Xiaoquan (Michael) Zhang

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1 RESEARCH ARTICLE HO DOES THE INTERNET AFFECT THE FINANCIAL MARKET? AN EQUILIBRIUM MODEL OF INTERNET-FACILITATED FEEDBACK TRADING Xiaoquan (Michael) Zhang School of Buine and Managemen, Hong Kong Unieriy of Science and Technology, Clear aer Bay, Kowloon, HONG KONG Lihong Zhang School of Economic and Managemen, Tinghua Unieriy, Beijing, CHINA Appendix A Proof Proof of Propoiion : e fir how ha for all, λ β. Plugging Equaion (3) and (6) ino Equaion (7) gie dp dp λd λ P d If λ β, hen hold for all > and >. Mahemaically, i incorrecly implie ha he Brownian moion i d P d deermined by a drif in ime. From a pracical poin of iew, i incorrecly implie ha informed rader bring only noie ino he marke. hen λ β, we hae ( ) dp λ P d λ d () Noe ha Equaion () i under filraion of F F. For a gien F, aking he condiional expecaion of Equaion () yield dp λ d (3) Thi i a ochaic differenial equaion of P under filraion F. To examine he properie of he price proce, we need o apply he filering lemma by Liper and Shiryae (977), which help anwer he following queion: Gien he oberaion of he ochaic proce P, wha i he be eimae of he ae baed on hee oberaion? Fir, le V E F and conider he filering of wih repec o {F } ##. By he filering lemma, we hae hu λ λ dv () d λ MIS Quarerly Vol. 39 No. Appendice/March 5 A

2 Zhang & Zhang/An Equilibrium Model of Inerne-Faciliaed Feedback Trading () dv d () where [ ] () ( ) aifie he following one-dimenional Riccai differenial equaion: d d Tha i, () E V F λ () () λ wih iniial alue () d d () () The oluion o hi equaion i () d (5) Plugging Equaion (5) ino Equaion () and uing he emi-rong efficiency condiion gie dp d d (6) Comparing coefficien of Equaion (3) and Equaion (6) yield λ d (7) Thu, d λ β (8) Since i ricly poiie, we can ee ha when he marke i emi-rong efficien he deph of he marke λ i alway greaer han β. Equaion (7) can be rewrien a Inegraing he aboe equaion wih repec o d yield ( ) d λ λ λ ( ) d ( ) (9) Again, ince i ricly poiie, i i eay o ee ha for all [, ] λ d > A MIS Quarerly Vol. 39 No. Appendice/March 5

3 Zhang & Zhang/An Equilibrium Model of Inerne-Faciliaed Feedback Trading Proof of Propoiion : By Schwarz inequaliy and he conrain, we now ha The equaliy hold if and only if for any [, ], λ λ d d λ (3) Proof of Theorem 3: Equaion (6) can be obained direcly from Equaion (3). Equaion (7) i obained by plugging Equaion (6) ino Equaion (9). Equaion (7) and (5) combined yield Equaion (8). Finally, Equaion (9) i obained by combining Equaion (5) and (3). Proof of Propoiion : Under he aumpion in our model, he profi earned by uninformed rader can be expreed by and E ( P) dxu E ( P) dx U E ( P)( dp d) β E ( βλ P ) ( P ) d d βλ E ( P ) d E ( ) ( P ) d βλ d E P dx q dx q d ( ) q I q U λ λ d E qdxu( q) d β λ βd E ( ) βd E ( ) d ( ) βd β d d βqλ λ q q q ( Pq ) dq d q d q q q q The reul i obained from Equaion (7), (), (6), and (8), and he ranformaion relaion beween he Iô and he ( ) ochaic inegraion. The la equaion aume ha i no a funcion of. If i indeed a funcion of, he reul i no changed: d imply meaure he aerage ariance of noie up o ime. heher i a funcion of ime doe no change any of our reul. MIS Quarerly Vol. 39 No. Appendice/March 5 A3

4 Zhang & Zhang/An Equilibrium Model of Inerne-Faciliaed Feedback Trading Proof of Theorem 5: ihou lo of generaliy, we uppoe P. The econd momen of he informed rader profi i E P dx E P d E d d E d d E I d d d d d Defining he fir erm by A, A d Inegraing by par (ochaic inegraion, generalized Iô formula), we can hae d d d By inerchangeabiliy of ordinary Riemann inegraion, we can calculae A d d d d d d d dd d d And he la erm A d d d d 3 The informed rader ariance of he profi i A MIS Quarerly Vol. 39 No. Appendice/March 5

5 Zhang & Zhang/An Equilibrium Model of Inerne-Faciliaed Feedback Trading [ 3 3 3] ( [ π () ] ) [ 3] [ π () ] EA A A E E A A A AA AA AA E E [ ] E [ ] E d E ( ) d d E[ ] E d E d d e hae ued he aumpion ha i independen of he Brownian moion, and he expecaion of i zero (i.e., P E ), and he la equaliy i obained from reul in Appendix B. [ ] e coninue o calculae he ariance of he uninformed rader profi. For impliciy, we uppoe ha β i a conan oer, denoed by β, he econd momen of he uninformed rader profi i The fir erm Thi la equaliy i obained from Appendix B. The econd erm E ( P) dx U E ( P) ( dp d ) β E ( P) dp ( P) d β E ( P) d ( P) ( ) d β β β E ( β P ) d E ( ) ( ) ( ) P d P β d E ( ) ( ) P β d B Eβ ( P ) d 5 E d E d d β 5 E d E d d β 7 β MIS Quarerly Vol. 39 No. Appendice/March 5 A5

6 Zhang & Zhang/An Equilibrium Model of Inerne-Faciliaed Feedback Trading The hird erm B ββ ( ) E ( P) d ( P) d ββ ( ) E ( A A A3) ( P) d d E ( A A A ) d d ββ 3 ββ ( ) ββ ( 3) ( ) E d E A A A dd ( ) ( ) ( ) 7 ββ E d E d d d d 3 ( ) β β E d E d d ( ) 7 dd 3 3β β ( ) E ( A A A3) d E( A A A3) dd E( A A A3) B3 ( ) E ( P) d β ( ) E ( P) d d β β E P d ( ) 3 ( β ) Here we hae ued he iomeric propery of he ochaic inegral. The uninformed rader ariance of profi i, herefore, ( ) β 3β ( β ) ( β ) ( ) B B B3 β Reference Liper, R., and Shiryae, A Saiic of Random Procee, Berlin: Springer-Verlag. Appendix B Calculaion of Expecaion Here we how how o calculae ome expecaion ueful for Appendix A. Fir, we calculae E d : A6 MIS Quarerly Vol. 39 No. Appendice/March 5

7 Zhang & Zhang/An Equilibrium Model of Inerne-Faciliaed Feedback Trading E d E d d E dd [ ] min, dd d d d d 3 In he following, we calculae Γ E d d dd ( ) Le I X Y d ( ) I d ( ) I d where I and Y are maringale. ha we wan i X Y Inegraing by par, XY XdY YdX Since Y i a maringale, Inegraing by par, we hae [ ] EXY E YdX E Y ( ) I d ( ) EYI [ ] d where <X, Y> denoe he quadraic ariaion proce of X and Y. herefore YI I dy YdI d YI < >, di IdI d< I, I> IdI ( ) d MIS Quarerly Vol. 39 No. Appendice/March 5 A7

8 Zhang & Zhang/An Equilibrium Model of Inerne-Faciliaed Feedback Trading EYI [ ] E Y ( ) [ ] EI d d ( ) I I d ( ) d d d ( ) ln Hence we obain [ ] [ ] EXY EYI d x ln( x) dx ( ) d 6 Γ 3 Nex, we calculae E ( ). Uing he ame noaion a aboe, wha we wan i. Inegraing by par, d d EX [ ] Hence, Inegraing by par, we hae [ ] X X XdX EX E XdX E X ( ) I d d ( ) where <X, Y> denoe he quadraic ariaion proce of X and Y. EXI [ ] XI I dx X di d X I <, > Id XdI di I di d < I, I > I di ( ) d herefore EXI [ ] E ( Id ) X ( ) ( EI ) [ ] d EX [ ] ( ) 3 ( ) ln ln( ) d d ( ) d ( ) ( ) ( ln( ) ) 3 d A8 MIS Quarerly Vol. 39 No. Appendice/March 5

9 Zhang & Zhang/An Equilibrium Model of Inerne-Faciliaed Feedback Trading 3 Here we hae ued he reul ha,, hen EI and EI. I N [ ] [ ] [ ] [ ] EX EXI d [ ] 5xln x dx5 d d d 3 d Appendix C Proof of Theorem 7 Proof of Theorem 7. Inering he uninformed rader new demand Equaion () o he pricing rule Equaion (7), we can obain ( γ ) λ dp λγ ( λ λ ε) P d d The logic of deriing he reul i he ame a before. Howeer, ince we hae one addiional dimenion of uncerainy coming from g, he filering proce need o deal wih he ecor. ε Conequenly, he deiaion of he price from he liquidaion alue a ime, (), i a marix () () () () () () ( ) () () ( )( ( )) [ ] wih E E F, ε ε E E F E F, and E ( ε) E ( ε) F. The ariance-coariance marix can be deried a [ ] [ ] d () () () () () () () () γ γ γ () () () () d (3) where he ymbol N denoe he ranpoe of he marix. Equaion (3) i a marix Riccai differenial equaion wih iniial alue The oluion o he equaion i () () () () () ε () ( () ) γ d γ γ d d d (3) Afer calculaion, we obain ha ( () ) ε ε ε ε MIS Quarerly Vol. 39 No. Appendice/March 5 A9

10 Zhang & Zhang/An Equilibrium Model of Inerne-Faciliaed Feedback Trading and () ε γ d γ γ d d d ε ε ε γ d ε () () γ d γ ε d d ε ε () The informed rader expeced profi a ime i ε d γ γ d d d ε ε ε ( ) ( I ) ( ) () d [ ] E P dx E P d E P d Same a before, he informed rader chooe o maximize her expeced profi. Tha i, () * arg max d Same a () in he baeline model, () doe no inole he feedback parameer β. So he informed rader maximizaion problem i independen of he feedback ineniy. Similarly, he uninformed rader wih imprecie informaion rie o maximize her profi a ime. The maximizaion problem i () γ * arg max γ d Since () doe no inole he feedback parameer β, he opimal γ i alo independen from he feedback ineniy. For he condiional expecaion, we hae d E EF [ ] [ ε F ] () d () γ [ ] γ γ γ E[ F ] ( ε) [ ε ] E F d Applying he emi-rong marke-efficiency condiion EF P, we hae for all and γ γ () () λ (33) (3) * * Inering he opimal alue and γ, we ge he reul abou λ. λ * * γ * * β γ (35) Thi reul i highly conien wih wha we hae obained in Theorem 3. The expreion of λ i ery imilar o ha in he baeline model. The only difference i ha he ariance of he liquidaion alue in he baeline model i replaced by he ariance and coariance of he liquidaion alue ogeher wih he error. Oerall, hi complee he proof ha, een if uninformed rader can obain imprecie ignal of informaion, feedback rading doe no affec informed rader raegy nor he marke price proce. A MIS Quarerly Vol. 39 No. Appendice/March 5

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