CED model for asset returns and fractal market hypothesis

COMPUTER

MODELLING PERGA~ON

Mathem~ical and Computer Modellmg 29 (1999) 23-36

CED Model for Asset Returns and Fractal Market Hypothesis S. T. RACHEV+ Department of Statistics and Applied Probability University of California Santa Barbara, CA 93106-3110, U.S.A. A. VERONA Hugo Steinhans Center for Stochastic Methods Wrodaw University of Technology 50-370 Wroclaw, Poland and Department of Statistics and Applied Probability University of California, Santa Barbara, CA 93106-3110, U.S.A. R. WERON* histitute of Mathematics, Wrocfaw University of Technology 50-370 Wrodaw, Poland Abstract-A

new generai model for asset returns is studied in the framework of the Fractal Market Hypothesis (FMH). To accommodate markets with arbitrage opportunities, it concerns capital market systems in which the Condit~on~ly Exponential Dependence (CED) property can be attached to each investor on the market. Employing the limit theorem for the CED systems, the universal characteristics for the distribution of asset returns are derived. This explains the special role of the Weibull distribution in modeling of global ssset returns for market with no arbitrage and the two-power laws property of the density of global returns, evident in the empirical data. Finally, the link with two-parameter Pareto distributions is established. @ 1999 Elsevier Science Ltd. All rights reserved. Keywords-Financial modeling, Asset returns, Fractal market hypothesis, Arbitrage, W&bull distribution, CED model, Two-parameter Pareto distributions.

1.

INTRODUCTION

It is well known starting from Mandelbrot [l] and Fama [Z], see [3], that market returns are not normally distribute, but this information has been downplays or rationalized away over the years to maintain the crucial assumption of the traditional Capital Market Theory (GMT). A variety of alternatives to the normal law can be found in literature and it is undeniable that as long as the distribution that is implied by these models is more leptokurtic than the Gaussian law, it will provide a better fit. The CMT approach had the advantage of offering a large body of tools for research. However, it has a very stringent requirement that the current change in price could not be inferred from tThe work of the first author wss supported by the Alexander von Humboldt Award for U.S. Scientists. tThe work of the second author was supported by the Fulbright Senior Grant No. 19736. *The work of the third author wss partially supported by the Foundation for Polish Science (FNP) scholarship. 06957177/99/g - see front matter. @ 1999 Elsevier Science Ltd. All rights ressrved. PII: s0895-7177(99)000~*~

Typeset by &&-T3$

24

S. T. RACHEV

et al.

previous changes. A step further took the Efficient Market Hypothesis (EMH) [4], by saying, in its “semistrong” form, that current prices reflected all public information-all past prices, published reports, and economic news-because of fundamental analysis. The current prices reflected this information because all investors had equal access to it, and, being “rational” they would, in their collective wisdom, value the security accordingly. According to the EMH, the frequency of price changes should be well represented by the normal distribution. The analysis of the frequency distributions of returns shows, however, that this is not the case. There are observed far too many large up-and-down changes at all frequencies for the normal curve to be fitted to these data. In a search for satisf~tory descriptive models of economic data, large numbers of distributions have been tried and many new distributions have been discovered. Entire classes of distributional types have been constructed and these often serve to direct the search process for a suitable choice, see [3] and references therein, where a variety of alternative distributions for asset returns is analyzed. In any particular case, it is always possible to find a distribution that fits the data well, provided one works within a suitably broad and flexible class of candidates. Some alternatives to the normal d~tribution, like the stable Pareto distribution 151,were often rejected even though they fit the data without modification. Why? Standard statistical analysis could not be applied using those distributions. Besides that, it is one thing to fit given data well through the choice of a “good distribution”, but it is an entirely different matter to explain return’s data through the use of a statistical model that predicts the data’s main characteristics. To deal with such a problem, this paper employs a new conditionally Exponential Dependence (CED) model, introduced recently in [S] to describe global distributional structure of asset returns. The EMH including Arbitrage Pricing Theory (APT) of Ross [7] and Capital Asset Pricing Model (CAPM) originally developed by Sharp [8], Lintner [9], and Mossin [lo], was very successful in making the mathematical environment easier, but unfortunately is not justified by the real data. Instead, there is a need to seek for a market hypothesis that fits the observed data better and takes into account why markets exist to begin with. In the EMH place, a new Fractal Market Hypothesis (FMH) has been recently proposed by Peters [ll]. Based on current developments of chaos theory and using the fractal objects whose disparate parts are self-similar, the FMH provides a new framework for more precisely modeling of the turbulence, discontinuity, and nonperiodicity that truly characterize today’s capital markets. The FMH seems to be a robust tool for understanding the conflicting market randomness and determinism we experience every trading and investing day. The reason to write this paper is to put the basic ideas of the FMH into a rigorous mathematical framework, so the further precise analysis can be done. For this purpose, we adopt here an another recent idea used by Jurlewicz et al. [12] to model asymptotic behavior of general complex systems with local conditionally exponential decay property. In EGARCH models, the effects of shocks on the conditional variance of a financial time series typically decay in an exponential manner and are long lasting 1131. The dependence in the CED systems measured by the conditional return excess decays similarly as in EGARCH models in an exponential way, but reflects both short as well as long range effects. This new probabilistic idea concerns systems in which the behavior of each individual entity strongly depends on its short- and long-range random interactions. The proposed static approach to the capital markets explains how the two contrary states: local randomness and global determinism coexist leading in a natural way to the universally observed non-Gaussi~ distribution of returns. This approach makes a signific~t step toward explaining features of the statistical mechanism of data generation and, moreover, it predicts the data’s main characteristics. In this framework the class of possible distributions, well representing the observations is uniquely determined by the general return equation, The structure of the paper is as follows. In Section 2 we summarize the FMH framework. The CED model is introduced and analysed in Section 3. We present there two theorems describing basic statistical characteristics of the market. Theorem 1 provides the global return equation and

CED Model presents

the density

with two-parameter in Section

of its solution Pareto

4. Finally,

in a useful form of two power-laws.

distributions.

in Section

The derivation

5 the empirical

2. THE FRACTAL The FMH emphasizes investors.

the impact

In traditional

also generic.

Basically,

of the available

finance

approach, impact

where information

all investors

The following

return That

HYPOTHESIS

is treated

horizons

as a generic

on the behavior item.

is also considered

Day trader’s

has a different

primary

activity

is

because

price-taker,

i.e., someone

and knows how to value current

information.

This generic

are general cases, implies that all types of information

were proposed

is made up of many individuals

2. Information

of

The investor

rational

by Peters

[ll] for his FMH.

with a large number

of different

The behavior of a day trader is quite different from that of a pension fund. the investment horizon is measured in minutes; in the later case, in years. FMH

is given

is where it fails.

five basic assumptions

FMH 1. The market horizons.

2 gives a link

equation

is any one who wants to buy, sell, or hold a security

and investors

equally.

return

of the CED model is discussed.

and investment

information

Theorem

of the global

MARKET

The investor

who always wants to maximize

evidence

of information

theory,

an investor

information.

25

impact

is trading.

on different A day trader

investment

investment

In the former case,

horizons.

will be more concerned

with technical

information. Most technicians have short investment horizon, and, within their frame, fundamental information is of little value. From the other side, most fundamental analysts and economists who also work in the markets have long investment horizons. They tend to think that technical trends are not of use to long-term investors. Only by assessing value can true investment returns be made. In the framework of the FMH, both technicians and fundamentalists are right because the impact of information is largely dependent on each individual’s investment horizon. FMH 3. The stability of the market is largely a matter of liquidity (balancing of supply and demand). Liquidity is available when the market is composed of many investors with many different

investment

horizons.

In that way, if a piece of information comes through that causes a severe drop in price at the short investment horizon, the longer-term investors will step in to buy, because they do not value the information investment long-term

as highly.

horizons, investors

When the market

then the market causes the entire

loses this structure,

becomes market

unstable to trade

because

and the investors

have the some

there is no liquidity.

based on the same information

The loss of set, which

is primarily technical (or a crowd behavior phenomenon). Typically, the market horizon becomes short-term when the long-term outlook becomes highly uncertain (often for political reasons). Thus, the market stability relies on diversification (fractal structure) of the investment horizons of the participants. The market is stable because the different horizons value the information flow differently, and can provide liquidity if there is a crash at one of the other investment horizons. This follows from the errortolerant fractal (self-similar) statistical structure of the global system. FMH 4. Prices reflect a combination valuation.

of short-term

technical

trading

and long-term

fundamental

Thus, short-term price changes are likely be more volatile, than long-term trades. The underlying trend in the market is reflective of changes in expected earnings, based on the changing economic environment. Short-term trends are more likely the result of crowd behavior. There is no reason to believe that the length of the short-term trends is related to the long-term economic trend. FMH 5. If a security has no tie to the economic cycle, then there nading, liquidity, and short-term information will dominate.

will be no long term

trend.

If a market because

is tied to economic

the economic

and bonds,

stock returns

have a short-term

However,

in physics.

during

information

handle

is valued

the FMH,

horizon.

When

markets

panics

over time

than trading

capital

superimposed

being a trading

Instability investment

accordingly occurs

horizon,

because

the transition

the market

markets,

activity, like stocks

over a long-term

market

behavior stable,

economic

only, have only the fractal

the EMH,

APT,

when

and market break

seems to

down,

like singularities

and the CAPM

are equilibrium

Unlike the EMH, the FMH says that

horizon

of the investor.

it has no characteristic

the market

price movements

the EMH and CAPM

those models

to turbulence.

to the investment

is stable when

are consider

and stampedes,

This is not unexpected,

They cannot

uniform

structure

Currencies,

risk will decrease

as well. Economic

of the FMH is to give a model of investor

fits our observations.

work fine.

under

less volatile

then

cycle is less volatile

structure.

The purpose

models.

over the long term, The economic

fractal statistical

cycle, which may be deterministic. statistical

growth

cycles dominates.

which makes long-term

that

et al.

S. T. RACHEV

26

loses its fractal

time

structure

The key is that

scale or investment and

assumes

a fairly

see [ll].

3. CED

MODEL

In classical finance theory, markets were assumed to be efficient; that is, prices already reflected all current information that could anticipate future events. Therefore, only the speculative, stochastic component could be modeled. If market returns are normally distributed white noise, then they are the same at all investment horizons. The risks to each are the same. Risk and return grow at a commiserative rate over time. There is no advantage of being a long-term investor. In contrast, it follows from the FMH, that both investment horizon and the information reflect the local randomness of the system. Both also impact the behavior of an individual investor. If such a point of view is taken into account,

the relation

between

different

sources of randomness

should

yield even more complex limit laws that need to be represented in terms of stochastic model. As yet, this more general probabilistic approach to modeling financial market data does not seem to have been studied distribution

by researches

interested

in the largely descriptive

issue of the “best” statistical

of asset returns.

Below, we propose a new statistical mechanism that explains the observed market local randomness and global determinism. We hope that this approach clarifies also the ideas of the FMH and provides a rigorous mathematical framework for further analysis. The distributional form of returns

on financial assets has important implications for theoretical and empirical analyses and finance. For example, asset, portfolio, and option pricing theories are typically

in economics

based on distributional assumptions. In empirical tests, statistical inference concerning the efficient market hypothesis, the excess volatility question or option pricing models may be sensitive to the distributional assumptions for the returns of the underlying assets. The stock market is made up of investors, from tick traders to long-term investors. Each has a different investment horizon that can be ordered in time. When all investors with different horizons are trading simultaneously the market is stable. The stability of the market relies, however, not only on a random diversification of the investment horizons of the participants but also on the fact that the different horizons value the importance of the information flow differently. Hence, both the information flow and the investment horizons should have their own contribution to the observed global market features. In general, the locally random markets have a global statistical structure that is nonrandom. Following Rachev et al. [6], we will assume that the model is a discrete finite number of trading dates from time 0 to time T, and its uncertainty on the market index daily returns on the interval [0, T]. (As a proxy of typically uses a stock index, for example, S&P or NASDAQ.) In the family let us identify those N who are acting on a given market described by

time economy with a has a global impact the market index, one of all world investors, a chosen index. Call

CED Model

27

INN. Let RON be the positive (or the absolute value of negative) part of the themIlN,IzN,..., ith investor’s return. The economy is populated by a finite, but a large number N of investors on the market. common

Each ith investor

complement

(inter-cluster) type

markets.

interactions

of interactions

by the random

is related

and is reflected

is imposed

risk factors

with a cluster

The influence

of agents

of this cluster

by a random

on the ith investor

Bj for all j # i.

acting

of agents

risk-aversion reflect

on

factor Ai. The long-range

by the intercluster

They

simultaneously

is of type of short-range relationship

manifested

how fast the information

flows to

ith investor. ASSUMPTION 1. For ith investor,

the following

CED property

holds:

1(3,b) = P (RN 2 r 1Ai = U, b,’ IlXlX(Bf, . . . ,Bf_l, B;+,, . . . , B;)

&,(,

= b)

(1)

= exp (- [u min(r, b)]“) , constants, bN is a suitable, positive normalizing constant, c is justified by the reversion tendency of the market.

where r, a, b are nonnegative The range of the exponent

and c 2 1.

The dependence in the CED model measured by the conditional return excess decays similarly as in EGARCH models in an exponential way, cf. (l), but reflects both short- as well as long-range effects. This new probabilistic idea concerns systems in which the behavior of each individual entity strongly depends on its short- and long-range random interactions. ASSUMPTION 2. We assume interaction”)

affected

horizon of the investor reffect the information

that

the investors

by a different

information

is reflected by the random flow to this investor.

have different

investment

set (“long-range

horizons

interaction”).

Ai, while {Bj,

variable

(“short-range The investment

j = 1,2,. . . , N, j # i)

The probability that the return RiN will be not less than r is conditioned by the value a taken by the random variable Ai and by the value b taken by the maximum of the set of random variables {B:, j = 1,2, . . . , N, j # i}. Therefore, equation (1) can be rewritten as follows:

&N(r

1a, b) 3

1,

for r = 0,

exp (-(ar)c),

for r < b,

{ exp (-(ab)c) i.e., the conditional exponent

return

c as r tends

assumption

excess &N(r

1 a, b) decays

to the value b. Then

,

(2)

for r 2 b,

exponentially

it takes a constant

with

a decay

rate

a and

value << 1. The basic statistical

is the following.

3. The random variables Al, A2,. . . and Bi, B&. . . form independent and convergent (with respect to addition and maximum, respectivelu 1121) sequences of nonnegative, independent, identically distributed &id.) random variables. The variables RON, . . . , RNN axe ASSUMPTION

also nonnegative,

i.i.d. for each N.

Let us stress, however, that the dependence on external conditions is expressed by the above relationship, equation (l), of each RiN with Ai and max(Bf, . . . , Bf_,, Bf,,, . . . , Bh). Assumption 3 can be partially justified by the following argument. Institutional trading is a major factor in the determination of security prices. If professional investment managers have similar beliefs, then the i.i.d. distributions assumption may hold as a first approximation. Professional managers are likely to have similar beliefs because they have access to a similar information sources. This uniformity of information over time would tend to generate similar beliefs. It is important to point out that the assumption of i.i.d. of the returns RiN is not as restrictive as it may appear. For example, Mittnik and Rachev [3] have shown that the assumption of i.i.d. random asset-price changes can be used to describe large classes of well-known financial models. The cut-off in the return excess equation (2) given by the value r = b determines the probability (indeed, very small) that the return of ith investor can reach any value greater than b. ‘rhe value

S. T. RACHEV et al.

28

of this probability is the smallest one, since a cut-off by any other value bi < b yields a greater probability than exp[-(ab)c]. This is a manifestation of the unlimited returns. Thus the market contains some arbitrage opportunities.

Note that equation (1) precisely defines the meaning of

random variables related by it. It does not hold for sets of any arbitrarily

chosen variables.

If

RN has to denote a return, then Ai = a has the sense of an individual risk aversion factor and bN_‘max(Rf

,...,

Rf_i,Rj+i

,...,

Rh)

=b

the sense of a submarket maximal risk factor given by hN(T

1 b) =

where FA is the common distribution

s

om d%N(r

1 a, b) @A(a),

function (but unknown!) of the sequence of random vari-

ables {Ai}. THEOREM 1. Let the global behavior of the asset market be given by

d(r) =

P

(liliwr, min(&NT --RNN)

where TN is a suitable, positive normaJjsjng constant. tion #(r), fulfills the Global Return Equation

2

Y

Under the above assumptions,

(1- exp (- y))

S(T)= -CAM-’

r)

the fnnc-

4(r),

where the parameters X > 0, k > 0, ~1> 0, and cx = Q’C (c > 1, 0 < CY’5 1) are determined the limiting procedure in (3). Moreover, the probability

density f(r)

= -v (Xr)+,

by

has the two power-laws property forXr<
‘(‘Io({ (X7-)-m-1, for Xr >> 1, only if p = (Y or p > CY. The above two cases are distinguished excess for large r and also by the long-range exponent m, (Y m=

Ic’ ( p-a,

by the behavior of the return

for p = (Y, (6)

forj.i>a.

The short-range exponent n does not depend on h, since we have n = 1 - cr. The above equality (3) defines the return excess P (ii 2 T) of a system as a whole, where R rep resents the global return. The derivative f(r) = - $$ (r) represents the frequency distribution (probability density) of a global market return. A more general type of the above equation has been recently studied in the context of stochastic systems by Jurlewicz et al. [12]. It turns out that the solution of equation (4) for p = cx has the following integral form:

f#(~) = exp [ -1 k J Ok(‘r)” (1 -exp

(-i))

The function 4(r) monotonically decreases from 4(O) = 1 to I of the global return exhibits the two power-laws

‘(r)

(XT)“-l,

for Xr < 1,

o( { (XT)-(~I”)-‘,

for Xr >> 1.

ds] . = 0. The probability density

(8)

CED Model Hence,

the Global

Return

(GR) distribution

GR((r, X, Ic). Here LYis the shape stress market;

k + 0 denotes

arbitrage

opportunities

is characterized

by the following

and X the scale parameter, k. It decides

the role of the parameter

29

how fast the information

the case when the long-range on the market.

respectively.

interaction

If Ic + 0, equation

three

parameters

At this point

flow is spread

is neglected.

let us

out in the

So, there

are no

(4) takes the form

$$r)= -a(hyl~(r) with the solution 4(r) = exp [-(XT)~]. in the case when k 4 0, the probability

Thus,

form of the Weibull

l

of the global return

obtains

the well-known

density f(r)

and the following

density

(10)

exp (-(XT)~) ,

= aX(Xr)“-’

(11)

specific cases are observed:

if (Y’ = 1 (Ai is a nonrandom

l

if cx = 1, then

l

tail for Xr > 1; if Q = 2, then f(r)

f(r)

(deterministic)

= Xexp(-XT)-the

variate),

density

= 2X2rexp(-(Xr)2)-the

then Q! = c;

of the global

density

return

of the global

has the exponential return

has the normal

tail for Xr > 1. In general,

the parameter

0 < Q’ 5 1 slows down, in comparison

with an individual

investor,

the

return rate aX(Xr)“-’ of the global market return distribution. Let us observe that the inclusion of arbitrage opportunities (k > 0) changes essentially the tail of the density f(r) for Xr > 1. Both solutions, equations (8) and (ll), have the same behavior for Xr < 1. Thus, the above discussion

explains

well the special

role of the Weibull

distribution

in modeling

of asset returns

for markets with no arbitrage, see [3]. However, our model leads one step further introducing the new type of return distribution completely described by GRE (4). It exhibits the two power-laws property (8) of the density of the global returns, evident in the empirical data. The form of $(T) given by (7) does not indicate directly any commonly known probability distribution function. However, it can be shown its close relationship with a two-parameter Pareto (also called Burr) distribution, see [14]. Indeed, taking into account two terms in the in the integrand, one obtains the approximate series expansion of the exponential term expx-’ form

h(r) = [l + k(Xr)a]-“k of the solution

of (4).

two-parameter Pareto lim T+m(4(r))/(&(r))

= 1 - FCllkjJr)

So, the GR distribution F(r) = 1 - 4(r) can be approximated by the Observe that lim,,o(4(r))/($,(r)) = 1 and distribution F(l/k),u(r). = exp[-(1 - y)/k], where y = 0.577216.. . is the Euler gamma constant.

Hence we have the following.

2. The GR distribution function F(r) determined by the following two-parameter Pareto distribution

THEOREM imated

k(Xrlal-l~k > ,, F(r) =

4.

DERIVATION

%/k),a(~L

for Xr CC 1,

e-(l-r)‘k)F~l,k),,(r),

for Xr z$ 1.

OF THE GLOBAL

PROOF OF THEOREM 1. The existence CED property (1) under we have the following:

Assumptions

by the GRE (4) is well approxfunction F~l/k),~(r) = 1 - [l +

RETURN

(12) EQUATION

of a sequence of i.i.d. random variables {Rg,r} with the 1 and 3 is proved in [12]. From the law of total probability 00

P(Ri~>r\Ai=a)=

s0

exp { - [a min(r, b)]“} ~FB,N(~),

(13)

30

S. T. RACHEV et al.

where PB,N(~) denotes the distribution

function of the random variable

bN-lmax(Bf,...,Bf_l,Bf+l,...,B~), i.e., the probability that this random variable has taken a value less than b. The above equality is justified by the assumption that Al,Az, . . . and Bi,Bz,. . . form independent sequences of random variables. By the assumption that Bj are i.i.d. r.v.s, we have N-l

,

where FE denotes the distribution function (but unknown) of each Bj. Assuming FB differentiable, we have FB,N differentiable, too, and fp

( >rrN 1 a)=[~-FB,N (k)] fexp [-(+-)‘I, R.

A.= z

tN -

where rN is a suitable normalizing constant. from the Lebesque Theorem, we obtain

From the law of total probability once again, and

(14) where L(FA=; (T/TN)‘) is the Laplace transform of the distribution function FA= at the point (~/TN)‘. Because RON are i.i.d. for each N, equation (3) can be written in the form

(15) On the other hand, it follows from (14) that

As it is well known from probability theory, the Nth power of the Laplace transform of a nondegenerate distribution function F converges to the nondengenerate limiting transform, as N tends to infinity, if and only if F belongs to the domain of attraction of the Levy stable law (see [15,16]). Hence, $h&

k (F,&; (&)‘)I”=eXp [-(Ar)a’C],

(17)

where A is a positive constant and 0 < (Y’ 5 1. The range (0, l] for the index of stability a’ follows from the nonnegativity of the random variables Ai, i = 1,2, . . . , N. The case Q’ = 1 corresponds to a degenerate limiting distribution of A = 1imN-+w CL(~/(~N)~ At the same time, FB,N

(

T G

= P[bN-‘max(Bf

,...,

Bf_,,Bf+,

,...,

BL)
)

tends to a nondegenerate distribution function of nonnegative random variable, as N tends to infinity, if and only if FB, the distribution function of each Bj, belongs to the domain of attraction of the max-stable law of type II, cf. [17]. For a suitable normalizing constant bN, we have

$m$wv

(&)

=exp(-y),

(18)

CED Model

31

for some positive constants p, k, and the constant A taken from (17). To obtain the limiting forms (17) and (18), we need not know the detailed nature of FA” and FB. In fact, those are unknown distribution functions. The limits in (17) and (18) are determined only by the behavior of the tail of Fp for large a and the tall of FB for large b, respectively. The necessary and sufficient conditions for the random variables Al, AZ,. . . and Br, Bz, . . . to have the limits in (17) and (18), respectively, can be expressed as the following self-similar properties: l

for any x > 0 and large a,

P (Ai 2 xa) = x-~“P l

(Ai 2 a),

for any x > 0 and large b,

P (Bi 2 zcb) = x-p’p (Bi > b) ) where 0 < (Y’5 1, c > 1, and /.L> 0. Taking into account equations (13)-(18), it is a straightforward cess $(r) fulfills the following equation if we substitute A = X

z(r) = -aX(Xr)“-1 (1-exp where cy = (Y’C and ~1 > 0. f(r)

= -F

(-y))

result that the return ex-

4(r),

(21)

It can be shown similarly as in 1181 that the probability density

has the two power-laws property (Xr)+, ‘(r)

a { (XT)+-‘,

for Xr < 1, for Xr >> 1,

(22)

only if p = cr or /A> (Y. The two cases are distinguished by the behavior of the return excess q%(r) for large r, namely, as r goes to infinity

(23) and also by the long-range exponent m,

m=

Q -9

k j.h- Q,

for p = (Y, for ~1> LY.

(24)

The short-range exponent n does not depend on p, since we have n = 1 - CL If we exclude the possibility, that the return excess 4(r) of the global return may be greater than 0 for large r what corresponds to a global arbitrage strategy, the case p = a will be the proper one to describe capital markets. Hence, the GRE takes form (4) with all the consequences discussed in the previous section. I

5. EMPIRICAL

ANALYSIS

EXAMPLE 1. STANDARD & POOR’S INDEX. The data set is the S&P 500 index which is a composite index based on the performance of the main 500 shares on the New York Stock Exchange. To illustrate the typical characteristics of the global market returns, we consider daily observations during the period from July 2, 1962 to December 31, 1991. We note that this period includes the stock market crash of October, 1987. Let S(t) be the daily observation of day t for the S&P 500 Index and R(t) be the daily return of day t. Then S(t) and R(t) are related by R(t) = log S(t) - logS(t - 1). The 7420 daily returns, calculated by above formula, for the

S. T. RACHEVet at.

32

0.1 0.05

0

I

-Q.o54.1 -0.15-0.2-025 0

1 I , woo2oOomoo4ooa!iooo801107ooo~

I

(a) The 7421 daily returns of the S&P 500 Index, July 2, 1962 to December 31, 1991.

10’

loo

(b) The Zipf plots (double ~og~ithmic plots) of empirical densities (kernel ~t~rnato~) of S&P composite daily returns vs. rank for positive and absolute value of negative returns. Figure 1.

S&P 500 Index are displayed in Figure la.

Note the large negative return due to the Octo-

ber, 1987 stock market crash. They are split to 3879 positive and 3495 negative returns; there are also 47 zero returns. The Zipf plots (double logarithmic plots) of empirical densities (kernel estimators) of S&P composite daily returns vs. rank are shown in Figure lb for positive and absolute value of negative returns. Observe that the Zipf plots demonstrate clearly the difference in the behavior of positive and negative returns and also visualize the two power-laws, see (5), for small and large returns, respectively. Applying the CED model via formula (8) separately to both data sets, we are getting the following parameters: ty+ = 0.8656,

X+ = 142.8571,

kf = 0.4977

o-

X- = 125.0000,

k- = 0.3392

and = 0.7405,

for positive and absolute value of negative S&P returns, respectively. These parameters are different than those provided by an earlier fitting study given in [6), where an approximation through the Weibull distribution was used.

CED Model

33

NASDAQ INDEX. The data set is the NASDAQ Index which is a composite index based on the performance of shares listed on the NASDAQ market. To illustrate the typical characteristics of the global market returns we consider daily observations during the period from December 14, 1972 to December 31, 1991. We note that thii period includes the stock market crash of October, 1987. The 4809 daily returns for the NASDAQ Index are displayed in Figure 2a. They are split to 2735 positive and 2067 negative returns; there are also seven zero returns. The Zipf plots of empirical densities of NASDAQ composite daily returns vs. rank are shown in Figure 2b for positive and absolute value of negative returns. EXAMPLE 2.

-0.151

0

loo0

3Ow

(a)The 4809 dally returns of the NASDAQ Index, December 14, 1972 to December 31, 1991. Id\

7 . ...I

I

1

..-...._.....,

10' :

lO':c

-iwthlmaulm ~~~~'nbmbv8hmd~mtung

;*

lo4

?04 10-l (b) The Zipf plots (double logarithmic plots) of empirical densities (kernel estimators) of NASDAQ composite daily returns vs. rank for positive and absolute value of negative returns. Figure 2.

Applying the CED model via formula (8) separately following parameters:

to both data sets, we are getting the

cy+ = 0.9228,

X+ = 166.6667,

k+ = 0.4230

(Y- = 0.8596,

X- = 181.8182,

k- = 0.6510

and

for positive and absolute value of negative NASDAQ returns, respectively.

S. T. RACHEVet al.

34

EXAMPLE 3. Dow JONES INDUSTRIALAVERAGEINDEX. The data set is the DJIA Index which

is a composite index based on the performance of the 30 main shares on the New York Stock Exchange. To illustrate the typical characteristics of the global market returns, we consider daily observations during the period from January 2, 1930 to April 30, 1991, The 16382 daily returns for the DJIA Index are displayed in Figure 3a. They are split to 8494 positive and 7802 negative returns; there are also 86 zero returns. The Zipf plots of empirical densities of DJIA composite daily returns vs. rank are shown in Figure 3b for positive and absolute value of negative returns, It is interesting to mention at this point close similarities between Figure 3b and related Figure 14.5 and Figure 17.7 in [ll], for daily DJIA and the Mackey-Glass equation.

-0.3’

I

2ooo

4ooo

I

L

omo

Eooo

loooo 12ooo

14ooo

16ooo

18oal

(a) The 16382 daily returns of the DJIA Index, January 2, 1930 to April 30, 1991. lop

-----I t

(b) The Zipf plots (double logarithmic plots) of empirical densities (kernel estimators) of DJIA composite daily returns vs. rank for positive and absolute value of negative returns. Figure 3. EXAMPLE 4. JPYfUSD EXCHANGE RATE. The data set is the JPY/USD Exchange Rate. To ihustrate the typical characteristics of the global market returns, we consider daily observations during the period from January 1, 1980 to December 7, 1990. The 2852 daily returns for the JPY/USD Exchange Rate are displayed in Figure 4a. They are split to 1426 positive and 1410 negative returns; there are also 16 zero returns. The Zipf plots of empirical densities of JPY/USD Exchange Rate daily returns vs. rank are shown in Figure 4b for positive and absolute value of negative returns, see also [19,20].

CED Model

-*

0

tom

1500

35

3im

(a) The 2852 daily returns of the JPY/USD Exchange Rate, January 1, 1980 to December 7, 1990.

to’

I-

7

10’

too

J lo-'

(b) The Zipf piots (double logarithmic plots) of empirical densities (kernel estimators) of JPY/USD Exchange Rate composite dally returns vs. rank for positive and absolute value of negative returns. Figure 4.

REVERENCES 1. B.B. Mandelbrot, New methods in statistical economics, Journal of Political Economy 71, 421-440 (1963). 2. E. Fama, The behavior of stock market prices, Journal of Business 38, 34-105 (1965). 3. S. Mittnik and S.T. Rachev, Modeling asset returns with alternative stable distributions, Econometric Reviews 12, 261-330 (1993); Reply to Comments, Econometric Reviews 12, 341-389 (1993). 4. R.A. Jarrow, Finunce Theory, Prentice-Hall, Englewood Cliffs, NJ, (1988). 5. J.H. McC!ulloch, Financial applications of stable distributions, In ~a~b~k of Statis~cs 14, (Edited by G.S. Maddala and CR. Rao), Elsevier, (1996). 6. ST. Ftachev, A. Weron and K. Weron, Conditionally exponential dependence model for asset returns, Appl. Math. Lett. 10 (l), 5-9 (1997). 7. S. Ross, The arbitrage theory of capital asset pricing, Journal of Economic Theory 13, 341-360 (1976). 8. W. Sharp, Capital asset price; A theory of market equilibrium under conditions of risk, Journal of Finance 19, 425-442 (1964). 9. J. Lintner, The valuation of risk sssets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics 47, 13-37 (1965). 10. J. Mossin, Equilibrium in a capital asset market, Econometrica 34, 766-783 (1966). 11. E.E. Peters, Fhzctal Market Analysis. Applying Chaos Theory to Investment d Economics, J. Wiley & Sons, New York, (1994). 12. A. Jurlewicz, A, Weron and K. Weron, Asymptotic behavior of stochastic systems with conditionally exponential decay property, App~~~t~on~ M~themuticae 23, 379-394 (1996).

S. T. RACHEV et al.

36

13. T. Bollerslev, R.Y. Chou and K.F. Kroner, ARCH modelfng in finance. A review of the theory and empirical evidence, Journal of Econometrics 52, 5-59 (1992). 14. R.E. Beard, T. Pentikainen and E. Pesonen, Risk Theory. The Stochastic Basis for Insurance, Chapman

and Hall, London, (1984). 15. G. Samorodnitsky and MS. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall, London, (1994). 16. A. Janicki and A. Weron, Simulation and Chaotic Behavior of Stable Stochastic Processes, Marcel Dekker,

New York, (1994). M.R. Leadbetter, G. Lindgren and H. Rootzen, Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, New York, (1983). 18. K. Weron and A. Jurlewicz, Two forms of self-similarity as a fundamental feature of the power-law dielectric relaxation, Journal of Physics A: Mathematical d General 26, 395-410 (1993). 19. A. Weron, Sz. Mercik and R. Weron, Origins of the scaling behaviour in the dynamics of financial data, 17.

20.

Physica A 264, 562-569 (1999). Sz. Mercik and R. Weron, Scaling in currency Physica A 267, 235-250 (1999).

exchange

= a conditionally

exponential

decay approach,