TABLE 7.1 Difference between lagged correlation for FX rates and gold.

The sample period is from June 6, 1973, (August 8, 1980 for gold) to February 1, 1995. The lags are measured in weeks and 3 hr (in #-time), respectively. The negative values indicate the predictability of finely defined volatility from coarse volatility.

Similar behavior of the lagged correlation is observed for other FX rates such as USD-JPY and GBP-USD, cross rates such as DEM-JPY, and gold (XAU-USD). Table 7.1 reports the difference q\ -q~\ and 52- q-2 for a set of these time series. The numbers are similar across the different rates (and also all of the investigated minor FX rates not shown here). The first lag difference is around -0.13 and the second lag difference is around -0.07.

The results with daily data also prevail in high-frequency and in intraday data. Every intra-day study requires an appropriate treatment of the strong intradaily seasonality of volatility. Here we use the predefined business time scale & presented in Chapter 6. A time series with regular intervals in ?-time is constructed by selecting the last quote before each point of a regular #-grid. As a basic time interval* in #-time, we choose 30 min. This means there is only some 7 min of physical time during the daily volatility peak in the European afternoon and American morning.6 Fine volatility is now the mean absolute half-hourly returns averaged over six observations, covering a 3-hr time interval. Coarse volatility is the absolute returns over a full 3-hr interval. All these time intervals are calculated in i?-time. An interval of 3 !?-hr is clearly smaller than the working day of an FX dealer. It often covers a time span with quite homogeneous market conditions.

Figure 7.8 (right panel) provides the lagged correlation function for USD-DEM in 8 years of half-hour returns. Although the half-hour data cover a shorter time span than the daily series, the number of observations is larger. The findings from the half-hourly data confirm the results from the daily series such that coarse volatility predicts fine volatility. We therefore conclude that these findings are independent of the data frequency.

The intradaily behavior of the lagged correlation is similar for other FX rates and gold (see Table 7.1). The empirical findings are similar across the different rates. The first lag difference is around -0.11 and the second lag difference is around -0.06, which are close to the corresponding values of Table 7.1. In the

6 In fact, a much higher frequency of the series should be avoided due to the fact that price changes observed over 5 min or less can be overly biased by microstructure effects (see Section 7.2).

0 8-j----;---0.8

-20 .10 0 10 20 -20 -10 0 10 20

Lag (in weeks) Lag (in multiples of 3 hr)

FIGURE 7.8 Asymmetric lagged correlation of fine and coarse volatilities for USD-DEM. The left figure is for working-daily return in a week. The right graph is for high resolution study with half-hourly returns within 3 hr (in ?-time). The negative lags indicate that the coarse volatility was lagged compared to the fine volatility. The thin curve indicates the asymmetry. The 95% confidence intervals are for identically and independently distributed observations. The sampling period for the left figure is 21 years and 8 months, from June 6, 1973, to February 1, 1995. The sampling period for the right figure is 8 years, from January 1, 1987, to January 1, 1995.

right panel of Figure 7.8, there is also a weak, rather wide local maximum around lag -11, corresponding to -33 hr in ?-time. This corresponds to a lag of about 1 working day (because a working day is 1/5 rather than 1/7 of a business week). The difference qt -q-t also has a significant (negative) peak around lag 11. This effect has been identified in the right panel of Figure 7.8 and discussed in Section 7.3. Following Engle et al. (1990), we call it a "heat wave" effect where traders have a better memory of the events approximately 1 working day ago (when they were active) than a broken number working days ago (when other traders on different continents, with different time zones, were active).

The peak around lag -11 can be explained by a residual seasonality that the i?-scale is unable to capture. However, the #-scale is well able to treat ordinary seasonality as indicated by the lack of an analogous peak around the positive lag 11. The heat wave effect is more than just seasonality and it cannot be eliminated by a simple time scale transformation. This can be interpreted such that volatility modeling should consider not only volatilities of different time resolutions but also volatilities with the selective memory of individual geographical markets and their time zones.

Assymetric lead-lag correlation is not only present in the FX market but also in the Eurofutures market as shown in Ballocchi etal. (1999a). Figure 7.9 presents the results of a lead/lag correlation analysis for forward rates implied from Euromark contracts on the London International Financial Futures Exchange (LIFFE). The asymmetry is highly significant for the first lag and for all maturities. At lag 1, again coarse volatility predicts fine volatility significantly better than the other way around. The study was conducted with a 3-hr grid in ?-time where the fine volatility is the mean absolute return measured every 3 hr over 3 days and the coarse volatility is the mean absolute return over the whole 3-day interval. The sample runs from April 1, 1992, to December 30, 1997, which constitutes 700 observations. The effect is rather robust with respect to changes in the definition of the fine and coarse volatilities. Moreover, it is interesting to note that the size of the effect seems to increase when increasing the time-to-start of the forward rate.

To explore this effect on a wider set of parameters, Gilles Zumbach suggested to the following quantities:

where T = 4 weeks and n and ri are the granularities of our volatility estimator. Then it is possible to compute a quantity / that depends on both n and ri:

which means that we look at the first lag difference where the lag is 4 weeks. This quantity should in principle be symmetric but we know from Figure 7.8 that it changes sign and is antisymmetric. Figure 7.10 presents the results of a study conducted by Zumbach (private communication), for the quantity / computed for values of n going from 2 to 12 over a period 7 of 4 weeks on the -time scale. This means that the returns are measured at frequencies as low as 2 weeks to frequencies as high as every 10 min in #-time. The FX rate is USD-CHF and the sampling period runs from June 1, 1987, to August 1, 1997. The asymmetry is striking and exists for all these different parameters. The maximum of the effect is obtained for n = 11 for the fine volatility and ri = 7 with differences as high as 0.29 between the two correlations, about two times more than in Table 7.1. Similar figures were also obtained for other FX rates like USD-DEM or USD-JPY.

7.4.3 Conditional Predictability

The conditional correlation studies of LeBaron (1992b,c) indicate that subsequent returnsare correlated in low-volatility periods and slightly anticorrelated in high-volatility periods. In continuous samples mixed from both low-volatility and high-volatility periods, this important effect indicating the forecastability of return does not exist unconditionally. It exists conditional to volatility. Thus, volatility is also an indicator for the persistence of trends. The idea is to compute the following triplet:

(7.10)

l(n,ri) = C(T,n,ri) - C(T, ri, n)

(7.11)

(v(t), r{t), r(t + At))