Simultaneous estimation of the n linear regression equations in (4.24) as described in Appendix 2 will give factor sensitivities P, and specific components

Now denote by oit the conditional standard deviation of asset / and by the conditional covariance between assets / and j. Assuming no conditional correlation between the market and specific components, taking variances and covariances of equation (4.24) yields

In the orthogonal A RCH model, the GARCH covariance matrices are obtained from univariate GARCH estimates of the variances of the principal components of the system

2 2 2 , 2

(4.25)

Thus all the GARCH variances and covariances of the assets in a portfolio are obtained from the GARCH variance of the market risk factor, and the GARCH variances and covariances of the stock-specific components. From a computational point of view there is much to be said for ignoring the covariance between specific components, the second term of the covariance equation. Then (4.25) gives the individual asset conditional variances and covariances in terms of univariate GARCH models only. It is not straightforward to generalize this framework to the case where the underlying factor models have more than one risk factor: firstly, generalized least squares should be used for the factor sensitivity and residual estimation (§A.3.3); and secondly, multivariate models for the covariances between risk factors will need to be employed.

The last model that allows GARCH covariance matrices to be generated using only univariate GARCH models is only mentioned here very briefly. In the orthogonal GARCH model, full details of which will be given in §7.4.3, the GARCH covariance matrices are obtained from univariate GARCH estimates of the variances of the principal components of the system. This method has many practical advantages, particularly for the generation of very large positive semi-definite covariance matrices (Alexander, 2001b).

Forecasting Volatility and Correlation

Previous chapters have described how volatility and correlation forecasts may be generated using different models. In some cases there are very noticeable differences between the various forecasts of the same underlying volatility or

correlation, and in some cases there are great similarities. It is a generic problem Which volatility is being with volatility forecasting that rather different results may be obtained, depend- forecast? ing on the model used and on the market conditions. Correlation forecasting is even more problematic because the inherent instability of some correlations compounds the difficulties.

Even if only one particular type of model were always used, the forecasts will depend on the parameters chosen. For example, in Figure 3.6 the exponentially weighted moving average volatility of the CAC at the end of 1997 could be 30% or 45% depending on whether the smoothing constant is chosen to be 0.96 or 0.84. Many other such examples have been encountered in previous chapters, such as the different historic forecasts of the rand 1-year swap rate volatilities in Figure 3.2. There are also differences between various types of GARCH correlation estimates, as shown in Figure 4.13.

The underlying market conditions will also affect results. When markets are stable, in that they appear to be bounded above and below or that they are trending with a relatively stable realized volatility, differences between the various forecasts are relatively small. It is following extreme market events that differences between forecasts tend to be greatest.

If one decides to approach the difficult problem of forecast evaluation, the first consideration is: which volatility is being forecast? For option pricing, portfolio optimization and risk management one needs a forecast of the volatility that governs the underlying price process until some future risk horizon. A geometric Brownian motion has constant volatility, so a forecast of the process volatility will be a constant whatever the risk horizon. Future volatility is an extremely difficult thing to forecast because the actual realization of the future process volatility will be influenced by events that happen in the future. If there is a large market movement at any time before the risk horizon but after t = 0, the forecast that is made at t = 0 will need to take this into account. Process volatility is not the only interesting volatility to forecast. In some cases one might wish to forecast implied volatilities, for

When markets are stable, in that they appear to be bounded above and below or that they are trending with a relatively stable realized volatility, differences between the various forecasts are relatively small

Which type of volatility should be used for the forecast?

When a number of independent forecasts of the same time series are available it is possible to pool them into a combined forecast that always predicts at least as well as any single component of that forecast

example in short-term volatility trades with positions such as straddles and butterflies (Fitzgerald, 1996).

The second consideration is the choice of a benchmark forecast. The benchmark volatility forecast could be anything, implied volatility or a long-term equally weighted average statistical volatility being the most common. If a sophisticated and technical model, such as GARCH, cannot forecast better than the implied or historical forecasts that are readily available from data suppliers (and very easily computed from the raw market data) then it may not be worth the time and expense for development and implementation.

A third consideration is, which type of volatility should be used for the forecast? Since both implied and statistical volatilities are forecasts of the same thing, either could be used. Thus a model could forecast implied volatility with either implied volatility or statistical volatility. Price process volatilities could be forecast by statistical or implied volatilities, or indeed both (some GARCH models use implied volatility in the conditional variance equation).

There is much to be said for developing models that use a combination of several volatility forecasts. When a number of independent forecasts of the same time series are available it is possible to pool them into a combined forecast that always predicts at least as well as any single component of that forecast (Granger and Ramanathan, 1984; Diebold and Lopez, 1996). The Granger-Ramanathan procedure will be described in §5.2.3; the focus of that discussion will be the generation of confidence intervals for volatility forecasts.

This chapter begins by outlining some standard measures of the accuracy of volatility and correlation forecasts. Statistical criteria, which are based on diagnostics such as root mean square forecasting error, out-of-sample likelihoods, or the correlation between forecasts and squared returns, are discussed in §5.1.1. Operational evaluation methods are more subjective because they depend on a trading or a risk management performance measure that is derived from the particular use of the forecast; these are reviewed in §5.1.2.

Any estimator of a parameter of the current or future return distribution has a distribution itself. A point forecast of volatility is (usually) just the expectation of the distribution of the volatility estimator,1 but in addition to this expectation one might also estimate the standard deviation of the distribution of the estimator, that is, the standard error of the volatility forecast. The standard error determines the width of a confidence interval for the forecast and indicates how reliable a forecast is considered to be. The

Similarly, a point forecast of correlation is just the expectation of the distribution of the correlation estimator.