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EPJ manuscript No. (will be inserted by the editor) 7 The market efficiency in the stock markets 0 0 2 n Jae-Suk Yang1, Wooseop Kwak1, Taisei Kaizoji2, and In-mook Kim1a a J 1 Department of Physics, Korea University,Seoul131-701, Republicof Korea 1 3 2 Division of Social Sciences, International Christian University,Osawa, Mitaka, Tokyo181-8585, Japan ] h p Received: February 2, 2008/ Revised version: date - c o s Abstract. We study the temporal evolution of the market efficiency in the stock markets using the com- . s c plexity, entropy density, standard deviation, autocorrelation function, and probability distribution of the i s y log return for Standard and Poor’s 500 (S&P 500), Nikkei stock average index, and Korean composition h p stockpriceindex(KOSPI).Basedonamicroscopicspinmodel,wealsofindthatthesestatisticalquantities [ 2 in stock marketsdepend on themarket efficiency. v 9 7 1 PACS. 89.65.Gh Economics;econophysics,financialmarkets,businessandmanagement–89.70.+c Infor- 1 0 mation theory and communication theory – 89.75.Fb Structuresand organization in complex systems 7 0 / s 1 Introduction ferentcharacteristicsbetweenmaturemarketsandemerg- c i s ing markets [11], market efficiency [12], and the relation y h Econophysics is one of the most active fields in interdis- between shape of PDFs and time lags [13] are studied us- p : v ciplinary research [1,2,3,4,5,6,7,8,9,10,11,12,13]. Time se- ing PDFs. Also it is used to distinguish between bubble i X ries analysis and agent based modelling have been stud- and anti-bubble [9,10]. r a ied by many researchers. There are many methodologies to analyze the financial time series. Observing probabil- Anothermethodiscomputationalmechanics[16].Com- ity distribution functions (FDFs) of log return is one of putational mechanics has been studied various fields of the simplest and the most popular methods. Many re- science [17,18,19], and it is applied to analyze the stock searchpapers aboutPDFs oflogreturnforstockmarkets market [7]. Computational mechanics is available to an- have already been published [10,11,12,13,14,15]. The dif- alyze complexity and structure quantitatively by finding Send offprint requests to: In-mook Kim a e-mail: [email protected] intrinsic causal structures of time series [20]. 2 Jae-Suk Yang et al.: The market efficiency in thestock markets Agent based modelling has been widely used in so- is defined as cial science and econophysics to construct artificial so- S(t) logY(t+∆t) logY(t), (1) cial and economic systems. Agent based models in econo- ≡ − physics are constructed using agents clustering [1], Ising- where Y(t) is the price at time t and ∆t is the time lag. like spin model [2,12], and Potts-like model [4]. Variation of PDFs shapes by traders’ characteristics [10] and infor- 2.1 Probability distribution and autocorrelation mation flow [12], and speculative activity explaining bub- bles and crashes in stock market [5] have been simulated The distribution of price changes are identified as non- by agent based model. Gaussian [12,13,14,15,21]. Especially, when the PDF has In this paper, we analyze the time series of Standard the powerlaw tail, the exponent ofpower at tail partcan and Poor’s (S&P 500), Nikkei stock average index, and be gotten from the PDF. That exponent is called as tail Korean composition stock price index (KOSPI) by time index. evolution of statistical measures such as PDFs of log return, autocorrelation function, complexity, entropy density, and scaling properties of the standard deviation of log return. Moreover, we construct the stock market us- s F D4 P ing microscopic spin model to simulate above time series of x e d n results. ail i t US 2 KR JP 1996 1998 2000 2002 time (year) 2 Empirical data and analysis 0.0010 US KR JP WeusetheS&P500datamainlyfortheperiodfrom1983 VACF0.0005 to 2006. Japanese data for the period from 1997 to 2005 and Korean data for the period from 1992 to 2003 are 0.0000 also used to support and confirm the results from S&P 1996 1998 2000 2002 time (year) 500.Thedataresolutionishighfrequency(1minute)data, andweuseonlyintra-dayreturnstoexcludediscontinuity Fig. 1. (a) Temporal evolution of tail index and (b) variance jumpsbetweenthepreviousday’scloseandthenextday’s of autocorrelation function for the S&P 500, the Nikkei stock open price due to the overnight effects. The price return index,and theKOSPI. Jae-Suk Yang et al.: The market efficiency in thestock markets 3 Fig.1ashowstemporalevolutionoftailindexinPDFs forS&P500,Nikkeistockindex,andKOSPI.Tailindexof US KR PDFs increases from around 2 to above 4 as time passes. 0.7 JP In2000s,theshapeofPDFbecomesnarrowerandthetail 0.6 part becomes thinner, while PDF has fatter tail and the 0.5 slope of tail part is more steep in 1990s. Autocorrelation 1990 2000 function is defined as follows: time (year) <S(t)S(t+τ)> R(τ)= , (2) σ2 Fig. 2. Temporal evolution of scaling properties of the stan- where σ is a standard deviation of S(t). Moreover, the dard deviation of log return. variance of autocorrelationfunction is defined as follows: V =<R(τ)2 >. (3) as a function of the time lag ∆t. The relation between ACF standard deviation and time lag is as follows: Fig. 1b shows the temporal evolution of variance of autocorrelation function. The increasing tendency for tail index is reverse to it for variance of autocorrelationfunc- σ(∆t) ∆tµ. (5) ∼ tion.Wecanguessthatthe reasonwhyprobabilitydistri- butions of log return are changed is related to autocorre- When µ is larger than 0.5, the time series has long range lation of log return time series. correlation,whilelongrangeanticorrelationwhenµ<0.5. Though the tendency is same for three stock markets, Thereis nocorrelationatµ=0.5andstrengthofcorrela- the value of tail index for the S&P 500 is larger than it tion (or anticorrelation) is proportional to the difference for the KOSPI and V for S&P 500 is smaller than it ACF between µ and 0.5. Fig. 2 shows the temporal evolution for the KOSPI in the 1990s. of scaling properties of the standard deviation of log return.Thevalueofµdecreasestoaround0.5.Untilthemid 2.2 Scaling property of standard deviation 1990s, time series of stock market index has strong long We investigate the long range memory of log return by range correlation.However,long range correlationpracti- observing the time evolution of scaling properties in the cally disappears in 2000s. standard deviation of log return [8]. The standard devia- In spite of the same tendency for temporal evolution tion of log return is defined as ofµ, the S&P 500is moreclose to 0.5 thanthe KOSPIin n (logY(t +∆t) logY(t ))2 σ(∆t)= qPi=1 i − i , (4) √n 1 the 1990s. − 4 Jae-Suk Yang et al.: The market efficiency in thestock markets 2.3 Entropy density and statistical complexity as a function of block length L where L=1,2,3, . En- ··· tropydensityismoreusefulbecauseitisnormalizedquan- We also analyze financial time series using computational tity while H(L) also increases as L increases. mechanics to find the statistical complexity and the en- In next, to calculate statistical complexity ǫ-machine tropy density. In order to calculate statistical complexity, hastobedefined.Aninfinitystring↔S canbedividedinto we used causal-state splitting reconstruction (CSSR) al- two semi-infinite parts such as a future →S and a history gorithm [16] to model ǫ-machine of the stock markets. ←S .Acausalstate isdefinedasasetofhistoriesthathave To calculate the entropy density and the statistical the same conditional distribution for all the futures. ǫ is complexity,weshouldsymbolizethetimeseriesasfollows: a function that maps each history to the sets of histories, each of which corresponds to a causal state: F(t) θ(Y(t+∆t) Y(t)), (6) ≡ − ǫ(←s )= ←s′ P(SL =sL ←S =←s )=P(SL =sL ←S =←s′ ), { | | | whereθ(x) is aHeavisidestep function.Thenthe original sL SL,←s′ ←S ,L Z+ . (10) ∈ ∈ ∈ } data Y(t) are changed into the binary time series F(t) with a countable set A= 0,1 .F(t) is 0 (or 1) when the ThetransitionprobabilityT(a)denotestheprobability { } ij next index has decreased (or increased). ofgeneratingasymbolawhenmakingthetransitionfrom Claude Shannon suggested the entropy of a discrete state Si to state Sj [23,24]. random variable X with a probability function P(x) [22] The combination of the function ǫ from histories to as follows: causalstateswiththelabelledtransitionprobabilitiesT(a) ij is called the ǫ-machine [23], which represents a computa- H[X]= P(x)log P(x). (7) −X 2 tional model underlying the given time series. x Given the ǫ-machine, statistical complexity is defined Let A be a countable set ofsymbols of time series and let as S be a random variable for A, and s is its realization. If C P(S )log P(S ). (11) a block of string with L consecutive variable is denoted µ ≡−X i 2 i {Si} as SL = S ,...,S , then Shannon entropy of length L is 1 L Fig.3showstemporalevolutionofstatisticalcomplex- defined as ity and entropy density. Statistical complexity decreases andentropydensity increasesin allthree marketsastime H[X]= P(s ,...,s )log P(s ,...,s ). − X ··· X 1 L 2 1 L s1∈A sL∈A passes. (8) Statisticalcomplexityisaround0whentimeserieshas Also entropy density for the finite length L is define as regularpatternoritis totallyrandom.Toclarifywhether h (L) H(L) H(L 1), (9) thetimeseriesisrandomorregular,theentropydensityis µ ≡ − − Jae-Suk Yang et al.: The market efficiency in thestock markets 5 characteristics for the stock price time series by modify- ingmicroscopicspinmodel[2].ThenumberofagentsisN, 6 5 and we consider i = 1,2,...,N agents with orientations y exit4 σ (t)= 1,correspondingtothedecisiontobuy(+1)and mpl i ± o3 c sell( 1)stockatdiscrete time-steps t.The orientationof 2 US − KR agent i at the next step, σ (t+1), depends on the local 1 JP i 1985 1990 1995 2000 2005 field: time (year) 1 Ipri(t)= A (t)σ (t)+h (t), (12) i N X ij j i j 1.00 whereA (t)representthetime-dependentinteractionstrength ij 0.95 sity among agents, and hi(t) is an external field reflecting the n de0.90 y effectoftheenvironment.Thetime-dependentinteraction op ntr0.85 e strength among agents is A (t) = Aξ(t) +aη (t) with US ij ij 0.80 KR JP ξ(t) and η (t) determined randomly in every step. A is ij 1985 1990 1995 2000 2005 time (year) anaverageinteractionstrengthandaisadeviationofthe individual interaction strength. The external field reflect- Fig. 3. Temporal evolution of (a) statistical complexity and ing the effect of the environment is h = hζ (t), where h i i (b) entropy density. is an information diffusion factor, and ζ (t) is an event i needed. Time series is totally random when entropy den- happening at time t and influencing the i-th agent. sity is around1 and entropy density is 0 when time series From the local field determined as above, agentantic- has periodic pattern because it is a measure of disorder. ipates log return of stock index as follows: Sowecanfindoutthatthe time seriesofstockmarketsis 2 xexp(t)= 1. (13) i 1+e−2Iipri(t) − gettingmorerandomlyandthepatternsinthetimeseries So, the local field on agent can be represented as follows: almost disappear in 2000s. Also the values of complexity and entropy density for the S&P 500 are different from Ii(t)=Iipri(t)+α(x(t−1)−xeixp(t−1)), (14) them for the KOSPI, though the inclination is same. where α is degree of adjustment. When α=0, agents determine their opinion from Ipri, while agents determine i their opinionfromthe price or log returnofprevious step 3 Model and results as well as information flowed into the market when α is We constructed the microscopic model of many interact- non-zero. For instance, in case positive α and x(t 1) > − ing agents to simulate the variation of some statistical xexp(t 1), agents determine their opinion by adding the i − 6 Jae-Suk Yang et al.: The market efficiency in thestock markets differencebetweenthemarketpricechangesandtheantic- Fig. 4a shows variance of autocorrelation function for ipatedpricechangesatthepreviousstep.Onthecontrary, various α. As α decreases, the tail is getting thinner and in case x(t 1) < xexp(t 1), agents subtract the differ- thinner, and the strength of autocorrelation is reduced. − i − encefromIpri toadjusttheirinexactinformation.Bythis Moreover, scaling exponents of standard deviation go to i wayagents refer to pastperformance,while agentsactby 0.5as α decreases[see Fig. 4b]. The generatedtime series fundamental expressed by Ipri in case of α=0. has long range correlationfor largerα, and Correlationis i From the local field determined as above, agent opin- almost disappeared for small value of α. ions in the next step are determined by: In Fig. 5, we can confirm the tendency of statistical complexity and entropy density for various α. As α de- +1with probability p σi(t+1)= , (15) creases,statistical complexity decreases and entropy den- 1with probability 1 p − − sityincreases.Fromthestatisticalcomplexity,thepattern where p = 1/(1 + exp 2I (t) ). In this model, price i {− } in time series is getting simpler or the degree of random- changes are: 1 ness of times series is larger for smaller α. What entropy x(t)= σ (t). (16) N X i density is 1 means the time series is practically random. Fig.5c is the relationbetweenentropydensity andstatis- tical complexity. From this relation we can distinguish if 0.003 the time series is random or regular. 0.002 VACF 0.001 4 Conclusions 0.000 0.8 0.4 0.0 We analyze the time series of stock index of U. S., Japan, and Korea using some statistical measures and simulate them by microscopic agent based spin model. 0.60 Timeserieshasafattailinlogreturndistributionand 0.55 a tail index is increased as time passes to present. Exis- 0.50 tence of pattern in the financial time series can be con- 0.8 0.4 0.0 firmed by autocorrelation function, entropy density and complexity. As time goes from past to present, entropy Fig. 4. (a) Variance of autocorrelation function for various density is increasedandcomplexityis decreased.Also au- and(b)Scalingexponentsofstandarddeviationforvariousα. tocorrelationisdecreased.Fromtheseresults,therelation Jae-Suk Yang et al.: The market efficiency in thestock markets 7 patepricechangesofnextstepwithadjustedinformation. In the past, the speed of information is slower and mar- ketis less efficient, so adjusting behavior is more effective 4 y exit and active in the same time interval compare to present. pl m o2 c Therefore,the pastmarketcorrespondingto higher α has long range correlationand vice versa. 0 1.5 1.0 0.5 0.0 Ipri(t) is generatedrandomly because its elements are i randomvariableswhileα(x(t 1) xexp(t 1))provides − − i − regularity to I (t) because effect of this term remains for 1.00 i y awhilelikeMarkovchain.Whenαis0,entropydensityis nsit0.95 e d y almost1andcomplexityis0becausetimeseriesforα=0 p o entr0.90 are almost random. As α is increased, entropy density is decreasedandcomplexityisincreasedbecausethepattern 0.85 1.5 1.0 0.5 0.0 is generated in the time series. The reason why these changes occur is that speed of information flow is becoming fast by the development of 4 infra for communication such as high speed internet, mo- y xit e pl m bile communicationand broadcastingsystems. So market o2 c hasbecomemoreefficient.Bytheefficientmarkethypoth- 0 esis(EMH),thespeedofinformationissofastthatagents 0.96 0.98 1.00 entropy density can not gain profit by superiority of information. We would like to thank Hang-Hyun Jo for helpful dis- Fig. 5. (a)Statistical complexity,(b)entropydensityfor var- cussions.ThisworkissupportedbytheSecondBrainKo- ious α, and (c) the relation between entropy density and sta- rea 21 project and also by the Grant No. R01-2004-000- tistical complexity. 10148-1from the Basic Research Programof KOSEF. between presentdata and past data is decreasing and the pattern in stock log return data disappears. References In the spin model, when α is non-zero, traders ad- just their opinion using the difference between their an- 1. V.M.Eguiluz,M.Zimmermann,Phys.Rev.Lett.85(2000) ticipated prices and real market prices, and they antici- 5659. 8 Jae-Suk Yang et al.: The market efficiency in thestock markets 2. A.Krawiecki,J.A.Hol yst,D.Helbing,Phys.Rev.Lett.89 20. R.W.Clarke,M.P.Freeman,N.W.Watkins,Phys.Rev. (2002) 158701. E 67 (2003) 016203. 3. D. Chowdhury,D.Stauffer, Eur. Phys.J. B 8 (1999) 477. 21. R. Vicente, C. M. de Toledo, V. B. P. Leite, N. Caticha, 4. T. Takaishi, Int.J. of Mod. Phys.C 16 (2005) 1311. 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We study the temporal evolution of the market efficiency in the stock markets using the com- . s c plexity, entropy density, standard deviation, autocorrelation function, and probability distribution of the i s y log return for Standard and Poor’s 500 (S&P 500), Nikkei stock average index, and Korean composition h p stockpriceindex(KOSPI).Basedonamicroscopicspinmodel,wealsofindthatthesestatisticalquantities [ 2 in stock marketsdepend on themarket efficiency. v 9 7 1 PACS. 89.65.Gh Economics;econophysics,financialmarkets,businessandmanagement–89.70.+c Infor- 1 0 mation theory and communication theory – 89.75.Fb Structuresand organization in complex systems 7 0 / s 1 Introduction teristics between mature markets and emerging markets c i s [?], market efficiency [?], and the relation between shape y h Econophysics is one of the most active fields in interdis- of PDFs and time lags [?] are studied using PDFs. Also p : v ciplinary research [?,?,?,?,?,?,?,?,?,?,?,?,?]. Time series it is used to distinguish between bubble and anti-bubble i X analysis and agent based modelling have been studied by [?,?]. r a many researchers.There are many methodologies to analyze the financial time series. Observing probability dis- Anothermethodiscomputationalmechanics[?].Com- tribution functions (FDFs) of log return is one of the putationalmechanicshasbeenstudiedvariousfieldsofsci- simplest and the most popular methods. Many research ence [?,?,?], and it is applied to analyze the stock market papers about PDFs of log return for stock markets have [?].Computationalmechanicsisavailabletoanalyzecom- alreadybeenpublished[?,?,?,?,?,?].Thedifferentcharac- plexity and structure quantitatively by finding intrinsic Send offprint requests to: In-mook Kim a e-mail: [email protected] causal structures of time series [?].