Days

-GBP -

DEM-

-JPY

16 15 14 13

12

>

11 10

- " < -- <1-10< >

Days

Figure 4.9

1995.

-GBP-----DEM-------------JPY

GARCH term structures of US dollar rates: (a) 2 March 1995; (b) 13 April

The speed of convergence of the GARCH(1,1) volatility term structure depends on the estimate of + p. The smaller this quantity the more rapid the convergence to the long-term volatility estimate that is determined by (4.16).20 Figure 4.9 illustrates some GARCH(1,1) term structure volatility forecasts. Using seven years of daily data on US dollar rates the model is estimated up to 2 March 1995, a period of low volatility (Figure 4.9a) and up to 13 April 1995, a period of high volatility (Figure 4.9b). In both cases a term structure of volatility is generated up to 200 days. The estimated parameters are given in Table 4.9.

The speed of convergence of the GARCH(IJ) volatility term structure depends on the estimate of a - fj

20The half-life of the return to the long-term average is 1/(1 - - P). So if a + P is estimated as 0.95 it is 20 da>s. and if + p is estimated as 0.99 it is 100 days.

Table 4.9: GARCH(1, 1) parameter estimates for US dollar rates

As expected (see §4.2.3) the estimates of a + (3 are high and the parameter estimates do not change very much between 2 March and 13 April. The GBP rate is least reactive and most persistent of the three, having the lowest estimate of a and the highest estimate of (3; the estimate a + J3 is about 0.985, so the half-life of the return to the long-term average is over 60 days. This is much shorter for the JPY and even shorter for the DEM. These rates have similar intensities of reaction to market events in both periods (the estimates of a increasing slightly in the second period) but the DEM volatility is more persistent than the JPY. The estimated long-term volatility level (approximately 1006/250, where a is given by (4.16)) changes hardly at all between the two dates, being approximately 10% for JPY, 10.7% for GBP and 11% for DEM.

The term structures in Figure 4.9 are quite different on 2 March and 13 April. On 2 March the GBP 1-day volatility forecast was very low, around 7.5%. Because of the slow convergence in GBP volatility, even the 200-day volatility was only 9.7%, well below its long-term average. The other two rates have higher short-term volatility, around 9%, and are much quicker to reach their average levels. GBP short-term volatility was much less than the other two on 13 April, in fact it was near its long-term average, whereas DEM volatility was rather high, at 16% in the short term. The 200-day forecast of DEM volatility on that day was 12%, a little more realistic than assuming it would be 16% for ever, as would be the case with a moving average model.

Convergence of GARCH Convergence of GARCH volatility term structures is typically rather slow in volatility term structures foreign exchange markets; often the GARCH models are close to being is typically rather slow in integrated. Equity markets often have more rapidly convergent volatility term foreign exchange markets structures, as is evident from the estimated GARCH parameters in Table 4.6.

For General Electric, for example, five time series of GARCH volatility forecasts, over the next 1, 10, 30, 60 and 120 days, are shown in Figure 4.10a.21

GARCH volatility term structures on every day from 16 October 1997 to 11 September on the FTSE 100 indices are shown in Figure 4.10b. A rolling

21In order to generate a time series of GARCH term structure volatility forecasts as in this figure one should really re-estimate the GARCH model every day, and each time record the GARCH term structure. But General Electric has such stable parameter estimates (Figure 4.6) that simply generating all the GARCH forecasts from one model gives a good indication of the history of the General Electric term structure.

Jan-96 Jul-96 Jan-97 Jul-97 Jan-98 Jul-98 Jan-99 Jul-99 Jan-00 Jul-00

- 1 -day - - 10-day - 30-day - 60-day 120-day

Oct-97 Dec-97 Feb-98 Apr-98 Jun-98 Aug-98

(b) - 1 day - - -1 week --- 2 weeks - -1 months 3 months - 6 months!

Figure 4.10 GARCH volatility term structure forecasts for: (a) General Electric; (b) FTSE 100.

window of 1 year of data has been used to estimate the models, much less than in other examples of this chapter, and 1 year is really the minimum historic period that one should use in a GARCH model. The volatility forecasts in Figure 4.10b reflect the average volatilities during the last year and they are a little higher than they would be if several years of data were used, since equity markets have been much more volatile since 1997 (see Figure 4.7e).

4.4.2 Option Pricing and Hedging

The classic paper of Hull and White (1987) examined the pricing of options when volatility is stochastic. The assumptions of the Black-Scholes model no