January 1999 February 1999

Time

FIGURE 1.3 Top panel: Annualized USD-JPY volatility computed with daily prices observed at 7 a.m. GMT (afternoon Japan, circles), 5 p.m. GMT (afternoon U.K., diamonds) and with high-frequency data (solid line). The data period is from January 1999 to February 1999. Bottom panel: The USD-JPY high-frequency price series from January 1999 to February 1999.

annualized volatility estimations by minimizing the influence of the random noise in the market.

Another aspect of this is the choice of model. With few data, one tends to favor the simpler models because they contain few parameters and because tests like the likelihood ratio test would strongly penalize the increase of parameters. Of course, simplicity is a desirable feature of theoretical models, but one should not seek simplicity at the cost of missing important features of the data-generating process. Sometimes, it is useful to explore more complicated (nonlinear) models, which may contain more parameters. This increasing complexity is strongly penalized when explored with low-frequency data because of the loss of degrees of freedom. In the case of high-frequency data, however, the penalty is relatively small because the abundance of the independently measured observations approximates an asymptotic environment.

data. Yet the challenge is still open to build models that are simple to implement and describe to a reasonable degree the empirical behavior of the data at all time scales.

1.4 new levels of significance

High-frequency data means a very large amount of data. The number of observations in one single day of a liquid market is equivalent to the number of daily data within 30 years. Statistically, the higher the number of independently measured observations, the higher is the degrees of freedom, which implies more precise estimators. The large amount of data allows us to distinguish between different models (model validation) with a higher statistical precision. New statistical methods become possible, for example, tail statistics to examine the probability of extreme events. Almost by definition, extreme events are rare and doing statistics on such extreme events is a challenge. With high-frequency data one can have samples with as many as 400,000 independent observations4 to study the 0.25% percentile and still have 1,000 observations with which to work. We shall see how important this is when we present the estimation of tail indices for return distributions. Similarly, when different models have to be ranked, the availability of a few hundred thousand observations allows us to find beyond a doubt which model provides the best description of the data-generating process (Miiller et al., 1997a).

Figure 1.3 demonstrates the importance of high-frequency data in model selection and inference within the context of Value-at-Risk (VaR) calculations. We report three different calculations all of which use the J. P. Morgan (1996) volatility model, which is in fact a 1 -day volatility forecast as further discussed in Section 9.2. The three calculations differ in terms of the sampling and the data frequency. The Japanese volatility calculations are based on prices observed daily at 7 a.m. GMT, which corresponds to the afternoon Japanese time. The U.K. volatility calculations are based on prices measured daily at 5 p.m. GMT, which is the afternoon in the U. K. The high-frequency volatility calculations are based on the high-frequency tick-by-tick data recorded continuously on a 24-hour cycle. The top panel in Figure 1.3 reports the annualized volatility calculations and the bottom panel shows the underlying prices for January and February 1999. The top panel demonstrates that volatility can be extremely different depending on the time of the day at which it is measured with daily data. If observations are picked randomly once a day, the underlying volatility can be as small as 15% or as large as 22% for a given day and for the same currency. In mid-January 1999, the U.S. Dollar - Japanese Yen (USD-JPY) investors in the U.K. are assumed to be facing the risk of losing 56,676,400 USD in a portfolio of a hundred million USD with a 1% probability. In Japan, this risk would be reduced to 38,643,000 USD for the same day and for the same currency, a difference of approximately 18,000,000 USD between the two geographical locations! The utilization of high frequency leads to more robust

4 This approximately corresponds to 10 years of returns measured over 10 minutes.

Researchers who want to use many observations with low-frequency data are using, for instance, daily observations of the Dow Jones Industrials from January 1897 like Ding et al. (1993) or LeBaron (1999a). In such a case, one is entitled to ask if the authors are actually analyzing the same market over the years. The huge technological changes that we experienced during this century have certainly affected the New York Stock Exchange and one is never sure, how this and any reconfiguration of the index has affected the results. To the contrary, high-frequency studies can be done for limited sampling periods with reasonably large samples. The market properties within such periods are nearly unchanged. The results are less affected by structural breaks or shifts in the overall economy than low-frequency studies with samples of many years. This is clearly an advantage when determining microstructure effects but also when examining the stability of some properties over time.

1.5 INTERRELATING DIFFERENT TIME SCALES

High-frequency data open the way for studying financial markets at very different time scales, from minutes to years. This represents an aggregation factor of four to five orders of magnitude.5 Some empirical properties are similar at different scales, leading to fractal behaviors. Stylized facts observed for daily or weekly data gain additional weight when also observed with high significance for intraday data. An example of this is the long memory effect in 20-min absolute returns studied by Dacorogna et al. (1993). At the time, similar hyperbolic decay of the autocorrelation function was observed on daily returns in Ding et al. (1993). It is very difficult to distinguish rigorously in the data between long memory effects and regime shifts. Many mathematicians are working precisely on this problem such as Mansfield et al. (1999) and Mikosch and Starica (1999). Yet the fact that hyperbolic decay is empirically found at time scales that differ by two orders of magnitude in aggregation is definitely a sign that the process must include some long range dependence or that there are regime shifts at all time scales, which is equivalent.

Scaling properties and scaling laws have been new objects of study since the early work of Mandelbrot (1963) on cotton prices. In 1990, the research group of O&A published empirical studies of scaling properties extending from a few minutes to a few years (Muller et al., 1990). These properties have shown remarkable stability over time (Guillaume et al, 1997) and were found in other financial instruments like interest rates (Piccinato et al, 1997). Mantegna and Stanley (1995) also found scaling behavior in the stock indices examined at high frequency. In a set of recent papers, Mandelbrot et al. (1997), Fisher et al. (1997) and Calvet et al. (1997) have derived a multifractal model based on the empirical scaling laws of different moments of the return distributions. Works on the scaling law of return

5 By order of magnitude we mean the number of times the time horizon must be multiplied by 10 to achieve the lower frequency. For instance, a weekly frequency is aggregated three orders of magnitude from 10 minutes data (one week is 1008 times 10 minutes).