6.3 A NEW BUSINESS TIME SCALE (#-SCALE)

17 = 17(0 =

The starting date fo chosen for the i?-scale is arbitrary. The activity is always positive, so its integral i?(0 is a monotonically increasing function. The §k represents in fact the business time scale of the kth market and is defined as

This quantity is informative in itself and can be used to model intramarket behavior. Because of the regular weekly pattern of , ? is predictable according to Equation 6.17; it may be computed also for the future. Due to normalization (see Equation 6.5), ?-time can be measured in the same units as physical time (e.g., hours, days, weeks); one full week in i?-time corresponds to one week in physical time.

The relative weight, Wk, of each market component can be defined with the help of the integral it> over a full week:

This is the share of the kth market in the & interval of one week. In Table 6.2, the relative weights of each component, as given by Equation 6.19, are presented together with the fitted parameters. These weights are in fact interesting pieces of information about the market shares of the components defined in Section 6.2.4 and Table 6.1. They are in line with the results of the market surveys regularly made by the (Bank for International Settlements, 1990, 1993, 1995).

The &-scale contracts periods of low activity and expands period of high activity. This is clearly seen on Figure 6.6 where the mapping function between i?-time and physical time is shown for USD-DEM over a week. Because the i?-time is normalized to physical time over 4 years (see the next section), the two scales almost coincide after a week but not exactly (i?-scale is slightly above 168 hr), because we have chosen the week of September 9 to October 1, 1995, where there was no market holiday. This figure shows that during the weekend, i?-time flows very slowly, compensating for the low activity during this period in physical time.

6.3.2 Adjustments of the #-Scale Definition

The &-scale defined in Section 6.3.1 reflects a rigid intraweekly pattern of expected market activity. However, there is more relevant information about the activity due to information on business holidays, daylight saving times, and scheduled events in general. In practice, for volatility forecasts, it is desirable to account for this information in the construction of the ?-scale. Such adjustments are carried out in Equation 6.17 by recalibrating the factor c* over the whole sample.

(6.18)

Vk(t + 1 week) -&k(t) # (r + 1 week) - #(0

&k(t + 1 week) - *(0 1 week

(6.19)

48 72 96 120 144

Physical Time (hours in a week)

FIGURE 6.6 The time mapping function between physical time and time. The week chosen to draw this mapping function is a week with no market holidays (September 25 to October 1, 1995). The thin line represents the flow of physical time.

It is difficult to take into account the different holidays of each market accurately.5 In the framework of the three markets of Table 6.1, our approach is an approximate solution. A holiday is considered if it is common to a large part of one of the three markets of the model. On such holidays, the activity a\ is set to zero for this market. The holiday is treated like a weekend day in Equation 6.10.

In some countries, there are half-day holidays. Their treatment would require the splitting of the daily activity functions into morning and afternoon parts. This splitting could also be used to model the few Saturday mornings in Japan (until 1989) when the banks were open. These modifications have not been made as they are beyond our objective of modeling the main features of the FX activity patterns.

The daylight saving time observed in two of the markets, Europe and America, has an influence on the activity pattern and thus on &. The presence of local markets depends on local time rather than on GMT. One way to deal with this is to convert the time constants of Table 6.2 from GMT to a typical local time scale of the

3 Future holidays are not always known in advance as, for instance, the Islamic holidays. Thus, might no longer be predictable in those special cases.

6.3 A NEW BUSINESS TIME SCALE (i?-SCALE)

Physical Time;

40 60

Time Interval in Days

FIGURE 6.7 The hourly returns for USD-DEM from June 3, 1996, 00:00:00 to September 11, 1996, 00:00:00 are plotted using the physical time scale and the #-time scale. Note also the extreme events that are clearly visible on both graphs.

market. This conversion yields different results for the local times in summer and in winter. The time constants are fixed to the mean of the summer and winter conversion results, reflecting the fact that the sample used in the activity fitting is composed of approximately half summer and half winter. The computation of the activity and # is then based on Equation 6.10 with these local time constants. A better algorithm, which takes into account the difference between summer and winter local market time and which allows a dynamic adaptation to changes in the activity pattern indicates substantial improvement (Breymann, 2000).

So far, volatility patterns with periods of more than one week have been neglected. Yet there may be patterns with longer periods caused by month-end effects, by the monthly or quarterly releases of certain important figures such as the American trade or unemployment figures, and by yearly effects. Moreover, there are long-term changes such as the overall volatility increase over the past 15 years as shown in Chapter 5. None of these effects has been found to be significant in a 4-year sample we studied.

Figure 6.7 illustrates the effect of the time transformation with the hourly returns of USD-DEM over 3 months both in physical and in #-time. It is easy to