Volatility models are workhorses of modern finance; notable examples include the GARCH/ARCH-type models. They have won their popularity for the ability to capture the tendency of the volatility of asset returns to be persistently high or persistently low (volatility clustering), and exhibit “fat tails” (excess kurtosis). In particular, volatility models are used for financial risk management, such as the value-at-risk of a portfolio of stocks.
However, as we explore in the following, forecasting the volatility of returns is not the same as forecasting the distribution of returns. Therefore, although we can forecast the volatility of returns, we cannot use it to determine the distribution of the returns. In turn, one should be very careful when calculating forecasted tail probabilities, such as forecasting the value-at-risk.
for fixed, , , and . Notice that the upper bound of ensures covariance stationarity. We are interested in obtaining the -step forecast of the volatility, which we define as the conditional standard deviation of the returns
where denotes the natural filtration.
We obtain the forecast for simply by substituting the expressions from the equations in (1), such that
Likewise, we obtain the forecast for ,
Observe that this finding holds for any distribution of the innovations that has unit variance, , and that as the forecast in (5) tends to the stationary variance . On a side note, we can also forecast the mean by a similar recursion, but it is of little interest for the ARCH(1) process, as it is zero for any value of .
However, when considering the above forecasts of the volatility, it is crucial to realize that the distribution of the process at time , for is no longer Gaussian! Consequently, the mean and variance for conditional on does not characterize the distribution at . The figure below illustrates this; it shows the forecasted densities for an ARCH(1) process with and starting in , for .
The density forecasts clearly illustrate how the tails of the distribution get “fatter” the further into the future we forecast. This reflects that the forecasts, as , will tend to the stationary distribution of the ARCH(1) process, which has fat tails.
The mathematical reason why the distribution of the forecasted returns is not Gaussian is quite straight-forward. If we consider the transition density for the ARCH(1) process,
it is clear that the variance is a non-linear function of (it is quadratic). If is known, then the density of is evidently Gaussian; as was the case with conditional on . Contrariwise, considering the density of conditional on , which can be written in terms of the integral
we see that the density function cannot be Gaussian. It is well-known that integrating out one of the parameters of the Gaussian density will yield a Gaussian density if the distribution of the parameter being integrated out is Gaussian. Moreover, it is also well-known that nonlinear transformations of Gaussian variables are not Gaussian. Thus, the nonlinear transformation of results in the distribution of being non-Gaussian, in turn rendering the distribution of conditional on non-Gaussian. In fact, there is not even a closed-form solution to the integral in (7). If we apply this finding to the transition density for , it is clear that the forecasted densities are no longer Gaussian beyond .
Concluding, if the objective is to forecast the value-at-risk, or an other tail-dependent statistic, then it will be insufficient to forecast the volatility (and the mean, if specified). Instead the distribution must be approximated by some form of Monte Carlo simulation or bootstrapping. The simplest way is to simulate some large number, say, realizations of length of the ARCH(1) process starting in , and produce a histogram of the realizations at . The above figure was generated this way by setting and applying a kernel density estimator to the realizations. There are, however, other and more efficient methods, such as efficient importance sampling.