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A nonextensive approach to the dynamics of financial observables S´ılvio M. Duarte Queirós,∗ Luis G. Moyano,† and Jeferson de Souza‡ Centro Brasileiro de Pesquisas F´ısicas, 150, 22290-180, Rio de Janeiro - RJ, Brazil Constantino Tsallis§ Santa Fe Institute, 1399 Hyde Park Road, Santa Fe - NM, USA and Centro Brasileiro de Pesquisas F´ısicas, 150, 22290-180, Rio de Janeiro - RJ, Brazil 6 (Dated: February 2, 2008) 0 We present results about financial market observables, specifically returns and traded volumes. 0 They are obtained within the current nonextensive statistical mechanical framework based on the 2 an entropy Sq = k1−1iWP−=1qpqi (q∈ℜ) (S1 ≡ SBG = −kiPW=1pi lnpi). More precisely, we present stochas- J tic dynamical mechanisms which mimic probability density functions empirically observed. These mechanismsprovidepossible interpretations fortheemergence of theentropicindices q in thetime 1 3 evolution of the corresponding observables. In addition to this, through multi-fractal analysis of return time series, we verify that the dual relation qstat+qsens = 2 is numerically satisfied, qstat ] and qsens being associated to the probability density function and to the sensitivity to initial con- n ditions respectively. This type of simple relation, whose understanding remains ellusive, has been a empirically verified in various other systems. - a t a I. INTRODUCTION where p is the probability of the microscopic configura- d i tion i [1]. The Boltzmann principle should be derivable . s In recent years statistical mechanics has enlarged its frommicroscopicdynamics,sinceitreferstomicroscopic c i originalassignment: the applicationof statistics to large states, but the implementation of such calculation has ys systems whose states are governed by some Hamilto- not been yet achieved. So, Boltzmann-Gibbs (BG) sta- h nianfunctional[1]. Itscapabilityforrelatingmicroscopic tisticalmechanicsis stillbasedonhypothesissuchasthe p statesofindividualconstituentsofasystemtoitsmacro- molecularchaos[4]andergodicity[5]. Inspiteofthelack [ scopic properties are nowadays used ubiquitously [2]. of an actual fundamental derivation, BG statistics has Certainly,themostimportantoftheseconnectionsstillis been undoubtedly successfulin the treatmentof systems 1 v the determination of thermodynamic properties through in which short spatio/temporal interactions dominate. 2 the correspondence between the entropy concept, origi- For such cases, ergodicity and (quasi-) independence are 2 nallyintroducedbyRudolfJulius EmmanuelClausiusin favoured and Khinchin’s approach to SBG is valid [5]. 2 1865 [3], and the number of allowed microscopic states, Therefore, it is entirely feasible that other physical en- 1 introduced by Ludwig Boltzmann around 1877 when he tropies, in addition to the BG one, can be defined in 0 was studying the approach to equilibrium of an ideal ordertoproperlytreatanomaloussystems,forwhichthe 6 gas [4]. This connection can be expressed as simplifyinghypothesisofergodicityand/orindependence 0 are not fulfilled. Examples are: metastable states in / s S =k lnW, (1) long-rangeinteracting Hamiltonian dynamics, metaequi- c libriumstatesinsmallsystems(i.e.,systemswhosenum- i where k is a positive constant, and W is the number s ber of particles is much smaller than Avogrado’s num- y of microstates compatible with the macroscopic state of ber),glassysystems,sometypesofdissipativedynamics, h an isolated system. This equation, known as Boltzmann p principle,isoneofthecornerstonesofstandardstatistical and other systems that in some way violate ergodicity. v: mechanics. Thisincludessystemswithnon-Markovianmemory(i.e., long-rangememory),likeitseemstobethecaseoffinan- i When the system is not isolated, but instead in con- X cial ones. Generically speaking, systems that may have tact to some large reservoir, it is possible to extend Eq. r (1),undersomeassumptions,andobtaintheBoltzmann- a multi-fractal, scale-free or hierarchicalstructure in the a occupancy of their phase space. Gibbs entropy Inspired by this kind of systems it was proposed in W 1988 the entropy [6] S =−k p lnp , (2) BG i i Xi=1 1− W pq i S =k iP=1 (q ∈ℜ), (3) q q−1 ∗e-mailaddress: [email protected] whichgeneralisesS (lim S =S ),asthebasisof †e-mailaddress: [email protected] BG q→1 q BG ‡e-mailaddress: [email protected] apossiblegeneralisationofBGstatisticalmechanics[7,9]. §e-mailaddress: [email protected] The value of the entropic index q for a specific system 2 is to be determined a priori from microscopic dynamics. JustlikeS ,S isnonnegative,concave,experimentally BG q [p(x)]q robust (or Lesche-stable [10]) (∀q >0), and leads to a (x−µ¯ )2 dx≡ (x−µ¯ )2 =σ¯2, (8) finiteentropy production per unit time [2,11]. Moreover, Z q [p(x)]qdx D q Eq q it has been recently shown [12] that it is also extensive, R correspondingtothegeneralised meanandvarianceofx, i.e., respectively [9]. N From the variational problem using (5) under the Sq(A1+A2+...+AN)≃ Sq(Ai), (4) above constraints, we obtain Xi=1 1 for special kinds of correlated systems, more precisely p(x)=A 1+(q−1)B (x−µ¯ )2 1−q , (q <3), q q q when the phase-space is occupied in a scale-invariant h i (9) form. By being extensive, for an appropriate value of where, q, S complies with Clausius’ concept on macroscopic q entropy, and with thermodynamics. sevSeirnaclerietssupltrsoipnosbaolt,henfutrnodpaym(e3n)tahlaasnbdeeanpptlhieedsopuhrycseicosf, A = ΓΓ[[2512−−−−23qqqq]]q1−πqBq ⇐ q <1 , (10) as well as in other scientific areas such as biology, chem- q  Γ[q−11] q−1B ⇐ q >1 istry, economics, geophysics and medicine [13]. Herein, Γ[23q−−q2]q π q we both review and present some new results concern-  and ing applications to the dynamics of financial market observables, namely the price fluctuations and traded vol- B = (3−q) σ¯2 −1. (11) q q umes. Specifically, we will introduce stochastic dynam- (cid:2) (cid:3) ical mechanisms which are able to reproduce some fea- Standard and generalised variances, σ¯2 and σ¯2 respec- q tures of quantities such as the probability density func- tively, are related by tions (PDFs) and the Kramer-Moyal moments. More- over, we will present some results concerning the return 5−3q σ¯2 =σ¯2 . (12) multi-fractal structure, and its relation to sensitivity to q 3−q initial conditions. Our dynamical proposals will be faced to empirical Defining the q-exponential function as analysis of 1 minute returns and traded volumes of the 30 companies that were used to compose the Dow Jones ex ≡[1+(1−q) x]1−1q (ex ≡ex), (13) q 1 Industrial Average (DJ30) between the 1st July and the 31st December 2004. In order to eliminate specious be- (exq =0 if 1+(1−q)x≤0) we can rewrite PDF (9) as haviourswe have removedthe well-knownintra-daypat- tern following a standard procedure [8]. After that, the p(x)=A e−Bq(x−µ¯q)2, (14) q q returnvalueswere subtractedfromits averagevalue and expressedinstandarddeviationunits,whereasthetraded hereafter referred to as q-Gaussian. volumes are expressed in mean traded volume units. For q = 3+m, the q-Gaussian form recovers the Stu- 1+m dent’s t-distribution with m degrees of freedom (m = 1,2,3,...) with finite moment up to order mth. So, II. VARIATIONAL PRINCIPLE USING THE for q > 1, PDF (14) presents an asymptotic power- ENTROPY Sq law behaviour. On the other hand, if q = n−4 with n−2 n = 3,4,5,..., p(x) recovers the r-distribution with n Before dealing with specific financial problems, let us degrees of freedom. Consistently, for q < 1, p(x) has analyse the probability density function which emerges a compact support which is defined by the condition when the variational principle is applied to Sq [9]. |x−µ¯ |≤ 3−q σ¯2. Let us consider its continuous version, i.e., q q1−q q 1− [p(x)]q dx S =k . (5) q R1−q III. APPLICATION TO MACROSCOPIC OBSERVABLES The natural constraints in the maximisation of (5) are A. Model for price changes p(x) dx=1, (6) Z TheGaussiandistribution,recoveredinthelimitq →1 corresponding to normalisation, and of expression (14), can be derived from various stand- [p(x)]q points. Besides the variational principle, it has been de- x dx≡hxi =µ¯ , (7) Z [p(x)]qdx q q rived, through dynamical arguments, by L. Bachelier in R 3 his1900workonpricechangesinParisstockmarket[14], Itisthisfactwhichallowedtheestablishmentofanalogies and also by A. Einstein in his 1905 article on Brownian (currently used in financial mimicry) between financial motion [15]. In particular, starting from a Langevin dy- markets dynamics and fluid turbulence [20]. namics, we are able to write the corresponding Fokker- Itisnoteworthythateq. (16)isstatisticallyequivalent Planck equation and, from it, to obtain as solution the to [18, 21] Gaussian distribution. Analogously, it is also possible, from certain classes of stochastic differential equations dr =−krdt+Aq(t) dWt+(q−1)Bq(r,t) dWt′, (20) and their associated Fokker-Planck equations, to obtain i.e., a stochastic differential equation with independent the distribution given by Eq. (14). additive and multiplicative noises. If eq. (15) allows an Inthissection,wewilldiscussadynamicalmechanism immediate heuristic relation between q and the response forreturns,r,whichisbasedonaLangevin-likeequation of the system to its own dynamics, eq. (20) permits thatleadstoaPDF(q-Gaussian)withasymptoticpower- a straighforward dynamical relation between q and the law behaviour [16, 17]. This equation is expressed as magnitude of multiplicative noise in sucha waythat, for q = 1, the Langevin equation is recovered as well as the dr =−krdt+ θ [p(r,t)](1−q)dW (q ≥1), (15) t Gaussian distribution. q InFig.(1)wepresentthetypicalPDFforthe1minute (inItoˆ convention)whereW is aregularWiener process t returnsofacompanyconstituentoftheDowJonesIndus- and p(r,t) is the intantaneous return PDF. In a return trialAverage30(upperpanel)presentingq =1.31±0.02, contextthedeterministictermofeq. (15)intendstorep- a time series generated by eq. (15) (middle panel), and resent internal mechanisms which tend to keep the mar- the U-shaped 2nd Kramers-Moyal moment for our data ket in some averagereturn or, in a analogousinterpreta- (lowerpanel). Asitcanbeseentheaccordanceusingthe tion, can be related to the eternal competition between simplestapproachisalreadyquite nice. Upgradesofthis speculative price and the actual worth of a company. In model can be obtained by taking into account the risk- our case, we use the simplest approach and write it as aversion effects, which induce asymmetry on the PDF, a restoring force, with a constant k, similar to the vis- and correlations on the volatility in a way which differs cousforceinthe regularLangevinequation. Inregardto from others previously proposed. The formulation pre- thestochasticterm,itaimstoreproducethemicroscopic sented herein has also the advantage of being aplicable response of the system to the return: θ is the volatility to systems which are not in a stationary state since the constant(intimately associatedtothe varianceofp(r,t)) time-dependent solutions of the Fokker-Planck equation and q, the nonextensive index, reflects the magnitude of are of the q-Gaussian type as well. that response. Since the largestunstabilities in the mar- ketareintroducedbythemostunexpectedreturnvalues, it is plausible that the stochastic term in Eq. (15) can B. Model for traded volumes have such inverse dependence on the PDF p(r,t). Fur- thermore, Eq. (15) presents a dynamical multiplicative Changes in the price of a certain equity are naturally noise structure given by, dependent on transactions ofthat equity and thus on its r(t)=Z t e−k(t−t′)qθ[p(r,t)](1−q) dWt′, (16) ttoratidcedpovwoelur-mlaew, vb.ehParveivouiorusofstturaddieesdpvroolvuemdethPeDaFsy[m22p]-, −∞ later extended for all values of v [23]. In this case it was where we have assumed r(−∞)=0. shownthatthetradedvolumePDFisverywelldescribed The associated Fokker-Planck equation to Eq. (15) is by the following ansatz distribution given by 1 v ρ v ∂p(r,t) = ∂ [krp(r,t)]+ 1 ∂2 θ [p(r,t)](2−q) , P (v)= Z (cid:18)ϕ(cid:19) expq(cid:18)−ϕ(cid:19), (21) ∂t ∂r 2∂r2 n o (17) where v represents the traded volume expressed in its and the long-term probability density function is[16, 18, mean value unit hVi, i.e., v = V/hVi, ρ and ϕ are pa- 19], rameters, and Z = ∞ v ρexp −v dv. 0 ϕ q ϕ 1 kr2 1−1q The probability dRensi(cid:16)ty(cid:17)function(cid:16)(21)(cid:17)was recently ob- p(r)= 1−(1−q) . (18) Z (cid:20) (2−q) Zq−1θ(cid:21) tained from a mesoscopic dynamical scenario [24] based inthefollowingmultiplicativenoisestochasticdifferential One of the most interesting features of eq. (15) is its equation aptitude to reproduce the celebrated U-shape of the 2nd (i.e., n=2) Kramers-Moyalmoment ω γ dv =−γ(v− )dt+ 2 vdW , (22) α r α t M (r,t,τ)= (r′−r)n P (r′,t+τ|r,t) dr′ ≈τθ [p(r,t)](1−q) . n Z where Wt is a regular Wiener process following a nor- (19) mal distribution, and v ≥ 0. The right-hand side terms 4 FIG. 2: Symbols represent the average correlation function forthe30timeseriesanalysedandthelinerepresentsadouble exponentialfit with characteristic times of T =13 and T = 1 2 332 yielding a ratio about 25 between the two time scales Eq. (24) (R2 = 0.991, χ2 = 9×10−6, and time is expressed in minutes). inverted Gamma stationary distribution: 1 v −α−2 ω f(v)= exp − . (23) ωΓ[α+1] ω v (cid:16) (cid:17) h i Consider now, that instead of being a constant, ω is a time dependent quantity which evolves on a time scale T larger than the time scale of order γ−1 required by eq. (22) to reach stationarity [25, 26]. This time dependence is, in the present model, associated to changes in thevolumeofactivity(numberoftradersthatperformed transactions) and empirically justified through the analysis of the self-correlationfunction for returns. In Fig. 2 wehaveverifiedthatthecorrelationfunctionisverywell described by C[v(t),v(t+τ)]=C e−τ/T1 +C e−τ/T2 (24) 1 2 with T = 332 ≫ T = 13. In other words, there is FIG. 1: Upper panel: Probability density function vs. r. 2 1 first a fast decay of C[v(t),v(t+τ)], related to local Symbols correspond to an average over the 30 equities used equilibrium, and then a much slower decay for larger τ. tobuiltDJ30andthelinerepresentsthePDFobtainedfrom Thisconstitutesanecessaryconditionfortheapplication a time series generated by eq. (16) which is presented on middle panel. Lower panel: 2nd Kramers-Moyal moment of a superstatistical model [25]. M ≈ τθ [p(r)](1−q) = τ k (5−3q)σ2+(q−1)r2 from If we assume that ω follows a Gamma PDF, i.e., 2 2−q which k parameter is obtained(cid:2)and where the stationa(cid:3)ry hy- pothesis is assumed (t0 = −∞ ≪ −k−1 ≪ 0). Parameter P(ω)= 1 ω δ−1exp −ω , (25) values: τ = 1min, k = 2.40±0.04, σ = 0.930±0.08 and λΓ[δ](cid:16)λ(cid:17) h λi q=1.31±0.02. then, the long-term distribution of v will be given by p(v)= f(v) P (ω) dω. This results in R 1 v −α−2 θ p(v)= exp − , (26) Z (cid:16)θ(cid:17) q(cid:20) v(cid:21) of eq. (22) represent inherent mechanisms of the system in order to keep v close to some “normal” value, where λ = θ(q−1), δ = q−11 −α−1. Bearing in mind ω/α, and to mimic microscopic effects on the evolution that, for q >1, of v, like a multiplicative noise commonly used in inter- 1 minigttFenotkkperro-Pcelsasnecsk. eTqhuiastdioynna[m18i]c,s,leaanddttohethceorfroelslopwonindg- xae−q xb =(cid:20)q−b 1(cid:21)q−1 xa−q−11 eq−b/(qx−1)2, (27) 5 Analogously, it was recently conjectured [28] that, for systemswhichcanbestudiedwithinnonextensivestatis- tical mechanics, the energy probability density function (associated to stationarity or (meta) equilibrium), the sensitivity to the initial conditions, and the relaxation would be described by three entropic indices q , q , stat sens and q , referred to as the q-triplet. The first physi- rel cal corroboration of such scenario has been made from the analysis of two sets of daily averages of the mag- netic field strength observed by Voyager 1 in the solar wind [29]. Others systems are currently on study (e.g., [30]). Ofcourse,if the system is non Hamiltonian, it has noenergydistribution, henceq cannotdefined inthis stat manner. We may howeverestimate it through a stationary state generalised Gaussian (which would generalise the Maxwellian distribution of velocities for a BG system in thermal equilibrium). In contrast, the other two indices, q and q , remain defined in the usual way. sens rel Let us focus now on the multi-fractal structure of return time series. It has been first conjectured, and later proved,foravarietyofnonextensiveone-dimensionalsys- tems, that the following relation holds [31]: 1 1 1 = − , (28) 1−q h h sens min max where h and h are respectively the minimal and FIG. 3: Upper panel: Excerpt of the time series generated min max maximal h-values of the associated multifractal spec- by our dynamical mechanism (simulation) to replicate 1 min trum f(h). In fig. 4 we depict the multifractal spec- traded volume of Citigroup stocks at NYSE (data). Lower trumof 1minute tradedvolumes,obtainedby the appli- panel: 1 min traded volume of Citigroup stocks probability densityfunctionvs. tradedvolume. Symbolsarefordata,and cation of the MF-DFA5 method [32]; h and f(h) have solidlineforthereplica. Parametervalues: θ=0.212±0.003, been obtained from averages of the empirical data of 30 ρ = 1.35 ± 0.02, and q = 1.15 ± 0.02 (χ2 = 3.6 × 10−4, companies. Through this analysis, we have determined R2 =0.994). h = 0.28± 0.04 and h = 0.83± 0.04. The use min max of Eq. (28) yields q = 0.58±0.10. Considering that sens the q value obtained for the return probability density wecanredefineourparametersandobtaintheq-Gamma function was q =1.31±0.02, we verify that the dual stat PDF (21). relation InFig.3wepresentacomparationbetweenthetraded volume of Citigroup (2004 world’s number one company q +q =2 (29) stat sens [27]) stocks, as well as a replica of that time series ob- tainedusingthisdynamicalproposal. Asitcanbeeasily isapproximatelysatisfiedwithintheerrorintervals. Tak- verified, the agreement is remarkable. ingintoaccountthe well-knownfastdecayofreturnself- correlations, we see that the price changes for a typical DJ30 stock may be essentially described by the q-triplet IV. THE NONEXTENSIVE q-TRIPLET AND {q ,q ,q }={0.58±0.10,1.31±0.02,1}. sen stat rel FINANCIAL OBSERVABLES Systems characterised by Boltzmann-Gibbs statistical V. FINAL REMARKS mechanicspresentthefollowingcharacteristics: (i)Their PDF for energies is proportional to an exponential func- In this article we have presented a nonextensive sta- tioninthepresenceofathermostat;(ii)Theyhavestrong tistical mechanics approach to the dynamics of finan- sensitivity to the initial conditions, i.e., this quantity in- cial markets observables, specifically the return and the creases exponentially with time (currently referred to as traded volume. With this approach we have been able strong chaos), being characterised by a positive maxi- to present mesoscopic dynamical interpretations for the mum Lyapunov exponent; (iii) They typically present, emergenceofthe entropicindexq frequentlyobtainedby for basic macroscopic quantities, an exponential decay a numerical adjustment for data PDF of eqs. (18) and with some relaxation time. In other words, these three (21). For the case of returns, q is related to the reaction behaviours exhibit exponential functions (i.e., q = 1). degree of the agents on the marketto fluctuations of the 6 the stationary state, the sensitivity to initial conditions, and the relaxation for nonextensive systems. The com- plete understanding of these connections remains ellusive. Forinstance,concerningrelaxationandtheq-triplet conjecture, a new question arise for price changes. It is well-known that the self-correlation for returns is of exponential kind, in contrast with the long-lasting correlations for the volatility (or returns magnitude) [33]. The latter is also considereda stylisedfact and it is compati- blewithaq-exponentialform. Inthisway,iftheefficient markethypothesisisconsideredthekeyelementinfinan- cialmarkets,thenitmakessensetoassumeq =1. But, rel if arbitrage on markets is considered as the fundamental feature instead, then the essentialrelaxationto be taken FIG.4: Multi-fractalspectrumf(h)vs. hfor1minutereturn averaged over the 30 equities with h = 0.28±0.04 and intoaccountmightbetheonerelatedtothevolatility,for min hmax =0.83±0.04. which qrel > 1. Progress is clearly still needed, at both the fundamentalandappliedlevels,inorderto achievea deep understanding of this complex system. observable,whileforthe caseoftradedvolumeitisasso- ciated to fluctuations on the (local) average traded volume. Along with these dynamical scenarios, and based We thank Olsen Data Services for the data provided on the multi-fractal nature of returns, we have verified and used herein. Financial support from PRONEX and thatthisquantityappearstoapproximativelysatisfythe CNPq (Brazilian agencies), FCT/MCES (Portuguese dual relation, q +q = 2, previously conjectured agency),andSIInternationalandAFRL(USAagencies), stat sens within the emergenceofthe q-tripletwhichcharacterises is also acknowledged. [1] K. Huang, Statistical Mechanics, (John Wiley & Sons, (2006), in press [cond-mat/0411073]; C. Tsallis, Milan NewYork, 1963). Journal of Mathematics 73, 145 (2005); C. Tsallis, M. [2] M. Gell-Mann and C. Tsallis (eds.), Nonextensive En- Gell-MannandY.Sato,Proc.Natl.Acad.Sci.USA102, tropy-InterdisciplinaryApplications,(OxfordUniversity 15377(2005);C.Tsallis, M.Gell-MannandY.Sato,Eu- Press, NewYork, 2004). rophysics News 36, 186 (2005). [3] E. Fermi, Thermodynamics, (Doubleday, New York, [13] Nonextensive Statistical Mechanics and its Applications, 1936). edited by S. Abe and Y. Okamoto, Lecture Notes in [4] L. Boltzmann, Lectures on Gas Theory, (Dover, New Physics 560 (Springer-Verlag, Heidelberg, 2001); Non- York,1995). Extensive Thermodynamics and Physical Applications, [5] A.I. Khinchin, Mathematical Foundations of Informa- edited by G. Kaniadakis, M. Lissia, and A. Rapisarda tion Theory (Dover,New York,1957) and Mathematical [Physica A 305 (2002)]; Anomalous Distributions, Non- Foundations of Satistical Mechanics (Dover, New York, linear Dynamics and Nonextensivity, edited by H. L. 1960). Swinney and C. Tsallis [Physica D 193 (2004)]; Nonex- [6] C. Tsallis, J. Stat. Phys. 52, 479 (1988). Bibliography tensive Entropy - Interdisciplinary Applications, edited URL:http://tsallis.cat.cbpf.br/biblio.htm. by M. Gell-Mann and C. Tsallis (Oxford University [7] E.M.F.CuradoandC.Tsallis,J.Phys.A24,L69(1991); Press, New York, 2004); Complexity, Metastability and Corrigenda: 24, 3187 (1991) and 25, 1019 (1992). Nonextensivity,editedbyC.Beck,G.Benedek,A.Rapis- [8] A.AdmatiandP.Pfleiderer,Rev.Fin.Stud.1,3(1988); arda and C. Tsallis (World Scientific, Singapore, 2005); Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Nonextensive statistical mechanics: newtrends, new per- Peng and H.E. Stanley,Phys. Rev.E 60, 1390 (1999). pectives, edited by J.P. Boon and C. Tsallis, Europhys. [9] C. Tsallis , R.S. Mendes and A.R. Plastino, Physica A News 36 (6) (2005). 261, 534 (1998). [14] L. Bachelier, Théorie de la spéculation, Ann. Sci. Ećole [10] B. Lesche, J. Stat.Phys. 27, 419 (1982) Norm. Sup.III-17, 21 (1900). [11] V. Latora and M. Baranger, Phys. Rev. Lett. 273, 97 [15] A. Einstein, Ann.der Phys. 17, 549 (1905). (1999). [16] L. Borland, Phys. Rev.E 57, 6634 (1998). [12] C.Tsallis,inComplexity,MetastabilityandNonextensiv- [17] S.M. Duarte Queirós, Quantit. Finance 5, 475 (2005); ity,editedbyC.Beck,G.Benedek,A.RapisardaandC. S.M. DuarteQueirós, working paper CBPF-NF-027/05. Tsallis (World Scientific, Singapore, 2005), page 13; Y. [18] H.Risken,The Fokker-Planck Equation: Methods of So- Sato and C. Tsallis, Proceedings of the Summer School lution and Applications, 2nd edition (Springer-Verlag, and Conference on Complexity (Patras and Olympia, Berlin, 1989). July 2004), edited by T. Bountis, G. Casati and I. Pro- [19] A.R. Plastino, A. Plastino, Physica A 222, 347 (1995); caccia, International Journal of Bifurcation and Chaos C.Tsallis,D.J.Bukman,Phys.Rev.E54,R2197(1996). 7 [20] S.Ghashghaie, W.Breymann,J.Peinke,P.Talknerand [26] S.M. DuarteQueirós, Europhys.Lett. 71, 339 (2005). Y.Dodge, Nature381, (1996) 767. [27] http://www.forbes.com [21] C. Anteneodo and C. Tsallis, J. Math. Phys. 44, 5194 [28] C. Tsallis, Physica A 340, 1 (2004). (2003). [29] L.F.BurlagaandA.F.-Vinãs,PhysicaA356,375(2005). [22] P.Gopikrishnan,V.Plerou,X.GabaixandH.E.Stanley, [30] B.M.BoghosianandE.P.Borges,privatecommunication Phys.Rev.E 62, R4493 (2000). (2005). [23] R. Osorio, L. Borland and C. Tsallis, in Nonextensive [31] M.L.LyraandC.Tsallis,Phys.Rev.Lett.80,53(1997). Entropy - Interdisciplinary Applications, M. Gell-Mann [32] J.W.Kantelhardt,S.A.Zschiegner,E.Koscielny-Bunde, andC.Tsallis(eds.)(OxfordUniversityPress,NewYork, S.Havlin,A.BundeandH.E.Stanley,PhysicaA316,87 2004). (2002). [24] J. de Souza, L.G. Moyano and S.M. Duarte [33] H.E. Stanley, L.A.N. Amaral, P. Gopikrishnan and V. Queirós, Eur. Phys. J. B (in press, 2006), preprint Plerou, Physica A 283, 31 (2000) arXiv:physics/0510112. [25] C. Beck and E.G.D. Cohen, Physica A 322, 267 (2003).