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116 noise. Yet, because longer time intervals cover a larger period, the Xeg values of longer time intervals are more stable against changing opening and closing hours, that is, their variance is clearly smaller than that of shorter intervals. One exception is model RTT for USDCHF where the 1hr time interval is best, but the models still have significant peaks at the 9hr interval (from 8:30 to 17:30,4.9%). Further evidence in support of conjecture 2 is given by the sixth column in Table 11.5, listing the Xejf values for 24hr trading for comparison with the best Xeff values attained for snorter trading intervals. The X(24hr) values are much lower than the best Xeg values for almost all rates and models. This failure of 24hr trading can be interpreted as an insufficiency of the models to deal with shortterm price movements, in particular the disruptive market behaviors arising when the main dealing activity shifts from one geographical location to another (with a different time zone). A 24hr trading interval leads to a dealing frequency higher than that of a 12hr interval. Contrary to the 24hr trading interval, the best 1hr intervals that coincide with the most active times of the day are seldom significantly worse than the rest of the intervals tested. In Table 11.5, there are also indications that conjecture 3 is valid. The USDJPY models show a strong tendency toward favoring opening hours early in the European morning (or closing times early in the afternoon), whereas GBPUSD and USDITL prefer opening times in the late morning. These results are in line with the time zones of the home markets of the currencies and must be related to market liquidity. For JPY, better results are obtained when its main market (Far East) is active, for GBP and ITL, when London (1hr behind the Zurich market) is active (ITL is traded more in London than in Milan). The results of the RTT model for USDNLG seem to contradict this conjecture, but it should be qualified. There is, in fact, another peak with an Xeg of 11.3% at a 12hr trading interval from 7:30 to 19:30. In conclusion, the systematic analysis of the influence of the trading hours on these models reveals some important facts. First of all, if we regard our model classes as representing mediumterm components in the market, we see that it is not useful to stay active 24hr a day. Without a much more sophisticated treatment of the intraday movements, it does not pay for a mediumterm trader to be active all the time. Second, it shows that, contrary to assumptions based on the classical efficient market hypothesis, a trading model is profitable when its active hours correspond to the most active hours of one of the main geographical components of the market. It is essential that the models execute their deals when the market is most liquid. This fact is illustrated by three empirical findings: The maxima of performance are clustered around opening hours when the main markets are active. The best active times are shifted for certain currencies to accommodate their main markets (Japan for JPY, London for GBP). * If the models are only allowed to trade for 1 hour, the best choice of this hour is usually around the peaks in the daily activity of the market.
The systematic variation of the business hours of the trading models again reveals the geographical structure of the FX market and its daily seasonality by the most profitable trading times being concentrated where the market is most liquid. 11.6.2 PriceGeneration Processes and Trading Models Instead of feeding the trading models with real data, we can use simulated data from different pricegeneration processes. The results indicate that the performance of the trading models with real FX data is much higher relative to the simulated price processes. This demonstrates that the trading models successfully exploit a certain predictability of returns that exists beyond the scope of the studied pricegeneration processes. The results also provide opportunity to compare different statistical price processes with each other. In the case of the RTT model described in Section 11.4.1, the outofsample test period is 7 years of highfrequency data on three major foreign exchange rates against the U.S. Dollar and one cross rate. From its launch in 1989 until the end of 1996, the model had not been reoptimized and was running on the original set of parameters estimated with data prior to 1989. This allows us a unique advantage that there is no socially determined coevolutionary relationship between our data set and the technical strategies used in implementing our specification tests. The trading model yields positive annualized returns (net of transaction costs) in all cases. Performance is measured by the annualized return, Xeg, Reg, deal frequency and maximum drawdown. Their simulated probability distributions are calculated with the three traditional processes, the random walk, GARCH, and ARGARCH, but also with an ARHARCH. The null hypothesis of whether the realtime performances of the foreign exchange series are consistent with these traditional processes is tested under the probability distributions of the performance measures. As expected from the discussions of the previous chapters, the results from the realtime trading model are not consistent with the random walk, GARCH(1,1) and ARGARCH(1,1) as the datagenerating processes. It is also the case with the ARHARCH processes. Simulation Methodology The distributions of the performance measures under various null processes are calculated by using a simulation methodology. In our trading model simulations, we use a 5min interval sampling of the prices in order to keep the computation within manageable bounds. It is a good compromise between efficient computation and realistic behavior when compared to the realtime trading model results generated from all ticks. The main information used by a trading model to update its indicators is the returns. The return between two consecutive selected ticks at time tj\ and tj is defined as = xixi\ and the corresponding elapsed #time (described in Section 6.2) between these two ticks is = 0j  0ji
By construction, in the sampled time series, the average elapsed #time between two ticks, AO, is nearly 5 min. Multiple time series from a given theoretical price generation process need to be generated. To keep the impact of special events like the data holes in the model behavior, we decided to replace the different bidask price values but always keep the recorded time values. As the different ticks are not exactly regularly spaced, even in #time, the average return corresponding to a 5min interval needs to be calculated. This is calculated by rescaling the observed return values where the exponent 1 / is called the drift exponent and it is set to 0.5 under the random walk process. To obtain meaningful results, a simulated time series should have the same average drift a and average variance a2 as the observed returns. This is done by generating returns, rj, corresponding to a 5min interval in #time. In the case of a random walk process, the returns rj are computed with where ey ~ N(0, a1). When the effective elapsed time between two ticks, A6j, is not exactly 5min, we scale again the generated return using the same scaling formula where AO is 5 min. If there is a data hole, the sum of the generated return r, is computed until the sum of the added 5min intervals is larger than the size of the data hole measured in #time. The sum of the returns is scaled with the same technique as individual returns. rj = a + cj The simulated logarithmic prices, xj, are computed by adding the generated returns rj to the first real logarithmic price value xo The bidask prices are computed by subtracting or adding half the average spread, that is, PbidJ exp   In the simulations, f is specified to be normally distributed. We also explored bootstrapping the residuals of the studied models. The main findings of the study remain unchanged between these two approaches.
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