Sergio Da Silva

English

Research Papers in Economics

Exponentially damped L´ evy ﬂights,
multiscaling and slow convergence in stock
markets.
Iram Gleria a,1 Annibal Figueiredo b Raul Matsushita c
Pushpa Rathie c Sergio da Silva d
aDepartamento de F´ ısica, Universidade Federal de Alagoas, Brazil.
bInstituto de F´ ısica, Universidade de Bras´ ılia, Brazil
cDepartamento de Estat´ ıstica, Universidade de Bras´ ılia, Brazil
dDepartamento de Economia, Universidade Federal do Rio Grande do Sul, Brazil
Abstract
We have previously examined the role of autocorrelations in the sum of stochastic
variables together with the existence of scaling power laws [1]. Here we employ such
an approach to analyze sluggish convergence [2] in data coming from the S&P500
index. We also employ our suggested exponentially damped L´ evy ﬂight [3] to assess
the multiscaling properties in the data.
Key words: L´ evy distributions; Econophysics; Multiscaling. 05.40.+j;02.50.-r
1 Introduction
We have previously put forward a connection between autocorrelation and
the presence of power laws [1]. We show that the typical scaling properties
of abruptly truncated L´ evy ﬂights (TLFs) [2] are linked to autocorrelation in
data. Here we move up and show how the property of sluggish convergence to
a Gaussian [2] can be explained by particular features of nonlinear autocorre-
lations.
Owing to its sharp truncation, the characteristic function (CF) of the TLF
is not inﬁnitely divisible [4–6]. To make the CF inﬁnitely divisible again, an
1 Corresponding author, e-mail:iram@df.ufal.br
Preprint submitted to Elsevier Preprint 8 January 2004exponentially damped L´ evy ﬂight (EDLF) has been suggested by some of us
[3] and used to analyze foreign exchange rate data [7]. This paper blends such
novel theoretical results with an analysis of actual data coming from the S&
P500 index. We consider daily closing values for the period from January/1980
to December/2002. The structure of the paper is as follows: section 2 examines
the problem of convergence to the Gaussian regime. Section 3 shows the EDLF
and its multiscaling properties, and section 4 concludes.
2 Slow convergence and nonlinear correlations
Consider the sum of random variables xi of zero mean given by
 n
i=1 xi.F o r
reduced variables, the CF ψ(z) of a process with ﬁnite second moments can
be written as ψ(z)=e−z2(1+w(z))/2 [8]. If w(z)=0 ,∀z one gets the CF of
the Gaussian distribution. In [1] we showed how a scaling power law in the
probability of return to the origin (P(0) = 0) emerges if the standard deviation
is also governed by a power law and wn(z) has some special properties. We
employ such an approach to exchange rate data [1] and get results that are
similar to the originally suggested by Mantegna and Stanley [2]. Here we
move up to extend the methodology and examine sluggish convergence to a
Gaussian. The statistical moments of order p of xi and Sn are µip and νip
respectively. A nonlinear autocorrelation can be tracked by
<p 1p2 ···pk >n=
n  
i1···ik
(<x
p1
i1 ···x
pk
ik > − <x
p1
i1 > ···<x
pk
ik >)( 1)
where p1p2 ···pk are positive integers, and i1  = i2  = ··· = ik. Our task is to
analyze the contribution of such nonlinear terms in the process of convergence.
If we take w(z)=wR + ıwI, and further carry out an expansion in series, it
can be shown that
wR = −
1
12
(µ
4 − 3)z
2 + O(z
4)=−
K
12
z
2 + O(z
4)( 2 )
wI =
1
3
µ
3z + O(z
3)=
Sk
3
z + O(z
3)( 3 )
where the kurtosis K and skewness Sk of the process are made explicit; actu-
ally they are the leading terms in the series expansion of w(z). For independent
and identically distributed (IID) variables, K and Sk approach zero at a pace
governed by ratios K → 1/n and Sk → 1/
√
n .S oa c c o r d i n gt ot h ec e n t r a l
limit theorem (CLT) [8],w(z) → 0, and the Gaussian regime is reached. Now
let us show how the nonlinear correlations in Eq. (1) can prevent the Gaussian
regime to be reached, which may also explain the slow convergence observed
2in real data [2]. If Ψ(z)=e−z2(1+Ω(z))/2is the CF of the reduced sum variable,
then from Eqs. (2) and (3) it can be shown that
ΩR = −
1
12
Knz
2 + O(z
4),
1
12
Kn =
n  
i=1
µ2
i2µi4
12ν2
n2
+
1
4
(1 −
σ2
n2
ν2
n2
) − (4)
1
12
< 1111 >n +6 < 112 >n +4 < 13 >n +3 < 22 >n
ν2
n2
(5)
≡
1
12
K
0
n + K
1
n + K
2
n (6)
and
ΩI =
1
3
Sknz + O(z
3), (7)
1
3
Skn =
n  
i=1
µ
3/2
i2 µi3
3ν
3/2
n2
+
< 111 >n +3 < 12 >n
ν
3/2
n2
≡
1
3
Sk
0
n + Sk
1
n (8)
where σnp = µ1p + ···µnp. Terms K0
n and Sk0
n are standard in the sum of
IID variables; they move toward zero according the CLT. But if at least one
of the terms K1
n,K2
n or Sk1
n, which involve the nonlinear autocorrelations, is
greater than nil, then Ω(z)=Ω R +ıΩI falls greater than nil. Here the process
cannot reach the Gaussian regime. It is worth to note that slow convergence
takes place even for short-range correlated processes, provided that either K2
n
or Sk1
n departures from zero (K1
n has only the usual autocorrelation function
< 11 >).
The sluggish convergence can be illustrated with data coming from daily vari-
ations of the S& P500 index. Fig. 1(a) displays the behavior of the kurtosis for
daily returns of the index. The process remains bounded at nonzero values for
several days and goes to Gaussian regime in an ultra slow pace. Compare with
previous works dealing with the S& P 500 [2] and the Ibovespa index (Brazil-
ian stock market) [9], where it is claimed that Gaussian regime is reached after
20-30 days. From our results, the crossover time seems to be greater.
The kurtosis and skewness does not decay to zero (at least in the time win-
dow considered). Fig. 1(b) shows K2
n, which involves nonlinear correlations of
fourth order. Thus splitting up the kurtosis makes it clear how K1
n (not shown)
and K2
n contribute to the nonzero kurtosis, thereby preventing the Gaussian
regime to be reached. And that happens thanks to the presence of the non-
linear autocorrelations. The same behavior is observed for the skewness (not
shown).
33Exponentially Damped L´ evy Flights and Multiscaling
Although our previous approach allows one to study a number of statistical
properties, it does not provide a model on its own for a stochastic process.
The nonstable, ﬁnite-variance TLF itself is such a model. Other modiﬁed
versions of the TLF are the smoothly truncated L´ evy ﬂight and the gradually
truncated L´ evy ﬂight [4–6]. The sharp cutoﬀ of the TLF renders its CF not
inﬁnitely divisible. Now we present our EDLF [3], for which the CF is inﬁnitely
divisible. Consider a symmetric L´ evy distribution of index α,t h a ti sL(S∆t)=
1
π
  ∞
0 e−γ∆tqα cos(qS∆t). Let z∆t = St −St−∆t (here we employ ∆t instead of n
). The EDLF is deﬁned as P(∆t)=ηL(z∆t)f(z∆t), where η is a normalizing
constant, and f(z∆t) is the change carried out on the distribution:
f( z∆t )=

     
     
1, z∆t  <l c
(∆t−1/α z∆t  + ϑ)β1eH(z∆t),l c <  z∆t  <l max
0, z∆t  >l max
(9)
where H(z∆t)=λ1 + λ2[1 −  z∆t /lmax]β2 + λ3( z∆t −lc)β3,a n dϑ,λ1,λ 2 ≤
0,β 1,β 2 and β3 are parameters describing the deviations from the L´ evy stable
distribution, lc is the size step at which the distribution begins to deviate from
the L´ evy, and lmax is the size step at which an abrupt truncation is carried
out. The TLF obtains if f(z∆t)=0 ,( z∆t  >l c), and f(z∆t)=1 ,( z∆t ≤lc).
One striking diﬀerence between the TLF and the EDLF is the presence of
multiscaling in the latter, which can also emerge in other modiﬁed L´ evy ﬂights
[4–6].
By scaling z∆t together with the truncation parameters, a distribution can be
collapsed onto the ∆t = 1distribution. Power laws for both the Kth absolute
moment and norm of the CF of the EDLF can then be found [7].
To examine multiscaling, let <  z∆t  >K= 1
n
 n
t=1  St−St−∆t K,a n d ˜ z K =
1
n
 n
t=1  St − St−1 K be the Kth sample mean of lagged absolute values of St
at time intervals ∆t and 1respectively. For moments that are low enough
(such as 0 <K<a ), P(z∆t) is expected to be approximated by L(z∆t), which
in turn does not depend on the truncation parameters. Thus we expect <
 z∆t K >≈ ∆t
K/α <  ˜ z K > to hold for lower moments. This means that ratio
R(K,∆t)= <  z∆t K >/< ˜ z K > scales with ∆t as R(K,∆t)=∆ t
K/α.
Here multiscaling is related to nonlinear behavior in the partition function
R(K,∆t).
The norm of the CF (φ(K))can also be used to assess parameter γ by taking
into account the same assumption that P(z∆t) ≈ L(z∆t) for low values of  K .
4Since φ(K) ≡ E[eiKz]=E[cos(Kz)+ısin(Kz)] = E[cos(Kz)] + E[ısin(Kz)]
then the squared norm of the CF gives  φ 2 = E2[cos(Kz)] + E2[sin(Kz)].
For some K and ∆t,  φ(K)  can be estimated by  ˜ φ(K) 2 =< cos(Kz) >2
+ < sin(Kz) >2. By assuming that ln[φ(K)] ≈− γ∆t K α for 0 <K<α ,
the estimated norm in logs of the CF is ln φL(K)  and then we can expect
that ln[˜ φL(K)] ≈− γ∆t K α
Fig. 2(a) displays sample ratios R(K,∆t) for K ranging from 0.0 (bottom
lines) to K =4 .0 (top lines) at intervals of 0.1in log-log plots of data coming
from daily returns of S& P 500. They exhibit power law dependence on ∆t.
By ﬁtting lnR(K,∆t)=ξ ln∆t for every K, we get the corresponding scaling
exponents shown in Fig. 2(b). The curves show linear dependence on K, for
0 <K<α . However scaling breaks down after K>α ,a n dan o n l i n e a r
behavior (multiscaling) steps in. Fig. 3(a) displays the sample logarithm of the
absolute CF versus ∆t for K ranging from 0.00 (upper lines) to 0.025 (bottom
lines) at intervals of 0.0005. A power law dependence on ∆t is clearcut for the
S&P500. By ﬁtting ln ˜ φL(K)  = ζ∆t for every K, the estimated values of
ζ versus  K α are plotted in Fig. 3(b). An approximate linear behavior for
all κ(α)= K α tracks mere single scaling, and a linear behavior for initial
values of κ(α) <α 0 followed by a nonlinear pattern after κ(α) >α 0 captures
multiscaling.
4 Conclusions
This paper shows that nonlinear autocorrelations may prevent convergence
to the Gaussian regime. It enlarges previous results obtained by some of the
authors [1]. The slow convergence is illustrated with data from the S&P500
index.
We further employ a modiﬁed version of the truncated L´ evy ﬂight whose
characteristic function is inﬁnitely divisible, namely our exponentially damped
L´ evy ﬂight. Then we examine the multiscaling property of our distribution,
and illustrate it with the S&P500.
5 Acknowledgements
We acknowledge the referees for fruitful suggestions and comments. Iram Gle-
ria acknowledges support from FAPEAL.
5Fig. 1. (a) Behavior of the kurtosis for daily returns of the S&P 500 index (black
circles), and an IID process (triangles) (n =days). (b) Contribution of K2
n.
Fig. 2. (a) Estimated R(K,∆t) for several K. (b) Estimated multiscaling exponents
ξ.
Fig. 3. (a) Estimated ratios ln φL(K)  for several K. (b) Estimated multiscaling
exponents ζ.
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7

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Exponentially damped L´ evy ﬂights, multiscaling, and exchange rates, Physica A 333c

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The gradually truncated L´ evy ﬂight for systems with power-law distributions,

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