the Student-r distribution with 3 degrees of freedom. The GARCH (1,1) model gives results closer to those of USD-DEM but still underestimates the risks by a significant amount. The HARCH model slightly overestimates the extreme risk. This is probably due to its long memory, which does not allow the process to modify sufficiently its tail behavior under aggregation. Obviously, further studies need to be pursued to assess how well models such as HARCH can predict extreme risks. In general, ARCH-type processes seem to capture the tail behavior of FX rates better than the simple unconditional distribution models. The advantage of having a model for the process equation is that it allows the use of a dynamic definition of the movements and it can hopefully provide an early warning in case of turbulent situations.

Paradoxically, in situations represented by the center of the distribution (nonex-treme quantiles), the usual Gaussian-based models would overestimate the risk. Our study is valid for the tails of the distribution, but it is known that far from the tails the normal distribution produces higher quantiles than actually seen in the data.

5.5 SCALING LAWS

In this section, we examine the behavior of the absolute size of returns as a function of the frequency at which they are measured. As already mentioned, there is no privileged time interval at which the data and the generating process should be investigated. Thus it is important to study how the different measures relate to each other. One way of doing this is to analyze the dependence of mean volatility on the time interval on which the returns are measured. For usual stochastic processes such as the random walk, this dependence gives rise to very simple scaling laws (Section 5.5.2). Since Muller et al. (1990) have empirically documented the existence of scaling laws in FX rates, there has been a large volume of work confirming these empirical findings, including Schnidrig and Wiirtz (1995); Fisher et al. (1997); Andersen et al. (2000) and Mantegna and Stanley (2000). This evidence is confirmed for other financial instruments,17 as reported by Mantegna and Stanley (1995) and Ballocchi et al. (1999b). The examination of the theoretical foundations of scaling laws are studied in Groenendijk et al. (1996); LeBaron (1999b), and Barndorff-Nielsen and Prause (1999).

Before discussing the literature, we present the empirical findings.

5.5.1 Empirical Evidence

The scaling law is empirically found for a wide range of financial data and time intervals in good approximation. It gives a direct relation between time intervals

17 Brock (1999) has extensive discussions of scaling in economics and finance. He points out that most o£ these regularities are unconditional objects and may have little power to discriminate between a broad class of stochastic processes. Brock (1999) points out that the main object of interest in economics and finance is the conditional predictive distribution as in Gallant et al. (1993). Scaling laws may help in restricting the acceptable classes of conditional predictive distributions,

At and the average volatility measured as a certain power p of the absolute returns observed over these intervals,

{£[rp]}Vp = c(p)AtD(P} (5.10)

where E is the expectation operator, and c(p) and D{p) are deterministic functions of p. We call D the drift exponent, which is similar to the characterization of Mandelbrot (1983, 1997). We choose this form for the left part of the equation in order to obtain, for a Gaussian random walk, a constant drift exponent of 0.5 whatever the choice of/?. A typical choice is p = 1, which corresponds to absolute returns.

Taking the logarithm of Equation 5.10, the estimation of and D can be carried out by an ordinary least squares regression. Linear regression is, in this case, an approximation. Strictly speaking, it should not be used because the E[\r\] values for different intervals / are not totally independent. The longer intervals are aggregates of shorter intervals. Consequently, the regression is applied here to slightly dependent observations. This approximation is acceptable because the factor between two neighboring At is at least 2 (sometimes more to get even values in minutes, hours, days, weeks, and mean months), and the total span of analyzed intervals is large: from 10 min to 2 months. In addition, we shall see in Chapters 7 and 8 that volatility measured at different frequencies carries asymmetric information. Thus we choose the standard regression technique,18 as Friedmann and Vandersteel (1982) and others do. The error terms used for E[\r\] take into account the approximate basic errors of our prices19 and the number of independent observations for each At.

The results presented here are computed for the cases p = 1 and p = 2. It is also interesting to compute instead of {£[ ]}1 the interquartile range as a function of At. Such a measure does not include the tails of the distribution. Recently, the development of the multifractal model (Fisher et al., 1997) has brought researchers to study a whole spectrum of (also noninteger) values for the exponent p. In the same line of thought, multicascade models (Frisch, 1995) are also mul-tifractals and involve different drift exponents for different measures of volatility. This feature is a typical signature of multifractality.

In Figure 5.8, we present empirical scaling laws for USD-JPY and GBP-USD for p - 1. Both the intervals / and the volatilities E[\r\] are plotted on a logarithmic scale. The sample includes 9 years of tick-by-tick data from January 1, 1987, to December 31, 1995. The empirical scaling law is indeed a power law as indicated by the straight line. It is well respected in a very wide range of time intervals from 10 min to 2 months. The standard errors of the exponents D are less then 1 %. The 1 -day point in Figure 5.8 can be identified only by the corresponding label, not by any change or break between the intraday and interday behaviors.

18 See Mosteller and Tukey (1977, Chapter 14).

19 In Section 5.5.3, we give a full treatment of this problem.

I I I :

6.0 8.0 10.0 12.0 14.0 16.0 6.0 8.0 10.0 12.0 14.0 16.0

log (time interval in seconds) log (time interval in seconds)

figure 5.8 Scaling law for USD-JPY (right) and GBP-USD (left). On the y-axis, the natural logarithm of the mean absolute return (p = 1 in Equation 5.10) is reported. The error bars correspond to the mode described in Section 5.5.3. The sample period is January I, 1987, to December 31, 1995.

Small deviations for the extreme interval sizes can be explained. The data errors grow on both sides, as discussed in Section 5.5.3. For long intervals, the number of observations in the sample becomes smaller and smaller, leading to a growing stochastic error. For very short intervals, the price uncertainty within the bid-ask spread becomes important. In fact, researchers such as Moody and Wu (1995) studied the scaling law at very high frequencies and obtained different exponents because they did not take into account the problem of price uncertainty. Recently, Fisher etal. (1997) also found a break of the scaling law around 2 hr. It is clear on both plots of Figure 5.8 that fortime intervals shorter than 1 hour the points start to depart from a straight line. This deviation can be treated in two ways. In a first approach, we treat the price uncertainty as a part of the measurement error, leading to error bars that are as wide as to easily include the observed deviation at short time intervals. In a second approach, the deviation is identified as a bias that can be explained, modeled, and even eliminated by a correction. Both approaches are presented in Section 5.5.3.

In Table 5.8, we report the values of the drift exponent for four of the major FX rates against the USD and for gold for three different measures of volatility. Each of these measures treats extreme events differently. The interquartile range completely ignores them. The measure with p - 2 gives more emphasis to the tails than p = 1. The scaling law exponents D are around 0.57 for all rates and for p - 1, very close to 0.5 for p = 2, and around 0.73 for the interquartile range.