Lg-Memy a Ode-Dve Make Blake LeBa ad Ryuch Yamam Ocbe 13, 2006 Absac Ths pape duces a de-dve make wh heegeeus vess, wh subm lm make des accdg he w adg ules. The adg ules ae epeaedly updaed va smple leag ad adapa f he vess. We aalyze makes wh ad whu leag ad adapa. The smula esuls shw ha u mdel wh leag ad adapa successfully eplcaes lg-memes adg vlume, sck eu vlaly, ad sgs f make des. We als dscuss why evluay dyamcs ae mpa geeag hese lg memy feaues. 1
1. Iduc Ths pape pvdes a smula aalyss f a de-dve make expla he fllwg empcal ppees. Fs, adg vlume s pesse ve me (Lba ad Velasc (2000)). Secd, sck eu vlaly has lg-memy, meag ha he sck pce flucuas have psve aucelas (Dg e al. (1993), Maega ad Saley (1995, 1997), Egle (1982), Paga (1996)). 1 Thd, hee s a psve cempaeus cela bewee vlaly ad vlume (Tauche ad Ps (1983)). Fally, he sgs f make des fllw a lg-memy pcess (Lll, Mke, ad Fame (2005)); ye he make s effce (Lll ad Fame (2004)). We mdfy a de-dve make wh heegeeus vess, whch was gally csuced by Chaella ad I (2002). I he de-dve make f Chaella ad I, vess se bds ad asks ad subm lm make des accdg exgeusly fxed ules. Ou key mdfca s ha he adg ules ae epeaedly updaed va leag ad adapa f he vess. A a gve p me, vess lk back a he pas pefmace f he pedcs. They ealze he pas msakes hese pedcs ad updae he pedc mehds ew es by mag saeges 1 Lg memy s deeced abslue eus f S&P 500 (Blleslev ad Mkkelse (1996)) ad vlaly f mal exchage aes (Balle, Blleslev ad Mkkelse (1996)). 2
f me successful vess. The success f hes s measued by he pas pedcve accuacy. Thus, vess lea fm he pas ad adap he behav mpve he fuue pefmaces he make. We cmpae he dyamcs ha emege fm hs ecmy wh ad whu ma. Fuhe, we exame hw he mdel ca expla he pevusly meed empcal feaues f he sck make. Ou make, wh ma adg saeges, successfully eplcaes he pevusly meed fu empcal egulaes he sck make. We fs fd ha vlume ad vlaly have lg-memy ly wh ma. Ths mples ha mave behav s celaed wh lg-memy pcesses. T epe hs esul, fs e ha ma f adg saeges mples ha ages subm smla ypes f des ve sme peds. Lage (small) umbes f make des ae fllwed by smla umbes f make des f he subseque peds. Sce u ecmy assumes ha ages ca ade e sck a a me, he umbes f make des ae us adg vlume. S, adg vlume becmes pesse. Theefe, smlay f adg behav s mpa udesad he lg-memy vlume. I wll emege a ecmy wh cea ypes f evluay behav, such as ma. We he demsae ha pessece he umbe f asacs s cucal udesadg lg-memy f vlaly. I de vesgae hs ela, we fs 3
shw a sgfca ad psve cempaeus cela bewee hem. The, we dscuss he lg em ela. We shw ha a psve cempaeus cela exss u de-dve make ly wh ma. I a ecmy wh ma, ages ae lkely mae he saeges f hes s ha he adg behav becmes smla amg ages. F example, as sme ages decde subm lm des buy, hes als ee he same ypes f des. Whe may ages place uexecued lm des, he pce des mve much because des ae msly lm des gg he de bk. Eve f sme des ae execued, pce chages wuld be small sce he lm de bk s geg hcke aud he bes pces. Whe ms ages subm me execug lm des, he umbe f asacs ceases, he de bk hs, ad pce chages cease. The smlay f adg behav mples he psve cela bewee asac fequecy ad vlaly. We he vesgae wha dves he lg-memy sck eu vlaly a make wh evlu. We pa he sample sub-peds, whch ca equal umbes f asacs. Ths se-up elmaes he mpac f lg-memy adg vlume pessece vlaly. The aucelas f vlaly ve such sub-peds ae examed. We fd ha he aucelas becme subsaally weake (bu sll exs wh sh lags) wh such dvss f he sample, dcag ha 4
pessece he asac fequecy s mpa f lg-memy feaues vlaly. Sgs f de flws ae als pesse u ecmes wh evlu. We shw ha he mave behav f vess geeaes pessece he sequeces f buye selle aed ades. Ths appeas as lg memy u sged de flw sees. I u make we shw ha hs lg age depedece ca cexs wh a elavely effce make ha eu sees ae elavely ucelaed. Seveal ece papes have aemped eplcae hese feaues a evme wh ze ellgece ages. 2 I hese mdels adm de flw s se he make cleag mechasm, ad he esulg me sees ae aalyzed. These adm behav mdels ae a mpa bechmak f udesadg whch me sees feaues culd be dve by sual feaues ale. By pug smple dsbual assumps de placemes hese mdels geeae shap ad esable pedcs abu fucal fms spead, pce dffus ae, ad de flw dyamcs (Fame, Paell, ad Zvk (2004)). Raaly-based hees aely make pedcs abu such fucal fms, ad as a esul, he pedcs ae hade 2 Thse sudes clude Daels, Fame, Gllem, I, ad Smh (2003), Fame, Paell, ad Zvk (2005), Fame, Gllem, Lll, Mke, ad Se (2004), Gllem, Fame, ad Lll (2005), ad I, Daels, Fame, Gllem, Kshamuy ad Smh (2003). 5
es. We mve fm he ze ellgece wld, e f ``lw ellgece'', buded aaly f seveal eass. Fs, we ae eesed havg sme ecmc mvas f u ages geeag buy ad sell des. Ths culd pec us fm esuls beg dve by wldly aal adg behav. Secd, we seek a edgeus explaa f he exeme pessece de sgs, vlume, ad lqudy. Fally, u ages ae mvaed by ecmc becves a leas wk had esue a effce make. We ae eesed seeg hw effce hese makes ca be ems f pces, whle geeag he he pesse me sees esuls. We wll d hs a sgle make wh smple, bu adapvely aal ages. The es f he pape pceeds as fllws. Sec 2 summazes pce deema mechasm f a acual de-dve make. Sec 3 peses he make sucue. Sec 4 descbes he sascal mehdlgy deec lg-memy a me sees. Sec 5 gves smula aalyses ad he esuls. The las sec ccludes. 6
2 A Bef Skech he Pcg Mechasm a Ode-Dve Make Befe ducg u afcal sck make, hs sec befly summazes he mechasm pce deema a de-dve make whu make makes. S, he fllwg explaa wuld be smla ha may makes wh aumaed cleag mechasms. We aalyze a de-dve make whee pce s deemed accdg he duble cuus auc. I he duble cuus auc, whch s a sadad pce fma mechasm facal makes, ages place w kds f des: lm des, ad make des. Lm des ae equess sell/buy a a specfc pce bee. They epese he lwes/hghes pce a whch he ves s wllg sell/buy. Make des ae equess buy sell a gve umbe f shaes mmedaely a he bes avalable pce he make. Make des ae execued me easly ha lm des. Lm des, whch ae execued mmedaely, ae sed he lm de bk. We call buy/sell lm des as bds/asks ( ffes). The dffeece bewee he lwes ask, ad he hghes bd s called he spead. Ivess place des a ay pce a 7
pe-specfed gd called he ck sze, whch s he ceme by whch pces ca mve. Make des ae mached wh lm des f he ppse sde he de bk, ad execued usg me py a a gve pce ad pce py acss pces. A asac emves a ask bd de whe hey ae mached wh cmg lm des make des. Lm des ae als emved afe hey ae caceled, a expa dae has passed. Hee s a example f hw he sck pce s deemed. 3 Le s csde he fllwg ables whch llusae he de bk. I Table 1, he bes bd s cuely 300 shaes a $178, ad he bes ask s 1000 shaes a $179. A ew de s mached wh he des aleady placed he bk. Suppse ha a make buy de f 100 shaes s eeed he bk. Ths s mached wh he sell de wh he hghes py, whch s f 1000 shaes a $179. S he 100 shaes ae bugh a $179, whch leave 900 shaes a $179 he ask sde f he bk. 3 Tky Sck Exchage (2004) explas he deals f a de-dve make. 8
Table1: A Example f Lm Ode Bk (1). Bd Pce Ask 100 Make Ode 180 400 179 1000 300 ##178 400 177 2000 176 Nex, suppse ha a buy de f 1000 shaes a a lm pce f $180 s placed as Table 2. Ths s mached wh he sell lm de f 900 shaes a a lm pce f $179. The emag 100 shaes ae he mached wh 400 ask a $180. The bk he lks lke Table 3, ad he pce s w $180. These asacs ae caed u cuusly. Table 2: A Example f Lm Ode Bk (2).!"# $%"&' ()* +,%*'-./%#'% 0111 021 311 44056 611 711 052 311 055 8111 059 9
Table 3: A Example f Lm Ode Bk (3).!"# $%"&' ()* +,%*'-./%#'% 00123 433 156 433 152 733 155 8333 159 3. Make Sucue Ths sec descbes a afcal sck make based he make uled Chaella ad I (2002). Chaella ad I (2002) duce a cuus duble auc make whee ages subm des he make sequeally ad a adm mes. Odes ae mached ad execued accdg he me ad pce pes. A sgle asse s aded he make, ad s fudameal value s assumed be csa. Ages ae assumed kw he fudameal value f p ad he pas hsy f he pces. The pce p s deemed a he level whee ay asac ccus. Whe hee s asac, a pxy f he pce s gve by he aveage f he lwes ask (he qued ask, a ) ad he q hghes bd (he qued bd, b ). We use he umbe f eves as a measue f me. q Ivess decde ee he make wh a pbably,!. They bseve all he 10
pas pce sees ad deeme he saeges based he expecas f fuue pces. They make he pedcs based he fllwg equa, f ( p ) p % (1) ˆ " g & + g L +! p #, + = 1 2 ' $ whee g 1, g 2, ad ae fudameals, chas, ad se-duced cmpes, especvely, ad ally hese ae admly assged accdg he fllwg dsbus, g ~ N(0,! 1), g2 ~ N(0, 2), 1! ~ N(0, 0 )! ~ N(0,1). The sg f g 2 epeses a ed chasg f g 2 >0, ad a caa saegy f g 2 <0. The value, L, s gve as: L 1 p (2) L =! L " " p p = 1 " " 1 " " 1. I s he aveage eu ve he eval, L, whch s admly daw fm a ufm ad depede dsbu acss ages ve he eval 1, L ). ( max Ages expec fuue pces a me +! accdg hse f he fuue eus as: (3) ˆ., p p e +! = +! ˆ Ths pape assumes ha all ages smulaeusly make fuue pce pedcs a a pacula p me, bu ade sequeally. 11
A age buys (sells) e u f sck a a pce b ( a ) whe she expecs a cease (decease) he pce. b ad a ae gve as fllws, (4) whee! b = p! a = p, + ", + " (1! k ), (1 + k ) k s admly ad ufmly assged acss ages he eval ( 0, k max), wh k! 1. max k s fxed ve me, bu vaes ve ages. Ages ae assumed subm a lm buy (sell) de a b ( a ) f e u, f b ( a ) s smalle (lage) ha he cue qued ask, a (bd, b ). q q If he bd(ask) pce s hghe(lwe) ha hey bes ask(bd) he he de execues mmedaely a cue bes qued pce he bk. The ade s execued f e shae, ad he de s cleaed fm he bk. Ths mmedaely execued lm de has sme cecs make des eal makes ha he age sees he cue pce, ad he decdes execue. Sce all asacs ae f e shae ly, hee s pssbly ha he buye selle wll walk up he bk. Evey lm de has a lfeme!, s ha afe ha ped, he lm des ae emved fm he bk. Eves hs afcal sck make pceed as fllws. Fs, all ages bseve he pas hsy f pces ad make pedcs fuue pces a me. Based he w pedcs, hey decde ee ee he make. These decss ae made 12
sequeally. Whe hey ee he make, hey subm des ad he pce s ecded. Afe all ages make sme decss eeg he make, hey evse he expecas fuue pce based he hsy f he pces, ad ahe adg ud sas aga. Ages adg saeges evlve ve me. Afe fve adg uds have bee cmpleed, all ages lk back he ecds he pas fecass, ad updae he fecas mehds ( pacula, he paamees equa (1)) mpve he fecasg ably. They ae me lkely selec fecas mehds, whch pefmed well he pas, make me accuae pedcs he fuue. Bee saeges ae lkely be maed by ages. Such leag ad adapa ae descbed wh he fllwg eal-valued geec alghm. Oce ages updae he paamees, he ex adg ud sas. Ths pcess, whch has fve adg uds f each GA u, s epeaed 50 mes. A smula csss f 250 adg uds al. The eal-valued GA: Sep 1) Selec Afe adg fve uds, all ages calculae he w fess, whch s gve by 13
( ) 2! " (5) fess = p E ( p ) 5 uds Equa (5) s he sum f squaed pedc es f age f fve uds. The pbables f beg cped he ex ud s gve by, (6) P = 1000! = 1 fess fess We selec 1000 paamees usg hese pbables. The paamees ha pduced bee fecass he pas ae me lkely be used by he ages. The bad saeges ae less lkely be cped f he ex geeas. The seleced paamees ae he subec he fllwg mua pea. Sep 2) Mua 4 Sep a) Selec admly a paamee se amg g 1, g 2, L,ad. Mua 4 Ths mua, whch eplaces he ld paamee wh he ew e wh a small pbably, s based ha Afvc ad Mas (1999). 14
(Expemea) akes place wh p m (=0.08). Sep b) A adm umbe ~ U[0,1]. (A adm umbe s daw fm a ufm dsbu he eval [0,1].) Sep c) If s less ha p m, a ew paamee value s daw fm a asscaed dsbu,.e., g1 ~ N(0,! 1), g2 ~ N(0,! 2), ~ N(0, 0), L U (1, L ) ad ha umbe eplaces he ld e. ~ max Sep d) Ohewse, he ves keeps he ld value. Ths pape aalyzes he lg-memy w ypes f he ecmes: ecmes wh ad whu evlu (e ha he make sucue s he same bh ecmes). The lg-memy s examed usg smple pls ad a sascal es, whch s descbed he ex sec. 4. A Sascal Mehd Deec Lg-Memy A adm pcess has lg-memy whe s aucela fuc has a asympcally decayg fm wh a pwe law fm, 15
2d # 1 (7) $ ( k) ~ ck as k "! whee d<0.5 ad c s a -ze csa em. A lg-memy pcess mples ha he fuue value s sgfcaly pedcable fm he pas value. Seveal papes shw ha may facal me sees ae chaacezed by lg memy pcesses. F example, lg memy s deeced abslue eus f he S&P 500 (Blleslev ad Mkkelse (1996)) ad vlaly f mal exchage aes (Balle, Blleslev ad Mkkelse (1996)). The escaled age age ve sadad deva (R/S) sasc, whch s gally develped Hus (1951) ca be used deec lg-memy. 5 6 I hs pape, we esmae L s mdfed escaled age (R/S) sasc (L (1991)) ad es s sgfcace. The L s mdfed R/S sasc s gve as fllws. Csde a me sees X 1, X 2,, X, ad le X be he sample mea, ( / )! = 1 X. Le 1 2! ˆ x ad!ˆ x be he sample vaace ad aucvaace esmas f X. 5 Hus (1951) fud pessece aucelas hydlgcal daa. 6 The Hus expe s fe calculaed descbe he memy. I s cfed wh a eval (0,1), ad epeed as fllws. A schasc pcess s a adm walk, whch s a sascally depede ad ucelaed pcess, f he Hus expe (H) s 0.5. 0.5 f H mples ha we ca pedc fuue values usg pas values f he sees. If H s geae ha 0.5 bu less ha equal 1, he sees mples he pessece. If H s less ha 0.5 bu geae ha equal 0, he sees s descbed wh a-pessece s ha has a ed eveg edecy. The epea f he Hus expe s well descbed Hamp (1996). 16
17 The he L s mdfed R/S sasc, deed by Q s defed as: (8) ( ) ( )!" # $ % & ' ' ' ( ) ) = * * = * * k k k k X X X X q Q 1 1 1 1 m max ) ( ˆ 1 + whee (9) ( ) ( )( ) q q q q X X X X q X X q q x q < +! " + = # $ % & ' (!! +! " ) ) ) ) = = + =! =, 1 1 ) (, ˆ ) ( 2 ˆ ) ( 2 1 ) ( ˆ 1 2 1 1 2 1 2 * + *, *,. The fs em backes equa (8) s he maxmum ve k f he paal sum f he fs k devas f he X fm he sample mea. The kh paal sum s defed as!( ) = k X 1, k = 1, 2,..,. The maxmum mus be psve sce he sum f all devas f he X s fm he mea s ze. The secd em equa (8) s he mmum ve k f he paal sums, s s always -psve. S, he R/S sasc s descbed wh he age f paal sums f devas f a me sees fm s mea, escaled by s sadad deva. The sasc s desged cmpae he maxmum ad mmum values f ug sums f devas fm he sample mea, e-malzed by he sample sadad deva. The devas ae geae a pesece f a lg-memy pcess ha wh lg-memy case. The L R/S sasc, Q, dffes fm he classcal R/S sasc f Hus ems f he dema, ) ( ˆ q!. ) ( ˆ q! he classcal sasc s he sample sadad
& 1 2 deva, $ (( X ' X ) % = 1 #! " 1/ 2. The L R/S sasc cludes he weghed aucvaace up lag q capue he effecs f sh-age depedece. A me sees s chaacezed have sh-age depedece f he maxmal depedece bewee eves a ay w daes becmes vally small as he me spa bewee hse w daes ceases. 7 L suggess a daa-depede fmula f calculag he value q: (10) q = [ k ] k & 3 # ( $! % 2 " 1/ 3 & 2 ˆ ) # $! % 1' ˆ ) " 2 / 3 whee [ k ] dees he geaes ege less ha equal k, ad!ˆ s he esmaed fs-de aucela ceffce f he daa. If he pcess has fe fuh mmes ad a sh-age depedece, a adm vaable, V! Q asympcally fllws:! " % $ k = 1 2 2 # 2 (11) = + ( # ) ( ) 2 k$ F ( ) 1 2 1 4k e $. The ccal value ude he ull hyphess ha he me sees s sh-age depede s fud wh hs fmula. We pefm e aled ess f he ull hyphess agas he aleave ha he me sees has lg memy. Fm Table II L (1991), f he sasc V s abve 1.747, we ca eec he ull hyphess wh 95% cfdece. 8 7 I mples ha he lg-age depedece pduces aucelas, whch decay vey slwly. Bh a-pesse ad pesse sees ae chaacezed as lg-age depedece sce bh have slwly-decayg aucelas alhugh hse f a-pesse sees cllapse ze. 8 We ca cduc w-aled es f u aleave hyphess s lg-age depedece. The 18
The L mdfed R/S sasc s kw have sme advaages ve he classcal R/S sasc. Fs, L pvdes he samplg hey s ha he mdfed R/S sasc ca be esed sascally wh s lmg dsbu. Secd, The L mdfed R/S sasc dsgushes bewee sh-age ad lg-age depedece a sees. Alhugh ece sck eu me sees have sh-age depedece, he classcal R/S sasc des csde such chaacescs, ad fe pduces sme based esuls he pesece f sh-age depedece. L shwed ha he pwe f he classcal R/S es s weak s ha fe des eec he ull hyphess f sh-memy. As L suggess, ay empcal vesga f lg-memy facal me sees mus accu f he pesece f hghe fequecy aucelas. 5. Smula Aalyses Lg-Memy Ths sec emphaszes ha a ecmy wh evlu ca pduce he fllwg ppees: (1) adg vlume s pesse, (2) vlaly f sck eus has lg-memy, (3) hee s a cempaeus psve cela bewee vlaly ad vlume, ad (4) sged de flw has lg-memy, bu he make s effce. We wll lwe al eec dcaes ha he aucelas cllapses ze alhugh decays vey slwly. 19
pese evdece ha suggess he ma f saeges s key udesadg hese empcal ppees a de-dve make. We u smulas w ypes f ecmes,.e., ecmes wh ad whu ma, wh paamee values Table 4. Table 4: Paamees Sd f fudamealss cmpe:! 1 1 Sd f chas cmpe:! 2 1.5 Sd f se-ade cmpe:! 0.5 Ode lfe:! 200 Pbably ee he make:! 0.5 k max 0.5 Fudameal value: p f 1000 Maxmum me hz chas 100 cmpe: L max Tck sze:! 0.1 The umbe f ages s 1000. The adg uds ae se 250. 9 I a ecmy wh ma, ages vke he GA evey 5 adg uds. We smply pl he aucela fucs, ad cduc he L mdfed R/S 9 We cu 1 adg ud whe all 1000 ages make sme decss,.e., eeg eeg he make. 20
es exame lg-memy pcesses. We csde dffee values f umbes f lags aucvaace equa (9), q = 4, 6, 8, ad 10. Vlume ad Vlaly: I u make, ages ca ade ly e u f sck pe asac. I de vesgae aucelas adg vlume, we eed assg a pseud-clck me. We cu e whe ages make a leas sme decss. Thse decss clude eeg he make, eeg ad submg make lm des. We csde 50 decss f e clck me. 10 Vlume s calculaed f each pseud-clck me. We measue eu vlaly smply by akg sadad devas pe pseud-clck me. The bld umbes Table 5 ad 6 gve he aveages f he R/S sasc ve 10 smulas. The symbl, **, dcaes ha we eec he ull hyphess f sh-age depedece a he 95% cfdece level a leas 70% f he smulas ve 10 smulas. The symbl, *, dcaes he eec f he ull wh 50% geae bu less ha 70% f he smulas ve 10 smulas. 10 We als vesgaed 25 ad 100 adg decss f a u f pseud-clck me. Hweve he esuls ae que smla. 21
Table 5: L s mdfed R/S ess f a Ecmy wh Evlu Bld umbes ae he aveages f R/S sasc ve 10 smulas. ** dcaes ha we eec he ull hyphess f sh-age depedece a he 95% cfdece level a leas 70% f he smulas ve 10 smulas. * dcaes he eec f he ull wh 50% geae bu less ha 70% f he smulas ve 10 smulas. Whe hee s sgs f ** * ex he bld umbe, dcaes ha we d eec he ull a all. Vlume Vlaly Sgs f Make Odes Reus q=4 7.919** 2.521** 4.660** 0.974 q=6 6.723** 2.137** 4.142** 1.047 q=8 5.946** 1.889* 3.773** 1.091 q=10 5.388** 1.710* 3.492** 1.121 Table 6: L s mdfed R/S ess f a Ecmy whu Evlu Bld umbes ae he aveages f R/S sasc ve 10 smulas. ** dcaes ha we eec he ull hyphess f sh-age depedece a he 95% cfdece level a leas 70% f he smulas ve 10 smulas. * dcaes he eec f he ull wh 50% geae bu less ha 70% f he smulas ve 10 smulas. Whe hee s sgs f ** * ex he bld umbe, dcaes ha we d eec he ull a all. Vlume Vlaly Sgs f Make Odes Reus q=4 0.702 0.458 2.785** 0.826 q=6 0.595 0.389 2.456** 0.885 q=8 0.525 0.344 2.226** 0.927 q=10 0.476 0.311 2.053** 0.957 The secd clums Table 5 ad 6 ae he esuls f vlume sees. The sh-age depedece s fe eeced f ay values f q a ecmy wh evlu, whle s eeced a ecmy whu evlu. Fgue 1 s he pl f aucela 22
fucs a ecmy wh evlu, whle Fgue 2 s ha whu evlu. The aucelas ae vey hgh eve wh lg lags he evluay ecmy, bu hey ae ea ze afe 50 lags a ecmy whu evlu. The fgues demsae ha vlume s pesse ve pseud-clck me ly a ecmy wh evlu. Sce ages ca ly ade e sck a a me, vlume us eflecs he umbe f asacs, execued lm des. Pessece vlume meas ha he umbes f asacs pe pseud-clck me ae au-celaed ve me. Sce de execu depeds ages expecas f fuue pces, 11 he umbe f asacs becmes pesse f he expecas ae smla amg a pacula umbe f ages ve sme peds. Ages ade sequeally u make, s smlay f he expecas wuld geeae pessece de ypes. Such smlay f expecas s me lkely as ages ae allwed mae each he. A ecmy wh evlu shws hghe ad lge aucelas vlume sce allws mave age 11 As dscussed Sec 3, ages subm a make de f e u f sck f b ( a ) s lage (smalle) ha he cue qued ask, a (bd, b ). q q b ad a deped he expecas as equa (4) as fllws. (4)! b = p! a = p, + ", + " (1! k ). (1 + k ) 23
behavs. Fgue 1: Fgue 2: Aucela Fucs f Tadg Aucela Fucs f Tadg Vlume wh Evlu Vlume whu Evlu 0.9 Aucela Fucs f Tadg Vlume WITH Evlu 0.9 Aucela Fucs f Tadg Vlume s a l e c u A 0.8 0.7 0.6 0.5 0.4 0.3 0.2 s a l e c u A 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0 0-0.1 0 50 100 150 200 250 300 Numbes f Lags -0.1 0 50 100 150 200 250 300 Numbes f Lags Fgue 3 ad 4 pese he aucela fucs f vlaly ecmes wh ad whu evlu, especvely. The hd clums Table 5 ad 6 ae he esuls f he L s R/S es f dffee values f q. The sascal esuls ad he fgues shw ha he lg-memy s fud ly a ecmy wh evlu. 24
Fgue 3: Fgue 4: Aucela Fucs f Vlaly Aucela Fucs f Vlaly wh Evlu whu Evlu s a l e c u A 0.35 0.3 0.25 0.2 0.15 0.1 Aucela Fucs f Vlaly f Reus WITH Evlu s a l e c u A 0.35 0.3 0.25 0.2 0.15 0.1 Aucela Fucs f Vlaly f Reus 0.05 0.05 0 0-0.05 0 50 100 150 200 250 300 Numbes f Lags -0.05 0 50 100 150 200 250 300 Numbes f Lags Nw we shw ha pessece he umbe f asacs s mpa udesad lg-memy vlaly. I de vesgae hs ela, le us fs vesgae whehe vlume s cempaeusly celaed wh vlaly. The we wll dscuss he lg em ela. Pevus empcal sudes demsaed ha hee s a psve cempaeus cela bewee vlaly, measued as abslue squaed pce chages, ad vlume. 12 Nw we shw ha such a cela exss u de-dve make ly wh evlu. Fs, we smply calculae he cela ceffces bewee he sadad devas f eus ad adg vlume f pseud-clck mes, whch ae 12 F example, Tauche ad Ps (1983) pese empcal evdece a psve cela bewee vlaly ad vlume. 25
gve Table 7. Bld umbes ae meas f he esmaes ve 10 us. Numbes paehess ae he sadad devas f he esmaes ve 10 us. We fud a sgfca cempaeus cela bewee vlaly ad vlume, abu 42%, a evluay ecmy. Hweve, hee s sascal evdece he cempaeus cela a ecmy whu evlu sce he esmae s 0.012 bu s sadad e s 0.03. Sce ages ca ly ade e sck a a me, vlume s gve by he umbe f asacs. Ou esul a ecmy wh evlu dcaes ha he umbe f asacs, adg vlume, has a psve cela wh vlaly. Ths s csse wh he fdgs such as Easley ad O Haa (1992). Table 7: Cempaeus celas (Vlaly vs. Vlume) Bld umbes ae meas f he esmaes ve 10 us. Numbes paehess ae he sadad devas f he esmaes ve 10 us. Wh Evlu Whu Evlu 0.418 (0.06) 0.012 (0.03) Ou bseva f hgh psve cela bewee vlume ad vlaly a ecmy wh evlu culd ccu because ages mae adg saeges. They decde he adg saeges based he expecas f fuue pces. The 26
expecas deped he pas pces ad paamees as equa (1). I a ecmy wh evlu, ages ae allwed chage he paamee values he expeca sucue, ad mae he weghs f hes. As a esul, hey ae lkely subm smla des ems f bd ask pces. Dug peds whch may des ae execued, he bk becmes spase aud he bes pces. As he bk becmes spase, he chage he execug pce wll be lage. Dug peds wh may uexecued lm des cmg, he bk becmes hcke as des say he bk ul hey ae emved. Pce chages hese peds ae small because f he make deph aud he bes pces. Theefe, smlay f he adg behav f ages explas he psve cela bewee adg vlume ad vlaly. A ecmy whu evlu ca pduce hs esul, sce such a ecmy des geeae he hmgeeus sequece f des ecessay f hs esul. We w demsae ha lg-memy he umbe f asacs s als ceced lg memy vlaly. The lg-em ela bewee adg vlume ad vlaly ca be aalyzed by emvg he effec f vlume aucela ha f vlaly. Ths ca be de by splg he sample sub-peds whch clude equal umbes f execued des pe sub-ped. Aucela fucs f he sadad devas f he eus ae calculaed ve he sub-peds. We aalyze he ela 27
bewee vlume ad vlaly ly he ecmy wh evlu sce Table 7 we shwed ha evlu was ecessay f he cec bewee vlume ad vlaly. Fgue 5 shws much weake aucelas ha Fgue 1. L s mdfed R/S es esuls ae peseed Table 8 whch shw evdece f lg memy vlaly afe adusg a adg vlume/asac me scale. I hs mdel speed f asac acvy appeas dve lg memy bh vlume ad vlaly. Fgue 5: Aucela Fucs f Vlaly f a Fxed Numbe f Tasacs wh Evlu 0.3 Aucela Fucs f Vlaly f a fxed umbe f asacs s a l e c u A 0.25 0.2 0.15 0.1 0.05 0-0.05-0.1 0 10 20 30 40 50 60 70 80 90 100 Numbes f Lags 28
Table 8: L s mdfed R/S ess f Vlaly elmag Vlume effec Bld umbes ae he aveages f R/S sasc ve 10 smulas. ** dcaes ha we eec he ull hyphess f sh-age depedece a he 95% cfdece level a leas 70% f he smulas ve 10 smulas. * dcaes he eec f he ull wh 50% geae bu less ha 70% f he smulas ve 10 smulas. Whe hee s sgs f ** * ex he bld umbe, dcaes ha we d eec he ull a all. Vlaly whu Vlume effec q=4 0.589 q=6 0.500 q=8 0.442 q=10 0.400 Fgue 5 des shw evdece f a weake sh age pessece he vlaly pcess whch he vlume me scale has emved. I s heefe pssble ha hee s a secd pece vlaly dyamcs ha s dve by smehg he ha vlume, bu des have lg memy pessece. Make Odes ad Effce Make: We als fd lg-memy sged de flw as demsaed Lll, Mke, ad Fame (2005). We aga dvde he sample pe pseud-clck me, ad dee he ped as +1 (-1) whe he may f des ae buye (selle) aed a ped. We fs pl he aucela fucs f sged des Fgue 6 ad 7. Fgue 6 ad 7 ae he aucelas wh ad whu evlu, especvely, whch shw sme memy f 29
make des bh ecmes. Hweve, ly he ecmy wh evlu dsplays psve aucelas ha d decay ze a lg lags. L s R/S sascs ae gve wh dffee values f q he fuh clums Table 5 ad 6. Lg-memy s deeced bh ecmes, bu s effec s sge a ecmy wh evlu ha whu evlu. The es sascs ae aud 3-4 f dffee values f q a evlu ecmy, bu ly aud 2 a ecmy whu evlu. Fgue 6: Fgue 7: Aucela Fucs f Sgs Aucela Fucs f Sgs f Make Odes wh Evlu f Make Odes whu Evlu Aucela Fuc f Sgs f Make Odes WITH Evlu Aucela Fuc f Sgs f Make Odes 0.35 0.35 s a l e c u A 0.3 0.25 0.2 0.15 0.1 s a l e c u A 0.3 0.25 0.2 0.15 0.1 0.05 0.05 0 0-0.05 0 20 40 60 80 100 120 140 160 180 200 Numbes f Lags -0.05 0 20 40 60 80 100 120 140 160 180 200 Numbes f Lags Wha causes hs dffeece esuls acss he ecmes? We ppse ha mave behav f ages s mpa geeag lg memy de flws. 30
Whe mave behav exss he make, gups f ages wll expec smla pce behav he fuue. Such mave behav pduces a sequece f buy sell des. Ou ecmy wh evlu ca geeae such mave behav. Fudamealss ad chass exs bh ecmes, ad he behav s dffee wh he dffee values f he paamees, g, 1 g 2, ad L equa (1) ad (2). 13 The values f hese paamees ae ally assged dffee adm values ha ae fxed ve me a ecmy whu evlu. Hweve, he values wll pssbly be smla whe ages mae each he, causg pesse de flw. The sgs f make des ae pesse, bu he make s acually effce. I de demsae hs esul, we pl he aucela fucs f eus wh a pa f 50 pseud-clck me Fgue 8 ad 9 ad shw he R/S es esuls he ffh clums Table 5 ad 6. 13 F cveece, we epea equas (1) ad (2) hee: (1) f L ( p ) p % ˆ " = g & + g L +! p # ad (2) 1 p " " p " " 1, + 1 2 L =! L = 1 p " " 1 ' $. 31
Fgue 8: Fgue 9: Aucela Fucs Aucela Fucs f Reus wh Evlu f Reus whu Evlu 0.2 Aucela Fucs f Reus WITH Evlu 0.2 Aucela Fucs f Reus s a l e c u A 0-0.2 s a l e c u A 0-0.2-0.4 0 50 100 150 200 250 300 Numbes f Lags -0.4 0 50 100 150 200 250 300 Numbes f Lags Fgue 8 ad 9 dcae ha he aucelas ae always aud ze, ad R/S es esuls shw ha we d always eec he ull hyphess f sh-age depedece bh ecmes. We cclude ha he eus have celas ve me s ha he make s clse effce. I ay facal seg wh adapve ages hee wll be a es bewee mave behav ad caa behav, whch beaks fee, ad akes advaage f he es f he hed. Ou make mus ca sme f bh f hese, maaed a balace hugh evlu. Saeges ha ely much hed chasg wll eveually fal, bu we kw hee mus be a l f hs gg sce s ceal dvg may f 32
u esuls. Hweve, evluay pessue appeas be sg eugh keep he gup behav he make fm geeag easly deecable lea pedcably u make eu sees. Ths s a eesg cas u bechmak expemes whu evlu. Thee, eu behav s als adm, bu admess hse makes mus be dve me hugh admess al saeges, ad hugh a adapve pcess wkg pedc fuue pces. The fac ha evlu ca maa a elavely effce make whle geeag he me sees feaues s mpa. 6. Cclus Ths pape aalyzes de-dve make wh ad whu evlu, ad shws ha a make wh evlu ca eplcae well lg-memy feaues sck eu vlaly, make des, ad vlume. We als fd ha a cempaeus psve ad sgfca cela bewee vlaly ad vlume a ecmy wh evlu, ad pessece he umbe f asacs s elaed lg-memy vlaly. Alhugh he sgs f make des ae pesse, we cclude ha he make s effce. We emphasze 33
ha mave behav f he ages adg saeges ae mpa udesad he dffeeces esuls fm he ecmes wh ad whu evlu. 34
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