Crossover from antipersistent to persistent behavior in time series possessing the generalyzed dynamic scaling law

PHYSICAL REVIEW E 69, 036121 ~2004!

Crossover from antipersistent to persistent behavior in time series possessingthe generalyzed dynamic scaling law

Alexander S. Balankin,1,2 Oswaldo Morales Matamoros,1 Ernesto Ga´lvez M.,3 and Alfonso Pe´rez A.21Seccioń de Posgrado e Investigacioń, ESIME, Instituto Politećnico Nacional, Me´xico D.F. 07738, Mexico

2Instituto Mexicano de Petro´leo, Mexico D.F. 07730, Mexico3Departamento de Economıá, Universidad de Sonora, Sonora 83000, Mexico

~Received 8 October 2003; revised manuscript received 11 December 2003; published 31 March 2004!

The behavior of crude oil price volatility is analyzed within a conceptual framework of kinetic rougheningof growing interfaces. We find that the persistent long-horizon volatilities satisfy the Family-Viscek dynamicscaling ansatz, whereas the mean-reverting in time short horizon volatilities obey the generalized scaling lawwith continuously varying scaling exponents. Furthermore we find that the crossover from antipersistent topersistent behavior is accompanied by a change in the type of volatility distribution. These phenomena areattributed to the complex avalanche dynamics of crude oil markets and so a similar behavior may be observedin a wide variety of physical systems governed by avalanche dynamics.

DOI: 10.1103/PhysRevE.69.036121 PACS number~s!: 89.75.Da, 89.65.Gh, 05.40.2a, 05.45.Tp

itsng

e.ble

orer

r--

ri-

le

s

o

s

iogatce

nu-

is-

mesa-ns

s alexin

ex-icegnm-

cingismeto

udy

c-ri-gentc-

tal

I. INTRODUCTION AND BACKGROUND

Dynamics of all realistic complex systems always exhibsome part of randomness, either due to internal reasons,cific for nonlinear dynamical systems, or caused by exterstochastic noise. Examples include many physical, biolocal, computer, social, and economic systems@1#. These sys-tems commonly exhibit dynamic scaling invariance, i.their behavior does not change under rescaling of varia~for example, space and time! combined with an appropriatrescaling of the observables~Z! and the control~X,t! param-eters@2,3#. In such a case, the randomness of spatiotempbehavior may be characterized by the fluctuations of obsable parameters, defined as s(D,t)5^^@Z(X,t)2^Z(X,t)&D#2&D&R

1/2, where^¯&D denotes the spatial aveage within a window of sizeD and ^¯&R denotes the average over different realizations.

Typically, it is expected that the dynamic scaling invaance implies that fluctuationss(D,t) satisfy the celebratedFamily-Viscek dynamic scaling ansatz@4#

s~D,t !}tb f „D/jz~ t !… ~1!

wherej(t)}t1/z is the correlation length of the ‘‘space’’ scaand the scaling function behaves asf (y)}yH, if y!1, orf (y)'1, if y@1; hereH is the so-called local randomnes~or Hurst! exponent;z is the dynamic exponent, andb5H/z is the growth exponent@5#. The Hurst exponent@6#gives an indication of whether the system behavior is rand(H50.5) or displays persistence (0.5,H,1) or antipersis-tence (0<H,0.5) @4#.

The scaling form~1! is valid for a large variety of systemfar from equilibrium, as well as for critical phenomena@1#.Specifically, the Family-Viscek dynamic scaling ansatzcommonly applied to describe the kinetic rougheninggrowing interfaces@4#. However, generally, a simple scalinlaw ~1! does not hold@2,3#, instead there is an evidence ththe logarithms of the parameters can be used to produdata collapse@3#. In this way, Sittler and Hinrichsen@3# have

1063-651X/2004/69~3!/036121~7!/$22.50 69 0361

spe-ali-

,es

alv-

m

sf

a

suggested the general dynamic scaling form with contiously varying scaling exponents@7#

s~D,t !}F~DH~D,t !,tb~D,t !!, ~2!

whereF is the scaling function. Examples of systems dplaying the general scaling dynamics~2! include certain self-organized critical sand pile models@8#, ion sputtering of sur-faces @9#, and DLA-related growth processes@10#. Thecontinuously varied scaling exponents were found in soexperiments in turbulence@11#, in paper wetting experiment@12#, in numerical analysis of Kuramoto-Sivashinsky eqution @13#, and also were observed in Monte Carlo simulatioof ion etch-front roughening@14#.

The dynamics of financial markets has recently becomefocus of interest to physicists because of its rich and compscaling behavior analogous to that commonly observedphysical systems with many interacting units@15#. Many sta-tistical properties of financial markets have already beenplored, and have revealed striking similarities between prvolatility dynamics and the kinetic roughening of growininterfaces @16#. Accordingly, physical models have beeshown to have wide application to understanding the dynaics of stock markets@17#. This allows us to use the dynamiscaling approach to study the kinetic roughening of growinterfaces@18#, as well as the financial time series. In thway, we expect that the scaling analysis of economic tiseries might yield novel results, providing new insights indynamics of very different complex systems@19#. In thiswork, the general dynamic scaling approach is used to stthe scaling properties of crude oil market.

II. GENERALIZED SCALING DYNAMICS OF CRUDEOIL MARKET

The world oil is a capital-intensive environment charaterized by complex interactions deriving from the wide vaety of products, transportation-storage issues, and strinenvironmental regulation. Crude oil is the world’s most atively traded commodity, accounting for about 10% of to

©2004 The American Physical Society21-1

ynncre

re

tca

e

I

tingex-la-

in-

toy

of

ofgg

he

in-nt

es,pt

be-

er-soer

po

oi

y

98

BALANKIN et al. PHYSICAL REVIEW E 69, 036121 ~2004!

world trade@20#. The crude oil market is characterized bextremely high levels of price volatility. Fluctuations icrude oil price are caused by supply and demand imbalaarising from events such as wars, changes in politicalgimes, economic crises, formation breakdown of trade agments, unexpected weather patterns, etc.@21# At the sametime, many of the price forecasting models are based onbelief that historical price series exhibit some statistiproperties that permit to predict future price movements@22#.

A. Data analyzed

To quantify the scaling dynamics of crude oil market, wstudied the daily records of the spot prices@23# P(t) @seeFigs. 1 and 2~a!# and the price volatilitiesVn(t) @see Figs.2~b!–2~d!# from the West Texas Intermediate@24# ~WTI!crude oil price listings@25#. Specifically, we analyze the WT

FIG. 1. Time records of West Texas Intermediate crude oil sprice in the current~1! and in the 1983 constant~2! dollars perbarrel, $/bbl~source: Bloomberg database@25#!. The graphs corre-spond to the period from 30 December 1983 to 23 June 2003.

FIG. 2. ~a! Time record of West Texas Intermediate crudespot price in the 1983 constant dollars per barrel~$/bbl!; and ~b!–~d! realized price volatilities for the period of 4096 business dafor different horizons:~b! n53, ~c! n58, and~d! n518 businessdays. All time series correspond to the period from 15 October 1to 23 December 2002.

03612

es-

e-

hel

crude oil price in constant 1983 US dollars@26# over theperiod from 30 December 1984–23 June 2003 represen5181 observations~weekends and business holidays arecluded!. Then we construct 699 time series of realized votility @27#

Vn~ t !5@^P2~ t !&n2^P~ t !&n2#1/2 ~3!

of lengthT54096 business days~about 16 business years!,for different time horizonsn52,3,...,700~from 2 businessdays to about 3 business years!, wheret is the business timeand ^¯&n denotes the business time average within a wdow of sizen. In this study, all records of volatility@see Figs.2~b!–2~d!# correspond to the period from 15 October 198623 December 2002@28#. One can see that the price volatilitchanges from day to day in such a way that time seriesvolatilities Vn(t) realized at different time intervalsn looksimilar.

B. Scaling analysis techniques

To detect and quantify the dynamic scaling behaviorprice volatility within a framework of interface roughenindynamics, here the volatility horizonn is treated as an analoof time variable~t!, while the business timet is treated as ananalog of lateral extent~X! of growing interface. Accord-ingly, the price volatility fluctuations are characterized by tanalog of interface height fluctuations@4#, defined as

s~n,t!5^@Vn~ t !2^Vn~ t !&t#2&R

1/2}F~t Hn,nb~t!!, ~4!

where^¯&t denotes the business time average within a wdow of sizet and ^¯&R denotes the average over differerealizations.

To characterize the scaling properties of time seriwithin a framework of the general dynamic scaling conce~2!, ~4! the volatility growth exponentb~t! can be deter-mined from the scaling behavior

s~n,D!}nb~t! ~5!

for different intervals of business timet. To test the correla-tions in business time scale (t<T54096) we studied theautocorrelation functions@29#

C~t!5^P~ t1t!P~ t !&T /^P2~ t !&T

and

Cn~t!5^Vn~ t1t!Vn~ t !&T /^Vn2~ t !&T , ~6!

where the angle brackets denote the time average. Thehaviors of correlation function att→0 andt→T→` char-acterize the stochastic ‘‘memory’’ of the time series. Furthmore, the scaling behavior of the price volatility was alinvestigated by calculating the structure factor or powspectrum

Sn~v!5^V& n~v!V& n~2v!&}v~2Hn11!Fn~vn1/z~n,t!!, ~7!

t

l

s

6

1-2

nsen

he

s

tleol-al

al

t

hedart

r

thnp

Thsal

retinea

eath

h

onenas-l tr

thlinth

eith

is

theat

ticf

sce

n

CROSSOVER FROM ANTIPERSISTENT TO PERSISTENT . . . PHYSICAL REVIEW E 69, 036121 ~2004!

where V& n(v)5T21/2*T@Vn(t)2^Vn(t)&T#exp(ivt)dt is theFourier transform of the volatility record~¯! and F(y) isthe scaling function, such thatFn(y)}yH, if y!1, orFn(y)'1, if y@1. The precise form of the autocorrelatiofunction is, however, not known. This makes difficult to uEqs.~6!, ~7! for the quantitative measure of Hurst exponefor finite time series.

To quantify the intensity of long range correlations, tlocal randomness~Hurst! exponents ofP(t), as well as ofeach time recordVn(t), were determined by five methodadopted from theBENOIT 1.3 software @30#: the rescaled-range analysis, the roughness length, the variogramm,power-spectrum, and the wavelet methods. The rescarange analysis is one of the oldest and best-known methfor determiningH. This method was proposed by Mandebrot and Wallis@31# and is based on previous hydrologicstudies of Hurst@32#. The rescaled rangeR/s is defined asthe ratio of the maximal range of the integrated signal~R!normalized to the standard deviation~4!. For time seriescharacterized by long-range correlations the expected vof rescaled range scales asR/s}tH. If the time record pos-sesses only short-range correlations then the log-log ploR/s is also a straight line with slope 0.5. Lo@33#, however,showed that this method may be significantly biased wthere is short-term dependence in the form of heterosketicity or autocorrelation. The roughness-length and vaogramm methods are based on the scaling behavior ofstandard deviationsn}(t)Hn and the semivariance Va}t2H. Problems with these methods are that the choicethe sampling interval, as well as the determination ofslope of the plot, may affect the result of Hurst exponeestimation. Power spectral methods are based on power stral analysis, which can be applied to time series data.power spectral density function~7! for random data describethe data in terms of spectral density of its mean square vfor different frequencies and scales asSn}t22Hn21. TheFourier transform uses cosines, sins and exponentials toresent a time record, and so it is more useful for represenlinear functions and it is less suitable for analyzing nonlinchaotic time series. Wavelets offer an alternative methodanalyze complex time series@34#. The Wavelet method isbased on the property that Wavelet transforms of the saffine traces have self-affine properties. This method ispropriate for analysis of nonstationary series, i.e., wherevariance does not remain constant with increasing lengtthe data set. Wavelets are similar to, but an extensionFourier analysis and the wavelet transform is computatially similar in principle to the fast Fourier transform. Thaim of the wavelet transform is to express an input siginto a series of coefficients of specified ‘‘energy.’’ The dicrete numbers associated with each coefficient contain alinformation needed to completely describe the series pvided one knows which analyzing wavelet was used fordecomposition. Wavelet transforms makes use of scafunctions that have the property of being localized in botime and frequency. A scaling coefficientw}aHn11/2 charac-terizes and measures the width of a wavelet, wherea denotesa scale parameter@35#. The statistical distributions of crudoil price and realized price volatilities were analyzed w

03612

t

hed-ds

ue

of

ns-

i-he

ofetec-e

ue

p-grto

lf-p-e

ofof-

l

heo-eg

the help of@RISK software@36#, which ranks the fitted dis-tributions using three test statistics:x2, Anderson-Darling,and Kolmogorov-Smirnov statistics@37#.

C. Results and discussion

We find that the autocorrelation function@Fig. 3~a!# ofcrude oil price record@Fig. 2~a!# decays exponentially@seeinset in Fig. 3~a!# asC}exp(t/t0), with a characteristic timet05120 business days~about the half business year!. Fur-thermore, we find that the Hurst exponent of price recordequal toH50.5060.02 @see Figs. 3~b!–3~d! and 4# @38#.This means that there are no long-range correlations incrude oil price record. This is consistent with the finding ththe crude oil spot price distribution is a symmetric logisdistribution@see Fig. 5~a!#, which provides the best fitting odata according to three test statistics mentioned above.

At the same time, we find that the realized volatilitie@Fig. 2~b!–2~d!# possess a statistical self-affine invarian

FIG. 3. ~a! Autocorrelation function of price record shown iFig. 2~a! @the inset showsC(t) in semilog coordinates# and~b!–~d!fractal graphs of price record obtained by~b! the rescaled range,~c!the roughness length, and~d! the power spectrum methods~timeinterval length in business days!.

FIG. 4. ~a! Price record as in Fig. 2~a! and ~b!–~d! three firstwavelets~time scale in business days,HW50.50).

1-3

ieselyol

is

a

a-

-

a

as

rudeng

rtial

g

tic-y

orq

an

n

ental


within wide ranges of business time scale@3,t,tC(n)#characterized by well defined Hurst exponentHn for eachhorizonn @39# @see Figs. 6~a!–6~d!#:

Hn50.0621n, whenn<12 ~8!

and

Hn50.8360.04, whenn>18. ~9!

Our finding means that the long horizon realized volatilit(n.8) are persistent, i.e., volatility increments are positivcorrelated in business time, whereas the short-horizon vtilities (n,8) are antipersistent, i.e.,Vn,8(t) displays nega-tive autocorrelations in business time.

Furthermore, we find that the transition from antiperstent to persistent volatility atn58 is accompanied by anabrupt change in the behavior of minn$Vn(T54096)% versusn@Fig. 7~a!#, nevertheless the time-average and the standdeviations ofVn(t) have no anomaly atn58 @see Fig. 7~b!#.Specifically, the time averaged volatility behaves^Vn(t)&t54096}n0.5 up to n5700, while the standard deviation of realized volatility behaves ass(t5T54096)}n0.25

up ton518, but it scales ass}n0.5, when the realized volatility is characterized by the constant Hurst exponentH50.8360.04.

Moreover, we find that the growth exponent behaves@see Figs. 7~b!, 7~c!#

b50.25tC /t if t,tC~n! andb50.25 if t.tC~n!,

FIG. 5. ~a! Conditional probability distribution of crude oil pricein $/bbl ~bins: experimental data, solid lines: fitting by the logisdistribution!. ~b! Conditional probability distributions of price volatility for horizon n53 @bins: experimental data, solid lines fitting bthe Pearson distribution~15! ~p value is 0.41!, inset: the distributiontail in the semilog coordinates#. ~c! Conditional probability distri-butions of price volatility for horizonn518 @bins: experimentaldata, solid lines: fitting by the log-logistic distribution~17! ~p valueis 0.38!, inset: the distribution tail in the log-log coordinates#. ~d!Cumulative distribution of normalized avalanches in log-log codinates@circles: experimental data, solid lines: the graph of E~19!; the squared coefficient of correlation between the datatheoretical line is equal toR250.9549].

03612

a-

-

rd

s

s

whenn,18, ~10!

while b50.5 for anyt, whenn.18. ~11!

We also find that the interval of correlationstC in the busi-ness time scale increases with the horizon of volatilitytC}nz(n,t), where the dynamic exponent is a function ofnand t if n,18, while z5H/b51.66 for long horizonsn.18 @see Fig. 7~d!#.

Accordingly, we find that the long-horizon volatility (n>18) satisfies the Family-Viscek dynamic scaling ansatz~1!,whereas for horizons smaller than 18 business days the coil price volatility satisfies the generalized dynamic scalilaw @3#

s~n,t!}F••

„~t!H~n!,nb~t!…, ~12!

with continuously varying scaling exponents~8! and ~10!,which are quasihomogeneous functions satisfying the padifferential equation@7#

] ln~Hn!

] ln~n!52

] ln~b!

] ln~t!51. ~13!

This implies that scaling functionF possesses a local scalininvariance@3#, i.e.,

F„~lt!H~n!,~ln!b~t!…5lF••

„~t!H~n!,nb~t!…, ~14!

wherel is a dilatation parameter andn,18, t,tC .

-.d

FIG. 6. ~a!–~c! Fractal graphs of price volatility records showin Figs. 2~b!–2~c! obtained by~a! the roughness length,~b! thevariogram, and~c! the rescaled-range methods@numbers correspondto different horizons:~1! corresponds ton53, ~2! corresponds ton58, ~3! corresponds ton520, and~4! corresponds ton560 busi-ness days#. ~d! Horizon dependence of the Hurst exponent~valuesof Hn are averaged through five methods! in the semilog coordi-nates@time scale in business days, circles and squares: experimdata, solid line: data fitting by Eq.~8!, R250.9882].

1-4

viicoas

t-

-

it

ofion

icea-up

to-

bev-

ofof

yore,be-isu-y-

s-

romin a

dy-

s-s ofof

dingng

ulA-

lan

:

l


The crossover from antipersistent to persistent behaindicates the existence of intrinsic horizon scale of prvolatility ~see Ref.@40#!, nC'8. To get a deeper insight intthe price volatility dynamics, we also perform the statisticanalysis of volatilitiesVn(t) and avalanches defined aQ(n,t)5Vn11(t)2Vn(t).

We find that for short time horizonsn<8, the conditionalprobability of realized volatility is best fitted by the lightailed Pearson distribution@Fig. 5~b!#

f ~Vn!5~Vn2f!2~g11!

r2gG~g!expS Vn2f

r D ~15!

with horizon dependent parameters

r50.48n034~R250.95! and f50.13n20.69~R250.97!

~16!

andg53.6601; G~g! is the gamma function.At the same time, for the larger time horizons,n.8, the

conditional probability of realized volatility is the loglogistic distribution@Fig. 5~c!#:

FIG. 7. Results of the scaling analysis of crude oil price votility. ~a! The minimum of realized volatility versus time horizo@points: experimental data, lines: power law fits~1! min(Vn)51025n3.95, R250.97 and~2! min(Vn)50.014n0.66, R250.993].~b!The mean~1! and the standard deviation~2,3! of realized volatility(t54096) versus time lag,n @points: experimental data, linespower law fittings.~1! ^Vn(t54096)&t51.5n0.5, R250.999, ~2!s$Vn(t54096)%50.18n0.26, R250.995, and~3! s$Vn(t54096)%50.1n0.45, R250.998; they axis labels indicate the quantity#. ~c!The volatility growth exponentb versus normalized time intervafor volatility horizonsn53 ~1,2! andn560 ~3!; points: results ofanalysis, lines: fitting by Eqs.~10! and ~11!. ~d! Horizon depen-dence of dynamic exponent for price volatility~time scale in busi-ness days!.

03612

ore

l

f ~Vn!5g@~Vn2f!/r#g21

r$11@~Vn2f!/r#g%2 , ~17!

whereg52.5560.07 @41# and

r50.104n0.385~R250.998!, f50.0082n0.744~R250.996!;

~18!

i.e., the long-horizon volatility distribution is fat tailed, butis well outside the stable Le´vy range 0,g,2 @42#.

Furthermore, we find that the statistical distributionavalanches is also best fitted by the log-logistic distributwith f50, r50.75n20.68 andg51.4660.02 ~the p value is0.48!; obeying a power law tail@see Fig. 5~d!#:

F}12$Q/median@Q~t!#%21.46 ~19!

with the scaling exponentg within the stable Le´vy range@41#.

This indicates that observed behavior of crude oil prvolatility can be interpreted in terms of scale-free avlanches, which define the intrinsic horizon scale and buildlong-range correlations in the price volatility@43#. Such abehavior can be modeled with the use of KuramoSivashinsky equation@13# with white or correlated noise.

III. CONCLUSIONS

We show that the time series of price volatility cananalyzed within a framework of kinetic roughening of moing interface, treating the length of series as an analogspatial variable and the horizon of volatility as an analogtime variable. In this way we find that crude oil volatilitpossesses a general dynamic scaling behavior. Furthermwe observe a transition from antipersistent to persistenthavior of the volatility in the horizon scale. This transitionaccompanied by the change in the type of volatility distribtion, which is light-tailed for short horizons and it is heavtailed for long horizons.

Our findings have potentially wide-ranging implicationin econophysics@44#, as well as in statistical physics of complex systems. Specifically, we expect that the crossover fantipersistent to persistent behavior should be observedwide variety of systems displaying generalized scalingnamics with continuously varying exponents@45#. The exis-tence of a ‘‘universal’’ mechanism which gives rise to crosover from antipersistent to persistent behavior in systemdifferent nature could provide a new insight to the physicscomplex systems governed by avalanche dynamics leato generalized scaling dynamics with continuously varyiexponents.

ACKNOWLEDGMENTS

The authors would like to thank J. M. Lopez for usefdiscussions. This work has been supported by the CONCYT of the Mexican Government~Project No. 3495-U!.

-

1-5

ed

ai

linerisb

apf

ngdth

tt

ev

tt.

v.

ic

ev

.

n,

g,

n-.

.

en

hys.

ail-sics

forpotted

e--

her

x-

eter.ar-

rd

09

.

ovewe


@1# P. Bak, How Nature Works: The Science of Self-OrganizCriticality ~Copernicus, New York, 1996!.

@2# J. J. Ramasco, J. M. Lo´pez, and M. A. Rodrı´guez, Phys. Rev.Lett. 84, 2199~2000!.

@3# L. Sittler and H. Hinrichsen, J. Phys. A35, 10 531~2002!.@4# A.-L. Barabasi and H. E. Stanley,Fractal Concepts in Surface

Growth ~Cambridge University Press, Cambridge, 1995!.@5# In the absence of any characteristic scale the fluctuations

expected to be self-affine. However, commonly the dynamscaling is characterized by three or more independent scaexponents. Specifically, according to the concept of gendynamic scaling@4#, the interface roughening dynamicscharacterized by six scaling exponents, four of which mayindependent.

@6# The Hurst exponent is related to the fractal dimension of grZ(X) asDF5d2H, whered is the topological dimension oembedded space@4#.

@7# It should be pointed out that the functional form of scaliexponents cannot be chosen freely; rather their functionalpendence is constrained by a group homomorphism linkingconcepts of generalized and ordinary scaling@3#.

@8# C. Tebaldi, M. De Menech, and A. L. Stella, Phys. Rev. Le83, 3952~1999!.

@9# J. T. Drotar, Y.-P. Zhao, T. M. Lu, and G.-C. Wang, Phys. RE 59, 177 ~1999!.

@10# B. Hinnemann, H. Hinrichsen, and D. Wolf, Phys. Rev. Le87, 135701–135704~2001!.

@11# X.-Z. Wu, L. Kadanoff, A. Libchaber, and M. Sano, Phys. ReLett. 64, 2140~1999!.

@12# A. S. Balankin and D. Morales Matamoros, inEmergent Na-ture, 1st ed., edited by M. M. Novak~World Scientific, Lon-don, 2001!, Vol. 1, pp. 345–356.

@13# J. Buceta, J. M. Pastor, M. A. Rubio, and F. J. Rubia, PhysD 113, 166 ~1988!.

@14# J. T. Drotar, Y.-P. Zhao, M.-T. Lu, and G.-C. Wang, Phys. RB 61, 3012~2000!.

@15# R. N. Mantegna and H. E. Stanley, Nature~London! 376, 45~1995!; J. P. Sethna, K. A. Dahmen, and Ch. R. Myers,ibid.410, 242 ~2001!; Y. Lee, L. A. Nunes Amaral, D. Canning, MMeyer, and H. E. Stanley, Phys. Rev. Lett.81, 3275~1998!; V.Plerou, P. Gopikrishnan, B. Rosenow, L. A. Nunes Amaral, aH. E. Stanley,ibid. 83, 1471 ~1999!; R. Friedrich, J. Peinkeand Ch. Renner,ibid. 84, 5224~2000!; V. Plerou, P. Gopikrish-nan, and H. E. Stanley, Nature~London! 421, 130 ~2003!.

@16# Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, Ch.-K. Penand H. E. Stanley, Phys. Rev. E60, 1390~1999!; V. Plerou, P.Gopikrishnan, L. A. Nunes Amaral, M. Meyer, and H. E. Staley, ibid. 60, 6519 ~1999!; P. Gopikrishnan, V. Plerou, L. ANunes Amaral, M. Meyer, and H. E. Stanley,ibid. 60, 5305~1999!; V. Plerou, P. Gopikrishnan, L. A. N. Amaral, XGabaix, and H. E. Stanley,ibid. 62, R3023~2000!; V. Plerou,P. Gopikrishnan, X. Gabaix, and H. E. Stanley,ibid. 66,027104~2002!; K. Matia, L. A. N. Amaral, S. P. Goodwin, andH. E. Stanley, ibid. 66, 045103 ~2002!; X. Gabalx, P.Gopikrishnan, P. Plerou, and H. E. Stanley, Nature~London!423, 267 ~2003!.

@17# B. B. Mandelbrot,Fractals and Scaling in Finance~Springer,New York, 1997!; J. P. Bouchaud and M. Potters,Theory ofFinancial Risk: From Statistical Physics to Risk Managem

03612

recg

ic

e

h

e-e

.

.

a

.

d

t

~Cambridge University Press, Cambridge, 2000!.@18# A. S. Balankin, O. Susarrey, and J. Marquez Gonzales, P

Rev. Lett.90, 096101~2003!; A. S. Balankin, D. Morales, andI. Campos, Philos. Mag. Lett.80, 165 ~2000!; A. S. Balankin,A. Bravo Ortega, and D. Morales Matamoros,ibid. 80, 503~2000!; A. S. Balankin and D. Morales Matamoros,ibid. 81,495 ~2001!.

@19# There is a huge amount of high frequency economic data avable, which can be treated with methods developed in phyof complex systems.

@20# S. Galina-Hidalgo, D. Romo, and A. Pe´rez, J. Energy Devel-opment28, 57–68~2002!.

@21# M. C. Lynch, J. Energy Development28, 107 ~2002!.@22# T. J. Considine and H. Eunnyeong, Energy Economics22, 527

~2000!.@23# In the cash market, purchases and sales of the commodity

immediate delivery occur at a price that is referred as the ‘‘sprice,’’ in contrast to the forward and future prices, associawith forward and future contracts, respectively.

@24# Crude oil prices are slightly different worldwide, however btween prices markers~Brent, Dubai, and West Texas Intermediate! are not large and they correlate closely which each ot@22,25#.

@25# URL: http://www.bloomberg.com@26# Crude oil has been traded on the New York Mercantile E

change~NYMEX ! since 1983.@27# Although it is common to talk about the price volatility, ther

is no single universally accepted definition of this parameVarious measures of price volatility can be constructed, in pticular the absolute values of price~or logarithms of price!returns, the standard deviation of price returns, etc.@5–7#. Infinance, it is typically characterized in terms of the standadeviation of prices at a particular time scale@see, for example,G. Bomer, Active Trader2, 86 ~2003!#.

@28# We also studied records corresponding to the period fromApril 1987 to 23 June 2003.

@29# K. Falconer,Techniques in Fractal Geometry~Wiley, NewYork, 1997!.

@30# W. Seffens, Science285, 1228~2000!.@31# B. B. Mandelbrot and J. R. Wallis, Water Resour. Res.4, 909

~1969!.@32# H. E. Hurst, Trans. Am. Soc. Civ. Eng.116, 770 ~1951!.@33# A. W. Lo, Econometrica59, 1279~1991!.@34# Z. R. Struzik, Physica A296, 307 ~2001!.@35# The mother wavelet inBENOIT 1.3 software is a step function@36# @RISK 4.5, URL http://palisade.com@37# W. J. Conover,Practical Nonparametric Statistics, 2nd ed.

~Wiley, New York, 1980!.@38# This value was determined by five methods mentioned ab

and it is not sensitive to the sampling period. Moreover,also determine the global scaling exponenta ~see Ref.@2#!from the scaling behaviorR5^maxtPT P(t)2mintPT P(t)&R}Ta

and find thata5H50.5060.03.@39# Different methods lead to slightly different values ofHn ; for

example, forn58 we obtainHR/s50.467,Hs50.508,HVar

50.492,HS50.477, andHW50.565, and soH850.5060.04@see also Figs. 3~b!–3~d!, 4, and 6~a!–6~c!#.

@40# Wen-Tau Juan and I. Lin, Phys. Rev. Lett.80, 3073~1998!.

1-6

r

tri

mto

is

inCrrcnt-

s,

la-p-

h-ye-

fromowsedenot


@41# We find thatg is independent of the horizon of volatility fotime lagsn.8.

@42# It should be pointed out that the parameters of volatility disbutions, as well as the scaling exponents in Eqs.~1! and ~14!,are not sensitive to the sampling period. Essentially the saresults were also obtained for the period from 9 April 198723 June 2003.

@43# Our findings suggest that the dynamics of price volatilitygoverned by self-organized criticality~SOC!, similar to thatwhich is commonly illustrated conceptually with avalanchesa pile of sand grains. The most essential feature of the SOthat the system jumps from one metastable state to anotheavalanche dynamics up to critical state without external foing. Accordingly, the price volatility evolves through transiestates at time horizonsn,18, which are not critical, to a dy

03612

-

e

isby-

namical attractor poised at criticality for long-time horizonwhen volatility obeys the Family-Viscek dynamic scaling~1!and could therefore be predictable in statistical sense.

@44# Accurate modeling and forecasting of commodity price votility is of paramount importance in risk management and otion pricing.

@45# We note that Monte Carlo simulations of reactive ion etcfront roughening@9# described by the Kuramoto-Sivashinskequation predict the generalized scaling dynamics with timdependent roughness exponent and continuous transitionantipersistent to persistent front roughness regime. This alla possibility to use a similar model for simulations of realizand employed volatility of crude oil prices. Unfortunately, thcrossover from antipersistent to persistent roughness wasdiscussed in Ref.@9#.

1-7

Documents

Crossover from antipersistent to persistent behavior in time series possessing the generalyzed dynamic scaling law