Multifractality of US Dollar/Deutsche Mark Exchange Rates - Studies2

intuition turns out to be correct, but misses a great deal of information. Mandelbrot (1989) shows
that an alternative renormalization is more informative. CFM neatly categorizes this interpretation
with two other common interpretations for f(α):
(D1) the limit of a renormalized histogram of coarse Hölder exponents,
(D2) the fractal dimension of the set of instants with Hölder exponent α,
(D3) the limit of k −1 log b P {αk >α} + 1 provided by Large Deviation Theory.
The second interpretation is a statement about fractal dimension, or Hausdorff-Besicovitch
dimension. In a partition of size △t, the number of intervals characterized by α takes the form
Nα(△t) ∼ (△t) −f(α) .
This form justifies calling f(α) a fractal dimension. 18 On the real line, fractal dimensions are
constrained to lie between zero and one. Fractal dimension and (unit renormalized) Lebesgue
measure coincide when both are either zero or one. On the other hand, all sets with positive
Lebesgue measure have a fractal dimension of one, and all sets with fractal dimension less than one
have zero Lebesgue measure. Thus, fractal dimension provides more information about some types
of sets. Standard continuous-time financial models have components with fractal dimension of one,
which applies to typical Itô diffusions, or zero, which applies to jump components. Thus, fractal
sets fill space in a manner that is midway between the standard cases of continuous time finance.
Definition (D3) is a statement about applying Large Deviation Theory (“LDT”) to multifractals
generated via multiplicative cascades. In the case of multifractals, the LDT provides far more
information about the generating mechanism than do the Strong Law of Large Numbers or Central
Limit Theorem.
These three definitions coincide when f(α) > 0, but clearly (D1) precludes negative values of
f(α). 19 Negative fractal dimensions are known as latent α’s, deriving from the fact that sets of
18 Note that the quantity − log Nα(△t)/ log(△t), which is f(α), does not depend on △t.
19 Definition (D2) is compatible with negative values for f(α) since Mandelbrot (1988, 1995) introduced the concept
of negative dimension as “degreee of emptiness of an empty set.” The idea is troubling at first, but has an interesting
probabilistic justification related to the embedding of a process or measure in a higher-dimensional space. For
example, in a one dimensional section drawn from a three dimensional space populated by stellar matter, the fractal
dimension of stellar matter is negative. An individual draw almost always has no matter, but by the generation of
numerous low-dimensional sections, called supersampling, one obtains information about the (positive) dimension of
stellar matter in the embedding space.
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