Jan-88 Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00

-IGARCH

EWMA

Jan-95 Jul-95 Jan-96 Jul-96 Jan-97 Jul-97 Jan-98 jul-98 Jan-99

-IGARCH

-EWMA

Figure 4.4 I-GARCH and EWMA volatitlies: (a) US dollar-sterling exchange rate; (b) CAC equity index.

Table 4.5: GARCH(1,1) parameters for crude oil

Notably, whatever the data period, the

estimated model is an I-GARCH model

Jul-88 Jul-90 Jul-92 Jul-94 Jul-96

Jul-98

-Spot --Future Figure 4.5 GARCH(1, 1) volatility estimates of WTI spot and prompt futures.

0.94. When the data period starts in July 1992 instead of July 1988, the alpha and beta estimates are quite different. If the whole data period were used the GARCH volatility would be spiky: highly reactive but not very persistent. However, when the Gulf war period is excluded from the data the resultant GARCH volatility is less reactive and more persistent. Notably, whatever the data period, the estimated model is an I-GARCH model.7 Thus, term structure forecasts will not converge and the estimated GARCH volatility will be approximately the same as an EWMA volatility series with a smoothing constant equal to the estimate of the GARCH beta coefficient.

When a GARCH model is estimated over a rolling data window, different long-term volatility levels will be estimated, corresponding to different estimates of the GARCH parameters (this will be discussed further, in §4.3.1). The components GARCH model extends this idea to allow variation of long-term volatility within the estimation period (Engle and Lee, 1993a, 1993b; Engle and Mezrich, 1995). It is most useful in currency and commodity markets, where GARCH models are often close to being integrated and so convergent term structures that fit the market implied volatility term structure cannot be generated. The components model is an attempt to regain the convergence in GARCH term structures in currency markets, by allowing for a time-varying long-term volatility.

The GARCH(l.l) conditional variance may be written in the form

07 = (1 - a - P)ct: + a£: , + Pct2 ! = cr + a(s: , - ) + ( 7 , - a2)

7Another example that demonstrates the I-GARCH nature of volatility in energy markets is given in §4.5.1.

where ct2 is defined by (4.4). In components GARCH ct2 is replaced by a time-varying permanent component given by

Therefore the conditional variance equation in the components GARCH model is

Equations (4.6) and (4.7) together define the components model. If p = 1, the permanent component to which long-term volatility forecasts mean-revert is just a random walk. While the components model has an attractive specification for currency markets, parameter estimation is not straightforward. Estimates may lack robustness and it seems difficult to recommend the use of the components model - except in the event that its specification has passed rigorous diagnostic tests.

4.2.4 Asymmetric GARCH

During the last few years the leverage effect (§4.1.2) has become quite noticeable, particularly in equity markets. For example, three out of the 20 stocks in Table 4.1 had pronounced asymmetry in their volatility clustering, so that volatility increased more when the stock price was falling than when it was rising by the same amount. It therefore came as no surprise that the application of symmetric GARCH to these three particular stocks produced some dubious parameter estimates in Table 4.2.

The vanilla GARCH model of §4.2.2 specifies a symmetric volatility response to market news. That is, the unexpected return e, always enters the conditional variance equation as a square, so it makes no difference whether it is positive or negative. However, there is an enormous literature on different specifications of the GARCH conditional variance equation to accommodate an asymmetric response.

The first asymmetric GARCH model that precipitated considerable academic interest was the exponential GARCH or E-GARCH model introduced by Nelson (1991).8 The conditional variance equation in the E-GARCH model is defined in terms of a standard normal variate zt:

q, = m + p(qt { - ) + C(e? i - 2 ,)

(4.6)

2,=qt + a(£2 ! - <?, ,) + P(ct2 , - qt {).

(4.7)

Inc.2 =a + g(zt i) + pincr2 i.

(4.8)

sThe continuous time limit of the AR(l)-exponential ARCH model gives a conditional variance process whose stationary distribution is lognormal. See Nelson (1990), where he also develops a class of diffusion approximations based on the exponential ARCH.