Multifractality of US Dollar/Deutsche Mark Exchange Rates - Studies2

6 at about two hours. For the remainder of this paper, we consider primarily the long range of
sampling intervals greater than two hours. These sampling intervals still span nearly three orders
of magnitude of △t. The absence of a sharp upper crossover in this region, at least up to 180 days,
is promising.
We conjecture that high-frequency crossover can be related to market frictions such as bid-ask
spreads, discreteness of trading units, and discontinuous trading. This topic will be investigated in
a forthcoming paper.
3.4.2 The Self-Affinity Index for BH(t)
A close up view of the slopes of the partition functions around the moments q = 2 is chosen in
Figure 6 for a particular purpose. The self-affinity index, H, of the Fractional Brownian Motion
BH(t) in the MMAR is identified by the condition that
where τP is the scaling function for the price process. 16
τP (1/H) =0, (6)
From Figure 6, the value of q with slope zero appears to be about 1.9, indicating H ≈ .53. This
implies small persistence in DM/USD exchange rates. At the same time, the estimated H is very
close to 1/2. Previous studies, based on well understood estimation methods, find white spectra
(implying H =1/2) for log increments of typical financial data. Thus, the martingale hypothesis
for log prices seems to be a reasonable approximation. At the same time, Section 5 shows strong
evidence of multifractality in trading time, implying long memory in squared returns.
3.4.3 Linearity of the Partition Functions
Linearity of the partition functions in Figure 6 indicates a multifractal scaling law that holds, at a
minimum, for close to three orders of magnitude of △t. Scaling is even more striking in Figure 7
because the moments are higher, reaching to q = 5. Once again, the lines are relatively straight,
consistent with the MMAR. Variability of the lines around their apparent slopes increases as q
increases, as is to be expected.
16 In a partition with fixed endpoints where all increments are restricted to have the same sign, the sum of increments
cannot depend on grid size. Intuitively, the total variation of an increasing process, estimated by finite increments,
does not depend on how we cut it up. Therefore, since θ(t) is increasing (being the c.d.f. of a random measure),
τθ(1) = 0. For a Fractional Brownian Motion, τ(q) =Hq − 1, and τP (q) =τθ(Hq) by Theorem 4.2 in MFC(1997).
This establishes that τP (1/H) = 0. See CFM and Mandelbrot (1997) for further discussion.
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