Differences of Opinion of Public Information and Speculative Trading in Stocks and Options

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1 Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons H. Henry Cao Cheung Kong Graduate School of Busness CKGSB Hu Ou-Yang Lehman Brothers and CKGSB We analyze the effects of dfferences of opnon on the dynamcs of tradng volume n stocks and optons. We fnd that dsagreements about the mean of the current- and next-perod publc nformaton lead to tradng n stocks n the current perod but have no effect on optons tradng. Wthout optons, we fnd that dsagreements about the precson of all past and current publc nformaton affect tradng n stocks n the current perod. Wth optons, only dsagreements about the precsons of the next- and current-perod nformaton affect stocks and optons tradng n the current perod. Our results suggest that optons tradng s concentrated around nformaton events that are lkely to cause dsagreements among nvestors, whereas tradng n stocks may be dffusve over many perods. JEL G1, G11, G1 Tradng n exchange-lsted securtes, such as stocks and ther optons, s extremely actve. In 000, the average daly tradng volume n the NYSE reached 1.04 bllon shares for 43.9 bllon US dollars. Tradng volume n optons s also huge. Optons tradng s now the world s bggest busness, wth an estmated daly turnover of over.5 trllon US dollars and an annual growth rate of around 14%. 1 Gven such a hgh tradng volume, the followng queston arses naturally: What drves nvestors tradng n the securtes market and the assocated optons market? Ths artcle analyzes the effects of dfferences of opnon regardng the mean and the precson of publc nformaton on tradng n stocks and optons and the effects of optons ntroducton on the tradng volume of the underlyng stock. In our model, nvestors have heterogeneous belefs even f they observe the same We are very grateful to an anonymous referee for offerng many nsghtful comments and suggestons that have mproved the artcle mmensely. We also thank Kerry Back, Pete Kyle, Hatao L, S. Vswanathan, We Xong, and semnar partcpants at Cheung Kong, Duke, Houston, Indana, NYU, Oklahoma, Tulane, and UNC, the 005 Amercan Fnance Assocaton Meetngs, the 005 Western Fnance Assocaton Meetngs, and the 004 Chna Internatonal Fnance Conference for helpful comments. Send correspondence to H. Henry Cao, Cheung Kong Graduate School of Busness, Bejng , Chna; e-mal: hncao@ckgsb.edu.cn or to Hu Ou-Yang, Lehman Brothers Asa-Pacfc Roppong Hlls Mor Towers, 31st Floor, , Roppong, Mnato-Ku, Tokyo , Japan. E-mal: houyang@lehman.com. 1 From Swan 000. Downloaded from at Pekng Unversty on September 7, 014 C The Author 008. Publshed by Oxford Unversty Press on behalf of the Socety for Fnancal Studes. All rghts reserved. For permssons, please e-mal: journals.permssons@oxfordjournals.org. do: /rfs/hhn00 Advance Access publcaton March 7, 008

2 The Revew of Fnancal Studes / v n publc sgnal. Kandel and Pearson 1995 have provded compellng evdence that nvestors may nterpret publc nformaton dfferently. The Mlgrom-Stokey 198 no-trade theorem does not apply when nvestors have dfferences of opnon regardng publc nformaton. More specfcally, n our model, nvestors have constant absolute rsk averson CARA utlty and beleve that the stock payoff dstrbuton s normal. After the frst round of trade, there are new publc sgnals about the fnal stock payoff arrvng at the market. Investors have the opportunty to trade agan n the market. These new publc sgnals create dfferences of opnon across nvestors because nvestors nterpret them dfferently. Followng Kandel and Pearson 1995; Danel, Hrshlefer, and Subrahmanyam 1998; and Hong and Sten 003, we assume that nvestors dsagree on the mean and the precson of sgnals. Dsagreements about the mean of a publc sgnal capture the nvestors condtonal optmsm and pessmsm about the asset value, whle dsagreements about the precson of a publc sgnal capture the heterogenety of the nvestors confdence level n the sgnal. For example, an nvestor, who has low expectatons about the publc sgnal n the next perod, tends to be more bearsh n the current perod. However, ths nvestor wll be relatvely more optmstc regardng the asset payoff n the next perod after the publc nformaton s announced and wll act more bullsh n the next perod. On the other hand, dfferences regardng the precson of the publc nformaton n the next perod do not nduce drectonal speculaton n the current perod and thus have no effect on stock tradng. An nvestor who overweghs the precson of the publc sgnal wll update more than the average nvestor, because hs posteror expectaton of the stock payoff s the precson-weghted average of pror expectatons and the sgnal. For a postve shock n the publc nformaton, such an nvestor wll beleve that the market has under-reacted to the publc nformaton and, as a result, he wll purchase more of the stock when the news s good and sell the stock when the news s bad. The tradng volume n the stock can be dvded nto four components. The frst comes from the dfferences n the nterpretaton of the mean of the current publc sgnal and the second comes from the dfferences of opnon regardng the mean of the publc sgnal n the next perod. Consequently, dfferences of opnon regardng the mean of a publc sgnal generate tradng n stocks both n the prevous perod and n the current perod. The thrd component arses from dfferences of opnon regardng the precson of the current publc sgnal and the fourth component arses from the dfferences of opnon about the precsons of all past publc sgnals. If nvestors dsagree on the precson of a publc sgnal once, they wll contnue to trade even f they agree on the sgnals n all future perods. Dsagreements about the precson of future publc nformaton do not generate tradng n the current stock market. We further establsh the followng tradng volume dynamcs and ther relatonshp wth the stock prce change: Tradng volume and absolute prce changes are postvely serally correlated; The tradng volume and absolute change of the precson-weghted average Downloaded from at Pekng Unversty on September 7,

3 Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons forecast are postvely correlated; Tradng volume s hgher when a publc sgnal s very nformatve; v Tradng volume ncreases wth the dsperson of belefs among nvestors. When optons are ntroduced, we show that nvestors who have hgher condtonal volatlty lower precson about the stock payoff take long postons n optons to synthesze convex payoffs, whereas nvestors who have lower condtonal volatlty take short postons n optons to acheve concave payoffs. Optons tradng volume can be decomposed nto two components. The frst part arses from the dsagreements about the precson of the current publc sgnal, whereas the second part comes from the dsagreements about the precson of the next-perod publc sgnal. The tradng volume n optons exhbts very dfferent temporal patterns than that n stocks. Dsagreements about the mean and dsagreements about the precson of past sgnals and sgnals after the next perod do not generate tradng n the optons market. Consequently, optons tradng should be clustered before and durng a bg, rare news event that s lkely to cause dfferences of opnon among nvestors, and tradng volume should declne quckly afterward. On the contrary, when the market s ncomplete, stock tradng should be actve at the tradng sesson when mportant publc nformaton s announced and persst thereafter. We further show that n the presence of optons, the tradng volume of the stock s related not only to ts prce change but also to lagged prce changes. Our model has the followng emprcal mplcatons regardng the tradng volumes of the stock and optons. Frst, tradng volumes n optons should be hgher around the dates of publc events, such as earnngs announcements, mergers and acqustons, and bond ratng changes, because publc nformaton generates dfferences of opnon. Second, when there are more dfferences of opnon about a stock s payoff, tradng volumes n both the stock and ts optons should be hgher because nvestors demands for optons depend on ther belefs about the volatlty of the stock payoff. 3 Thrd, tradng volumes are hgher for stocks wth optons because nvestors use underlyng stocks to hedge ther postons n the optons market. Fourth, tradng volume for optoned stocks should be more responsve to concurrent and lagged prce changes than nonoptoned stocks, due to addtonal hedgng demands assocated wth optons. Downloaded from at Pekng Unversty on September 7, 014 Leland 1980 shows that nvestors wth hgher expected returns buy portfolo nsurance, but he does not consder the case of dfferent volatltes. 3 One proxy for dfferences of opnon s the dsperson of belefs among fnancal analysts. It would be nformatve to determne whether tradng volume n optons s hgher for stocks wth more dsperson n fnancal analysts forecasts. Another proxy s to use open nterest n futures markets as a measure for dfferences of opnon. Bessembnder, Chan, and Segun 1996 fnd that tradng volume n stock ndex futures s correlated wth open nterest n the ndex futures market. Whle our paper has focused on tradng volume n stocks and optons, the model can also be used to analyze the relatonshp between stock returns and nvestor heterogenety. Anderson, Ghysels, and Juergens 005 estmate a consumpton-based model that ncorporates dsperson and bases n analysts forecasts and demonstrate emprcally that heterogenety explans a porton of expected returns and volatlty. 301

4 The Revew of Fnancal Studes / v n We fnd that the ntroducton of certan optons can make the market complete. 4 Consequently, we show that the prces of all opton clams on the underlyng stock satsfy the rsk-neutral prcng property of the Black-Scholes 1973 model and that the prces of all assets are determned as f there exsted a representatve agent. The representatve agent s belef s equal to the precson-weghted average of all nvestors belefs. We further extend the model to a multple-stock settng and show that the tradng volume of a stock depends not only on ts own stock prce change but also on the prce changes of related stocks. It s also shown that even f there are no dfferences of opnon or no sgnals about a stock s payoff, there may stll be tradng n that stock due to dfferences of opnon about the payoffs of other related stocks. These results may shed lght on the emprcal fndngs of Kandel and Pearson 1995 and Huberman and Regev 001. For example, the Kandel-Pearson result that the tradng volume of a stock can be postve even f ts prce does not change arses n our multstock model, as well as n our one-stock model wth optons. Agan, the equlbrum asset prces are equal to the prces that would arse n a representatve-agent economy, 5 and the representatve agent s belef s equal to the precson-weghted average belef of all nvestors. We show that the expected asset returns for the representatve agent follow the Captal Asset Prcng Model CAPM and that ths agent holds the market portfolo, whch s on hs effcent fronter. Suppose that the data represent the true dstrbuton of the stock payoff. Our result suggests that the CAPM wll not be the correct descrpton of the data unless the average belef happens to be the correct one. Ths artcle s related to Harrs and Ravv 1993 and Kandel and Pearson Both papers use dfferences of opnon to generate trades for a stock n the absence of optons. They show that wth two types of nvestors, dfferences of opnon can generate tradng patterns consstent wth stylzed emprcal evdence. The man dfference between our work and these studes s that we focus on the tradng volume of optons, as well as on the tradng volume of a stock n a multple-stock envronment, whch are not consdered n those models. Our model also makes dfferent predctons regardng the tradng volume of the underlyng stock. For example, Harrs and Ravv predct that tradng can occur only when nvestors have dfferent nterpretatons about publc sgnals n every perod, whereas we requre that nvestors nterpret sgnals dfferently n only one perod to generate sustaned tradng. The presence of many types of nvestors generates addtonal emprcal mplcatons. Kandel and Pearson consder a two-perod model and do not analyze the dynamc changes of tradng volume of the underlyng stock. Other studes that employ dfferences of opnon to generate trades nclude Harrson and Kreps 1978; Varan 1989; Detemple Downloaded from at Pekng Unversty on September 7, Yet tradng occurs n every perod as nvestors nterpret new sgnals dfferently n every perod. 5 Our aggregaton result wth dfferences of opnon s smlar to those of DeMarzo and Skadas 1998 and Bas, Bossaerts, and Spatt 003 wth asymmetrc nformaton. 30

5 Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons and Murthy 1994; Kraus and Smth 1996; Morrs 1996; Bas and Bossaerts 1998; Odean 1998; Zapatero 1998; Basak 000; Vswanathan 001; Brav and Heaton 00; Duffe, Garleanu, and Pederson 00; Kyle and Ln 00; Burasch and Jltsov 005; Davd 003; Hong and Sten 003; Qu, Starks, and Yan 003; and Schenkman and Xong 003. Our artcle s also related to studes that employ nose traders/random endowments and asymmetrc nformaton to generate trades. These nclude Pflederer 1984; Kyle 1985; Admat and Pflederer 1988; Brown and Jennngs 1989; Grundy and McNchols 1989; Km and Verreccha 1991; Holden and Subrahmanayam 199; Spegel and Subrahmanayam 199; Back 1993; Foster and Vswanathan 1993; Shalen 1993; Bas and Hllon 1994; Wang 1994; He and Wang 1995; Brennan and Cao 1996; and Easley, O Hara, and Srnvas In partcular, Brennan and Cao consder an equlbrum model wth optons. They focus on the mpact of ntroducng optons on nvestors welfare rather than on tradng volume n the optons market. They show that wth the ntroducton of an approprate opton securty, Pareto effcency can be acheved n only a sngle round of tradng, 6 and, as a result, nvestors wll no longer trade n ether the underlyng stock or the opton securty n future rounds. To generate addtonal tradng wth the arrval of new publc nformaton, Brennan and Cao and all other works under asymmetrc nformaton rely on the ntroducton of addtonal nose/lqudty tradng. A potental problem wth ths approach s that the argument to explan the tradng volume s crcular: t essentally requres new exogenous supply shocks to the stock to generate tradng volume. In ths sense, tradng s mposed onto the economy rather than endogenously generated. For example, to generate tradng around the earnngs announcement dates, these studes need nose traders for the equlbrum to be partally revealng. However, Kandel and Pearson 1995 fnd no evdence that nose tradng s partcularly hgh around earnngs announcements, and Pan and Poteshman 006 fnd no asymmetrcally nformed tradng n the ndex optons market. On the other hand, tradng n our model s drven endogenously by dfferental nterpretaton of publc sgnals wthout the need to ntroduce exogenous nose traders. Models usng the asymmetrc nformaton paradgm make very dfferent testable predctons regardng the nteracton between optons and the underlyng stock. For example, n the absence of addtonal nose tradng, Brennan and Cao 1996 predct that the ntroducton of optons reduces the tradng volume of the underlyng stock to zero and that there wll be no tradng volume n optons wth the arrval of new publc nformaton n future perods. 7 On the contrary, our model predcts that tradng volumes n the optons market Downloaded from at Pekng Unversty on September 7, Back 1993, Bas and Hllon 1994, and Easley, O Hara, and Srnvas 1998 also analyze the effects of optons when nformaton s asymmetrc. Detemple and Selden 1991 analyze the effects of optons under symmetrc nformaton. 7 These predctons are not supported by emprcal evdence. See, for example, Cao 1999 for a dscusson of stylzed emprcal results regardng the mpact of optons on the tradng volume of the underlyng stock. 303

6 The Revew of Fnancal Studes / v n should be clustered before and around the dates of publc announcements and are postvely related to the degrees of dsperson of belefs among nvestors. Moreover, our model predcts that optons tradng makes the tradng volumes of the underlyng stocks both hgher and more senstve to stock prce changes. The rest of ths artcle s organzed as follows. Secton 1 consders the economc model. Secton analyzes tradng volume n stocks and optons. Secton 3 develops a multstock equlbrum model, and Secton 4 concludes the artcle. The Appendx contans techncal proofs. 1. Economc Model In ths secton, we consder tradng n a stock due to dfferences of opnon n the absence of optons. It s assumed that the fnancal market conssts of a contnuum of nvestors, each ndexed by where 0, 1].Attme0, each nvestor s endowed wth x unts of the stock and, to avod unnecessary notaton, we assume that ndvdual endowments of the rskless bond are zero. Wthout loss of generalty, the rskless nterest rate s taken as zero. The rsky stock pays off at tme T an amount u, where u s normally dstrbuted wth mean ū and precson h. The per capta supply of the stock s postve and denoted as x.investor has a negatve exponental utlty functon defned over the tme T wealth, UW = exp W /τ, where τ represents the rsk tolerance of nvestors. As an ntroducton, we frst consder the basc sngle-perod model. 1.1 A statc one-perod model Let P 0 be the prce of the stock and D0 be the demand of nvestor for the stock. Investor s tme 1 wealth s gven by W = P 0 x + u P 0 D0.Its well known that n ths settng Sharpe, 1964 a lnear equlbrum exsts n whch P 0 = ū x τh, 1 D0 = τhū P 0] = x. Equaton 1 expresses the equlbrum prce as the expected payoff less a rsk premum that depends on the per capta stock supply x. We assume that the values of ū, x, τ, and h are such that P 0 s postve. The expected utlty of nvestor condtonal on hs endowment s gven by ūx EU = exp τ + x τ h x x ]. 3 τ h Downloaded from at Pekng Unversty on September 7, 014 Investor s wealth at tme 1, W s a lnear functon of the stock payoff u and can be wrtten as W u = P 0 x + xu P 0. The margnal rate of substtuton 304

7 Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons for nvestor between wealth contngent on u = u l and u = u k s gven by Mkl = exp{ hu k ū / W u k /τ} exp{ hu l ū / W u l /τ} { = exp 1 } hu k u l u k + u l P 0. 4 Because the margnal rate of substtuton s the same for all nvestors, ths equlbrum s Pareto effcent. 1. A dynamc model wth dfferences of opnon In ths subsecton, we extend the sngle-perod model to allow for addtonal market sessons between tme 0 and tme T, at whch pont the stock payoff s realzed and consumpton occurs. Immedately before each market sesson, a publc sgnal about the stock payoff arrves. Note that the one-perod equlbrum allocaton s Pareto effcent. Accordng to Mlgrom and Stokey 198, there should be no more tradng after the frst round when new nformaton about the fnal stock payoff becomes publcly avalable. However, the Mlgrom and Stokey theorem holds only when nvestors have essentally concordant belefs about the publc nformaton. When nvestors belefs are not essentally concordant, tradng among nvestors can occur wth the arrval of publc nformaton Equlbrum prce and demand Consder a settng n whch nformaton about the fnal payoff u s made avalable gradually by a seres of publc sgnals y t at tme t = 1,...,T 1. To generate optons tradng n a tractable manner, we assume that nvestors dsagree about how to nterpret the relatonshp between y t and u. In partcular, nvestor beleves that y t = u + η t, u Nū, 1/h, η t N m t, 1/n t. As a result, nvestors dsagree about both the mean and the precson of y t. Wthout loss of generalty, we assume that n t m t d = 0.8 Investor beleves that η t has a mean m t and a precson n t.letn t 1 0 n t denote the average precson of the publc sgnal. We defne the concepts of hgh confdence and low confdence as follows. Downloaded from at Pekng Unversty on September 7, 014 Defnton 1. Let ρ t n t /n t. When ρ t > 1 ρ t < 1, we defne that nvestor has hgh low confdence about the publc sgnal at tme t. 8 If n t mt d 0, then let n t = n t d, m t = n t m t d/n t, We can redefne a new publc sgnal ŷ t = y t m t = u + ˆη t, ˆη t N ˆm t, 1/n t, ˆm t m t m t. We then have n t ˆm t d = n t m t n t m t d = n t m t n t m t =

8 The Revew of Fnancal Studes / v n After each sgnal the market opens for tradng, and at tme T, the payoff of the stock s realzed and consumpton occurs. Let P t denote the prce of the stock at tme t. Trader s optmal demand for the stock at tme t s denoted by Dt. A dynamc equlbrum s descrbed n the followng theorem. Its proof and all other proofs are gven n the Appendx. Theorem 1. In an economy wth T tradng sessons, there exsts a dynamc equlbrum n whch prces, and demands for the stock, are gven by P t = µ t x/τk t, 5 where Dt = τkt ] µ τk t n t P t + t+1 m t+1 n t+1 = τ hū + µ t hū + t j=0 n j y j h + tj=0 n j K t Var t u] 1 = h + t j=0 n j y j K t P t + K tn t+1 m t+1 n t+1, µ t hū + t j=0 j n y j + m j t n j, and j=0 h + t j=0 n j K t 1 0, 6, K t d = h + t n j. Here µ t and µ t denote the condtonal expectatons of the stock fnal payoff u for the average nvestor and nvestor at tme t, respectvely. Kt denotes the condtonal precson of u for nvestor and K t denotes the average precson of all nvestors. Because nvestor s rsk averse, hs demand for the rsky stock ncreases wth hs precson about the sgnal, as well as hs condtonal mean of the fnal payoff, as n a typcal mean-varance framework. An nterestng feature of the equlbrum s that the prce depends on the average nvestor s condtonal mean and condtonal precson of the stock payoff. The average nvestor does not buy or sell n equlbrum. We next examne the equlbrum stock demands and prces when optons are added to the market. j=0 Downloaded from at Pekng Unversty on September 7, A dynamc model wth optons The fnancal market s ncomplete wth one rsky stock and one rsk-free asset. Breeden and Ltzenberger 1978 have shown that the market can be completed by the ntroducton of a complete set of optons. We complete the market by ntroducng all call optons wth postve strke prces and all put optons wth negatve strke prces. All optons are n zero net supply. Any dervatve asset 306

9 Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons wth a twce dfferentable prce functon f u can be syntheszed usng a collecton of optons: 0 f u = f 0 + f 0u + Z u + f ZdZ + 0 u Z + f ZdZ, 7 where Z denotes the strke prces. Let P CZt denote the prce of a call opton wth strke prce Z n tradng sesson t. LetDCZt denote nvestor s demand densty of optons at strke prce Z, that s, the holdngs of call optons wth strke prce Z to Z + dz s gven by DCZt dz. Defne the prce and demand for the put optons smlarly as P PZt and D PZt. We consder a set of bounded, contnuous strateges for nvestor, {Dt, D CZt, D PZt }. We show n the proof of Theorem that the profts derved from the nvestors equlbrum holdngs n the stock and optons are ndeed bounded under the proposed equlbrum prces. Theoretcally, we should also consder dscrete holdngs n optons. For example, an nvestor may hold 100 contracts at the strke prce of 0. It can be shown that the strateges wth dscrete holdngs are not optmal n equlbrum. We can then gnore such strateges wthout loss of generalty. Moreover, due to the put-call party, we consder nonzero demands for calls wth postve strke prces and nonzero demands for puts wth negatve strke prces. The next theorem establshes the exstence of an equlbrum wth optons and descrbes the demands and prces for both stock and optons. Theorem. There exsts a sequental dynamc equlbrum n whch P t = µ t x/τk t, 8 Dt = τkt ] µ τk t n t P t + t+1 m t+1 1 τ ρ K ] t t+1 + K t Kt P t, n t+1 n t+1 9 P CZt = P t ZNP t Z K t + 1 np t Z K t, Z 0, 10 Kt Downloaded from at Pekng Unversty on September 7, 014 P PZt = Z P t NZ P t K t + 1 Kt nz P t K t, Z < 0, 11 1 DCZt = τ ρ Kt t+1 ], Z 0, 1 + K t Kt n t+1 1 D PZt = τ ρ K ] t t+1 + K t Kt, Z < 0, 13 n t+1 307

10 The Revew of Fnancal Studes / v n where µ t hū + t j=0 n j y j h + tj=0 n j K t Var t u] 1 = h +, µ t hū + t j=0 j n y j + m j t n j, and j=0 h + t j=0 n j K t 1 0, K t d = h + t n j. j=0 Theorem shows that optons are not redundant securtes. In ths equlbrum, nvestors wth hgh confdence take short postons n the optons whle nvestors wth low confdence take long postons n the optons. 9 Intutvely, nvestors wth hgh confdence perceve a lower volatlty for the stock, so they beleve that optons are overvalued. As a result, they take short postons on optons. Smlarly, nvestors wth low confdence perceve optons to be undervalued, so they take long postons. Although nvestors acheve the Pareto optmal allocaton, those whose precson about the publc sgnal s dfferent from that of the average nvestor wll trade n optons n every perod. In the presence of optons, nvestors wll trade n the underlyng stock to hedge opton postons even f the prce of the underlyng stock does not change. Indeed, t wll be shown that the tradng volume of the underlyng stock s postve even f the stock prce remans unchanged. Note that the average nvestor serves as the representatve agent who prces both the stock and the optons based on the average belef. Ths nvestor does not buy and sell the stock and holds no optons n equlbrum. Wth normal stock payoff dstrbuton and CARA utlty, the Pareto effcent allocaton s a quadratc functon of the fnal payoff of the stock u. In equlbrum, nvestors use optons to synthesze the approprate payoffs that are quadratc functons of the stock payoff. Consequently, t s not necessary to ntroduce a contnuum of call and put optons to complete the market. We next show that a dervatve asset wth a payoff of Qu = u can complete the market, yeldng the followng result. 10 Theorem 3. Let P Qt denote the prce of the quadratc dervatve asset and D Qt denote the demand for the quadratc dervatve asset by nvestor n tradng sesson t. Then there exsts a dynamc equlbrum n whch Downloaded from at Pekng Unversty on September 7, 014 P t = µ t x/τk t, 14 Dt = τkt ] µ τk t n t P t + t+1 m t+1 D Qt n P t, t Assumng that, for each nvestor, ρ t 1 has the same sgn for all t. 10 Under asymmetrc nformaton, Brennan and Cao 1996 also show that a quadratc opton can complete the market. 308

11 Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons P Qt = Kt 1 + Pt, 16 D Qt = τ 1 ρ K ] t t+1 + K t Kt, 17 n t+1 where K t and K t are the same as n Theorem. Lke Theorem, ths theorem also shows that nvestors wth hgh confdence take short postons n optons and nvestors wth low confdence take long postons n optons. In addton, there wll be tradng volume n the underlyng stock even f ts prce does not change. Note that n one-perod models, when markets are complete, there exsts a representatve agent who prces all assets accordng to hs belef and margnal utlty. 11 Interestngly, Theorems and 3 show that there also exsts a representatve agent n our dynamc tradng model wth dfferental nterpretaton of publc sgnals. 1 To make a connecton wth the one-perod models, we next demonstrate the exstence of a representatve agent under dfferences of opnon by solvng a socal plannng problem. Let W T denote the per captal aggregate wealth and W T the wealth of nvestor at the fnal date T. As n Rubnsten 1974 and Brennan and Kraus 1978, the socal plannng problem s to maxmze the followng objectve functon: max π expw T /τ]d, 18 W T subject to the wealth constrant W T d = W T, 19 where π s a postve coeffcent. The frst-order condton mples that The Pareto optmal allocaton s then gven by π exp W T /τ/τ = λ. 0 W T = τlnτλ/π ]. 1 Downloaded from at Pekng Unversty on September 7, 014 Consequently, we have ] W T = τ lnτλ/π d. 11 See Rubnsten 1974, Brennan and Kraus 1978, and Brennan See also Joun and Napp 006 for the exstence of a representatve agent under dfferences of opnon among nvestors n a model wth consumpton n each perod. 309

12 The Revew of Fnancal Studes / v n The socal planner s utlty functon s thus gven by ] π expw T /τ]d = τλ = exp ln π d exp W T /τ. 3 It can now be seen that the utlty functon of the representatve nvestor s also exponental wth a rsk tolerance coeffcent τ. To derve the representatve agent s probablty belef, we note that Pareto optmalty mples that where θ s a constant. Ths leads to π exp W T /τ π exp W T /τ = θ, 4 W T = W T + τlnπ lnπ τ lnθ ]. 5 Integratng both sdes of Equaton 5 wth respect to yelds lnπ = lnπ d + lnθ d = T lnπ + 1 lnh 1 hu where + 1 T 1 t=1 ln n 1 t d = T lnπ + 1 lnh 1 hu n tη d 1 n t m d t + n t η m d t + lnθ d T 1 ln n 1 t d lnn t t=1 lnθ d = T lnπ + 1 lnh 1 hu + 1 T 1 lnn t 1 n tη, 6 lnθ d = 1 T 1 t=1 t=1 ln n 1 t d + lnn 1 t + n t m, t Downloaded from at Pekng Unversty on September 7, 014 because π ntegrates to one. As a result, the representatve nvestor has a belef that η t N0, n 1 t, and hs condtonal belef at tme t about u s gven by N µ t, K 1 t, where µ t = K 1 t hū + t j=0 n j y j and K t = h + t j=0 n j. 13 The next proposton summarzes the result. 13 Gven the constructon of the representatve agent, we can construct nvestor s optmal demand usng expresson

13 Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons Proposton 1. When markets are completed by addng ether a contnuum of call and put optons or a sngle quadratc opton, there exsts a representatve nvestor wth rsk tolerance τ and belef N µ t, K 1 t, where µ t = K 1 t hū + t j=0 n j y j. It s ndeed qute strkng that markets are effectvely complete wth the ntroducton of a sngle dervatve, and as a result, the prces of all contngent clams behave as f there exsted a representatve nvestor. Tradng n optons and the underlyng stock, however, s actve among nvestors at all tradng sessons due to dfferences of opnon among nvestors n every perod. Interestngly, even n the absence of optons, the representatve nvestor prcng stll works, that s, equlbrum prces depend only on K t and µ t.to understand ths result, we start wth a market beng completed by a quadratc opton. Consder the last tradng perod t = T 1. In ths complete market, Theorem 3 states that there exsts a representatve nvestor and hs probablty dstrbuton of u s N µ T 1, K 1 T 1. The frst-order condton for nvestor wth respect to hs stock holdngs s gven by ET 1 u P T 1 exp D T 1 u P ] T 1 + D QT 1 u P QT 1 = 0. τ 7 Note that we can rewrte the payoff of the quadratc asset as u = u P t 1 + P T 1 u P T 1 + PT 1. The frst-order condton then reduces to ET 1 u P T 1 exp ˆD T 1 u P ] T 1 + D QT 1 u P T 1 = 0, τ 8 where and ˆD t 1 = ˆD T 1 = D T 1 + D QT 1 P T 1, D T 1 d + P T 1 D QT 1 d = x + 0 = x. 9 Downloaded from at Pekng Unversty on September 7, 014 It means that holdng D T 1 shares of the stock and D QT 1 shares of the quadratc asset u s equvalent to holdng ˆD T 1 shares of the stock and D QT 1 shares of a new quadratc asset u P T 1, plus a bond wth a face value of D QT 1 P T

14 The Revew of Fnancal Studes / v n Substtutng the probablty dstrbuton of nvestor nto Equaton 8, we have exp ˆD T 1 τk T 1 µ T 1 P T 1]u P T 1 τ u P T 1 du = DQT exp 1 u P T 1 τ + u P T 1 K T 1 In Equaton 30, the ntegrand s u P T 1 tmes a fracton. The denomnator of the fracton s an even functon of u P T 1. At the optmum, the stock holdng s gven by ˆD T 1 τk T 1 µ T 1 P T 1] = 0 so that the numerator of the fracton s 1. As a result, the ntegrand s an odd functon and the ntegral s thus zero. Note that u P T 1 s also an even functon of u P T 1. Changng the shares of the quadratc asset u P T 1 does not change the fact that the denomnator of the fracton remans an even functon of u P T 1. In partcular, when nvestors cannot trade n the quadratc asset,.e., D QT 1 = 0, the Euler condton stll holds at ˆD T 1. As a result, even n an ncomplete market wthout dervatve assets, ˆD T 1 s stll the optmal demand for nvestor at the representatve-nvestor prce P T 1. In addton, ˆD T 1 clears the market, whch means that ˆD T 1 and P T 1 consttute an equlbrum n the ncomplete market wthout optons. In other words, equlbrum prces depend only on K t and µ t.. Tradng Volume n Stocks and Optons In ths secton, we use the results obtaned n Theorems 1 3 to analyze the nvestors tradng strateges and tradng volume n stocks and optons. In Sectons 3.1 through 3.4, we dscuss tradng strateges and prce dynamcs n the economy wthout optons..1 Tradng strateges and prce dynamcs Let P t P t P t 1 denote the prce change and D t D t D t 1 denote the tradng sze of nvestor n tradng sesson t. In equlbrum, Equaton 6 yelds: ] Dt Kt n t+1 = τ m t+1 K t 1n t m t n t n t+1 n m t t n t + τ Kt 1 n t Kt 1 Kt 1 P t K t 1 n t ] = τk t Kt n t+1 m t+1 n t+1 K tn t m t n t + τ n t K t 1 n t n t n t K t 1 K t 1] P t. 31 Downloaded from at Pekng Unversty on September 7,

15 Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons Equaton 31 llustrates that the tradng sze of nvestor can be dvded nto four components. The frst two components come from dsagreements about the mean of the next-perod publc sgnal and the current publc sgnal. The remanng two components come from the dsagreements about the precsons of the current sgnal and past cumulatve sgnals. Tradng can occur due to dsagreements about past publc nformaton, current publc nformaton, and the next-perod publc nformaton. However, dsagreements about the precsons of future publc nformaton and dsagreements about the means of past publc nformaton and future publc nformaton beyond the next perod do not generate tradng n stocks. We thus have the followng result regardng how dsagreements about the mean and precson of publc nformaton affect stock tradng dfferently. Proposton. Dsagreements about the mean of a publc sgnal result n stock tradng n only the current perod and the prevous perod. Dsagreements about the precson of a publc sgnal result n stock tradng n the current perod and all future perods. Ths proposton ndcates that to generate tradng, nvestors need to dsagree on the precson of publc sgnals only once. Thereafter, they wll trade even f they agree on all sgnals n future perods.. Tradng volume and dsagreements about mean We frst dscuss tradng volume due to dsagreements about the mean of y t alone. We assume n ths subsecton that n t = n t for all t and. Then the tradng of nvestor at perod t reduces to D t = τ K t m t+1 K t 1m t n tm t] = τkt m t+1 m t]. Note that at tme t 1, when nvestor has postve expectaton negatve about the mean of η t, m t > 0< 0, he wll be more bullsh bearsh about the next-perod prce change. As a result, the nvestor holds more stock relatve to the average nvestor at tme t 1, gven by τk t 1 m t. However, at tme t, gven the observaton of y t,nvestor expects to be relatvely pessmstc about future realzatons of the asset payoff. Investor wll unwnd part of hs holdngs at tme t, n the amount of τk t 1 m t. Moreover, as the nvestor s relatvely pessmstc at tme t, he further sells n the amount of τn t m t due to the dfference of condtonal expectaton of asset payoff. Thus tradng at tme t nduced by m t s to sell n the amount of τk tm t. In addton, f nvestor also has postve expectaton about η t+1, m t+1 > 0, he wll buy relatvely more stock compared to the average nvestor n the amount of τk t m t+1 ]. The combned effects of m t and m t+1 yeld the tradng amount of τk tm t+1 m t ]. The dfferences about the mean of publc nformaton affect tradng only n two perods: the current perod and the prevous adjacent perod. Downloaded from at Pekng Unversty on September 7,

16 The Revew of Fnancal Studes / v n Tradng volume and dsagreements about precson To analyze tradng due to dfferences n precson, we assume that m t = 0, and ρ t s the same for all t, that s, ρ t = ρ for ths subsecton and the next subsecton. Usng Equatons 5 and 6, we have D t+1 D t = τρ 1hP t+1 P t. 3 Investors who have hgh confdence ρ > 1 about a publc sgnal put more weght on the sgnal and thus trade n the drecton of the sgnal. If the publc sgnal s very postve, the prce wll go up. But nvestors wth hgh confdence stll beleve that the prce has not fully ncorporated the postve sgnal, due to the presence of nvestors wth low confdence. Hence, nvestors wth hgh confdence beleve that the stock prce wll go up even further and demand more shares of the stock. On the other hand, nvestors who have low confdence ρ < 1 about a publc sgnal put less weght on t. When the stock prce goes up, they beleve that the prce s overreactng to the publc sgnal due to the presence of nvestors wth hgh confdence. Hence, nvestors wth low confdence sell the stock. The followng proposton characterzes the relatons between trades and prce changes from dfferent nvestors perspectves. Proposton 3. The trades of nvestors wth hgh confdence ρ > 1 at tme t are postvely correlated wth the prce change at tme t; The trades of nvestors wth low confdence ρ < 1 at tme t are negatvely correlated wth the prce change at tme t; Investors wth the average confdence level ρ = 1 do not trade. The next proposton summarzes the prce dynamcs based on dfferent nvestors perspectves. Proposton 4. For an nvestor wth hgh confdence ρ > 1, the prce change at tme t + 1 s postvely correlated wth the prce at tme t and s postvely correlated wth the prce change at tme t, that s, Downloaded from at Pekng Unversty on September 7, 014 Cov P t+1 P t, P t > 0, Cov P t+1 P t, P t P t 1 > For an nvestor wth low confdence ρ < 1, the prce change at tme t + 1 s negatvely correlated wth the prce at tme t and s negatvely correlated wth the prce change at tme t, that s, Cov P t+1 P t, P t < 0, Cov P t+1 P t, P t P t 1 <

17 Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons For the average nvestor ρ = 1, the prce change at tme t + 1 s not correlated wth the prce at tme t and s not correlated wth the prce change at tme t, that s, Cov P t+1 P t, P t = 0, Cov P t+1 P t, P t P t 1 = We next apply Theorem 1 to study the equlbrum tradng volume of the stock..4 Tradng volume and prce changes Many emprcal studes have examned the contemporaneous behavor of volume and absolute prce changes and found a postve correlaton between the two e.g., Karpoff, Snce the dynamcs of prce volatlty and tradng volume can only be studed n a multple tradng economy, ths subsecton presents addtonal results on the autocorrelaton propertes of tradng volume, as well as the relaton between tradng volume and the number of tradng sessons between tme 0 and tme T. Let P t = P t P t 1 denote the prce change at tme t, where t = 1,...,T 1. Let tradng volume at tme t, V t, be defned as one-half the sum of all purchases and sales, that s, V t = Dt D t 1 d = 1 = K t τk t n t K P t d t n t τh ρ 1 P t d. 36 Note that there s no hedgng demand for the stock n our model and that all trades are due to the dfferences of opnon about publc sgnals. As a result, we obtan a smple result that the tradng volume n each perod s proportonal to the product of the absolute prce change and the dsperson of nvestors precsons n publc nformaton. In Proposton 4, we have shown that prce changes are serally correlated f the true precson s dfferent from the average precson. Snce the correlaton coeffcent between the absolute values of two normally dstrbuted varables x and y wth correlaton rx, y and means of zero s gven by Downloaded from at Pekng Unversty on September 7, 014 Corr x, y = π the followng lemma s mmedate. 14 rx,y 0 arcsn tdt> 0, 14 Harrs and Ravv 1993 derve results,, and based on dfferences of opnon wth rsk-neutral nvestors. Brennan and Cao 1996 also obtan predctons andv, usng a partally revealng ratonal expectatons model wth dfferentally nformed nvestors. 315

18 The Revew of Fnancal Studes / v n Lemma 1. Tradng volume and absolute prce changes are postvely correlated. Tradng volume and absolute change of the precson-weghted average forecast are postvely correlated. Tradng volume s hgher when the publc sgnal s very nformatve a hgh value of n t. v Tradng volume ncreases wth the dsperson of belefs among nvestors. v For any nvestor whose precson s dfferent from the average precson, the absolute prce change and tradng volume are postvely serally correlated. The frst three mplcatons are consstent wth the emprcal evdence summarzed n Karpoff Implcaton v mples that tradng volume may be related to the dsperson among fnancal analysts forecasts. Emprcally, Frankel and Froot 1990 examne foregn exchange data and fnd a postve relaton between volume and dsperson, and Ajnkya, Atase, and Gft 1991 also obtan a postve relaton between stock volume of tradng and the dsperson n fnancal analysts earnngs forecasts, both of whch support our predcton v. Implcaton v may be tested usng survey data of nvestors belefs about the stock prce changes..5 Tradng volume n optons Ths subsecton consders tradng volume n optons. Let DCZt denote nvestor s amount of tradng for a call opton wth strke prce Z, whch, accordng to Theorem, s gven by nt DCZt = τ n t 1 K t 1 n + ] n t+1 n Kt t+1 t n. 37 t+1 Tradng n optons can be dvded nto two parts. The frst part comes from the dsagreements about current nformaton whle the second part comes from the dsagreements about the next-perod nformaton. Suppose that nvestors dsagree about the publc nformaton only at tme t. Then nvestor wll trade n the optons market wth a holdng of τn t n t K t 1 /n t at tradng sesson t 1. At tradng sesson t,nvestor wll partally unwnd the holdngs n sesson t 1 to acheve a poston of τn t n t, and there wll be no more tradng n future perods. Past dsagreements about precsons and dsagreements about means do not generate tradng n the optons market, contrary to the results obtaned n the stock market. The followng proposton summarzes ths result. Downloaded from at Pekng Unversty on September 7, 014 Proposton 5. Dsagreements about the precson of the current- and nextperod nformaton generate tradng n the optons market. Dsagreements about precsons of past nformaton and nformaton after the next perod do 316

19 Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons not generate tradng n the optons market. Moreover, dsagreements about mean do not generate tradng n the optons market. It s nterestng to contrast the results on tradng strateges n stocks and optons. We have shown that past dsagreements and current agreements n precson generate tradng n the stock market, as do the dsagreements n the mean n the current perod and next perod. In contrast, tradng n optons depends only on current dsagreements and next-perod dsagreements about precsons. In a bg rare event, t s more lkely to have dfferences of opnon. Our results suggest that we should observe more clustered tradng n optons market just before and durng the event. Let the tradng volume n optons, V CZt, be defned as half of the sum of the absolute trades. Let O CZt denote the open nterest n optons. We now analyze how dsperson of belefs n precsons and average precson n t affect optons tradng and open nterest. To smplfy the analyss, we assume that m t = 0 and ρ t = ρ for all t n the rest of ths secton. We have V CZt = τ K t K t+1 K t 1K t 1 n t+1 n 1 ρ d t 0 = τ 1 Var P t ] 1 1 Var P t+1 ] 1 ρ d, O CZT = D d K CZt = τ 1 ρ ] d t + K t h. 39 n t+1 0 Because the average of ρ s one, ρ d s a measure of nvestors dsperson of belefs. Equatons 38 and 39 then lead to the followng result regardng tradng volume n optons. 15 Proposton 6. Investors open nterest n optons ncreases wth the average perceved precson, n t, of a publc sgnal and nvestors tradng volume and open nterest n optons are hgher when they have hgher dsperson of belefs. Wth a hgher dsperson of belefs, nvestors dsagree on the volatlty of the stock payoff more; equvalently, they dsagree on the value of optons more, and the tradng volume for optons naturally ncreases. When publc nformaton s very nformatve at tme t, orn t s so large that Var P t ] Var P t+1 ] > 0, the volume of trades n optons ncreases wth the average precson of publc nformaton n that sesson. Thus, tradng n optons wll also be more actve before and durng nformatve publc announcement dates. These predctons may be tested usng event studes to analyze the behavor of open nterest and tradng volume n optons around nformatve publc events. Downloaded from at Pekng Unversty on September 7, The results for put optons can be obtaned smlarly. 317

20 The Revew of Fnancal Studes / v n Stock tradng volume n the presence of optons The lterature has provded some emprcal evdence that the ntroducton of optons tends to ncrease the tradng volume of the underlyng stock. See, for example, Sknner 1990 and Kumar, Sarn, and Shastr In ths subsecton, we examne the effects of optons on the tradng volume of the underlyng stock: ] ] Dt Kt n t+1 = τ m t+1 K tn t m t n t + τ n t+1 n t Kt 1 n t Kt 1 Kt 1 P t K t 1 n t 1 τ ρ Kt t+1 P t + ] 1 ρ Kt 1 t P t 1 n t+1 n t ] Kt n t+1 = τ m t+1 K tn t m t n t+1 n t 1 τ ρ Kt t+1 P t + 1 ρ t Kt P t K t 1 P t 1 n t+1 n t ]. 40 Note that n the presence of optons, stock tradng no longer responds to past dsagreements about precsons. Also note that the market s completed by the ntroducton of optons. If there wll be no more dsagreements n the future, then nvestors would have traded to a Pareto optmal allocaton after tradng n the current perod and there wll be no more tradng. Investors wll trade only f there are dsagreements n the future. We thus have the followng results. Proposton 7. In the presence of optons, dsagreements about the mean and the precson of a publc sgnal result n stock tradng n only the current perod and the prevous perod. To smplfy our analyss further, we consder a specal case n whch m t = 0, and ρ t s the same for all t, that s, ρ t = ρ. Followng the results of Theorem, we have D t = τ n t y t K t P t + K t 1 Pt 1 + ρ 1 Kt P t K t 1 P t 1 n t+1 n t = τρ Kt K t+1 1 P t K ] t 1K t P t n t+1 n t ] Downloaded from at Pekng Unversty on September 7, Mayhew and Mhov 005, however, argue that the prevous lterature does not take nto account the endogenety ssues n optons lstng approprately. Usng a matched control sample to avod the endogenety ssue, they reconfrmed earler results that optons tradng ncreases the tradng volume of the underlyng stock. 318

21 Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons The tradng volume of the underlyng stock s then gven by V t 1 1 D d K t K t+1 t = τ P t K t 1K t P t 1 n t+1 n ρ 1 d. 4 t 0 It s clear that the tradng volume can be postve even f the stock prce remans the same, that s, P t = 0. The reason s that n the presence of optons, nvestors trade n the stock to hedge optons. Even f the stock prce remans unchanged, there may stll be a need to hedge optons because opton prces may change due to dfferences of opnon about publc sgnals. Our result ndcates that tradng volume s related not only to the current prce change but also to the past prce changes. In addton, the coeffcent on lagged prce change should be larger for stocks wth optons than for stocks wthout optons. To offer sharper emprcal predctons, we next assume that the volatlty of the stock-prce change s statonary across tradng sessons for the average nvestor, that s, Var P t ] = Var P t+1 ] for all t. Ths serves as a suffcent condton for the result regardng the expected tradng volume of the underlyng stock gven n the followng proposton. Proposton 8. When Var P t ] = Var P t+1 ], the ntroducton of optons ncreases the expected tradng volume of the underlyng stock. Moreover, the expected tradng volume s more senstve both to the prce changes of the stock and to the dsperson of forecasts among nvestors. The results of ths proposton are due to nvestors hedgng demands for optons. For example, wth optons, a change n the stock prce affects the propertes of the optons assocated wth the stock, whch requres more hedgng for optons. As a result, the expected tradng volume of the stock s more senstve to stock prce changes n the presence of optons. To test the mplcatons of Proposton 8, one may take two approaches. The frst approach s to conduct an event study to analyze the amount of tradng volume before and after the ntroducton of optons. The second approach s to perform a cross-sectonal study to compare tradng volume and ts senstvty to prce changes and the dsperson of forecasts among nvestors between stocks wth optons and those wthout optons. Downloaded from at Pekng Unversty on September 7, Multple Stocks We have shown that the tradng volume s related to the prce change n a sngle-stock model. Emprcal studes have shown that the tradng volume of a stock s related to the prce changes of not only that stock but also those wth related payoffs Huberman and Regev, 001. We next consder a multstock dynamc model and examne the relatonshp between tradng volume and prce changes. For tractablty, we omt optons n ths model. 319

22 The Revew of Fnancal Studes / v n The payoffs of the M stocks are realzed at tme 1, and are represented by an M 1 normally dstrbuted random vector Ũ wth mean Ū and precson matrx H. Each nvestor, 0, 1], s endowed at tme 0 wth rsky assets denoted by the vector X ; nvestors are characterzed by negatve exponental utlty functons as defned earler. The vector of the aggregate per capta supply of the rsky assets s X. Immedately pror to tradng sesson t, a vector of publc sgnals s released. The publc sgnals are represented by the M 1 vector Ỹ t, where Ỹ t = Ũ + η t. Investors have dfferental nterpretaton about the publc sgnals. For each nvestor, η t s normally dstrbuted wth mean Mt and precson matrx Nt.17 The average precson matrx for the publc sgnal s N t N t d. Let P t denote the vector of equlbrum rsky asset prces, Dt the vector of nvestor s demands for the rsky assets, and F t the publc nformaton set ncludng the prces P t, all at tradng sesson t. The followng theorem descrbes the asset prces and the nvestors asset demands at each market sesson n a sequental equlbrum. Theorem 4. There exsts a sequental equlbrum. The vectors of rsky asset prces, P t, and nvestor s asset demands, Dt, are gven by P t = Kt 1 K t µ t τx], Dt = τkt µ t P t ] + τk t N 1 t+1 N t+1 M t+1, 43 where t 1 µ t E 1 t U] = Kt HŪ + N j Ỹ, µ t Kt 1 Kt µ t d, j=0 0 j t 1 t Kt Var t Ũ]] 1 = H + N j, K t K t d = H + N j. j=1 The optmal tradng strategy of nvestor, Dt, and the tradng volume of the stocks, V t, are gven by Dt Dt D t 1 = τ K t Kt Pt P t 1 ] + τ K t N 1 t+1 N t+1 M t+1 K t 1Nt 1 Nt M t Nt ] M t = τk t Kt P t P t 1 ] + τk t N 1 t+1 N t+1 M t+1 N t 1 Nt M t ], 44 0 j=0 Downloaded from at Pekng Unversty on September 7, We assume that N 0 = O and N 1 T = O, where O denotes the zero matrx. Ths assumpton s consstent wth the earler assumpton that there s no publc nformaton at tme 0 and that all rsky asset returns are realzed at sesson T. 30