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118 from the simulated random walk processes are 53.79, 63.68, 47.68, and 53.49 for the USDDEM, USDCHF, USDFRF, and DEMJPY, respectively. The mean of the simulated maximum drawdowns are three or four times larger than the actual maximum drawdowns. The deal frequencies are 1.68, 1.29, 1.05, and 2.14 per week for the four currency pairs from the actual data. The deal frequencies indicate that the RTT model trades on average no more than two trades per week although the data feed is at the 5min frequency. The mean simulated deal frequencies are 2.46, 1.98, 1.65, and 3.08, which are significantly larger than the actual ones. The values for the maximum drawdown and the deal frequency indicate that the random walk simulation will yield larger maximum drawdown and deal frequency values relative to the values of these statistics from the actual data. In other words, the random walk simulations deal more frequently and result in more volatile equity curves on average relative to the equity curve from the actual data. Correspondingly, the pvalues indicate that the random walk process cannot be representative of the actual foreign exchange series under these two performance measures. The summary statistics of the simulated performance measures have negligible skewness and statistically insignificant excess kurtosis. This indicates that the distributions of the performance measures are symmetric and do not exhibit fat tails. The simulation results with the random walk process demonstrate that the realtime trading model is a consistent model. In other words, a process with no mean and a homoskedastic variance should only perform to generate an average return that would match the mean transaction costs. This consistency property is an essential ingredient of a trading model and the realtime trading model passes this consistency test. The means of the simulations indicate that the distributions are correctly centered at the average transaction costs, which is expected under the random walk process. For instance, the mean simulated deal frequency of the USDDEM is 2.46 deals per week or 127.92 (2.46 x 52) deals peryear. The relative spread for the USDDEM is 0.00025, which in turn indicates an average transaction cost of 3.20% per year. Given that the mean of the simulated annualized return is 3.44, we can conclude that the mean of the simulated annualized return distribution is centered around the mean transaction cost. The behavior of the performance measures across 7day, 29day, 117day and 301day horizons is also investigated with Xeg and Reg. The importance of the performance analysis at various horizons is that it permits a more detailed analysis of the equity curve at the predetermined points in time. These horizons correspond approximately to a week, a month, 4 months and a years performance. The Xeg and Reff values indicate that the RTT model performance improves over longer time horizons. This is in accordance with the low dealing frequency of the RTT model. In all horizons, the pvalues for the \eg and Reg are less than a half a percent for USDDEM, USDFRF, and DEMJPY. For USDCHF, the pvalues are less than 2.4% for all horizons. Overall, the multihorizon analysis indicates that the random walk process is not consistent with the datagenerating process of the foreign exchange returns.
TABLE 11.8 GARCH( 1,1) parameter estimates. The sample is 5min returns from 19901996.  USDDEM  USDCHF  USDFRF  DEMJPY  «0  4.95 (4.23)  0.11 (0.12)  9.38 (7.09)  2.97 (4.03)  «1  0.1111 (0.0005)  0.1032 (0.0007)  0.1572 (0.0007)  0.0910 (0.0005)   0.8622 (0.0007)  0.8578 (0.0009)  0.8137 (0.0009)  0.8988 (0.0006)   6.45  6.17  6.29  6.34  G(12)  4810  4201  4256  3089   1.04  1.03  1.07  1.05   0.07  0.03  0.05  0.16   11.73  7.28  22.93  27.73 
GARCH( 1,1) Process A more realistic process for the foreign exchange returns is the GARCH(1,1) process, which allows for conditional heteroskedasticity. The GARCH(1,1) estimation results are presented in Table 11.8. The numbers in parentheses are the robust standard errors and the GARCH(l.l) parameters are statistically significant at the 5% level for all currency pairs. The LjungBox statistic is calculated up to 12 lags for the standardized residuals and it is distributed with x2 with 12 degrees of freedom. The LjungBox statistics indicate serial correlation for the USDDEM. The variances of the normalized residuals are near one. There is no evidence of skewness but the excess kurtosis remains large for the residuals. In Table 11.9, the simulation results with the GARCH(1,1) process are presented for the USDDEM rate. Because GARCH(1,1) allows for conditional heteroskedasticity, it is expected that the simulated performance of the RTT model would yield higher pvalues and retain the null hypothesis that GARCH(1,1) is consistent with the datagenerating process of the foreign exchange returns. The results, however, indicate smaller values, which is in favor of a stronger rejection of this process relative to the random walk process. One important reason for the rejection of the GARCH(1,1) process as well as the random walk model is that these are pure volatility processes without predictability of the direction of returns, which matters for trading models. Another reason is the aggregation property of the GARCH( 1,1) process. The GARCH( 1,1) process behaves more like a homoskedastic process as the frequency is reduced from high to low frequency. Because the RTT model trading frequency is less than two deals per week, the trading model does not pick up the 5min level heteroskedastic structure at the weekly frequency. Rather, the heteroskedastic structure behaves as if it is measurement noise where the model takes positions, and this leads to the stronger rejection of the GARCH(1,1) as a candidate for the foreign exchange datagenerating process.
TABLE 11.9 GARCH( 1,1) simulations for USDDEM. Description  Historical  pvalue  Percentile  Mean  St.Dev  Skew.  Kurt.   realization  (in %)  (%, 95%)      Annual return  9.63   11.14,5.12  3.27  4.90  0.08  0.01  Xeffective  3.78   20.40,3.16  11.88  5.18  0.07  0.11  Reflective  4.43   26.60. 6.37  16.50  6.10  0.14  0.05  Max drawdown  11.02  100.0  24.17,93.96  53.33  21.50  0.73  0.30  Deal frequency  1.68  100.0  2.14,2.64  2.39  0.15  0.02  0.15  Horizon:  7 days        Xeffective  3.47   19.56, 3.49  11.64  4.90  0.06  0.03  R effective  1.80   24.19,6.58  15.37  5.38  0.08  0.05  Horizon:  29 days        Xeffective  3.27   19.95, 3.29  11.86  5.00  0.12  0.04  Rcffcctivc  2.16   26.92, 6.75  17.20  6.04  0.20  0.03  Horizon:  117 days        Xeffective  4.07   21.24,2.77  11.91  5.56  0.03  0.28  Reffective  5.10   30.17, 5.44  17.57  7.60  0.29  0.31  Horizon:  301 days        Xeffective  4.62   22.73,1.48  11.73  6.42  0.28  0.16  Reffective  6.83   27.64, 2.01  14.28  7.72  0.26  0.38 
In a GARCH process, the conditional heteroskedasticity is captured at the frequency that the data have been generated. As it is moved away from this frequency to lower frequencies, the heteroskedastic structure slowly dies away leaving itself to a more homogeneous structure in time. More elaborate processes, such as the multiple horizon ARCH models (as in the HARCH process of Muller etal., 1997a), possess conditionally heteroskedastic structure at all frequencies in general. The existence of a multiple frequency heteroskedastic structure seems to be more in line with the heterogeneous structure of the foreign exchange markets. Table 11.6 we presented a summary of the pvalues of the annualized return for the USDDEM, USDCHF, USDFRF and DEMJPY. In the case of the GARCH(l,l) simulation, they are 0.4, 8.4, 0.9, and 1.2%, respectively. All four currency pairs except USDCHF yield pvalues, which are smaller than 1.3%. The Xeg and Reff are 0.1 and 0.0% for USDDEM, 1.4 and 0.9 percent for USDCHF, 0.1 and 0.1% for USDFRF, and 0.4 and 0.4 percent for DEMJPY. The historical maximum drawdown and deal frequency of the RTT model is smaller than those generated from the simulated data. The maximum drawdowns for the USDDEM, USDCHF, USDFRF, and DEMJPY are 11.02, 16.08, 11.36, and 12.03 for the four currencies. The mean simulated drawdowns are 53.33, 60.58, 46.00, and 48.77 for the four currencies. The mean simulated maximum drawdowns are three to four times larger than the historical ones. The historical deal frequencies are 1.68, 1.29, 1.05, and 2.14. The mean simulated deal frequencies are 2.39,1.87, 1.59, and 2.66 for the four currencies. The differences between
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