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77 exponential declines with widely differing time constants comes close to a hyperbolic decline. 2. In a homogeneous market, the more agents are present, the faster the price should converge to the "real market value" on which all agents have "rational expectations." Thus, the volatility should by negatively correlated with market presence and activity. In a heterogeneous market, different market actors are likely to settle for different prices and decide to execute their transactions in different market situations. In other words, they create volatility. This is reflected in the empirically found, positive correlation of volatility and market presence. 3. The market is also heterogeneous in the geographic location of the participants. This immediately explains the "heat wave" effect. In Section 7.3.1, we indicated that the memory in the volatility process is relatively weak at time lags of about \ or 1 j business days when market actors on opposite sides of the globe are related to each other and relatively strong at time lags of about 1 or 2 business days when identical groups of participants are considered. The market participants of the heterogeneous market hypothesis differ also in other aspects beyond the time horizons and the geographical locations. They may have different degrees of risk aversion, face different institutional constraints, and transaction costs. 7.4.1 Volatilities of Different Time Resolutions The heterogeneous market hypothesis presented in the previous section is associated with fractal phenomena in the empirical behavior of FX markets. A scaling law relating time horizon and size of price movements (volatility) was identified in Chapter 5. This relation is used here to explain why the perception of volatility differs for market agents with different time horizons. Shortterm traders are constantly watching the market to reevaluate their current positions and execute transactions at a high frequency. Longterm traders may look at the market only once a day or less frequently. A quick price increase of 0.5% followed by a quick decrease of the same size, for example, is a major event for an FX intraday trader but a nonevent for central banks and longterm investors.4 Longterm traders are interested only in large price movements and these normally happen only over long time intervals (see the scaling law of Miiller et al, 1990). Therefore, longterm traders with open positions have no need to watch the market every minute.5 In other words, they judge the market, its prices, and also its volatility with a coarse time grid. A coarse time grid reflects the view of a longterm trader and a fine time grid that of a shortterm trader. Bjorn (1994) follows similar methodologies for building an automatic trading model. 4 Small, shortterm price moves may sometimes have a certain influence on the timing of longterm traders transactions but not on their investment decisions. 5 They have other means to limit the risk of rare large price movements by stoploss limits or options.
The time grid in which real traders watch the market is not strictly regular. In the following lagged correlation study, however, we measure volatilities over different but regularly spaced grids. These volatilities are defined in terms of absolute returns. We prefer mean absolute values to roots of mean squares here because they are statistically less dominated by extreme observations, which are rather important in FX markets with their fattailed unconditional distribution functions. The convergence of the fourth momenta requirement for many types of analysis such as the autocorrelation of squared returnsis not guaranteed for empirical returns. In Chapter 5, we demonstrated that the autocorrelations of the returns indicate a stronger signal for powers around one. This argument is reinforced in Dacorogna et al (2001a), where the autocorrelation of absolute returns is also shown to be much more stable under sample size changes than that of the squared returns. Other studies, such as Ding et al. (1993), also find absolute returns to be optimal in the autocorrelation studies. The volatility based on absolute returns has two essential timing parameters (Guillaume etal, 1997): The interval size of the time grid in which returns are observed The total size of the sample over which it is computed (the number of grid intervals considered) For exploring the behavior of volatilities of different time resolution, we define two types of volatility. The "coarse" volatility, vc, and the "fine" volatility, vj, are defined by vc(ti) = \Yjr{At\ti\ +jAt)\ and vI(ti) = YJ\r{bt,ti +jAt)\ 7 = 1 7 = 1 (7.9) where At = At In. Figure 7.7 illustrates this definition where at every time point, = ] + 6At, both quantities are simultaneously defined. In this way, the two synchronous time series are obtained whose relation can be explored. 7.4.2 Asymmetric LeadLag Correlation of Volatilities Analyzing the correlation between two time series, such as fine and coarse volatilities, is a standard technique used in empirical finance where the correlation coefficient measures the linear dependence of the two time series. Lagged correlation is a more powerful tool to investigate the relation between two time series. The lagged correlation function considers the two series not only simultaneously (at lag 0) but also with a time shift. The correlation coefficient qt of one time series and another one shifted by a positive or negative time lag is measured and plotted against the value of the lag. The lagged correlation study of this section follows Muller etal (1997a). Lagged correlation reveals causal relations and information flow structures in the sense of Granger causality. If two time series were generated on the basis of
Fine vf(t\) Vf(t) \n\ + \n\ + 1 1 + + Irsl + Coarse Veil  1) 411111 \n + + + 4 + r5 + r6 I VcU) + IEo1 FIGURE 7.7 The coarse volatility, vc(t), captures the view and actions of longterm traders while the fine volatility, Vf(t), captures the view and actions of shortterm traders. The two volatilities are calculated at the same time points and are synchronized. a synchronous information flow, they would have a symmetric lagged correlation function, Qx = qt The symmetry would be violated only by insignificantly small, purely stochastic deviations. As soon as the deviations between qt and Qr become significant, there is asymmetry in the information flow and a causal relation that requires an explanation. In a first analysis, we consider a workingdaily time series where weekends are omitted. The variables under study are the "fine volatility" and the "coarse volatility." Fine volatility is the mean absolute workingdaily returns averaged over five observations, so covering a full (working) week. Coarse volatility is the absolute return over a full weekly interval. The correlation between fine volatility and coarse volatility is a function of the number of lags. When the number of lags is zero, the fine and coarse volatilities are completely identical. In the case of first positive or negative lag, the two intervals do not overlap but follow each other immediately. The panel on the left hand side of Figure 7.8 shows the lagged correlation function for the USDDEM in a sample longer than 21 years. The correlation maximum is found at lag zero, which is expected. For the nonzero lags, there is an asymmetry where the coarse volatility predicts fine volatility better than the other way around. The asymmetry is significant for the first two lags where the difference qz  qt , represented by the thin curve in Figure 7.8, is distinctly outside the confidence interval for identically and independently distributed observations. This result can be explained in terms of the heterogeneous market hypothesis presented earlier in this section. For shortterm traders, the level of coarse volatility matters because it determines the expected size of trends and thus the scope of trading opportunities. On one hand, shortterm traders react to clusters of coarse volatility by changing their trading behavior and so causing clusters of fine volatility. On the other hand, the level of fine volatility does not affect the trading strategies of longterm traders (who often act according to the "fundamentals" of the market).
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