Both human traders and algorithmic

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1 Shuhao Chen s a Ph.D. canddae n sascs a Rugers Unversy n Pscaaway, NJ. bhmchen@sa.rugers.edu Rong Chen s a professor of Rugers Unversy n Pscaaway, NJ and Peng Unversy, n Bejng, Chna. rongchen@sa.rugers.edu Gary Ardell s he head of Fnancal Engneerng and Advanced Tradng Soluon a he ConvergEx Group n Boson, MA. gardell@convergex.com Bquan Ln s n Fnancal Engneerng and Advanced Tradng Soluon a he ConvergEx Group n Iseln, NJ. bln@convergex.com End-of-Day Soc Tradng Volume Predcon wh a Two-Componen Herarchcal Model Shuhao Chen, Rong Chen, Gary Ardell, and Bquan Ln Boh human raders and algorhmc radng engne desgners have a profound neres n he hgh-qualy predcon of he volume ha wll be raded n he remander of he radng day. Ths volume represens he lqudy agans whch orders can be ransaced whle he avalable lqudy deermnes he mare mpac of worng any order. Indeed, wh nadequae lqudy, may no be feasble or may be oo expensve o execue a large order n he remander of he radng day. The commonly used guaraneed execuon algorhms, such as he naon prce, face he challenge of worng paren orders across he day. To mnmze mare mpac, he algorhm mus eep s volume parcpaon rae as low as possble. On he oher hand, he algorhm mus also ensure ha s volume parcpaon rae s hgh enough n order o complee he order whn he day. Thus, s crucal o have an accurae predcon of he volume n he remander of he day o effecvely execue such algorhms. Remander day s volume predcon also plays an mporan role n analyzng ransacon cos. Berowz [988] nroduced he concep of daly volume weghed average prce (VWAP) and used he dfference beween he average execuon prce and he recorded VWAP o measure he cos of each rade. Mnmzng such cos s among crcal goals for nsuonal nvesors and havng beer nowledge of remnder day s volume would defnely help o mae an effcen execuon decson and, hus, favorable cos. Recenly, boh academa and praconers have been pursung beer models forecasng he radng volumes. Lo and Wang [000] analyzed behavor of equy radng volume usng he capal asse prcng model (CAPM). Hausch [00] modeled he nraday volume acvy based on volume duraons usng auoregressve condonal duraon (ACD) models, whch were orgnally nroduced by Engle and Russell [998]. Darolles and Le Fol [003] proposed a mehodology of decomposng radng volume and Balows, Darolles, and Le Fol [008] exended he prevous wor no nraday daa, decomposng nraday radng volume no wo componens: ) reflecs volume change assocaed wh mare evoluons, and ) represens soc-specfc volume paern. I used he hsorcal VWAP curve o esmae he mare componen and he auoregressve movng average (ARMA) and he selfexcng auoregressve (SETAR) models o esmae soc specfc componen. Alhough our am s o predc he volume o be raded n he remnder of he day, we nsead nvesgae he oal volume accrued hroughou he day, or he endof-day volume, for smplcy. Gven he fac ha he volume ha has been accumulaed Auhor Draf For Revew Only Summer 0 The Journal of Tradng JOT-CHEN.ndd 5/6/ :3:56 PM

2 from he begnnng of he day o he me of he predcon s nown o us, he wo volumes are equvalen. Inuvely, here are wo sources of useful nformaon for projecng he end-of-day volume. Snce we have observed he paral volume observed up o he me of predcon, and f he dsrbuon of oal volume hroughou he day s relavely sable and can be esmaed usng hsorcal daa, hen he oal end-of-day volume predcon can be made wh he paral volume and he esmaed proporon should assume for he oal volume. We call hs mehod nraday predcon. The second source of nformaon s he dynamcs of daly volume changng over me. If such dynamcs s properly modeled, daly volume can be predced as well. We denoe hs mehod as daly predcon. Snce he nraday mehod ulzes he volume accrued durng he radng day whle he daly mehod uses daly volume seres, hose wo mehods provde ndependen predcons. They can be mproved by combnng boh sources of nformaon. In hs arcle, we propose a herarchcal model for such a combnaon. I exends o he sable seasonal paern model of Chen and Fomby [999], where he model was used o predc end-of-year oal number of ourss. A smlar dea was used by Olver [999] as well. Ths approach s dfferen from ha of Balows, Darolles, and Le Fol [008] who decomposed he radng volume no wo componens ha reflec volume changes due o mare evoluons and he soc-specfc volume paern. Ths arcle s organzed as follows: In he nex secon, he wo-componen herarchcal model s presened. Is predcon procedures and some exensons are shown n he hrd secon. The fourh secon shows an emprcal sudy usng Dow Jones Indusral Average componen socs, comparng ou-of-sample predcon performance of dfferen mehods. TWO-COMPONENT HIERARCHICAL MODEL Consder a radng volume seres, x, x, x d, where =,,, n corresponds o dfferen radng days and d denoes he number of radng perods used each day. In hs arcle, we use d = 3, correspondng o hreen 30-mnue radng nervals of he U.S. equy mare. The end-of-day volume for day s hen gven by y = x + +x d. In Equaon (), we assume ha he daly paern s sable across dfferen days, whch s o say ha gven he end-of-day volume oal y, (x, x, x d ) are condonally ndependen for =,,, n. Namely px [,, x,, x,, x y,, y ] d n nd n n = px [,, x y ] () = d where p[ ] denoes he condonal densy. Ths assumpon allows us o model he seres, x, x,, x d, wh a wo-componen herarchcal srucure, shown n Equaon (). px [,, x y ], M ( y, θ) and d { y} M ( y,, β ), () where he nraday model M s a (d )-dmensonal dsrbuon and he daly model M s a me seres model for he daly volume seres {y }. θ and β are parameers o be esmaed. There are several mporan characerscs assocaed wh hs herarchcal model. Frs, s assumed ha he dependency beween dfferen days s only hrough he oal volume {y }, no he ndvdual observaons whn he days. Second, gven he oal volume, he nraday dsrbuon of he oal volume hroughou he day s he same. Thrd, he wo models, one for nraday dsrbuon and he oher for he daly volume seres, can be modeled separaely. Those hree properes are deal n erms of mang nraday predcons of he end-of-day volume. In he followng seps, we dscuss he possble models for M and M. Snce {x } s neger-valued, one mmedae choce for model M s he mulnomal dsrbuon, wh he oal beng he daly oal volume. However, radng volume s ofen large enough o be reaed as a connuous varable. Treang as a connuous varable also provdes he advanage of more flexbly n he modelng of oal volume seres y. Neverheless, we sll wan s dsrbuon o possess he proporonal nerpreaon of he mulnomal dsrbuon. Chen and Fomby [999] proposed a connuous analogue of he mulnomal dsrbuon called he Gaussan mulnomal dsrbuon. Specfcally, as shown n Equaon (3), a d-dmensonal random varable (x, x,, x d ) s sad o follow Gaussan-mulnomal(G-MN) dsrbuon (y, θ,, θ d, ) f Auhor Draf For Revew Only End-of -Day Soc Tradng Volume Predcon wh a Two -Componen Herarchcal Model Summer 0 JOT-CHEN.ndd 5/6/ 0:7:3 PM

3 d px [,, x ( d )] N( µ, ) and x = y x d, d where 0< θ <, θ = and θ y θ y µ =, θ y d = = (3) θ ( θ ) θ θ θ θ d θθ θ ( θ ) θθ d = θθ θθ θ ( θ ) d d d d Noe ha µ and do no depend on, due o our assumpon of a sable daly paern. Ths dsrbuon can be vewed as a connuous verson of he mulnomal dsrbuon. The mean and correlaon coeffcen marx are he same as he mulnomal dsrbuon, excep for he exra varance parameer. In he mulnomal dsrbuon, he varance vares wh he oal volume n a relavely resrced way. In he Gaussan mulnomal dsrbuon, he varance s consan. I s possble o allow for varyng varance (dependng on observable varables) smlar o he weghed regresson seng or a GARCH ype of heeroscedascy. Here we choose o assume consan varance. I can be shown ha he Gaussan mulnomal dsrbuon also has he combnaon propery of he mulnomal dsrbuon. For example, f (x,, x d ) G MN(y, θ,, θ d, ), hen (x + x, x 3,, x d ) G MN(y, θ + θ,, θ d, ). More crcally, hs propery mples ha for <d x ~ N( γ y, γ ( γ )), where γ = θ = = Ths feaure allows us o esmae γ much more easly han esmang θ,, θ n he hgh dmensonal space. The parameer γ s of crcal neres as our predcon procedure nvolves only γ, nsead of he ndvdual parameers θ,, θ. In Equaon (4), he maxmum lelhood esmaor of γ n our seng s ( ) n x y ˆγ = = = (4) n y = wh sandard error / = [ γˆ ( γˆ )/ y ]. The maxmum lelhood esmaor for s θ θ = n ( x ˆ y ) ˆ ( x ˆ y ) = n In Equaon (5), o model he daly volume dynamcs, we use a Gaussan model where py [ y ]~ N( µ, ) (5) where y = (y,, y ), m = f(y, θ), and s = g(y, θ). Ths model ncludes all he sandard auoregressve negraed movng average (ARIMA) models of Box and Jenns [976] and Brocwell and Davs [986], wh or whou GARCH errors (Bollerslev [986]). PREDICTION BASED ON THE TWO-COMPONENT HIERARCHICAL MODELS A a gven me of he radng day, we are neresed n esmang he end-of-day volume y usng volume accumulaed up o me, namely (x, x,, x ) and he hsorcal daly volume seres y,, y. ( ) Specfcally, le x = ( x, x,, x ) and = ( V ), j where V j = θθ j for j and v = θ ( θ ). Then, where and µ, py [ x,, x y ] p[ x,, x y ],, py [ y θ Σ θ = ]~ N( µ, ),, γ + = ( γ ) ( ) x Σ θ µ Σ x = + = =, ( γ ) Auhor Draf For Revew Only µ + +, (6) Hence, he leas squares predcon of y s m,. In fac, leng c = /, Equaon (6) can be wren as Equaon (7) Summer 0 The Journal of Tradng 3 JOT-CHEN.ndd 3 5/6/ 0:7:36 PM

4 µ, c = x + ( γ ) µ γ + ( γ ) c γ + ( γ ) c w = x = x = + ( w ) µ (7) γ Here, Σ and µ γ are he predcons of he end-of-day volume based on M model alone and M model alone, respecvely. The weghs ω and ω dcae he conrbuon of models M and M o he predcon of he oal volume. Noe ha, when he model for he oal volume seres and he model for volumes whn he day have equal precson, ha s, =, hen d µ = Σ x + ( γ ) µ = Σ x + Σ θµ. Hence, he, = = = adjusmen procedure s smply o replace he predced conrbuon of he ndvdual observaons θ µ by her observed values x. When c <, Σ x =, bears more wegh n predcon, because he ndvdual volume observaon has less varance (more accuracy) han he daly volume model. An alernave s o rea he combnaon weghs ω and ω n Equaon (7) as unnown parameers. If one assumes homoscedascy n he daly volume seres, he weghs are consan over me. They can be esmaed wh leas squares as f he daly oal volume follows a regresson model wh Σ x = and µ as explanaory varables. Ths model has a nonparamerc modelng flavor as bypasses he assumpons of M and M and opmzes he lnear combnaon drecly. We call hs model he regresson approach. One can furher exend hs model by ncludng oher nformave varables n he combnaon, such as mare radng volume up o perod. EMPIRICAL STUDY In hs secon, we apply he wo-componen herarchcal model o he hsorcal volume profle of he 30 seleced socs n he Dow Jones Indusral Average from January 00 o Sepember 00. Dealed resuls are shown usng he volume profle of Apple, Inc. There are 85 radng days n he sudy. We use he frs 55 days for modelng and parameer esmaon and he las 30 days for ou-of-sample predcon performance comparson. Exhb shows he volume seres of Apple, Inc. The volumes o he lef of vercal dash lne are E x h b Apple, Inc., Volume Seres used n fng, whle he rgh-sde volumes are used for predcon and esng purposes. To avod dealng wh mcro-srucure nose and for he ease of compuaon, we aggregae he mnueby-mnue volume daa o 30-mnue nervals wh 3 perods per day. Here we use :00 p.m. as our me for he end-of-day volume predcon, ha s, we have observed he radng volume up o :00 p.m. (hus, = 7) and wan o esmae he end-of-day volume usng he accumulaed volume ha day and he hsorcal daly volumes before ha day. We use he Gaussan mulnomal dsrbuon for Model M. Exhb depcs he esmaed θ, 3 for Apple, Inc., for he nraday dsrbuon of he oal volume no he 3 nraday perods. Regulary of he E x h b Inraday Volume Dsrbuon of Dfferen Socs Auhor Draf For Revew Only Noe: (x-axs corresponds o he 3 radng perods and y-axs represens he volume percenage θ. Volumes would accumulae o 00%.) 4 End-of -Day Soc Tradng Volume Predcon wh a Two -Componen Herarchcal Model Summer 0 JOT-CHEN.ndd 4 5/6/ :4:05 PM

5 paern for dfferen socs s also demonsraed and verfed. Furhermore, Exhb 3 shows he esmaes and sandard errors of γ Σ θ =. We f he hsorcal daly volume seres usng Gaussan ARMA models and ARMA-GARCH models. The auocorrelaon funcon (ACF) and paral auocorrelaon funcon (PACF) of Apple, Inc. s daly volume seres are shown n Exhb 4. They appear o be que saonary. Furher sudy usng he exended auocorrelaon funcon of Tsay and Tao [984] and he adjused exended auocorrelaon funcon of Chen, Mn, and Chen [00] and he AIC creron of Aae [974] denfes an AR() model as an approprae model for he daly volume seres of Apple, Inc. To nclude heeroscedascy, we also model he error erm of he AR() model as a GARCH(, 0) process, afer carryng ou a model selecon procedure. Exhb 5 shows he esmaed parameers for AR() and AR() GARCH(, 0) model. The GARCH componen s margnally sgnfcan. We also f oher volume seres ndvdually and denfy he bes ARMA and ARMA GARCH models. They are used as Model M n he predcon exercses. Gven he esmaed parameers of M and M, we compare followng sx dfferen predcon mehods. E x h b 3 Maxmum Lelhood Esmaes (sandard errors) of he Gamma E x h b 4 ACF and PACF for Apple, Inc. s Daly Volume Seres E x h b 5 Fng Resuls for Canddae ARMA Models. ARMA:Predcon usng he daly volume seres only, under he ARMA model;. GARCH: Predcon usng he daly volume seres only under ARMA GARCH model; 3. Inraday: Predcon usng nraday model M only; 4. New-ARMA: Predcon usng he new herarchcal model wh ARMA as model M ; 5. New-GARCH: Predcon usng he new herarchcal model wh ARMA GARCH as model M ; 6. Reg: Predcon usng combnaon (7) wh leas squares opmzed weghs. We frs show he dealed resuls of Apple, Inc. Exhb 6 numeraes he esmaed varance of he nraday Gaussan Mulnomal dsrbuon (M ) and for he ARMA model (M ) and he correspondng weghs ω and ω n Equaon (7). Noe ha we assume ha he ARMA models for M, c and ω are ndeed ndependen of me. In hs case, he predcon provded by he daly seres s relavely more accurae (smaller varance), hence, havng a large wegh n he combned predcon. The esmaed weghs under leas square creron s also presened and ends o pu relavely more wegh on nraday predcon, whch s no opmal n hs case. Auhor Draf For Revew Only E x h b 6 Varance Comparson I Summer 0 The Journal of Tradng 5 JOT-CHEN.ndd 5 5/6/ :4:08 PM

6 Exhb 7 shows he esmaed varance under he ARMA GARCH model for he 30 days n our predcon perod and s correspondng rao c, comparng o he esmaed lsed n Exhb 6. I does change que sgnfcanly, from 0. 3 o almos 3. Exhb 8 shows he predcons of ARMA, GARCH, nraday, reg, new-arma and new- GARCH, wh he rue observaons mared as dos. Day 8 s an unusual observaon. The acual volume s sgnfcanly larger han normal. The new predcon mehod s able o capure such a large movemen, whle E x h b 7 Varance Comparson II E x h b 9 Predcon Resuls from Varous Mehods (n mllons of shares) he daly model underpredcs and he nraday model overpredcs. Exhb 9 shows he acual predcon of he volume (n mllons of shares) of he 30 day predcon perod, under dfferen mehods. The rue observaon, labelled as real s shown a he boom lne for comparson. Exhb 0 summarzes he predcon performance by showng he roo-mean-squared predcon error n RMSE = Σ ( Pred True ). = E x h b 8 Comparson of Dfferen Predcon Mehods Auhor Draf For Revew Only 6 End-of -Day Soc Tradng Volume Predcon wh a Two -Componen Herarchcal Model Summer 0 JOT-CHEN.ndd 6 5/6/ :4: PM

7 In he followng, we presen he predcon performance comparson for he 30 socs n he Dow Jones Indusral Average Index. The frs wo columns of Exhb provde he company nformaon; he followng wo columns sore he ARMA and GARCH orders of he daly volume seres. Mean-squared predcon errors of he sx predcon mehods are hen lsed, followed by he rao comparson sascs. We mae he followng observaons:. In mos cases, he new wo-componen herarchcal model performs much beer han usng he nraday model alone and usng he daly seres dynamcs alone. In more han half of he cases, E x h b 0 Roo Mean Square Error from Varous Mehods he mprovemen s over 0%, some sgnfcanly hgher.. Alhough he ARMA GARCH model may be margnally beer han ARMA model for modelng he daly volume seres dynamcs (as n he case of Apple, Inc. ), he ARMA model wors almos as well as ARMA GARCH model n erms of predcon. 3. The alernave combnaon mehod wh he leas-squares-opmzed weghs ouperforms he nraday model alone and he daly seres alone. Is overall performance s no as good as he one based on he herarchcal model. E x h b Predcon Comparson on Dow Jones Indusral Average Componens Based on Predcon Mean Square Error (n mllons of shares) Auhor Draf For Revew Only Summer 0 The Journal of Tradng 7 JOT-CHEN.ndd 7 5/6/ :4:4 PM

8 4. The phenomenon observed on he 8h day of he Apple, Inc., volume seres s que mporan. I shows ha he herarchcal model can ndeed effecvely combne wo ndependen sources of nformaon and produce a more accurae predcon. Ths phenomenon s also observed n oher socs as well. 5. The dea of ncludng oher facors, such as mare volume, n he combnaon has been red, whou success. More research needs o be done o fnd more approprae facors. In concluson, our emprcal sudy shows ha he proposed wo-componen herarchcal model and s assocaed predcon mehod are effecve n mang predcons of he end-of-day volume, and s more accurae compared o ha usng he nraday model alone and daly volume seres alone. Endnoes Arbrarly chosen, no ncluded n DJIA hough. Apple, Inc., soc rades under AAPL on he NASDAQ. REFERENCES Aae, H. A New Loo a he Sascal Model Idenfcaon. IEEE Transacons on Auomac Conrol, Vol. 9, No. 6 (974), pp Berowz, S.A., D.E. Logue, and E.A.J. Noser. The Toal Cos of Transacon a he NYSE, Journal of Fnance, Vol. 43, No. (988), pp Bersmas, D., and A.W. Lo. Opmal Conrol of Execuon Coss, Journal of Fnancal Mares, (998), pp Balows, J., S. Darolles, and G. Le Fol. Improvng VWAP Sraeges: A Dynamc Volume Approach. Journal of Banng and Fnance, Vol. 3, No. 9 (Sepember 008), pp Bollerslev, T. Generalzed Auoregressve Condonal Heerosedascy. Journal of Economercs, 3 (986), pp Box, G.E.P., and G.M. Jenns. Tme Seres Analyss: Forecasng and Conrol, San Francsco, CA: Holden-Day, 976. Brocwell, P.J., and R.A. Davs. Tme Seres: Theory and Mehods, New Yor: Sprnger-Verlag, 986. Chen, R., and T.B. Fomby. Forcasng wh Saable Seasonal Paern Models wh an Applcaon o Hawaan Toursm Daa. Journal of Busness and Economc Sascs, Vol. 7, No. 4 (Ocober 999). Chen, S., W. Mn, and R. Chen. Model Indenfcaon for Tme Seres wh Dependen Innovaons, Techncal repor, Deparmen of Sascs, Rugers Unversy, (00). Darolles, S., and G. Le Fol. Tradng Volume and Arbrage. Worng paper, CREST, 003. Engle, R.F. Auoregressve Condonal Heeroscedascy wh Esmaes of he Varance of Uned Kngdom Inflaon. Economerca, Vol. 50, No. 4 (98), pp Engle, R.F., and J. Russell. Auoregreesve Condonal Duraon: A New Model for Irregularly Spaced Transacon Daa. Economerca, 66 (998), pp Hausch, N. Modellng Inraday Tradng Acvy Usng Box-Cox ACD Models. Worng paper, Cener of Fnance and Economercs, Unversy of Konsanz, February 005. Lo, A., and J. Wang. Tradng Volume: Defnon, Daa Analyss, and Implcaon of Porfolo Theory. Revew of Fnancal Sudes, 3 (000), pp Madhavan, A. Vwap Sraeges. Transacon Performance: The Changng Face of Tradng Invesmen Gudes Seres, New Yor: Insuonal Invesor, Inc., 00. Olver, R.M. Bayesan Forecasng wh Sable Seasonal Paerns. Journal of Busness and Economc Sascs, Vol. 5 (Ocober 999), pp Tsay, R.S., and G.C. Tao. Conssen Esmaes of Auoregressve Parameers and Exended Sample Auocorrelaon Funcon for Saonary and Nonsaonary ARMA Models. Journal of Amercan Sascal Assocaon, Vol. 79, No. 385 (March 984). Auhor Draf For Revew Only To order reprns of hs arcle, please conac Dewey Palmer a dpalmer@journals. com or End-of -Day Soc Tradng Volume Predcon wh a Two -Componen Herarchcal Model Summer 0 JOT-CHEN.ndd 8 5/6/ :4:4 PM