CAPITAL MARKET EFFICIENCY AND TSALLIS ENTROPY

Transcription

1 CAPTAL MARKET EFFCENCY AND TALL ENTROPY Miloslav Vošvrda nsiue of nformaion Theory and Auomaion of he A CR Economerics vosvrda@uiacascz Absrac The concep of he capial marke efficiency is a cenral noion in he financial markes heory This noion is generally useful o describe a capial marke in which is relevan informaion f relevan informaion is compleely processed by he capial marke price mechanism hen such capial marke is called o be efficien Thus he capial marke efficiency accenuaes he informaional efficiency of capial markes means ha in he efficien capial marke invesors canno expec achieving of enormous reurns by long ime n oher words he capial marke is efficien if he flucuaions of reurns in ime are unpredicable Thus a ime series of such flucuaions can be generaed by some derivaions of Brownian moion Considering long range macroeconomic forces which are nonexensive we use Tsallis s approach for maximum enropy formulaing By his way i is possible o obain fa ailed disribuions by power laws opimizing Tsallis s enropy and o give some heoreical base o some sylized facs on capial markes For he esimaing of he capial marke efficiency is used he noion Tsallis enropy (informaion gain) Keywords Capial marke efficiency expecaion Famma s approach Tsallis enropy macroeconomic euilibrium JEL C1 C13 C46 1 nroducion The capial marke efficiency (CME) has roos daing back o he urn of he las cenury The erm CME is used o inroduce a capial marke in which relevan informaion is absorbed ino is price generaing sysem o obain he price of capial asses n his definiion is formerly emphasized he informaional CME way f capial markes are sofisicaed and compeiive hen economics indicaors indicae ha invesors canno expec achieving superior profis f i is assumed ha a capial marke is in euilibrium sae hen i is expeced ha for his capial marke he euilibrium price of securiy will be fair price This consrucion is a base for rus esablishing in fair funcioning of he capial marke mechanism The CME is mainly consruced on he probabiliy calculus The price analyzing of differen securiies on capial markes are execued by very sofisicaed mehods The base for analyzing a ime series of prices is he random walk model Thus shor-erm reurns have flucuaions wih a random naure means ha he securiy prices have incorporaed he marke relevan informaion a any momen of ime immediaely Afer informaion processing and risk assessing he marke mechanism generaes euilibrium prices Any deviaions of hese euilibrium prices should be lock-up and unpredicable Afer many heoreical and empirical invesigaions i is sured o admi he random walk model for he ime series of securiy prices ([3][4][6][9]) Efficien capial marke is closely relaed o he concep of he marke informaion The marke informaion is clearly essenial in financial aciviies or rading This one is arguably he mos imporan deerminan of success in he financial life We shall suppose ha sraegic invesor decisions are formulaed and execued on he basis of price informaion in he public domain and available o all Le us assume ha invesors are considered raional We shall furher assume ha informaion once known remains known - no forgeing - and can be accessed in real ime The abiliy o reain informaion organize i and access i uickly is one of he main facors which will discriminae beween he abiliies of differen economic agens o reac on changing marke condiions We resric ourselves o very simple siuaion no differeniaing beween agens on he basis of heir informaion processing abiliies Thus as ime passes new informaion becomes available o all agens who coninually updae heir informaion We shall use riples ( Ω P) for expressing of a probabiliy space and he following expression E X for he condiional expecaions The { ω Ω } is a se of he elemenary marke siuaions The is some σ -algebra of he subses of Ω and P is a probabiliy measure on he This srucure gives us all he machinery for saic siuaions involving randomness For dynamic siuaions involving randomness over ime we need a seuence of σ - algebras { } which are increasing + 1 for all wih represening he informaion available o us a ime represens an iniial informaion (if here is none We always suppose ha all σ -algebras o be complee Thus = { Ω }) On he oher hand a siuaion ha all we ever shall know is represened by he following expression 34

2 = lim uch a family { } is called a filraion; a probabiliy space endowed wih such a filraion ( Ω { } P) is called a filered probabiliy space which is also called a sochasic basis uppose ha ( ) a measurable space { } X ( Ω { } P) aking values in ( A) and having he disribuion P = P funcions from ( A) o ( ) where sands for he Borel σ -algebra of he subses of A is X ω is a seuence of independen random elemens defined on a probabiliy space Capial Marke Efficiency and Expecaions C P Le = ( Ω { } ) be a capial marke wih disinguished flows { } space We also call { } an informaion flow and an expression { } Definiion 1 A capial marke ( { } P) { } Le M be he se of all measurable of σ-algebras filered probabiliy M is a securiy price process Ω is called an efficien if here exiss P such ha each securiy price seuence = saisfies he following condiion: he seuence is a P-maringale ie he variables are { } (31) - measurable and EP < EP + 1 = ξ is he seuence of independen random variables such ha E P ξ < ξ = = = and hen evidenly he securiy price seuence f a seuence { } 1 ξ ξ EP [ ξ ] 1 σ ( ξ1 ξ ) { } = { } where is a maringale wih respec o { } ξ ξ = = ξ1 + + ξ for 1 and = and E E ξ P + 1 = + P + 1 f a seuence { } 1 is a maringale wih respec o he filraion { } { ξ} 1 is a maringale difference ie ξ is -measurable EP ξ < EP ξ 1 = E ξ ξ + = for each and 1 We have [ ] < E ξ k k ie he variables { ξ } (3) (33) and = ξ1 + + ξ wih ξ = hen (34) are uncorrelaed provided ha for 1 n oher words suare-inegrable maringale belongs o he class of random seuences wih orhogonal incremens: E [ + k ] = (35) where 1 = ξ and + k = ξ+ k Thus he capial marke efficiency is nohing else han he maringal propery of securiy price processes in i 3 Capial Marke Efficiency and Maximum Enropy Consider now where as a securiy price process ha is represened by he following form = X (41) X is a soluion of he nonlinear Fokker-Plank euaion [1] 341

3 σ σ wih boh a( ) and g x of g x a x g x b x g x = ( ) + ( ) x x (4) b as a drif force and diffusion coefficien respecively and a probabiliy densiy funcion X a ime Le us consider by Borland s modificaion [14] a sochasic diffusion coefficien in euaion (4) by he following form = + c x = b x g x (43) This modificaion yields he modificaion of (4) as follows c ( x ) 1 g ( x) a( x ) g ( x) g ( x) x x (44) A sochasic differenial euaion for X has he following form c ( X ) dx = a X d + dw ω where W ( ω ) is a sandard Wiener process Definiion A generalized enropy following indicaor H (Tsallis enropy) of he capial marke ( { } P) H = 1 x X g 1 x dx (45) Ω a ime is inroduced by he A parameer is called an enropic index A maximizaion of (46) wih he following consrains 3 g ( x) g ( x) dx = 1 x dx = σ 5 3 g y dy wih x g σ = x dx being he second-order momen leads o he following form of he probabiliy densiy funcion g ( x ) [6]: g ( x) 1 1/ Γ ( 1 ) x = 1 ( 5 3) π ( 5 3) 3 Γ The expression (48) is denoed as a -Gaussian because for 1 1 1/ ( 1 ) (46) (47) (48) he Gaussian disribuion is recovered The parameer can be used as a measure of non-gaussianiy The probabiliy densiy funcion () g is for > 1 lepokuric ie a disribuion wih long ails relaive o Gaussian The 3-D versions of he -Gaussian are presened in Fig1 and Fig for σ=1 and =1 respecively Nex le F and G be wo probabiliy disribuions on wih f and g as heir probabiliy densiy funcions respecively An informaion gain is a non-commuaive measure of he difference beween wo probabiliy disribuions F andg Generalized informaion divergence associaed wih wo probabiliy disribuions F and G is defined as follows 1 1 g x f x ( F G ) = f ( x ) dx 1 Afer some operaions on (411) we ge he following expression (411) 34

4 1 f ( F G ) = 1 g ( x) dx 1 g x ( x) (41) Noe: For =1 he imporan uaniy (41) is relaed o he hannon enropy which is called Kullback-Leibler divergence The Tsallis enropy is appropriae for non-euilibrium sysems replacing exponenial Bolzmann facors by power-law disribuion[13] Because ( F G ) ( G F ) unless F G adjused on he following expression = he generalized informaion divergence ( ) ( F G ) ( F G ) ( G F ) / Values of he crierion ( F G ) for any F G belong o 1 f ( ) F G is hus = + (413) F G is eual hen F = G (414) ie he sochasic processes have he same informaion gain Le X T in period T has a F be an uniform probabiliy disribuion on f analyzed sochasic proces { } probabiliy disribuion G on and ( F G ) = hen dependence inside { X T} As an illusraion of he behavior of his crierion ( ) G is an informaion-gain-free abou an inerior F G le us pu boh he uniform probabiliy disribuion on 55 for F and he G o be -Gaussian for each on 55 presened in he following Fig 3 for differen levels of = (11 16) and σ = (1 5) Definion 3 Le G be a probabiliy disribuion of he proces { d T} d capial marke is called an efficien if for any d holds Theorem 1 A capial marke C is he efficien if and only if any Proof ( ) Le C be an efficien Any reurn G G = for T and herefore W d { X T} ( ) Draw any is he efficien d C T and G Gd d d This one is Le F be an uniform disribuion on The T and for > 1 he following expression ( F G d ) = d 4 imulaion and Empirical Analysis C for all T and G d = F C T is by (45) a sandard Wiener process Hence has no any informaion gain abou he dependence in = F Then ( d ) F G = for any d C T Hence C Now we generae realizaions of he size 5 from he following heoreical models on he one hand wih and on he oher wihou a correlaion dependence inside Considered heoreical models are Normal Noise wih µ= and σ = 1 Random Walk ( ω ) = 1 ( ω ) + ε Random -Walks ε ω = ω ω + ε 1 Moving Average () ( ω ) = ε + 1 ε 1 4 ε Auoregressive Model () 7 3 ω = ω ω + ε 1 343

5 Normal Noise wih µ= and σ = 3 3 ε AuoRegressive Moving Average () Deerminisic Process ( ω ) = 9 ( ω ) ω ω + 5 ω = ε + 7 ε 3 ε 1 1 Empirical daa were obained from he following daily observaions of marke indices PX5 DAX BUX AX WG &P 5 GX Nikkei 5 NADAQ and Dow Jones ndusriel Average for he period ill 813 The securiy price reurns model ln ( ω ) / 1 ( ω ) is used for a ransformaion of he observaions Le F be a uniform disribuion on 1 1 A smooh esimaion of he empirical probabiliy disribuions by Bernsein polynomials [15] is used We esimae crierions ( F G ) and σ=1 Resuls are inroduced in Table 1 and in Fig 4 and Fig 5 F G for paricular models wih =1 and 1 Tab 1 Resuls of he crierions ( F G ) and 1 ( ) The Crierions ( F G ) and ( F G ) model F F G for differen models 1 G ( ) F G ( F G ) 1 1 Uniform Uniform Uniform Random Walk Uniform Random -Walks Uniform MA() Uniform AR() Uniform Normal Noise wih µ= and σ = Uniform ARMA() Uniform Deerminisic Process Uniform PX5 Prague Czech Republic Uniform DAX Frankfur Germany Uniform BUX Budapes Hungary Uniform AX Braislava lovak Uniform WG Warsaw Poland Uniform &P 5 UA Uniform GXingapore ingapore Uniform Nikkei 5 Tokyo Japan Uniform NADAQ UA Uniform Dow Jones ndusriel Average UA Concluding Remarks Resuls show ha he curren non-exensive approach could be more useful ool in he analysis of CME This approach is more sensiive on deecing of any dependencies in he capial marke processes is demonsraed ha F G By ( F G ) as a measure of he divergence on capial markes pursues his fac beer han 1 ( ) comparison resuls of ( ) F G from empirical and simulaion daa i is shown ha capial markes have had a remarkable fail in an efficiency of he incorporaing of he marke relevan informaion is possible o obain any farher more accurae resuls afer esimaing of he enropy index Conseuenly he nex imporan sep is o consruc an esimaor ˆ for he enropy index The esimaor ˆ will give a possibiliy o involve more pregnanly he lepokuriciy of he probabiliy disribuions ino he measure ˆ Acknowledgmens Financial suppor from Czech cience Foundaion 4/9/965 and MMT LC675 is acknowledged References [1] BORLAND L Microscopic dynamics of he nonlinear Fokker-Plank euaion: Phenomenological model Phys Rev E

6 [] KRAFT J Changes of macroeconomic environmen a he urn of cenury as deerminan facor for operaion of sae adminisraion companies Aca economia No 7 Ekonomická eória a praxa - dnes a zajra Bánská Bysrica EF UMB sr BN [3] COOTNER PH The Random Characer of ock Marke Price MT Press Cambridge MA 1964 [4] FAMA E The Behavior of ock Marke Prices Journal of Business pp [5] KENDALLl M The Analysis of Economic Time erie Journal of he Royal aisical ociey eries A pp 11-5 [6] QUERO DUARTE M On non-gaussianiy and dependence in financial ime series: a nonexensive approach Quaniaive Finance Vol 5 No 5 Ocober 5 pp [7] OBORNE MFM Brownian Moion in he ock Marke Operaions Research pp [8] PLATENE Arbirage in coninuous complee markes Adv Appl Probab34 pp [9] RKENHThe Fokker-Planck Euaion: Mehods of oluion and Applicaions nd sd 1989 pringer-verlag Berlin [1] ROBERT H ock Marke `Paerns and Financial Analysis: Mehodological uggesion Journal of Finance pp 1-1 [11] HRYAEV AN Probabiliy nd ed pringer-verlag New York 1995 [1] TALL C Bukmann DJ Anomalous diffusion in he presence of exernal forces Phys Rev E R197 [13] TALL C Possible Generalizaion of Bolzmann-Gibbs aisics Journal of aisical Physics Vol 5 Nos ½ 1988 [14] TALL C ANTENEODO C BORLAND L OORO R Nonexensive aisical Mechanics and Economics Physica A [15] BABU GJBOYARKY A CHAUBEY YP GORA P A New aisical Mehods for Filering and Enropy Esimaion of a Chaoic Map from Noisy Daa nernaional Journal of Bifurcaion and Chaos Vol14 No Fig 1 -Gaussian σ=1 Fig -Gaussian =1 Fig3 3D-figure of he crierion Fig 4 Tsallis divergence among differen models and uniform model Fig5 Kullback-Leibler divergence among differen models and uniform model 345