Does the correlation between stocks in an index affect the implied volatility of the index?

Question

Does the correlation between stocks in a sector or between sectors in the S&P 500 have an impact, all else equal, on the S&P's implied volatility / the VIX?

My guess is that the correlation between stocks does not have a direct impact on the implied volatility. See e.g. this answer on how to approximate the implied volatility in a Black-Scholes framework; it does not seem to me that a higher or lower degree of correlation between the stocks would directly affect this approximation. (Of course I understand that a high correlation between stocks might be the result of complacency, which would also lead to a low implied volatility. However, that's not a direct link between the two variables.)

On a higher level, where I am coming from is that I am curious if indiscriminate buying by index funds, which should lower the correlation between the stocks in the S&P 500 or at least between the stocks in a sector, could lead to a VIX of x today not being comparable to a VIX of x 10 years ago.

Answer 1

Concerning your specific question:

Does the correlation between stocks in a sector or between sectors in the S&P 500 have an impact, all else equal, on the S&P's implied volatility / the VIX?

As of Chicago Board Options Exchange's (CBOE) documentation, page 4, the VIX index is calculated as follows:

$$ \begin{align} & \text{VIX} = 100 \times \sigma_{I} \\[12pt] & \sigma_{I} = \frac{2}{T}\sum_ie^{rT}\frac{\Delta K_i}{K_i}Q(K_i)-\frac{1}{T}\left(\frac{F}{K_0}-1\right)^2 \end{align} $$

To answer the question at hand, we only need to look at the variable $Q(K_i)$ which is "the midpoint of the bid-ask spread for each option with strike $K_i$". So, simplifying, it is the price of an option written on the S&P Index.

The correlation between stocks $-$ and hence between sectors $-$ will without doubt have an effect on this option price: consider a toy example with a simple index $I$ made up of 2 stocks $S_1$ and $S_2$ such that:

$$ I(t) = S_1(t) + S_2(t) $$

Assume that both stocks have the same variance $\sigma_S$. Now let us take 2 extreme cases:

• If the correlation is $-1$, the index will not move, so any OTM option $-$ note that the VIX value is calculated with OTM options $-$ will have $0$ value:

$$ \mathbb{Var}[I(t)] = \sigma_S^2 + \sigma_S^2 - 2\sigma_S\sigma_S = 0 $$

• If the correlation is $1$, $S_1$ and $S_2$ will move in unison as any price change in any of the 2 stocks will be "compounded" by an equally intense price change from the other stock. In this case, the index price $I(t)$ might end up below OTM strikes hence OTM options cannot be worthless $-$ as it would constitute an arbitrage opportunity:

$$ \mathbb{Var}[I(t)] = \sigma_S^2 + \sigma_S^2 + 2\sigma_S\sigma_S = 4\sigma_S^2 $$

Given pairwise stock correlations have an impact on one of the elements of the VIX formula, they must have an effect on the VIX value.

Answer 2

As Quantuple said, the variance of an index is related to the pairwise correlations $\rho_{ij}$ between stocks in the index

$\sigma^2_p = \sum_i^N\sigma_i^2w_i^2+2\sum_{i=1}^{N}\sum_{j>i}^N w_i\rho_{ij} \sigma_i \sigma_j$

If we assume that all rho's are close to a "typical" or "average" value $\bar{\rho}$ then this expression simplifies considerably. Tierens and Anadu (2004) showed that (as you can easily check) we can then derive the following equation

$\bar{\rho} = \frac{\sigma_p^2-\sum_{i=1}^N w_i^2\sigma_i^2}{2\sum_{i=1}^{N}\sum_{j>i}^N w_i \sigma_i \sigma_j}$

We know the weights $w_i$ of the S&P index, and we can observe in the option market the implied volatilties $\sigma_p,\sigma_i$ of the index and the constituent stocks. Therefore we can compute $\bar{\rho}$, sometimes called the market implied correlation among the stocks.