 Another model that generates the power law distributions is the
stochastic Lotka-Volterra system (see  and references therein).
The generic Lotka-Volterra system is used for describing different
phenomena, particularly the population dynamics with the predator-
prey interactions. For our discussion, it is important that some agent-
based models of financial markets (see Chapter 12) can be reduced to
the Lotka-Volterra system . The discrete Lotka-Volterra system
has the form
w
i
(t þ 1) ¼ l(t)w
i
(t) aW(t) bw
i
(t)W(t), W(t) ¼
1
N
X
N
i ¼1
w
i
(t) (8:2:1)
where w
i
is an individual characteristic (e.g., wealth of an investor i;
i ¼ 1, 2, ..., N), a and b are the model parameters, and l(t) is a
random variable. The components of this system evolve spontan-
eously into the power law distribution f(w, t) w
(1þa)
.Inthe
mean time, evolution of W(t) exhibits intermittent fluctuations that
can be parameterized using the truncated Levy distribution with the
same index a .
Seeking universal properties of the financial market crashes is
another interesting problem explored by Sornette and others (see
 for details). The main idea here is that financial crashes are
caused by collective trader behavior (dumping stocks in panic),
which resembles the critical phenomena in hierarchical systems.
Within this analogy, the asymptotic behavior of the asset price S(t)
has the log-periodic form
S(t) ¼ A þ B(t
c
t)
a
{1 þ C cos [w ln (t
c
t) w]} (8:2:2)
where t
c
is the crash time; A, B, C, w, a , and w are the fitting
parameters. There has been some success in describing several market
crashes with the log-periodic asymptotes . Criticism of this ap-
proach is given in  and references therein.
8.3 NEW DEVELOPMENTS
So, do the findings listed in the preceding section solve the problem
of scaling in finance? This remains to be seen. First, B. LeBaron has
shown how the price distributions that seem to have the power-law
form can be generated by a mix of the normal distributions with
90 Scaling in Financial Time Series different time scales . In this work, the daily returns are assumed
to have the form
R(t) ¼ exp [gx(t) þ m]e(t) (8:3:1)
where e(t) is an independent random normal variable with zero mean
and unit variance. The function x(t) is the sum of three processes with
different characteristic times
x(t) ¼ a
1
y
1
(t) þ a
2
y
2
(t) þ a
3
y
3
(t) (8:3:2)
The first process y
1
(t) is an AR(1) process
y
1
(t þ 1) ¼ r
1
y
1
(t) þ Z
1
(t þ 1) (8:3:3)
where r
1
¼ 0:999 and Z
1
(t) is an independent Gaussian adjusted so
that var[y
1
(t)] ¼ 1. While AR(1) yields exponential decay, the chosen
value of r
1
gives a long-range half-life of about 2.7 years. Similarly,
y
2
(t þ 1) ¼ r
2
y
2
(t) þ Z
2
(t þ 1) (8:3:4)
where Z
2
(t) is an independent Gaussian adjusted so that
var[y
2
(t)] ¼ 1. The chosen value r
2
¼ 0:95 gives a half-life of about
2.5 weeks. The process y
3
(t) is an independent Gaussian with unit
variance and zero mean, which retains volatility shock for one day.
The normalization rule is applied to the coefficients a
i
a
1
2
þ a
2
2
þ a
3
2
¼ 1: (8:3:5)
The parameters a
1
,a
2
, g, and m are chosen to adjust the empirical data.
This model was used for analysis of the Dow returns for 100 years
(from 1900 to 2000). The surprising outcome of this analysis is retrieval
of the power law with the index in the range of 2.98 to 3.33 for the data
aggregation ranges of 1 to 20 days. Then there are generic comments by
T. Lux on spurious scaling laws that may be extracted from finite
financial data samples . Some reservation has also been expressed
about the graphical inference method widely used in the empirical
research. In this method, the linear regression equations are recovered
from the log - log plots. While such an approach may provide correct
asymptotes, at times it does not stand up to more rigorous statistical
hypothesis testing. A case in point is the distribution in the form
f(x) ¼ x
a
L(x) (8:3:6)
where L(x) is a slowly-varying function that determines behavior of
the distribution in the short-range region. Obviously, the ‘‘universal’’
Scaling in Financial Time Series 91

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