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the booms and the market crashes in financial markets have been an object of study in economics and a history of economy for a long time . economists @xcite , and economic historians @xcite , @xcite , @xcite have often suggested the importance of psychological factors and irrational factors in explaining historical financial euphoria . as keynes @xcite , a famous economist and outstandingly successful investor , acutely pointed out in his book , _ the general theory of employment , interest and money _ , stock price changes have their origin in the collective crowd behaviour of many interacting agents rather than fundamental values which can be derived from careful analysis of present conditions and future prospects of firms . in a recent paper published in the economic journal , lux @xcite modelled the idea explicitly and proposed a new theoretical framework to explain bubbles and subsequent crashes which links market crashes to the phase transitions studied in statistical physics . in his model the emergence of bubbles and crashes is formalised as a self - organising process of infections among heterogeneous traders . in recent independent works , several groups of physicists @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite proposed and demonstrated empirically that large stock market crashes , such as the 1929 and the 1987 crashes , are analogous to critical points . they have claimed that financial crashes can be predicted using the idea of log - periodic oscillations or by other methods inspired by the physics of critical phenomena . in this paper we present an interacting - agent model of speculative activity explaining bubbles and crashes in stock markets . we describe stock markets through an infinite - range ising model to formulate the tendency of traders getting influenced by the investment attitude of other traders . bubbles and crashes are understood and described qualitatively and quantitatively in terms of the classical phase transitions . although the interacting - agent hypothesis @xcite is advocated as an alternative approach to the efficient market hypothesis ( or rational expectation hypothesis ) @xcite , little attention has been given to the point how probabilistic rules , that agents switch their investment attitude , are connected with their decision - making or their expectation formations . our interacting - agent model follows the line of lux @xcite , but differs from his work in the respect that we model speculative activity here from a viewpoint of traders decision - making . the decision - making of interacting - agents will be formalised by _ minimum energy principle _ , and the stationary probability distribution on traders investment attitudes will be derived . next , the stationary states of the system and speculative dynamics are analysed by using the mean field approximation . it is suggested that the mean field approximation can be considered as a mathematical formularisation of _ keynes beauty contest_. there are three basic stationary states in the system : a bull market equilibrium , a bear market equilibrium , and a fundamental equilibrium . we show that the variation of parameters like the bandwagon effect or the investment environment , which corresponds to the external field , can change the size of cluster of traders investment attitude or make the system jump to another market phase . when the bandwagon effect reaches some critical value , a second - order phase transition and critical behaviour can be observed . there is a symmetry breaking at the fundamental equilibrium , and two stable equilibria , the bull market equilibrium and bear market equilibrium appear . when the system stays in the bull market equilibrium , speculative bubble occurs in the stock market . for a certain range of the investment environment multistability and hysteresis phenomena are observed . when the investment environment reaches some critical values , the rapid changes ( the first - order phase transitions ) in the distribution of investment attitude are caused . the phase transition from a bull market phase to a bear market phase is considered as a stock market crash . then , we estimate the parameters of the interacting - agent model using an actual financial data . as an example of large crashes we will analyse the japan crisis ( bubble and crash in japanese stock market in 1987 - 1992 ) . the estimated equation attempts to explain japan crisis over 6 year period 1987 - 1992 , and was constructed using monthly adjusted data for the first difference of topix and the investment environment which is defined below . results of estimation suggest that the traders in the japanese stock market stayed the bull market equilibria , so that the speculative bubbles were caused by the strong bandwagon effect and betterment of the investment environment in 3 year period 1987 - 1989 , but a turn for the worse of the investment environment in 1990 gave cause to the first - order phase transition from a bull market phase to a bear market phase . we will demonstrate that the market - phase transition occurred in march 1990 . in section 2 we construct the model . in section 3 we investigate the relationship between crashes and the phase transitions . we implement an empirical study of japan crisis in section 4 . we give some concluding remarks in section 5 . we think of the stock market that large numbers of traders participate in trading . the stock market consists of n traders ( members of a trader group ) . traders are indexed by @xmath0 . we assume that each of them can share one of two investment attitudes , buyer or seller , and buy or sell a fixed amount of stock ( q ) in a period . @xmath1 denotes the investment attitude of trader @xmath2 at a period . the investment attitude @xmath1 is defined as follows : if trader @xmath2 is the buyer of the stock at a period , then @xmath3 . if trader @xmath2 , in contrast , is the seller of the stock at a period , then @xmath4 .      in the stock market the price changes are subject to the law of demand and supply , that the price rises when there is excess demand , and the price falls when there is excess supply . it seems natural to assume that the price raises if the number of the buyer exceeds the number of the seller because there may be excess demand , and the price falls if the number of seller exceeds the number of the seller because there may be excess supply . thus a trader , who expects a certain exchange profit through trading , will predict every other traders behaviour , and will choose the same behaviour as the other traders behaviour as thoroughly as possible he could . the decision - making of traders will be also influenced by changes of the firm s fundamental value , which can be derived from analysis of present conditions and future prospects of the firm , and the return on the alternative asset ( e.g. bonds ) . for simplicity of an empirical analysis we will use the ratio of ordinary profits to total capital that is a typical measure of investment , as a proxy for changes of the fundamental value , and the long - term interest rate as a proxy for changes of the return on the alternative asset . furthermore we define the investment environment as    _ investment environment = ratio of ordinary profits to total capital - long - term interest rate_.    when the investment environment increases ( decreases ) a trader may think that now is the time for him to buy ( sell ) the stock . formally let us assume that the investment attitude of trader @xmath2 is determined by minimisation of the following _ disagreement _ function @xmath5 , @xmath6 where @xmath7 denotes the strength of trader @xmath8 s influence on trader @xmath2 , and @xmath9 denotes the strength of the reaction of trader @xmath2 upon the change of the investment environment @xmath10 which may be interpreted as an external field , and @xmath11 denotes the vector of investment attitude @xmath12 . the optimisation problem that should be solved for every trader to achieve minimisation of their disagreement functions @xmath5 at the same time is formalised by @xmath13 now let us assume that trader s decision making is subject to a probabilistic rule . the summation over all possible configurations of agents investment attitude @xmath14 is computationally explosive with size of the number of trader @xmath15.therefore under the circumstance that a large number of traders participates into trading , a probabilistic setting may be one of best means to analyse the collective behaviour of the many interacting traders . let us introduce a random variable @xmath16 , @xmath17 . the state of the agents investment attitude @xmath18 occur with probability @xmath19 with the requirement @xmath20 and @xmath21 . we define the amount of uncertainty before the occurrence of the state @xmath18 with probability @xmath22 as the logarithmic function : @xmath23 . under these assumptions the above optimisation problem is formalised by @xmath24 subject to @xmath25 ,    where @xmath26 . @xmath18 is a state , and @xmath27 is information entropy . @xmath22 is the relative frequency the occurrence of the state @xmath18 . the well - known solutions of the above optimisation problem is @xmath28 where the parameter @xmath29 may be interested as a _ market temperature _ describing a degree of randomness in the behaviour of traders . the probability distribution @xmath22 is called the _ boltzmann distribution _ where @xmath22 is the probability that the traders investment attitude is in the state @xmath30 with the function @xmath31 , and @xmath32 is the partition function . we call the optimising behaviour of the traders with interaction among the other traders a _ relative expectation formation_.      the trading volume should depends upon the investment attitudes of all traders . since traders are supposed to either buy or sell a fixed amount of stock ( q ) in a period , the aggregate excess demand for stock at a period is given by @xmath33 . we assume the existence of a market - maker whose function is to adjust the price . if the excess demand @xmath34 is positive ( negative ) , the market maker raises ( reduces ) the stock price for the next period . precisely , the new price is calculated as the previous price @xmath35 plus some fraction of the excess demand of the previous period according : @xmath36 , where @xmath37 denotes the investment attitude of trader @xmath2 at period @xmath38 , @xmath39 the price change from the current period to the next period , i.e. @xmath40 , and the parameter @xmath41 represents the speed of adjustment of the market price . at the equilibrium prices that clear the market there should exist an equal number of buyers or sellers , i.e. : @xmath42 . using the boltzmann distribution ( [ eqn : a4 ] ) the mean value of the price changes @xmath43 is given by @xmath44 . we can derive analytically the stationary states of the traders investment attitude using a mean field approximation which is a well known technique in statistical physics . let us replace the discrete summation of the investment attitude with the mean field variable , @xmath45 . we additionally assume that the parameter @xmath7 is equal to @xmath46 @xmath47 and @xmath48 for every trader . in this case our interacting - agent model reduces to the ising model with long - range interactions . then the function ( [ eqn : a2 ] ) is approximated by @xmath49 . since the mean value of the stationary distribution is given by @xmath50 , the mean field @xmath51 is @xmath52 the first term of the right side of eq . ( [ eqn : a5 ] ) represents that traders tend to adopt the same investment attitude as prediction of the average investment attitude and the second term represents the influence of the change of investment environment to traders investment attitude . the first term may be interpreted as a mathematical formularisation of _ keynes beauty contest_. keynes @xcite argued that stock prices are not only determined by the firm s fundamental value , but in addition mass psychology and investors expectations influence financial markets significantly . it was his opinion that professional investors prefer to devote their energies not to estimating fundamental values , but rather to analysing how the crowd of investors is likely to behave in the future . as a result , he said , most persons are _ largely concerned , not with making superior long - term forecasts of the probable yield of an investment over its whole life , but with foreseeing changes in the conventional basis of valuation a short time ahead of the general public_. keynes used his famous _ beauty contest _ as a parable to stock markets . in order to predict the winner of a beauty contest , objective beauty is not much important , but knowledge or prediction of others predictions of beauty is much more relevant . in keynes view , the optimal strategy is not to pick those faces the player thinks prettiest , but those the other players are likely to be about what the average opinion will be , or to proceed even further along this sequence . we assume that the traders investment attitude changes simultaneously ( synchronous dynamics ) in discrete time steps . we represent the dynamics of the investment attitude using a straightforward iteration of equation ( [ eqn : a5 ] ) , such that : @xmath53 we call @xmath54 the bandwagon coefficient and @xmath55 the investment environment coefficient below because @xmath54 may be interpreted as a parameter that denotes the strength that traders chase the price trend . next , let us approximate the adjustment process of the price using the mean field @xmath56 . we can get the following speculative price dynamics , @xmath57 inserting the mean field of the price change , @xmath58 into eq . [ eqn : a6 ] , the dynamics of the mean field of @xmath56 can be rewritten as @xmath59 . this equation shows that the traders result in basing their trading decisions on an analysis of the price trend @xmath60 as well as the change of the investment environment @xmath61 . the stationary states of the mean field @xmath51 satisfy the following @xmath62 eq . [ eqn : a8 ] has a unique solution @xmath51 for arbitrary @xmath63 , when @xmath54 is less than @xmath64 . the stationary state @xmath65 with @xmath66 is called as the _ fundamental equilibrium _ which is stable and corresponds to the maximum of the stationary distribution of the investment attitude @xmath67 . in this stable equilibrium there is equal numbers of traders sharing both investment attitudes on average .    moving in the parameter space @xmath68 and starting from different configurations one have several possible scenarios of market - phase transitions . to begin with , let us consider the case that the investment environment @xmath10 is equal to zero . in theory , when the bandwagon coefficient @xmath54 is increased starting from the phase at @xmath69 , there is a symmetry breaking at critical point , @xmath70 , and two different market phases appear , that is , the _ bear market _ and the _ bull market_. in analogy to physical systems the transitions may be called the second - order phase transition . fig . 1 illustrates the two graphical solutions of the eq . ( [ eqn : a8 ] ) for @xmath71 and @xmath72 and for @xmath69 and @xmath72 . the figure shows that for @xmath69 and @xmath72 the fundamental equilibrium is unique and stable , but for @xmath71 and @xmath72 the fundamental equilibrium @xmath73 becomes unstable , and the two new equilibria , the bull market equilibrium @xmath46 and the bear market equilibrium @xmath74 are stable . at the bull market equilibrium more than half number of traders is buyer , so that the speculative bubbles occur in the stock market . then , let us consider the effect of changes of the investment environment @xmath10 in the case with a weak bandwagon coefficient , @xmath69 and a positive investment environment coefficient , @xmath75 . 2 shows that as the investment environment @xmath10 changes for the better ( @xmath76 ) , the system shifts to the bull market phase . by contrast , as the investment environment @xmath10 changes for the worse ( @xmath77 ) , the system shifts to the bear market phase . therefore when the bandwagon effect is weak , the stock price go up or down slowly according rises or falls in the investment environment . finally , let us consider the effect of changes of @xmath63 in the case with a strong bandwagon coefficient @xmath71 . in this case multistability and hysteresis phenomena in the distribution of investment attitude , as well as market - phase transitions are observed . the system has three equilibria , when @xmath71 and @xmath78 , where @xmath79 is determined by the equation , @xmath80 = \alpha . \label{eqn : a9}\ ] ] two maximum of the stationary distribution on the investment attitude are found at @xmath81 and @xmath82 and one minimum at @xmath83 . for @xmath84 and @xmath71 , two of the three equilibria coincide at @xmath85 for @xmath86 and @xmath71 the two of three equilibria vanishes and only one equilibrium remains . in analogy to physical systems the transitions may be called first - order phase transitions . fig . 3 illustrates the effect of changes in the investment environment in the case with a strong bandwagon effect @xmath71 . in this case speculative bubbles and market crashes occur . when the investment environment changes for the better ( worse ) , the curve that denotes @xmath87 , shifts upward ( downward ) . as the investment environment @xmath10 keeps on rising , and when it reach a critical value , that is , @xmath84 , the bear market equilibrium and the fundamental equilibrium vanish , and the bull market equilibrium becomes an unique equilibrium of the system , so that the speculative bubble occurs in the stock market . even though the investment environment changes for the worse , the hysteresis phenomena are observed in a range of the investment environment that can be calculated by solving eq . ( [ eqn : a9 ] ) . in other words the speculative bubble continues in a range of the investment environment . however when the negative impact of the investment environment reaches a critical value , that is , @xmath84 , the bull market vanish . further decrease of @xmath10 cause the market - phase transition from a bull market to a bear market . in a bear market more than half number of trader is seller on average , so that the stock price continues to fall on average . thus this market phase transition may be considered as the stock market crash . perhaps the most spectacular boom and bust of the late twentieth century involved the japan s stock markets in 1987 - 1992 . stock prices increased from 1982 to 1989 . at their peak in december 1989 , japanese stocks had a total market value of about 4 trillion , almost 1.5 times the value of all u.s . equities and close to 45 percent of the world s equity market capitalisation . the fall was almost as extreme as the u.s . stock - market crash from the end of 1929 to mid-1932 . the japanese ( nikkei ) stock - market index reached a high of almost 40,000 yen in the end of 1989 . by mid - august 1992 , the index had declined to 14,309 yen , a drop of about 63 percent .    in this section we will estimate the parameter vector @xmath68 for the japan s stock market in 1987 - 1992 . we use monthly adjusted data for the first difference of tokyo stock price index ( topix ) and the investment environment over the period 1988 to 1992 . in order to get the mean value of the stock price changes , topix is adjusted through the use of a simple centred 12 point moving average , and are normalised into the range of @xmath88 to @xmath89 by dividing by the maximum value of the absolute value of the price changes . the normalised price change is defined as @xmath90 . 4 and fig . 5 show the normalised stock price changes and the investment environment respectively . this normalised stock price change @xmath90 is used as a proxy of the mean field @xmath56 . inserting @xmath90 for eq . ( [ eqn : a8 ] ) , we get @xmath91 .      the bandwagon coefficient @xmath54 and the investment environment coefficient @xmath55 should be estimated using any one of estimation technique . since we can not get the analytical solution because the function @xmath92 is non - linear , we use the _ gradient - descent algorithm _ for the parameter estimation . the gradient - descent algorithm is a stable and robust procedure for minimising the following one - step - prediction error function @xmath93 ^ 2 . \label{eqn : a33}\ ] ] more specifically , the gradient - descent algorithm changes the parameter vector @xmath94 to satisfy the following condition : @xmath95 where @xmath96 , @xmath97 , and @xmath98 . to accomplish this the gradient - descent algorithm adjusts each parameter @xmath99 and @xmath100 by amounts @xmath101 and @xmath102 proportional to the negative of the gradient of @xmath103 at the current location : @xmath104 where @xmath105 is a learning rate . the gradient - descent rule necessarily decreases the error with a small value of @xmath105 . if the error function ( [ eqn : a33 ] ) , thus , has a single minimum at @xmath106 , then the parameters @xmath94 approaches the optimal values with enough iterations . the error function @xmath107 , however , is non - linear , and hence it is possible that the error function ( [ eqn : a33 ] ) may have _ local minima _ besides the global minimum at @xmath106 . in this case the gradient - descent rule may become stuck at such a local minimum . to check convergence of the gradient - descent method to the global minimum we estimate the parameters for a variety of alternative start up parameters @xmath108 with @xmath109 . as a consequence the gradient - descent rule estimated the same values of the parameters , @xmath110 with enough iterations . the fit of the estimated equation can be seen graphically in fig . ( 6 ) , which compare the actual and forecasted series over one period , respectively . the correlation coefficient that indicates goodness of fit is @xmath111 . the capacity of the model to forecast adequately in comparison with competing models is an important element in the evaluation of its overall performance . let us consider the linear regression model @xmath112 . the regression results are @xmath113 . the correlation coefficient of the linear regression model is equal to 0.93 , and is lower that that of the interacting - agent model . root mean squared errors ( rmses ) of the interacting - agent model is equal to 0.05 , and that of the linear regression model is equal to 0.148 over the period 1987 - 1992 . the interacting - agent model has a smaller rmse than that of the linear regression model . by the traditional rmse criterion the interacting - agent model is , thus , superior to the linear regression model in term of their forecasting performance . in conclusion we can say that the model not only explains the japan crisis but also correctly forecasts changing the stock price over the period 1987 - 1992 . given that the set of parameters ( @xmath114 ) is equal to ( 1.04 , 0.5 ) , the critical point of the investment environment at which the phase transition are caused from the bull market to bear market are calculated from the theoretical results in the preceding section . one can say that the phase transition from the bull market to the bear market occurs when the investment environment is below @xmath115 under @xmath116 . by contrast the phase transition from the bear market to the bull market occurs when the investment environment is beyond @xmath117 under @xmath116 . .the stock market crash in the japanese stock market [ cols="^,^,^,^",options="header " , ]     from theoretical view point one may say that bursting speculative bubbles will begin when the investment environment is below @xmath115 , and the process of the collapse of the stock market continues till the investment environment is beyond @xmath117 .    in the real world the investment environment became below @xmath118 at march 1990 for the first time and continued to be negative values since then over the period 1987 - 1992 . the fall of the actual price in the japan s stock market began at march 1990 , and the stock prices continued to fall over the period 1990 - 1992 . on these grounds we have come to the conclusion that theory and practice are in a perfect harmony . ( see table [ tab1 ] . ) this paper presents an interacting - agent model of speculative activity explaining the bubbles and crashes in stock markets in terms of a mean field approximation . we show theoretically that these phenomena are easy to understand as the market- phase transitions . bubbles and crashes are corresponding to the second - order transitions and the first - order transitions respectively . furthermore we estimate the parameters of the model for the japanese stock market in 1987 - 1992 . the empirical results demonstrate that theory and practice are in a perfect harmony . this fact justifies our model that bubbles and crashes have their origin in the collective crowd behaviour of many interacting agents . i would like to thank masao mori and thomas lux for helpful comments and suggestions to an earlier version of this paper . 99 j. m. keynes , _ the general theory of employment , interest and money _ . harcourt , 1934 .    c. mackay , _ extraordinary popular delusions and the madness of crowds _ the noonday press , 1932 .    c. p. kindlberger , _ manias , panics , and crashes : a history of financial crises _ , macmillan , london , 1989 . j. k. galbraith , _ a short history of financial euphoria _ . whittle direct books , tennessee , u.s.a . , t. lux,_the economic journal _ , * 105 * ( 1995 ) 881 - 896 . t. kaizoji , _ simulation _ ( in japanese ) , 12 , 2 ( 1993 ) 67 - 74 . d. sornette , a. johansen , j.p . bouchaud , _ j. phys . i. france _ * 6 * ( 1996 ) 167 - 175 . d. sornette , a. johansen , _ physica a _ * 245 * ( 1997 ) 411 . d. sornette and a. johansen , _ physica a _ * 261 * ( nos . 3 - 4 ) ( 1998 )    a. johansen , d. sornette , _ eur.phys.j . b _ * 1 * ( 1998 ) 141 - 143 . a. johansen , d. sornette , _ int . c _ * 10 * ( 1999 ) 563 - 575 . a. johansen , d. sornette , _ int . c _ * 11 * ( 2000 ) 359 - 364 . j. a. feigenbaum , p.g.o . freund , _ int . b _ * 10 * ( 1996 ) 3737 - 3745 . j. a. feigenbaum , p.g.o . freund , _ int . b _ * 12 * ( 1998 ) 57 - 60 . n. vandewalle , ph . boveroux , a. minguet , m. ausloos , _ physica a _ * 225 * ( 1998 ) 201 - 210 . n. vandewalle , m. ausloos , ph . boveroux , a. minguet , _ eur . j. b _ * 4 * ( 1998 ) 139 - 141 . s. gluzman , and v. i. yukalov , _ mod . b _ * 12 * ( 1998 ) 75 - 84 . l. laloux , m. potters , r. cont , j .- aguilar and j .- bouchaud , _ europhysics letters _ , * 45 * , 1 ( 1999 ) 1 - 5 . d. chowhury , d. stauffer _ eur . * 8 * , ( 1999 ) 477 . j. a. holyst , k. kacperski , f. schweitzer , submetted in _ physica a _ , ( 2000 ) . t. lux , and m. marchesi , _ nature _ , * 297 * ( 1999 ) 498 - 500 . e. fama , _ journal of finance _ * 46 * ( 1970 ) 1575 - 1618 . d. e. rumelhart , g. e. hinton , r. j. williams , in : _ parallel distributed processing _ * 1 * , chap . 8 , d. e. rumelhart , j. l. mcclelland , et al . , eds . cambridge , ma : mit press , 1986 . figure 1 . the second - order phase transition . numerical solution of eq.([eqn : a8 ] ) for @xmath72 . the curve is the rhs of [ eqn : a8 ] plotted for different values of @xmath54 : the aa line - @xmath119 , the bb line - @xmath120 . + figure 2 . numerical solution of eq.([eqn : a8 ] ) for @xmath121 and @xmath122 . @xmath87 is the rhs of ( 8) plotted for different values of @xmath54 : the aa line - @xmath123 , the bb line - @xmath124 . + figure 3 . the first - order transition : speculative bubble and crash . numerical solution of eq . ( [ eqn : a8 ] ) for @xmath125 and @xmath122 . the curves is @xmath87 plotted for different values of the investment environment : the aa line - @xmath126 , the bb line - @xmath127 . + figure 4 . the adjusted stock price change , @xmath128 : january 1987 - december 1992 . + figure 5 . the japan s investment environment @xmath61 : january 1987 - december 1992 . + figure 6 . one - step - forecast of the stock price changes : january 1987 - december 1992 .
in this paper we present an interacting - agent model of speculative activity explaining bubbles and crashes in stock markets . 
 we describe stock markets through an infinite - range ising model to formulate the tendency of traders getting influenced by the investment attitude of other traders . 
 bubbles and crashes are understood and described qualitatively and quantitatively in terms of the classical phase transitions . 
 when the interactions among traders become stronger and reach some critical values , a second - order phase transition and critical behaviour can be observed , and a bull market phase and a bear market phase appear . 
 when the system stays at the bull market phase , speculative bubbles occur in the stock market . for a certain range of the external field that we shall call the investment environment , multistability and hysteresis phenomena 
 are observed . 
 when the investment environment reaches some critical values , the rapid changes in the distribution of investment attitude are caused . 
 the first - order phase transition from a bull market phase to a bear market phase is considered as a stock market crash . 
 furthermore we estimate the parameters of the model using the actual financial data . as an example of large crashes we analyse japan crisis ( the bubble and the subsequent crash in the japanese stock market in 1987 - 1992 ) , and 
 show that the good quality of the fits , as well as the consistency of the values of the parameters are obtained from japan crisis . 
 the results of the empirical study demonstrate that japan crisis can be explained quite naturally by the model that bubbles and crashes have their origin in the collective crowd behaviour of many interacting agents . 
 + * keyword * speculative bubbles ; stock market crash ; phase transition ; mean field approximation ; japan crisis