SENTIMENT PATTERNS IN THE HETEROGENEOUS AGENT MODEL

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1 SENTIMENT PATTERNS IN THE HETEROGENEOUS AGENT MODEL Lukas Vaca,, Jozef Barunik,, Miloslav Vosvrda, Úsav eorie informace a auomaizace, Akademie věd České republiky, v.v.i, Insiu ekonomickýc sudií, FSV UK Praa Absrac In is paper we exended e original model of eerogeneous agen model by inroducing smar raders and canges in e agens senimen o e model. Te idea of e smar raders is based on endeavor of e marke agens o esimae e fuure price movemens. By adding smar raders and senimen canges we ry o improve e original eerogeneous agens model so i will be able o come o closer descripion of e real markes. Te main resul of e simulaions is a probabiliy disribuion funcions of e price deviaion canges significanly wi adding smar raders o e model, and i also canges significanly wi inroducion of e senimen canges. We use also Hurs exponen o measure e persisence of e price deviaions and we find a e Hurs exponen is significanly increasing wi smar raders in simulaions. Tis means a e inroducion of smar raders concep ino e model resuls in significanly iger persisence of e simulaed price deviaions. On e oer and, inroducion of canging senimen in e proposed form does no cange e persisence of simulaed prices significanly. Keywords: eerogeneous agen model, marke srucure, smar raders, Hurs exponen JEL: C5, D84, G4. Inroducion Modern finance is undergoing an imporan cange in e percepion of economic agens, i.e., from a represenaive raional agen approac owards a beavioral, agen-based approac wi markes boundedly raional, were eerogeneous agens apply rule of umb sraegies. Te radiional approac, resed on a simple analyically racable models wi a represenaive, perfecly raional agen and maemaics as e main ool of analysis. Te new beavioral approac fis muc beer wi agen-based simulaion models and compuaional and numerical meods ave become an imporan ool of analysis, Hommes (005). Te new beavioral, eerogeneous agens approac callenges e radiional represenaive raional agen framework. Heerogeneiy in expecaions can lead o marke insabiliy and

2 complicaed price dynamics. Prices are driven by endogenous marke forces. Typically, in e eerogeneous agens model (HAM), wo ypes of agens are disinguised: fundamenaliss and cariss. Fundamenaliss base eir expecaions abou fuure asse prices and eir rading sraegies on marke fundamenals and economic facors, suc as dividends, earnings, macroeconomic grow, unemploymen raes, ec. Cariss or ecnical analyss ry o exrapolae observed price paerns, suc as rends, and exploi ese paerns in eir invesmen decisions. One suc model was developed by Brock, Hommes (998). In our previous papers, Vosvrda, Vaca (00, 003), we inroduced a memory and some learning scemes o e model of Brock and Hommes. We also inroduced furer exensions, suc as socasic formaion of beliefs and parameers including e memory leng. Anoer exension is e applicaion of e Wors Ou Algorim (WOA), wic periodically replaces e rading sraegies wi e lowes performance, Vaca, Vosvrda (005, 007). In Vaca, Vosvrda (005) we sowed ow memory leng disribuion in e agens' performance measure affecs e persisence of e generaed price ime series. In is paper we inroduce a new concep of smar predicors. Te idea of e smar raders is based on endeavor of e marke agens o esimae e fuure price movemens. By adding smar raders we ry o improve e original eerogeneous agens model so i will be able o beer approximae e real markes. Te smar raders are designed o forecas e fuure rend parameer of e price deviaions using e informaion se consising of pas deviaions. Tey are modeled o assume a e deviaions are AR() process and ey use maximum likeliood esimaion meod for forecasing. Tus in our model we use groups of raders, smar raders and group of socasically generaed rading sraegies wic are moreover seleced by Wors Ou Algorim. Furermore we inroduce e canges in senimen wic we define as a sif of e beliefs abou fuure rend of newly incoming invesor sraegy o e marke. Tis allows us o model rend-followers and conrarians. In is paper we use only e form of jumps of senimen. Our main expecaion is a inroducion of smar raders and canges in senimen o e marke will cange e simulaed marke prices significanly. Te firs par of e paper concerns a eerogeneous agen model, wic is an exension of e Brock and Hommes (998) model. Te second par briefly inroduces e implemenaion of e smar raders ino e eerogeneous agen model framework. Te las par of e paper invesigaes ow e presence of smar predicors and senimen jumps qualiaively canges e marke srucure.

3 . Model Capial markes are perceived as sysems of ineracing agens wo immediaely process new informaion. Agens adap eir predicions by coosing from a limied number of beliefs (predicors or rading sraegies). Eac belief is appreciaed by a performance measure. Agens on e capial marke use is performance measure o make a raional coice wic depends on e eerogeneiy in agen informaion and subsequen decisions of e agen eier as a fundamenalis or as a caris, Ciarella, He (000). Te model presens a form of evoluionary dynamics, called Adapive Belief Sysem, in a simple presen discouned value (PDV) pricing model. Te firs par of is model was elaboraed by Brock and Hommes (998). Consider an asse pricing model wi one risky asse and one risk free asse. Le denoe e price (ex dividend) per sare of e risky asse a ime, ( random variables a ime + are denoed in bold), and le { y } be a socasic dividend process of e risky asse. Te supply of e risk free asse is perfecly elasic a e gross risk free ineres rae R, wic is equal o + r, were r is e ineres rae. Ten e dynamics of e weal is defined as: ( ) W RW p y Rp z + = , (..) p were z denoes e number of sares of e asse purcased a ime. Le us furer consider E and V as e condiional expecaion and condiional variance operaors based on e se of publicly available informaion consising of pas prices and dividends, i.e., F = { p, p,?; y, y,?} E, V, on e informaion se. Le and denoe beliefs (or forecas) of ype invesor abou e condiional expecaion and condiional variance. Invesors are supposed o be a myopic mean-variance maximizers so a e demand risky asse is obained by a solving of e following crierion were e risk aversion coefficien, demand z, a { E, [ + ] V, [ + ] } z, max W W, (..) z, a > 0 of ype for risky asse as e following form [ ], + + +, = 0, for, is assumed o be e same for all raders. Tus e E p y Rp aσ z (.3.) 3

4 z E [ p + y Rp ], + +, =, aσ (.4.) assuming a e condiional variance of excess reurns is a consan for all invesor ypes ( + ) = = V p y Rp σ σ (.5.), + +. s Le be e supply of ouside risky sares. Le n be a fracion of ype invesor a z, ime. Te equilibrium of e demand and supply is were H [ p + y Rp ] E = H, + + s n, z = aσ, (.6.) is e number of differen rader ypes. In e case of zero supply of ouside sares s, i.e., z = 0, e marke equilibrium is as follows H { [ ]} Rp = n E p + + y +. (.7.),, = In a marke, were all agens ave raional expecaions, e asse price is deermined by economic fundamenals. Te price is given by e discouned sum of fuure dividends Te fundamenal price p k= [ + k] k ( + r ) E = y. (.8.) p depends upon e socasic dividend process y. In e special case wen e dividend process { y } is an independen, idenically-disribued (IID) process, wi consan mean E{ y } = y, en we ave e fundamenal price given by p y y = =. k (.9.) r k= ( + r ).. Heerogeneous Beliefs Tis par deals wi raders' expecaions abou fuure prices. As in Brock, Hommes (998), we assume beliefs abou fuure dividends o be e same for all rader ypes and equal o e rue condiional expecaion, i.e. [ ] [ ] E,,...,,, y+ = E y+ = H (.0.) In e case wen e dividend process { y } is an independen, idenically-disribued (IID) process, E{ y + } = y, en all raders are able o derive e fundamenal price p a would dominae in a perfecly raional world. Abandoning e idea of raionaliy and moving 4

5 o e real world we allow prices o deviae from eir fundamenal value p. For our purpose, i is convenien o work wi a deviaion p, i.e., x from e bencmark fundamenal price x = p p. (..) In general, beliefs abou e fuure price E, [ ] = + ( x ) [ ] + E p, + E p f x,...,, L for all,, p ave e following form (..) were f ( x,..., x ) represens a model of e marke. Te rader ype believes a L e marke price will deviae from is fundamenal value p. Te eerogeneous agen marke equilibrium, defined in Eg. (.7.), can be reformulaed in deviaions from e bencmark fundamenal as H [ ] Rx = n E x n f. (.3.),, +,, = = H.. Selecion of Sraegies Beliefs are updaed evoluionary, e selecion is conrolled by e endogenous marke forces, Brock and Hommes (997). Te fracions of rader ypes on e marke n,, are given by e mulinomial logi probabiliies of discree coice n ( βu ) = exp / Z,,, Z H j = ( βu, ) (.4.) = exp, (.5.) were U is e finess measure of sraegy evaluaed a e beginning of period., As a finess measure of rading sraegies we use moving averages of realized profis, were m denoes e leng of e moving average filer. In a real marke is parameer can be inerpreed as a memory leng or evaluaion orizon for a rading sraegy a ime. Finess measure U, is defined as m U, = x Rx. (.6.) m k= aσ ( ) ( f ), Rx 5

6 3. Trading Sraegies In Brock and Hommes (998) ave invesigaed simple linear rules wi one lag wi fixed g : f, = gx + b, (3..) wic served as a basic framework for e eerogeneous agen models. In is paper we enric e model by inroducing e smar raders wo are able o forecas ese linear rules and us are able o forecas fuure rend parameer wic is varian in ime. In our model we ave wo groups of rading sraegies. Te firs group consiss of smar rading g, sraegies wic use simple linear regression predicions of f,, and e second group comprises rading sraegies wic are generaed socasically and ey are seleced wi e Wors Ou Algorim (WOA) during simulaions. Tese wo groups defined below. f and, f will be, 3..Smar Traders Te idea of e smar raders is based on endeavor of e marke agens o esimae e fuure price movemens. By adding smar raders we ry o improve e original eerogeneous agens model so i will be able o come o closer descripion of e real markes. Te simples way ow o implemen is ype of marke beavior ino e Brock and Hommes model is o use e simple linear forecasing ecniques. Te smar rading sraegies us use e maximum likeliood esimaion of AR() process o esimae e rend parameer g, for e nex period. Smar raders us assume a e deviaions x follows an AR() process, and ey base eir forecass of x + on e informaion se F = { x, x,?, x }. k Ten e rading sraegies of e smar raders are defined as follows: f, + = f = ϕ, x, (3..) were ϕ is a esimaed rend g. In e simulaions, we use various ypes of smar raders, wi differen leng of e informaion se k. 6

7 3.. Socasic Beliefs Trading sraegies of e second group, f, are generaed socasically. Te rend, parameer g and e bias parameer b of e rader ype are realizaions from e normal disribuion N(0, σ ). In is paper we use N (0, 0.6) and N(0,0.09) respecively. Te memory parameer m of e rading sraegy f is a realizaion from e, uniform disribuion, specifically U (,00). Te memory parameer can be inerpreed as an evaluaion orizon for e rading sraegy. Furer on, e WOA periodically replaces e rading sraegies a ave e lowes performance level of e sraegies presened on e marke by new ones. Wiou loss of generaliy, is algorim is consruced o evaluae and rank e performance of all e sraegies from e second group afer every 40 ieraion in descending order. Te four sraegies wi e lowes performance are en replaced by e newly generaed sraegies. Te use of e WOA in simulaions can significanly cange price ime series parameers and modify e beavior of invesors on e simulaed marke, see Vaca, Vosvrda (005, 007). Wen m = for all ypes, we ge e Brock and Hommes model. If b = 0 and g > 0, e invesor is called a pure rend caser. If b = 0 and g < 0, e invesor is called a conrarian. Moreover, if g = 0, and e b > 0 ( b < 0 ), e invesor is said o ave an upward (downward) bias in beliefs. In e special case of g = b = 0, e invesor is fundamenalis, i.e., e invesor believes a e price always reurn o is fundamenal value Socasic Beliefs wi Cange of Senimen Furer on, we invesigae e impac of senimen cange of e marke on e simulaed price. We define e cange of e senimen as a sif of e beliefs abou e fuure rend g,. In is paper we will focus on e direc jump paern of e agens' senimen. Tis can be seen as jumps of e senimen from opimism o pessimism and vice versa. We model canges in senimen by jumps in e rend parameer incoming invesor sraegy o e marke. More preciselly, e rend parameer g of newly g jumps beween e realizaions from e normal disribuions N (0.4,0.6) and N( 0.4,0.6) eac 4000 ieraions. Canges in senimen can be observed in Figure in e nex secion. 7

8 4. Simulaions Resuls Tis secion describes e meodology of our simulaions and summarizes e main resuls. Te main purpose of e simulaions is o examine influence of e proposed smar raders concep and senimen canges on e simulaed marke prices. We compare e iniial model wiou smar raders (0ST) wi e model wi 5 smar raders (5ST) and e model wi 5 smar raders in e firs group and senimen canges in e second group (5STS). Alogeer we consider 40 rading sraegies for eac simulaion. For e model wiou smar raders, all e sraegies are second group sraegies f. For e simulaions wi 5 smar raders, ere are 5 firs group sraegies f, and 35 second group sraegies f. For e simulaions wi 5 smar raders and cange of senimen, ere is 5 firs group,, sraegies f and 35 second group sraegies, f were e rend parameer g jumps, beween e realizaions from e normal disribuions N (0.4,0.6) and N( 0.4, 0.6) eac 4000 ieraions. Figure sows e empirical probabiliy densiy funcion of e rend parameer observed a e simulaed marke. I is e crossecion roug e ieraions, and canges of senimen can be clearly observed. Figure : Empirical PDF of e rend parameers g roug ieraions Te lengs of e informaion se, defined by k, used for e rend parameer esimaion by various ypes of smar raders is 5 { k } = = {80,60,40,0,5} i i for bo models. Oer parameers of e simulaion are: β = 300, number of all ieraions N = 5000, ασ =, R =.. Eac of e six models as been simulaed 36 imes o acieve robus 8

9 resuls. Table sows e descripive saisics of e simulaed deviaions x. Furer on, Figure sows e kernel esimaion of e probabiliy densiy funcions (PDFs). Table : Descripive saisics Saisics 0 ST 5 ST 5 STS Mean Median Variance S. Dev Skewness Kurosis Min Max Figure : Empirical PDF of x for simulaed models wiou smar raders, wi 5 smar raders and wi 5 smar raders and canging senimen 0ST 5ST 5STS 0 We begin e resuls summary wi descripive saisics of x. We can see a e means and variances of e x are no canging wi increasing number of smar raders. Te models wi smar raders produce e lepokuric disribuions of x, wile e model wiou smar raders produces playkuric disribuion. Wile e values of skewness and kurosis are e arimeic means of all e simulaions, we use Kruskal-Wallis (95) es o compare e disribuions of simulaed x. Tis es does no assume normal disribuion of compared ses of daa as analogous analysis of variance. Te null ypoesis of e es is equal populaion of medians agains alernaive of unequal populaion of medians. Te we use Epanecnikov kernel wic is of following form: K(u) = 3 4 ( u ) u. 9

10 Kruskal-Wallis es rejecs e null ypoesis of equal medians of e ses on e 0% significance level, us e skewness and kurosis of all 3 models are significanly differen. We can conclude a adding e smar raders as well as adding cange of senimen o e original model significanly canges e simulaed disribuions of x. We coninue our analysis wi esimaion of e Hurs (95) exponen for all e simulaed models. Our expecaion is a inroducion of e smar raders and senimen cange o e simulaed marke will increase also e Hurs exponen significanly. We again use e Kruskal-Wallis es wic rejeced e null ypoesis of equal medians of esimaed Hurs exponens on e % significance level for compared models wiou smar raders and wi smar raders. Tus we conclude a implemening smar raders ino e model significanly increases e Hurs exponen and us increases e persisence of simulaed marke. On e oer and, Kruskal-Wallis es does no rejec e null ypoesis of equal medians wen comparing e models 5ST and 5STS. Tis means a inroducion of canging senimen in e proposed form does no cange e persisence of simulaed prices significanly. 5. Conclusion In is paper we exended e original model of eerogeneous agen model by inroducing smar raders concep. Te idea of e smar raders is based on endeavor of e marke agens o esimae e fuure price movemens. By adding smar raders we ry o improve e original eerogeneous agens model so i will be able o come o closer descripion of e real markes. Smar raders are able o forecas e rend parameer of e price deviaions using e informaion se consising of e pas deviaions. Tey are modeled o assume a e deviaions are AR() process and ey use maximum likeliood esimaion meod for forecasing. Tus in our model we use groups of raders, smar raders and group of socasically generaed rading sraegies wic are moreover seleced by Wors Ou Algorim. Furermore, we invesigae e impac of senimen cange of e marke paricipans on e simulaed price. Te cange of e senimen is defined as a sif of e beliefs abou e fuure rend value. Te main resul of e simulaions is a probabiliy disribuion funcions of e price deviaion canges significanly wi increasing number of smar raders in e model, and i also canges significanly wi inroducion of e senimen canges. We use also Hurs exponen o measure e persisence of e price deviaions and we find a e Hurs 0

11 exponen is significanly increasing wi adding smar raders ino e simulaions. Tis means a e inroducion of smar raders concep ino e model resuls in significanly iger persisence of e simulaed price deviaions. On e oer and, inroducion of canging senimen in e proposed form does no cange e persisence of simulaed prices significanly. Tese are e preliminary resuls as more forms of senimen canges needs o be esed o make more general implicaions for e real markes. As is paper inroduces e new concep for modeling eerogeneous agens, i also opens large space of possibiliies for furer researc. Te mos ineresing par is o sow e impac of e differen senimen canges on e marke price. Acknowledgemen A suppor from e Czec Science Foundaion under gran 40/08/P07, GAUK 4608, and and Minisry of Educaion MSM is graefully acknowledged. References Brock WA, Hommes CH (997) A Raional Roue o Randomness. Economerica 66: Brock WA, Hommes CH (998) Heerogeneous Beliefs and Roues o Caos in a Simple Asse Pricing Model. Journal of Economic Dynamics and Conrol :35-74 Ciarella C, He X (000) Heerogeneous Beliefs, Risk and Learning in a Simple Asse Pricing Model wi a Marke Maker. Quaniaive Finance Researc Group. Universiy of Tecnology Sydney, Researc Paper No~35 Hommes CH (005) Heerogeneous Agen Models in Economics and Finance, In: Tesfasion L, Judd KL (eds.), Handbook of Compuaional Economics, Agen-Based Compuaional Economics, Elsevier Science B.V. : Hurs HE (95) Long-erm sorage capaciy of reservoirs. Transacions of e American Sociey of Civil Engineers 6: Kruskal WH, Wallis WA (95) Use of ranks in one crierion variance analysis. Journal of e American Saisical Associaion 47(60), Vaca L, Vosvrda M (005) Dynamical Agens' Sraegies and e Fracal Marke Hypoesis Prague Economic Papers 4:7--79 Vaca L, Vosvrda M (007) Wavele Decomposiion of e Financial Marke, Prague Economic Papers 6:38-54

12 Vosvrda M, Vaca L (00) Heerogeneous Agen Model and Numerical Analysis of Learning, Bullein of e Czec Economeric Sociey, 7:5-- Vosvrda M, Vaca L (003) Heerogeneous Agen Model wi Memory and Asse Price Beaviour, Prague Economic Papers :55--68