scaling exponent a ¼log [f(x)]= log (x) is as accurate as L(x) is close
to a constant. This problem is relevant also to the multifractal scaling
analysis that has become another ‘‘hot’’ direction in the field.
The multifractal patterns have been found in several financial time
series (see, e.g., [20, 21] and references therein). The multifractal
framework has been further advanced by Mandelbrot and others.
They proposed compound stochastic process in which a multifractal
cascade is used for time transformations [22]. Namely, it was assumed
that the price returns R(t) are described as
R(t) ¼ B
H
[u(t)] (8:3:7)
where B
H
[] is the fractional Brownian motion with index H and u(t) is
a distribution function of multifractal measure (see Section 6.2). Both
stochastic components of the compound process are assumed inde-
pendent. The function u(t) has a sense of ‘‘trading time’’ that reflects
intensity of the trading process. Current research in this direction
shows some promising results [23–26]. In particular, it was shown
that both the binomial cascade and the lognormal cascade embedded
into the Wiener process (i.e., into B
H
[] with H ¼ 0:5) may yield a more
accurate description of several financial time series than the GARCH
model [23]. Nevertheless, this chapter remains ‘‘unfinished’’ as new
findings in empirical research continue to pose new challenges for
theoreticians.
8.4 REFERENCES FOR FURTHER READING
Early research of scaling in finance is described in [2, 6, 7, 9, 17].
For recent findings in this field, readers may consult [10–13, 23–26].
8.5 EXERCISES
**1. Verify how a sum of Gaussians can reproduce a distribution
with the power-law tails in the spirit of [18].
**2. Discuss the recent polemics on the power-law tails of stock
prices [27–29].
**3. Discuss the scaling properties of financial time series reported
in [30].
92 Scaling in Financial Time Series

Get Quantitative Finance for Physicists now with the O’Reilly learning platform.

O’Reilly members experience live online training, plus books, videos, and digital content from nearly 200 publishers.