Having obtained a GARCH conditional variance for the next observation period, its square root may be taken and it may be annualized in the usual way to obtain a GARCH volatility estimate. However, this current estimate is not taken to be the forecast of volatility over all future time horizons, as it is in the moving average methods described in Chapter 3. Instead, mean-reverting volatility term structures that converge to the long-term average volatility are derived analytically from the estimated GARCH model parameters, as explained in §4.4.1.

GARCH volatility forecasts are very flexible and can be adapted to any time period. The forward volatilities that are generated by GARCH models have many applications. Valuing path-dependent options or volatility options, measuring risk capital requirements, calibration of binomial trees - all of these require forecasts of forward volatilities that have a proper mean-reverting property.

Perhaps the most important of all the advantages of GARCH models is that they are based on a statistical theory that is justified by empirical evidence.

There is no need to hold Unlike constant volatility models, there is no need to hold a GARCH model

a GARCH model together together with bits of string and sticky plaster, to force it into a framework that with bits of string and is inconsistent with its basic assumptions. This coherency has led to many

sticky plaster, to force it applications of GARCH models to the pricing and hedging of options.

into a framework that is

inconsistent with its basic Some GARCH models have diffusion limits that will provide a model for assumptions stochastic volatility.16 In §4.4.2 the motivation behind their use to price and hedge any type of option, path-dependent or otherwise, is explained. GARCH models have also been used to fit and forecast implied volatility smiles, as described in §4.4.3.

4.4.1 GARCH Volatility Term Structures

Long-term (steady-state) forecasts of volatility from a GARCH(1,1) model have already been referred to in §4.3.1. There it was also noted that one can fix this long-term volatility at a level that reflects any reasonable scenario, and use the formulation (4.13) to estimate the GARCH reaction and persistence parameters using historic data.

Whether the long-term volatility is fixed or estimated freely, the real strength of the GARCH model is that volatility forecasts for any maturity may be obtained from the one estimated model. Forecasts are very simple to construct in many GARCH models because they take a simple analytic form. No approximations or lengthy simulations are necessary. Term structure forecasts that are constructed from GARCH models mean-revert to the long-term level

16In his PhD dissertation. Nelson (1988) made the bold conjecture that it was not possible to find a conditionally heteroscedastic diffusion limit for a GARCH process. He later corrected this mistake (Nelson, 1990).

Volatility forecasts for any maturity may be obtained from the one estimated model

of volatility at a speed that is determined by the estimated GARCH parameters. This is the great advantage of GARCH over moving average methods, which are based on the less realistic assumption of constant volatility term structures (§3.3).

The first step is to construct forecasts of instantaneous forward volatilities - that is, the volatility of rl+J, made at time t for every step ahead j. For example, in the GARCH(1,1) model the 1-day forward variance forecast is

g2+1 = cb + ae? + Pa2 (4.16)

and the j-step ahead forecasts are computed iteratively as17

g2+; = cb + (a+P)g2+,„,.

Putting o2+J - g2 for all j gives the steady-state variance estimate

2 = cb/(l -a- p) (4.17)

and this determines the long-term volatility level to which GARCH(1,1) term structure forecasts converge if ci + P < 1.

The forecasts from an A-GARCH model (4.9) also have a simple analytic form. The one-step-ahead forecast is

g2+1 = cb + a(e, - X)2 + pG?

(4.18)

and the steady-state variance estimate is

G2 = (a + oU2)/(l -a-P).

(4.19)

Comparison of (4.17) and (4.19) shows that the leverage coefficient X would have the effect of increasing long-term volatility forecasts, ceteris paribus. That is, if the , a and p estimates were not changed very much by moving from a symmetric GARCH(1,1) to an A-GARCH(1,1) model, the long-term volatility forecasts from the A-GARCH model would be higher than those from a symmetric GARCH model. However, there will be a change in the , a and P estimates and the steady-state variance estimate in (4.19) should not differ from the GARCH(1,1) steady state (4.17).

The most noticeable differences between the forecasts made by symmetric and asymmetric GARCH models are in the short-term volatility forecasts following a large fall in market price

The most noticeable differences between the forecasts made by symmetric and asymmetric GARCH models are in the short-term volatility forecasts following a large fall in market price. Comparison of (4.16) with (4.18) shows that the difference between one-step ahead variance forecasts will be dominated by the term aX(X - 2e(). Differences in volatility forecasts may be considerable if a very large unexpected negative return is experienced at time t, as shown in Table 4.8.

17The unexpected return at time <+j is unknown for j>0. But E(sj+J) = dj

We construct h-day forecasts by adding the ystep-ahead GARCH variance forecasts for

j=/,...,h

To construct a term structure of volatility forecasts from any GARCH model, first note that the log return at time t over the next h days is

Er+/-

Since

(4.20)

the GARCH forecast of / -period variance is the sum of the instantaneous GARCH forecast variances, plus the double sum of the forecast autocovar-iances between returns. This double sum will be very small compared to the first sum on the right-hand side of (4.20) - indeed, in the majority of cases the conditional mean equation in a GARCH model is simply a constant, so the double sum is zero.18 Hence we ignore the second term and construct / -day forecasts simply by adding the j-step-ahead GARCH variance forecasts. These are square-rooted and annualized with the appropriate factor as in (1.2) to give GARCH / -day volatility forecasts.19

Since all 1-day forward variance forecasts are computed it is also a simple matter to generate / -day forward volatility forecasts at any future date. For example, the average volatility over the next h days in n months time may be used to price path-dependent options. This is a feature of MBRMs Universal Add-in for GARCH that is on the CD.

Even in an AR(1)-GARCH(1,1) model with autocorrelation coefficient p in the conditional mean equation, (2.31) becomes a2 = a2+l + of [p(l - p")/(l - p)]2, and the first term clearly dominates the second.

19For a GARCH model that is based on daily returns with A daily returns per year, the annualizing factor for the /i-day forecast is A/h.

Table 4.8: Approximate increase in 1-day volatility forecast due to X