# Just what are you implying?

Tracy Alloway points us to another fun indicator:

Type JCJ GP on your Bloomberg.

Youâ€™ll get a chart that looks like this:

Itâ€™s the CBOEâ€™s S&P 500 Implied Correlation Index â€” based on options expiring in January 2011. Itâ€™s a basic measure of the correlation of stocks within the S&P 500, and you can see that itâ€™s been approaching the end of day record it hit back in July.

Unfortunately, I do not have a Bloomberg terminal. But my Goog-Fu is strong, so I bring you the free version:

CBOE SPX implied correlation index Jan 2011

So what is this thing, exactly? The CBOE’s page is a little vague, but their white paper provides details.

The basic idea is this. According to the standard Black-Scholes options pricing model, option prices depend on the “volatility” of the underlying asset’s price. By examining option prices and then inverting the Black-Scholes formula, you can infer the market’s estimate of the volatility of the underlying. This is called the “implied volatility”.

Now, options trade not only on individual stocks, but also on indices. (You may have heard of the VIX). Since an index is just a basket of stocks, there should be some relationship between the volatility implied by an option on an index and the volatilities implied by options on the stocks in the index.

And there is. For example, suppose you have two stocks with equal volatilities and you construct a portfolio containing those two stocks in equal weight. Then the volatility of the portfolio — aka. “the standard deviation of the average of two random variables” — will be 1/sqrt(2) times the volatility of either stock.

…if the stocks are totally uncorrelated, that is. If instead the stocks are perfectly correlated — i.e., they move together exactly all the time — then the volatility of the portfolio will obviously be identical to the volatility of either stock. And if, say, the stocks are perfectly anti-correlated — i.e., they move exactly opposite to each other all the time — then the volatility of the portfolio will obviously be zero, since the portfolio’s value will never change.

In other words, by examining the implied volatility of an index and comparing it to the implied volatility of the stocks in the index, we can deduce the market’s implied correlation for the stocks in the index. The math for more than two stocks and unequal weights is a little more complicated, but should be familiar if you ever took an undergraduate class in statistics; see the white paper for details.

(The correlation between two random variables is quantified by their “correlation coefficient”. Perfectly correlated variables have a correlation coefficient of 1; perfectly anti-correlated, -1; totally uncorrelated, 0. This index calculates the implied average correlation coefficient for the S&P 500.)

One caveat: The CBOE implied correlation index only examines the implied volatility of the top 50 stocks in the S&P 500. From the white paper:

On May 29, 2009, the total capitalization of the 50-stock basket was $4.15 trillion. The weight of Exxon Mobil Corp (XOM), the largest component in the 50-stock basket, was 8.3% ($343 billion / \$4.15 trillion), compared to 4.3% in the S&P 500 Index.

Which is a convoluted way of saying the top 50 stocks represent 4.3/8.3 = 52% of the market cap of the entire S&P 500. If for some reason the correlation of those stocks is not representative of all of the stocks in the index, the implied correlation calculation could give the wrong answer. (I have not yet tried to quantify this. Maybe later.)

Summary: This indicator provides the market’s expectation for the correlation of S&P 500 stocks between now and Jan 2011. And it has been spiking recently, which means the market is expecting everything to move together, one way or another…

She was always using the word “infer”
When she obviously meant “imply”
And I know some guys would put up with that kind of thing
But frankly, I can’t imagine why
W.A. Yankovic

### 6 comments to Just what are you implying?

• JoeLeTaxi

But there is a logarithm. Vol index is the expected standard dev of the first diff of the log of the index. Vol stock is is the expected standard dev of the first diff of the log of the stock price. And if the index is a linear comb of the stock prices, the log index is not a linear comb of the log stock prices. So there should be some sort of a corrective term – unless the construction of the CBOE takes correctly into account the log by computing the correlations of the first diffs of the stock, but in this case your example with the two stocks independant,correlated, or anti-correlated, is only approximatively true, and you should have the some sort of a corrective term.

• Nemo

JoeLeTaxi —

Very good point. Read the white paper for yourself, especially the formula at the top of page 2.

Plug in zero for the implied correlation and 0.5 for the weights, and you get the usual 1/sqrt(2) factor. So I think my description is an accurate summary of the CBOE’s methodology.

I see no corrective term in the rest of their white paper either. Do you? How much of an effect is this likely to have?

Thanks for the great comment. Definitely food for thought.

• jaguaracer

The “corrective” term is correlation and is calculated by log of first difference: 1-log(day2/day1). std dev is always calculated this way. std dev of index = sum over weights^2* vol_stocks + sum over weights*variance*correlation

pg 2 of white paper

Dispersion trades were big about 5 years ago, but sort of went away as liquidity decreased, making execution more expensive (trade requires a *bunch* of options obviously). Something like this could theoretically increase the attractiveness of these trades I guess.

google “Dispersion – A Guide for the clueless”

• >Summary: This indicator provides the marketâ€™s expectation for the correlation of S&P 500 stocks between now and Jan 2011. And >it has been spiking recently, which means the market is expecting everything to move together, one way or anotherâ€¦

There is an old saying ‘in decreasing markets everything decreases, but correlation’, so I’d rather infer a market’s guess for decreasing markets from this statistic. To be even more correct, rather than market’s guess on its own direction, it’s a no-arbitrage value of average correlation in top-50 stocks. Does it have any predictive power? Hard to tell without running a regression. Time will tell if this ‘prediction’ comes true.

(on the sidelines, the ‘old saying’ tells us, that classical diversification does not work in decreasing markets)

• newfire

I believe you meant KCJ GP

• Nemo

Actually, I meant “JCJ GP” because this post is from 2010.