Before we get into index options (in another paper) I need to formally introduce the concept of correlation. This is essential in understanding the volatility of the entire market as a function of the sum (a better term is aggregation) of the volatilities of individual stocks.
R-squared is a statistical calculation that measures the correlation between two assets. It is a linear calculation (employing historical information extrapolated into the future) used to describe a non-linear world (markets), so it is best to use it only as a general guideline. The base correlation of R-squared is covariance, a mathematical measurement of the tendency of two assets to move or vary together, i.e. co-vary. Where the variance is the average of the squared deviation of an asset from its mean, the covariance is the average of the products of the deviations of the assets from their means.
Don't think about this too hard, just know that a high covariance means two variables move in the same direction by a similar percentage amount. A covariance near zero means that there is almost no correlation or predictive value of one asset on another, and a high negative covariance means that two things move in opposite directions by a similar percentage amount. Covariance is not "normalized", meaning that we cannot compare the correlation of two assets to another two assets because the standard deviations (or volatility) of one group may be different than the other. So to normalize covariance we divide it by the product of the standard deviations of the two variables. This gives us R, called the correlation coefficient: it has maximum values of 1 (high correlation) and -1 (high negative correlation). Because what we really care about is only if two things are correlated or not, we can square this number and come up finally with R-squared, which has maximum values of 0 (no correlation) and 1 (perfect correlation).
You can get the calculations for these statistics in any statistical text book, but it is more important to just understand them conceptually. The R-squared between two highly correlated stocks like Fanny Mae (FNM:NYSE) and Freddie Mac (FRE:NYSE) is currently .64: this number is fairly high for two individual stocks (although it has come down recently from a higher level of .74). The R-squared between the SPX and the Dow industrials is .96. Intuitively it makes sense that the correlation between any two stocks would be lower than that of the correlation between indexes. The stock prices of individual companies are predominately driven by unsystematic or company specific risks, which can vary greatly from company to company, even if the companies are in the same industry. Indexes have these risks smoothed away (by the nature of an index, which is a mean of a group of stocks) and are predominately driven by macro forces, which are common among all companies.
The most important concept in understanding the volatility of the market as a whole (indexes and their options) is the correlation of its underlying stocks. When the correlation between all stocks goes up (usually because of some type of macro event like a terrorist attack, or a big move in oil prices, or a big move in interest rates) systematic risk becomes more prevalent in driving stock prices than before, while stock specific influences become less prevalent. This will cause the volatility of the index to go up. For example, before a big move in rates, stock A may go up the same day that stock B goes down (they have a negative correlation). This has the affect of keeping the underlying index relatively unchanged and volatility is low. After a big move in rates, both stocks go down because of the macro affect on all stocks. The correlation between the stocks becomes more positive and the index moves down quite a bit. So the volatility of the index is affected by the correlation of the underlying stocks: as correlation becomes more positive, volatility increases.
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