TABLE 6.1 Definition of the three generic markets.

Grouping of the different countries appearing in the multicontributor pages or record from Reuters according to the three components of the worldwide market.

Index Component Countries

1 East Asia Australia, Hong Kong, India, Indonesia, Japan, South Korea,

Malaysia, New Zealand, Singapore

2 Europe Austria, Bahrain, Belgium, Germany, Denmark, Finland,

France, Great Britain, Greece, Ireland, Italy, Israel, Jordan, Kuwait, Luxembourg, Netherlands, Norway, Saudi Arabia, South Africa, Spain, Sweden, Switzerland, Turkey, United Arab Emirates

3 America Argentina, Brazil, Canada, Mexico, United States

of price quotes originating from each of the three markets defined by Table 6.1 reveals two aspects on market presence:

A market has opening times that are longer than those of a particular submarket (e.g., an individual bank in one financial center such as Tokyo, Paris, or Chicago). The market opening time is the union of the opening times of all relevant institutions of the market.

Two markets (East Asia and Europe) have a local price quote frequency minimum in the middle of their working day, corresponding to a noon break. This local minimum is very pronounced in East Asia and moderate in Europe. In America, there is no minimum around noon. These differences reflect the well-known, different business habits concerning lunch breaks.

Each of the three markets is modeled to have two basic states, either open or closed. The activity does not completely go to zero when the market is closed because it is defined in terms of returns. The activity during the closing hours is modeled to stay on a small constant base level ao,*- During the opening hours, a much stronger, varying, positive activity adds to the base level,

a{t) = J}ao,*+au(01 = a0 + ]Tai.*(0 > 0 (6.7)

*=1 k=\

The joint base level ao is regarded as one model parameter. There is no need to analyze components ao,*-

The activity during opening hours, ai, is modeled with a polynomial with smooth transition to the constant behavior of the closing hours. This choice is mathematically convenient because such functions are easily differentiable and

analytically integrable. For parsimony, the number of parameters of this polynomial is kept at a minimum to model the smooth transitions, the lunch break, and the skewness to account for the relative weights of morning and afternoon hours.

In the subsequent analysis, the statistical week is considered from t - 0 on Monday 00:00 to / = 168 hr on Sunday 24:00 (GMT), as shown in Figure 6.1. In order to define the opening and closing conditions of the markets in a convenient form, an auxiliary time scale Tk is introduced. Essentially, it is GMT time; the following market-dependent transformations are only done for technical convenience:

Tk = [(t + Atk) modulo (24 hr)] - Atk (6.8)

where Atk has the value of 9 hr for East Asia, 0 for Europe, and -5 hr for America. (The result of the modulo operator is the left-hand side argument minus the nearest lower integer multiple of the right-hand side argument.) The weekend condition (WEC) also depends on the market:3

(t + Atk) modulo (168 hr) > 120 hr (6.9)

Now the model for an individual market component can be formulated by

(t, \ 0 if Tk < ok or > ck or (WEC)

aiM) ~~ j OopcnMO if ok < < ck and not (WEC) ;

where Ok and Ck are the parameters for the opening and closing hours, respectively. The polynomial function is

CQfc 9 9 9 9

aoperuO*) = 0k+Ck -(Tk-Ok) (Tk-Ck) {Tk - Sk)[(Tk -mk) + dk]

(6. )

where Wk represents the scale factor of the kth market, Sk the skewness of the activity curve, mk fixes the place of the relative minimum around the noon break, and dk determines the depth of this minimum. The special form of the first factor is chosen to avoid too strong a dependence of the scale factor on Sk-

In Figure 6.2, the panel on the left illustrates the shape of the geographical seasonality in the European market. The opening and closing times are where the activity level is zero. These parameters are illustrated with "<?" and "c" signs. The seasonality has two peaks with the second peak higher than the former. The relative minimum between the two peaks is the lunch break effect. The location and depth of this relative minimum are controlled by the parameters "m" and "d" of the last term of Equation 6.11. The activity starts to peak with the opening

3 The Japanese markets were open on some Saturday mornings according to certain rules in earlier years. These Saturdays, which are noticeable in Figure 6.1, are neglected here, but discussed in Section 6.3.2.

6 9 12 15 18 12 15 18 21 24

Hour of the Day (in GMT) Hour of the Day (in GMT)

FIGURE 6.2 The geographical seasonality patterns. The panel on the left illustrates the shape of the geographical seasonality in the European market. The seasonality has two peaks with the second peak higher than the former. The relative minimum between the two peaks is the lunch break effect. In the right panel, the North American geographical seasonality is plotted. It has no lunch break effect.

of the market in the morning, slows down during the lunch break and it peaks in the afternoon again. As the market closing time approaches, the level of activity gradually goes down and reaches zero. In the right panel of Figure 6.2, the North American geographical seasonality is plotted, which has no lunch break effect. The parameter s controls the asymmetry of the peaks for the European market, whereas in the case of the North American market, it controls the skewness of the overall pattern.

This polynomial model applies to all markets. The European and Asian markets (k = 1,2) have finite a* values in the fitting process, but for the American one, the parameter 3 always diverges to very high values. This reflects the missing noon break in this market, which has already been found in the tick frequency statistics.

The Equation 6.11 for America thus degenerates to a simpler form with no local activity minimum

. = 03+c3 (T3 - o3)2 ( 3 - c3)2 ( 3 - s3) (6.12)

2 «3

Some of the model parameters, the opening and closing times, are already known from the quote frequency statistics. For the other parameters, there are constraints.