# Does the correlation between stocks in an index affect the implied volatility of the index? [closed]

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.

## closed as off-topic by Quantuple, LocalVolatility, Alex C, Bob Jansen♦Jun 2 '17 at 10:50

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Basic financial questions are off-topic as they are assumed to be common knowledge for those studying or working in the field of quantitative finance." – Quantuple, LocalVolatility, Alex C, Bob Jansen
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• Do pairwise correlations between the individual returns of assets making up a portfolio affect that portfolio returns' variance? – Quantuple Jun 1 '17 at 6:50
• I don't see why this is closed as a "Basic financial question." Implied correlation and dispersion trading are hot topics in quant finance. – user217285 May 21 '18 at 6:06

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.

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.