EXECUTION COSTS IN FINANCIAL MARKETS WITH SEVERAL INSTITUTIONAL INVESTORS

Transcription

1 EXECUION COSS IN FINANCIAL MARKES WIH SEVERAL INSIUIONAL INVESORS Somayeh Moazen, Yuyng L, Kae Larson Cheron School of Compuer Scence Unversy of Waerloo, Waerloo, ON, Canada emal: {smoazen, yuyng, ABSRAC We sudy mul-perod radng sraeges of nsuonal nvesors who plan o rade he same secury durng some fne me horzons Invesors who rade large volumes face a prce mpac ha depends on her radng volumes smulaneously, and s usually represened as a funcon, he so called prce-mpac funcon We show hrough a numercal example ha a radng sraegy, opmal for radng n solaon, may become subopmal n he presence of oher nsuonal nvesors who rade he same secury a he same me hus, he radng acves of oher nvesors should no be gnored n pracce and need o be modeled properly Under he assumpons ha he number of oher nvesors and her radng volumes are known, he problem can be modeled as a smulaneous game We nvesgae he properes of he equlbrum radng sraeges and prove ha, under mld assumpons on he prce-mpac funcon, an equlbrum unquely exss and can be compued effcenly Parcularly, when shor sellng s allowed, he equlbrum s found by solvng a sysem of lnear equaons Fnally, we evaluae he expeced execuon cos of he equlbrum radng sraegy hrough smulaons and demonsrae ha even when oher nvesors choose her radng sraeges a random, he expeced execuon cos of he equlbrum radng sraegy s lkely o be less han he expeced execuon cos of he radng sraegy ha was opmal n he absence of oher nvesors KEY WORDS Algorhmc radng, Execuon Cos, radng Sraegy, Mul-nvesor Markes, Insuonal Invesors 1 Inroducon Large radng volumes of nsuonal nvesors exer a nonneglgble mpac on he execuon coss of her rades A consderable proporon of hs mpac arses from lqudy coss and nformaon effecs derved from large szes of he rades Lqudy coss nclude he cos pad by a purchase naor sell naor o denfy poenal sellers buyers and resul n an nsananeous mpac on he execuon prce ha s called he emporary mpac Furhermore, he mbalance beween supply and demand usually ransms some nformaon o he marke ha may cause a permanen mpac on he fuure execuon prces he sum of he emporary and permanen mpacs deermnes he oal prce mpac ncurred by an nsuonal nvesor Dsncons beween emporary and permanen prce mpacs and her characerscs have been addressed broadly n he leraure 10 A common resul s ha he magnude of he prce mpac s a funcon of he radng volume, he so called prce-mpac funcon Insuonal nvesors do recognze he prce mpac of her rades and s dependence on he radng volume In order o reduce hese prce mpacs, ypcally nsuonal nvesors spl her rades no several smaller paral orders, packages, and subm he orders durng some fxed fne number of perods A sequence of orders submed durng he perods s called a radng sraegy here are many possble radng sraeges o execue a rade For a gven prce-mpac funcon, he execuon cos problem deals wh fndng a radng sraegy ha mnmzes he expeced execuon cos of he rade When only one block of equy s raded n he marke, he execuon cos ha should be pad by he nvesor only depends on hs own rade Consequenly, he execuon cos problem s reduced o a sngle-agen decson makng problem, e, an opmzaon problem However, nsuonal radng does no occur n solaon and all of he nsuonal rades collecvely nfluence he curren and fuure execuon prces A radng sraegy aken by an nvesor no only nduces hs own execuon cos bu also affecs oher nvesors execuon coss In Secon we show ha oher nsuonal rades may cause a radng sraegy, ha used o be opmal n he absence of oher nvesors, o become subopmal Whle each nsuonal nvesor s execuon cos depends on oher nvesors radng processes, he s no wellnformed abou hem Hence, from each nsuonal nvesor s pon of vew oher nvesors radng sraeges are unceran and he execuon cos problem urns no a mulagen decson makng problem hs movaed many researchers o nvesgae he nsuonal radng managemen n a game-heorec seng and use he language of game heory o dscuss he problem Mos of hese works model he problem as a dynamc game n whch decsons are made gradually over an nfne number of perods Moreover, hese models usually make some assumpons abou radng parners In hs paper, we model he execuon cos problem

2 n a mul-nvesor marke as a smulaneous game whou makng any assumpon abou he nvesors radng parners In conras o he models usng dynamc games, n our model decsons on he szes of he packages are made smulaneously before sarng o rade hs s of neres parcularly when radng me horzons are shor We analyze he equlbrum radng sraeges of he generaed game and show ha for many prce-mpac funcons, he equlbrum s unque and can be compued effcenly We prove ha he equlbrum of he generaed game s robus wh respec o oher nsuonal nvesors acons for sraegc nvesors Fnally, we evaluae he poenal performance of he equlbrum radng sraegy by a se of smulaons No only s he equlbrum radng sraegy he bes response o oher sraegc nvesors, also performs well agans random nvesors who place orders of random amouns a each perod In hs work, we focus solely on he sngle-secury case and leave he nvesgaon of radng porfolos n mul-nvesor markes for fuure work hroughou hs paper, by an nvesor we mean an nsuonal nvesor he paper s organzed as follows Secon nroduces he execuon cos problem n sngle-nvesor markes as an opmzaon problem Secon examnes how he resuls for he sngle-agen models may change when some oher nvesors are radng he same secury n he marke A game-heorec model of he problem n a fxed fne me horzon s presened n Secon Some properes of he equlbrum radng sraeges are nvesgaed n Secon and her performance are evaluaed n Secon 6 hs paper s concluded n Secon 7 Execuon Coss n Sngle-Invesor Markes Consder a fnancal marke n whch an nvesor plans o rade S shares of some secury durng a gven fne me horzon he nvesor begns hs rade a me 0 and hs program mus be compleed by me hroughou, whou loss of generaly, we assume ha me s measured n dscree nervals of un lengh herefore, we may consder ha he secury s raded over perods Alhough here s an asymmery n he overall mpac of buys and sells for nsance see 11, her mahemacal models are smlar Here, we assume ha he nvesor s goal s o purchase he block of equy Denoe he number of shares raded a he perod by S Posve negave S mples ha he secury has been bough sold n perod A radng sraegy S =S 1,, S s feasble f S = S whch guaranees ha he rade s fnshed a he end of he me horzon Le P be he execuon prce of one share of he secury a perod, and hs usually follows a sochasc process he deermnsc nal secury prce per share before he rade begns execung s denoed by P 0, compued from he laes quoe us precedng he frs prce mpac he execuon prce a each perod s deermned hrough some prce-mpac funcon In hs paper, we buld on prce-mpac funcons ha are lnear n he radng volume Lnear prce-mpac funcons are well-suded n he leraure, for nsance see,, 6, 10 Furhermore, Huberman e al 10 demonsrae ha only lnear permanen prce-mpac funcons rule ou arbrage hese reasons movaed us o use lnear prce-mpac funcons fs = =1 β S for our nvesgaon herefore, he execuon prce dynamc model s P S =P 0 β S σ =1 ξ for =1,,, 1 =1 where σ represens he volaly of he secury and ξ s are ndependen zero-mean Gaussan random varables For every =1,, and 1, nonnegave β quanfes he permanen prce mpac and he coeffcen β 0 measures he mporance of he emporary mpac Larger magnude of he emporary mpac relave o he sze of he permanen mpac whch has been observed n emprcal sudes mples ha for every = 1,, and 1, β β herefore n sngle-nvesor markes, an nvesor may fnd an execuon cos effcen radng sraegy by mnmzng he expeced value of he execuon cos hen he nvesor s opmzaon problem s mn E P SS s S = S, S 0 =1,,, S where he nonnegavy consrans appear only when shor sellng s no allowed Some oher echncal consrans may also be added o he problem, and some measure of rsk, eg, varance of coss, can be ncorporaed n he obecve funcon For general lnear prce-mpac funcons, he opmzaon problem s a quadrac programmng problem ha can be solved by avalable opmzaon sofware When shor sellng s allowed and for every =1,,, β = β and for every <, β = α, for some consans α β, he unque opmal soluon of Problem s he Nave sraegy, ha s dvdng he oal order no dencal packages, e, S = S for =1,, For a dealed dscusson abou he exsence and unqueness of he opmal radng sraegy for he execuon prce dynamc model 1 see 9 hroughou hs paper, we call an opmal soluon of Problem an opmal radng sraegy We use hs ermnology n he conex of mul-nvesor markes o refer o a radng sraegy ha has been obaned by gnorng he prce mpac of oher nvesors Mos of he exsng leraure on he execuon cos problem focus on markes where only one nvesor rades for nsance see,,, 6, 8, 9 hese works analyze he marke dynamcs hrough he opmzaon problem whou solvng for an equlbrum o fnd an expeced execuon cos effcen radng sraegy, an nvesor may smply gnore he effec of oher nvesors and solve a sngle-agen decson makng problem However as we show n Secon an opmal radng sraegy, obaned hrough Problem, may no reman opmal when oher nvesors ener he marke and sar o rade n he same secury I s worhy of noce when he nvesor knows he probably dsrbuons of oher nvesors radng sraeges, he can ncorporae ha

3 x 10 Shor sellng No Allowed; New S=Sbar/*max6,0 Nave Sraegy New Sraegy x 10 Shor sellng Allowed; New S=Sbar/*10*1 K 1 Execued Cos for Frs rader 1 1 Execued Cos for Frs rader Nave Sraegy New Sraegy Number of raders Number of raders FIGURE 1: Expeced execuon cos of he Nave sraegy versus he sraegy S = max6, 0 for = 1 1,,, 1, when shor sellng s no allowed FIGURE : Expeced execuon cos of he Nave sraegy versus he sraegy S = for = 1 K 1 1,,, 1, when shor sellng s allowed nformaon no he problem In hs case, a cos effcen radng sraegy can sll be found by means of an opmzaon problem smlar o Problem, alhough oher acve nvesors do exs n he marke However, s unlkely ha an nvesor has such exac nformaon abou he behavor of oher nvesors Execuon Coss n Mul-Invesor Markes In hs secon, we show hrough an example ha he opmal soluon of Problem may be subopmal when oher nvesors rade he same secury a he same me Consder a marke n whch K 1 nvesors are smulaneously submng orders o buy 10, 000 shares of some secury, whose curren execuon prce s 0$/share, durng 1 perods he magnude of he permanen and emporary mpacs are dencal for all of he nvesors over he perods, e, β = 10 $/share for =1,, 1 and Assume each of hese nvesors follows he opmal soluon of Problem, he Nave sraegy Now assume anoher nvesor arrves a he marke o nae o buy 10, 000 shares of he same secury durng 1 perods Fgures 1 and llusrae when oher acve nvesors do exs n he marke, e, K, he expeced execuon cos of anoher radng sraegy s less han he expeced execuon cos of he Nave sraegy hus he radng acves of oher nvesors may consderably reduce he performance of he radng sraegy ha was opmal n he absence of oher nvesors he man reason s ha n mul-nvesor markes, he prce movemen depends smulaneously on all of he nvesors acons hese acons should be aken no accoun when an ndvdual nvesor seeks a radng sraegy wh an effcen expeced execuon cos radng sraeges and dynamc behavor of nvesors n mul-nvesor sengs have been wdely consdered as a dynamc game n he leraure In hese frameworks sraegcally neracng nvesors choose radng sraeges ha affec curren and fuure prces of he secures and varous equlbrum conceps have been nvesgaed For a ler- aure revew on he avalable equlbrum approaches, see 1 Mos of hese equlbra hold under he assumpon ha he me horzon s nfne Moreover hese approaches usually se some resrcons on he radng parners In Secon, we formulae he execuon cos problem as a non-cooperave smulaneous game n whch nvesors rade durng a fne number of perods and choose a radng sraegy before sarng o rade he model allows any number of nvesors o he bes of our knowledge, hs s he frs sac model of he execuon cos problem durng a fne number of perods n a mul-nvesor marke Robus Equlbrum for he Execuon Cos Problem Le here be K nvesors gong o subm marke orders o rade S 1, S,, S K shares of he same equy smulaneously For k = 1,, K, suppose he radng program of he kh nvesor mus be compleed by me consequenly durng perods Denoe he kh nvesor s radng sraegy by ha s he -uple 1,, Sk For he sake of smplcy n expresson, we denoe he radng sraeges chosen by all of he nvesors excep he kh nvesor by S k Smlar o 1 for he lnear prce-mpac funcon fs 1,, S K = K m=1 =1 β msm he execuon prce dynamc model n he mul-nvesor seng can be saed as P S 1,, S K =P 0 K m=1 =1 β ms m σ ξ, 1 where ξ s are zero-mean random varables and σ represens he volaly of he underlyng secury he coeffcen β k quanfes he effec of he kh nvesor s rade, execued n perod, on he execuon prce a perod If he kh nvesor s goal s o buy sell he secury, hen β k 0 β k 0 Hence, he expeced execuon cos of he kh nvesor s rade, E k P S 1,, S K, equals =1

4 k Sk P 0 K m=1 =1 β msm, ha s denoed by ϕ k S 1,,,, S K or for smplcy ϕ k,s k herefore he problem of buyng a block of equy durng perods n he presence of K 1 oher nvesors can be vewed as a non-cooperave smulaneous game whose players are nvesors and her radng sraeges are couned as players sraeges hus he acon space of he kh player nvesor s he se of all feasble radng sraeges avalable o hm: M k := { 1,, : k = }, ha poenally has nfnely many elemens he se M k may also nclude some nonnegavy consrans f shor sellng s no allowed In hs smulaneous game, nvesors choose her radng sraeges whou knowng hose seleced by oher nvesors Moreover he game s played only once and a decson on a radng sraegy s made before submng he frs paral order For modelng purposes n hs paper, we assume ha he marke characerscs ncludng he number of nvesors K, he szes of her blocks and he execuon prce dynamc model 1, ha depends on all of he nvesors radng szes smulaneously, are known by every nvesor hs assumpon s no resrcve We make hs assumpon snce our goal here s o model he problem as a smulaneous game As soon as he problem s modeled as a game, exsng resuls n game heory abou paral nformaon games can be appled whenever he marke characerscs are no fully or correcly known by some of he nvesors For a few of hese resuls, applcable n our seng, see 1, 1 he equlbrum whose propery we nvesgae n hs paper s defned as follows: Defnon 1 he collecon of radng sraeges Ŝ1,, Ŝk,, ŜK s a Nash equlbrum of he game f for every nvesor k =1,,, K ϕ k Ŝk, Ŝ k ϕ k, Ŝ k, for any radng sraegy n M k hroughou, we refer o he radng sraeges a Nash equlbrum as he equlbrum radng sraeges As we show n Secon under some condons on he coeffcens of he prce-mpac funcon he equlbrum s unque A equlbrum no sraegc nvesor has ncenve o change hs own radng sraegy unlaerally gven ha no oher nvesor changes s sraegy Alhough we are more neresed n demonsrang ha equlbrum radng sraeges ouperform he Nave radng sraegy when oher nvesors rade accordng o he Nave radng sraegy or even a random see Secon 6, nce properes of he equlbrum when nvesors rade sraegcally should no be gnored he followng proposon shows ha under he assumpon ha nvesors rade sraegcally whch s known o all of hem, he equlbrum radng sraeges ncur he leas expeced execuon cos among he radng sraeges ha are robus wh respec o oher nvesors acons For s proof, see Appendx A As he proof shows he saemen of Proposon 1 remans rue for every convex prce-mpac funcon Proposon 1 Le he equlbrum radng sraegy be unque and assume nvesors ac sraegcally hen he equlbrum radng sraegy has he leas expeced execuon cos among he sraeges ha are robus wh respec o oher nvesors radng sraeges, e, for every k =1,,, K ϕ k Ŝk, Ŝ k mn max ϕ M k S k k,s k, =k M where Ŝ1,, ŜK s he equlbrum radng sraegy o conclude hs secon, we wan o emphasze ha he model and he soluon concep do no rely on he assumpons ha nvesors rade solely wh each oher or he res of he nvesors canno move he prce Moreover, n conras o he dynamc game models, a smulaneous game model s able o offer a radng sraegy over all of he perods before submng he frs paral order Compung he Equlbrum radng Sraeges One of he concerns abou any equlbrum concep s how effcenly can be compued In hs secon, we analyze he exsence and unqueness of he equlbrum for he execuon prce dynamc model 1 and show how can be compued Our dscusson explos he convexy of he underlyng prce-mpac funcon hroughou, we defne β m =0whenever > m Frs assume ha none of he nvesors s neresed n shor sellng durng hs radng program herefore he se of possble radng sraeges for he kh nvesor, M k, s convex, bounded and ncludes s boundary he followng proposon provdes a condon under whch he expeced execuon cos funcon of each nvesor s convex n hs own radng sraegy Proposon 1 Le he execuon prce a perod come from Equaon 1 hen for every nvesor k = 1,,, K and for every fxed value of S k, he expeced execuon cos funcon ϕ k,s k s convex n f and only f he marx β1 1 k β1k β1k β 1 k β1k β k βk β Ω k := k 1 β 1 k β k β k β k s posve semdefne, e, v Ω k v 0, for every - uple real vecor v Moreover, ϕ k,s k s srcly convex n f and only f Ω k s posve defne Proof he marx Ω k s he Hessan of he funcon ϕ k,s k Snce ϕ k,s k s a quadrac funcon, s convex wh respec o f and only f he Hessan of he funcon ϕ k,s k s posve semdefne hus ϕ k,s k s convex f and only f Ω k s posve semdefne

5 Noe ha for many values of prce-mpac parameers, he marces Ω k s are posve defne For nsance, for = 1,,, le β k =βk and for 1, β k =αk, where βk and αk are some consan parameers so ha βk αk βk >αk, hen he marx Ω k s posve semdefne posve defne Convexy and smoohness of he funcon ϕ k,s k for k = 1,, K along wh he fac ha every M k s convex, bounded and ncludes s boundary mply ha he generaed game n he presence of shor sellng consrans belong n a famous class of games, namely convex games 1 I was proven ha every convex game does have a leas one Nash equlbrum and under some condons s unque For a dealed argumen abou hese condons for he execuon prce dynamc model 1 see Appendx B he followng corollary s a drec consequence of he dscusson n Appendx B for a specal case Corollary 1 Le all of he nvesors have he same me horzon and affec he execuon prce wh he same magnude, e, β k =β for every =1,, and hen he posve defneness of Ω k mples ha he equlbrum unquely exss hs unque equlbrum can be found by he proeced graden mehod for convex mahemacal programmng problems 1 We would lke o emphasze ha fndng necessary condons on he parameers of he prcempac funcon under whch he equlbrum s unque remans a subec of ongong research Now suppose ha nvesors are allowed o shor sell hus, for each nvesor k, M k s unbounded We proec M k ono R 1 by seng = 1 herefore, ϕ k,s k can be resaed n erms of he frs 1 order szes: ϕ k S 1,,,, S K 1 K = P 0 1 β k =1 1 m=1 =1 1 =1 =1 K β ms m 1 m=1 =1 P 0 β ms m K β ms m m=1,m k Applyng hs expresson of ϕ k s, we may use he followng well known resul n game heory, namely Equlbrum es for nsance see : Proposon Le he collecon of radng sraeges Ŝ =Ŝ1,, ŜK sasfy he followng condons: 1 For every nvesor k, he graden of he expeced execuon cos funcon wh respec o he kh nvesor s order sze n perod, a pon Ŝ s zero, e, ϕ k Ŝ =0for =1,, 1 For every k, Ŝ k s he unque radng sraegy ha mnmzes he expeced execuon cos of he kh nvesor, when oher nvesors follow he radng sraeges Ŝ k he Hessan marces of he expeced execuon cos funcons ϕ k s a pon Ŝ are posve defne hen Ŝ s he unque Nash equlbrum of he generaed game For every nvesor k, when oher nvesors follow radng sraegy S k he problem of mnmzng ϕ k,s k wh respec o M k s a quadrac programmng problem Hence, posve defneness of Ω k mples ha s opmal mnma s unque herefore, when for every k =1,, K he marx Ω k s posve defne, he condon of Proposon s sasfed Moreover when Ω k s are posve defne, he condon holds a every S 1,, S K hus he soluon of he sysem ϕ k,s k =0, k =1,,K, =1,, k, s he unque Nash equlbrum for he generaed game when shor sellng s allowed Applyng he model 1 he above sysem s reduced o a sysem of lnear equaons hs dscusson s summarzed n he followng proposon For convenence, we sae he followng proposon for he case ha nvesors have dencal me horzons Proposon For every k =1,, K, le = and shor sellng be allowed Moreover assume he execuon prce a every perod =1,, comes from 1 so ha he marces Ω k s are posve defne hen he unque equlbrum s derved from he followng sysem: 1 β k β k β kβ k =1 β k β kβ k 1 = 1 K βk ī β k β kβ k m=1,m k =1 K 1 m=1,m k = 1 = S k β k β k β mβ m β m β m β m S m K m=1,m k for every k =1,,, K, =1,,, 1 S m β m S m, Proof A se of radng sraeges S 1,, S K sasfes Equaon f and only f sasfes hus when nvesors move he prce wh he same magnude, e, β k = β for every k = 1,, K, all an nvesor needs o know n order o compue he Nash equlbrum s he oal radng volume of oher nvesors We use he resul of Proposon n our smulaon

6 7 x 107 Shor Sellng Allowed, Oher raders Follow Opmal Sraegy 11 x 107 Shor Sellng Allowed, Oher raders Choose radng Sraeges A Random Opmal SraegyNave Equlbrum Sraegy 10 Opmal SraegyNave Equlbrum Sraegy 6 Expeced Execuon Cos of he Frs rader Expeced Execuon Cos of he Frs rader Number of raders K Number of raders K FIGURE 61: Expeced execuon cos of he equlbrum sraegy versus he Nave sraegy when k 1 oher nsuonal nvesors follow he Nave sraegy 6 Evaluaon of he Equlbrum radng Sraeges In hs secon, we compare he expeced execuon cos of he equlbrum radng sraegy wh he opmal soluon of Problem, he Nave sraegy, hrough smulaed nvesors We resrc our smulaon o he case ha shor sellng s allowed he seup of he smulaon s as follows We assume K = 10 nvesors commence buyng a block of some secury whose curren execuon prce s P 0 = 0$/share For k =1,, K, we se = 100, 000 shares, =1, σ =09 $/share/un of me, and for =1,,,, 1 we defne he emporary mpac β k = 10 $/share and he permanen mpac β k = 10 $/share herefore, he execuon prce follows he followng sochasc process: P = 0 10 K 1 10 k=1 =1 K k=1 σ ξ =1 When each nvesor gnores he radng acves of oher nvesors, he follows he opmal soluon of Problem, wh he execuon prce dynamc model 1, and hence places he paral orders accordng o he Nave sraegy In he frs example, we assume he nvesors k =,, K follow he Nave sraegy As Fgure 61 depcs under hs assumpon abou oher nvesors, he frs nvesor ncurs less expeced execuon cos by followng he equlbrum radng sraegy han ha of he Nave sraegy Sudyng he economc behavor of agens under he assumpon ha he agens acons are randomly dsrbued s a farly sandard approach n many dynamc models 7 of he real world markes hs movaed us o carry ou our second se of expermens under he assumpon ha oher nvesors choose her radng sraeges a random he seup of he marke s he same as before We ran M smulaons A a sngle smulaon, we generaed K 1 random permuaons R k of he number of perods,, hen for k =,, K, we defned he random radng sraegy = l rand randn, where l = for 1 R k, l =for 6 R k 10 and l = for 6 R k 1, wh = 1 =1 Sk FIGURE 6: Expeced execuon cos of he equlbrum sraegy versus he Nave sraegy when k 1 oher nsuonal nvesors rade a random for 1000 smulaons For each se of K 1 generaed radng sraeges, we compued he expeced execuon cos of he frs nvesor, when he follows he Nave sraegy, and hs expeced execuon cos of he equlbrum radng sraegy hen we compued he average of he obaned expeced execuon coss over M smulaons Fgure 6 llusraes he average expeced execuon coss for M = 1000 We ran anoher se of smulaons when nvesors rade a random wh Gaussan dsrbuon, e, for k =,, K, wese = randn for =1,, 1 and = 1 =1 Sk Fgure 71 depcs he frs nvesor s average expeced execuon coss over 1000 sngle smulaons by followng he Nave sraegy versus he random radng sraegy As he graphs demonsrae, he average of he frs nvesor s expeced execuon coss over M realzaons of he marke s less when he follows he equlbrum radng sraegy compared o ha of he Nave sraegy herefore, as far as he expeced execuon cos s concerned, s lkely ha he performance of he equlbrum radng sraegy domnaes he performance of he Nave sraegy 7 Concluson and Some Drecons for Fuure Work We suded he execuon cos problem for rsk neural nvesors n mul-nvesor markes wh lnear prce-mpac funcons We showed hrough an example ha each nvesor mus ake no accoun oher nvesors oal radng volume when decdng on a radng sraegy, oherwse a subopmal radng sraegy may be chosen We apply game heory o propose a model for hs suaon he Nash equlbrum of he generaed game s numercally compuable Our smulaons llusrae ha he expeced execuon cos of he equlbrum radng sraegy s lkely o be less han he expeced execuon cos of he opmal radng sraegy here remans a need o srenghen he heorecal aspecs of he properes of he Nash equlbrum for more general prce-mpac funcons Moreover explorng he performance of oher soluon conceps n game heory may 6

7 11 x Shor Sellng Allowed, Oher raders Choose radng Sraeges A Random Opmal SraegyNave Equlbrum Sraegy 1 M Marnacc, Ambguous games, Games and Economc Behavor Expeced Execuon Cos of he Frs rader M Prsker, Large nvesors: mplcaons for equlbrum asse reurns, Fnance and Economcs Dscusson Seres, Washngon, DC, J B Rosen, Exsence and Unqueness of Equlbrum Pons for Concave N-Person Games, Economerca Number of raders K FIGURE 71: Expeced execuon cos of he equlbrum sraegy versus he Nave sraegy when k 1 oher nsuonal nvesors rade a random wh Gaussan dsrbuon for 1000 smulaons resul n more neresng conrbuons, parcularly hose soluon conceps ha rely on weaker assumpons abou he nvesors We also hope o exend our work o porfolo radng Appendx A Here we prove Proposon 1 Le Ŝ1,, ŜK be he unque equlbrum radng sraegy hus for every M k, maxŝ k K =1, =k M ϕ k,s k s no less han ϕ k, Ŝ k herefore, mn max M k Ŝ k ϕ k,s k s greaer K =1, =k M han or equal o mn M ϕ k, k Ŝ k hs nequaly along wh he fac ha Ŝ1,, ŜK s he unque equlbrum, e, mn M ϕ k, k Ŝ k = ϕ k Ŝk, Ŝ k proves he saemen of Proposon 1 References 1 M Aghass, D Bersmas, Robus game heory, Mahemacal Programmng C D Alprans, S K Chakrabar, Games and decson makng, Oxford Unversy Press, 000 R Almgren, N Chrss, Opmal execuon of porfolo ransacons, he Journal of Rsk 000/001 9 R Almgren, Opmal execuon wh nonlnear mpac funcons and radng-enhanced rsk, Appled Mahemacal Fnance D Bersmas, A W Lo, Opmal conrol of execuon coss, Journal of Fnancal Markes D Bersmas, A W Lo, P Hummel, Opmal conrol of execuon coss for porfolos, IEEE Compuaonal Scence and Engneerng J D Farmer, I Zovko, he power of paence: a behavoral regulary n lm order placemen, Quanave Fnance H He, H Mamaysky, Dynamc radng polces wh prce mpac, Journal of Economc Dynamcs and Conrol G Huberman, W Sanzl, Opmal lqudy radng, Revew of Fnance G Huberman, W Sanzl, Prce manpulaon and quasarbrage, Economerca R w Holhausen, R W Lefwch, D Mayers, he effec of large block ransacons on secury prces, a crossseconal analyss, Journal of Fnancal Economcs Appendx B Le S =S 1,,,, S K be a collecon of radng sraeges for he nvesors For each fxed K-uple real vecor r 1,, r K, defne he funcon gs, r 1,, r K n erms of he gradens k ϕ k,s k as follows: gs, r 1,, r K = r 1 1 ϕ 1 S 1,S 1 r ϕ S,S r K K ϕ K S K,S K Denoe he Jacoban of gs, r 1,, r K wh respec o S wh GS, r 1,, r K Rosen 1 proves ha f he marx GS, r 1,, r K G S, r 1,, r K s posve defne for every S, hen he equlbrum sasfyng 1 s unque Accordng o he noaons n Equaon 1, he K m=1 m K m=1 m marx GS, r 1,, r K equals r 1Ω 1 r 1Λ 1, r 1Λ 1, r 1Λ 1,K r Λ,1 r Ω r Λ, r Λ,K, 91 r KΛ K,1 r KΛ K, r KΛ K, r KΩ K where he marx Λ, s defned as follows: β β 1 β 0 0 Λ, = β 1 β β β When he assumpons n Corollary 1 hold, he marces Λ, are dencal, e, Ω k =Ωfor some fxed marx Ω herefore, seng r 1 = r = = r K =1, he marx GS, r 1,, r K G S, r 1,, r K s posve defne f and only f Ω s posve defne 7