A proper understanding of cross-correlations between different stocks, or more generally between different financial assets, is of crucial importance for risk management and, as will be discussed in the Chapter 12, for portfolio optimization. In fact, much as the tail of the distributions of individual returns is of particular relevance for risk management, it is the correlations between extreme events that must be faithfully described.

As we have discussed in depth in Chapter 7, the volatility of each asset is in general not constant in time, but fluctuates randomly. It is therefore tempting to think that the whole covariance matrix (of which the diagonal contains the squared volatilities) also randomly fluctuates in time. The simplest case would be that the correlation matrix has a fixed structure, and that the variations of the covariance matrix only reflects that of the volatilities. However, it is a common belief in the financial industry that cross-correlations between stocks themselves fluctuate in time, and in fact increase substantially in a period of high market volatility. If true, this would mean that the possibility of risk diversification is much reduced, since in the periods that really matter as far as risk is concerned, all stocks would behave similarly.