Dyamical Agets Strategies ad the Fractal Market Hypothesis Lukas Vacha, Miloslav Vosvrda Departmet of Ecoometrics Istitute of Iformatio Theory ad Automatio Academy of Scieces of the Czech Republic Prague Abstract: It has bee show i may papers ([], [8] etc.) that the Efficiet Market Hypothesis (EMH) fails as a valid model of fiacial markets. The Fractal Market Hypothesis (FMH) is i a place as a more geeral alterative way to the EMH. The FMH ca be formed o the followig parameter: agets ivestmet horizos. It lead us to coclude that a fiacial market is more stable whe we adopt this fractal character i structures of aget s ivestmet horizos. For computer simulatios, the Brock ad Hommes model [2] is modified. This adjusted model shows that various frequecy distributios o agets ivestmet horizos lead to differet returs behavior. The FMH focuses o matchig of demad ad supply of agets ivestmet horizos i the fiacial market. It is the corerstoe that holds fiacial markets together. The EMH assumes the market is at equilibrium. The FMH o the other had asserts that ivestors have a iformatio differetly based o temporal attributes. Sice all ivestors i the market have differet time ivestmet horizos, the market remais stable. Our simulatios of probability distributios of agets ivestmet horizos demostrate that may ivestmet horizos esure stability of the fiacial market. The behavior of the model uder dyamical chages of agets tradig strategies is aalyzed. Keywords: Efficiet Market Hypothesis, Fractal Market Hypothesis, agets ivestmet horizos, agets tradig strategies JEL:C6; G4; D84 A fiacial support from Grat Agecy of Charles Uiversity, Prague uder the grat 454/24/ A EK/ FSV ad from Grat Agecy of the Czech Republic uder the grat 42/3/H57 ad Metrostav a.s. is gratefully ackowledged.
. Itroductio The EMH was paradigm of ecoomic ad fiace theory for the last twety years. After empirical data aalysis o fiacial markets ad after theoretical ecoomic ad fiace progress this paradigm is gotte over. There are pheomea observed i real data collected from fiacial markets that caot be explaied by the recet ecoomic ad fiace theories. Oe paradigm of recet ecoomic ad fiace theory asserts that sources of risk ad ecoomic fluctuatios are exogeous. Therefore the ecoomic system would coverge to a steady-state path, which is determied by fudametals ad there are o opportuities for speculative profits i the absece of exteral shocks prices. It meas that the other factors play importat role i a costructio of real market forces as heterogeeous expectatios. Sice agets o have sufficiet kowledge of a structure of the ecoomy to form correct theoretical expectatios, it is impossible for ay formal theory to postulate uique value expectatios that would be held by all agets [6]. Prices are partly determied by fudametals ad partly by the observed fluctuatios edogeously caused by o-liear market forces. This implies that techical tradig rules eed ot be systematically bad ad may help i predictig future price chages. Developmets i the theory of o-liear dyamic systems have cotributed to ew approaches i ecoomics ad fiace theory [3]. Itroducig o-liearity i these models may improve research of a mechaism geeratig the observed movemets i the real fiacial data. The fiacial market exhibits local radomess but also global determiism. Thus fiacial markets ca be cosidered as oliear dyamic systems of the iteractig agets processig ew iformatio immediately. Ivestors with the same ivestmet horizos, ad holdig similar positios i the market may utilize this iformatio differetly. Therefore a fiacial market has a fractal structure i ivestmet horizos. For a aalyzig of behaviour of such market, a adjusted versio of the model, itroduced by Vacha ad Vosvrda i the [4] with two mai types of traders, i.e., fudametalists, ad techical traders, is used. Techical traders ted to put little faith i strict efficiet markets. Fudametalists rely o their model employig fudametal iformatio basis for forecastig of the ext price period. The traders determie whether curret coditios call for the acquisitio of fudametal iformatio i a forward lookig maers, rather tha relyig o post performace. This approach relies o heterogeeity i the aget iformatio ad subsequet decisios either as fudametalists or as chartists. Chagig of the chartist s profitability ad fudametalist s positios is a
basis of the cycles behaviour. A more detailed aalysis is itroduced i the Brock ad Hommes model. This model with memory was aalyzed i [5]. The model is preseted i a form of evolutioary dyamics of price model. The fudametalists are cosidered as traders with more rich structure of memory for price predictio. The chartists are cosidered as traders with more simple structure of memory for a price predictio. A simulatio aalysis of this model uder chagig probability properties of memory shows coectios betwee EMH ad FMH. Sectio 2 is devoted to dyamics of fractios of differet traders. Aget s ivestmet horizos with a differet form of memory structures i the performace measure are aalyzed. Fractal structure of fiacial markets is show i Sectio 3. Results of the simulatio aalysis are itroduced i Sectio 4.
2. Dyamics of Traders Let us cocetrate o dyamics of the fractios h,t of differet h-trader types, i.e. (,..., ) a x = f x x f t h, t h t t L h, t h, t h h t+ t+ pt, (2.) where h,t- deotes the fractio of trader type h at the begiig of period t, before tha the equilibrium price x t has bee observed ad a deotes a gross retur of a risk free asset which is perfectly elastically supplied, i.e., a > ad L is a umber of lags. Now the realized excess retur over period t to the period t+ is computed, where x t = p t p t *, ad p t * is the price correspodig to the itersectio poit of demad ad supply, by Z = p a (2.2) Z = x + p a x a p (2.3) * * t+ t+ t+ t t ( ) ( ) Z = x a x + p E p + E p a p (2.4) * * * * t+ t+ t t+ t t+ t t+ t From the equatio (2.4) we get * * * * ( + ) = ad δ t + pt + Et( pt + ) E p a p t t t is a martigale differece sequece with respect to ( ) Et δ t + I t =, for all t. So the Eq.(2.4) ca be writte as follows Zt+ = xt+ a xt + δt + = (2.5) I t i.e., (2.6) The decompositio of the equatio (2.6) as separatig the explaatio part of realized excess returs Z t+ ito the cotributio x t+ - a x t ad the additioal part δt+. Let a performace measure π(z t+,ρ h,t )be defied by π ht, = π ( Zt+, ρht, ) = Zt+ zt ρht, (2.7) π = x a x + δ z ρ where ( ) ht, t+ t t+ t ht, ( ) ρ ht, = Eht, Zt+ = fht, a xt ad z t deotes the umber of shares of the asset purchased at time t. So the h-performace is give by the realized profits for the h- trader.
Let the updated fractios h,t be give by the discrete choice probability ( β π ) Y ( h, t ) = exp / (2.8) ht, ht, t Yt = exp β π h (2.9) The parameter β is the itesity of choice measurig how fast agets switch betwee differet predictors. The parameter β is a measure of trader s ratioality. The variable Y t is just a ormalizatio so that fractios h,t sum up to. If the itesity of choice is ifiite (β = + ), the etire mass of traders uses the strategy that has the highest fitess. If the itesity of choice is zero, the mass of traders distributes itself evely across the set of available strategies.
3. Memory i the performace measure The performace measure is give by summatio m-values of the lagged h-performace measures i the followig form m ht, = exp β πht, p / Yt, (3.) m p= m Yt = exp β π,, h m h t p (3.2) p= where the m deotes the memory legth, ad the η is the realizatio of the radom vector of predictor memory (tradig horizos). We assume that the expressio m = E h [η] holds. All beliefs or the formatio of expectatios with differet lag legths will be of the followig form f = g x + b (3.3) ht, h t h where g h deotes the tred, b h the bias of trader type h. If b h =, the aget h is called a pure tred chaser if g > (strog tred chaser if g > a) ad a cotraria if g < (strog cotraria if g < -a). If gh =, type h trader is said to be purely biased. He is upward (dowward) biased if b h > (b h < ). I the special case g h = b h =, type h trader is called fudametalist i.e., the trader is believig that prices retur to their fudametal value. Fudametalists do have all past prices ad divideds i their iformatio set, but they do ot kow the fractios h,t of the other belief types.
4. R/S Aalysis of Aget Ivestmet Strategies For estimatig ad aalysig of correlatio structures o capital markets, a oparametric method of Hurst type is used. H. E. Hurst discovered very robust oparametric methodology which is called rescaled rage, or R/S aalysis. The R/S aalysis was used for distiguishig radom ad o-radom systems, the persistece of treds, ad duratio of cycles. This method is very coveiet for distiguishig radom time series from fractal time series as well. Startig poit for the Hurst s coefficiet was the Browia motio as a primary model for radom walk processes. For computatio of the R/S coefficiets we have to divide the time series of legth T, ito N itervals of the legth, where N=T. Values {(R/S) } are defied i the followig form where ( R S) = R N N S (4.) / k k R = max xi x xi x k k i= i= where is the rage ad ( ) mi ( ) (4.2) S i = ( ) 2 x = x (4.3) i is the sample stadard deviatio. The Hurst expoet H ca be approximated by the followig equatio log (( R/ S) ) = log( c) + H log ( ), (4.4) where c is a costat, ad = 9,,4. If a system of radom variables {(R/S) } is i.i.d, the H =.5 The values of Hurst expoet belogig to < H<.5 sigifies atipersistet system of variables coverig less space tha radom oes. Such a system must reverse itself more frequetly tha a radom process. With the assumptio of a stable mea (which we do ot impose here) we ca equate this behaviour to a mea-revertig process. Values.5 < H < show persistet process that is characterized by log memory effects. This log memory occurs regardless of time scale, i.e., there is o characteristic time scale which is the key characteristic of fractal time series []. For obtaiig of the expected {(R/S) } values we have used the follow-
ig equatio []: ER ( / S).5 π 2 ( )( ) = 2 r = r r (3.3) Moreover, we have still used the statistics V which estimates the breaks i the R/S plot i a easier way. This oe is usually used as a good measure of cycle legth i the presece of oise []. This statistics is defied as V R/S (3.4) A plot of V versus log() would be flat if the system of radom variables was a idepedet system, i.e., the R/S statistic was scalig with the square root of time. For H >.5 (persistet) the R/S was scalig faster tha the square root of time so the plot would be upward slopig. O the other had for H <.5 (atipersistet) the graph would be dowward slopig. Data used i the aalysis are geerated by the model outlied above i part 2 ad 3. I all the cases we used with twety belief types (traders), which has the tred g ad bias b geerated by radom umbers geerator with the ormal distributio N ~ (,.4) ad N ~ (,.3). The model geerates four thousads observatios, but to avoid problems with trasiets we do ot use the first four hudreds observatios. The itesity of choice (β = 8) is the same for all simulatios. Memory or tradig horizos are also radom umbers geerated by three distributios (ormal, uiform, weibull), the fourth is a give, fixed, value for all traders of memory i a particular simulatio. With each probability distributio of predictor memory, we make step by step the four simulatios with differet expected memory as follows (5,, 2, 4).To miimize a liear depedecy of raw returs, which ca bias estimates of the Hurst expoet sigificatly [], we have used AR () residuals of these oes. This procedure elimiates serial correlatio. Results of those experimets are demostrated i Tables (4.-4.4). I these tables there are two estimates of the Hurst expoet. The first oe used data from period = 9 to 4. The secod oe (i.e., Hurst mod) used data from period = 9 to break-eve poit of the V- statistics. Figures (4. 4.4) demostrate R/S ad V statistic for four differet memory distributios with the same expected memory (E[η] = 2). There are visible differeces i the positio of maximal value of the V- statistic (break-eve poit). Figures (4., 4.5-4.7), where differet memory legths are show for ormal distributio, make obvious the importace of the memory legth for the pricig behavior of the market.
Table 4.. Estimated Hurst coefficiets ad statistic of raw returs for memory mea 5 E(η) = 5 Normal (5,.25) Uiform (,) Fixed (5) Weibull (.3) Hurst.2.22.7.28 Hurst mod.858 (9-2).636 (9-2).85 (9-2).542 (9-25) Var (x).44.7.48.69 Kurtosis (x).29.9 -.39.93 Skewess (x).25.97 -.35 -.378 Table 4.2. Estimated Hurst coefficiets ad statistic of raw returs for memory mea E(η) = Normal (,2.5) Uiform (,2) Fixed () Weibull (.3) Hurst.86.64.239.46 Hurst mod.78 (9-36).727 (9-2).789 (9-36).594 (9-25) Var (x).9.29.584.27 Kurtosis (x).642.95.37.379 Skewess (x).86.4.96 -.63 Table 4.3. Estimated Hurst coefficiets ad statistic of raw returs for memory mea 2 E(η) = 2 Normal (2,5) Uiform (,4) Fixed (2) Weibull (.3) Hurst.387.3.383.38 Hurst mod.677 (9-4).732 (9-2).7 (9-45).733 (9-8) Var (x).438.732.55.778 Kurtosis (x).58 8.8 3.66 9.54 Skewess (x) -.57 -.339 -.55.67 Table 4.4. Estimated Hurst coefficiets ad statistic of raw returs for memory mea 4 E(η) = 4 Normal (4,) Uiform (,8) Fixed (4) Weibull (.3) Hurst.462.42.47.399 Hurst mod.6 (9-72).556 (9-72).66 (9-72).587 (9-45) Var (x).398.52.354.267 Kurtosis (x).245 24.652.23 7.795 Skewess (x) -.53 2.328 -.895-2.392
2 R/S aalysis.4 V - statistic.5.2 Log (R/S) V Statistic.8.5.6.5.5 2 2.5 3 3.5.4.5.5 2 2.5 3 3.5 Fig. 4.. R/S aalysis ad V statistic for memory distributio N ~ (2,5). 2 R/S aalysis.4 V - statistic.5.2 Log (R/S) V Statistic.8.5.6.5.5 2 2.5 3 3.5.4.5.5 2 2.5 3 3.5 Fig. 4.2. R/S aalysis ad V statistic for memory distributio Uiform (,4). 2 R/S aalysis.4 V - statistic.5.2 Log (R/S) V Statistic.8.5.6.5.5 2 2.5 3 3.5.4.5.5 2 2.5 3 3.5
Fig. 4.3. R/S aalysis ad V statistic for memory distributio Fixed (2). 2 R/S aalysis.4 V - statistic.5.2 Log (R/S) V Statistic.8.5.6.5.5 2 2.5 3 3.5.4.5.5 2 2.5 3 3.5 Fig. 4.4. R/S aalysis ad V statistic for memory distributio Weibull (.3), E(η) = 2. 2 R/S aalysis.5 V - statistic Log (R/S).5.5 V Statistic.5.5.5 2 2.5 3 3.5.5.5 2 2.5 3 3.5 Fig. 4.5. R/S aalysis ad V statistic for memory distributio N ~ (5,.25) 2 R/S aalysis.5 V - statistic Log (R/S).5.5 V Statistic.5.5.5 2 2.5 3 3.5.5.5 2 2.5 3 3.5
Fig. 4.6. R/S aalysis ad V statistic for memory distributio N ~ (,2.5) 2 R/S aalysis.4 V - statistic.5.2 Log (R/S) V Statistic.5.8.5.5 2 2.5 3 3.5.6.5.5 2 2.5 3 3.5 Fig. 4.7. R/S aalysis ad V statistic for memory distributio N ~ (4,) 3. 4 Weibull Distributio Number of obs. 2. 4. 4 2 4 6 8 Memory legth Fig. 4.8. Weibull distributio for memory legth with E[m] = 2.
Coclusios Short memories of predictors i.e., short aget s ivestmet horizos cause more volatile of price realizatios o capital markets, but by values of the Hurst coefficiets there exist possibilities of the price predictios due to the persistece of the fudametal strategy structures. Log memories of predictors i.e., log aget s ivestmet horizos cause more stable behaviour of price realizatios o capital markets (see Tables 4. 4.4). These tables demostrate depedecies amog aget s ivestmet horizos ad both local radomess ad global determiism. The FMH is a more geeral otatio tha the EMH. The FMH ad the EMH are equivalet i the break-eve poit of the V-statistics. Here are equivalet the Browia motio ad the fractioal Browia motio. Therefore fiacial markets are oliear systems with a fractal structure of aget s ivestmet horizos. These markets are upredictable i the log-term period, but predictable i the short-term period. The key features- the legths of memory ad probability distributio i memory- ifluece self-similarity properties i aget s ivestmet horizos are demostrated i tables (4. 4.4) ad figures (4. 4.7).
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[6] Vacha L, Vosvrda M (23) Learig i Heterogeeous Aget with the WOA. 6 th Iteratioal Coferece o Applicatrios of Mathematics ad Statistics i Ecoomy, Báská Bystrica, Slovakia [7] Zeema EC (974) The ustable behaviour of stock exchage. Joural of Mathematical Ecoomics [8] Zhou WX, Sorette D (23) Predictability of Large Future Chages i Major Fiacial Idices http://xxxlalgov/ps_cache/cod-mat/pdf/34/346pdf