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1. Consider the quadratic map Xk Â¼ XkÃ€1 2 Ã¾ C, where C > 0.

(a) Prove that C Â¼ 0:25 is a bifurcation point.

(b) Find fixed points for C Â¼ 0:125. Define what point is an

attractor and what is its attraction basin for X > 0.

2. Verify the equilibrium points of the Lorenz model (7.4.3).

*3. Calculate the Lyapunov exponent of the logistic map as a

function of the parameter A.

*4. Implement the algorithm for simulating the Lorenz model.

(a) Reproduce the â€˜â€˜butterflyâ€™â€™ trajectories depicted in Figure

7.8.

(b) Verify existence of the periodicity window at r Â¼ 150.

(c) Verify existence of the limit cycle at r Â¼ 350.

Hint: Use a simple algorithm: Xk Â¼ XkÃ€1 Ã¾ tF(XkÃ€1 , YkÃ€1 , ZkÃ€1 )

where the time step t can be assigned 0.01.

Chapter 8

Scaling in Financial Time

Series

8.1 INTRODUCTION

Two well-documented findings motivate further analysis of financial

time series. First, the probability distributions of returns often deviate

significantly from the normal distribution by having fat tails and excess

kurtosis. Secondly, returns exhibit volatility clustering. The latter effect

has led to the development of the GARCH models described in Section

5.3.1 In this chapter, we shall focus on scaling in the probability distri-

butions of returns, the concept that has attracted significant attention

from economists and physicists alike.

Alas, as the leading experts in Econophysics, H. E. Stanley and

R. Mantegna acknowledged [2]:

â€˜â€˜No model exists for the stochastic process describing the

time evolution of the logarithm of price that is accepted by

all researchers.â€™â€™

There are several reasons for the status quo.2 First, different financial

time series may have varying non-stationary components. Indeed, the

stock price reflects not only the current value of a companyâ€™s assets

but also the expectations of the companyâ€™s growth. Yet, there is no

general pattern for evolution of a business enterprise.3 Therefore,

87

88 Scaling in Financial Time Series

empirical research often concentrates on the average economic in-

dexes, such as the S&P 500. Averaging over a large number of

companies certainly smoothes noise. Yet, the composition of these

indicators is dynamic: Companies may be added to or dropped from

indexes, and the companyâ€™s contribution to the economic index usu-

ally depends on its ever-changing market capitalization.

Foreign exchange rates are another object frequently used in empir-

ical research.4 Unfortunately, many of the findings accumulated during

the 1990s have become somewhat irrelevant, as several European cur-

rencies ceased to exist after the birth of the Euro in 1999. In any case, the

foreign exchange rates, being a measure of relative currency strength,

may have statistical features that differ among themselves and in com-

parison with the economic indicators of single countries.

Another problem is data granularity. Low granularity may under-

estimate the contributions of market rallies and crashes. On the other

hand, high-frequency data are extremely noisy. Hence, one may

expect that universal properties of financial time series (if any exist)

have both short-range and long-range time limitations.

The current theoretical framework might be too simplistic to ac-

curately describe the real world. Yet, important advances in under-

standing of scaling in finance have been made in recent years. In the

next section, the asymptotic power laws that may be recovered from

the financial time series are discussed. In Section 8.3, the recent

developments including the multifractal approach are outlined.

8.2 POWER LAWS IN FINANCIAL DATA

The importance of long-range dependencies in the financial time series

was shown first by B. Mandelbrot [6]. Using the R/S analysis (see Section

6.1), Mandelbrot and others have found multiple deviations of the

empirical probability distributions from the normal distribution [7].

Early research of universality in the financial time series [6] was

based on the stable distributions (see Section 3.3). This approach,

however, has fallen out of favor because the stable distributions have

infinite volatility, which is unacceptable for many financial applica-

tions [8]. The truncated Levy flights that satisfy the requirement for

finite volatility have been used as a way around this problem [2, 9, 10].

One disadvantage of the truncated Levy flights is that the truncating

89

Scaling in Financial Time Series

distance yields an additional fitting parameter. More importantly,

the recent research by H. Stanley and others indicates that the asymp-

totic probability distributions of several typical financial time series

resemble the power law with the index a close to three [11â€“13]. This

means that the probability distributions examined by Stanleyâ€™s team

are not stable at all (recall that the stable distributions satisfy the

condition 0 < a 2). Let us provide more details about these interest-

ing findings.

In [11], returns of the S&P 500 index were studied for the period

1984â€“1996 with the time scales Dt varying from 1 minute to 1 month.

It was found that the probability distributions at Dt < 4 days were

consistent with the power-law asymptotic behavior with the index

a % 3. At Dt > 4 days, the distributions slowly converge to the

normal distribution. Similar results were obtained for daily returns

of the NIKKEI index and the Hang-Seng index. These results are

complemented by another work [12] where the returns of several

thousand U.S. companies were analyzed for Dt in the range from

five minutes to about four years. It was found that the returns of

individual companies at Dt < 16 days are also described with the

power-law distribution having the index a % 3. At longer Dt, the

probability distributions slowly approach the normal form. It was

also shown that the probability distributions of the S&P 500 index

and of individual companies have the same asymptotic behavior due

to the strong cross-correlations of the companiesâ€™ returns. When these

cross-correlations were destroyed with randomization of the time

series, the probability distributions converged to normal at a much

faster pace.

The theoretical model offered in [13] may provide some explan-

ation to the power-law distribution of returns with the index a % 3.

This model is based on two observations: (a) the distribution of the

trading volumes obeys the power law with an index of about 1.5; and

(b) the distribution of the number of trades is a power law with an

index of about three (in fact, it is close to 3.4). Two assumptions were

made to derive the index a of three. First, it was assumed that the

price movements were caused primarily by the activity of large mutual

funds whose size distribution is the power law with index of one (so-

called Zipfâ€™s law [4]). In addition, it was assumed that the mutual fund

managers trade in an optimal way.

90 Scaling in Financial Time Series

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

1X N

wi (t) (8:2:1)

wi (t Ã¾ 1) Â¼ l(t)wi (t) Ã€ aW(t) Ã€ bwi (t)W(t), W(t) Â¼

N iÂ¼1

where wi is an individual characteristic (e.g., wealth of an investor i;

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