there seems to be some contradiction between the work of Goodhart and Figliuoli (1991), which claims that the fat tails start to decrease at these frequencies, and the paper of Bollerslev and Domowitz (1993), which gives some evidence of a still increasing fat-tailedness. One can show, however, that both results hold depending on whether one uses the linear interpolation method or the previous tick to obtain price values at fixed time intervals at such frequencies. This is an example of the difficulty of making reliable analyses of quoted prices at frequencies higher than 10 min. The divergence of the fourth moment explains why absolute values of the returns are often found to be the best choice of a definition of the volatility (i.e., the one that exhibits the strongest structures).4 Indeed, because the fourth moment of the distribution enters the computation of the autocorrelation function of the variance, the autocorrelation values will systematically decrease with a growing number of observations.

To complement Tables 5.1 and 5.2, we plot on Figure 5.6 the cumulative frequency of USD-JPY for returns measured at 10 min, 1 day, and 1 week on the scale of the cumulative Gaussian probability distribution. Normal distributions have the form of a straight line, which is approximately the case for the weekly returns with a moderate (excess) kurtosis of approximately 1.3. The distribution of 10-min returns, however, has a distinctly fat-tailed form and its kurtosis in Table 5.1 is very high. If the data-generating process was a random walk with increments from a stable distribution, which is defined by the law that scaled returns r/{At)y for a certain have the same distribution irrespective of the measurement interval At, we would obtain a uniform distribution with identical moments within the significance limits.5 Considering all the presented results, this is clearly not the case. This instability of distributions was also found by other authors. McFarland et al. (1982) and Boothe and Glassman (1987) suggest that distributions are composed of reactions to different flows of information. Calderon-Rossel and Ben-Horim (1982) are in agreement with our findings and claim that the returns cannot be accurately described by a unique type of stable distribution.

5.4.2 The Tail Index of Return Distributions

The tails of all possible distributions can be classified into three categories:6

i. Thin-tailed distributions for which all moments are finite and whose cumulative distribution function declines exponentially in the tails

ii. Fat-tailed distributions whose cumulative distribution function declines with a power in the tails

iii. Bounded distributions which have no tails

4 We shall see some evidence of this in Section 5.6.1 and in Chapter 7.

5 Here there is no need to further characterize stable distributions in addition to the described scaling behavior. Section 5.5.2 has a definition and discussion of stable distributions.

6 The interested reader will find the full development of the theory in Leadbetter et al. (1983), and de Haan (1990).

FIGURE 5.6 The cumulative distributions for 10-min, I-day, and I-week USD-JPY returns shown against the Gaussian probability on the y-axis. On the x-axis the returns normalized to their mean absolute value are shown. The mean absolute return for 10 min is 2.62 x 10-4, for I day 3.76 x 10-3, and for I week 1.14 x 10-2. The three curves are S-shaped as typical of fat-tailed distributions. The S-shapes of the three curves are very differently pronounced.

A nice result is that these categories can be distinguished by the use of only one parameter, the tail index a with a = oo for distributions of category (i), a > 0 for category (ii), and a < 0 for category (iii). The empirical estimation of the tail index and its variance crucially depends on the size of the sample (Pictet et al, 1998). Only a well chosen set of the most extreme observations should be used. The very large sample size available for intradaily data ensures that enough "tail observations" are present in the sample. An important result is that the tails of a fat-tailed distribution are invariant under addition although the distribution as a whole may vary according to temporal aggregation (Feller, 1971). That is, if weekly returns are Student-? identically and independently distributed, then monthly returns are not Student-? distributed.7 Yet the tails of the monthly return

7 This is an implication of the central limit theorem.

distribution are like the tails of the weekly returns, with the same exponent a, but the real tail might be very far out and not even seen in data samples of limited size.8 Another important result in the case of fat-tailed distributions concerns the finiteness of the moments of the distribution. From

where X is the observed variable, Mo is the part of the moment due to the center of the distribution (up to s), is a scale variable and a is the tail index. It is easily seen that only the first k-moments, < a, are bounded.

How heavy are the tails of financial asset returns? The answer to this question is not only the key to evaluating risk in financial markets but also to accurately modeling the process of price formation. Evidence of heavy tails presence in financial asset return distributions is plentiful (Koedijk et al., 1990; Hols and De Vries, 1991; Loretan and Phillips, 1994; Ghose and Kroner, 1995; Miiller et al., 1998) ever since the seminal work of Mandelbrot on cotton prices (Mandelbrot, 1963). He advanced the hypothesis of a stable distribution on the basis of an observed invariance of the return distribution across different frequencies and the apparent heavy tails of the distribution. A controversy has long been going on in the financial research community as to whether the second moment of the distribution of returns converges. This question is central to many models in finance, which heavily rely on the finiteness of the variance of returns. The risk in financial markets has often been associated with the variance of returns since portfolio theory was developed. From option pricing models (Black and Scholes, 1973) to the Sharpe ratio (Sharpe, 1994) used for measuring portfolio performance, the volatility variable is always present.

Another important motivation of this study is the need to evaluate extreme risks in financial markets. Recently, the problem of risk in these markets has become topical following few unexpected big losses like in the case of Barings or Daiwa. The Bank for International Settlements has set rules to be followed by banks to control their risks, but most of the current models for assessing risks are based on the assumption that financial assets are distributed according to a normal distribution. In the Gaussian model the evaluation of extreme risks is directly related to the variance, but in the case of fat-tailed distributions this is no longer the case.

Computing the tail index is a demanding task but with the help of high frequency data it is possible to achieve reasonable accuracy (see Pictet et al., 1998; Dacorogna et al., 2001a), where a theorem is proved which explicitly shows that more data improve the estimation of the tail index. Here, we present the main framework9.

8 See, for instance, the simulations done in Pictet etal. (1998) where for a high enough aggregation level, it is not possible to recover the theoretical tail index for Student-? distributions even if one can use 128 years of 10-min data.

9 The interested reader can find the details in two recent papers by Pictet etal. (1998) and Dacorogna et al. (2001a).

(5.2)