Quantitative Finance for Physicists

Another model that generates the power law distributions is the

stochastic Lotka-Volterra system (see [14] 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 [15]. 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 [14].

Seeking universal properties of the financial market crashes is

another interesting problem explored by Sornette and others (see

[16] 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 [16]. Criticism of this ap-

proach is given in [17] 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 [18]. 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 [19]. 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