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68

To ensure positive activities, an and must be positive and sk outside the opening hours,

an > 0 , > 0 , St < Ok St > Ck (6.13)

The parameter rtik in Equation 6.11 should be within the opening hours as it models the noon break:

Ok < mk < ck (6.14)

The functions ajr) must be fitted to the results of the statistics, astat(f), by minimizing the integral of the weighted square deviation of a(r) from astat(r). A continuous function astat(0 is not available but rather the hourly series astat,, from Equation 6.4. Therefore, the sum over the intraweekly sample is used instead of the integral:

mm , ,- = (i - -) hr (6.15)

The hourly intervals are represented by their middle points in this approximation. The least square model has 11 parameters, three coks, three sts, twomjs, two dks, and the base activity an. The values of opening, ok, and closing, Ck, are subject to random measurement error originating from the price quote frequency statistics. Therefore, these values are allowed to vary slightly for adjusting the fit. The minimization problem of Equation 6.15 is nonlinear in some of the parameters. It can be solved by the Levenberg-Marquardt method (see Press et al., 1986, Section 14.4), but in complex cases a simple genetic search algorithm provides the optimum parameters much more efficiently.

The main American and European markets observe daylight saving time during summer, whereas the main East Asian markets do not. This fact is ignored for the fitting. Only the GMT scale is used. A posterior daylight saving time correction is proposed in Section 6.3.2.

The resulting parameter estimates for four major FX rates and gold (XAU-USD) are presented in Table 6.2 together with the relative weights of the different markets (to be defined in Section 6.3.1). In the top panel of Figure 6.3, the resulting activity model together with the statistical activity for the USD-JPY is shown, and the bottom panel of Figure 6.3 shows the same quantity for the USD-CHF. Figure 6.4 displays the activity model over 48 hr (outside the weekend) with its different components for the same rates.

6.2.6 Interpretation of the Activity Modeling Results

The resulting parameters of the activity model and Figures 6.3 and 6.4 confirm the close relation between market presence and intraweekly volatility patterns. The market-specific tick frequency analysis and the activity fitting results compare favorably taking into account the Reuters coverage and the limitations of our model.

168 [astat,/ -ao- Ya=\ ai,*0i)]

a2 .

i=\ error,;

TABLE 6.2 The parameter estimates for the three generic markets.

The parameter estimates for the major FX rates and gold (XAU-USD) with the corresponding market weights. The sum of the market weights is less than 100 percent. The rest is accounted for by the basic activity oq. The residual activity ao, the scale factor , and the parameter d, which determines the depth of the minimum at lunch time, are dimensionless numbers. The values are a factor of 10-4.

 Rate "0 Market Weight USD-DEM 0.03 bast Asian 24.1 % 1.69 -3:32 8:24 3:33 0.97 -3:33 European 38.5% 1.07 5:54 18:39 11:07 2.06 20:21 American 34.1% 12.46 11:24 23:25 40:44 USD-JPY 0.03 East Asian 35.4% 1.40 8:43 3:35 1.01 -4:17 European 27.6% 5.37 6:55 16:40 11:02 1.51 17:23 American 33.4% 18.73 11:48 22:50 34:55 GBP-USD 0.02 East Asian 24.3% 1.05 -3:48 8:59 3:40 1.08 -4:02 European 39.1% 0.98 6:00 18:19 11:13 2.85 20:05 American 34.0% 13.88 11:24 23:11 31:43 USD-CHF 0.0! East Asian 22.0% 1.12 -4:00 9:00 3:40 1.06 -4:00 European 45.1% 1.04 5:00 18:00 11:23 2.45 -4:45 American 31.6% 13.71 12:00 24:00 24:00 XAU-USD 0.02 East Asian 9.7% 0.14 -3:43 9:36 4:05 3.17 -4:15 European 54.8% 2.98 5:36 17:19 11:10 1.54 2:42 American 33.8% 354.9 15:21 21:30 21:32

In both cases and for all FX rates, the local minima around noon have the following properties: they are pronounced in East Asia, moderate in Europe, and do not exist in America.

The USD-DEM and the USD-CHF have close parameter values as would be expected with a larger weight for Europe in the case of the USD-CHF, whereas the USD-DEM shows a higher weight for the American market. Gold (XAU-USD) has a very small East Asian market, which extends late because it is mainly traded with Europe. In general, its active trading periods in the individual markets seem to be less extended than for the FX rates. A similar effect is detected with silver. The USD-JPY has a strong East Asian component with a strong overlap with the American market. It is for this rate that the earliest opening of the East Asian market is found. The first example in Figure 6.4 (USD-JPY) has its main market in East Asia. The second example in Figure 6.4 (USD-CHF) has it in Europe, in line with the common sense expectation.

An alternative measure of market activity could also be based on the frequency of price quotes. According to the study in Chapter 5 (Table 5.13), this variable is highly correlated to the volatility. Yet we do not recommend it as an activity measure for two reasons.

1. This number depends on the coverage of the FX market by Reuters and its policy to publish prices on its FXFX page. For instance, a new price was

0 12 24 36 48 60 72 84 96 108 120 132 144 156 168

Intraweek Hourly Index

FIGURE 6.1 Histogram of the average hourly activity (as defined in Equation 6.4 for a statistical week (over 4 years) for the USD-DEM rate.

The scaling law, Equation 6.2, is applied to the ith hourly subsample instead of the full sample and mathematically transformed to

*. = -A (6.3)

From Chapter 5, we know that r, can strongly vary for the different hours of a week. The time interval At = 1 hr (for the hourly sampling) is nevertheless constant. Therefore, it is replaced by the interval #,- on the new time scale &. The size of A&j is no longer constant, but reflects the typical volatility of the ith hour. The constant c* is essentially the of Equation 6.2, but can differ slightly as it is calibrated by a normalization condition presented later.

The activity of the ith hourly subsample directly follows from Equation 6.1,

1 fE[\ri\]\l/D

«su,u = - -~ , Af = 1 hr (6.4)

Af V c J

This is the volatility-based activity definition used in the following analysis. The constant c* is calibrated to satisfy the following, straightforward normalization condition:

168

X>taU = 1 (6.5)

6.2.4 Geographical Components of Market Activity

In Figure 6.1, the histogram of the average hourly activity defined by Equation 6.4 is plotted for the USD-DEM rate. Although the activity definition is based only

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