I 1

0.00 10.00 20.00 30.00 40.00 50.00 60.00

Time Lag (in minutes)

FIGURE 5.1 The autocorrelation function for the USD-DEM returns is plotted for different time lags in minutes up to 60 min. The returns are computed with prices interpolated using the previous tick interpolation method (see Chapter 3). The two horizontal lines represent the 95% confidence interval of an i.i.d. Gaussian process. The sampling period runs from January 5, 1987, to January 5, 1993. The autocorrelation is significantly negative up to a time lag of 4 min.

Wasserfallen and Zimmermann (1985); Feinstone (1987); Ito and Roley (1987), and Wasserfallen (1989). These papers analyze intradaily samples restricted to particular local markets and their local business hours. Recently the group of Barndorff-Nielsen has come up with the normal inverse Gaussian distribution that seems to capture some of the features we describe here, as observed in Eberlein et al. (1998); Barndorff-Nielsen (1998), and Barndorff-Nielsen and Prause (1999).

5.2 PRICE FORMATION PROCESS

The following three facts pertain to the short-term (less than 10 min) behavior of the foreign exchange intradaily returns. They highlight the difficulties inherent in tick-by-tick analysis.

5.2.1 Negative First-Order Autocorrelation of Returns

Goodhart (1989) and Goodhart and Figliuoli (1991) first reported the existence of negative first-order autocorrelation of returns at the highest frequencies, which disappears once the price formation process is over. In Figure 5.1, the autocorrelation function of returns measured at a 1 min interval is plotted against its lags. The returns are computed using the previous tick interpolation. There is significant

autocorrelation up to a lag of 4 min. For longer lags, the autocorrelations mainly lie within the 95% confidence interval of an identical and independent (i.i.d.) Gaussian distribution. Goodhart (1989) also demonstrated that this negative autocorrelation is not affected by the presence (or absence) of major news announcements. Finally, Goodhart and Figliuoli (1992) showed that the resulting oscillations of the prices are not caused by bouncing prices between different geographical areas with different information sets. In Figure 5.1, negative autocorrelation is observed not only at the first lag (1 min) but also at further lags up to about 3 or 4 min. This is due to irregular spacing of ticks. If tick time is taken (i.e., an artificial time scale that moves by one unit with every tick), the negative autocorrelation is observed only at the first lag and rarely at larger lags, thus justifying the term "first-order." This behavior is characteristic if individual ticks randomly deviate from the market average while return clusters of longer duration are absent.

A first explanation of this fact is that traders have diverging opinions about the impact of news on the direction of prices-contrary to the conventional assumption that the FX market is composed of homogeneous traders who would share the same views about the effect of news so that no negative correlation of the returns would be observed. A second-and complementary-explanation for this negative autocorrelation is the tendency of market makers to skew the spread in a particular direction when they have order imbalances (Bollerslev and Domowitz, 1993; Flood, 1994). A third explanation is that even without order imbalances or diverging opinions on the price, certain banks systematically publish higher bid-ask spreads. This could also cause the prices to bounce back and forth between banks (Bollerslev and Domowitz, 1993). An early model for this bid/ask bounce was proposed by Roll (1984) in modeling transaction data in the stock market. The idea is that the two prices, bid and ask, can be hit randomly according to the number of buyers and sellers in the market. If the number of buyers is equal the number of sellers, which is the case most of the time in the market without exo-geneous news, this model will produce a negative autocorrelation of transaction returns at high-frequency.

This negative autocorrelation is also seen in FX-rate transaction prices (Goodhart et al, 1995) and in Eurofutures contracts (Ballocchi et al, 1999b). For some stock indices such as the S&P 500, Bouchaud and Potters (2000) finds the autocorrelation of returns to be positive while it is not found in stock returns themselves or in futures contracts on indices (Ahn et al, 2000). The explanation for the positive autocorrelation of stock indices is that some of them are constructed from equities that have very different liquidity. The model is called the lagged adjustment model (Ahn et al, 2000). In this model one group of stocks reacts more slowly to aggregate information than another group of stocks. Because the autocovariance of a well-diversified portfolio is just the average cross-covariance of the stocks that make up the portfolio, this results in positive autocorrelations. In any case, the autocorrelation of returns is directly related to microstructure effects

moo

Quoted Spreads

Transaction Spreads

Basis Points

Basis Points

FIGURE 5.2 The figure on the left presents the spread size frequencies for USD-DEM quotes during June 16, 1993, collected from Reuters FXFX page. The figure on the right presents the spread size frequencies for USD-DEM transactions duringjune 16, 1993, from an analysis of Reuters Dealing 2000-2 by Goodhart et al. (1995).

in the market and should be carefully considered before using data at very high frequency.

The negative first-order autocorrelation can be seen as unwanted noise to be removed in a further study. An effective price can be defined in a way to eliminate the negative autocorrelation, as already discussed at the end of Section 3.2.2.

5.2.2 Discreteness of Quoted Spreads

Bid-ask spreads have discrete values. For studying this, we use the spread in its raw form, defined as ask price minus bid price, rather than the relative spread defined by Equation 3.12. In the example of Figure 5.2, bid-ask spreads of FX quotes are discretely distributed with the major peak at 5 basis points, followed by peaks at 10 and 7 basis points. A basis point is the smallest quoted decimal digit, which is 0.0001 German Marks per U.S. Dollar in the case of USD-DEM. In other, longer sampling periods and for other FX rates, we additionally observe spreads of 3, 8, 15, and 20 basis points with noticeable frequency. In a sample investigated by Bollerslev and Melvin (1994), the peaks at 5, 7, 10, and 15 basis points account for more than 97% of the distribution. These conventional spread values have evolved over the years, depending on the markets. For USD-DEM and some other major FX rates, the highest spread frequency peak shifted from 10 to 5 basis points during the 1990s, partly because