Multifractality of US Dollar/Deutsche Mark Exchange Rates - Studies2

The measure in an interval is the product
µ(η1...ηk) =M1(η1)...Mk(ηk)Ω(η1...ηk).
Mj denotes the multiplier at the j th level of the cascade. The r.v. Ω is called the high-frequency
component, and captures changes in total mass of the interval caused by stages beyond k. It can
be shown that Ω has the same distribution for all levels of the cascade, and thus has the same
distribution as the limiting mass in the interval [0, 1]. 30
For convenience, denote
V = − log b M ∼ N(λ, σ 2 ).
CFM shows that the multifractal spectrum of a multiplicative cascade with lognormal multipliers
is given by
fθ(α) =1−
(α − λ)2
2σ2 . (11)
log b
We can generate a further restriction by requiring average conservation of mass, so that
leading to
1
b = E(M) =E(e−V log b ),
log b =
2(λ − 1)
σ2 .
Substituting this into (11) eliminates both b and σ 2 , giving
fθ(α) =1−
(α − λ)2
4(λ − 1) .
We now have a single parameter expression for the multifractal spectrum of trading time; however,
we directly estimate the price spectrum which is obtained by applying equation (10) :
fP (α) =1−
(α − Hλ)2
4H 2 (λ − 1)
Finally, using the substitution λ = α0/H allows us to write the spectrum in terms of α0 and H:
fP (α) =1−
(α − α0) 2
4H(α0 − H) .
30
Convergence of mass to a random variable is guaranteed by the convergence of a positive martingale, or alternatively,
by the ergodic theorem.
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