Dynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, Abstract

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1 Sock Marke Insider Trading in Coninuous Time wih Imperfec Dynamic Informaion Albina Danilova Deparmen of Mahemaical Sciences Carnegie Mellon Universiy Sepember 6, 28 Absrac This paper sudies he equilibrium pricing of asse shares in he presence of dynamic privae informaion. The marke consiss of a risk-neural informed agen who observes he firm value, noise raders, and compeiive marke makers who se share prices using he oal order flow as a noisy signal of he insider s informaion. I provide a characerizaion of all opimal sraegies, and prove exisence of boh Markovian and non Markovian equilibria by deriving closed form soluions for he opimal order process of he informed rader and he opimal pricing rule of he marke maker. The consideraion of non Markovian equilibrium is relevan since he marke maker migh decide o re-weigh pas informaion afer receiving a new signal. Also, I show ha a here is a unique Markovian equilibrium price process which allows he insider o rade undeeced, and ha b he presence of an insider increases he marke informaional efficiency, in paricular for imes close o dividend paymen. I benefied from helpful commens from Peer Bank, Rene Carmona, Chrisian Julliard, Dmiry Kramkov, Michael Monoyios, Andrew Ng, Bern Øksendal and seminar and workshop paricipans a 4h Oxford - Princeon Workshop, 4h Mahemaics and Economics Workshop Universiy of Oslo, Sochasic Filering and Conrol Workshop Warwick Universiy and Warwick Business School. Deparmen of Mahemaical Sciences, Carnegie Mellon Universiy, Pisburgh, PA , USA, danilova@andrew.cmu.edu

2 Inroducion Alhough financial markes wih informaional asymmeries have been widely discussed in he marke microsrucure lieraure see O Hara 995 and Brunnermeier 2 for a review, he characerizaion of he opimal rading sraegy of an invesor who posses superior informaion has been, unil laely, largely unaddressed by he mahemaical finance lieraure. In recen years, wih he developmen of enlargemen of filraions heory see Mansuy and Yor 26, models of so called insider rading have been gaining aenion in mahemaical finance as well see e.g. Karazas and Pikovsky 996, Biagini and Oksendal 25, Ankirchner and Imkeller 26. The salien assumpions of hese models are ha i he informaional advanage of he insider is a funcional of he sock price process e.g. he insider migh know in advance he maximum value he sock price will achieve, and ha ii he insider does no affec he sock price dynamics. Bu in fac, since equilibrium sock prices should clear he marke, and hus depend on he fuure random demand of marke paricipans, assuming ha he informaional advanage of he insider is a funcional of he price process implies ha she eiher knows he fuure demand processes of all marke paricipans, or she knows ha he price will exogenously converge o a fundamenal ha is known o her. Since he assumpion of an omniscien insider is unrealisic, one would have o assume he laer. Neverheless, since he presence of an insider by assumpion in hese models does no affec he price process, his raises he quesion of wha makes he price converge o is fundamenal value wihou informaion being released o he marke. Thus, from he marke microsrucure poin of view, hese modelling assumpions ranslae ino i imposing srong efficiency of he markes even wihou an insider providing, hrough her rading, informaion o he marke ha is, assuming a priori ha he price will converge o he fundamenal value and ha ii he less informed agens are no fully raional, since hey do no ry o infer he insider s privae signal from marke daa since here is no feedback from insider rading o equilibrium price. Par of he mahemaical finance lieraure has ried o address hese shorcomings by considering he informaional conen of sock prices, and opimal informaion-based rading, in a 2

3 raional expecaions equilibrium framework see e.g. Back 992, Cho 23. In hese models o preserve racabiliy he privae informaion of he insider has been generally assumed o be saic. For example, in Back 992 and in Cho 23 he insider knows ex ane he final value of he firm, and in Campi and Cein 27 she knows ex ane he ime of defaul of he company issuing he asse. This lieraure has shown ha i he presence of an insider on he marke does no necessarily lead o arbirage i.e. he value funcion of he insider is finie, and ha ii he presence of insiders migh be considered beneficial o he marke, in he sense ha i leads o higher informaion efficiency of he equilibrium price process. Neverheless, he assumpion of insider s perfec foresigh is unrealisic, since he fundamenal value of he firm should be conneced o elemens like fuure cash-flows, produciviy, sales ec. ha have inrinsically an aleaory componen. Tha is, a more naural assumpion would be ha he fundamenal value is in iself a sochasic process, and ha he insider can observe i direcly or a leas observe i in a less noisy way han he oher agens on he marke. Thus, in his paper I relax he assumpion of saic insider informaion, and sudy he equilibrium rading and price processes, as well as marke efficiency, in a seing wih dynamic privae informaion. The model I consider in his paper is a generalizaion of he saic informaion seing of Back 992. An earlier aemp o generalize his framework o include dynamic informaion is in Back and Pedersen 998. This laer paper considers a much smaller se of admissible rading sraegies and pricing rules, and has much more sringen assumpions on he parameers, han he ones considered in my work. Moreover, i shows he exisence of one possible Markovian equilibrium, while my work characerizes all opimal sraegies and esablishes ha here is a unique Markovian inconspicuous equilibrium price process, i.e. an equilibrium price ha allows he insider o rade undeeced and depends only on he oal order process. Moreover, I idenify his Markovian equilibrium in closed form, and show ha he presence of an insider increases he marke informaional efficiency for imes close o dividend paymen. Furhermore, I show ha even when he marke parameers do no saisfy he condiions for he exisence of a Markovian equilibrium, here exiss a non Markovian inconspicuous equilibrium which I also idenify in closed 3

4 form. Addiionally, I give characerizaion of all opimal rading sraegies for he equilibrium price process. I show, based on his characerizaion, ha in he case of non Markovian price process i is opimal for he insider o reveal her privae informaion no only a he erminal ime, bu also a some predefined inerim imes hus bringing he marke o higher efficiency han in he case of Markovian price process. The remainder of he paper is organized as follows. Secion 2 presens he model and he assumpions. Exisence of Markovian equilibrium, and uniqueness of he inconspicuous Markovian equilibrium price process, are proved in Secion 3. Exisence of equilibrium for more general pricing funcionals is demonsraed in he Secion 4. Secion 5 concludes. 2 The Model Seup Consider a sock issued by a company wih fundamenal value given by he process Z, defined on Ω, F, F, P, and saisfying Z = v + σ z sdb s where B is a sandard Brownian moion on F, v is N, σ independen of F B σ z s a is deerminisic funcion. for any, and Then, if he firm value is observable, he fair sock price should be a funcion of Z and. However, he assumpion of he company value being discernable by he whole marke in coninuous ime is counerfacual, and i will be more realisic o assume ha his informaion is revealed o he marke only a given ime inervals such as dividend paymens imes or when balance shees are publicized. In his model I herefore assume, wihou loss of generaliy, ha he ime of he nex informaion release is =, and he marke erminaes afer ha. Hence, in his seing he sock can be viewed as a European opion on he firm value wih mauriy T = and payoff fz. In addiion o his risky asse, here is a riskless asse ha yields an ineres rae normalized o zero for simpliciy This is wihou loss of generaliy, since he exension o muliple informaion release imes is sraighforward. 4

5 of exposiion. In wha follows i is assumed ha all random variables are defined on he same sochasic basis Ω, F, F, P. The microsrucure of he marke, and he ineracion of marke paricipans, is modelled as a generalizaion of Back 992. There are hree ypes of agens: noisy/liquidiy raders, an informed rader insider, and compeiive marke makers, all of whom are risk neural. The agens differ in heir informaion ses, and objecives, as follows. Noisy/liquidiy raders rade for liquidiy reasons, and heir oal demand a ime is given by a sandard Brownian moion B 2 independen of B and v. Marke makers observe only he oal marke order process Y = θ + B 2, where θ is he oal order of he insider, i.e. heir filraion is F M = F Y. Since hey are compeiive and risk neural, on he basis of he observed informaion hey se he price as P Y [,], = P = E [ fz F M ]. 2. As in Cho 23, I assume ha marke makers se he price as a funcion of weighed oal order process a ime, i.e. I consider pricing funcionals P Y [,], of he following form P Y [,], = H wsdy s,. where ws is some posiive deerminisic funcion. The informed invesor observes he price process P = H wsdy s, and he rue firm value Z, i.e. her filraion is given by F I o maximize he expeced final wealh, i.e. = Z,P F. Since she is risk-neural, her objecive is [ ] [ sup E X θ = sup E fz P θ + θ AH,w θ AH,w θ s dp s ] 2.2 where AH, w is he se of admissible rading sraegies for he given price funcional H wsdy s, which will be defined laer. Tha is, he insider maximizes he expeced 5

6 value of her final wealh X θ, where he firs erm on he righ hand side of equaion 2.2 is he conribuion o he final wealh due o a poenial differenial beween price and fundamenal a he ime of informaion release, and he second erm is he conribuion o final wealh coming from he rading aciviy. Noe ha seing σ z, he resuling marke would be he saic informaion one considered by Back 992. Noe also ha he above marke srucure implies ha he insider s opimal rading sraegy akes ino accoun he feedback effec i.e. he ha prices reac o her rading sraegy according o equaion 2.. Idenifying he opimal insider s sraegy is equivalen o he problem of finding he raional expecaions equilibrium of his marke, i.e. a pair consising of an admissible price funcional and an admissible rading sraegy such ha: a given he price funcional he rading sraegy is opimal, and b given he rading sraegy he price funcional saisfies 2.. To formalize his definiion, we firs need o define he ses of admissible pricing rules and rading sraegies. Alhough i is sandard in he insider rading lieraure o limi he se of admissible sraegies o absoluely coninuous ones, in wha follows I consider a much broader class of sraegies given by he se of semimaringales saisfying some sandard echnical condiions ha eliminae doubling sraegies. The formal definiion of he se of admissible rading sraegies is summarized in he following definiion. Definiion 2. An insider s rading sraegy, θ, is admissible for a given pricing rule Hy,, w θ AH, w if θ is F I adaped semimaringale, and no doubling sraegies are allowed i.e. [ E H 2 wsdθ s + ] wsdbs 2, d <. 2.3 Moreover, we call he insider s rading sraegy inconspicuous if Y = θ + B 2 is a Brownian moion on is own filraion F Y since in his case he presence of he insider is undeecable. Remark 2. An equilibrium in which he opimal insider s rading sraegy is inconspicuous is a 6

7 desirable feaure of any insider rading model, and I will show ha in his seing such an equilibrium exiss. In fac, given he poenially high cos associaed wih being idenified as an insider, i migh be reasonable o consider only his ype of equilibrium. The definiion of admissible pricing rules is a generalizaion of he one in Back wih addiional regulariy condiion 5 below which insures ha, given he marke maker s filraion, he oal order process has finie variance. This generalizaion allows he marke maker o re-weigh her pas informaion. Definiion 2.2 A pair of measurable funcions, H C 2, R [, ], H : R [, ] R and w : [, ] R + \{}, is an admissible pricing rule H, w H if and only if:. The weighing funcion, w, is a piecewise posiive consan funcion given by w = n σy i { i, i ]} 2.4 i= where = < <... < n = and n i= σi y 2 =. This condiion doesn cause loss of generaliy because: a i was shown by Cho 23, in he saic privae informaion case, ha in he equilibrium w = and b i is always possible o re-scale w o have n i= σi y 2 =. [ 2. E H2 ] wsdb2 s, d <. [ 3. E H 2 ] wsdb2 s, <. The wo condiions above, ogeher wih equaion2.3, rule ou doubling sraegies. 4. y Hy, is increasing for each fixed, ha is he price increases if he sock demand increases. 5. E [ h [ i E fz F Z ] ] 2 i < where h i is he inverse of Hy, i. 2 Seing ws will make condiions 2-4 exacly he same as in Back 992 7

8 Moreover, H is a raional pricing rule if, for a given θ, i saisfies H wsdy s, = E [ fz F M ]. Remark 2.2 Due o condiion 4 on he admissible pricing rules, he insider can infer he oal order process from he price process by invering H wsdy s, = P. Therefore, since I will be considering raional expecaions equilibria, and because ws is sricly posiive, she can infer he oal order process Y and, since she knows her own oal order process θ, she can deduce B 2 = Y θ from i. As a consequence, he filraion of he insider can be wrien as F I = F B2,Z = F B2,B σv, where σv is he sigma algebra generaed by he random variable v. Given hese definiions of admissible pricing rules and rading sraegies, i is now possible o formally define he marke equilibrium as follows. Definiion 2.3 A pair H, w, θ is an equilibrium if H, w is an admissible pricing rule, θ is admissible sraegy, and:. Given θ, H, w is a raional pricing rule, i.e. i saisfies H wsdy s, = E [ fz F M ]. 2. Given H, w, θ solves he opimizaion problem [ sup θ AH,w E fz P θ + θ s dp s ] Moreover, a pricing rule H y,, w is an inconspicuous equilibrium pricing rule if here exiss an inconspicuous insider rading sraegy θ such ha H, w, θ is an equilibrium. Addiionally, o define a well behaved problem I impose he following echnical condiions on he model parameers. 8

9 Assumpion 2. The fundamenal value of he risky sock, F z,, given by F Z, = E [ fz F Z ] 2.5 is well defined and is a square inegrable maringale, i.e. E [ f 2 Z ] <, 2.6 and f. is an increasing funcion. Assumpion 2.2 The variance of he firm value, Σ z = σ2 zsds, is finie for any. Remark 2.3 Since he final payoff of he sock is given by fz, he above assumpion implies ha i is always possible o redefine he funcion f so ha σ 2 = Σ z. 2.7 In wha follows, I will always assume ha his equaliy holds. 3 The Markovian Equilibrium In his secion I address he problem of exisence and uniqueness of an equilibrium given by Definiion 2.3 in he case of Markovian pricing rule i.e. I consider w. Before saing he main resul of his secion, I need o impose addiional condiions on he model o insure ha he problem is well-posed. Assumpion 3. For any [, we have Σz s + σ 2 s 2 ds < 3. and eiher Σz s + σ 2 s 2 ds < 3.2 9

10 or lim Σ z s + σ 2 ds = 3.3 s The above assumpion is needed for he filering problem of he marke maker o be well defined. Assumpion 3.2 There exiss a [, such ha > σ 2 zsds 3.4 for any and σ z is coninuous on [, ]. Moreover, for all [, ] we have Σ z + σ This assumpion insures ha: a close o he marke erminal ime, he insider s signal is more [ Z precise han he marke maker s i.e. E E [ Z F M ] ] [ 2 Z F M > E E [ Z F I ] ] 2 F I, and b ha he insider s signal is always a leas as precise as he marke maker s. Remark 3. Noice ha Assumpions 2.2, 3. and 3.2 guaranee ha when condiion 3.2 is no saisfied { } λ = exp Σ z s + σ 2 s ds, and ha if Ξ = +σzs 2 ds, hen λ 2 s lim λ2 Ξ log log Ξ =. 3.6 Furhermore, assumpion 3.2 can be relaxed by replacing 3.4 wih condiion 3.6. Proof. See Appendix A Now we are in he posiion o sae he main resul of his secion which is summarized in he nex heorem.

11 Theorem 3. Suppose ha Assumpions 2., 2.2, 3. and 3.2 are saisfied. Then he pair H, θ, where H saisfies H y, + 2 H yyy, = 3.7 Hy, = fy, 3.8 i.e. H y, = E [ fy + B 2 B2 ] and θ = Z s Y s ds, 3.9 Σ z s s + σ2 is an equilibrium. Moreover, he pricing rule H is he unique inconspicuous equilibrium pricing rule in H. Furhermore, given his pricing rule H, he rading sraegy θ is opimal in AH for he insider if and only if. The process θ is coninuous and has bounded variaion. 2. The oal order, Y = θ + B 2, saisfies Y = Z. Therefore, when he parameers of he marke saisfy he saed assumpions, here exiss a unique Markovian pricing rule such ha: a a leas one opimal rading sraegy of he insider, given by 3.9, is increasing marke efficiency during all rading periods since he insider pushes he price o he fundamenal value of he sock, b due o, he variance of he risky asse is no influenced by insider s rading if she rades opimally, and c he insider presence increases marke efficiency close o he marke erminaion ime due o 2. I will prove his heorem in hree proposiions ha focus on differen aspecs of he equilibrium. In paricular, he proposiions will address: a characerizaion of he opimal insider rading sraegy, b exisence of he equilibrium, and c uniqueness of he inconspicuous pricing rule. The conclusion of Theorem 3. is driven by he following resul: for any pricing rule in H saisfying equaion 3.7, here exiss a finie upper bound on he informed agen s value funcion which is aained by a rading sraegy which is no deecable by he marke maker, no locally correlaed wih noisy rades, and such ha all he privae informaion is revealed only a ime =.

12 Thus, his resul gives he characerizaion of he opimal insider s rading sraegy in a slighly more general form han saed in Theorem 3.. This is summarized in he following proposiion. Proposiion 3. Suppose ha Assumpions 2., 2.2, 3. and 3.2 are saisfied. Then, given an admissible pricing rule H H saisfying he parial differenial equaion PDE 3.7, an admissible rading sraegy θ AH is opimal for he insider if and only if:. The process θ is coninuous and has bounded variaion. 2. The oal order, Y = θ + B 2, saisfies h Y = HY, = f Z, 3. where hy = Hy, and fz is he final payoff of he asse. Proof. Sufficiency For any admissible rading sraegy, by using inegraion by pars for semimaringales Proer 24, Corollary II.6.2, p. 68, we have [ ] [ E X θ = E F Z s, s HY s, sdθ s + θ s df Z s, s + [θ, F Z, HY, ] ]. By applying Iô formula for semimaringales Proer 24, Theorem II.6.33, p. 8 o Hy, and F z,, and using he fac ha F Z, is a rue maringale, we ge F Z, = F z, + HY, = H, F z Z s, sdz s H y Y s, sdy s + H yy Y s, sd [Y ] s + s H Y s, sds [ HY s, s H y Y s, s Y s ]. Since Y = θ + B 2, we have ha [Y ] = + θ c + 2 θ c, B 2 + s θ s 2. Therefore, using 2

13 he fac ha Hy, saisfies equaion 3.7, we have ha HY, = H, + + H y Y s, sdy c s + 2 H yy Y s, sd θ c, B 2 s + s HY s, s. H yy Y s, sd θ c s Therefore, by Theorem 26.6 of Kallenberg 2, and Theorem II.6.29 of Proer 24, we have noice ha Z s and B 2 are coninuous [θ, F Z, ] = [θ, HY, ] = F z Z s, sd [θ c, Z] s H y Y s, sd [θ c ] s + H y Y s, sd [ θ c, B 2] s + s HY s, s θ s. On he oher hand, consider a funcion Jy, z = y z y fz Hx, dx, where y z is he soluion of Hy z, = fz. Le [ V y, z, = E J y + B 2 B 2, z + σ z sdb s ]. 3. This funcion is well defined i is easy o check ha E [ JB 2, Z ] < and saisfies he parial differenial equaion V y, z, + 2 V yyy, z, + σ2 z 2 V zzy, z, = 3.2 wih erminal condiion V y, z, = Jy, z. Therefore V y, z, V y z, z, = for any fixed z and any y y z. Moreover, since Hy, is a nondecreasing coninuous funcion of 3

14 y, we can use he monoone convergence heorem o obain V y +, z, V y, z, lim + [ y+b 2 E B 2 fz + ] y+ +B = lim 2 B2 σ zsdbs Hx, dx + = F z, E lim + y+b 2 B 2 y+ +B 2 B2 Hx, dx = E [ Hy + B 2 B 2, ] F z,. Thus, due o he definiion of an admissible pricing rule, we have V y +, z, V y, z, lim + F z, Hy, = The same argumen can be applied o he lef derivaive of V wih respec o y o obain V y y, z, + F z, Hy, =. 3.4 As a consequence, we can express E [ X θ ] in erms of V as noice ha B2 df Z, is a maringale [ ] E X θ [ = E V y Y s, Z s, sdθ s V yy Y s, Z s, sd [θ c ] s V z Y s, Z s, sdz s V zy Y s, Z s, sd [θ c, Z] s V yy Y s, Z s, sd [ θ c, B 2] s V y Y s, Z s, s θ s. s On he oher hand, by applying he Iô formula for semimaringales o V direcly Proer 24, Theorem II.6.33, p. 8 we ge [ E [V Y, Z, ] = E V, Z, X θ + 2 V yy Y s, Z s, sd [θ c ] s + s V y Y s, Z s, sdb 2 s [ V Y s, Z s, s V y Y s, Z s, s Y s ]. Noice ha, due o he definiion of he fundamenal value F and of admissible pricing rule 4

15 [ ] H, we have E V zy s, Z s, sdbs 2 =. Therefore [ ] E X θ [ = E V, Z, V Y, Z, 2 V yy Y s, Z s, sd [θ c ] s + s [ V Y s, Z s, s V y Y s, Z s, s Y s ]. Moreover, due o he properies of V we have V Y s, Z s, s V y Y s, Z s, s Y s, 3.5 s V yy Y s, Z s, s d [θ c ] 2 s, 3.6 V, Y, Z V, y Z, Z. 3.7 The above inequaliies become equaliies if and only if he following condiions hold: θ = for equaion 3.5; [θ c ] = for equaion 3.6; HY, = f Z for equaion 3.7. Therefore, for any funcion V saisfying equaions 3.2, 3.4 and he final condiion given by V y, z, V y z, z, = for every z and any y y z where y z is he soluion of Hy z, = fz, we have ha [ ] E X θ V, Z,. This expression holds wih equaliy if and only if θ is coninuous and condiion 3. is saisfied. Hence, if θ is such ha hese condiions are saisfied, hen i is opimal. Necessiy Consider he coninuous maringale given by X = GZ, = E [ h fz F I ]. This maringale is well defined since H is an admissible pricing rule. Consider θ = X s Y s s ds. In his case, we can solve he sochasic differenial equaion for 5

16 Y o ge Y = X v + s dx s s db2 s. Noice ha Y is coninuous, herefore θ has bounded variaion almos surely. Moreover, HY, = fz almos surely, hence his choice of θ gives [ ] E X θ = V, Z,. Since, by he sufficiency proof, we have ha for any θ which is eiher no coninuous or does no saisfy equaion 3. ] [ ] E [X θ < V, Z, = E X θ, we know ha any such θ is no opimal. From his characerizaion resul, i follows ha he θ given by 3.9 is an opimal insider rading sraegy given an admissible pricing rule H saisfying 3.7 and 3.8. Esablishing he raionaliy of he pricing rule H, on he oher hand, is no so direc. Therefore, o se up he sage for proving ha he H, θ given in Theorem 3. is indeed an equilibrium, we firs need o demonsrae he following lemma. Lemma 3. Consider he process Y saisfying he sochasic differenial equaion dy s = Z s Y s Σ z s s + σ 2 ds + db2 s, wih Z = v + σ z sdb s, where B and B 2 are wo independen sandard Brownian moions, v is N, σ independen of F B,B 2 and Σ z = σ2 zsds. Suppose ha σ, σ z and Σ z saisfy Assumpions 2., 2.2, 3. and 3.2. Then, on he filraion F Y, he process Y is a sandard Brownian moion and Y = Z. 6

17 Proof. Fix any T [,. From Theorem.3 of Lipser and Shiryaev 2 noe ha, due o Assumpion 3., he condiions of he heorem are saisfied, we have ha on he filraion F Y he sochasic differenial equaion for Y is T dy s = m s Y s Σ z s s + σ 2 ds + dby s, wih where B Y dm s = γ s Σ z s s + σ 2 dby s, is Brownian moion on F Y, and γ s saisfies he following ordinary differenial equaion ODE γ s = σzs 2 γs 2 Σ z s s + σ 2 2 wih iniial condiion γ = σ 2. Noice ha γ s = Σ z s s + σ 2 is he unique soluion of his ODE and iniial condiion. Therefore on F Y, he process Y saisfies T dy s = Bs Y Y s Σ z s s + σ 2 ds + dby s. The unique srong soluion of his sochasic differenial equaion on [, T ] is Y s = B Y s see Karazas and Shreve 99, Example Hence, on he inerval [,, he process Y is a Brownian moion on is own compleed filraion. By coninuiy of Y, his process is a Brownian moion on [, ]. To prove ha Y = Z, noice ha Y { where λ = exp Σ zs+σ }. 2 s ds Noe ha a random variable variance σ z s = Z + λ v + λs db2 s λs db s λs db2 s σ zs λs db s is normally disribued wih mean and +σzs 2 ds. Therefore, due o he Assumpion 3. and in paricular condiion 3.3, if λ 2 s +σz lim 2s ds <, hen Y λ 2 s = Z. 7

18 On he oher hand, if lim X = and a change of ime τ given by +σz 2s ds =, consider he process λ 2 s σ z s λs db2 s λs db s, τ + σzs 2 λ 2 ds =. s Then, W s = X τs is a Brownian moion. Hence, we can use he law of ieraed logarihm o ge lim sup s lim inf s W s 2s log log s = W s 2s log log s = or, in he original ime, lim sup lim inf X 2Ξ log logξ = X 2Ξ log logξ = where Ξ = +σz 2s ds. Since, due o he Assumpions 2.2, 3. and 3.2, we have λ 2 s lim λ2 Ξ log log Ξ =, i follows ha lim λx =, herefore Y = Z. Wih his lemma a hand, esablishing ha he pair H, θ given in he Theorem 3. is indeed an equilibrium is sraighforward, as he following proposiion demonsraes. Proposiion 3.2 Suppose ha Assumpions 2., 2.2, 3. and 3.2 are saisfied. Then he pair H, θ, where H y, saisfies he parial differenial equaion PDE 3.7 wih erminal condiion 3.8, and he process θ is given by 3.9, is an equilibrium. 8

19 Proof. Due o Proposiion 3., he θ defined by 3.9 is he opimal rading sraegy given he admissible pricing rule H y, which saisfies equaion 3.7 and 3.8, if and only if: a θ is coninuous wih bounded variaion, and b Y = θ + B 2 saisfies Y = Z. Due o Lemma 3., we have ha θ is coninuous wih bounded variaion, and Y = Z. Therefore θ is an opimal rading sraegy given he pricing rule H y,. On he oher hand, due o he Lemma 3., for θ given by 3.9, Y is a Brownian moion wih Y = Z. Therefore, he raional pricing rule given θ should be Hy, = E [ fy + B 2 B 2 ]. This pricing rule saisfies he PDE 3.7 wih erminal condiion 3.8. Therefore, H y, = Hy, is a raional pricing rule. Hence, he pair H, θ given in his proposiion is an equilibrium. To complee he proof of Theorem 3., we need o show uniqueness of he inconspicuous pricing rule in H. Proposiion 3.3 The pricing rule H y, which saisfies he PDE 3.7, wih erminal condiion 3.8, is he unique inconspicuous pricing rule. Proof. From Proposiion 3.3 and Lemma 3., i direcly follows ha H y, saisfying he PDE 3.7 wih erminal condiion 3.8 is an inconspicuous equilibrium pricing rule. To prove uniqueness, consider some equilibrium inconspicuous pricing rule H. By definiion, here exiss a rading sraegy θ AH such ha he H, θ is an equilibrium, and he oal order process Y = θ + B 2 is a Brownian moion on F M. By he definiion of equilibrium, HY, = E [ fz F M ] = E [ HY, F M ]. Since Y is a Brownian moion on F M, and given he definiion of admissible pricing rule, H mus saisfy he PDE 3.7 wih erminal condiion Hy, = hy, for some nondecreasing funcion h wih E [ h 2 Y ] <. Hence, o show uniqueness of H we need o demonsrae ha h = f almos everywhere. Due o Proposiion 3., i follows from he opimaliy of θ ha fz = hy and, 9

20 since θ is inconspicuous, Y N,. Since Z N, by definiion, one can have fz = hy if and only if f = h almos everywhere, hence H is indeed a unique inconspicuous pricing rule. 4 Non Markovian equilibrium In his secion I address he problem of exisence of an equilibrium given by Definiion 2.3 in he more general case of non Markovian pricing rule, i.e. I consider general weighing funcions w saisfying Definiion 2.2 hus allowing he marke maker o assign differen weighs o he informaion she receives. As in he case of Markovian pricing rule, he exisence of an equilibrium resul is driven by he exisence of a finie upper bound on he informed agen s value funcion, and he characerizaion of he rading sraegies which aain i. This characerizaion is summarized in he following proposiion. Proposiion 4. Suppose ha Assumpions 2. and 2.2 are saisfied. Then, given an admissible pricing rule w defined by 2.4 wih σ i y < σ i+ y parial differenial equaion for any i and H, w H wih H saisfying he H y, + w2 H yy y, = 4. 2 an admissible rading sraegy θ AH, w is opimal for insider if and only if:. The process θ is coninuous and has bounded variaion. 2. The weighed oal order, ξ = wsdθ s + wsdb2 s saisfies h i ξ i = Hξ i, i = F Z i, i. 4.2 Therefore, as in he case of Markovian pricing rule, he opimal sraegy of he insider does no aler quadraic variaion of oal order process, does no add jumps o i and is uncorrelaed wih i. Bu, differenly from he Markovian case, i follows from 4.2 ha in his seing i is opimal for 2

21 he insider o reveal her informaion no only a he marke erminal ime, bu also in he inerim imes whenever he marke maker changes her weighing funcion. Proof. Sufficiency As in he proof of Proposiion 3., for any admissible rading sraegy we have [ ] [ E X θ = E F Z s, s Hξ s, sdθ s + H ξ ξ s, swsd [θ c ] s θ s df Z s, s + F z Z s, sd [θ c, Z] s H ξ ξ s, swsd [ θ c, B 2] s Hξ s, s θ s. s On he oher hand, consider he funcions J i ξ, z = ξ z ξ F z, i Hx, i dx, where ξ i z is he soluion of Hξ i z, i = F z, i. For i le i i ] V i ξ, z, = E [J i ξ + wsdbs 2, z + σ z sdbs. [ These funcions are well defined i is easy o check ha E J i ] i wsdb2 s, Z i < and saisfy he parial differenial equaion V i ξ, z, + w2 s Vξξ i 2 ξ, z, + σ2 z 2 V zzξ, i z, = wih erminal condiion V i ξ, z, i = J i ξ, z. Therefore, V i ξ, z, i V ξ i z, z, i = for any fixed z and any ξ ξi z. Moreover, since Hξ, is a nondecreasing coninuous funcion of ξ, and due o he definiion of an admissible pricing rule, we have Vξ i ξ, z, + F z, Hξ, =. 2

22 Define he funcion V as V ξ, z, = i<n: i σ i y σy i+ V i ξ, z, + σy n V n ξ, z, Noice ha, due o he properies of he funcions V i, we have ha V is well defined and saisfies he parial differenial equaion wih condiions V ξ, z, i = σ n y J n ξ, z. Moreover, we have V ξ, z, + w2 s V ξξ ξ, z, + σ2 z 2 2 V zzξ, z, =, 4.3 σ i y J i ξ, z + V ξ, z, σy i+ i + if i < n and V ξ, z, n = V ξ ξ, z, + F z, Hξ, w =. 4.4 As a consequence, we can express E [ X θ is a maringale ] u in erms of V as noice ha wsdb2 s df Z u, u [ ] E X θ [ = E V ξ ξ s, Z s, swsdθ s V zξ ξ s, Z s, swsd [θ c, Z] s V z ξ s, Z s, sdz s V ξξ ξ s, Z s, sw 2 sd [θ c ] s V ξξ ξ s, Z s, sw 2 sd [ θ c, B 2] s wsv ξ ξ s, Z s, s θ s. s On he oher hand, by applying he Iô formula for semimaringales o V direcly Proer 24, Theorem II.6.33, p. 8 and removing maringale erms we ge [ ] E X θ [ n = E V, Z, 2 i= σ i y V ξξ ξ s, Z s, sw 2 sd [θ c ] s + s σy i+ J i ξ i, Z i, i σy n J n ξ i, Z i, i [ V ξ s, Z s, s V ξ ξ s, Z s, s ξ s ]. 22

23 Moreover, due o he properies of V we have V ξ s, Z s, s V ξ ξ s, Z s, s ξ s, 4.5 s V ξξ ξ s, Z s, sw 2 s d [θ c ] 2 s, 4.6 J i i, ξ i, Z i. 4.7 The above inequaliies become equaliies if and only if he following condiions hold: θ = for equaion 4.5; [θ c ] = for equaion 4.6; Hξ i, i = F Z i, i for equaions 4.7. Therefore, we have ha [ ] E X θ V, Z,. This expression holds wih equaliy if and only if θ is coninuous wih bounded variaion and condiion 4.2 is saisfied. Hence, if θ is such ha hese condiions are saisfied, hen i is opimal. Necessiy Consider he process given by X = GZ, = n i= E [ h i F Z i, i F I ] { i, i ]} wih X = E [ h F Z, F I ] where h i is inverse of Hy, i. This process is well defined since H is an admissible pricing rule. Consider he rading sraegy given by θ = and dθ = n X ξ i= σy i i s { i, i ]}d. In his case, we can solve he sochasic differenial equaion for ξ on each inerval [ i, i ] o ge ξ = X i Xi ξ i + i i i i s dx s i σi y i s db2 s. Noice ha ξ is finie almos surely, herefore θ has bounded variaion almos surely. More- 23

24 over, Hξ i, = F Z i, i almos surely, hence his choice of θ gives [ ] E X θ = V, Z,. Since, by he sufficiency proof, we have ha for any θ which is eiher no coninuous or does no saisfy equaion 3. ] [ ] E [X θ < V, Z, = E X θ, we know ha any such θ is no opimal. From his characerizaion follows he exisence of equilibrium resul, as he nex heorem demonsraes. Theorem 4. Suppose ha σ z and σ are such ha here exiss a piecewise consan funcion g = n i= α i { i, i ]} wih = <... < n =, < α i < α i+ for any i and n i= α2 i =, saisfying he following condiions: Σ z + σ 2 Σ z i + σ 2 i g 2 sds > for all [, ]\{ i } n i=, 4.8 g 2 sds = for all i, 4.9 i Σz s + σ 2 s g2 udu 2 ds < for all [ i, i and any i n, 4. lim i i Σ z s + σ 2 s ds =. 4. g2 udu Then here exiss an equilibrium and i is given by he weighing funcion w s = gs, he pricing [ rule H ξ, = E f ξ + ] gsdb2 s, and he rading sraegy θ saisfying θ = and n dθ Z α Y α = {, ]} Σ z + σ 2 g2 sds d + i= { i, i+ ]} Z Z i α i+ Y Y i α i+ Σ z + σ 2 d. g2 sds 24

25 This heorem implies ha if here are imes i such ha Σ z i + σ 2 i w2 sds =, and he inensiy of privae informaion arrival is fas enough a hese poins i.e. 4. is saisfied, hen i is: a raional for he marke maker o change her weighing funcion a hese poins and b i is opimal for he insider o reveal her informaion a hese imes. Moreover, noice ha his equilibrium exiss even when assumpions 3. and 3.2 insuring exisence of Markovian equilibrium are no saisfied. Tha is, even if Σ z s + σ 2 s < for some s, here is a non Markovian equilibrium as long as here exiss a piecewise linear increasing funcion g, he inegral of which is bounding he realized variance of he insider signal Σ z σ 2 from below and saisfies he condiions of he heorem. The proof of his heorem relies on linear filering and deerminisic ime change. Proof. To demonsrae ha H, w, θ is an equilibrium, i is enough o show ha Y Y i is a Brownian moion on [ i, i+ ] in is own filraion, 4.2 where Y = θ + B 2 and α i+ Yi+ Y i = Zi+ Z i. 4.3 Indeed, if hese wo condiions are saisfied, since Σ z i + σ 2 i g2 sds =, we will have ha H ξ i, i = F Z i, i, herefore H is an admissible and raional pricing rule. Moreover, if condiion 4.2 is saisfied, hen θ is coninuous wih bounded variaion. Therefore i follows from Proposiion 4. ha θ is opimal if condiion 4.3 holds noice ha H saisfies PDE 4.. Thus, o show ha Y saisfies 4.2 and 4.3 is he nex goal. The proof is by inducion. I Consider he inerval [, ]. A = we have Y =, Z = v and Y saisfies he following sochasic differenial equaion on [, ]: dy = Z α Y α Σ z + σ 2 α 2 d + db2 wih dz = σ z db. From Theorem.3 of Lipser and Shiryaev 2 noe ha due o 25

26 4., he condiions of he heorem are saisfied, we have ha on he filraion F Y he sochasic differenial equaion for Y is < dy s = m s α Y s α Σ z + σ 2 α 2 ds + dby s, wih where B Y dm s = γ s α Σ z + σ 2 α 2 dby s, is Brownian moion on F Y, and γ s saisfies he following ODE γ s = σzs 2 γs 2 α 2 Σz + σ 2 α 2 2 wih iniial condiion γ = σ 2. Noice ha γ s = Σ z + σ 2 α 2 is he unique soluion of his ODE and iniial condiion. Therefore on F Y, he process Y saisfies dy s = B Y s Y s α 2 Σ z + σ 2 α 2 ds + dby s. The unique srong soluion of his sochasic differenial equaion on [, is Y s = B Y s see Karazas and Shreve 99, Example Hence, on he inerval [,, he process Y is a Brownian moion on is own compleed filraion. By coninuiy of Y, his process is a Brownian moion on [, ]. To prove ha α Y = Z, noice ha α Y { where λ = exp } α Σ z+σ 2 α 2ds. Noe ha a random variable and variance hen ξ = α Y = Z. α σ z s = Z + λ v + λs db2 s λs db s α λs db2 s σ zs λs db s is normally disribued wih mean α 2 +σ2 zs α ds. Therefore, due o condiion 4.9, if lim 2 λ 2 +σ2 zs s ds <, λ 2 s 26

27 On he oher hand, if lim X = α 2 +σ2 z s ds =, consider he process λ 2 s α σ z s λs db2 s λs db s, and a change of ime τ given by τ α 2 + σ2 zs λ 2 ds =. s Then, W s = X τs is a Brownian moion. Hence, we can use he law of ieraed logarihm o ge lim sup s lim inf s W s 2s log log s = W s 2s log log s = or, in he original ime, lim sup lim inf X 2Ξ log logξ = X 2Ξ log logξ = where Ξ = α 2 +σ2 z s ds. Due o he condiions in his case we have λ 2 s lim λ 2 Ξ log log Ξ =, herefore i follows ha lim λx =, hus ξ = α Y = Z. II Suppose α j Y j Y j = Z j Z j for any j i. Consider he inerval [ i, i+ ]. A = i we have ξ i = Z i, and Ỹ = Y Y i saisfies he following sochasic differenial equaion on [ i, i+ ]: dỹ = Z α i+ Ỹ α i+ Σ z Σ z i α 2 i+ i d + db2 27

28 wih Z = Z Z i, hus d Z = σ z db and Z i =. From Theorem.3 of Lipser and Shiryaev 2 noe ha, due o 4., he condiions of he heorem are saisfied, we have ha on he filraion F Y he sochasic differenial equaion for Y is dỹs = [ i, i+ m s α i+ Ỹ s α i+ Σ z Σ z i α 2 i+ i ds + dby s, wih where B Y dm s = γ s α i+ Σ z Σ z i α 2 i+ i dby s, is Brownian moion on F Y, and γ s saisfies he following ODE γ s = σzs 2 γs 2 αi+ 2 Σz Σ z i αi+ 2 i 2 wih iniial condiion γ i =. Noice ha γ s = Σ z Σ z i αi+ 2 i is he unique soluion of his ODE and iniial condiion. Therefore on F Y, he process Y saisfies [ i, i+ dy s = B Y s Y s α 2 i+ Σ z Σ z i α 2 i+ i ds + dby s. The unique srong soluion of his sochasic differenial equaion on [ i, i+ is Y s = B Y s see Karazas and Shreve 99, Example Hence, on he inerval [ i, i+, he process Y is a Brownian moion on is own compleed filraion. By coninuiy of Y, his process is a Brownian moion on [ i, i+ ]. To prove ha α i+ Ỹ i+ = Z i+, noice ha α i+ Y { where λ = exp } α i+ i Σ z Σ z i α 2 i+ i ds. α i+ σ z s = Z + λ i λs db2 s i λs db s Therefore, due o condiions we have, exacly as in he previous case, α i+ Ỹ i+ = Z i+. 28

29 By he principle of mahemaical inducion, condiions 4.2 and 4.3 hold for any i. 5 Conclusion This paper demonsraes ha, in he presence of dynamic privae informaion of he insider, and under minimal resricions on he admissible rading sraegies, an equilibrium exiss and here is a unique Markovian pricing rule as a funcion of oal order process ha admis inconspicuous equilibrium. Moreover, he opimal insider rading sraegy is based on he marke esimaes of he fundamenals, raher han on he sock price: he insider buys he sock when he marke overesimaes he fundamenal value, and sells i oherwise, hus leading o higher informaiveness of he sock price. Furhermore, his induces convergence of he price o he fundamenal value a he erminal ime in he case of Markovian pricing rule and a some some se of imes which include he erminal ime in he case of non Markovian pricing rule. Fuure research can be conduced along he following direcions: assumpions on he marke parameers could be furher relaxed, and a more general model of he oal order of he noisy raders could be considered. The model can also be generalized furher by allowing for poenial bankrupcy of he firm issuing he sock, wih he ime of bankrupcy defined as he random ime a which he underlying process governing he firm value his a given barrier. 29

30 References [] Ankirchner, S., Imkeller, P., 26: Financial markes wih asymmeric informaion: informaion drif, addiional uiliy and enropy, mimeo. [2] Back, K. 992: Insider rading in coninuous ime. The Review of Financial Sudies, 53, pp [3] Back, K., Pedersen, H. 998: Long-lived informaion and inraday paerns. Journal of Financial Markes, 3-4, pp [4] Biagini, F,, Øksendal, B., 25: Minimal variance hedging for insider rading, mimeo. [5] Brunnermeier, M. K., 2: Asse pricing under asymmeric informaion - bubbles, crashes, echnical analysis and herding, Oxford Universiy Press. [6] Campi, L., Cein, U., 27: Insider rading in an equilibrium model wih defaul: a passage from reduced-form o srucural modelling, To appear in Finance and Sochasics. [7] Cho, K.-H., 23: Coninuous aucions and insider rading, Finance and Sochasics 7, [8] Kallenberg, O. 2: Foundaions of modern probabiliy, Springer-Verlag. [9] Karazas, I., Pikovsky, I. 996: Anicipaive sochasic opimizaion, Advances in Applied Probabiliy 28, [] Karazas, I., S.E. Shreve 99: Brownian moion and sochasic calculus, Springer-Verlag. [] Mansuy, R., Yor M., 26: Random Times And Enlargemens of Filraions in a Brownian Seing, Springer Verlag. [2] Lipser, R.S., Shiryaev, A. N. 2: Saisics of Random Processes, Springer-Verlag. [3] O Hara, M., 995: Marke microsrucure heory, Blackwell Publishing Ld. [4] Proer, P. E. 24: Sochasic inegraion and differenial equaions, Springer-Verlag. 3

31 A Proof of Remark 3. Suppose Assumpion 2.2 is saisfied, lim Ξ = and condiions 3.3 and 3.4 hold. Then L Hôpial rule will give noice ha due o 3.4 and coninuiy of σ z in he viciniy of we have lim + σ 2 z < 2 lim λ2 Ξ log log Ξ = 2 lim + σ 2 z lim Σz + σ 2 log log Ξ. Since by L Hôpial rule we have i follows ha lim λ2 Ξ =, lim λ 2 Ξ log log Ξ 2 lim + σ 2 z lim Σz + σ 2 log log λ 2 = 2 lim + σ 2 z lim Σz + σ 2 log Σ z s + σ 2 s ds = 2 lim + σ 2 z lim where f = need o show ha for some α > o esablish 3.6. logf f, Σ zs+σ 2 s ds and lim logx f =. Since lim x x α Consider any α, and denoe by lim sup f α f < =, for any α > we A.4 < g = f α f, hen for we have f α = α gs ds + c where c is some posiive consan. Due o his expression and since lim f =, α < and, 3

32 due o 3., f < for any [, we mus have lim g =. Thus A.4 holds and herefore 3.6 is esablished. 32