FIGURE 5.12 Hourly intraday and intraweek distribution of the absolute return, the spread and the tick frequency: a sampling interval of At = I hour is chosen. The day is subdivided into 24 hours from 0:00 - 1:00 to 23:00 - 24:00 (GMT) and the week is subdivided into 168 hours from Monday 0:00 - 1:00 to Sunday 23:00 - 24:00 (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 the correct index /. The sample pattern does not account for bank holidays and daylight saving time. The FX rate is USD-DEM and the sampling period covers the 6 years from 1987 to 1992.

TABLE 5.12 Average volatility.

Average volatility for each day of the week (including weekends) for the USD against DEM, JPY, CHF, and GBP and XAU (gold) against USD; for the period from January I, 1987, to December 31, 1993. The volatility figures have to be multiplied by 10-3. They refer to one day. Corresponding annualized volatility figures are obtained through another multiplication by the factor V365.25 % 19.11.

volatility is roughly four times higher than the minimum. The patterns can be explained by considering the structure of the world market, which consists of three main parts with different time zones: America, Europe, and East Asia. Even the lunch-break familiar to the European and East Asian markets, but not to the American one, can be detected in the form of the two minima of the histogram for USD-DEM. The main daily maximum occurs when both the American and the European markets are active. Other markets have similar patterns with characteristic differences in the weights of the markets, such as a higher volatility for the USD-JPY when the East Asian markets are active (as to be shown in Chapter 6). The patterns for USD-CHF and USD-DEM are similar, as expected. The pattern for XAU-USD reflects the well-known fact that the East Asian gold market is less active than the European and American ones. Different volatilities across the American, European, and Japanese markets were also detected by Ito and Roley (1987),26 in their intraday study of the Japanese Yen.

Table 5.12 shows quite similar volatilities for the working days of the week. It does not confirm the weekend effect found by McFarland etal. (1982) with systematically lower volatilities on Fridays.27 Their analysis was however different by taking daily changes at 18:00 GMT and putting together Saturdays, Sundays, and Mondays. The volatility is low on weekends, but, for FX rates, higher on Sundays than on Saturdays. This is due to the early Monday mornings in East Asia and in Australia, which coincide with Sunday nights in GMT.

The intraweek volatilities of Table 5.12 are correlated with the activities measured in terms of the number of ticks (Table 5.11). The analogous correlation

26 For these authors volatility is measured by both the standard deviation and the mean absolute returns.

27 The effect seems to be the reverse on the stock market (high returns on Fridays).

TABLE 5.13 Correlation coefficients for activity measures.

Correlation coefficients computed for the different intraday analyses for the USD against DEM, JPY, CHF, and GBP and XAU (gold) against USD. Sampling period: from January I, 1987, to December 31,1993.

coefficients between the intradaily histograms of Figure 5.12 are also positive, as explicitly shown in the first line of Table 5.13. We conjecture that both variables are positively correlated to a third one, the worldwide intraday transaction volume, which is not known for the FX market. Transaction volume figures are, however, available for the stock market; their positive correlation to squared returns (and hence the volatility) has been found by Harris (1987) and other authors. Recently, Hasbrouck (1999) examined the data of the New York Stock Exchange and found similar correlations as in Table 5.13 for his transaction data, but the correlations did not uniformly increase when the data were aggregated.

The statistics show that an analysis of return distributions that neglects the large differences between the hours of a day and the days of the week is inappropriate. In Chapter 6, we will introduce a new time scale to solve this problem.

5.6.3 Seasonal Volatility: U-Shaped for Exchange Traded Instruments

Intradaily seasonalities were also found in the stock markets by Ghysels and Jasiak (1995), Andersen and Bollerslev (1997b) and Hasbrouck (1999). Unlike the FX market, stock exchanges and money market exchanges are active less than 24 hr a day. Thus the shape of the seasonality is different. It is called the U-shape because the high volatility of the opening is followed by a decrease, which is in turn followed by an increase of volatility just before closing. Ballocchi et al. (1999b) study the Eurofutures markets and find the expected intraday seasonality. For all contracts traded on LIFFE the hourly tick activity displays the U-shape with its minimum around 11 a.m. to 1 p.m. (GMT) and a clustering of activity around the beginning and the end of the trading day. There are differences among Eurofutures between the levels and widths of the peaks and the level of the minimum. The Eurodollar (a contract type traded on CME, see section 2.4.1) displays similar behavior but the activity in the first half of the working day, which takes place when the European markets are still open, is higher than during the second half of the day, when European markets have already closed and Asian markets are not yet open.