surprises. If |i is positive, then a large market return at time T induces an upward revision in the forecast of volatility for time T + 1.

Does the GARCH estimator share any similarities with our variance estimator, which uses a set of decaying weights on the return data? The answer to this question is easily found by rearranging equation (16.4) as follows:13

varT+1|f(i)| = (l-"/T)varT|r(i)| +wTtj (16.11')

Clearly, our estimator is a restricted version of a GARCH(1,1) process, in which the parameter m is set equal to zero, and a and |3 are restricted to add up to 1 (so-called integrated GARCH process). At first one may conclude that our specification, although more parsimonious, is too restrictive. In practice, the benefit of parsimony becomes apparent when dealing with multiple assets. In fact, the proliferation of parameters in a multivariate GARCH process without restrictions makes it often very hard if not impossible to estimate.

The covariance matrix decomposition in equation (16.8) provides a great opportunity to use relatively unrestricted GARCH processes even when dealing with large sets of assets. In fact, as long as the specification of the correlation matrix is kept simple (e.g., a slow-moving correlation matrix like the one proposed earlier in this chapter), the volatility process for each asset can be modeled separately and estimated as a univariate process, without altering the positive semidefinite nature of the covariance matrix.

Implied Volatilities

In recent years, with the increasing popularity of derivatives markets, researchers have focused their interest on volatility measures implied by traded options. This is essentially an exercise in reverse engineering. Since volatility is one of the key inputs into the Black-Scholes option pricing model (and its variations), one can infer the volatility perceived by market participants by using option prices and recovering the implied volatility from a standard option pricing model. These estimates are based on prevailing market prices rather than on the past history of returns and, therefore, they are forward-looking measures of volatility.

Unfortunately, although the idea sounds appealing, this approach has some limitations. First, the number of liquid markets on derivatives products is still very limited compared to the number of assets for which we may be interested in building a risk model. Second, most derivatives can be used to infer implied volatilities, but very few products exist whose price depends on the correlation between two assets. This means that, for most assets, we are still far from being able to estimate implied correlations from observed market prices.

For the time being, we believe that the evidence from implied volatilities can be used in a productive way under special circumstances. For example, in the presence of extreme events, one may want to measure the change in implied "Although equation (16.4) defines the covariance between two assets, the formula for the variance is obtained by assuming that assets i and / coincide.