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63

6 12 18

Hour of the Day (GMT)

- 1.2-

- 1.0-c

0.6-

TTTT

6 12 18

Hour of the Day (GMT)

FIGURE 5.13 Intraday analysis of Short Sterling in position two. The intraday tick activity (left histogram) displays the average number of ticks occurring in each hour of the day whereas the intraday volatility (right histogram) shows the mean absolute return. Both plots display similar U-shapes, the only difference being that the minimum appears one hour later for intraday returns. The time scale is GMT (not UKT, the local time used by LIFFE in London). The sampling period starts on January 1, 1994, and ends on April 15, 1997. The total number of ticks is 184,360.

Intraday returns follow a pattern similar to that presented by intraday tick activity. In general, opening hours show the highest price variation (the difference with respect to the average of the other hours is around one basis point); only in some cases does the largest return occur toward closing time (usually in the last positions). Differences occur in some positions28 for Short Sterling, Eurolira, and Three-Month Ecu,29 which display the minimum of the U-curve 1 hour later than in the tick-activity case. This can be seen in Figure 5.13, which displays intraday tick activity and intraday returns for Short Sterling in position two. Note that the U-shapes in this figure are blurred by the fact that Greenwich Mean Time (GMT) is used. The observations do not only cover winter months but also summers where the time scale used by LIFFE in London is shifted by 1 hour (daylight saving time). If the time scale was local time (UKT) instead of GMT, the U-shapes would be more pronounced with clearer peaks at opening and closing.

The first two positions of the Euromark display less regularity in the intraday return behavior. This behavior is confirmed also by correlation results: on the whole, the correlation between hourly tick activity and hourly returns is above 0.96; only Euromark for the first two positions and Three Month-Ecu for the fourth position show a lower correlation around 0.90. In general, for Eurodollar, Euromark and Short Sterling, hourly returns tend to increase from position 1

28 For an explanation of the word "position," see Section 2.1.2.

29 Short Sterling, Eurolira, Three Month-Ecu, and Euromark are names of LIFFE contracts, all with an underlying 3-month deposit. Ecu is the European Currency Unit that preceded the Euro.



0 90 180 270 360

Time to Maturity (Days)

FIGURE 5.14 Volatility as a function of time to expiry. The volatility values are daily averages over 36 contracts (9 for Eurodollar, 9 for Euromark, 9 for Short Sterling, and 9 for Eurolira). The abscissa corresponds to the time to expiry: the farther on the right-hand side, the farther away from expiry.

to position 4; Eurodollar and Short Sterling display a decrease for some hours in position 4.

Looking at intraweek tick activity, there is evidence of a day-of-the-week effect. In general, the level of activity displays a minimum on Monday and a maximum on the last two working days of the week, usually on Thursday for LIFFE contracts and on Friday for CME contracts. The difference is definitely significant for the Eurodollar; in fact, for positions 1 and 2 the tick activity on Friday is almost double that on Monday and it becomes more than double for positions 3 and 4. In general, there is a gradual increase from Monday to Friday.

5.6.4 Deterministic Volatility in Eurofutures Contracts

Ballocchi et al. (2001) provide evidence that the volatility of futures prices systematically depends on the time interval left until contract expiry. We call these systematic volatility patterns deterministic, as opposed to the also existing stochastic fluctuations of volatility. In order to probe the existence of a seasonality related to contract expiry, a sample consisting of many futures contracts is needed. For several Eurofutures contact type (Eurodollar, Euromark, Short Sterling, and Eurolira) and for each contract expiry, we build a series of hourly returns using linear interpolation. Then we compute daily volatilities taking the mean absolute value of hourly returns from 00:00 to 24:00 (GMT) of each working day (weekends and holidays are excluded). These daily volatilities are plotted against time to expiry. The result is shown in Figure 5.14. The vertical axis represents the mean volatility



computed from all Eurofutures and all contracts together. The horizontal axis represents the time left to expiry, as we move towards the left the number of days to expiry decreases.

Figure 5.14 spans a period of about 360 days because only within that period we are able to compute our mean volatility based on a full set of contracts. Some contracts have bad data coverage for times to expiry exceeding 360 days. The results obtained are quite interesting. There is a downward trend in volatility as the time to expiry decreases (moving from right to left in Figure 5.14). This downward trend is weak between about 300 and 180 days before expiry but becomes strong as we move toward the expiry date. There is also an unexpected behavior consisting of oscillatory movements with peaks every 90 days corresponding to rollover activities near the ending of contracts. These results are confirmed also by a deterministic volatility study on each single Eurofutures type-except Eurolira, which displays an increment in volatility as we move toward expiry. Eurodollar, Euromark, and Short Sterling show a decreasing volatility at least for the last 300 days before expiry. All Eurofutures display oscillatory movements with peaks around expiry dates (this appears particularly evident for Short Sterling).

A possible explanation for this effect is that these markets are all "cash settled" and therefore have no "delivery risk"; this means there is no risk of holding these futures on expiry day. Due to transaction costs, it is cheaper to take the cash at expiry than to close the position and realize the cash the day before. In other future markets such as the Deutsche Termin-Borse or the commodity markets, people who hold long positions to expiry actually take physical delivery of the underlying commodity or bond. There is a risk as expiry approaches as to which bond or type of commodity will be delivered. This may cause an increase in volatility as expiry approaches-a behavior opposite to that of Figure 5.14.

5.6.5 Bid-Ask Spreads

The bid-ask spread reflects many factors such as transaction costs, the market makers profit, and the compensation against risk for the market maker, see (Glass-man, 1987; Glosten, 1987). The subject of the intraday and intraweek analysis is the relative spread sj (Equation 3.12). It is usually below or around 0.1%, and its distribution is not symmetric. Negative changes are bounded as spreads are always positive, but the spread can exceed 0.5% in times of low market activity. The arithmetic mean of sj weights these low-activity spreads too strongly and therefore we choose the geometric mean as a more appropriate measure:

The index i indicates the hour of a day (or a week) or the day of the week, depending on the analysis. The total number of ticks that belong to the ih interval is /, j is the index and sjj the spread of these ticks.

(5.39)



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