Multifractality of US Dollar/Deutsche Mark Exchange Rates - Studies2

H is estimated by equation (6), which gives
H =1.88 −1
for the daily data. This leaves us with only one parameter to fit. 31 Using
α0 = .589
produces the parabola shown by circled stars in Figure 12. The fit with the actual data appears quite
remarkable. The next section uses Monte Carlo methods to asses the variability of the spectrum.
5.4 Monte Carlo Simulation
The methods used for simulation are quite simple. (See MFC for other simulation examples).
We approximate Fractional Brownian Motion with a Fractional Gaussian Noise, as in Section 2.3.
For trading time, a finite multiplicative cascade with lognormal multipliers approximates a limit
lognormal multifractal measure. The values of λ and σ 2 are recovered from H and α0 by the
equations above.
Several simulated processes, with corresponding log first differences, are shown in Figure 13.
The results are promising, with a variety of large asymmetric price changes, apparent trends, and
other characteristics typical of real data.
Another goal of simulation is to test the partition function methodology. We calculated partition
functions for twenty simulations with sample size identical to the data. Replication of the scaling
properties of the data constitutes a joint test of the partition function methodology and simulation
methods. About six or seven simulations followed a scaling law close to the data. Figure 14 shows
four of these graphs. The plots are relatively straight, and vary somewhat around their predicted
values. Some of the remaining simulations had nonlinear partition functions. The others had
straight partition functions, but with different slope than the data.
Rejection of the MMAR is thus difficult, in a problematic sense, because of apparent low
power. At the same time, the results are promising. Although variability is high, the simulated
partition functions tend to follow their predicted lines. Improvements in either partition function
31 Joint estimation of parameters will generally improve fit, by whatever metric is chosen. This point is relatively
unimportant since sequential estimation produces good results.
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