Stock market as a dynamic game with continuum of players 2


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1 Stock market as a dynamc game wth contnuum of players Agneszka WsznewskaMatyszkel Insttute of Appled Mathematcs and Mechancs Warsaw Unversty emal: Abstract. Ths paper contans a gametheoretc model descrbng the behavour of nvestors at a stock exchange. The model presented s developed to reect the actual market mcrostructure. The players consttute a nonunform contnuum, derng, among others, by the plannng horzon, the external ow of money whch can be nvested, formaton of expectatons about future prces, whch, brey, dvdes the nvestors nto the followng groups: fundamental analysts, chartst, users of varous econometrc models, users of Captal Asset Prcng Model, and players observng a random exogenous sgnal. Prces are determned by orders and the equlbratng mechansm of the stock exchange. The mechansm presented s the actual sngleprce aucton system used, among others, at Warsaw Stock Exchange. One of the man ssues are selfverfyng belefs. Results of numercal smulatons of stock exchange based on the model are also ncluded. Keywords: multstage games, contnuum of players, Nash equlbrum, belefdstorted Nash equlbrum AMS subject classcaton 91A80, 91B26, 91A13, 91A10, 91A50, 91B52 Scentfc work fnanced by funds for scence n years , grant # 1 H02B Introducton The stock exchange, startng from a place where buyers and sellers could face each other and even negotate prces, evolved to a place, also n the vrtual meanng, n whch anonymous masses of nvestors buy or sell at prces dctated by the equlbratng mechansm. Durng ths process of evoluton, as the anonymty ncreased, varous models predctng future prces were developed, among others: fundamental analyss, techncal analyss, varous econometrc models and the Captal Asset Prcng model. In ths paper the author tres to present a model of stock exchange reectng ts actual mcrostructure. In ths model each sngle player has a neglgble mpact on the aggregated values, such as the market demand and supply, and ther functons, ncludng the market prce. Nevertheless, prces are determned by the equlbratng 1
2 Stock market as a dynamc game wth contnuum of players 2 mechansm of the stock exchange usng only players' orders. Each player has strateges dependng on nformaton about past prces and values of other avalable varables ltered by the prognostc technque nherent to hs type of formng expectatons. Such games, called games wth dstorted nformaton were formally ntroduced by the author n WsznewskaMatyszkel [42] and developed n WsznewskaMatyszkel [43] n the form more applcable for modellng nancal markets. In order to make the model realstc an actual market mechansm of a real stock exchange was mplemented t s sngleprce aucton system taken from Warsaw Stock Exchange WSE) but smlar mechansms are used at many stock exchanges. Actual formaton of prces s, as n the real lfe, fully determnstc: prces are determned by orders and the equlbratng mechansm of the stock exchange. The model consdered n ths paper contnues the research contaned n Wsznewska Matyszkel [40]. A contnuum of players s used n order to model a mature stock exchange: there are many agents, each of them nsgncant. Each sngle player s conscous that hs order cannot aect prces and ths reects real stuatons. On the other hand, prces are eect of agents' orders. Dependng on szes of types, even very abstract belefs can become selfverfyng at least to some extent. In the paper there are examples of such selfverfyng belefs: some of fundamental nature, techncal sgnals of changes of trends and an absolutely abstract formaton of cat. Ths formaton has not exsted by now and emprcal data does not conrm t. It s explaned n a quaspsychologcal way whch s frequently used by authors of textbooks on techncal analyss. Moreover, ths formaton, f t was popularzed among nvestors, would become selfverfyng. Ths cat s an example of selfverfyng character of some technques of foreseeng future prces. The paper starts by a short descrpton of some models of prce formaton subsecton 1.2). The model s formulated n secton 2. We state some results about equlbra n secton 3; those concernng threshold prces and weak domnance n subgames wth dstorted nformaton are n subsecton 3.2. In secton 4 we examne the ssue of selfvercaton of varous prognostc approaches. Some of them are selfverfyng when used by a strong group of players but not the whole populaton), e.g. players usng fundamental analyss cause fast convergence to a prce close to the fundamental value of a share subsecton 4.1), whle some others are selffalsfyng subsecton 4.2). The results of numercal smulatons are n secton Games wth a contnuum of players. Models wth contnuum of players were rst ntroduced by Aumann [3] and [2] and Vnd [26] to model compettve markets. By then t was very dcult to model nsgncance of each sngle player.
3 Stock market as a dynamc game wth contnuum of players 3 Games wth contnuum of players was formally dened by Schmedler [21], and afterwards the general theory of such games was extensvely studed n, among others, MasColell [17], Balder [5], Weczorek [27] and [28], Weczorek and Wsznewska [29] or WsznewskaMatyszkel [31]. Dynamc games wth contnuum of players are qute new some examples of applcatons of such games are Karatzas, Shubk and Sudderth [8], WsznewskaMatyszkel [30] and [32], [41], and general theory of such games n WsznewskaMatyszkel [33], [34], and [36]). An nterestng ssue s the problem of convergence of parameters of equlbra n ntelymanyplayers counterparts of a dynamc game wth a contnuum of players to the parameters of equlbra n ths game see WsznewskaMatyszkel [38] and [37] Some models of prces of shares. In ths subsecton we shortly present some models and technques used for foreseeng future prces of shares. Fundamental analyss. The fundamental analyss approach s based on calculaton of the actual value of a share, called ts fundamental value. The most obvous denton s a dscounted value of the nnte seres of expected future dvdends. Gven the nterest rate r and the sequence of expected at tme t 0 ) dvdends of company, {A t } t=t0,t 0 +1,..., the fundamental value at tme t 0 equals F t 0 ) = 1 t t0 t=t 0 1+r) A t. However, at WSE most companes do not pay dvdends. In such a case the fundamental value of a share should reect the fracton of the value of the company correspondng to ths share. Investors usng fundamental analyss assume that the prce should be close to the fundamental value and any dstorton s caused by speculatons and t can preval only n a short perod the prces on the stock exchange should reect the fundamental value. Techncal analyss. The basc assumpton of techncal analyss s opposte to that of fundamental analyss: the prces move n trends. The real processes n the economy are perceved as secondary to the behavour of prces and volumes of shares n the past. Techncal analysts explan ths counterntutve assumpton by sayng that prces of shares contan nformaton of future state of the economy, even ths whch s not explctly known to the nvestors e.g. Prng [19]). The explanatons are based on varous socologcal, psychologcal and economc terms, but n fact, techncal analyss reduces to analyss of past prces and volumes. Formerly t was manly analyss of charts, therefore ts users are called chartsts. Although t s usually dsregarded by scentsts, t s taught at many departments of economc scences and t s now the most popular way of predctng prces by prvate nvestors at WSE. Therefore t may really nuence the prces as t s descrbed n the paper).
4 Stock market as a dynamc game wth contnuum of players 4 Probablstc models. In ths subsecton we can descrbe varous models wth one common feature: all of them treat prces of shares as a realzaton of a stochastc process. Portfolo analyss and Captal Assets Prcng Model CAPM). Portfolo analyss, started by Markowtz [13] and [15], see also [14] and [16]), was rst a normatve theory of nvestment n rsky assets. It reduced the problem to an analyss of the mean and varance of the asset return. It was converted nto a descrpton of the behavour of nvestors by Lntner [12], Mossn [18], Sharpe [23], and Fama e.g. [6]) and s known as CAPM Captal Asset Prcng Model). The parameters of the model mean and the covarance matrx, and, consequently, so called β coecents) are estmated on the bass on emprcal data taken from the stock exchange. Accordng to ths model, at equlbrum the prce of an asset should be such that the expected return fulls the equaton R = r + β R M r), where R s the expected return of asset, β ts βcoecent, r the nterest rate of the rsk free asset and R M the expected return from the market portfolo usually the stock exchange ndex). Ths model s statc, but after a slght modcaton t can be appled for predctng prces at a stock exchange. Econometrc models. Ths wde genre of models encompasses all prognostc methods based on data analyss usng varous econometrc technques, startng from the smplest lnear regresson. In such models, we can consder dependence on past prces and volumes, day of the week, or some external data. Prcng of dervatve securtes. A model of prce formaton on a stock exchange s necessary not only for usual nvestors tryng to make money on buyng and sellng shares, but also for nancal nsttutons sellng dervatve securtes based on assets sold at the stock exchange. Prcng of dervatve securtes requres knowledge about the form of the equaton descrbng future prces. It s usually a stochastc derental equaton. In the model of Black and Scholes [4] t s dp t) = P t) bt)dt + σt)dw t)), where W s a standard Wener process. See also Karatzas [7] for an extensve descrpton of the theory of prcng of dervatves. Besde the prces of dervatves, the hedgng strateges depend on the equaton descrbng the evoluton of prces of shares. We are not gong to model such nvestors, manly because ther strateges of buyng or sellng shares depend on the contract they want to hedge, whch s exogenous to the model consdered.
5 Stock market as a dynamc game wth contnuum of players 5 No model. There are also nvestors who do not form expectatons about prces. They ether choose a strategy from some smple nvestors manuals e.g. constant sum, constant relaton or constat reacton), and belevng that they turn out to be frutful, or decde at random by openng the bble or vstng a fortune teller. Both knds of players may turn out to be successful. However, the rst type cannot be nontrvally modelled by a gametheoretc model, snce ther strateges are xed and no optmzaton takes place. The latter type can be encompassed by our model of a stock exchange. Moreover, they can mprove the operaton of the stock exchange. Prevous models of stock exchange based on optmzaton of ndependent agents. The model presented n ths paper, as well as the earler authors papers on nancal markets [35] and [40], are not the rst models, consderng a mcroeconomc approach to the behavour of the players. The man ssue n agentbased models was the nuence of players expectatons about prce behavour on actual prces. There were so called models of artcal stock exchange, n whch players tended to maxmze ther payos gven some expectatons. One of them was the model and a computer smulaton programme called Santa Fe Artcal Stock Market. In ths models there s a share wth a stochastc dvdend and a rsk free asset. Player estmate the expected value of future dvdends. A market clearng condton was added. Players adjust ther expectatons durng the game. See e.g. Arthur et al. [1], LeBaron [9] and [10] or LeBaron, Arthur and Palmer [11] for more detals. 2. Formulaton of the model In ths secton we formulate the game theoretc model of a stock exchange. A game G s dened by specfyng the set of players, the sets of players strateges and the payo functons. Here we consder a dynamc game, therefore the strategy speces choces of decsons at every tme nstant durng the game and the response of the whole system to these decsons. The rst object to dene s the set of players. We consder a model of a mature stock exchange,.e. such that a sngle player has a neglgble mpact on prces the set of players s the unt nterval Ω = [0, 1] wth the Lebesgue measure λ. In our model of stock exchange we consder n + 2 types of assets. Frstly, there are shares of n companes sold at the stock exchange. Shares n our model are not assumed to pay any dvdends. Secondly, there s a rsk free but not fully lqud asset of postve nterest rate r, for smplcty called bonds. And nally, money, whch are rsk free and lqud but of nterest rate 0. The game s dynamc, t starts at t 0 ntal tme and termnates at +, but each player has hs own termnal tme T ω +, dentcal for players of the same
6 Stock market as a dynamc game wth contnuum of players 6 type. We shall denote the set of possble tme nstants {t 0, t 0 + 1,...} by T, whle the symbol T ω denotes {t 0, t 0 + 1,..., T ω + 1} f T ω s nte, T otherwse. The set of possble stock prces P s a dscrete subset of R + \{0}. There are some restrctons on prces at tme t t should be n the nterval [1 h) pt 1), 1 + h) pt 1)], where h > 0 denotes the maxmal rate of varablty. Besde the money earned at the stock exchange, the players can nvest money from an external ow of captal or be forced to wthdraw some money). For player ω t wll be represented by a functon M ω : T R. The players have to pay a commsson for any transacton, but they do not have to pay addtonal commsson for orders. For smplcty of calculatons we shall assume a constant commsson rate C << 1. The same commsson s also pad for buyng or sellng bonds. Portfolo of a player, denoted by x, s an n+2tuple wth coordnates correspondng to shares of n companes, bonds and money. Therefore x R n+2 +. At the begnnng of the game player ω s assgned an ntal portfolo x ω. Players' decsons at each tme nstant conssts of: an order to sell S a par p S, q S ) P n R n +, two orders to buy BM a par p BM, q BM ) P n R n + buy for money) and BB a par p BB, q BB ) P n R n + buy for bonds), and the part of value non nvested n shares whch s hold n cash: e. In each case p. denotes the vector of prce lmts for all shares, q. the vector of amounts. Prce lmts coordnates of p. ) are n P, amounts are nonnegatve, and the rato of lqud money e [0, 1]. Besdes the general form of the orders we want to be able to llustrate the fact, that some players do not nvest n some knd of companes, some players never keep cash or that some players never keep bonds. Therefore the set of decsons of player ω D ω s a subset of the set D = { BM, BB, S, e) : BM, BB, S P n R n +, e [0, 1] }. These sets D ω have the form D ω = P n Γ ω ) 3 E ω, where Γ ω R n + s a product of real semlnes startng from 0 and sngletons {0}. We also have to dene the noton of physcal admssblty of a decson, dependng on the portfolo. The symbol D ω x ω ) D ω wll denote the set of decsons of player ω avalable at hs portfolo x ω. It s dened by the constrants n =1 1 + C) pbm q BM x ω n+2 where x ω n+2 denotes money; ths reads as a player 1 + C) pbm q BM 1 C)x ω n+1 cannot pay more money than he possesses), n =1 where x ω n+1 denotes value of bonds) and q S x ω.e. shortsellng s forbdden) for each share = 1,..., n. If x = {x ω } ω Ω represent a famly of portfolos of the players, then any measurable functon δ : Ω D such that for every ω δω) D ω x ω ) s called a statc prole avalable at x. The set of all statc proles avalable at x wll be denoted by Σx), whle Σ wll denote the set of all statc proles.
7 Stock market as a dynamc game wth contnuum of players 7 A statc prole together wth the past prce determnes the market prce as explaned below. Aggregated demand, aggregated supply and the market mechansm Let us consder the market for shares of company at a xed tme nstant t and players porfolos x. Gven a statc prole avalable at x p BM ω), q BM ω)), p BB ω), q BB ω)), p S ω), q S ω)), eω) ), the market supply of share AS : P R + s equal to AS p ) = Ω Ω q S ω)1 p S ω) p dλω), whle the market demand for share AD : P R + s equal to AD p ) = q BM ω) 1 p BM ω) p + q BB ω) 1 p BB ω) p dλω), where 1 condton s equal to 1 when the condton s fullled and 0 otherwse. In order to calculate the market prce fo share, the market mechansm consdered n the paper rst returns returns the prce maxmzng a lexcographc order wth crtera, startng from the most mportant: 1. maxmzng volume.e. the functon mnad p ), AS p )); 2. mnmzng dsequlbrum.e. the functon AD p ) AS p ) ; 3. mnmzng the number of shares n sellng orders wth prce lmt less then the market prce and buyng orders wth prce lmts hgher than the market prce ; 4. mnmzng the absolute value of the derence between the calculated prce and the reference prce.e. p p t 1). The result s projected on the set [1 h) pt 1), 1 + h) pt 1)] P and t consttutes the market prce p t). A smlar procedure s used at WSE see [20]). The derences are caused by obvous mstakes and nconsstences of the regulatons of WSE. The problem of these mperfectons was studed n WsznewskaMatyszkel [39]. Evoluton of portfolos, strateges, and dynamc proles The portfolo of player ω at tme t s denoted by X ω t). If player ω chooses at tme t a decson BM, BB, S, e) D ω X ω t)) and the prce at tme t s pt), then: X ω t + 1) = X ω t) + q BM 1 p BM p t) + q BB = 1,..., n, Xn+1t ω + 1) = 1 + r) + 1 e X ω 1+C n+2t) n =1 1 + C) q BM )) 1 C) q S p S 1 p S p t), Xn+1t) ω n 1+C =1 1 C qbb p BB p BM 1 p BB p t) q S 1 p S p t) for t t 0, 1 p BM p t) 1 p BB p t)+
8 Stock market as a dynamc game wth contnuum of players 8 Xn+2t ω + 1) = M ω t + 1) + e Xn+2t) ω n =1 1 + C) q BM p BM 1 p BM p t) )) 1 C) q S p S 1 p S p t). A strategy of player ω s a functon denng choces of decsons at all tme nstants t s a functon ω : T D ω wth ω t) D ω X ω t)), where X ω denotes the trajectory of portfolo of player ω, whch s dened by the above evoluton equaton wth the ntal condton X ω t 0 ) = x ω. The set of strateges of player ω wll be denoted by S. If for a choce of players' strateges = { ω } ω Ω for every t the functon ω ω t) s measurable, then s called a dynamc prole. The trajectory correspondng to wll be denoted by X and the sequence of market prces p. The set of all dynamc proles wll be denoted by Σ. Players' payos and expected payos If T ω s nte, then the payo of a player, gven hs strateges and a sequence of market prces along the prole s dened n the obvous way as the present value of the portfolo at tme T ω + 1, V Tω+1,Xω T ω+1)), where V : T 1+r) Tω+1 t 0 ω R n+2 + R denotes any functon representng the value of the portfolo. Here we consder V t, x) = x n+1 + x n+2 + n =1 p t) x. Elementary calculatons show that the payo can be equvalently expressed as Tω V t+1,x ω t+1)) 1+r) V t,x ω t)) t=t 0, snce subtractng a constant does not change choces 1+r) t+1 t 0 of the players. Ths denton of payo can be obvously extended to T ω = + f the sum s well dened t can attan nnte values. Formally, the payo functon of player ω Π ω : Σ R dened by Π ω ) = Tω t=t 0 V t+1,x ) ω t+1)) 1+r) V t,x ) ω t)) 1+r) t+1 t 0 for V t, x) = x n+1 + x n+2 + n =1 p t) x. Ths ends the denton of our actual game G. As n the context of more general games wth dstorted nformaton, dened n WsznewskaMatyszkel [42] and [43], we can also dene the expected payo of player ω at tme t gven hs belef correspondence based on hs observaton of the hstory of the game. It represents the supremum over future decsons of player ω of hs payo assumng the belef correspondence  the player assumes that n future he s gong to behave optmally and consders hs guaranteed payo the payo correspondng to the worst future hstory of the system n hs belef correspondence. In ths paper, n order to avod a complcated notaton, we shall ncorporate the belef correspondence nto the expected payo functon. Whle analyzng decson makng processes of stock exchange nvestors we have to take nto account what nformaton they can use durng the decson process. Ths nformaton s used to estmate the behavour of future prces of underlyng assets, and, consequently, players expected payos. In order to buld a model we have to formalze all descrptons of formaton of
9 Stock market as a dynamc game wth contnuum of players 9 expectatons. When ths ssue s concerned, we shall consder ve general types of players: fundamental, techncal, econometrc, portfolo and stochastc, and the rst letter wll be used as a type ndex k. The symbol kω) denotes the type of formaton of expectatons of player ω. We shall dene the expected utlty functon of players of type k U k : I k P D k R, where I k s a specc form of processed nformaton used by type k. The form of ths functon depends on type snce the form and nterpretaton of nformaton changes. The nformaton used by type k durng the game consttutes a functon I k : Σ T I k such that I k, t) s ndependent of s) for s t. The specc form of nformaton, general constrants on the strategy sets and the expected payo functons for ve types of formaton of expectaton are as follows. 1. Fundamental players. Ther nformaton s a vector of fundamental values of n shares f R n +, whch s not based on prces of share. They are the knd of players watng for results n a long tme horzon, therefore they do not keep lqud money they nvest only n bonds and shares.e. e 0 a constrant on ther avalable decsons' set ). The expected payo s dened by U f f, p, BM, BB, S, e)) = f p 1 + C) 2) q BB =1,...,n p f p 1 C) 2) ) q S 1 p S p 1 p BB p + + f p ) q BM 1 p BM. The rst part corresponds to buyngforbonds order, therefore the commsson s pad twce, the second s buyngformoney, therefore no commsson s subtracted otherwse fundamental players wll also have to pay t n order to buy bonds, n the sellng order commsson s pad twce agan snce fundamental players wll have to buy bonds for money: n ths case for each share we get prot compared to the fundamental value) p C p C 1 C)p ) f whch equals f p 1 C) 2). Ths explans the general rule of denng payos the expected payo of each order s the derence between ths order and dong nothng wth nterpretaton specc to ths type. Smlarly we dene the remanng payos. 2. Techncal players. They use some technques of techncal analyss, based on past prces and volumes. Ther nformaton n our model wll be represented as the vector p R n of expected changes of prce of n shares) of mnmal absolute value. Techncal players look for short perod trends, therefore n our model they do not nvest n bonds they want to have lqud money to react at once snce sellng bonds s costly), whch s represented by e 1. U t p, p, BM, BB, S, e)) = = ) p t 1) + p p 1 + C)) q BM 1 p BM p + q BB 1 p BB p + =1,...,n p t 1) + p p 1 C)) q S 1 p S p.
10 Stock market as a dynamc game wth contnuum of players Econometrc players. We do not assume that a consderable porton of stock exchange nvestors have economc or mathematcal educaton sucent to buld an econometrc model. Ths type of players use a ready programme usng an econometrc model, and they do not reestmate t durng the game. The programme predcts prces P t + j) for τ perods wth declared accuracy w. Econmetrc players n ths model treat w as the number that has to be subtracted from the estmated future prce when they consder a buyng order and added to the estmated prce when they consder a sellng order. Ther nformaton s a vector of maxmal dscounted prces for the prognoss perod p = max P b t+j) j=1,...,τ. As fundamental players they do not keep 1+r) j lqud money they nvest only n bonds and shares: e 0. U e p, p, BM, BB, S, e)) = = =1,...,n p w p 1 + C) 2) q BB + p w p ) q BM 1 p BB p + 1 p BM p p + w p 1 C) 2) q S 1 p S p ). 4. Portfolo players. They know models of portfolo analyss, ncludng CAPM and they try to use t for predctng prces. The problem s that n CAPM the dstrbuton of future prce s known, especally the expected return R. In our model the players know the varance of returns as well βcoecent for all shares, and consequently, the vector of expected returns accordng to CAPM, denoted by ρ. At each stage of the game they calculate the average return for last l perods R for each share whch consttute ther nformaton R) and compare t wth ρ. As fundamental and econometrc players they do not keep lqud money they nvest only n bonds and shares: e 0. U p R, p, BM, BB, S, e) ) = = 1 ) ) 2 + R p t 1) p 1 + C) 2 ρ p q BB 1 p BB p + =1,...,n 1 ) ) R p t 1) p ρ p q BM 1 p BM p + 1 ) ) 2 + R p t 1) p 1 C) 2 ρ p q S 1 p S p ). 5. Stochastc players. In our model t wll be a type descrbng all knds of fortunetellers clents. Stochastc players obtan only clear sgnals buyng +1, sellng 1 or no sgnal 0) whch are realzaton of some random varables. These random varables n common consttute a Young measure see e.g. Valader [25]), whch mples that the set of players obtanng the same sgnal at each tme nstant s measurable. We do not assume that the sgnals observed by varous stochastc players are ndependent. We only assume that the measures of sets of players obtanng buyng and sellng sgnals are postve wth probablty 1 and wth hgh probablty detached from 0 and that sgnals obtaned n derent tme nstants are ndependent. Ther nformaton s the sgnal s they obtaned. As techncal players, they do not nvest n
11 Stock market as a dynamc game wth contnuum of players 11 bonds: e 1. For smplcty each type of stochastc players wll nvest n only one company. U s s, p, BM, BB, S, e)) = = =1,...,n 2 h s p t 1) C p ) q BM ) 1 p BM p + q BB 1 p BB p q S 1 p S p. For a prole we ntroduce the symbol G t for the game wth the same set of players, players strategy sets D ) ω X ω ) t), and payo functons Πω p, d) = U kω) I kω), t), p, d). Ths game s called subgame wth dstorted nformaton of our game G. 3. Results Here we present two concepts of equlbra wth applcatons to our model Nash equlbra and belefdstorted Nash equlbra. The basc concept of game theory s Nash equlbrum. Denton 1. A prole s a Nash equlbrum f for a.e. ω Ω, for every prole such that ν) = ν) for ν ω we have Π ω ) Π ω ). However, all Nash equlbra n our game are not very nterestng and they are far from realty at a Nash equlbrum the stock exchange cannot operate. Theorem 1. Consder a game n whch players have dentcal avalable strategy sets and T ω. If C > 0 and the maxmal payo that can be attaned by the players durng the game s nte, then at every Nash equlbrum for {1,..., n} and every t T the volume s 0. If, moreover, esssup ω Ω,q BM esssup ω Ω,q BB ω,t)>0 p BB then esssup ω Ω,q BM esssup ω Ω,q BB ω,t)>0 p BM ω,t)>0 p BB ω,t)>0 p BM ω, t), essnf ω Ω,q S ω,t)>0 p S ω, t) and ω, t) are n the nterval [1 h) p t 1), 1 + h) p t 1)] ω, t) < essnf ω Ω,q S ω,t)>0 p S ω, t) and ω, t) < essnf ω Ω,q S ω,t)>0 p S ω, t). Proof. Let us consder a Nash equlbrum prole wth a trajectory of prces p. Let us assume that at tme t player ω sells a postve amount q S ω, t).e. he has p S ω, t) p t)) whle player ν buys q BM ν, t) > 0 for money.e. he has p BM ν, t) p t)). Frst let us show that at equlbrum t s mpossble that a player outsde a set of measure 0) both buys and sells shares at the same tme nstant,.e. that such a stuaton s mpossble for ν = ω. Let us assume the converse and let us denote by q the mnmum of q BM q S ω, t). If player ω decreases both q BM ω, t) and ω, t) and q S ω, t) by q, then he ncreases hs
12 Stock market as a dynamc game wth contnuum of players 12 nstantaneous payo at tme t by q 1 + C) p t) q 1 C) p t) = 2 C p t) > 0. At equlbrum the set of players who do not maxmze ther payos s of measure 0. Now let consder two players ω and ν. Now let us consder a change of strategy of player ω such that nstead of sellng share at tme t, he repeats the part of strategy of player ν resultng from buyng t, multpled by a coecent q = qs ω,t). In order q BM ν,t) to precse what we mean, we label the money obtaned from sellng t by player ν, bonds or shares bought for ths money and so on, recursvely. Ths labellng does not have to be unque, but t exsts. The part of payo of player ν resultng from the labelled transactons dscounted for t 0 ), V ν, has to fulll V ν p t) q BM ν,t), snce 1+r) t t 0 otherwse t s better for player ν not to buy share but stay wth money f t s avalable n hs strategy set) or buy bonds nstead. Now let us explan what we mean by repeatng the labelled part of strategy of player ν by player ω. Let us consder the orders for any share j. At tme t we change only q S ω, t) to 0. For any tme s > t for whch p S j ν, s) > p j s), p BM j ν, s) < p j s) or p BB j ν, s) < p j s) we do not change the correspondng orders for share j. Otherwse, we have the followng stuatons. 1. The prce lmt n the sellng order fulls p S j ν, s) p j s). Let q denote the labelled part of qj S ν, s). If p S j ω, s) p j s), then we change only qj S ω, s) to qj S ω, s) + q q. Otherwse, we change p S j ω, s) to p j s) and qj S ω, s) to q q. 2. The prce lmt n the BM order fulls p BM j ν, s). labelled part of q BM j If p BM j we change p BM j ω, s) p j s), then we change only q BM j ω, s) to p j s) and q BM j 3. The prce lmt fulls p BB j qj S ν, s). If p BB j ω, s) p j s), then we change only q BB j ν, s) p j s). Let q denote the ω, s) to qj BM ω, s)+q q. Otherwse, ω, s) to q q. ν, s) p j s). Let q denote the labelled part of ω, s) to q BB j ω, s)+q q. Otherwse, we change p BB j ω, s) to p j s) and qj BB ω, s) to q q. The payo of player ω ncreases by V ν q but decreases by the payo correspondng to the the part of strategy resultng from sellng share at tme t dscounted for t 0, V ω 1 C), whch we dene analogously, by labellng the part of strategy of player ω resultng from the money obtaned for share. Now we assume that player ν, nstead of buyng share for money at tme t repeats the labelled transactons of player ω, multpled by, analogously to the form we have dened for player ω. By ths he 1 q ncreases hs payo by Vω q wthout multplyng by 1 C) snce he does not have to pay commsson) but decreases t by V v. At equlbrum the set of players that can mprove ther payos by changng ther decson s of measure 0, therefore for a.e. such ω and ν, we have both V ν q V ω 1 C) 0 and Vω q V ν 0, whch s
13 Stock market as a dynamc game wth contnuum of players 13 mpossble for C 0, 1). For q BB ω, t) > 0, the reasonng s analogous. Snce Nash equlbrum seems unrealstc n the context of a stock exchange, as n WsznewskaMatyszkel [42] and [43], we ntroduce another concept of equlbrum, takng the dstorted nformaton structure nto account. Denton 2. A prole s a belefdstorted Nash equlbrum f for every t T, a.e. ω Ω and every d D ω X ) ω t) ) we have U kω) I kω), t), p t), ω t) ) U kω) I kω), t), p t), d ). Note that for a belefdstorted Nash equlbrum, all statc proles t) are Nash equlbra n G t, correspondngly. Theorem 2. If C > 0 and a.e. player ω s of the same type of formaton of expectatons, then at every belefdstorted Nash equlbrum for every t the volume s 0. If, moreover, esssup ω Ω,q BM esssup ω Ω,q BB ω,t)>0 p BB then esssup ω Ω,q BM esssup ω Ω,q BB ω,t)>0 p BM ω,t)>0 p BB ω,t)>0 p BM ω, t), essnf ω Ω,q S ω,t)>0 p S ω, t) and ω, t) are n the nterval [1 h) p t 1), 1 + h) p t 1)] ω, t) < essnf ω Ω,q S ω,t)>0 p S ω, t) and ω, t) < essnf ω Ω,q S ω,t)>0 p S ω, t). Proof. After substtutng the specc form of the expected utlty functon for every type of formaton of expectatons t becomes an easy calculaton. In WsznewskaMatyszkel [42] and [43] equvalence theorems were stated between Nash equlbra and belefdstorted Nash equlbra along the perfect foresght path. In ths paper a smlar result can be proven. However, t requres an explct formulaton of the belef correspondence, omtted here for concson Threshold prces and weak domnance. We start our nvestgaton of the model by denng a mnmal protable prce n a sellng order ps k I) gven nformaton I as well as maxmal protable prce n both buyng orders pbm k I) for buyng for money and pbb k I) buyng for bonds. Denton 3. a) A prce ps k I) s the threshold prce for sellng order for players of type k at nformaton I f for every strategy δ wth p S = ps k I) and q S postve, and
14 Stock market as a dynamc game wth contnuum of players 14 a strategy δ derng from δ only by p S and wth p S < ps k I) we have U k I, p, δ ) > U k I, p, δ) for some p P n and U k I, p, δ ) U k I, p, δ) for all p P n. b) A prce pbm k I) s the threshold prce for buyng for money order for players of type k at nformaton I f for every strategy δ wth p BM = pbm k I) and q BM postve, and a strategy δ derng from δ only by p BM and wth p BM > pbm k I) we have U k I, p, δ ) > U k I, p, δ) for some p P n and U k I, p, δ ) U k I, p, δ) for all p P n. c) A prce pbb k I) s the threshold prce for buyng for bonds order for players of type k at nformaton I f for every strategy δ wth p BB = pbb k I) and q BB postve, and a strategy δ derng from δ only by p BB and wth p BB > pbb k I) we have U k I, p, δ ) > U k I, p, δ) for some p P n and U k I, p, δ ) U k I, p, δ) for all p P n. Now we shall calculate the threshold prces for all types of players, gven ther nformaton. In order to smplfy the notaton, we shall ntroduce two symbols: f a s a nonnegatve real then by succa) = mn p P,p a p and by preda) = max p P,p a p. Proposton 3. Threshold prces gven nformaton of the form correspondng to the type are as follows. a) For fundamental players ps f f f ) = succ ), 1 C) 2 pbm f f ) = pred f ), pbb f f f ) = pred ). 1+C) 2 ) b) For techncal players ps t p s, p, p ) = succ + p, 1 C) ) pbm t s, p, p ) = pbb t p f ) = pred + p. 1+C) ) c) For stochastc players ps s p s, p ) = succ +2hs, 1 C) ) pbm s s, p ) = pbb s p s) = pred +2hs. 1+C) d) For econometrc players ps e bp p ) = succ +w ), 1 C) 2 pbm e p ) = pred p w), pbb e bp p ) = pred w ). 1+C) 2 e) For portfolo players ps p R 1+ R) 2 p t 1), p t 1)) = succ 1 C) 2 +ρ ), pbm p R, p t 1)) = pred 1+ R) 2 p t 1) 1+ρ ), pbb p R, p t 1)) = pred 1+ R) 2 p t 1) 1+C) 2 +ρ ). Proof. We shall state the proof for fundamental players. For the remanng players t s analogous.
15 Stock market as a dynamc game wth contnuum of players 15 Frst let us consder the part of the expected payo correspondng to the sellng order for the th share f p 1 C) 2) q S 1 p S p for postve q S. It ncreases wth p S for p S p and s 0 for p S > p. If we restrct our attenton to comparng decsons derng only by the prce n ths order, the remanng parts of the expected payo do not change. Ths part s nonnegatve f f p 1 C) 2) 0,.e. p f. The 1 C) 2 f lowest prce at whch t s satsed s succ ). Let us take a decson d wth 1 C) ) 2 p S f = succ and d derng from d only by p S f 1 C) 2 < succ ). If the actual 1 C) 2 f prce p succ ), then both orders wll be admssble and for the decson d 1 C) 2 the correspondng part of the expected payo wll be nonnegatve, whle for d t wll f be negatve. If the actual prce p < succ ), then the correspondng part of 1 C) 2 the expected payo for d wll be 0, whle for d t wll be nonpostve. Therefore the threshold prce n sellng order s ps f f f ) = succ ). 1 C) 2 To get nonnegatvty of the correspondng part of the expected payo for BM order we take f p 0, therefore the prce lmt wll be pred f ). f For BB order, analogously, we get pred 1+C) 2 ). The noton of threshold prce mples the followng weak domnance results. Proposton 4. Assume that at tme nstant t for a past realsaton of a prole player ω of type k has portfolo x ω wth nonzero x ω and hs nformaton s I. a) If ps k I) [1 h) p t 1), 1 + h) p t 1)], then every strategy such that p S ps k I) or q S < x ω s weakly domnated n G t. b) If ps k I) < 1 h) p t 1), then every strategy such that p S > succ 1 h) p t 1)) or q S < x ω s weakly domnated n G t. Proof. a) As whle calculatng the threshold prces, we compare strateges n G t derng only by the prce and amount n the sellng order for share and the correspondng part of the payo functon. In all cases the payo s constructed such that ths part may be consdered separately. Note that for a strategy d wth p S = ps k I) and q S > 0 t s always nonnegatve, whle for any market prce hgher than ps k I) t s strctly postve. For a strategy d derng only by p S wth p S > ps k I) at the market prce lower than p S the order wll not be executed, therefore ths part of the payo wll be 0
16 Stock market as a dynamc game wth contnuum of players 16 less than for d), whle at the market prce hgher than p S payos for d and d wll be dentcal. For a strategy d derng only by p S wth p S < ps k I) at the market prce greater or equal to p S, the correspondng part of the payo wll be negatve, whle for d t s nonnegatve. At the market prce less than p S the correspondng part of the payo for both strateges wll be 0. Ths completes the proof that not sayng the threshold prce n sellng order s weakly domnated. Now we compare d wth a strategy d such that p S = ps k I) and q S < x ω. The coecent at q S s always nonnegatve and at some prces postve, therefore the maxmum s obtaned at the constrant q S = x ω. b) An analogous reasonng holds for the threshold prce below the lower varablty lmt. It s the result of the fact that the market prce must be at least 1 h) p t 1). The analogous fact for buyng orders does not hold. One of the reasons s that money or bonds can be used for buyng all knds of shares. Even f we assume that a player nvests only n shares of one company or ts money and bonds are labeled n the sense that the fracton of them that can be nvested n shares of each company s prevously dened, such a fact wll not hold. The reason s the constrant: sayng a lower prce players can buy more shares, f the market prce happens to be less or equal to the prce lmt. However, we have to remember the fact that our order can be not executable and we shall get nothng for ths order. So we have to compare two opposte eect: moderate ncrease of the payo by ncreasng the amount and consderable ncrease of rsk of loosng sure prot. The prot from tellng a lower prce grows wth the derence, and t s the hghest, when we say the lower varablty lmt whle our threshold prce s equal to the upper varablty lmt. The threshold prce s equal to the upper lmt of varablty when we expect a consderable growth of prces. In such a stuaton tellng the least possble prce s a nonsense, and ratonal nvestors at a stock exchange surely do not behave ths way. Therefore, from now on, we add ths assumpton to the descrpton of players' strateges. Denton 4. We say that the set of avalable strateges of player ω s constraned wth respect to nformaton I f p BM pbm kω) I) and p BB pbb kω) I). Proposton 5. Assume that a tme nstant t gven the past realsaton of a prole player ω of type k has nformaton I. a) If player's portfolo x ω has postve x ω n+2 and pbm k I) [1 h) p t 1), 1+h) p t 1)] and s the only share consdered by ω such that pbm k j I) 1 h) p j t 1),
17 Stock market as a dynamc game wth contnuum of players 17 then each strategy of ω wth p BM pbm k I) or q BM < xω n+2 s weakly domnated p BM 1+C) n G t wth the set of strateges of ω constraned wth respect to I. b) If player's portfolo x ω has postve x ω n+2 and pbm k I) > 1 + h) p t 1) and s the only share consdered by ω such that pbm k j I) 1 h) p j t 1), then each strategy of ω wth p BM < pred 1 + h) p t 1)) or q BM < xω n+2 s weakly p BM 1+C) domnated n G t wth the set of strateges of ω constraned wth respect to I. c) If player's portfolo x ω has postve x ω n+1 and pbb k I) [1 h) p t 1), 1+h) p t 1)] and t s the only share consdered by ω such that pbb k j I) 1 h) p j t 1), then each strategy of ω wth p BB pbb k I) or q BB < 1 C) xω n+1 s weakly domnated p BB 1+C) n G t wth the set of strateges of ω constraned wth respect to I. d) If player's portfolo x ω has postve x ω n+1 and pbb k I) > 1 + h) p t 1) and t s the only share consdered by ω such that pbb k j I) 1 h) p j t 1), then each strategy of ω wth p BB < pred 1 + h) p t 1)) or q BB < 1 C) xω n+1 s weakly p BB 1+C) domnated n G t wth the set of strateges of ω constraned wth respect to I. Proof. Analogous to the proof of 4. Proposton 6. Assume that a tme nstant t gven the past realsaton of a pro le player ω of type k nvestng only n share and havng constant e has nformaton I. If player's ω portfolo x ω has postve x ω n+1 and x ω n+2, the threshold prces pbm k I) and pbb k I) are greater or equal to the lower lmt of varablty and ps k I) s less or equal to the upper lmt of varablty, ) than the strategy of ω pbm k x I), ω n+2 ), p BM 1+C) pbbk I), 1 C) xω n+1 ), p BB 1+C) psk I), x ω ), e s weakly domnant n G t wth the set of strateges of ω constraned wth respect to I. Proof. As of proof Implcatons for predcton From now on we shall assume that players use only strateges consstent wth ther nformaton. We shall answer the queston, what may happen f a strong.e. large and havng a consderable porton of assets) group of players uses the same prognostc technque and they obtan the same nformaton.
18 Stock market as a dynamc game wth contnuum of players 18 We assume that there s at least a small group of stochastc players. The reason s that n the case when all players have dentcal prognostc technque, the stock exchange cannot work we need at least a small fracton of players havng expectatons to some extent opposte than the majorty Selfverfyng belefs. It s obvous from ths model, but also from the real lfe, that the belefs can nuence prces. In ths context, the most nterestng thng to consder s the queston, whether and to what extent the ways of predctng prces can force the prces behave accordng to the belefs we have to match the abstract nformaton the players obtan wth ther nterpretaton of future prces. Fundamental analyss. The smplest example of selfverfyng belefs s fundamental analyss. We shall consder a game startng at tme t 0 wth a vector of reference prces pt 0 1). Assume that there s a strong group of fundamental players wth dentcal {F t)}, and assume that there s also a small group of stochastc players nvestng n, possessng as well as bonds or money. Consder any tme nstant t such that reachng the fundamental value s theoretcally possble,.e. F t) P [1 h) t t0 p t 1), 1 + h) t t0 p t 1)]. Frst, we have to dene what we understand by a strong group of players n G t a group that can domnate the market. Denton 5. We call a set of players Ω Ω strong n G t a) n share for = 1,..., n) f Ω1 h) p t 1) X ω t)dλω) Ω\ Ω Xω n+1t) 1 C) + Xn+2t)dλω); ω b) n bonds f Ω Xn+1t) 1 C)dλω) ω n c) n money f Ω X ω n+2t)dλω) n =1 =1 1+h) p Ω\ Ω t 1) X ω t)dλω); 1 + h) p Ω\ Ω t 1) X ω t)dλω); 1 + h) d) n rsk free assets Ω Xn+1t) ω 1 C) + Xn+2t)dλω) ω n =1 p t 1) X ω t)dλω). Ω\ Ω Denton 6. A set of players Ω Ω s strong n assets) f for every t sup ω Ω T ω and every prole the set Ω s strong n G t n assets). Proposton 7. Let Ω be a set of fundamental players wth dentcal F t) and let be a belef dstorted Nash equlbrum. a) If Ω s strong n n G t, then p t) wll not exceed maxps f F t)), 1 h) p t 1)). b) If Ω s strong n money n G t, then p t) wll not be less than mnpbm f F t)), 1+ h) p t 1)), c) If Ω s strong n bonds n G t, then p t) wll not be less than mnpbb f F t), 1+ h) p t 1))).
19 Stock market as a dynamc game wth contnuum of players 19 Proof. The probablty that a set of stochastc players possessng shares of postve measure wll get a sellng sgnal and the probablty that a set of stochastc players of postve measure possessng money or bonds wll get a buyng sgnal are equal to 1. Let us note that the threshold sellng prce for stochastc players gettng sellng sgnal s below 1 h) p t 1). Therefore we shall have some sellng orders wth the prce lmt greater or equal to lower lmt of varablty as well as some buyng orders wth the prce lmt greater or equal to upper lmt of varablty. Therefore at each prce greater or equal to Ps f F t)) the volume s equal to the demand, whch s nonncreasng. Assume that prce of at tme t s equal to p > Ps f F t)). Ths would mply the demand s constant at the nterval [Ps f F t)), p ], as well as the dsequlbrum. Now let us check crteron 3. In our case we want to mnmse the number of shares n sellng order wth prce lmt greater than the market prce. The mnmum cannot be attaned at p, only n Ps f F t)), whch contradcts our assumpton. b) and c) are proven analogously. Frst we assume that a lower prce was chosen. In ths case the volume s equal to the supply. Thus t s constant at the correspondng nterval, as well as dsequlbrum, but then crteron 3 s not satsed. Thus we get fast convergence to qute a narrow nterval of prces. Techncal analyss. Smlar selfvercaton results can be proven for techncal analyss. Nevertheless, they cannot be treated as a proof of valdty of techncal analyss as a cognton devce. Formaton of cat. In order to show how techncal analyss can make the prces behave as t predcts we shall show an abstract formaton, prevously dened n WsznewskaMatyszkel [40], and consder the results of ts popularzaton among nvestors. Ths formaton has not exsted n techncal analyss and s not reected by data. It wll be formulated as n textbooks on techncal analyss and explaned by a smlar quassocologcal explanaton see e.g. Prng [19]) and t wll turn out to be approxmately selfverfyng. Formaton of Cat starts by a moderate ncrease of prces of shares back of the neck), then prces rapdly rce, and afterwards fall left ear), then there s a at summt crown of the head) and the thrd summt smlar to the rst one rght ear), endng by a moderate fall of prce forehead) startng from the base of the rght ear and lastng at least as long as the rght ear. The volumes at the crown of the head are always low. If the volume at the top of the rght ear s less than at the top of the left ear, then the cat s lookng down, f the converse holds, the cat s lookng up. Snce cats
20 Stock market as a dynamc game wth contnuum of players 20 are contrary anmals, cats lookng up forecast fall of prces, whle cat lookng down forecast rse of prces, and the absolute value of changes s at least one and a half of the heght of the ears. Fgure 1 Now we construct a quassocologcal explanaton as from textbooks on techncal analyss. A moderate but qute stable ncrease of prces causes an exaggerated optmsm among players, whch ncreases the demand. At the top of left ear strong better nformed) players sell ther shares to weak worse nformed) players, consttutng majorty. Then there s a correcton and weak players sell ther shares. When the prce reaches the level of the end of the back of the neck, players observe the market watng for sgnals, therefore the volume s low. If the optmsm wns, the rght ear s formed. Hgh volume at rght ear means strong dstrbuton : strong players sell ther shares to weak players, whch are prone to panc n the case of fall of prces. Low volumes at rght ear mean that the majorty of shares s n the hands of strong players, whch usually do not panc, snce by ther nformaton they expect ncrease of prces. To smplfy the analyss, we assume that we consder only players nvestng n share. We shall denote the heght of the ears by U. Assume that techncal players usng
21 Stock market as a dynamc game wth contnuum of players 21 the cat formaton ether have no further sgnals or treat them as less mportant than the cat formaton and that there s also a small set of stochastc players possessng shares and rsk free assets. Proposton 8. Let be a relsaton of a prole and let t be a tme nstant at whch the cat formed as a result of playng up to t. If the set Ω of techncal players belevng n cat formaton s strong n rsk free assets n G t, then at every belef dstorted Nash equlbrum ) the cat lookng down mples an ncrease of prce of at p t 1)+ least to pred 3 2 U, whle f Ω s strong n n G 1+C) t, then the cat lookng up ) p t 1) mples a decrease of prces at least to succ 3 2 U. 1 C) Proof. Let us consder the cat lookng down. Snce techncal players expect ncrease ) of p t 1)+ prce, ther threshold prce at each tme nstant s equal to pred 3 2 U. Frst 1+C) t can be above the upper varablty lmt. In each of such tme nstants t the prce lmt n buyng orders of techncal players wll be equal to pred p t 1) 1 + h)). As n the proof of proposton 7, we get that the market prce s equal to the prce lmt of the strongest group of players. Fnally, techncal ) players wll have the prce p t 1)+ lmt equal to the threshold prce pred 3 2 U, whch wll be the market prce. 1+C) The reasonng for the cat lookng up s analogous. Strong sgnals n techncal analyss. In the case of strong sgnals n techncal analyss, especally when techncal players expect a change of the trend, they expect changes of prces of large absolute value. Proposton 9. Let be a belef dstorted Nash equlbrum and let t be a tme nstant at whch a strong sgnal was observed and ndentcally nterpreted as p by a set Ω of techncal players. a) Assume p << h p t 1) a sellng sgnal). If Ω s strong n n G t and there s a set of stochastc players of postve measure nvestng n ths company stll possessng rsk free assets at t, then wth probablty 1 prces of share wll fall and the fall wll be to at least succ p t 1)+ p 1 C) b) Assume p >> h p t 1) a buyng sgnal). If Ω nvests only n company or for other companes j consdered by players from Ω pbm k j I) < 1 h) p j t 1) and f Ω s strong n rsk free assets n G t and there s a set of stochastc players of postve measure stll possessng at t, then wth probablty ) 1 prces of wll grow and the ncrease wll be to at least pred p t 1)+ p. 1+C) ).
22 Stock market as a dynamc game wth contnuum of players 22 Proof. The proof s analogous to that of the cat formaton Selffalsfyng belefs. Here we want to show that not all belefs are selfverfyng. To smplfy the analyss, we agan consder players nvestng n share only, and money or bonds, and assume that they consder strategy sets constraned wth respect to nformaton. CAPM. Now we shall consder the case n whch there s a strong group of portfolo players and a small group of stochastc players. We also assume that C s small. The basc result n the papers about CAPM cted n the ntroducton, s that prces adjust such that the return of each asset s equal to ts theoretcal ρ. However, there was assumpton that there s an equlbrum and no dynamcs was consdered. We get the result, that n the case of startng from aggregate returns derng from ρ, we do not have to converge to t. Conversely, rather dvergence can be expected. Proposton 10. Let be a relsaton of a prole, let t be a tme nstant and let Ω be a set of porfolo players. Portfolo analyss s selffalsfyng n the sense, that a) f R s essentally greater than ρ, Ω s strong n money n G t and there s a set of stochastc players of postve measure nvestng n stll possessng at t, then R t) wll be greater than R ; b) f R s greater than ρ + C 2 + 2C, Ω s strong n rsk free assets n G t and there s a set of stochastc players of postve measure nvestng n stll possessng at t, then R t) wll be greater than R ; c) f R s essentally less than ρ + C 2 2C, Ω s strong n n G t and there s a set of stochastc players of postve measure nvestng n stll possessng rsk free assets at t, then R t) wll be less than R. Proof. a) Here R > ρ and portfolo players are strong n money. In ths case we shall calculate ther return n the case when the market prce equals ther threshold prce pbm p R, p t 1)). Then the return at tme) t fullls 1+ R) 2 p t 1) pred 1+ρ p t 1) R t) = 1+ R ) 2 1+ρ p t 1) 1 ε p, where ε s a small t 1) number denng the precson of prce representaton n the part of P under consderaton,.e. such a number that for p = 1+ R ) 2 p t 1) 1+ρ pred p ) p ε. If the
23 Stock market as a dynamc game wth contnuum of players 23 derence between R and ρ s large enough, then 1+ R ) 2 1+ρ ε > 1+ R ) 2 p t 1) 1+ R = R, therefore R t) > R. In the buyng for money orders of portfolo players the prce lmt s equal to the threshold prce. As n the proof of proposton 7, we get that the market prce s greater or equal to the prce lmt n the buyng orders of the strongest group of players, n ths case the threshold prce pbm p R, p t 1)) for the porfolo players. b) Now let us assume a greater derence R > ρ + C 2 + 2C and let us assume that portfolo players are strong n bonds. If the market prce equals ) the threshold prce pbb p R, p t 1)), then 1+ R) 2 p t 1) pred 1+C) 2 +ρ p t 1) R t) = p t 1) 1+ R ) 2 1+C 2 +2C+ρ 1 ε p. If the derence t 1) between R and ρ +C C s large enough, then R ) 2 1+C 2 +2C+ρ ε > 1+ R ) 2 p t 1) 1+ R = R, therefore R t) > R. The market prce wll be greater or equal ether to pbb p R, p t 1)) or pbm p R, p t 1)) f Ω Xn+2t)dλω) ω > 0), for whch we have already proven the nequalty. c) Now let us consder the case when C 2 2C + ρ > R and ) Ω s strong n. The threshold prce ps p R 1+ R) 2 p t 1), p t 1)) s succ, therefore f the R t) = 1 C) 2 +ρ ) market prce s equal to ths ) threshold prce, the return fullls 1+ R) 2 p t 1) succ 1 C) 2 +ρ p t 1) p t 1) 1+ R ) 2 1 C) 2 +ρ 1 + ε = 1+ R ) 2 p t 1) 1+C 2 2C+ρ 1 + ε, for ε such that for p p t 1) = 1+ R ) 2 p t 1) 1 C) 2 +ρ succ p ) p +ε. If the derence between ρ +C 2 2C and R 1+ s large enough, then R ) 2 1+C 2 2C+ρ + ε < 1+ R ) 2 p t 1) 1+ R = R, therefore R t) < R. Analogously to the reasonng for the buyng orders, the market prce s less or equal to the prce lmt of sellng order of portfolo players ps p R, p t 1)). The facts stated n proposton may lead to trends of acceleratng ncreases or acceleratng decreases of prces. Econometrc models. We cannot state anythng precse about econometrc models n general. Dependng on the specc type of the model they can be ether approxmately selfverfyng or selffalsfyng. If we treat them lterally, they wll be usually selffalsfyng: ncreases and decreases of prces are pror to the moment they were prognosed for. Nevertheless, econometrc models used as tools to foresee general
24 Stock market as a dynamc game wth contnuum of players 24 tendences are approxmately selfverfyng. 5. Numercal smulatons The followng smulatons were made wth the programme SGPW [22]. In each of them we assumed exstence of a small group of stochastc players wth constant ow of money and possessng a small fracton of shares consdered Convergence to the fundamental value. The gures below llustrate convergence to the fundamental value gven the ntal prce of a share from WSE) n the game wth a large group of fundamental analysts. Fgure 2 Fgure Trends caused by chartsts. A group of chartst and trends caused by them gven varous ntal values form WSE: Fgure 4 Fgure 5
25 Stock market as a dynamc game wth contnuum of players 25 For comparson, f we consder stochastc players only, we get somethng smlar to a random walk: at each tme nstant we ether go up the upper varablty lmt f the measure of the set of players obtanng sellng sgnal s less than the measure of the set of players obtanng the buyng sgnal or to the lower varablty lmt f the measure of the set of players obtanng sellng sgnal s greater than the measure of the set of players obtanng the buyng sgnal Trends caused by portfolo players. For the case of a strong group of portfolo players the results are exactly as stated n the model ether an exponental growth of the prces or an exponental decrease Some econometrc models. In ths case we present two econometrc models: one of them consderng lnear trend and snusodal weekly perodcty and length of prognoss 2, and the other one wth the average of some of past prces. The former one s approxmately selfverfyng only because the lnear trend domnates. However, the oscllatons are translated. The latter one becomes selfverfyng after a perod of transton. Fgure 6 Fgure 7 6. Conclusons The paper presents a model of stock exchange as a game wth a contnuum of players takng nto account varous prognostc technques. The contnuum was used to model nsgncance of any sngle player, whle prces, and consequently, players payos are solely a result of players decsons. One of the results of the paper s that usually the strateges of tellng the actual threshold prces are weakly domnant, whle strateges of not tellng the actual threshold prces are weakly domnated n a sequence of subgames wth dstorted nformaton along the prole, therefore they consttute a belef dstorted Nash equlbrum.
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