2+2\sum_{i<j}\rho_{i,j}p_ip_j\sigma_i\sigma_j\Big). \end{equation} This is exactly the formulation of CBOE ESPX. The square in Equation $(2)$ implies that \begin{equation}\label{eq:D^2posi} \frac{\mathbf E[D^2]}{dt}\ge0. \tag3 \end{equation}

Let us now proceed to examine the following sequence of inequalities. \begin{equation}\label{eq:portVolIneq} \sum_ip_i^2\sigma_i^2\le\sum_i(p_i\sigma_i)^2+2\sum_{i<j}\rho_{i,j}p_ip_j\sigma_i\sigma_j\le\Big(\sum_ip_i\sigma_i\Big)^2 \le\sum_ip_i\sigma_i^2 \end{equation} for $$\rho_{i,j}\in[0,1],\,\forall i,j.$$ We make this assumption because almost all the stocks in SPX are almost always positively correlated.

The first and inequalities are simple consequence of the positivity of all the numbers and the quadratic expansion. The last inequality follows either from Equation $(3)$ which is $\forall\rho_{i,j}$ or when $\rho_{i,j}=1,\,\forall i,j$. It also follows from the Cauchy-Schwartz inequality as follows. $$\Big(\sum_ip_i\sigma_i\Big)^2= \Big(\sum_i(\sqrt p_i\sigma_i)\sqrt p_i\Big)^2\le \Big(\sum_i p_i\sigma_i^2\Big) \sum_ip_i=\sum_i p_i\sigma_i^2.$$ Or more simply and equivalently by Jensen's inequality and the convexity of the square function $$\Big(\sum_ip_i\sigma_i\Big)^2\le \sum_i p_i\sigma_i^2.$$ The equality is achieved only when $\sigma_i$'s are all the same.

The first two inequalities say the index variance is bounded by the variances of the fictitious portfolios of completely uncorrelated and completely correlated constituents. The last inequality says that DSPX overestimates the variance of the index even when the constituents are completely correlated or in other words when the constituents are not at all dispersed.

Why do they use this relaxed upper bound rather than the tight upper bound $\big(\sum_ip_i\sigma_i\big)^2$? I suppose the advantage of this relaxed upper bound is that it can be constructed from traded instruments, since the variances $\sigma_i^2$ can be constructed from strips of single name options just as VIX$^2$ is whereas we need to use more exotic options such as spread options to trade $\big(\sum_ip_i\sigma_i\big)^2$. We can construct the lower bound the same way only with the weights squared. So we may well use this lower bound to created another index to give a more complete picture of the dispersion.