The explosive growth of interest rate trading, together with the deregulation and globalization of the financial industry, have fueled the market for interest rate derivatives since the late 1970s and early 1980s. Although there has been extensive academic research on U.S. Treasury bill futures, Eurofutures markets have not been extensively studied.

In 1997, the daily volume for the Chicago Mercantile Exchange (CME) Eurodollar was about 450 times larger than for CME U.S. Treasury bill futures. This takes into account neither the Eurodollar futures traded in other markets, nor Eurofutures for many other currencies, while the CME accounts for the entire U.S. Treasury bill volume.

The bid-ask spread on the CME Eurodollar can be as small as half a basis point,(n1) compared with quoted spreads on Eurodollar deposits that are at least two to twenty-five times higher. Due to the size of the small bid-ask spreads, even tiny moves are tradable in the Eurofutures markets, while they would not be in the Eurodeposit market. This means that the Eurofutures markets act as a very sensitive measuring device for interest rate expectations. They are also transparent markets because transaction prices are public knowledge, which is almost never the case for over-the-counter (OTC) markets. The Eurofutures markets are typically the most liquid markets for interest rate instruments, playing a crucial role in the price discovery mechanism. They yield high-quality intraday data consisting of transaction prices, firm bid and ask quotes, and sometimes volume information. Because of their liquidity and data availability, they are one of the best laboratories in which to investigate market microstructure.

Eurofutures are exchange-traded contracts and entail several differences with respect to over-the-counter instruments. First, Eurofutures are linked to a specific exchange, except when a fungibility agreement is in effect.(n2) Second, the trading is typically geographically localized, and there is no twenty-four-hour trading, unlike the foreign exchange market, although the trend is to effectively lengthen trading hours with after-hours sessions.

Given that futures contracts are exchange-traded and each transaction is recorded centrally, futures markets offer a high price transparency. The historical data always include tick-by-tick transaction prices and, depending on the data source, bid and ask quotes, bid and ask sizes, and actual volumes for each transaction are also included. This allows us to conduct the first full investigation of transaction prices,(n3) a comparison between the transaction prices and the bid-ask quotes for the outcry sessions.

Each futures contract has a specific expiration date and normally begins to be traded one year or more in advance. Eurofutures for an underlying three-month deposit have four expiration months in a year, March, June, September, and December, which are known as quarterly expirations. There are also serial expiration contracts such as contracts expiring in months that do not correspond to the quarterly sequence. The serial expiration contracts are not considered here because they typically exhibit lower liquidity.

We define the first position at a given time as the contract that expires next in the quarterly sequence, and the second position as the second contract to expire in the same quarterly sequence. The third and the fourth positions are constructed similarly. For each Eurofutures, we build two different types of price time series, by position and by contract. The position time series is not interrupted by contract expirations, and consists of different contracts. At each quarterly expiration, it switches to a new contract. The contract time series starts on the opening date of the contract and stops when the contract expires.

In the recent literature, Harvey and Huang [1991] indicate that interest rate and foreign exchange futures prices are much more volatile during the first sixty to seventy minutes of trading on Thursdays and Fridays than at any other hour over the trading week. Harvey and Huang [1991] postulate that this pattern is due to the fact that many macroeconomic announcements occur during the first hour of the trading on these two days and it is not due to the opening itself.

These findings are supported by Ederington and Lee [1993,1995], who analyze the impact of scheduled macroeconomic announcements on interest rate and foreign exchange futures markets. They find that these announcements are responsible for most of the observed time-of-day and day-of-the-week effects. In addition, Ederington and Lee [1993] investigate the speed at which markets adjust to news releases. They find that major price adjustments occur within one minute of the release and the direction of the subsequent price adjustments is independent of the first minute's price change. The price continues to be more volatile than normal for fifteen minutes and slightly more volatile for several hours. This adjustment rate is more than Patell and Wolfson [1984] observed in the securities markets.

Our study of the CME and the London International Financial Futures Exchange (LIFFE) focuses on Eurodollar (traded at CME), Euroswiss, Euromark, short sterling, and three-month ECU (all traded at LIFFE). Our results indicate the following:

• Eurodollar, Euromark, and short sterling display a high liquidity(n4) in all positions so that the number of intraday ticks(n5) reaches a maximum in the second position and then decreases. Euroswiss and three-month ECU, on the contrary, display a decreasing level of liquidity from Position I to Position 4. For Euroswiss, this reduction in the level of liquidity is seven times lower. For all positions, three-month ECU shows limited tick activity compared to all the other Eurofutures.
• For all contracts traded on LIFFE, the hourly tick activity displays a U-shape, with its minimum between 11 am-1 pm (GMT), and a clustering of activity around the beginning and the end of the day.
• Intraweek tick activity exhibits evidence of a day-of-the-week effect. The level of activity displays a minimum on Monday and a maximum on the last two working days of the week. The difference is significant for the Eurodollar. For Positions 1 and 2, the tick activity on Friday is almost twice as much as on Monday, and it becomes more significant in Positions 3 and 4.
• Intraday volatility follows similar patterns to that presented by intraday tick activity. In general, opening hours show the highest price variation; only in some cases does the largest price change occur toward closing time, which is usually in the last position. Short sterling and three-month ECU display the minimum of the U-curve one hour later than in the tick activity.
• For most contracts, the correlation between hourly tick activity and hourly volatility is above 96%. Euromark for the first two positions and three-month ECU for the fourth position show a lower correlation, around 90%.
• In general, Eurodollar, Euromark, and short sterling hourly volatility tend to increase from Position 1 to Position 4. For Euroswiss and three-month ECU, volatility shows an increase from Position 1 to Position 2 and decreases afterward.
• Eurodollar, Euroswiss, and three-month ECU display bid-ask tick activity higher than transaction activity. In particular, Eurodollar tick activity for bid-ask quotes is about five times higher than for tick activity of transaction prices for all positions. For the Euromark and short sterling tick activity, transaction prices are on the average double that for bid-ask quotes.
• During business days, bid-ask price volatility is usually higher than transaction price volatility, and the difference is often less than 1 basis point. Over weekends, with very few exceptions, bid-ask price changes are higher than transaction prices, and even higher on Sunday than on Saturday.
• The return series exhibit serial correlation that provides evidence for predictability and timing ability. This predictability does not translate into net profitability in the short horizons, such as three-minute data. However, the net profitability opportunities exist at the lower frequencies, such as the thirty-minute horizon, after taking trading costs into account.

We next examine the analysis per position. For all Eurofutures and all positions, the studied sample covers January 1, 1994-April 15, 1997, more than three years of tick-by-tick data. Later in the article we carry out the analysis per contract on Eurodollar (traded at CME), Euromark, and short sterling (all traded at LIFFE). For each Eurofuture, nine contracts associated with the following expirations are considered: March 1995, June 1995, September 1995, December 1995, March 1996, June 1996, September 1996, December 1996, and March 1997. Between different contracts, sample lengths differ but at least one year of data for each contract is available.

Lastly, we present scaling law analysis, as well as the persistence properties of the return, volume, and tick dynamics. We conclude afterward.

Studying futures by position as we propose can be justified on the basis of how this market works. Traders (both strategic traders and hedgers) holding close-to-expiration contracts need to roll their positions forward into the next expiration contract in order to stay in the market. By doing so, they are constructing a time series by position, extending beyond the expiration of each contract.

The study by contract, on the other hand, provides evidence that properties that depend on the contract lifetime (e.g., the deterministic volatility pattern) are a function of the time left before expiration. From the tick-by-tick data, the logarithmic price, P(t_i), is calculated by a simple linear interpolation so that

(1) p(t_i) = p(t_{p) + [p(t[sub s}) - p(t_p)] t_i -t_p/t_s - t_p

where t_p and t_s are the most recent previous and subsequent ticks relative to t_i. The change of price, or return at time t_i, r(t_i), is defined as

(2) r(t_i) equivalent to r (Delta t; t_i) equivalent to [p(t_i) - p(t_i - Delta t)]

where p(t_i) is the sequence of equally spaced in time logarithmic price, and Delta t is the fixed time interval, such as ten minutes, one hour, one day, etc. The volatility at time t_i, v(t_i), is defined as

(3) [Multiple line equation(s) cannot be represented in ASCII text]

where S is the sample period on which the volatility is computed. In Equation (3), the absolute value of the returns is preferred to the more usual squared value. This is because the former quantity better captures the autocorrelation and the seasonality of the data (Taylor [1988], Muller et al. [1990], and Granger and Ding [1993]). The tick frequency at time t_i, f(t_i), is defined as

(4) f(t_i) equivalent to f(S;t_i) equivalent to 1/S N ({x(t_j)|t_j)|t_j is an element [t_i - S, t_i]})

The log tick frequency at time t_i, log f(t_i), is defined as

(5) log f(t_i) equivalent to log f (S; t_i)

where N({x(t_j)}) is the counting function and S is the sample period on which the counting is computed.

The time series behavior of the Eurofutures indicates at least two different kinds of volatility. One is the spread existing between minimum and maximum prices occurring during the period under consideration (long-term volatility), and the other is the daily or weekly change in prices (short-term volatility). From both points of view, all Eurofutures display an increase in volatility going from Position 1 to Position 4.

In Exhibit 1, the time series behavior of Eurodollar and Euromark prices for Positions 1 and 4 are plotted. Both Eurodollars and Euromarks show an increase in volatility from Positions 1 to 4, both over short and long periods of time. This is representative of all the other Eurofutures series.

The intraday statistics use a uniform time grid with twenty-four-hour intervals; intraweek statistics use a uniform seven-day interval grid (from Monday to Sunday) in Greenwich Mean Time (GMT). Both measure volatility and tick activity, which can be considered as approximations of the level of market activity.

The intraday tick activity per position characterizing CME and LIFFE futures prices are summarized in Exhibit 2, which indicates that Eurodollar, Euromark, and short sterling display a high liquidity in all positions. The number of ticks reaches a maximum in Position 2 and then decreases. Position 4 has at most as many ticks as Position 1.

Euroswiss and three-month ECU, on the contrary, display a decreasing level of liquidity from Position I to Position 4. For Euroswiss, this reduction in the level of liquidity is seven times lower. For all positions, three-month ECU shows a limited tick activity compared to all the other Eurofutures.

Exhibit 3 shows the intraday tick activity and intraday price changes for short sterling in Position 2. The intraday tick activity displays the average number of ticks occurring in each hour of the day, while intraday volatility shows the mean absolute change in the logarithm of the price. Both plots display similar U-shapes, with the difference being that the minimum appears one hour later for intraday price changes. The sampling period starts on January 1, 1994, and ends on April 15, 1997. The total number of ticks in Exhibit 3 is 184,360.

For all contracts traded on LIFFE, the hourly tick activity displays a U-shape with its minimum around 11 am-1 pm (GMT) and a clustering of activity at the beginning and the end of the day.(n6) Intraday volatility follows similar patterns. In general, opening hours show the highest price variation,(n7) and only in some cases the largest price change occurs toward the closing time, usually in the last positions. Short sterling and three-month ECU display the minimum of the U-curve one hour later than in the tick activity.

Hourly and daily volatility is defined(n8) as the mean absolute change of logarithmic prices. For most contracts, the correlation between hourly tick activity and hourly volatility is above 96%; only Euromarks for Positions 1 and 2 and three-month ECU for Position 4 show a lower correlation, around 90%. In general, Eurodollar, Euromark, and short sterling hourly volatility tend to increase from Position 1 to Position 4. For Euroswiss and three-month ECU, volatility shows an increment from Position 1 to Position 2, and then they decrease again.

Values of daily and weekly volatility for each Eurofuture and each position are reported in Exhibit 4. Results for daily volatility confirm the results from the intraweek volatility analysis. For Eurodollars, Euromarks and short sterling, the level of variation tends to increase from Position 1 to Position 4. Euroswiss show an increment from Position 1 to Position 2, and then start to decrease. Three-month ECU displays a slight tendency to increase.

On a weekly basis, for all Eurofutures, price variations tend to increase passing from Position 1 to Position 4. Note that the ratio between the daily and weekly volatility almost always exceeds square root of 7, the latter being the ratio predicted by the aggregation properties of the Gaussian random walk.

Intraweek tick activity exhibits evidence of a day-of-the-week effect. The level of activity displays a minimum on Mondays and a maximum on Thursdays for LIFFE contracts and on Fridays 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. Exhibit 5 illustrates the intraweek tick activity for Eurodollars and Euromarks in Positions 1 and 4. They display considerable day-of-the-week differences.

The correlation between intraweek tick activity and intraweek price changes is high, but only Eurodollar intraweek volatility follows a pattern similar to that displayed by intraweek tick activity. In fact, except for Position 1, the maximum price variation for Eurodollars is reached on Friday and the volatility between Monday's and Friday's price changes tends to increase from Positions 2 to 4. Position 1 shows the highest level of variation on Monday and presents the lowest value of the correlation coefficient among all the Eurofutures. In addition, the spread of price changes on Friday is on average about 2 basis points higher than on other days.

Three-month ECU, Euroswiss, and Euromarks reach the maximum spread on Mondays and Thursdays (usually higher on Mondays) and short sterling on Wednesdays. In the case of short sterling, note that, at least for Positions 1 to 3, price changes on Wednesdays are about 2 basis points higher than on the other days (see Exhibit 6A, in which Position 2's results are displayed). This could be explained by the fact that the relevant news releases in the U.K. occur on Tuesdays and Wednesdays.

Now we compare the results from the transaction prices with the bid-ask quotes, with the objective of examining differences between transaction prices and bid-ask quotes, and in-hours and after-hours trading sessions. This section uses data from the same Eurofutures, but from a shorter period of December 2, 1996, to April 15, 1997. The comparison indicates that Eurodollars, Euroswiss and three-month ECU display bid-ask tick activity higher than transaction activity. In particular, Eurodollar tick activity for bid-ask quotes is about five times higher than tick activity for transaction prices for all positions.

For the Euromark and short sterling tick activity, transaction prices are on the average double that of bid-ask quotes. During business days, bid-ask price changes are usually higher than transaction price changes and the difference is often less than 1 basis point. Over weekends, with very few exceptions, bid-ask price changes are higher than transaction prices, and even higher on Sunday than on Saturday, on which there are practically no transaction price changes.

In Exhibits 6B and 6C, intraweek tick activity for transaction prices and intraweek tick activity for bid-ask quotes are presented. Both are computed for short sterling in Position 2. Activity on transaction prices appears slightly higher than activity on bid-ask.

For all Eurofutures traded on LIFFE, intraday tick activity displays a U-shape for both transaction prices and bid-ask quotes. In general, both transaction and bid-ask tick activities are concentrated between 6 am-7 am and 5 pm-6 pm (GMT). The hourly tick activity for the Eurodollar bid-ask is on average four times higher than for transaction data, and tick activity for transaction data is mostly concentrated between 11 am and 9 pm (GMT). For the bid-ask series, there are a significant number of ticks for the entire twenty-four hours.

For both transaction prices and bid-ask quotes, all Eurofutures traded on LIFFE display significant intraday volatility between 6 am-7 am and 5 pm-6 pm (GMT), which is similar to the intraday tick activity behavior. In general, hourly volatility is higher for bid-ask quotes than for transaction prices of all positions. For Eurodollars, transaction price volatility is mostly concentrated between 6 am and 1 am (GMT), while bid-ask volatility is spread over the whole day, with important peaks between 12 pm and 8 pm.

Exhibit 7 shows hourly volatility for transaction prices and bid-ask quotes. Both are computed for Eurodollars and Euromarks in Position 1. In both cases, the difference in price variation between bid-ask quotes and transaction prices is higher during open market hours than when the market is closed.

The purpose of this section is to analyze the volatility behavior of futures prices per contract. As shown before, futures contracts exhibit volatility seasonalities like those reported in the literature for foreign exchange data (Muller et al. [1990]) and also for equity markets (Andersen and Bollerslev [1997], Ghysels and Jasiak [1995]). Those seasonalities are attributed to dealing patterns such as market presence (Dacorogna et al. [1993]).

However, the characteristic feature of the futures markets when compared to foreign exchange or equity markets is that each contract has an expiration. We show that this leads to another seasonality, depending on the time left to expiration. On a daffy or weekly basis, there are indications that Eurodollar, Euromark, and short sterling display a decreasing volatility toward the expiration. We define the deterministic volatility pattern as the relation between volatility and time left to expiration. In order to probe the existence of a seasonality related to contract expiration, a sample consisting of many futures contracts is needed.

For each Eurofuture (Eurodollar, Euromark, short sterling) and for each contract, we build a series of hourly price differences determined by linear interpolation. Then we compute daffy volatilities, taking the mean absolute value of hourly price differences from 00:00 to 24:00 (GMT) of each working day (weekends and holidays excluded). The daffy volatilities are then plotted against time to expiration. The result is shown in Exhibit 8.

The vertical axis represents the deterministic volatility computed on all Eurofutures and all contracts together. The horizontal axis represents the time left to expiration, and, as we move toward the origin, the number of days before expiration decreases.(n9) Exhibit 8 spans a period of about 300 days, because only within that period were we able to compute our deterministic volatility on at least thirty-five contracts.

After 300 days, only a small number of contracts displays any real activity, so the average is affected by the volatility, which characterizes only a few Eurofutures, and it becomes less significant. The results indicate that there is a downward trend in volatility as the time left before expiration decreases. There are also unexpected oscillatory movements with peaks every ninety days corresponding to rollover activities near the end of contracts.

These results are also confirmed by a deterministic volatility study of each single Eurofuture. Eurodollars, Euromarks, and short sterling show a decreasing volatility for at least the 300 days before expiration. All Eurofutures display oscillatory movements with peaks around expiration dates.

Another test of a random walk process is given by the existence of an empirical scaling law, which relates the mean absolute price changes to the time interval on which the change is measured. This test was proposed in Muller et al. [1990], and empirically studied for foreign exchange rates. The idea behind the scaling law is straightforward. The random walk model is

(6) x_t = x_t-1 + Epsilon_t

where x_t is the logarithm of a financial price series and Epsilon_t are independently and identically distributed following a normal distribution with mean zero and variance Sigma². Let r^{n, sub t} denote the n-period return at time t, i.e., r^{n, sub t} = x_t - x_t-n so that

(7) [Multiple line equation(s) cannot be represented in ASCII text]

Due to the independence of the Epsilon_t, the variance of r^{n, sub t} is equal to Sigma^{2, sub rw} = n Sigma². The plot of the logarithm of the variance of the n-period random walk return [log(n Sigma²)] against the logarithm of the horizon n[log(n)] leads to a linear function with slope equal to 1 [or 0.5 when log(Sigma_rw) is plotted against log(n)]. This simple calculation provides an intuitive basis for a test of the random walk hypothesis. The idea is to plot the logarithm of the n-period return variance -- or some other measure of volatility (here the mean absolute price change) -- against the return period and to test whether the relationship is linear and the slope coefficient is the one implied by the random walk process. The scaling law provides a simultaneous test of many different time horizons (Guillaume et al. [1997]) for a particular process.

An empirical scaling law(n10) examines the relationship between intervals At and the average absolute price changes

(8) |Delta x| = c Delta t^1/E

where the bar over the |Delta x| indicates the average over the whole sample period, c is a constant, and 1/E is the drift exponent, which is 0.5 for the Gaussian random walk process. Results for drift exponents of different Eurofutures contracts, using overlapping observations, are shown in Exhibit 9. The drift exponents are all significantly above 0.5. The time intervals we consider for the absolute returns vary from below one day to half a year.

In a second step, we repeat the scaling law analysis on an average of contracts. We average the mean absolute values (associated with each time interval) on the number of contracts. When the analysis referred to single Eurofutures, the average was computed on nine contracts; when it referred to all the Eurofutures and all contracts together, the average was computed on thirty-six contracts. Then we performed a linear regression for the logarithm of the computed averages against the corresponding logarithm of time intervals, taking the following time intervals: one day, two days, one week, two weeks, four weeks, eight weeks, and half a year.

The scaling law analysis and the regression results are also presented in Exhibit 10. The confidence interval around the ordinary least squares (OLS) represents two standard deviations from the regression fit. The results indicate that the OLS model fits fairly well to the data implied by the empirical scaling law. Exhibit 10 also demonstrates that the scaling law implied by the random walk model stays outside the confidence intervals, implying rejection of the Gaussian random walk model as the data generation process.

The results in Exhibit 11 for single Eurofutures indicate that short sterling on short periods of time (one and two days) displays a slightly higher volatility than Eurodollars, while on the remaining intervals the opposite is true. In fact, short sterling's drift exponent is lower than that of the Eurodollar. Eurodollars and short sterling show higher volatility than Euromarks for all intervals except the longest one, which corresponds to twenty-six weeks or more. Euromark's drift exponent is the highest among all Eurofutures.

The drift exponent for all the Eurofutures are remarkably similar to those obtained on foreign exchange rates (Guillaume et al. [1997]) and on interbank money market rates (Ballocchi and Dacorogna [1996]).

This proves further that, in addition to being a test of the random walk hypothesis, the drift exponent is measuring something related to the nature of the market rather than the behavior of a particular asset. Guillaume et al. [1997] show that the drift exponent for regulated markets like the European Monetary System (EMS) is smaller than 0.5, and that it reverts to values larger than 0.5 when the EMS bands are relaxed. Here we have the confirmation [after the interbank interest rate and the stock market (Mantegna and Stanley [1995])] that, for liquid markets, the exponent is around 0.6, independent of which asset is traded.

The time series properties of the returns, volatility, and tick dynamics are investigated for the three- and thirty-minute series by position. In Exhibits 12-15, the three-minute USD series are investigated from Position 1 to Position 4. The return series exhibit a significant negative skewness with a large excess kurtosis. The skewness figures for the returns are -28.28, -13.72, -2.66, and -7.61 for Positions 1 to 4.

The size of the skewness decreases as it is moves from Position I to Position 4. The kurtosis figures are 3,455.90, 1,147.54, 1,985.30, and 476.13 for Positions 1 to 4, respectively. There is a gradual decrease of excess kurtosis as it is moves from Position 1 to Position 4, although the kurtosis of Position 3 is slightly higher than that of Position 2.

Similar to the returns, the volatility exhibits significant skewness and excess kurtosis relative to the Gaussian distribution. The tick series are also highly skewed and have excess kurtosis, although their skewness and kurtosis relative to the volatility dynamics are much less pronounced.

For the returns, the first four autocorrelations of Position 1, the first three autocorrelations of Position 2, the first two autocorrelations of Position 3, and the first six autocorrelations of Position 4 are statistically significant and negative. The statistical significance of the autocorrelations are measured by the Bartlett standard errors and the Ljung-Box-Pierce statistic. The size of an average movement of the Eurofutures in basis points can be measured from the mean absolute volatility.

For Position 1 in Exhibit 12, the average movement in basis points for the USD is about 0.247 basis points. An average twelve-minute movement is 0.494 for Position 1. Given that the trading cost for Eurofutures is about 0.5 basis point for transaction costs (the roundtrip execution cost; one way is 0.25 basis points) and 0.5 basis point for the spread, the size of the movement in basis points for a twelve-minute interval stays below the level of profitability.

The average movement in basis points gets larger from Position 1 toward Position 4. For Position 4 in Exhibit 15, the average movement for the three-minute data is 0.329 basis points. The persistence of the return dynamics lasts six consecutive three minutes, and the average movement in basis points is 0.806 for eighteen minutes. The size of the trading cost is 1 basis point for a roundtrip trade and 0.75 for a one-way trade.

For Position 4, the predictability gets closer to profitability if the trade is one-way, but not if it is roundtrip. One interpretation of these results is that conditional mean predictability can be interpreted as a timing advantage. Those who plan to buy may wait a bit longer given that the autocorrelations are negative, which may provide a price advantage when the market is moving down. For those who plan to sell, it indicates that they have to move quickly with their trade.

For the volatility and the tick dynamics, the first ten autocorrelations are statistically significant and positive. In particular, the serial persistence in the tick dynamics is highly significant. For instance, the Ljung-Box-Pierce statistics for volatility and the tick dynamics for Position 1 are 4,230 and 71,200 where the Chi²(10) at the 5% level is 18.307. The day-of-the-week effects are reported at the lower part of Exhibits 12-15. The p-values of t-statistics for the day-of-the-week dummies indicate that the day-of-the-week effects are statistically insignificant for the returns. For the volatility and the tick dynamics, there are significant day-of-the-week effects and this is highly significant for the tick dynamics.

The same analysis is also studied for the DEM series by position(n11) for the three-minute data. Similar to the USD Eurofutures, the returns, volatility, and tick series are highly skewed and have excess kurtosis. The return series have negative skewness for all positions; this is also true for the USD Eurofutures. There is also a gradual decline in the excess kurtosis as it is moves from Position 1 to Position 4.

Although we do not report the analysis with the GBP and JPY Eurofutures here, similar findings also prevail for these series. Overall, all the Eurofutures studied here are highly skewed where the skewness is negative on the return series and positive for the volatility and tick activity. All positions have large excess kurtosis and the size of the kurtosis declines from Position 1 to Position 4.

For the DEM Eurofutures, the return series have statistically significant negative serial correlations. For Positions 1 and 2, the length of the persistence prevails up to the ninth lag, which accounts for twenty-seven minutes of persistence. For Positions 3 and 4, the serial persistence lasts up to the sixth lag. For Position 1, the mean movement is 0.156 basis points, and it is 0.468 basis points for twenty-seven minutes. For Position 2, the mean movement for the basis points is 0.212, and it is 0.636 for the twenty-seven-minute window.

The volatility and the tick activity of the DEM position series are highly serially correlated and the serial correlation is more prominent in the tick activity. The day-of-the-week effect indicates that the returns do not have significant day-of-the-week seasonalities. The volatility and the tick activity, however, do exhibit statistically significant day-of-the-week effects, similar to the USD Eurofutures analysis.

Similar results also hold for the GBP and the JPY Eurofutures, although they are not reported here. The qualitative results here support the histogram analysis for the day-of-the-week effects earlier.

We further analyze the USD and the DEM Euro-futures with the thirty-minute data.(n12) The skewness and the excess kurtosis properties are similar to those of the three-minute data, although the size of the kurtosis is relatively smaller. The kurtosis for the returns for USD are 756.53, 271.52, 269.63, and 89.98 for Positions I to 4. The serial correlations indicate that the first-order serial correlations are negative and statistically significant.

For Position 2, the serial correlations for the first, third, and fourth lags are statistically significant. For Position 3, the first two autocorrelations are statistically significant. For Position 4, the first four lags have statistically significant serial correlations. In the case of Position 4, the average movement is 1.181 basis points in thirty minutes. This is approximately a 2.362-basis point move in a two-hour interval, and indicates the existence of net profit opportunities.

Similar findings prevail for the DEM series with the thirty-minute data. For all four positions, the results indicate that the first two autocorrelations for all four positions are negative and statistically significant. The mean movement in basis points increases from Position 1 to Position 4.

This is also the case for the USD series. The mean movement for DEM in Position 4 is 0.959 basis points. For a one-hour horizon, this is approximately 1.36 basis points. The GBP and the JPY series with the thirty-minute data also have similar characteristics but are not reported here.

The time series properties of the USD and DEM Eurofutures for the March and September 1995 contracts are studied here with the three- and thirty-minute data. The return series for contract data exhibit rather small skewness but the excess kurtosis remains large. The volatility and tick activity are highly skewed and have excess kurtosis for the three-minute as well as the thirty-minute series.

The serial correlation analysis indicates that USD futures have statistically significant autocorrelations at the first, fourth, fifth, and seventh to tenth lags with the three-minute data. The LBP test statistic on the returns is 1,210, where the 5% level for the x2 is 18.307. For the thirty-minute series, the first two, and the fourth to the ninth lags of the autocorrelations are statistically significant. The mean volatility for the thirty-minute series is 0.0001219, which indicates that the average movement for these series is 1.2 basis points. For a two-hour move, this is approximately a 2.258-basis point move and may be profitable after the trading costs.

For the three-minute data, the first six autocorrelations of the DEM Eurofutures are statistically significant for the return series. Given that the size of the average move is 0.23 basis points, this persistence may not be profitable at the three-minute frequency. The volatility and the tick activity display strong persistence, but the persistence is more prominent in the tick activity.

For the thirty-minute series, the first two autocorrelations are statistically significant. Given that the mean volatility is 0.87 basis points, this is approximately 1.23 basis points movements in an hour. This may be at the margin for net profitability.

We further analyzed the other contracts, including the 1996 and 1997 contracts on USD, DEM, GBP, and JPY Eurofutures. The results are similar in that the persistence properties of returns can be used as a timing indicator and may lead to pockets of predictability after taking the trading costs into account.

We present a statistical study of fundamental properties characterizing the Eurofutures markets. Intraday price changes and tick activity present a high positive correlation and display a U-shape, confirming the existence of intraday seasonality. The activity and volatility peak at the opening and closing times. There is evidence of intraweek seasonalities so that 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). There is practically no activity during the weekends.

A more detailed analysis of bid-ask quotes and transaction prices displays two additional features. The first is that Eurodollar, Euroswiss, and three-month ECU intraday tick activity is higher for bid-ask quotes than for transaction prices, except over weekends, when bid-ask tick activity prevails for all Eurofutures. The second is that the difference between the hourly volatility of the transaction prices the bid-ask quotes is very small (below 1 basis point).

For Eurodollars, the difference is much less than 1 basis point when the market is fully active. This difference increases when the market is not active. For futures traded on LIFFE, the difference is more important during active hours and it reaches a maximum during the first working hours of the day.

Our analysis shows the existence of a scaling law, which relates the volatility to the corresponding time intervals. The scaling law displays a drift exponent significantly larger than that expected for a Gaussian random walk and very close to the values obtained for foreign exchange rates. For data per contract, we show that price volatility displays a dependence on the time left to expiration. On average, volatility tends to decrease as we move toward expiration, and peak approximately every ninety days near quarterly expiration.

This study shows that, on the one hand, the Eurofutures markets are remarkably similar to the other markets studied so far with high-frequency data (Guillaume et al. [1997], Andersen and Bollerslev [1997], and Ghysels and Jasiak [1995]) in terms of intraday/intraweek seasonalities and scaling law for absolute price changes. On the other hand, we have found new properties that depend on the fact that futures contracts have predefined expiration dates. A striking example of such a property is the deterministic volatility that varies with contract expiration.

Our results also indicate that the return series exhibit serial correlation, which provides evidence for predictability and timing ability. This predictability does not translate into net profitability in the short horizons such as three-minute data. However, net profitability opportunities do exist at lower frequencies such as the thirty-minute horizon after taking trading costs into account.

This article would not have been possible without a team effort. The authors thank Bernard Hechinger for interpreting the data, providing information about the futures markets, and sharing his trading experience, and Rakhal Dave for data preparation and data access tools. We also thank the software engineers Beat Christen, Silvano Maffeis, Dan Smith, and Leena Tirkkonen for making the data accessible. Thanks are also due to Paul MacGregor from the Market Data Services Department of the London International Financial Futures Exchange for his kind help in providing and interpreting data from LIFFE. Part of this article was written while Gencay was visiting Olsen & Associates as a research scholar. He thanks them for their generosity and for providing an excellent research platform for high-frequency finance research.

(n1) basis point corresponds to 1/100%, and its monetary value (in the case of Eurodollar futures) is $25. The minimum price movement for the CME Eurodollar is 1/2 basis point.

(n2) An example of a fungibility agreement is the mutual offset system between the Chicago Mercantile Exchange (CME) and the Singapore International Monetary Exchange (SIMEX), through which contracts opened in one exchange can be liquidated on the other.

(n3) In the intraday studies of the foreign exchange markets, Muller et al. [1990], Dacorogna et al. [1993], and Guillaume et al. [1997] studied bid-ask quotes.

(n4) Although liquidity has been used in many contexts in the literature, we use tick activity as a measure of liquidity as it is a proxy for market activity.

(n5) Differences in tick activity across different futures positions are also studied in Ballocchi et al. [1998].

(n6) Similar forms of intraday seasonality are also reported in Andersen and Bollerslev [1997] and Ghysels and Jasiak [1995] with different data sets of exchange-traded instruments.

(n7) The difference with respect to the other hours is an average of 1 basis point.

(n8) For completeness, the other definition of the volatility as the square root of the variance computed is also calculated. There are no significant differences between the two.

(n9) The smoothing in Exhibit 8 is calculated by the maximum overlap discrete wavelet transformation, which is a time series multiresolution technique and is applicable to non-stationary time series (Percival and Mofjeld [1997]).

(n10) An application of the empirical scaling law within the context of foreign exchange markets is studied in Muller et al. [1990].

(n11) Tables for DEM Eurofutures are not presented here for the sake of brevity and can be requested from the authors.

(n12) Tables for thirty-minute data are not presented here and can be requested from the authors.

GRAPHS: EXHIBIT 1; Time Series Behavior of Eurodollar and Euromark Prices

Legend for Chart:

B - Eurodollar
C - Euromark
D - Sterling
E - Euroswiss
F - ECU

A          B      C      D     E     F

Position 1    104    161    151    84    41
Position 2    180    227    214    81    30
Position 3    151    215    158    29    14
Position 4    117    175    110    12     9

Notes: The intraday tick activity per position characterizes
CME and LIFFE futures prices. Eurodollars, Euromarks, and
short sterling display a high liquidity in all positions.
The number of ticks reaches a maximum in Position 2 and
then decreases. Position 4 has at most as many ticks as
Position 1.

GRAPH: EXHIBIT 3A; Intraday Tick Activity for Short Sterling

GRAPH: EXHIBIT 3B; Intraday Volatility Activity for Short Sterling

Legend for Chart:

B - Eudollar Day
C - Eudollar Week
D - Euromark Day
E - Euromark Week
F - Sterling Day
G - Sterling Week
H - Euroswiss Day
I - Euroswiss Week
J - ECU Day
K - ECU Week

A           B       C       D       E      F
G       H       I       J      K

Position 1     5.8    16.3     4.6    12.1     3.3
8.2     6.5    17.8     5.2    14.1

Position 2     6.7    18.9     5.4    13.7     4.6
11.2     7.1    19.7     5.2    13.5

Position 3     9.5    24.2     5.4    14.5     5.7
14.0     7.4    21.2     5.2    14.5

Position 4     7.1    21.0     5.2    14.9     6.1
16.2     7.6    23.0     5.4    15.8

Notes: Values of daily and weekly volatility for each
Eurofutures and each position. Results for daily volatility
confirm the results from the intraweek volatility analysis.
For Eurodollars, Euromarks, and short sterling, the level
of variation tends to increase from Position 1 to Position
4. Euroswiss shows an increment from Position 1 to Position
2, and then starts to decrease. Three-month ECU displays a
slight tendency to increase. The basis points are 10[sup -4.

GRAPHS: EXHIBIT 5; Intraweek Tick Activity for Eurodollars and Euromarks in Positions 1 and 4

GRAPHS: EXHIBIT 6

GRAPH: Panel A. Hourly Eurodollar Volatility, Transaction Prices (Position 1)

GRAPH: Panel B. Hourly Eurodollar Volatility Bid-Ask Prices (Position 1)

GRAPH: Panel C. Hourly Euromark Volatility, Transaction Prices (Position 1)

GRAPH: Panel D. Hourly Euromark Volatility Bid-Ask Prices (Position 1)

GRAPH: EXHIBIT 8; Volatility as a Function of Time to Expiration

Legend for Chart:

A - Expiration
B - Eurodollar
C - Euromark
D - Sterling

A                 B                C                D

March 1995        0.60 +/- 0.02    0.60 +/- 0.01    0.61 +/- 0.02
June 1995         0.66 +/- 0.02    0.65 +/- 0.01    0.62 +/- 0.02
September 1995    0.68 +/- 0.02    0.66 +/- 0.01    0.62 +/- 0.02
December 1995     0.64 +/- 0.02    0.66 +/- 0.01    0.64 +/- 0.02
March 1996        0.57 +/- 0.03    0.66 +/- 0.01    0.63 +/- 0.02
June 1996         0.70 +/- 0.01    0.62 +/- 0.01    0.62 +/- 0.02
September 1996    0.70 +/- 0.01    0.65 +/- 0.01    0.62 +/- 0.01
December 1996     0.69 +/- 0.01    0.63 +/- 0.02    0.60 +/- 0.02
March 1997        0.66 +/- 0.02    0.62 +/- 0.02    0.63 +/- 0.02

Notes: An empirical scaling law examines the relationship between
intervals At and the average absolute price changes, |Deltax| =
cDeltat1/E, where the bar over the |Deltax| indicates the
average over the whole sample period, c is a constant, and 1/E
is the drift exponent, which is 0.5 for the Gaussian process.
The drift exponents are all significantly above 0.5, which
indicates deviation from the Gaussian random walk.

GRAPH: EXHIBIT 10; Scaling Law on All Contracts

Legend for Chart:

B - All Eurofutures
C - Eurodollars
D - Euromarks
E - Sterling

A             B        C        D        E

Drift Exponent    0.599    0.634    0.664    0.599
Standard Error    0.007    0.008    0.023    0.013

Notes: This table reports the drift exponents for all contracts
and single contracts for Eurodollars, Euromarks, and short
sterling. For single Eurofutures, the average is computed on
nine contracts. For all Eurofutures, the average is computed
on thirty-six contracts.

Legend for Chart:

A - Description
B - Return
C - Volatility
D - Tick

A                     B                  C
D

Sample Size                      124,838         124,838
124,838

Mean                          -0.0000002       0.0000247
0.63

Standard                       0.00007         0.00007
1.30

Skewness                     -28.28           43.66
3.99

Kurtosis                    3455.90         4333.64
32.69

Max                            0.0047          0.0087
40

Min                           -0.0087          0.0000
0

Rho1                    -0.1416          0.1212
0.4522

Rho2                    -0.0245          0.0792
0.3591

Rho3                    -0.0109          0.0682
0.3199

Rho4                    -0.0126          0.0712
0.2778

Rho5                     0.0027          0.0563
0.2400

Rho6                    -0.0028          0.0571
0.2157

Rho7                    -0.0027          0.0464
0.1968

Rho8                    -0.0034          0.0405
0.1757

Rho9                    -0.0008          0.0406
0.1615

Rho10                   -0.0038          0.0376
0.1457

Bartlett Std. Errors           0.0028          0.0028
0.0028

LBP                             2,610           4,230
71,200

Chi2, sub 0.05 (10)     18.307

BetaConstant             0.059           0.000
0.000

BetaTuesday              0.053           0.023
0.000

BetaWednesday            0.041           0.036
0.000

BetaThursday             0.057           0.028
0.000

BetaFriday               0.125           0.000
0.000

F (zero slopes)                0.056           0.000
0.000

R2                       0.0004          0.008
0.064

Notes: Rho1, ..., Rho10 are the first ten
autocorrelations of each series. LBP refers to the
Ljung-Box-Pierce statistic and it is distributed Chi2
(10) under the null hypothesis of identical and independent
observations.

BetaConstant, BetaTuesday, BetaWednesday,
BetaThursday, and BetaFriday are the p-values of
the day-of-the-week dummies.

Legend for Chart:

A - Description
B - Return
C - Volatility
D - Tick

A                      B                   C
D

Sample Size                       147,257          147,257
147,257

Mean                           -0.0000002        0.0000319
1.03

Standard                        0.00008          0.00007
1.85

Skewness                      -13.72            21.30
3.29

Kurtosis                     1147.54          1623.98
18.10

Max                             0.0022           0.0075
34

Min                            -0.0075           0.0000
0

Rho1                     -0.1635           0.1910
0.5264

Rho2                     -0.0217           0.1348
0.4338

Rho3                     -0.0135           0.1189
0.3893

Rho4                     -0.0013           0.1239
0.3471

Rho5                      0.0007           0.1009
0.3044

Rho6                     -0.0003           0.0965
0.2768

Rho7                      0.0001           0.0929
0.2578

Rho8                     -0.0077           0.0819
0.2339

Rho9                      0.0010           0.0756
0.2147

Rho10                     0.0035           0.0721
0.2007

Bartlett Std. Errors            0.0026           0.0026
0.0026

LBP                              4,040           19,100
164,000

Chi2, sub 0.05 (10)      18.307

BetaConstant              0.009            0.000
0.000

BetaTuesday               0.013            0.000
0.000

BetaWednesday             0.014            0.000
0.000

BetaThursday              0.019            0.000
0.000

BetaFriday                0.056            0.000
0.000

F (zero slopes)                 0.037            0.000
0.000

R2                        0.0007           0.003
0.012

Notes: Rho1, ..., Rho10 are the first ten
autocorrelations of each series. LBP refers to the
Ljung-Box-Pierce statistic and it is distributed
Chi2 (10) under the null hypothesis of identical
and independent observations.

Legend for Chart:

A - Description
B - Return
C - Volatility
D - Tick

A                       B                    C
D

Sample Size                        146,052           146,052
146,052

Mean                            -0.0000002         0.0000328
0.87

Standard                         0.00011           0.00011
1.66

Skewness                        -2.66             18.18
4.13

Kurtosis                     1,985.30          2,349.12
36.64

Max                              0.0220            0.0042
50

Min                             -0.0222            0.0000
0

Rho1                      -0.0608            0.0956
0.5171

Rho2                      -0.0116            0.0728
0.4313

Rho3                      -0.0022            0.0631
0.3866

Rho4                      -0.0022            0.0670
0.3533

Rho5                       0.0002            0.0534
0.3036

Rho6                       0.0009            0.0478
0.2793

Rho7                      -0.0008            0.0441
0.2584

Rho8                      -0.0022            0.0418
0.2361

Rho9                      -0.0006            0.0375
0.2253

Rho10                      0.0024            0.0361
0.2061

Bartlett Std. Errors             0.0026            0.0026
0.0026

LBP                                 562             5,030
163,000

Chi2, sub 0.05 (10)       18.307

BetaConstant               0.579             0.000
0.000

BetaTuesday                0.493             0.000
0.000

BetaWednesday              0.482             0.000
0.000

BetaThursday               0.619             0.000
0.000

BetaFriday                 0.804             0.000
0.000

F (zero slopes)                  0.764             0.000
0.000

R2                         0.0001            0.003
0.015

Notes: Rho1, ..., Rho10 are the first ten
autocorrelations of each series. LBP refers to the
Ljung-Box-Pierce statistic and it is distributed Chi2
(10) under the null hypothesis of identical and independent
observations.

BetaConstant, BetaTuesday,  BetaWednesday,
BetaThursday, and  BetaFriday are the p-values
of the day-of-the week dummies.

Legend for Chart:

A - Description
B - Return
C - Volatility
D - Tick

A                      B                 C
D

Sample Size                      126,605         126,605
126,605

Mean                          -0.0000002       0.0000329
0.78

Std.                           0.00008         0.00008
1.50

Skewness                      -7.61           13.93
3.66

Kurtosis                     476.13          648.90
25.55

Max                            0.0022          0.0056
39

Min                           -0.0056          0.0000
0

Rho1                    -0.0910          0.2115
0.4924

Rho2                    -0.0127          0.1632
0.4059

Rho3                    -0.0031          0.1476
0.3665

Rho4                    -0.0028          0.1546
0.3338

Rho5                     0.0076          0.1259
0.2863

Rho6                    -0.0044          0.1117
0.2662

Rho7                     0.0004          0.0958
0.2423

Rho8                     0.0019          0.0922
0.2213

Rho9                     0.0058          0.0900
0.2087

Rho10                    0.0036          0.0858
0.1941

Bartlett Std. Errors           0.0028          0.0028
0.0028

LBP                             1,094          22,600
126,000

Chi2, sub 0.05 (10)      18.307

BetaConstant             0.008           0.000
0.000

BetaTuesday              0.016           0.000
0.000

BetaWednesday            0.035           0.000
0.000

BetaThursday             0.022           0.000
0.000

BetaFriday               0.334           0.000
0.000

F (zero slopes)                0.146           0.000
0.000

R2                       0.0005          0.007
0.018

Notes: Rho1, ..., Rho10 are the first ten
autocorrelations of each series. LBP refers to the
Ljung-Box-Pierce statistic and it is distributed Chi2
(10) under the null hypothesis of identical and independent
observations.

BetaConstant, BetaTuesday,  BetaWednesday,
BetaThursday, and  BetaFriday are the p-values
of the day-of-the week dummies.