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66 Because the #scale fully accounts for the seasonality of x, x* has no seasonal volatility patterns. The process x* may however have nonseasonal volatility patterns; it may be conditionally heteroskedastic. No attempt is made in this chapter to determine its exact nature. The time scale &(t) is a strictly monotonic function of physical time t. Any time interval from t\ to t2(> t\) corresponds to a #time interval of the positive size #2  #1 • The new activity variable a is defined as the ratio of the interval sizes on the different scales, a\,i =  (6.1) tit\ This activity reflects the seasonal volatility patterns. Its relation to other "activity" variables such as market presence or transaction volume was mentioned in Section 6.2.1 and is discussed below.2 6.2.3 Market Activity and Scaling Law The volatilitybased activity defined by Equation 6.1 can be computed with the empirical scaling law (see Chapter 5) for returns, which relates (for p = 1) (I Ax I), the mean absolute returns over a time interval to the size of this interval, At, E[\r\] = cAtD (6.2) where E is the expectation operator, is a constant depending of the specific time series. D is the drift exponent, which determines the scaling properties of the underlying process across different data frequencies. The drift exponent D is about 0.6 for major FX rates, whereas the pure Gaussian random walk model would imply D = 0.5. The scaling law expressed in Equation 6.2 holds for all time series studied and for a wide variety of time intervals ranging from 10 min to more than a year. The scaling law is applied to subsamples in a socalled intraweek analysis that allows us to study the daily seasonality (open periods of the main markets around the world) as well as the weekly seasonality (working days  weekend). Here, we choose a sampling granularity of At  1 hr. The week is subdivided into 168 hr from Monday 0:00  1:00 to Sunday 23:00  24:00 (Greenwich Mean Time, GMT) with index i. Each observation of the analyzed variable is made in one of these hourly intervals and is assigned to the corresponding subsample with index i. The 168 subsamples together constitute the full 4year sample. The sample pattern is independent of bank holidays and daylight saving time. A typical intraday and intraweek pattern across the 168 hr of a typical week is shown in Figure 6.1. 2 In skipping Saturdays and Sundays, other researchers use an implicit activity model with zero activity on the weekends.
3.0  4.0  FIGURE 6.3 The histograms of the average hourly activity for a statistical week (over 4 years) for the USDJPY (above) and USDCHF (below) rates and the modeled activity. shown for a particular rate on this page at maximum one price every 6 sec. Some relevant price revisions were therefore lost because of limitations of the data supplier. Whereas the price revisions depend directly on the data suppliers coverage or policy, the prices are issued by market makers who closely follow the real market value and have many data sources available. Thus published prices are conditioned more by other simultaneously available prices, which do not necessarily appear on this data source. In order to provide some empirical evidence of this dependence, we compare the hourly shares of the weekly number of price revisions in the 168 hr of the statistical week (see Section 6.2.1) of two different data suppli
on return statistics, the histogram exhibits clear structures where there is very low activity over the weekends and strongly oscillating activity patterns on normal business days. The most active period is the afternoon (GMT) when the European and American markets are open simultaneously. We have varied the At granularity of this analysis from 15 min to 4 hr and found no systematic deviations of the resulting activity patterns from the hourly ones. Furthermore, the activity patterns are remarkably stable for each of the 4 years of the total sample. The strong relation between return activity and market presence leads to the explanation of activity as the sum of geographical components. Although the FX market is worldwide, the actual transactions are executed and entered in the bookkeeping of particular market centers, the main ones being London, New York, and Tokyo. These centers contribute to the total activity of the market during different market hours that sometimes overlap. Goodhart and Figliuoli (1992) have explored the geographical nature of the FX market to look for what they call the island hypothesis. They studied the possibility that the price bounces back and forth from different centers when special news occurs before finally adjusting to it. Along the same idea, Engle et al. (1990), in a study with daily opening and closing USDJPY prices in the New York and Tokyo markets and a marketspecific GARCH model, investigate the interaction between markets. They use the terms heat wave hypothesis for a purely marketdependent interaction and meteor shower hypothesis for a marketindependent autocorrelation. TJiey find empirical evidence in favor of the latter hypothesis. Both studies havenot found peculiar behavior for different markets. This encourages us to model the activity with geographical components exhibiting similar behavior. The activity patterns shown in Figure 6.1 and the results reported in Chapter 5 suggest that the worldwide market can be divided into three continental components: East Asia, Europe, and America. The grouping of the countries appearing on the Reuters pages in our three components can be found in Table 6.1. This division into three components is quite natural and some empirical evidence supporting it will be presented in Chapter 7. The model activity of a particular geographical component is called a*(t); the sum of the three additive component activities is a(t): This total activity should model the intraweekly pattern of the statistical activity astat,( as closely as possible. Unlike astat, which has relatively complex behavior (see Figure 6.1), the components ak(t) should have a simple form, in line with known opening and closing hours and activity peaks of the market centers. 6.2.5 A Model of Intraweek Market Activity Each of the three markets has its activity function (t). For modeling this, we use quantitative information on market presence. A statistical analysis of the number (6.6)
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