Trading Volume, Price Autocorrelation and Volatility under Proportional Transaction Costs

Transcription

1 Tradng Volume, Prce Autocorrelaton and Volatlty under Proortonal Transacton Costs Hua Cheng Unversty of Pars Dauhne - Deartment of Economcs SDF (Fnancal Strateges & Dynamcs), P33A, Deartment of Economcs, Unversty of ParsDauhne,PlacedeMaréchaldeLattredeTassgny ParsCedex6.Tel: Emal address: hua.cheng@dauhne.fr.

2 Abstract We develo a dynamc model n whch traders have dfferental nformaton about the true value of the rsky asset and trade the rsky asset wth roortonal transacton costs. We show that wthout addtonal assumton, tradng volume can not totally remove the nose n the rcng equaton. However, because tradng volume ncreases n the absolute value of nosy er cata suly change, t rovdes useful nformaton on the asset fundamental value whch cannot be nferred from the equlbrum rce. We further nvestgate the relaton between tradng volume, rce autocorrelaton, return volatlty and roortonal transacton costs. Frstly, tradng volume decreases n roortonal transacton costs and the nfluence of roortonal transacton costs decreases at the margn. Secondly, rce autocorrelaton can be generated by roortonal transacton costs: under no transacton costs, the equlbrum rces at date and are not correlated; however under roortonal transacton costs, they are correlated - the hgher (lower) the equlbrum rce at date, the lower (hgher) the equlbrum rce at date. Thrdly, we show that return volatlty may be ncreasng n roortonal transacton costs, whch s contrary to Stgltz 989, Summers & Summers 989 s reasonng but s consstent wth Umlauf 993 and Jones & Segun 997 s emrcal results. Keywords: Tradng Volume, Autocorrelaton, Volatlty JEL codes: G, G4, D8

3 In the standard ratonal exectaton equlbrum model wth aggregate suly uncertanty, f tradng volume lays any role, t s manly to resolve the nosy suly from the equlbrum rce equaton. All traders observng tradng volume know the aggregate suly and thus there s a revealng rce. Whle t s very dffcult for traders to submt ther orders condtonal on rce and tradng volume n ractce, we nvestgate the role of tradng volume n the equlbrum n whch traders make ther decsons condtonal u to but not ncludng the market statstc resultng from ther desred trade. Transacton costs are an mortant factor n determnng the tradng behavor of market artcants. On the consequence, transactons costs should affect tradng volume, asset rces and ther tme seres features. Whle consderable attenton has focused on the effects of transacton costs on asset rces, there are very few models whch nvestgate ther nfluences on rce autocorrelaton and return volatlty 3. Increased transacton costs are usually thought to reduce the ncentve to traders and therefore roduce a thnner market. Thn tradng tends to nduce or ncrease autocorrelaton, as n Boudoukh, Rchardson & Whtelaw 994. Stgltz 989 and Summers & Summers 989 argue that transacton costs should dmnsh return volatlty. The reasonng undernnng ths clam stems from the belef that nose trades, whch are not based on nformaton about underlyng values, may move rces away from the ntrnsc value and ncrease volatlty 4. However, ths reasonng s not consstent wth the emrcal results (Umlauf Ths aroach s frst suggested by Hellwg 98. Blume & Easley 984 use ths aroach to examne the nformaton content of ast market rces. Blume, Easley & O Hara 994 use t to nvestgate the role of volume. Cheng 005a uses t to study the role of volume n an economy wth roortonal transacton costs and the nfluence of transacton costs n a statc model. Transacton costs fall nto two broad categores. Frst are the drect enunary costs of tradng. These nclude the market-maker s sread, the broker s fees, and any transacton taxes, such as stam dutes. Second are the ndrect costs. These nclude the costs of acqurng and rocessng nformaton about asset values, comanes, market movements and any other nformaton whch may be relevant to the decson to buy and sell assets. We defne transacton costs n a narrow way and they nclude only the frst n our model. 3 The drecton and magntude of the affects of transacton costs on asset rces are stll subject to consderable controversy and debate. Constantndes 986, Heaton & Lucas 996, and Huang 003 show that transacton costs have only a small mact on asset rces. Amhud & Mendelson 986 and Lo, Mamaysky & Wang 004 fnd that the lqudty dscount of transacton costs can be substantal, deste relatvely small transacton costs. Whle those models argue that there exsts always the lqudty remum of transacton costs, Vayanos 998 and Cheng 005a suggest that asset rce may ncrease n ts transacton costs. 4 They defne nose traders broadly. Ths defnton ncludes (but s not lmted to) ortfolo nsurers and other so-called ostve feedback traders, snce ther trades are based only on changes n reorted rces rather than on ntrnsc value, as well as others who beleve (rratonally) that tradng systems, horoscoes, etc., are benefcal n forecastng rces. 3

4 993, Jones & Segun 997, Green, Maggon & Murnde 000, and Hau 006, etc.). In the end of ther aer, Jones & Segun 997 conclude that "Our results, when combned wth those of Umlauf 993, suggest that the logc of ncreasng transacton taxes to reduce the mact of nose traders and therefore, to reduce volatlty, does not wthstand emrcal scrutny. Indeed, our results ndcate that ncreasng transacton costs through any avenue may well have an effect exactly ooste from that ntended." We develo a two erods model n whch traders have dfferental nformaton about the true value of the rsky asset and there are roortonal transacton costs on the rsky asset. We show that wthout addtonal assumton, tradng volume can not totally remove the nose n the rcng equaton. However, because tradng volume ncreases n the absolute value of nosy er cata suly change, t rovdes useful nformaton on the asset fundamental value whch cannot be nferred from the equlbrum rce. Our second result s that tradng volume decreases n roortonal transacton costs and the nfluence of roortonal transacton costs decreases at the margn. The thrd result s about how rce autocorrelaton can be generated only by roortonal transacton costs: under no transacton costs, the equlbrum rces at date and are not correlated (ndeendent); however under roortonal transacton costs, they are correlated - for all the other arameters gven, the hgher the equlbrum rce (common nformaton) at date, the lower the equlbrum rce at date. Our fourth result s to show that return volatlty may be ncreasng n roortonal transacton costs, whch s contrary to Stgltz 989, Summers & Summers 989 s reasonng but s consstent wth most emrcal results (Umlauf 993, Jones & Segun 997, Green, Maggon & Murnde 000, and Hau 006, etc.). Therestoftheartclesstructuredasfollows: nsectonwedevelothe basc model. In secton we study the equlbrums at date and and the relaton between the nosy er cata suly change and tradng volume. In secton 3 we analyze the relatonsh between tradng volume, rce autocorrelaton, return volatlty and roortonal transacton costs. In secton 4, we contrast our results to several relevant emrcal aers and argue that our argument s consstent wth the avalable evdence. We resent conclusons n secton 5. 4

5 Economy We consder a smle economy wth two assets n the economy: a rskless asset and a rsky asset. The rskless asset s assumed to have an nfntely elastc suly and the suly of the rsky asset s random. Let x and x denote the random er cata sules of the rsky asset at dates and, resectvely 5. The nterest rate of rskless asset s smlfed to be 0 and ts rce s normalzed to one. All assets are traded n a comettve market. There are two tyes of rsk-averse traders n the economy and we dvde the traders nto two grous wth N I un traders n grou and N U ( u)n traders n grou. The two classes of nvestors are dfferent n ther nformaton about the state of the economy whch s further defned as follows. We focus on the results wth un and ( u)n, namely the results n the large economy. u s suosed to be tme ndeendent and nformed (unnformed) traders at date are stll nformed (unnformed) at date.. Preference All traders have constant absolute rsk averson (CARA). Each trader maxmzes hs exected utlty of consumton at date 3 h h E U(w j 3, z ) j t, E ex( Rw j 3 z ) j t, where w j 3, s trader j n grou s wealth at date 3, the common absolute rsk averson R s smlfed to, and z j t, s the nformaton set avalable to trader j n grou at date t.. Informaton Structure Traders are a ror dentcal: at date 0, each trader enters the frst erod wth an endowment of z 0 unts of the rskless asset and has the dentcal belefs about the ayoff of the rsky asset ψ, whch s assumed to be a normal dstrbuton N(ψ 0, /ρ 0 ). Just ror to the oenng of the market at date and, each trader receves 5 Uncertanty n the er cata suly reflect uncertan order flowbasedonnosetraders. In our model, nose traders s norrowly defned and nclude only those who trade for lqudty reasons. Then t s logcal that ther trades are not (or lttle) nfluenced by transacton costs. Whether there s transacton cost or not at date, the nosy er cata suly s dstrbuted nthesameway. 5

6 hs rvate sgnal and then traders trade wth each other. The rvate sgnals that each grou receves are dentcally dstrbuted but the dstrbutons for two grous are dfferent. More recsely y j t, ψ + w t + e j t for nformed trader j n grou y j t, ψ + w t + ε j t for unnformed trader j n grou,t, where w t s a common error term dstrbuted N(0, /ρ w t ), e j t and ε j t are dosyncratc errors for traders n grou and grou whch are dstrbuted N(0, /ρ t ) and N(0, /ρ t ), resectvely. Assume further that ρ t >ρ t. Snce the nformaton that traders n grou receve s more relable than that traders n grou receve, we refer to traders n grou as nformed traders and traders n grou as unnformed traders. At the end of the second erod (at date 3), the value of the rsky asset s revealed and all traders consume ther wealth. To smlfy the notaton, we wrte each trader s nformaton structures of the rsky asset after recevng the rvate sgnal n the followng way where ρ s t y j t, N(ψ, /ρ s t ), for nformed trader j n grou y j t, N(ψ, /ρ s t ), for unnformed trader j n grou ρ t ρ w t / ρ t + ρ w t and ρ s t ρ t ρ w t / ρ t + ρ w t. Condtonal on wt y j t w t N(θ t, /ρ t ), for nformed trader j n grou y j t w t N(θ t, /ρ t ), for unnformed trader j n grou where θ t ψ + w t denotes common nformaton to all traders at date t. Soby the Strong Law of Large Numbers, the mean sgnal n each grou, y t and y t converges almost surely to θ t as N. In the large economy, the mean sgnal s almost surely equal to the ture value lus the common error. All the random varables (w t,e j t,ε j t, and x t ) are suosed to be mutually ndeendent..3 Otmzaton Ideally, each trader would redct the stochastc rocess of rces condtonal on hs nformaton set, and solve the ntertemoral decson roblem and take otental catal gans nto account. However, ths roblem s tractable f and only f future rces are normally dstrbuted. It s not the case n our model for two reasons. Frstly, the recson of the common and rvate nformaton 6

7 are random and we don t know ther future value. Secondly, even though we suose that the recson of the common and rvate nformaton are constant or determnant, because of transacton costs, the equlbrum rce n the second erod wll never be normally dstrbuted. Followng Brown & Jennng 989 and Blume, Easley & O Hara 994, we assume that traders have myoc, or nave, demands so that each trader chooses hs demand to maxmze hs exected utlty at date 3 wthout takng ossble future transactons nto account. We suose that each trader makes hs decson condtonal on all nformaton u to but not ncludng the market statstcs from ther desred trades (the equlbrum rce and tradng volume). Then the nformaton sets of each trader are n o z j, ψ 0,y j,,sgn(x ) at date n o z j, ψ 0,y j,,,v,sgn(x ),y j,,sgn(x x ) at date The reason why we suose the sgns of the nosy er cata suly (change) are known to all traders s exlaned n the begnnng of subsecton.. Ths assumton does not affect the equlbrum n the frst erod but does affect that n the second erod..4 Transacton Costs To examne the effect of transacton costs on the equlbrum rce and tradng volume, we ntroduce a er-share fee, c, for each share bought or sold of the rsky asset n the second erod. Thus the total transacton costs ncrease wth the number of shares traded 6 7. Ths s consstent wth most theoretcal models, e.g., Vayanos 998, Barron & Karoff 004, and emrcal evdence, e.g.,brennan & Chorda Includng a transacton cost at date comlcates our analyss, but we do not exect that t affects our man results. 7 Our results requre that the total transacton costs are not decreasng wth the number of shares traded and that they are the same for the same shares bought or sold. For examle, the retults retan f transacton costs consst only of a fxed, or lum-sum, comonent. However, f we suose that for each transacton, the buyer and seller have to ay a combned exoggenous fxed cost and the allocaton of ths fxed cost between buyer and sell s determned endogenously as n Lo, Mamaysky & Wang 004, the results change. 7

8 .5 Market Clearng We can nvestgate the market clearng n a "stock" sense and n a "flow" sense. The market for rsky asset clears n a "stock" sense f total holdngs at a gven ont n tme are equal to the nosy er cata suly. The market for rsky asset clears n a "flow" sense f the net number of shares bought and sold s equal to the change of the nosy er cata suly. In the frst erod, these two senses of market clearng are the same. However, although we can also comute the equlbrum rce n ether sense n the second erod, t s relatvely more convenent to comute the equlbrum rce n a "stock" sense because of rce autocorrelaton nvestgated n secton 3. Equlbrum A feasble tradng strategy requres that lanned asset holdngs be measurable wth resect to the trader s avalable nformaton set and satsfy the ndvdual s budget at each tradng date. Let d j, denote trader j n grou s tme holdng of the rsky asset. Then the ayoff at date 3 s d j, ψ + z 0 d j, at date and d j, ψ +dj, +z 0 d j, d j, d j, dj, c at date. The otmal tradng strategy s determned by solvng h ³ ³ J, ³d j, max ex d j d j, ψ + z 0 d j, j z, at date, and J, ³d j, h ³ ³ max ex z 0 + d j d j, (ψ ), +d j, ( ) d j, dj j, c z, at date An equlbrum s a ar of demand functons (d j,,dj, ) for each trader and a ar of the equlbrum rce functon (, )thattogethersatsfythefollowng condtons. Frst, the equlbrum rce t are functons of z t through ther deendence on traders demands and the nosy er cata sules. Second, each trader s strategy s feasble and solves above otmzaton equaton. Fnally, traders strateges and the equlbrum rces are such that market clears. The market-clearng condton n a "stock" or "flow" senses are wrtten resectvely 8

9 as d t d t N d t N NX j NX d j t, x t n a stock sensee j ³ d j t, dj t, x t x t n a flow sense We use er catal tradng volume as a measure of tradng volume 8 V t N unx d j t, j + NX jun+ d j t,. Equlbrum n the Frst Perod + x t x t Prooston In the frst erod wthout transacton costs, () the equlbrum rce s ρ 0ψ 0 + uρ s +( u) ρ s θ x ρ 0 + uρ s +( u) ρs () the demand of each trader s ³ d (y j, )ρ 0 (ψ 0 )+ρ s y j, (3) and, gven θ and x, tradng volume s V u V + ( u) V + x where V ρs ρ φ Ã! δ ρ ρ s à δ Φ δ ρ s! à ρ + δ Φ δ ρ ρ s! δ ρ 0 (ψ 0 )+ρ s (θ ) 8 A number of measures of volume have been roosed and studed: aggregate share volume, ndvdual share volume, aggregate dollar volume, ndvdual dollar volume, ndvdual turnover, aggregate turnover, etc. See Lo & Wang 000 for an excellent analyss on these dfferent measures. 9

10 where φ s the standard normal densty, and Φ s the standard normal cumulatve dstrbuton functon 9. Three remarks are n order. Frstly, the demand of each trader does not deend on the nosy er cata suly drectly and deends on t only ndrectly va the equlbrum rce. Secondly, for the same common nformaton, the hgher the nosy er cata suly, the lower the equlbrum rce. Because the nosy er cata suly s the counterart of the nosy er cata demand of lqudty traders, the hgh ostve er cata suly means hgh sell order from lqudty traders. Ths sell ressure ushes down the equlbrum rce. For the same reason, the hgh negatve er cata suly ushs u the equlbrum rce. Thrdly, because nether nformed nor unnformed trader knows ths nosy er cata suly, nether of them knows the common nformaton from only the equlbrum rce. For examle, f the equlbrum rce s equal to traders dentcal ror exectaton ψ 0, t may be due to a ostve common nformaton wth a ostve er cata nosy suly, or a negatve common nformaton wth a negatve er cata nosy suly, or a neutral common nformaton wth zero nosy er cata suly. Both of them now need look at tradng volume. Prooston In the frst erod wthout transacton costs, for the equlbrum rce gven, tradng volume s a decreasng functon of the nosy er cata suly when x < 0 and an ncreasng functon of nosy er cata suly when x > 0 and acheves ts mnmum at x 0. At the frst glance, ths result s easly exlaned: the second art of Prooston says that the demand of each trader deends only on hs rvate nformaton and the equlbrum rce. Thus the frst and second terms n tradng volume formula, whch stands for tradng volume nduced by both nformed and unnformed traders, do not change wth the nosy er cata suly. The art of tradng volume nduced by lqudty trader s obvously an ncreasng functon of the nosy er cata suly when x > 0 and a decreasng functon of the nosy er cata suly when x < 0. However, ths conjecture s not correct. From the equlbrum rce equaton exressed n the frst art of Prooston, for the equlbrum rce gven, the value of common nformaton should change whenever the nosy er cata 9 If we suose that the nosy er cata suly s zero and that nformed traders know the recson of unnformed traders rvate nformatons recsons ρ but the latters do not the formers recsons ρ, we get Blume, Easley & O Hara 994 s framework. It s not surrsng that the results are the same. 0

11 suly changes. More recsely, when the nosy er cata suly ncreases, common nformaton should ncrease to kee the equlbrum rce uρ s +( u)ρs unchanged. In other words, whle the varances of the rvate nformaton for both nformed and unnformed trader retan, ther robabltes change and so do the means of the rvate nformaton. Recall that V θ ue hd (y j, ) θ where y j, θ N h θ, /ρ and V ue d (y j, ) θ where y j, θ N θ, /ρ. The art of tradng volume nduced by both unnformed and nformed traders change wth the nosy er cata suly. Whether tradng volume nduced by the unnformed and nformed traders ncreases or decreases wth the nosy er cata suly deends on the nosy er cata suly and other arameters of the market. However, the change of tradng volume nduced by lqudty traders s so strong that ths effect always domnates and that the total tradng volume always ncreases wth the absolute value of the nosy er cata suly. From Prooston, f traders look at only the equlbrum rce, there are nfnte soluton of the nosy er cata suly and common nformaton whch verfy the equlbrum rce equaton. If they look at both the equlbrum rce and tradng volume, there are only two ossble solutons. When ρ w t,ρ t ρ w, then ρ s t ρ t ρ t, θ t ψ, andwehavethe Brown & Jennng 989 framework. In ther orgnal aer, they suose that traders look at only the equlbrum rce and use the nformaton contaned n the equlbrum rce. They show that the equlbrum s not revealng. The seres of the equlbrum rces hel traders to know the true value of the rsky asset and the techncal analyss s useful. As Blume, Easley & O Hara 994 argue, n the Brown & Jennng 989 framework f traders do know the equlbrum rce and tradng volume and use the nformaton conveyed by the equlbrum rce and tradng volume, there s a revealng equlbrum: every trader wll demand the same amount of the rsky asset d l t d j t d t for all l and j and tradng volume wll be ( d t + x t x t ). In ths settng, even though traders receve the rvate nformaton wth dfference qualty ρ t 6 ρ t, there exsts always the same revealng equlbrum. In our model, each trader makes hs decson condtonal on all nformaton u to but not ncludng the market statstcs from ther desred trades. The above calculaton shows that wthout further assumton, traders can not nfer the exact underlyng suly uncertanty from tradng volume and the equlbrum rce s not revealng. The dfferental nformaton that each trader owns

12 does matter. Every trader makes hs own decson accordng to hs nformaton and has hs own dfferent demand. Even though all traders receve ther rvate nformaton wth the same qualty as n Brown & Jennng 989 framework, ther demands stll dffer.. Equlbrum n the Second Perod In ths secton, we nvestgate the equlbrum n the second erod. To make the analyss tractable, we want the equlbrum n the frst erod to be revealng. The analyss n the last secton show that f traders know the sgn of the nosy er cata suly, they can nfer the nosy er cata suly and common nformaton from both the equlbrum rce and tradng volume. We assume that traders do know the sgn of the nosy er cata suly although they do not know ts value. At the end of the frst erod, all traders have the same ror exectaton agan wth mean θ ρw θ +ρ 0 ψ 0 ρ 0 +ρ and varance (bρ w ) (ρ 0 + ρ w ). Informed traders n grou then receve sgnals y j ψ + w + e j where the recson of the e j dstrbuton s ρ. Smlarly, unnformed traders n grou receve sgnals y j ψ+w +ε j where the recson of the εj dstrbuton s ρ ρ >ρ. Condtonal hs ror exectaton and rvate nformaton, each trader comutes hs exectaton on the ayoff of the rsky asset, maxmzes hs exected wealth at date 3 and consders how to change hs exosure of the rsky asset. The followng Prooston shows the equlbrum rce and the demand of each trader n the second erod. Prooston 3 In the second erod wth roortonal transacton costs, () the equlbrum rce s gven by Bθ Cc + D x A

13 where A u bρ + ρ s ( Φ (m )+Φ(n )) +( u) bρ + ρ s ( Φ (m )+Φ(n )) B uρ s ( Φ (m )+Φ (n )) + ( u) ρ s ( Φ (m )+Φ (n )) C u bρ + ρ s ( Φ (m ) Φ (n )) +( u) bρ + ρ s ( Φ (m ) Φ (n )) D bρ θ ( u (Φ (m ) Φ (n )) ( u)(φ (m ) Φ (n ))) + uρ s (Φ (m ) Φ (n )) + ( u) ρ s (Φ (m ) Φ (n )) θ u ρ 0 + ρ s (Φ (m ) Φ (n )) ( u) ρ 0 + ρ s (Φ (m ) Φ (n )) m +(u (Φ (m ) Φ (n )) + ( u)(φ (m ) Φ (n ))) ρ 0 ψ 0 + u (φ (m χ ) φ (n )) + u (φ (m χ ) φ (n )) q χ q n χ χ ρ s θ ρ s χ Ã ρ s bρ + c θ + ρ s ( + c)+ρ 0 (ψ 0 ) ρ s χ ρ bρ c θ + ρ s ( c)+ρ 0 (ψ 0 ) ρ s χ θ + ρ s ρ! where c s transacton cost er share, φ s the standard normal densty, Φ s the standard normal cumulatve dstrbuton functon, χ s the weghted change of the mean of the rvate nformaton for traders n grou, and χ s the recson of the weghted change of the mean of the rvate nformaton for traders n grou ; () the demand for the rsky asset for each trader s gven by d (y j, ) ( max(bρ θ + ρ s (y j, )+ bρ + ρ s c, d (y j, )) forbuyer mn(bρ θ + ρ s (y j, ) bρ + ρ s c, d (y j, )) for seller The frstartreflects nvestor s ror nformaton, the second reflects the nformaton surrse and the last reflects the effect of transacton costs. When there are no transacton costs n the second erod, m n,a u bρ + ρ s +( u) bρ + ρ s,b uρ s +( u) ρ s, and D bρ θ. The 3

14 equlbrum rce smlfes to ρ θ +(uρ s +( u)ρs trader smlfes to bρ θ + ρ s the frst erod. )θ x u( ρ +ρ s )+( u)( ρ +ρ s ) ³ y j, and the demand of each, whch are smlar as those n It s very nterestng to comare the equlbrum rce and the demand of each trader under roortonal transacton costs wth those under no transacton costs. As to the demand of each trader, the man dfference s whether hs demand n the second erod deends on hs demand n the frst erod. Under no transacton costs, the demand of each trader s totally determned by hs rvate nformaton, hs ror (dentcal) exectaton, and the equlbrum rce. Because of the free transacton, t s almost always otmal to trade. However, under roortonal transacton costs, for each share bought or sold, a transacton cost c has to be ad. If the gan from the change of hs oston s not hgh enough to comensate transacton costs, he refers not to trade and hs demand n the second erod s that n the frst erod. If the gan s hgh enough to comensate the transacton costs, hs demand changes and s exressed n the frst term of the second art of Prooston. Thus hs demand n the second erod deends on that n the frst erod. The nterretaton of each term n the demand of each trader s smlar as that n Cheng 005a s statc model. The dfference of the demand of each trader between under no transacton costs and under roortonal transacton costs leads to the dfference of the equlbrum rce between these two cases. When there are no transacton costs, the equlbrum rce n the second erod deends on all the arameters n the second erod ρ s,ρ s,θ,x and only one art of the arameters n the frst erod: the ntal dentcal exectaton mean (ψ 0 ) and recson (ρ 0 ),thevalue of common nformaton (θ ) and ts recson (ρ w ), and the nosy er cata suly (x ). The recson of the rvate nformaton are of no mortance. Because these recson are necessary to know the equlbrum rce n the frst erod, the equlbrum rce n the second erod does not deend on that n the frst erod. When there are roortonal transacton costs, because the demand of each trader n the second erod deends on that n the frst erod and the latter deends all the arameters n the frst erod, the equlbrum rce n the second erod deends on all the arameters both n the second erod and n the frst erod. The equlbrum rce n the frst erod nfluences that n the second va m and n. 4

15 The transacton of each trader s the dfference of hs demand n the second erod and that n the frst erod. From the second arts of Prooston 3 and of Prooston, t can be exressed as ³ 4 j, (yj,,yj, )4j, Y j ³ bρ θ + ρ s y j, bρ + ρ s c d, (y j ) for buyer 0 for others ³ bρ θ + ρ s y j, + bρ + ρ s c d, (y j ) for seller where Y j bρ θ + Y j ρ s + ρ s bρ + ρ s c ρ0 (ψ 0 ) 0 f others bρ θ + Y j ρ s + ρ s + bρ + ρ s c ρ0 (ψ 0 ) 0 f Y j > m + χ χ 0 f Y j < n + χ χ s the weghted change of trader j s rvate nformaton (Y j ρ s y j, + χ are the u crtcal onts of the weghted change of the χ ρ s y j, ), m rvate nformaton above whch traders n grou wll buy and n + χ χ are the down crtcal onts of the weghted change of the rvate nformaton below whch traders n grou wll sell. Tradng volume n the second erod s the weghted sum of the above transactons for both nformed and nformed traders lus the nosy er cata suly change. Prooston 4 In the second erod wth roortonal transacton costs, gven θ and x x,tradngvolumes V u V + ( u) V + x x where V χ (φ ( m ) m Φ ( m )+φ (n )+n Φ (n )) When there are no transacton costs, tradng volume smlfes a lttle and V becomes V µ q µ q µ q φ δ χ χ δ Φ δ χ + δ Φ δ χ 5

16 where δ bρ θ + ρ0 (ψ 0 )+ρ s ( θ ) ρ s ( θ ) It s worth notng that whether there are transacton costs or not, tradng volume n the second erods always deends on all the arameters of the market n both erods. It s not surrsng snce each trader s transacton s always nduced by the change of hs rvate nformaton together wth the nosy er cata suly change and common nformaton. Prooston 5 In the second erod wth roortonal transacton costs, () the equlbrum rce s ncreasng n common nformaton θ B A > 0 () for the equlbrum rce gven, tradng volume s a decreasng functon of the nosy er cata suly change when x x < 0 and an ncreasng functon of the nosy er cata suly change when x x > 0 and acheves ts mnmum at x x 0. Three remarks are n order. Frstly, although transacton costs change each trader s behavor and the equlbrum rce, t does not change the economc logc between common nformaton and the equlbrum rce: the better common nformaton s, the hgher the traders mean exectaton on the ayoff of the rsky asset s, and the hgher the equlbrum rce s. Secondly, t s the nosy er cata suly change not the nosy er cata suly tself whch matters to tradng volume. Thrdly, comarson of Prooston 5 wth Prooston shows that transacton costs do not change the relaton between tradng volume, the nosy er cata suly change and the equlbrum rce. Thus both nformed and unnformed traders nfer the value of common nformaton and the nosy er cata suly change from the equlbrum rce and tradng volume n the second erod. 3 Tradng Volume, Prce Autocorrelaton and Volatlty wth Transacton Costs As shown n the revous secton, transacton costs nfluence each trader s behavor and the equlbrum rce. Transacton costs nfluence the equlbrum 6

17 rce at date n two ways: the deendence of the demand of each trader at date on that at date and the nterval of no transacton wth the length bρ + ρ s c. The nfluence n the frst way makes the equlbrum rces at date andcorrelatedandthenfluence n the both ways changes return volatlty. The nfluence of transacton costs on tradng volume s nvestgated n the next subsecton and the nfluences on rce autocorrelaton and return volatlty are dscussed n the second and thrd subsecton, resectvely. 3. Transacton Costs and Tradng Volume Prooston 6 In the second erod wth roortonal transacton costs, tradng volume (V ) s a decreasng convex functon of transacton costs V c u u < 0 ρ ρ s V c u + u > 0 bρ + ρ s (Φ ( m )+Φ(n )) bρ + ρ s (Φ ( m )+Φ(n )) bρ + ρ s (φ (m )+φ(n )) ρ ρ s bρ + ρ s (φ (m )+φ(n )) Tradng volume decreases because traders do not trade at all f the weghted changes of ther rvate sgnals that the traders receve n both erods are not sgnfcant enough ( n + χ χ <Y j < m + χ χ ). Even for those traders whose rvate sgnals weghted changes are sgnfcant enough (Y j < n + χ χ or Y j > m + χ χ ), they buy or sell less wth the amount of bρ + ρ s c. Only traders who receve ther rvate nformaton y j, y, ρ 0 ( ψ 0 )+ρs n the ρ s frst erod and y j, y, ρ ( θ )+ρ s n the second erod at the same ρ s tme do not change ther demands whch are, however, always zero regards of transacton costs. The convexty of tradng volume wth resect to transacton costs means that when transacton costs are near to zero, the nfluence of transacton costs on tradng volume s the most sgnfcant. On the contrary, when transacton costs are hgh, ther margnal nfluence on tradng volume s much smaller. 7

18 V When transacton costs are zero, c acheves ts mnmum u bρ + ρ s u bρ + ρ s. In the extreme case of nfnte transacton costs, V c tends to 0. Transacton costs are so hgh that t forbds any transacton. In fact, when m transacton costs are nfnte, +χ χ tends to ostve nfnte, n +χ χ tends to negatve nfnte, and the ntervals of no transacton m n c bρ χ + ρ s are also nfnte. In ths case, nobody transacts n the second erod and every trader holds hs oston n the frst erod. 3. Transacton Costs and Prce Autocorrelaton After Prooston 3 n the second secton, we show that when there s no transacton costs, the equlbrum rce n the second erod does not deend on that n the frst erod and that when there s roorton transacton costs, the equlbrum rce n the second does deend on that n the frst erod. In ths subsecton, we nvestgate the relaton between transacton costs and rce autocorrelaton. When there are no transacton costs, the autocorrelaton between the equlbrum rce at date and that at date s equal to corr à uρ s +( u) ρ s θ x ρ 0 + uρ s +( u) ρs,! uρ s +( u) ρ s θ x u (bρ + ρ s )+( u)(bρ + ρ s ) Recall that θ ψ + w,θ ψ+ w and that common error (w,w ) and the nosy er cata suly (x,x ) are suosed to be ndeendent, ths autocorrelaton s obvously zero. When there are transacton costs, ths correlaton s no longer zero. Because we can not get the exlct soluton to the equlbrum equaton n the second erod, t s mossble to exress ths autocorrelaton exlctly. To understand the nfluence of roortonal transacton costs on the autocorrelaton, we calculate the artal dervatve of wth resect to u ρ0 + ρ s (Φ (m ) Φ (n )) + ( u) ρ 0 + ρ s (Φ (m ) Φ (n )) A Snce m n, Φ (m ) Φ (n ). Together wth A>0, the sgn of ths artal dervatve s never ostve. The equalty holds when there are no transacton costs (m n ), whch s consstent wth our analyss above. Otherwse, t s always negatve. 8

19 Prooston 7 In the large economy above, () when there are no transacton costs, there s no rce autocorrelaton; () when there are roortonal transacton costs, there s rce autocorrelaton. For common nformaton at date gven, the hgher (lower) the equlbrum rce at date, the lower (hgher) the equlbrum rce at date. n Recall that + χ m χ, + χ χ are the ntervals of no transacton and m +n + χ χ are the medums of the ntervals for traders n grou. Our calculaton show that these medums are decreasng functons of the equlbrum rce n the frst erod ( ) µ m +n + χ χ bρ + ρ s ρ 0 + ρ s bρ + ρ s ρ 0 + ρ s < 0 θ bρ ρ w (ρ 0 + ρ w ) θ It means that the hgher the equlbrum rce n the frst erod, the lower the medums of the ntervals of no transacton. Note that the thrd term s dentcal to both nformed and unnformed traders. Snce ρ s t >ρ s t,thenfluence of the equlbrum rce at date s more sgnfcant to nformed traders than to unnformed traders. Ths move of the medums makes the dfference between total cancelled buy orders and total sell orders less (more) sgnfcant n case of ostve (negatve) common nformaton and thus decreases the equlbrum rce at date. To get an ntutonal dea, let us look at a secal case wth θ ψ 0 and x t 0. In ths secal case, θ ψ 0 and x 0 θ ψ 0. Then m and n smlfy to χ bρ ± c θ + ρ s ( ± c θ ). If θ θ, the medums of the ntervals smly to 0. All cancelled buy orders are equal to all cancelled sell orders and the equlbrum rce under roortonal transacton costs s equal to that under no transacton cots ( θ θ ). If θ > θ (θ < θ ), the medums of the ntervals are nferor (sueror) to 0. Then all canceled buy (sell) orders are sueror to all cancelled sell (buy) orders and the equlbrum rce when θ > θ (θ < θ ) s lower (hgher) than that when θ θ. 9

20 3.3 Transacton Costs and Volatlty Proortonal transacton costs cancel both nformed and unnformed traders transacton. For each grou of traders, not only buy orders but sell orders are reduced. If the total cancelled buy orders are always equal to the total cancelled sell orders regardless of roortonal transacton costs, the equlbrum rce should not change and volatlty s ndeendent on transacton costs. If not, they should be the functons of roortonal transacton costs. Prooston 8 In the second erod wth roortonal transacton costs, the equlbrum rce may be an ncreasng (C <0) or decreasng (C >0) functon of roortonal transacton costs. From the equlbrum rce equaton exressed n Prooston 3, the artal artal dervatve of the equlbrum rce wth resect to transacton costs can be calculated and s equal to C A by Imlct Functon Theorem. Snce A s always ostve, the sgh of ths µ artal dervatve deends on that of C. m We know that + χ n χ + χ χ are the u (down) crtcal onts of the change of the rvate nformaton above (below) whch traders n grou wll buy (sell) and m +n + χ χ are the medums of these two crtcal onts (nterval of no transacton). Whle the medums deend on arameters both at date and at date, the lengths of ntervals bρ + ρ s c deend only on the arameters at date. The deendence of the medums on the arameters at date nduces rce autocorrelaton, whch has been studed n µ last subsecton. Our calculaton shows that the medum m +n + χ m +n χ + χ χ s a decreasng (ncreasng) functon of common nformaton θ. Because traders n grou () receve more (less) relable nformaton, t s economcally logcal that the hgher common nformaton, the lower (hgher) the crtcal ont of traders n grou () to buy. Note that C denote the cancelled buy orders mnus the cancelled sell orders of all traders who stll transact under roortonal transacton costs (Refer to the roof of Prooston 3 n the Aendx). Because the cancelled sell orders and the cancelled buy orders are symmetrc wth resect to the medums of no transacton ntervals, the dfference of the cancelled buy orders mnus the cancelled sell orders of traders who do not transact under transacton costs have thesamesgnasc. Then the ostve sgn of C means that transacton costs 0

21 elmnates more buy orders than sell orders n total. Thus the equlbrum rce under roortonal transacton costs s lower than that under no transacton costs when C s ostve 0. In our model, volatlty s measured by the varance of return. Although the analytcal result s not avalable because of the lack of closed-form soluton of the equlbrum rce at date, a secal case hels us to understand the nfluence on return volatlty A Secal Case wth θ ψ 0 and x t 0 Prooston 9 In the secal case wth θ ψ 0 and x 0, condtonal on the equlbrum rce θ ψ 0 and ror dentcal exectaton θ, () the equlbrum rce under roortonal transacton costs and that under no transacton costs have the same mean : E c θ, θ E n θ, θ θ, where c and n stand for the equlbrum rces under and under no transacton costs; () return volatlty under roortonal transacton costs may be hgher or lower than that under no transacton costs. More recsely, ³ ³ c Var n θ, θ >Var θ, θ when u < u ³ ³ c Var n θ, θ <Var θ, θ when u > u where u ( ρ +ρs )( Φ(m ) Φ(n )) ( ρ +ρ s )( Φ(m ) Φ(n )) The condtonal mean of the exected equlbrum rce under roortonal transacton costs does not change because the nfluence of transacton costs s symmetrc when common nformaton s ostve or negatve. Whether return volatlty under roortonal transacton costs s hgher or lower than that under no transacton costs deends on whether the equlbrum rce under roortonal transacton costs s hgher or lower than that under no transacton costs. The latter deends on the sgn of C as shown above. It s easy to show that 0 In Cheng (005) s statc model wth a smlar settng wthout the nosy suly shock, they have even stronger results: f u s relatvely small, the equlbrum rce wth roortonal transacton costs s hgher (lower) than that wthout transacton cost when common nformaton s ostve (negatve). Normally, rce volatlty s measured by the varance of the rce and return volatlty s measured by the varance of the rce ercent change. Snce rce volatlty wth defferent rce means can not be drectly comarable, we refer return volatlty n our model.

22 when u<u,c<0 f θ > θ, and C>0 f θ < θ General Case Generally, we exect that return volatlty s hgher than that n the secal case above for two reasons. Frst, the coeffcent of the er cata nosy suly s bgger under roortonal transacton costs than that under no transacton costs. By Imlct Functon Theorem, we calculate the artal dervatve of the equlbrum rce wth resect to the er cata suly x A. Snce m >n, A<u bρ + ρ s +( u) bρ + ρ s. It means that for the one unt change of the er cata suly, ts nfluence on the equlbrum rce under roortonal transacton costs s more mortant than that under no transacton costs. Second, there s rce autocorrelaton under roortonal transacton costs. From Prooston 7, we know that for common nformaton at date gven, the hgher the equlbrum rce at date, the lower the equlbrum rce at date. Ths autocorrelaton nduces hgher return volatlty. For examle, for θ θ, f ψ 0, then θ ; f >ψ 0, then < θ ; f <ψ 0, then > θ. Whenever 6 ψ 0, theabsolutevalueoftheercentrcechangeshgherthan that when ψ 0. 4 Related Emrcal Research Some relevant emrcal research ncludes Umlauf 993, Jones & Segun 997, Green, Maggon & Murnde 000, and Hau 006, among others. Sweden n the 980s rovdes an excellent settng for a controlled laboratory-style exerment to determne how transacton costs (taxes) affect stock market behavor. Sweden stock market began wthout transacton taxes. In 984 a % round-tr tax was mosed on equty transactons and two years later the equty transacton tax rate was ncreased to %. Umlauf 993 studes the effects of transacton taxes on the behavor of Swedsh equty returns and the man results are: () ncreasng transacton taxes resulted n a shar dro n tradng volume; () weekly to daly returns varance ratos declned durng hgh-tax regmes, suggestng taxes nduced greater negatve autocorrelaton n return; (3) all else beng equal, taxes ncrease volatlty. For more detals about how roortonal transacton costs nfluence the net cancelled orders of nformed and unnformed traders n the very dfferenct way, lease refer to Cheng 005a.

23 Another smlar examle s n Amercan fnancal market. On May, 975, the lower and negotated commssons on U.S. natonal stock exchange are ntroduced. Jones & Segun 997 use the data of New York Stock Exchange (NYSE), the Amercan Stock Exchange (AMEX) and Natonal Assocaton of Securtes Dealers Automatc Quotatons (NASDAQ) to nvestgate the relaton between the volatlty and ths reducton n transacton costs 3. Two man results of Jones & Segun 997 s emrcal aer are: () the aggregate NYSE/AMEX ortfolo exhbts relably less volatlty after the reducton n transacton costs; () the results are smlar for all but the smallest sze-based ortfolo. By usng the data n the London Stock Exchange from 870 to 986, Green, Maggon & Murnde 000 study the mact of transacton costs on market volatlty and fnd that the sgn of the relatonsh between transacton costs and market volatlty s ostve. Ths relatonsh n French stock between 995 and 999 s studed by Hau 006. Durng ths erod, French stocks were subject to an mortant transacton cost ncrease whenever ther rce moved above the French franc (FF) 500 rce threshold. Above FF 500, the mnmal tck sze for quotes ncreased by a factor of 0 from FF 0. to FF, whch consttutes an exogenous cost comonent nduced by rcng grd of the electronc order book. He concludes that the effect of transacton costs on volatlty s ostve and sgnfcant, both statstcally and economcally. Our model can exlan these results qute well. The facts that transacton costs nduce shar dro of tradng volume and (greater) negatve autocorrelaton s consstent wth our analyss n subsectons 3. and 3.. From the second art of Prooston 9, we know that when ρ ρ ( ρ +ρ, s )( Φ(m ) Φ(n )), ( ρ +ρ s )( Φ(m ) Φ(n )) thus u. It means that the condton n whch return volatlty under roortonal transacton costs are hgher than that under no transacton costs s always verfed. For bg-sze (hgh stock rce from FF 400 to FF 600) comanes, so many market artcants search the relevant nformatons and the dfferences of the recson between nformed and unnformed traders are small. Consequently, ther return volatlty under transacton costs s hgher than that under no transacton costs for these bg-sze comanes, as shown n Jones & Segun 997 s emrcal aer. In fact, Jones & Segun 997 and Umlauf 993 s frst result show that n general, the dfferences of recson between nformed and unnformed traders are qute small that the whole market return volatlty 3 Regulated commssons are smlar to transacton taxes snce both are fxednamount and leved on artes whenever a securtes transacton occurs. Thus the event mentoned above s analogous to a one-tme reducton n a tax on equty transactons. 3

24 ncrease n transacton costs. 5 Conclusons We develo a two erods model n whch traders have dfferental nformaton about the true value of the rsky asset and trade the rsky asset wth roortonal transacton costs. We assume that traders maxmze ther wealth n the end condtonal on all nformaton u to but not ncludng the market statstcs resultng from ther desred trade. We show that wthout addtonal assumton, tradng volume can not totally remove the nose n the rcng equaton. However, because tradng volume ncreases n the absolute value of nosy er cata suly change, t rovdes useful nformaton on the asset fundamental value whch cannot be nferred from the equlbrum rce. We then nvestgate the relaton between tradng volume, rce autocorrelaton, return volatlty and roortonal transacton costs. Tradng volume decreases n roortonal transacton costs and the nfluence of roortonal transacton costs decreases at the margn. The result that rce autocorrelaton can be generated only by roortonal transacton costs s nterestng: under no transacton costs, the equlbrum rces at date and are not correlated (ndeendent); however under roortonal transacton costs, they are correlated - the hgher the equlbrum rce at date, the lower the equlbrum rce at date. Contrary to "conventonal wsdom" on the relaton between return volatlty and transacton costs, we show that return volatlty may be and n general s ncreasng wth roortonal transacton costs, whch s consstent wth avable emrcal evdence. 4

25 6 Aendx Proof. of Prooston : At date, each trader maxmzes by choce of d j, the followng functon h ³ ³ max E ex w j 3, µ ³ max ex E ³ Var w j 3, z j, w j 3, z j, where the wealth at date 3 s w j 3, dj, (ψ )+z 0 The frst order condton for a maxmum gves ³ E ψ d j, ³ V ψ z j, z j, By Bayes Rule, the condtonal exectaton and condtonal varance at date are ³ E ψ ³ V ψ z j, z j, ρ 0ψ 0 + ρ s y j, ρ 0 + ρ s ρ 0 + ρ s Substtutng the condtonal exectaton and condtonal varance gves the demand at date exressed n the second art of Prooston. The total demand of the market s NX un d (y j, ) X d (y j, )+ j j unx j + ³ N X jun+ ρ 0 (ψ 0 )+ρ s NX jun+ ³ d (y j, ) ρ 0 (ψ 0 )+ρ s ³ y j, ³ y j, By the Strong Law of Large Numbers as un and ( u)n,thetotal 5

26 demand of the market can be rewrtten as NX d (y j, ) Nρ 0ψ 0 N ρ 0 + uρ s +( u) ρ s j +N uρ s +( u) ρ s θ Nx In the last equalty, the fact that the total demand of the market should be equal to the nosy er cata suly n the equlbrum s used. After several arrangements, we have the equlbrum rce exressed n the frst art of Prooston. Recall that n our model we defne er catal volume as V N unx j d (y j, ) + N X jun+ d (y j, ) + x By the Strong Law of Large Numbers as un and ( u)n, ths sequence of tradng volume seres converges almost surely to ³ ue d (y j, ) +( u)e d (y j, ) + x Usng the lemma roved n Cheng 005a wth x j α + βy α ρ 0 (ψ 0 ) ρ s β ρ s ³ y N θ, ρ we get tradng volume exressed n the thrd art of Prooston mmedately. Proof. of Prooston : We frst rove the case x < 0. Dfferentatng tradng volume exressed n the thrd art of Prooston wth resect to the nosy er cata suly gves V µ V + V x x x 6

27 where Ã!Ã! V ρs δ φ ρ x ρ ρ s δ ρ ρ δ ρ s ρ s x + δ à Ã! Ã!! δ Φ ρ x ρ s Φ δ ρ ρ s Ã!à δ φ δ ρ! ρ δ ρ s ρ s x Ã!Ã! +δ φ δ ρ ρ δ ρ s ρ s x δ à Ã! Ã!! δ Φ ρ x ρ s Φ δ ρ ρ s From the equlbrum rce, we can exress θ by θ Then δ can be wrtten n another way δ ρ 0 (ψ 0 )+ρ s (θ ) ρ 0 (ψ 0 )+ρ s ρ0 + uρ +( u) ρ ρ 0 ψ 0 + x uρ +( u) ρ à ρ0 + uρ s +( u) ρ s (uρ s µ ρ 0 (ψ 0 )+ρ s ρ0 ( ψ 0 )+x Thus we have (uρ s ρ 0 (ψ 0 )( u)(ρ s ρs )+ρ s x uρ s +( u)ρs ρ 0 (ψ 0 )u(ρ s ρs )+ρ s x uρ s +( u)ρs V x + u + u uρ s uρ s ρ s +( u) ρs ρ s +( u) ρs +( u) ρs )! ρ 0 ψ 0 + x +( u) ρs ) for for à Ã! à δ Φ ρ Φ Ã Ã Φ ρ s δ ρ ρ s! Ã Φ ρ δ ρ s ρ δ ρ s!!!! 7

28 < + u + 0 uρ s ρ s +( u) ρs uρ s +( u) ρ s uρ s +( u) ρs + u uρ s ρ s +( u) ρs Inthesameway,wecanrovethatwhenx > 0, V x > 0. Proof. of Prooston 3: At date, each trader buys or sells accordng to the common ror and hs rvate nformaton at date. The wealth constrant n ths erod s Hs wealth at date 3 s d j, + z j, + d j, dj, c d j, + z j, w j 3 d j, ψ + zj, d j, ψ + dj, + z j, dj, d j, dj, c ³ d j, ψ + dj, + z j, dj, d j, dj, cforbuyeratdate ³ d j, ψ + dj, + z j, dj, + d j, dj, c for seller at date Each trader maxmzes hs exected utlty of wealth h ³ ³ max E ex w j 3, µ ³ max ex E ³ Var w j 3, z j, w j 3, z j, As n the frst erods, the frst order condton for a maxmum gves d j, E(ψ z j, ) c V (ψ z j, ) E(ψ z j, ) +c V (ψ z j, ) for buyeratdate for selleratdate Recall that both nformed and unnformed traders know all the nformaton from the market n the frst erod and agan have a common ror exectaton on the rsky asset s true value. By Bayes Rule, the condtonal exectaton and 8

29 condtonal varance at date are ³ E ψ ³ V ψ z j, z j, bρ θ + ρ s y j, bρ + ρ s bρ + ρ s Substtutng the condtonal exectaton and condtonal varance gves ³ d j, bρ θ + ρ s y j, bρ + ρ s cforbuyeratdate ³ bρ θ + ρ s y j, + bρ + ρ s cforselleratdate Thus the demand for the rsky asset d, (y j,,yj, ) s the maxmum of dj, and d j, for buyer n the second erod and the mnmum of dj, and dj, for seller n the second erod. To calculate the equlbrum rce, we ntroduce a functon + f ρ s y j, ρs f(y j,,yj, ) where y j, >yu ρ (θ ) ρ 0 (ψ 0 )+ρ s yj, ρs yj, ρs +ρs c( ρ +ρ s ) f others f ρ s y j, ρs y j, <ydown y u bρ + c θ + ρ s ( + c)+ρ 0 (ψ 0 ) ρ s y down bρ c θ + ρ s ( c)+ρ 0 (ψ 0 ) ρ s and ³ ρ s y j, ρs y j, N χ, χ Ã N ρ s θ ρ s θ, ρ s ρ + ρ s ρ! Then the demand of each trader can be exressed n the followng way ³ d, (y j,,yj, ) bρ θ + ρ s yj, bρ + ρ s ³ )c + f(y j,,yj, 9

30 The total demand of the market s NX un d (y j,,yj, ) X d, (y j,,yj, )+ j j unx j + N X jun+ d, (y j,,yj, ) ³ bρ θ + ρ s yj, bρ + ρ s ³ )c + f(y j,,yj, NX jun+ ³ bρ θ + ρ s y j, bρ + ρ s ³ )c + f(y j,,yj, By the Strong Law of Large Numbers as un and ( u)n,thetotal demand of the market can be rewrtten as NX d (y j,,yj, ) Nbρ θ N u bρ + ρ s +( u) bρ + ρ s j +N uρ s +( u) ρ s Nu bρ + ρ s E hf(y j,,yj, ) c N ( u) bρ + ρ s E hf(y j,,yj, ) c θ By the market clearng condton n the second erod, the equlbrum rce converges almost surely to bρ + uρ s +( u) ρs ³ u bρ + ρ s bρ θ + uρ s +( u) ρ s θ x E hf(y j,,yj, ) +( u) bρ + ρ s E hf(y j,,yj, ) c where E hf(y j,,yj, ) Z y down Z + + y u Z y u µq ( ) qχ φ χ x χ dx µq (+) qχ φ χ x χ dx bρ θ ρ0 (ψ 0 ) ρ s + ρ s + χ c(bρ + ρ s ) + y down µq qχ φ χ x χ dx 30

31 Z y u + x χ y down c(bρ + ρ s µq Φ µqχ φ χ ) qχ φ µq χ x χ dx y down χ + Φ µq µqχ φ χ y u χ + c(bρ + ρ s ) bρ θ bρ + ρ s ρ 0 ψ 0 + ρ 0 + ρ s + χ µ µq Φ χ y u χ µq Φ χ y down χ µ µq c(bρ + ρ s ) φ χ χ y u χ µq φ χ y down χ Substtutng these two exectaton gves the coeffcents A, B, C, and D and the equlbrum rce follows after several arrangements. Proof. of Prooston 4: From the second arts of Prooston 3 and of Prooston, the transacton of each trader n the second erod s where 4 j, (yj,,yj, ) bρ θ + ρ s ³ y j, ³ bρ θ + ρ s y j, bρ θ + Y j ρ s + ρ s bρ + ρ s c ρ0 (ψ 0 ) 0 f others bρ θ + Y j ρ s + ρ s + bρ + ρ s c ρ0 (ψ 0 ) Y j bρ + ρ s c d (y j, ) for buyer 0 for others + bρ + ρ s c d (y j, ) for seller ρ s y j, ρs y j, 0 f Y j 0 f Y j >y u <y down Then tradng volume s V N ³ ue unx j 4 j, (Y j )+ N X ³ 4 j, (Y j ) z j, j+un 4 j, (Y j +( u) E uv +( u) V x x + ) + x x ³ 4 j, (Y j z ) j, + x x 3

32 where Z y down V Z 4 j, (xj ) f (x j )dxj + 4 j, (xj ) f (x j )dxj Usng the lemma roved n Cheng 005a wth y u x j α + βy α bρ θ ρ s ± bρ + ρ s c ρ0 (ψ 0 )+ρ s β y N ³ χ, χ we have tradng volume exressed n Prooston 4. Proof. of Prooston 5: Because we could not get the exlct soluton for the equlbrum rce, we need to calculate frst F θ and F where F A Bθ + Cc D + x 0. where F A B θ + C c D B θ θ θ θ θ + A u bρ θ + ρ s µ φ (m ) m φ (n ) n θ θ B θ uρ s θ ( u) bρ + ρ s µ φ (m ) m µ φ (m ) m +( u) ρ s + C c u bρ θ + ρ s D θ ubρ θ φ (n ) n θ θ µ φ (m ) m φ (n ) n θ θ θ φ (n ) n θ θ µ φ (m ) m + φ (n ) n θ θ ( u) bρ + ρ s µ φ (m ) m µ φ (m ) m θ φ (n ) n θ θ + φ (n ) n θ θ +( u) bρ θ µ φ (m ) m θ φ (n ) n θ 3