Possibilities and Limitations of Deterministic Nonlinear Dynamic Model of the Stock Market

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1 Possibiliies nd Limiions of Deerminisic Nonliner Dynmic Model of he Sock Mrke ANDREY DMITRIEV, VITALY SILCHEV, VICTOR DMITRIEV School of Business Informics Nionl Reserch Universiy Higher School of Economics 33 Kirichny Sr., Moscow h:// bi.hse.ru/en/ Absrc: - This er rooses nonliner dynmicl model of sock mrke. Dynmicl vribles of he model re he vriion of sk nd bid rice relive o equilibrium vlues nd difference beween numbers of mrke gens in -se nd -se. A riculr mrke gen being in -se hs mximum moun of vluble informion bou finncil sse nd hs minimum informion being in -se. This model exlins he imossibiliy of exisence of n equilibrium se of he mrke, shows he resence of deerminisic chos in sock mrke nd frcl finncil ime series. The resuls of he nonliner dynmicl nlysis nd sisicl nlysis of he emiricl finncil ime series re resened. We show he resuls of nonliner nlysis for he model s n oen nonequilibrium sysem, s well s comrison wih emiricl resuls. Key-Words: - sock mrke model, sock mrke indexes, deerminisic chos, finncil ime series, correlion dimension, frcl dimension, 1/f noise, long memory, q-gussin disribuion. 1 Inroducion From he second hlf of he XX cenury he generl rend of science develomen is he enerion of he ides nd mehods of hysics ino oher nurl nd humnirin discilines. Mehods of hysicl modelling re ofen used in sciences such s demogrhy, sociology nd linguisics. An inerdiscilinry reserch field, known s econohysics, ws formed in he middle 199s s n roch o solve vrious roblems in economics, such s unceriny or sochsic rocesses nd nonliner dynmics, by lying heories nd mehods originlly develoed by hysiciss. The erm econohysics ws coined by H. Eugene Snley in order o describe he lrge number of ers wrien by hysiciss in he roblems of (sock nd oher) mrkes (for econohysics reviews see refs. [1-4]). Curren se of heoreicl economics llows one o effecively use dvnced mehods of hysicomhemicl modelling for economicl sysem. A remrkble exmle is lying nonliner dynmics o nlysis of finncil ime series [5,6]. Moreover, in 1963 Benoi B. Mndelbro[7] during his reserch of coon rices found ou h he rices follows scled disribuion in ime. Th discovery origined new roch in mrke reserch clled frcl mrke nlysis [8]. A sysemic reserch of deerminisic chos in finncil mrkes sred from works of Rober Svi [9]. By he end of 2 h cenury here were formed wo lines of reserch of deerminisic chos in finncil mrkes. The firs one is reled o discovery nd nlysis of deerminisic chos in he srucure of finncil mrkes. Sudies of h kind re usully bsed on quliive chrcerisic nd quniive mesures of chos [1], nd heir resuls show conclusively h deerminisic chos exiss in finncil mrkes [11-19]. The second line conneced o rerievl of exlici form of such dynmicl sysems. Definiely, consrucion of such models is fr more comlex roblem. Th is why he number of relevn ublicions on his oic is relively smll. The mos comrehensive survey of mhemicl models of finncil mrkes cn be found in he book of R.J. Ellio nd P.E. Ko [2]. Alhough he book nd oher relevn ublicions conin numerous conceul models, we hve no found ny econohysicl model of sock mrke h cn exlin is fundmenl funcioning mechnisms. Thus, he urose of his work is building of econohysicl model of sock mrke using rllels beween mrke funcioning nd hysicl rinciles of lser oerion nd is defining ossibiliies nd limiions of deerminisic nonliner dynmic model of he sock mrke. E-ISSN: Volume 14, 217

2 This er is orgnized s follows. In secion 2 we resen he simle econohysicl model of sock mrke s n oen nonequilibrium sysem, including he definiion of he dynmic vribles nd rmeers, nd he resuls nd discussions, including he definiion of he regulr nd he choic ses of he sock mrke. In secion 3 we resen he resuls of nonliner dynmicl nd sisicl nlysis of emiricl finncil ime series (sock mrke indexes). In secion 4 we conclude his er, defining ossibiliies nd limiions of deerminisic nonliner dynmic model of he sock mrke. 2 Consrucion of he Model 2.1 Clss of hermodynmicl sysems In generl, heoreicl economics consider he roblem of consrucing mhemicl conces for modeling of economic rocesses, which consis of numerous inernl elemens nd incorores severl influenil forces. Therefore, heoreicl economics rovide bsic ides, roosiions nd mehods s ground for creing mhemicl model of economic dynmics h clerly reflecs he evoluion of comlex economic sysems ccording o secific economicl rinciles. The key roery of such sysems is synergy h mkes oneself eviden in he fc h he whole sysem obins some chrcerisics h is elemens do no originlly hve. Alhough he economic dynmics is new nd develoing field of sudy, i origines from nurl sciences where he mehods of dynmicl modeling hve lredy been successfully lied o vriey of roblems. However, mhemicl conces, which work fine in elecrodynmics or hysics of liquids nd gs, cnno be lied o economic objecs s is. On he conrry, he bes resuls cn be chieved only when he model is creed wih due considerion of he roeries of he objec being modeled. In hysics, here is clss of he models h ws consruced ccording o he guidelines exlined bove. These sysems re clled hermodynmicl sysems. Originlly, he sysems of his clss were develoed for modeling ggreged gs nd fluid chrcerisics like emerure or ressure, bu bsed on chrcerisics of gs (or fluid) ricles, which re simly single molecules h do no hve hese chrcerisics. Th is why we do no roose o use lredy buil hysicl models of he box, bu o use he sme rinciles in consrucing cusom economic dynmicl models. Comlex hermodynmicl sysems re he sysems h consis of numerous elemens, similr o ech oher ( oms ). So, hese oms inercion follows some deermined rules. As resul, he evoluion of he whole sysem deends in comliced wy from he evoluion of every om. In hermodynmicl sysems, here re hree levels of deil: Locl micro-dynmics S : his level inercion of n om wih ohers is considered. I is n onogeneic level of he sysem, which forms ll oher dynmicl effecs. Meso-dynmics S 1 : his convolued level considers he verged moion chrcerisics of ech om. Mcro-dynmics S 2 : observed indicors s funcion of ggreged sysem ses from wo revious levels. Le us denoe n observed vrible wih index α he momen of observion T s x α. Therefore, he ule of simulneously observed vribles which secify he globl mcro se of he sysem is 1, 2 x x x,..., x α,.... (1) ( ) Assume h every single om i in Λ belongs o bsic se of sysem elemens nd hs is own sce of inernl ses. We denoe he inernl se i of elemen i ; i= 1,..., N he momen s ω. i By ghering ll ω, we hve hse micro ses for he whole sysem he momen : 1, 2,..., N ω ω ω ω. (2) ( ) Imorn remrk: here we consider big, bu finie sysems, so he se of oms Λ is lrge enough. Le Φ be he union of ll ossible mcro ses nd Ω union of ll micro ses. Φ Xω, Ω. (3) Then Ω is he underlying sce of he sysem. I is worh o noice, h Ω is n bsrc mhemicl consruc, since, in generl, one cnno observe he micro ses of he sysem. So, one cn only mke n ssumion on Ω nd on kind of inercion beween he oms, from which he bsrc sochsic dynmics of micro level is derived. Furher, he globl dynmics on mcro level cn be reconsruced by lying he convoluion echniques o micro-dynmicl E-ISSN: Volume 14, 217

3 informion of he sysem, which is yicl sk of sochsic dynmics. I is fc h every sochsic dynmicl sysem being wihou exernl influence ends o he neres locl equilibrium se, boh on micro nd mcro levels. The rocess of sysem rnsiion o equilibrium is clled relxion. From he reserch rnsiionl rocesses in big sochsic dynmicl sysems, he effec of self orgnizion nd creion of ordered srucures ws discovered. The necessry condiions for new dynmicl informion re srong deviion from equilibrium ses nd nonliner inercion beween he elemens. This effec is clled synergy, nd i works in vrious comlex hermodynmicl sysems. Therefore, synergy shows iself in economics due o he fc, h he big economicl sysems like globl mrkes cn lso be modeled wih his kind of dynmicl sysems. Bck o he micro level of he sysem, ssume, i i i h for every om Λ, A = ω is n inernl i hse sce for. Then i i i A= ξ, A, i= 1,2,..., N. (4) ( ) is srucure of he micro level. I includes ll sysem oms nd heir ossible inernl ses. Moving furher, le us consider he microdynmics of he sysem. Agin, ssume h momen he sysem is in se ω (2). Then Lˆ 1 Lˆ f 1 Lˆ f f 1 2 ω ω... ω ω (5) is generlized rjecory of sysem evoluion. Here he rjecory srs wih iniil sysem se ω f ime = nd reches he se ω ime = f. In his model, locl micro movemens k 1 Lˆ k k ω ω re deermined by sochsic dynmicl oerors L ˆk. Thus, he roblem of finding he micro dynmics of he sysem reduces o definiion of such oerors. In generl, for comlex dynmicl sysems, oerors L ˆk re nonliner nd no locl. Moreover, hey hve funcionl deendence on he revious sysem ses or, in oher words, re funcions wih memory. Due o hese chrcerisics, hese funcions enlce he rjecories of ossible sysem evoluions. Therefore, he srucure S ( ALξ, ˆ ( )) (6) describes he globl informion dynmics in he sysem. Lˆ ˆ 1 1 L k f f f k = ω ω... ω ω (7) k= ϕ 1 1 ϕ k f f f k = i i... i i k= ω ω ω ω (8) Equions (7) nd (8) reresen he segmens of globl sysem rjecory nd rjecories of single oms resecively. Here we inroduce he evoluion oerors ( ) Tˆ, ω : = f h urn he sysem from iniil se ω ino ω. Finlly, he se of such oerors form he informionl nd meril flows in he sysem. Then, king his se of evoluion oerors wih nd sysem hse sce Ω, we ge he Liouville flows. The Liouville flows is well known mhemicl bsrcion, h llows us o connec he micro level of he sysem wih is meso- nd mcro-levels, by convolving nd ggreging dynmicl informion. The key conce of he Liouville flows is robbiliy densiy funcion of he sysem ρ ( x,) being in some se ner he oin 1 1 N N x x = ω,..., x = ω some momen of ime ( ) T. Assume, h dx N N = dx i= 1 i is n elemen of volume in vrible sce x. Then N dw ( x, ) = ρ ( x, ) dx (9) is he robbiliy h sysem will be in some se ner he oin x. To convolve he informion in funcion ρ ( x,) we inegre i using ll vribles exce x s. As resul, we hve n omic se funcion for om s 1 ϕs( x, ) = ρ( xs, ) dxs K, (1) where 1 K is normlizing consn. Thus, funcions ϕ s ( x, ) become he generlized vribles h describe sysem se on meso-dynmicl level. In fc, hey re he resul of ggreged influence on elemen s from he res of he sysem. Le us ϕ x, s denoe he se of omic se funcion s ( ) Φ( x,). Togeher omic se funcions defines mesodynmics of he sysem. (, ) = F( (, ) ) Φ x Φ x (11) The equions (11) in hermodynmicl sysems re clled kineic equions. Nex se is o move from meso-dynmics o mcro dynmics by convolving ggreged omic se funcions ino ggreged globl sysem se i f E-ISSN: Volume 14, 217

4 funcions nd binding hem wih observed indicors. No need o sy, h hese globl sysem se funcions deend in riculr wy on globl dynmicl vribles, which, in urn, re defined by omic se funcions of meso-dynmicl level. 2.2 Model ssumions Sock mrke is mcroscoic sysem Assume h he sock mrke is dynmicl sysem h consiss of numerous mrke gens (invesors) ( N >> 1) (fig. 1). Modeling of such sysems does no require deiled nlysis of inercions beween he gens on he micro-level. For mcrosysem descriion we use mcroscoic rmeers nd dynmicl vribles of he sysem. For mcroscoic dynmicl vribles we hve chosen ggreged flows of sk nd bid rice chnges nd dynmicl difference of mrke gens in secific ses. A riculr mrke gen being in -se hs mximum moun of vluble informion bou finncil sse ( I ) nd hs minimum informion ( I ) being in -se. The gen being in -se is ble o genere locl demnd on del wih he sse nd send n sk-qunum o oher gens. If he gen is in -se (he/she does no hve enough vluble informion bou he sse), hen he gen's rionl decision is do no genere demnd on del ( bid-qunum ). Moreover, for he gen in - se genering of del offer deends on he gen's recion on received sk-qunum (fig. 2) or cn be his or her own decision (fig. 2b). Generl ern in sock mrkes is h locl sk wves ( qun ) induce locl bid wves ( qun ). Figure 2. Agens genere «sk-qun» (red) nd «bid-qun» (green). (а) Forsed generion of n «bid-qunum». (b) Sonneous generion of n «sk-qunum». Figure 1. The sock mrke s nonequilibrium oen sysem Sock mrke is oin uonomous dynmicl sysem xφ = x β,, (12) ( ) m where β R is m -dimensionl vecor of d rmeers, x = x. d This semen goes wihou sying, s i deends only on chosen modelling roch nd objecives. However, he choice of (12) s he bse model is mde for reson. I is bsed on ess h he consruced mhemicl model grees wih emiricl (observed) d, for which we used vilble finncil ime series of sk/bid sock rices Every mrke gen cn be in one of wo ossible ses: cive ( -se) or ssive ( - se) Sock mrke is nonequilibrium oen sysem Indeed, sock mrke is n oen sysem h coninuously exchnges informion nd money flows wih he exernl world. Sources of exernl informion include corore finncil reors, finncil news feeds, sock-icker d nd ohers. This informion flow, in some sense, um u he sock mrke, mking inverse oulion of mrke gens: N >> N, where N is he number of gens being in -se, N is he number of gens being in -se. Wih cceble ccurcy, he disribuion of number of gens by heir ses cn be reresened s follows: I I N = N ex, (13) θ where θ is verge inensiy of sochsic inercions beween mrke gens Simle nlysis of equion (13) llows o idenify wo mcroscoic ses of he mrke: sble equilibrium se nd nonequilibrium se. If I I >> θ, hen E-ISSN: Volume 14, 217

5 N << N. In his cse, he sysem is in sble equilibrium se. Oherwise, if I I >> θ, hen N >> N. This cse corresonds o nonequilibrium se of he sysem. Tking ino ccoun coninuous informion uming, sock mrke is lwys funcioning in nonequilibrium se, mking vlnches of sk nd bid qun. Due o informion uming, he equilibrium se is lmos unrechble. I is crucilly imorn, h exisence of choic ses is fundmenl roery of nonequilibrium oen sysems [24, 25]. 2.3 Dynmicl vribles of he model nd relionshi beween hem Dynmicl vribles of he models Le us define dynmicl vribles in equion (12) for consrucing nonliner dynmicl model of sock mrke: x eq is he vriion of sk rice ( ) relive o equilibrium vlue ( eq ) is he sk rice in equilibrium se); y b b eq is he vriion of bid rice ( b ) relive o equilibrium vlue ( b eq ) is he bid rice in equilibrium se); z N N insnneous difference ( ) ( ) beween numbers of gens in -se nd - se. The choice of hese dynmicl vribles resonds o ossibiliy o es wheher he consruced dynmicl sysem grees wih emiricl d, for which we hve chosen vilble ime series of sk nd bid rices from rel sock mrkes. However, i is imossible o comre he hird dynmicl vrible wih cul d, due o vilble dses does no conin hese vlues. Thus, he fi es cn be erformed only wih wo dynmicl vribles. Le us esblish connecions beween dynmicl vribles nd heir chnge res Ask rice dynmics Vriion re of he sk rices is defined by concurrency of wo fcors: re decrese due o mrke relxion ( α x ) nd re increse due o growing vriion of bid rices ( + β y ): x = αx + βy (14) Term α x in (3) is necessry due o relxion of nonequilibrium sysem. According o Le Chelier's rincile [26], when he sysem equilibrium is subjeced o chnge by exernl force, hen he sysem redjuss iself o counerc (rilly) he effec of he lied chnge. Indeed, wihou erm + β y he equion (14) hs he following form: x = α x. (15) A soluion of differenil equion (15) is funcion of form x = Aex( α). Therefore, eq when (sock mrke end o sble equilibrium). In equion (15) α relxion rmeer, reled o relxion ime ( τ 1 ) ccording o: α = 1 τ1. Term + β y in (14) refers o he fc, h increse of bid rice vriion leds o increse of vriion re of sk rices Bid rice dynmics Vriion re of he bid rices is defined by concurrency of wo fcors: re decrese due o mrke relxion ( γ y ) nd re increse due o + cxz. y = γ y + cxz (16) Presence of he firs erm in (16) is exlined by Le Chelier's rincile. Term + cxz is exlined s follows: bid qunum, on which every mrke gen recs considering sk qun flow, is roorionl o sk rice vriion nd deends on he gen s curren se ( -se or -se) Dynmics of difference beween numbers of mrke gens in -se nd -se z = ε ( I z) + kxy (17) Agin, erm ε z is in equion (17) due o Le Chelier's rincile. Prmeer I refers o inensiy of exernl informion uming, so insnneous difference beween numbers of gens in -se nd -se grows wih increse of I. Term + kxy reresens he ower h he ggreged sk rice vriion sends on creion of he ggreged bid rice vriion. 2.4 Modeling resuls nd heir inerreion E-ISSN: Volume 14, 217

6 The sysem of differenil equions (14), (16) и (17) reresens he well-known Lorenz Hken equions [27]: x = αx + βy y = γ y + cxz (18) z = ε ( I z) + kxy Sysem (18) is one of he mos sudied 3- dimensionl dynmicl sysems. Generl roeries of (18) re resened in works [28, 29]. Le us consider (18) s sysem wih one conrol rmeer I. From chnging conrol rmeer s vlue, we cn mke wo imorn conclusions bou sysem (7). If βc αγ 1 nd < I < 1, hen eq, b b N N s. In cse eq nd ( ) ( ) of relively smll inensiy of exernl informion uming, he sock mrke ends o sble equilibrium (fig. 3). However, rciclly his sble equilibrium se cnno be reched, since he mrke is n oen sysem wih ermnen eernl informion uming. Figure 3. Asymoiclly sbiliy soluion of (7). If βc αγ 1 nd I 28, hen he sock mrke funcions s n oen nonequilibrium sysem wih deerminisic chos (fig. 4). I is worh o menion h such behviour is yicl for finncil mrke wih considerbly inense exernl informion. Figure 4. Choic soluion of (7). 3-dimensionl dynmicl model (7) exlins some roeries of he sock mrke funcioning such s frcliy (frcl dimension ( D F ) equls 1.497), choic nure (correlion dimension ( D C ) equls 1.896) nd bsence of memory (Hurs exonen ( H ) equls.528) of finncil ime series (FTS) [28]. Generlly, dynmicl sysem cn be defined s σ in follows [8]: is se ime is oin ( ) some E -dimensionl hse sce R, nd he rjecory beween he imes nd + is deermined by rules in which does no ener exlicily. Ech oin in hse sce cn be ken s he iniil se S ( ) =, nd is followed by n orbi defined by he S( ), >. A dynmicl sysem is sid o hve n rcor if E here exiss roer subse A R, such h S nd lrge enough, lmos ll sring oins ( ) S( ) is close o some oin of A. An rcor which hs frcl (no ineger) dimension is clled srnge rcor. So, he erm choic dynmicl sysem usully refers o dynmicl sysem wih srnge rcor. However, someimes he word choic my be used jus s oosie o sochsic, in order o emhsize he difference beween rndom nd deermined sysems. The wekness of his model lies in significn discrency beween emiricl nd heoreicl rjecories of FTS. Moreover, i is imossible o fi heoreicl rjecories o observed d by vrying conrol rmeers (in rnge of choic se) of he dynmicl sysem. The dynmicl sysem (7) hs 3 equilibrium oins for ny vlues of conrol rmeers in rnge of choic se. Therefore, heoreicl robbiliy densiy funcion (PDF) is hree-modl disribuion (hree mxim of he PDF) (fig. 5) nd x is whie rndom rocess. This PDF is no f-iled disribuion [3]. The whie noise signl is no signl of csrohic evens. E E-ISSN: Volume 14, 217

7 Figure 5. Probbiliy densiy funcion of he heoreicl FTS. 3 Anlysis of Emiricl Finncil Time Series For nlysis of emiricl FTS, we chose he following sock mrke indexes: 3Y T-Bond INT Res (1), ASX Ausrli (2), BOVESPA Brzil (3), CAC 4 (4), DAX 3 (5), DJ Indusril (6), FTSE 1 (7), ISEX Sin (8), NASDAQ Com (9), Nikkei 225 (1), S&P 5 (11), Shnghi Comosie (12), USD Index (13), VIX Voliliy Index (14). 3.1 Nonliner nlysis The nonliner nlysis ws conduced for ll chosen TTS. Such mesures s correlion dimension ( DC ), Hurs exonens ( H ), ower of ower secrl densiy ( β = 2H + 1) nd frcl dimension ( D F ) were clculed (ble 1). Sock mrke index Tble 1. Resuls of nonliner FTS nlysis D C H β D F (1) (2) (3) (4) (5) (6) (7) (8) (9) (1) (11) (12) (13) (14) Deerminion of he correlion dimension [31] for suosed choic rocess direcly from exerimenl ime series is ofen used o ge informion bou he nure of he underlying dynmics (see, for exmle, conribuions in ref. [32]). In riculr, such nlysis hs been mde o suor he hyohesis h he ime series re genered from he inherenly low-dimensionl choic rocess [32]. The geomery of choic rcors cn be comlex nd difficul o describe. I is herefore useful o undersnd quniive chrcerizions of such geomericl objecs. One of hese chrcerizions is D C. D C of he rcor of dynmicl sysem cn be esimed using he Grssberger Proccci lgorihm [31]. D C hs severl dvnges in comrison o he oher dimensionl mesures: 1. if D C is finie, hen FTS is choic ime series (genered by dynmicl sysem); 2. if D C is infinie, hen FTS is sochsic ime series (genered by urely rndom rocess). For clculion of D F we used he lgorihm described in er [33]. If D F > D T ( D T is oologicl dimension of he FTS, h equls 1 for ll ime series), hen he FTS is rndom frcl. A vlue of H = 2 DF llows o give noise clssificion (1 f -clssificion, where f is signl frequency) of he FTS [34]: 1. if < H <.5, hen he FTS is chrcerized by ni-ersisence (he ime series chnges he endency more ofen, hn series of indeenden rndom vribles) nd reresens rocess wih 1 f noise or ink noise; 2. if.5 < H < 1, hen he FTS is chrcerized by ersisence (he ime series is chrcerized by he effec of he long memory nd hs n inclinion o follow he rends) nd reresens rocess wih 1 f β ( β > 2 ) noise or blck noise; 3. if H =.5, hen FTS reresens rocess wih he bsence of memory or whie noise. Emiricl FTS is blck rndom rocess. Mos of he ime series, which cn be observed in exisence, cn usully be reled o one of he bove-menioned clsses [35,36]. Thus, he ime series observed in urbulence rocesses, show he bes correlion wih he ink noise. The blck noises cn be regisered in floods, solr civiy, sisics of he nurl nd induced csrohes. The blck noise indices long erm ersisence nd long memory. There is simle scling relion, connecing β nd H [34]: β = 2H + 1. The resuls for β re shown in he ble Sisicl nlysis E-ISSN: Volume 14, 217

8 Poin nd inervl esimes of he rmeers of emiricl PDF re he quesions of sisicl nlysis. The PDFs of reurns nd voliliies of mny finncil ime series hve ower low ils [37]: ( 1+ γ ) ( x) ~ x, γ >. (19) The PDF (8) belongs o he clss of he ower lw PDFs or he f-iled PDFs. The fundmenl difference of he PDF (19) from he comc disribuions is he fc, h hose evens, which fll on he disribuion il re, ke lce no so rrely o be negleced. Figure 6 rovides PDF for emiricl NASDAQ ime series of reurns. The PDFs of oher sock mrke indexes hve similr form. Visully, hese PDFs corresond o he Gussin (comc disribuion) or o he generlized Gussin f-iled PDF (19), for exmle, he q- Gussin disribuion [38-4]: χ 2 ρ( x) = ex q ( χx ), (2) Cq 1 ( 1 ) where exq ( x ) 1 ( 1 q ) x q +, C q normlizion fcor. Figure 6. PDF of he NASDAQ ime series of reurns. The disribuion (2) is wo-rmeer generlizion ( q < 3 is she rmeer, χ > is re rmeer) of he one-rmeer Gussin disribuion. In he limi q 1, PDF (2) recovers he usul Gussin disribuion, so q 1 indices derure from Gussin sisics. For q > 1, he ils of q- Gussin decrese s ower lws [4], 2 ( q 1 ρ ) ( x) ~ x. (21) Tble 2 conins he esimed vlues for rmeers of PDF (2) obined by he mximum likelihood mehod [41]. Tble 2. Poin nd inervl esimions of he PDF (2) rmeers nd γ-rmeers of he PDF (19) Sock mrke q χ γ index (1) 2.55± ± (2) 2.675± ± (3) 2.734± ± (4) 2.35± ± (5) 2.63± ± (6) 2.532± ± (7) 2.665± ± (8) 2.689± ± (9) 2.432± ± (1) 2.638± ± (11) 2.531± ± (12) 2.682± ± (13) 2.331± ± (14) 2.834± ± Thus, ccording o he oin nd inervl vlues of he PDF (2), shown in he ble 1, he following conclusion cn be mde: PDF is f-iled PDF (( 1+ γ )-vlues vry from 1.91 o 1.533). q -Gussin disribuion kes lce by he mximizion of he Tsllis enroy [42] considering definie limiions. Tsllis enroy s nonddiive generlizion of he Bolzmnn Gibbs enroy hs he following form: 1 q q Tq = 1 i q 1. (22) i= 1 The robbiliy i = Ni N( ε ) cn be esimed in much he sme wy s h one used in he Renyi enroy: N i is number of sysem elemens for he i -elemen of he ε -riion; N ( ε ) is full number of elemens of he given ε -cover. In conrs o ll enroy yes, he Tsllis enroy is nonddiive. Being lied o he finncil mrke (such s, for exmle, sock mrke) i gives ossibiliy o correcly describe finncil mrke, where ny mrke gen inercs no only wih he neres mrke gen or severl neres mrke gens, bu lso wih he whole mrke or some of is rs. Besides, from (22) i follows h T q is concve by q > nd convex by q <. Thus, he enroy descriion of he sock mrke, bsed on Tsllis sisics is rorie for sudying of evoluion of he sock mrke h conins lrge number of mrke gens who inerc wih ech oher in riculr wy nd secificlly every mrke gen cn inerc no only wih his or her neres neighbors bu lso wih remoe mrke gens (for exmle, see refs [43,44]). The biliy of he sysem o hve long memory for is s, nd he biliy of he sysem E-ISSN: Volume 14, 217

9 elemens o feel ech oher wide-r, cn be considered by he emergen behvior. A he sisicl level he emergen behviors re usully reled o he long-rnge ime nd sce correlions. The mer concerns he ime-series wih he long memory, or hose ime-series, uocorrelion funcion (ACF) of which decreses slowly. The exression long-rnge deendence, which is someimes used o refer o noise, hs lso been used in he oher conexs wih somewh differen menings. Long memory nd oher vrins re lso someimes used in he sme wy. Figure 7 rovides he ACF of heoreicl FTS. Figure 7. ACF of he heoreicl FTS. Figure 8 rovides he ACF of NASDAQ FTS. The ACFs of oher sock mrke indexes hve similr form. Figure 7. ACF of he NASDAQ FTS. Thus, heoreicl FTS reresens rocess wih he bsence of memory. The emiricl FTS reresens rocesses wih he long memory. 4 Conclusion The consruced simle econohysicl model of sock mrke s n oen nonequilibrium sysem llows o exlin he following henomen: 1. Unrelizbiliy of equilibrium se of he mrke s well. 2. Aernce of deerminisic chos in he mrke ( D C is finie). 3. There exiss frcl finncil ime series ( DF > DT ). These henomen re exlined only by quniive chrcerisics of exernl informion uming s conrol rmeer of he sysem. However, his simle model cnno exlin severl oher imorn henomen, such s hevy-iled disribuion of finncil ime series ((1+ γ )-vlues vry from 1.91 o 1.533, see ble 2); finncil ime series s blck rndom rocess ( 1 f β noise, where β vries from 2.44 o 2.292, see ble 1); finncil ime series s rndom rocess wih long memory (see fig. 7). We suose, h exlnion of hevy-iled disribuion of finncil ime series requires modificion of dynmicl sysem by inroducing he noise of secific kind (such s rmeric noise, for exmle, inensiy of exernl informion uming is rndom vrible wih q -exonenil disribuion). Priculrly, he form of ime series wih hevyiled disribuion cn be chieved by including ower-lw mulilicive noise [43-45], wh is he subjec of our furher reserch. References: [1] Chkrbori, A., Toke, I., Prirc, V., Abergel, F., Econohysics Review: I. Emiricl Fcs Quniive Finnce, Qun. Fin., Vol. 11, 211, [2] Chkrbori, A., Toke, I., Prirc, V., Abergel, F., Econohysics Review: II. Agenbsed Models, Qun. Fin., Vol. 11, 211, [3] Richmond, P., Mimkes, J., Huzler, S., Econohysics nd Physicl Economics, Oxford Universiy Press, 213. [4] Svoiu, G., Econohysics. Bckground nd Alicions in Economics, Finnce, nd Sociohysics, Elsevier, 213. [5] Hsieh, D.A., Chos nd Nonliner Dynmics: Alicion o Finncil Mrkes, J. Fin., Vol. 46, 1991, [6] Smll, M., Tse, C.K., Deerminism in Finncil Time Series, Sud. Nonlin. Dyn. Econom., Vol. 7, 23, [7] Mndelbro, B.B., The Vriion of Cerin Seculive Prices, J. Bus. Univ. Chicgo, Vol. 36, 1963, [8] Hudson, R.L., Mndelbro, B.B., The (Mis)Behvior of Mrkes: A Frcl View of Risk, Ruin, nd Rewrd, Bsic Books, New York, 24. [9] Svi, R., When Rndom is no Rndom: An Inroducion o Chos in Mrke Prices, J. Fu. Mrk., Vol. 8, 1988, [1] Li, Y.C., Ye, N., Recen Develomens in Choic Time Series Anlysis, In. J. Bif. Chos, Vol. 13, 23, E-ISSN: Volume 14, 217

10 [11] Murry, F., Sengos, T., Mesuring he Srngeness of Gold nd Silver Res of Reurn, Rev. Econom. Sud., Vol. 56, 1989, [12] Blnk, S., Chos in Fuures mrkes? A Nonliner Dynmicl Anlysis, J. Fu. Mrk., Vol. 11, 1991, [13] Decoser, G.P., Lbys, W.C., Michell, D.W., Evidence of Chos in Commodiy Fuures Prices, J. Fu. Mrk., Vol. 12, 1992, [14] Abhynkr, A., Coelnd, L.S., Wong, W., Nonliner Dynmics in Rel-Time Equiy Mrke Indices: Evidence from he Unied Kingdom, Econom. J., Vol. 15, 1995, [15] Andreou, A.S., Pvlides, G., Kryinos, A., Nonliner Time-Series Anlysis of he Greek Exchnge-Re Mrke, In. J. Bif. Chos, Vol. 1, 2, [16] Pns, E., Ninni, V., Are Oil Mrkes Choic? A Non-Liner Dynmic Anlysis, Energy Economics, Vol. 22, 2, [17] Anoniou, A., Vorlow, C.E., Price Clusering nd Discreeness: Is here Chos behind he Noise? Physic A, Vol. 348, 25, [18] Hfner, C.M., Reznikov, O., On he esimion of dynmic condiionl correlion models, Com. S. D Anl., Vol. 56, 212, [19] Urrui, J.L., Gronewoller, P., Hoque, M., Nonlineriy nd Low Deerminisic Choic Behvior in Insurnce Porfolio Sock Reurns, J. Risk Insur., Vol. 69, 22, [2] Ellio, R.J., Ko, P.E., Mhemics of he Finncil Mrkes, Sringer Berlin Heidelberg, 25. [21] Ci, G., Hung, J., A New Finnce Choic Arcor, In. J. Nonlin. Sci., Vol. 3, 27, [22] Chen, W.C., Dynmics nd Conrol of Finncil Sysem wih Time-delyed Feedbcks, Chos, Solions nd Frcls, Vol. 37, 28, [23] Holys, J.A., Zebrowsk, M., Urbnowicz, K., Observions of he Deerminisic Chos in Finncil Time Series by Recurrence Plos, Cn One Conrol Choic Economy? Euro. Phys. J. B, Vol. 2, 21, [24] Loskuov, A.Yu., Dynmicl Chos: Sysems of Clssicl Mechnics, Physics Usekhi, Vol. 177, 27, [25] Loskuov, A.Yu., Fscinion of Chos, Physics Usekhi, Vol. 18, 21, [26] Akins, P.W., The Elemens of Physicl Chemisry, Oxford Universiy Press, [27] Hken, H., Anlogy beween Higher Insbiliies in Fluids nd Lsers, Phys. Le. A, Vol. 53, 1975, [28] Srrow, C., The Lorenz Equions: Bifurcions, Chos nd Srnge Arcors, Sringer, [29] Hilborn, R.C., Chos nd Nonliner Dynmics: An Inroducion for Scieniss nd Engineers, Oxford Universiy Press, 2. [3] Mndelbro, B., Frcls nd Scling in Finnce: Disconinuiy, Concenrion, Risk, Sringer, [31] Grssberger, P., Proccci, I., Mesuring he rngeness of srnge rcors, Physic D, Vol. 9, 1983, [32] Ding, M., Grebogi, C., O, E., Suer, T., Yorke, J., Esiming correlion dimension from choic ime series: when does leu onse occur? Physic D, Vol. 69, 1993, [33] Dubovikov, M.M., Srchenko, N.S., Dubovikov, M.S., Dimension of he miniml cover nd frcl nlysis of ime series, Physic A, Vol. 339, 24, [34] Mndelbro, B.B., Ness, V., Frcionl Brownin moions, frcionl noises nd licions, SIAM Rev., Vol. 1, 1968, [35] Cmbel, A.B., Alied chos heory: A rdigm for comlexiy, Acdemic Press, [36] Peers, E.E., Chos nd order in he cil mrkes, John Willey & Sons, [37] Schmid, A.B., Quniive finnce for hysicis: n inroducion, Elsevier, 25. [38] Tsllis, C., Wh re he numbers h exerimens rovide? Quimic Nov, Vol. 17, 1994, [39] Tsllis, C., Nonddiive enroy nd nonexensive sisicl mechnics - n overview fer 2 yers, Brz. J. Phys., Vol. 39, 29, [4] Picoli, S., Mendes, R.S., Mlcrne, L.C., Snos, R.P.B., q-disribuions in comlex sysems: brief review, Brz. J. Phys., Vol. 39, 29, [41] Zhng, F., Shi, Y., Ng, H., Wng, R., Tsllis sisics in relibiliy nlysis: Theory nd mehods, Eur. Phys. J. Plus, Vol. 131, 216, E-ISSN: Volume 14, 217

11 [42] Tsllis, C., Possible generlizion of Bolzmnn-Gibbs sisics, Journl of Sisicl Physics, Vol. 52, 1998, [43] Rusecks, J., Gonis, V., Kulkys, B., Nonexensive sisicl mechnics disribuions nd dynmics of finncil observbles from he nonliner sochsic differenil equions, Advnces in Comlex Sysems, Vol. 15, 212, [44] Gonis, V., Kulkys, B., Rusecks, J., Nonliner sochsic differenil equion s he bckground of finncil flucuions, Proceedings of 2h Inernionl Conference Noise nd Flucuions, 29, [45] Kulkys, B., Alburd, M., Modeling Scled Processes nd 1/f β Noise using Nonliner Sochsic Differenil Equions, J. S. Mech., 29, P251, E-ISSN: Volume 14, 217