A Risk-Averse Insider and Asset Pricing in Continuous Time

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1 Managemen Science and Financial Engineering Vol 9, No, May 3, pp-6 ISSN EISSN hp://dxdoiorg/7737/msfe39 3 KORMS A Rik-Avere Inider and Ae Pricing in oninuou Time Byung Hwa Lim Graduae School of Financial Eng, The Univeriy of Suwon (Received: Ocober 5, / Revied: Ocober 5, / Acceped: November, ) ABSTRAT Thi paper derive an equilibrium ae price when here exi hree kind of rader in financial marke: a rik-avere informed rader, noie rader, and rik neural marke maker Thi paper i an exended verion of Kyle (985, Economerica) coninuou ime model by inroducing inider rik averion We obain no only he equilibrium ae pricing and marke deph parameer bu alo inider value funcion and opimal inider rading raegy explicily The comparaive aic how ha he marke deph (he reciprocal of marke preure) increae wih ime and volailiy of noie rader rading Keyword: Rik-Averion, Informed Trader, Ae Pricing, Marke Deph, Dynamic Programming Principle orreponding Auhor, byunghwalim@uwonackr INTRODUTION Many kind of rader who have differen informaion exi in he financial marke Each rader ha differen moive for paricipaing in ae rading and ome rader who have monopoly power wih heir privae informaion may crucially affec he equilibrium ae pricing Thi feaure exi in he mo real financial marke uch a equiy and derivaive marke So i i imporan o include he aymmeric informaion rucure in ae pricing and differen kind of rader hould be incorporaed in he model The lieraure of ae pricing under aymmeric informaion have been developed from 98 and now regard a one of he mo imporan rand for underanding he equilibrium ae price Kyle (985) i a pioneer o model he equenial acion under aymmeric informaion Hi model coni of wo age In he fir age, a ingle informed rader and noie rader ubmi heir marke order imulaneouly and hen he marke maker e he equilibrium price given he marke order Since he marke maker e he price afer receiving oal marke order of informed rader and noie rader, he equenial model i called ignaling model in which he marke maker eimae he liquidaion value of he ock from he marke order Thi model i compared wih he creening model in which he marke maker e he price fir Thi model i fir conidered by Gloen (989), In ha paper marke maker offer he menu of conrac in he fir age Then he informed rader chooe hi demand funcion o maximize hi profi The Kyle model i exended ino variou apec Back (99) develop coninuou ime model in he general eing and Back and Pederen (998) include long-lived informaion wih ime varying volailiy of noie rading The rik averion of informed rader i included in Subrahmanyam (99) and exended ino dicree ime eing by Holden and Subrahmanyam (994) Baruch () udie coninuou ime model wih a ingle rik-avere informed rader Hi model i an exended verion of Back (99) and Hoden and Subrahmanyam (994) The model developed in hi paper i imilar o Baruch () However, we apply dynamic programming principle o derive he equilibrium oucome o ha he marke deph, opimal demand funcion for informed rader, and hi value funcion are obained explicily The equilibrium oucome derived in hi paper give

2 Lim: Managemen Science and Financial Engineering Vol 9, No, May 3, pp-6, 3 KORMS imilar implicaion wih previou udie and addiional comparaive aic are alo given In ummary, when here i a ingle rik-avere informed rader, while he marke deph increae a hi rik averion increae near he end of rading, he parameer decreae wih rik averion in he beginning of he rading Moreover he marke deph alo increae wih ime and he volailiy of he noie rading Thi paper i organized a follow In ecion, we decribe he financial marke and he equilibrium definiion and oucome are given in ecion 3 Then he comparaive aic reul are repreened in ecion 4 Secion 5 conclude wih a ummary and fuure reearch THE FINANIAL MARKET We conider he coninuou marke microrucure in which here are hree kind of rader: a ingle rikavere informed rader, noie rader, and rik neural marke maker The informed rader ha privae informaion of he liquidaion value of a ock o he rade becaue of he informaional reaon The noie rader rading moive i ouide marke and marke maker e he equilibrium price o clear he marke order All marke paricipan rade or e marke price coninuouly o ha hi model i neiher ignaling nor creening model There i one racable ock in he marke and le v be he Ex po liquidaion value of he riky ae, which i a normal random variable wih mean p and variance Σ When he liquidaion value v realize, he informed rader receive he realized full informaion v and he ue hi informaional power o maximize hi expeced uiliy (or expeced profi when he i rik neural) The noie rader rading a ime i denoed by z which i aumed o evolve dz db where B i a andard Brownian moion and repreen a conan volailiy of noie rading If we denoe he informed rader cumulaive rading quaniy and he equilibrium price a ime a x and p repecively, he informed rader rading raegy i given by dx β ( v p ) d, where β i an informed rader eniiviy of rade Thu he informed rader remaining profi Π i given by β Π ( v p ) dx ( v p ) d So an informed rader who ha ARA uiliy funcion wih a coefficien of rik averion γ i defined by d U ( Π ( x, p) ) e γ Π There are many compeiive rik neural marke maker in he marke They average zero profi and heir role i o e he equilibrium ae price by he marke efficien condiion Therefore, he kech of rading progre i a follow: he informed rader and noie rader ubmi heir marke order and marke maker e he equilibrium ae price o clear he marke order afer receiving all marke order, which i called order flow (dw dx +dz ) Since our model i coninuou ime model, he price hould alo be conained in he informed rader informaion and he can deduce he order flow from he marke price While paricipaing marke, he informed rader objecive i o maximize hi expeced uiliy by uing hi privae informaion and he marke price he marke maker e To make profi a high a poible, he hould preend no o have any informaion a noie rader So he may diribue hi privae informaion during he rading period 3 EQUILIBRIUM The compeiion among rik neural marke maker induce ha he equilibrium ae price equal o he marke maker opimal forecaing of he liquidaion value of a ock from receiving he order flow, ie, p E[ vw, < ] () Therefore he equilibrium i defined by a pair (x, p), which aifie he following wo condiion: i) (profi maximizaion) given he equilibrium price p, he informed rader raegy x maximize E[ U( Π ( x, p) ) F ] where F i he -algebra generaed by v v and { p :< } ii) (marke efficiency) given informed rader rading raegy x, he equilibrium price rule i compeiive o ha i i e by () From (), he pricing rule i given by p p + λ dw, where λ i ome deerminiic funcion which i defined a he marke preure in Kyle (985) Therefore given x β ( v p ) d, he linear pricing rule evolve dp λdw λβ ( v p ) d + λdb () The reciprocal of marke preure / λ i called

3 A Rik-Avere Inider and Ae Pricing in oninuou Time Vol 9, No, May 3, pp-6, 3 KORMS 3 marke deph ince i repreen he required order flow o change price a a given amoun Since he liquidaion value v and he volailiy of noie rade volume are given exogenouly, he equilibrium i equal o find a pair ( β, λ) a a naural conequence ompared o Baruch (), ince he conider elaic noie demand funcion he variance of noie rading quaniy i no conan any more Thi i he main difference and main facor ha make i hard o obain he informed rader value funcion explicily We alo adop differen approach o ge he marke preure λ and he eniiviy β of he informed rader raegy The informed rader problem can be rewrien a ubjec o he pricing rule p given in () wih wo boundary condiion V(, p ) - and λ d Σ d V(, P ) max e γ E Π F β ( v p) dx max E e γ β F (3) Thee wo boundary condiion are naural The fir condiion i rivial condiion from he definiion of he uiliy funcion and he econd condiion mean ha he informaion i fully revealed afer end of he rading Since he monopoli rik-avere informed rader wan o maximize hi expeced uiliy funcion, i i opimal o ue hi informaional advanage unil he end of he rading a much a poible Then he informed rader value funcion can be derived by he dynamic programming principle and we have he Hamilon-Jacobi-Bellman (HJB) equaion in he nex propoiion Propoiion 3: The HJB equaion of he informed rader value funcion (3) i deermined by V V + p λ V + λ γ β p wih wo condiion max ( v p) V( v p), β V(, p ) and λ d Σ Proof: Le β be an opimal value of he problem afer ome fixed ime and define he informed rader eniiviy a β, if β β, if, Then he value funcion become γ ( v p) dx max E e β F max E V(, p ) e β γ ( v p) dx F γ ( H H ) max E[ V(, p ) e ] β F max [ V(, p ) φ F ] V(, p ), β E ( H H) where dh : ( v p) βd, φ : e γ If we define q V(, p) φ, hen he Io formula give and dq φ dv + V dφ +< dφ, dv > (4) dv + Vd q q φ φ (5) The hird erm in (4) hould be vanihed becaue he funcion φ i deerminiic Furhermore, he differenial form of he value funcion i given by V V V d + dp + d dv ( ) p p λ + + p V + λ db p V V V λβ ( v p) ( λ ) d Therefore, we can rewrie he relaion in (5) a q V + + ( ) + ( ) p p V + φλ db γ V φ( v p ) β d p V V V φ λ β v p λ d If we ake a condiional expecaion on boh ide wih he filraion F, he previou equaion repreened by E[q F] V E { HJB} d, φ F and he dynamic programming principle give he HJB equaion Thi complee he proof I i eaily checked ha he opimal eniiviy β of informed rader raegy i arbirary Thi mean here are many opima for he informed rader and hi i he ame reul wih Back (99) in which he informed rader i rik neural Even hough here are many opima, we can derive a unique equilibrium in hi cae The opimal eniiviy parameer in HJB equaion i derived from he informed rader problem only bu he

4 Lim: Managemen Science and Financial Engineering Vol 9, No, May 3, pp-6, 3 KORMS 4 marke efficien condiion, which i he econd condiion of equilibrium, give oher implicaion for he opimal parameer In Propoiion 3, he HJB equaion i linear in β o we have wo differenial equaion wih boundary condiion Thee equaion deermine he marke preure λ and he informed rader value funcion Theorem 3: The informed rader value funcion which aifie he HJB equaion in (3) i obained by equaion which deermine he marke preure parameer explicily a (6) From he hird ODE, he deerminiic funcion g() can be induced by v e γ γ g() +, where i alo an arbirary conan Then he coefficien and are oluion o he equaion induced by he wo condiion aed in Propoiion 3 where v+ ( + )( vp p ) V (, p) γ+ e γ γ γ and are given by γ + γ + / Σ, γ v e γ+ The price preure parameer i alo obained from λ γ + (6) Proof: The opimal β of he HJB equaion in Propoiion 3 implie he following differenial equaion whoe oluion are he value funcion and marke preure λ V V λ + p V λ p ( v p) γv( v p) The econd equaion i an ordinary differenial equaion (ODE) which ha a oluion of he form γ γ vp p λ λ V(, p ) ge ( ), where g() i a deerminiic funcion If we ubiue hi V(, p ) ino he econd equaion, he following yem of ODE i derived ' γ γ λ ' γ v γ v λ ' g () γλ ( γv ) g(), (7) The fir and econd equaion in (6) are ame γ+, e γ v Σ + γ The coefficien ha wo real oluion bu he poiiviy of marke preure λ a ime make uniquely When he informed rader i rik neural he marke preure i conan expreed by Σ / In fac, hi value i he exac ame wih original Kyle coninuou ime model Thi mean ha informed rader co of rading i conan over he rading period o ha he hould ue hi privae informaion a a conan rae Furhermore, he explici marke preure acually ha he imilar form wih ha of Baruch () He derive he yem of differenial equaion for marke preure and condiional variance Σ of v given he order flow unil ime Since he conider he elaic noie demand he volailiy of noie rading i no conan any more So he marke preure ha differen value correponding o he form of elaic noie demand I i eaily checked ha, however, he oluion o ODE for marke preure ha he ame form of he marke preure (6) when he volailiy of he noie rading i conan In addiion, he figure ou ha he marke preure behavior, which i decreaing in ime, i independen of he elaiciy of he noie demand funcion and we alo have he imilar characeriic for he marke deph onra o ime variable, he behavior of marke preure i no clear for he variaion of rik averion and volailiy of noie rading The more pecific reul are given in he nex ecion The explici value funcion can alo give ome inereing implicaion The informed rader expeced profi a ime i deduced from he value funcion and he impac of informed rader rik aking behavior on hi profi can be analyzed Bu i i hard o give explici explanaion becaue he variaion of rik averion affec he marke preure and he equilibrium price a he ame ime Alhough we have an explici value funcion, he opimal raegy of informed rader i no deermined uniquely a menioned in Propoiion 3 The marke efficien condiion of equilibrium condiion lead o

5 A Rik-Avere Inider and Ae Pricing in oninuou Time Vol 9, No, May 3, pp-6, 3 KORMS 5 anoher expreion for he marke preure and i deermine he informed rader demand funcion uniquely Theorem 3: When he monopoli inider ha a ARA uiliy funcion wih rik averion γ, hi inananeou opimal demand i given by dx β ( v p ) d, where he opimal eniiviy i derived by β γ+ Furhermore, he condiional variance Σ i alo given by γ+ γ+ Σ where he conan coefficien i deermined in Theorem 3 Proof: Since he condiional variance hould have he ame value wih he volailiy of he price () i can be calculaed by λ d γ + Σ d Moreover, he marke efficien condiion make he equilibrium price be wrien a I can be eaily verified ha Σ Σ and Σ The fir condiion i naural, and he econd condiion mean ha i i opimal for he informed rader o ue hi privae informaion fully ju before he public announcemen Thi i becaue ha he condiional variance Σ meaure he informaion which ha no been revealed in he price marke maker e The full informaion revelaion by he end of he rading i he propery which i independen of informed rader rik averene a Kyle (985) and Baruch () The fac ha he opimal eniiviy of informed rader demand i increaing wih ime mean he informaional reveal peed i up prior o he public announcemen So he equilibrium price converge o he liquidaion value of he ock and he convergence peed alo increae wih ime Moreover, ince near he public announcemen he marke preure decreae wih informed rader rik averion he informed rader wih higher rik averion rade more near he end of he rading 4 OMPARATIVE STATIS The equilibrium oucome obained in previou ecion are affeced by he rik averion of he informed rader To he ae pricing under aymmeric informaion, he informaional advanage and informed rader uage are he imporan facor o underanding he equilibrium price When he i rik avere he hould conider he rik aking behavior a well In addiion o preending he i noie rader, he wan o maximize hi expec uiliy bu he i more cauiou han rik neural rader In hi ecion, we examine he impac of rik averion on he equilibrium parameer uch a marke preure, informed rader eniiviy, and condiional variance which i relaed o he unrevealed informaion Epecially we focu on he marke deph which i defined a he abiliy of he marke o aborb he rading quaniie wihou large price impac The reciprocal of he marke deph i he marke preure λ and a menioned, i repreen he co of rading o informed rader and he degree of informaion revelaion of equilibrium price marke maker e In oher word, he marke deph repreen he marke characeriic relaed o he price p E [ vw, < ] p + λ dw Here he marke preure hould have he ame value wih he value derived from filering heorem and i i deermined from β Σ λ Therefore we have he opimal eniiviy explicily Propoiion 4: Le he reciprocal of marke preure be marke deph (/ λ ) Then he following comparaive reul hold: (i) /λ decreae a rik averion γ increae; (ii) /λ increae a rik averion γ increae; (iii) / λ increae a volailiy of noie rading increae; (iv) / λ increae wih ime Proof: Omied The fir wo reul are inuiive If he informed rader i more rik avere a he beginning of he rading, he may ac more aggreively Then hi privae informaion would be incorporae ino he equilibrium price and he price become more volaile On he oher hand, a he informed rader ake more rik, he unveiled informaion reduce near he end of rading becaue of he inene releae in he beginning The hird relaion i alo inuiive When he noie rading i more volaile, he fundamenal volailiy of he

6 Lim: Managemen Science and Financial Engineering Vol 9, No, May 3, pp-6, 3 KORMS 6 equilibrium price i alo high So he informed rader ha more room o ac raegically The la reul mean ha he abiliy o aborb he marke order increae wih ime o ha he informed rader rade more acively o ge he more profi or expeced uiliy The conequence of he informed rader opimal eniiviy parameer β i auomaically obained The rik averion and ime affec ame way o he marke deph Thi i naural becaue he deeper marke deph he more informed rader rading volume 5 ONLUSIONS The ae pricing model when he marke coni of ingle rik-avere informed rader, noie rader and marke maker i conidered in coninuou ime Thi paper i he exenion verion of Kyle (985) by incorporaing a rik averene of an informed rader ompared o he original Kyle coninuou ime model, our equilibrium reul give ome addiional implicaion The rik averene of he monopoli informed rader induce he ime varying marke preure and more rading volume near he end of he rading Thi would be he imilar o he reul in Baruch () bu we apply he differen approach o ge he explici oluion and obain he informed rader value funcion and he opimal raegy explicily, which are no derived in Baruch () Furhermore, we have comparaive aic reul in he oher apec uch a volailiy of he noie rading Thi work can be exended ino variou direcion If here are many informed rader in he marke, hey hould concern wih he oher informed rader behavior More raegic reul may be induced In addiion, he marke maker rading moive can be changed ino profi maximizing while hey e price Then he lo of he noie rader i diribued o informed rader and marke maker Thi give more realiic feaure when financial iniuion rade wih ome privae informaion Laly, our model can alo be applied o pecific financial ae like derivaive when marke paricipan have differen informaional moive REFERENES Back, K, Inider rading in coninuou ime, Review of Financial Sudie 5, (99), Back, K and H Pederen, Long-lived informaion and inraday paern, Journal of Financial Marke, (998), Baruch, S, Inider rading and rik averion, Journal of Financial Marke 5, (), Gloen, L, Inider rading, liquidiy, and he role of he monopoli peciali, Journal of Buine 6 (989), -35 Holden, W and A Subrahmanyam, Rik averion, imperfec compeiion and long lived informaion, Economic Leer 44, (994), 89-9 Kyle, A S, oninuou aucion and inider rading, Economerica 53, (985), Subrahmanyam, A, Rik averion, marke liquidiy, and price efficiency, Review of Financial Sudie 4, (99), 47-44