Portfolio optimization for a large investor under partial information and price impact

Transcription

1 Mah Meh Oper Res DOI 1.17/s x Porfolio opimizaion for a large invesor under parial informaion and price impac Zehra Eksi 1 Hyejin Ku Received: 4 Ocober 16 / Acceped: 1 April 17 Springer-Verlag Berlin Heidelberg 17 Absrac This paper sudies porfolio opimizaion problems in a marke wih parial informaion and price impac. We consider a large invesor wih an objecive of expeced uiliy maximizaion from erminal wealh. The drif of he underlying price process is modeled as a diffusion affeced by a coninuous-ime Markov chain and he acions of he large invesor. Using he sochasic filering heory, we reduce he opimal conrol problem under parial informaion o he one wih complee observaion. For logarihmic and power uiliy cases we solve he uiliy maximizaion problem explicily and we obain opimal invesmen sraegies in he feedback form. We compare he value funcions o hose for he case wihou price impac in Bäuerle and Rieder (IEEE Trans Auom Conrol 49(3):44 447, 4) and Bäuerle and Rieder (J Appl Prob , 5). I urns ou ha he invesor would be beer off due o he presence of a price impac boh in complee-informaion and parial-informaion seings. Moreover, he presence of he price impac resuls in a shif, which depends on he disance o final ime and on he sae of he filer, on he opimal conrol sraegy. The auhors would like o hank he Isaac Newon Insiue for Mahemaical Sciences, Cambridge, for suppor and hospialiy during he programme Sysemic Risk: Mahemaical Modelling and Inerdisciplinary Approaches where work on his paper was underaken. The work of Zehra Eksi was suppored by EPSRC Gran No. EP/K38/1. The work of Hyejin Ku was parially suppored by Naural Sciences and Engineering Research Council of Canada. B Hyejin Ku hku@mahsa.yorku.ca Zehra Eksi zehra.eksi@wu.ac.a 1 Insiue for Saisics and Mahemaics, WU-Universiy of Economics and Business, Welhandelsplaz 1, 1 Vienna, Ausria Deparmen of Mahemaics and Saisics, York Universiy, 47 Keele S., Torono, ON, Canada

2 Z. Eksi, H. Ku Keywords Porfolio opimizaion Markov-modulaion Sochasic conrol Parial informaion Large invesor Price impac Filering 1 Inroducion In he Meron s classical porfolio opimizaion problem he underlying sock price process is assumed o be independen of he acions of he invesor. When he invesor is large, such as hedge funds, muual funds or insurance companies, his may no hold, as he acion of he invesor may influence he risky price process hrough differen channels. This gives us a moivaion o consider sock price dynamics whose drif is affeced by he decisions of he large invesor. Moreover, during long invesmen periods drif of he price process may change in accordance wih he changing marke condiions. This phenomenon can be refleced by a regime-swiching drif. Invesors may or may no observe he par of he drif which is governed by he curren marke condiions. In his paper, we deal wih a finie-ime porfolio opimizaion problem for a large invesor, where he underlying price process is a diffusion affeced by he sae of he marke and he acions of he invesor. The sae of he marke is represened by a finie-sae Markov chain allowing for he drif of he price process o change in accordance wih he changing marke condiions for long invesmen periods. We also le he drif change according o he invesmen decisions of he large invesor. In fac, he drif of he price process is aken o be a funcion of he fracion of he wealh invesed in he risky asse by he large invesor. Hence, he porfolio decomposiion of he large invesor migh be considered as anoher facor governing he drif of he price process. We firs assume ha he sae of he economy is observable (complee- informaion case) by he invesor. By allowing for he price impac, we exend he seing given in Bäuerle and Rieder (4). Under he full informaion seing we obain resuls for general impac and uiliy funcion (logarihmic and power) choices. For any sufficienly regular impac funcion and for he choice of logarihmic uiliy, he corresponding opimizaion problem can be solved direcly and opimal invesmen sraegies are characerized explicily. In he case of power uiliy, we address he problem by using dynamic programming mehods. In paricular, he case wih linear impac funcion yields an opimal conrol in he feedback form as well as a probabilisic represenaion for he corresponding value funcion. We show ha for boh logarihmic and power uiliy preferences and linear impac funcion choice he resuling value funcion dominaes he value funcion corresponding o he uiliy maximizaion problem in a seing wihou price impac. Secondly, we repea our analysis for he seing where he sae of he economy is no direcly observable by he invesor (parial-informaion case). Technically, his resuls in an opimizaion problem under parial informaion. To solve such a problem we derive an equivalen conrol problem under full informaion via he so-called reducion approach (see, e.g., Fleming and Pardoux 198). This requires he derivaion of he filering equaion for he unobservable sae variable. In order o obain he filering equaions we address innovaions approach o non-linear filering (see, e.g., Ellio

3 Porfolio opimizaion for large invesor under parial 198). Then, we inroduce he filer for he Markov chain as an addiional sae variable for he opimizaion problem. This reduces our problem o a conrol problem wih complee informaion. We rea logarihmic and power uiliy cases in he same way explained above. In paricular, for he case wih linear impac funcion we obain an opimal conrol in he feedback form and we provide a probabilisic represenaion for he corresponding value funcion boh for logarihmic uiliy and power uiliy preferences. We obain he resul for he power uiliy case by employing a power change of variable approach (see, e.g., Zariphopoulou 1). Also, we derive noarbirage condiions on he impac funcion. Overall, we find ha he invesor benefis from he presence of he price impac in he sense ha he resuling value funcions corresponding o full-informaion and parial-informaion seings dominae he ones given in Bäuerle and Rieder (4) and Bäuerle and Rieder (5), respecively. This resul, of course, would be counerinuiive for he problem of an opimal order execuion, where he invesor is allowed o rade only in one direcion, e.g. only buy or sell orders. Moreover, our numerical resuls based on a wo-sae Markov chain sugges ha he presence of he price impac yields a shif on he opimal conrol sraegies. In paricular, for he case of parial informaion he magniude of he shif depends on he disance o final ime and on he sae of he filer. There is an ample amoun of lieraure concerning he porfolio opimizaion problems wih Markov chain modulaed price dynamics under complee and parialinformaion. The case wih complee informaion has been addressed in Bäuerle and Rieder (4), in which he problem of expeced uiliy maximizaion from erminal wealh is solved by sochasic conrol mehods for differen uiliy funcions. Sass and Haussmann (4) and Haussmann and Sass (4) have reaed he porfolio opimizaion problem in a muli-asse seing under parial informaion and hey obained he opimal porfolio sraegy by using he maringale approach. On he oher hand, Bäuerle and Rieder (5) has addressed he porfolio opimizaion problem wih unobservable Markov chain modulaed drif process by using a dynamic programming approach. Björk e al. (1) considers a relaively general seing and i provides explici represenaions of he opimal wealh and invesmen processes for he uiliy maximizaion problem under parial informaion by using he maringale approach. Sener (4) sudies he risk-sensiive porfolio maximizaion problem when he dynamics of he asse prices depend on some economical facors, which are compleely or parially observed. Frey e al. (1) solves he porfolio opimizaion problem under parial informaion by including exper opinions in he analysis as a second source of informaion. Concerning porfolio opimizaion problems under parial informaion we also refer o Pham (11) which gives a very broad overview of previous sudies in he subjec. The lieraure relaed wih he large invesor wih price impac mainly considers direc impac on he underlying sock price process. In paricular, Cvianić e al.(1996), Cuoco and Cvianić (1998) and Kraf and Kühn (11) assume ha he large invesor has an influence on he drif and volailiy of he price process via he dollar amoun invesed in he sock. Also, Ku and Zhang (16) assume ha he drif is affeced by he speed of he invesor s rading acion. In his respec, our model deviaes from he exising lieraure as we assume ha he impac on he drif of he price process is a

4 Z. Eksi, H. Ku funcion of he fracion of wealh invesed in he sock. Busch e al. (13) has sudied an opimal consumpion and invesmen problem in which he price process follows a regime-swiching jump-diffusion. By modeling he inensiy of he regime swich as a funcion of he fracion of wealh invesed in he sock, hey allow he large invesor o have an indirec, persisen effec. Noe ha here is also a growing lieraure on price impac models used in he conex of opimal order execuion problems where he sock price process is driven by a diffusion whose drif is a funcion of he volume or speed of rading (see, e.g., Almgren and Chriss 1). For a deailed overview of price impac models in he conex of opimal order execuion, we refer o he surveys Gökay e al. (11) and Gaheral and Schied (13). This paper is srucured as follows. In Sec., we inroduce he modeling framework and he opimizaion problem. In Sec. 3, we solve he opimizaion problem under complee informaion for he case of logarihmic and power uiliy preferences and linear impac funcion. In Sec. 4, we inroduce he parial informaion seing, derive he corresponding filering equaions and solve he opimizaion problem for he case of logarihmic and power uiliy and linear impac funcions. Secion 5 illusraes numerical resuls and Sec. 6 concludes. Financial marke model We consider a finie ime inerval [, T ] and a coninuous-ime finie-sae Markov chain Y defined on he filered probabiliy space (Ω, G, G, P), where G = (G ) saisfies he usual condiions; all processes we consider here are assumed o be G- adaped. Y represens he uncerainy of he sae of he marke. We denoe by E = {e 1, e,..., e K } he sae space where, wihou loss of generaliy, we assume ha e k is he basis column vecor of R K. Y has he generaor Q = (Q ij ) and is iniial disribuion is denoed by Π = (Π 1,,Π K ). We have a large invesor wih given iniial wealh x R + and whose objecive is o find self-financing invesmen sraegies ha maximize expeced uiliy from erminal wealh. We consider a risk-free bond and a risky asse as he available insrumens in he marke. The bond price process has he dynamics db = rb d,, where r > is he risk-free rae. Le h R denoes he fracion of he wealh ha is invesed in he risky asse a ime. Then, 1 h denoes he fracion of he wealh invesed in he bond a ime. In order o avoid echnical difficulies we assume A1 h [ L, L], L R + for all [, T ]. Noe ha in A1, L migh be chosen large enough o guaranee ha he opimal soluion is an inerior one. The price of he risky asse evolves according o a diffusion whose drif is a funcion of he curren sae of he marke and he fracion of he wealh invesed in risky asse by he large invesor. Tha is,

5 Porfolio opimizaion for large invesor under parial ds S = (μ(y ) + g(h )) d + σ dw, S = s, (1) where W isa G-Brownian moion independen of Y and he funcion g represens he impac of he large invesor on he drif of he price process. Also noe ha μ(y ) = MY wih M k = μ(e k ),1 k K. This is due o he finie-sae propery of he Markov chain. In he sequel we are going o use boh noaions inerchangeably. The dynamics in (1) suggess ha he porfolio choice of he large invesor migh be aken as a signal by he res of he marke. Tha is, he porfolio choice of he large invesor acs as a facor governing he drif of he sock price. Throughou his paper we cover he case wih no impac on volailiy. Noe ha in his case he dynamics in (1) admis a unique weak soluion provided ha he funcion g is sufficienly regular. On he oher hand, he case wih an impac on volailiy, i.e., wih σ(h ), under parial informaion would bring us o an ineresing seing where acions of he invesor creae a rade-off beween he increase in he conrolled par of he drif and he decrease in he precision of he esimaes of he unobserved par of he drif. However, his seing is also echnically more delicae and is lef for fuure research. The self-financing porfolio propery assumpion implies ha he dynamics of he wealh of he invesor saisfies dx (h) X (h) = h ds S + (1 h ) db B. Tha is, dx (h) X (h) = (h (μ(y ) + g(h )) + (1 h )r) d + h σ dw, () X (h) >. In order o ensure ha he wealh process is well defined, we consider invesmen sraegies ha saisfy ( ) ds < almos surely. A T h s X (h) s Recall ha one of he assumpions in he classical seing is ha he invesors are price akers. In he curren seing his assumpion is violaed as we allow he invesor o have price impac. Hence one can no rely on he no-arbirage condiion provided for he classical seing. In his conex, in he nex heorem we derive he no-arbirage condiion on he impac funcion g. Theorem 1 (No-arbirage) Le S be given by he SDE (1), A1, A areassumedo hold and he funcion g saisfies g(h ) C h for a posiive consan C. Then, he marke is arbirage free. Proof For an admissible sraegy h,le θ() = μ(y ) r, T. σ

6 Z. Eksi, H. Ku Since Y is adaped and σ is consan, θ() is a G -adaped process. By Girsanov heorem, here is an equivalen probabiliy measure P under which W = W + θ(s) ds is a Brownian moion. We noe ha θ() μ(y ) +r σ for T and he Novikov condiion is saisfied. We also have he Radon Nikodym derivaive given by ( d P T dp = exp θ() dw 1 T ) θ () d. Define, for any posiive α, where R = L = exp ( αr ), h s (μ(y s ) + g(h s ) r) ds + h s σ dw s. Then we have ( ( )) L = exp α h s g(h s ) ds + h s σ d W s. By Iô s formula, dl = L (( αh g(h ) + 1 α h σ ) d αh σ d W ). If α> C σ, hen he d erm becomes negaive (noe ha L is negaive). Considering he inegrabiliy condiion on L we obain L is a P-supermaringale, hus Ẽ(L T ) Ẽ(L ) = 1 (3) where Ẽ represens expecaions under probabiliy measure P. Le h be an admissible sraegy ha saisfies P{e rt X (h) T X (h) }=1, which is P{R T } =1. Since P and P are equivalen, P{R T } =1 and using equaion (3) we obain P{R T = } =1. This implies ha P{e rt X (h) T = X (h) }=1. Therefore, h canno be an arbirage sraegy. Suppose we are given a concave, increasing and wice coninuously differeniable uiliy funcion U : R + R. Then, he problem of he large invesor is o

7 Porfolio opimizaion for large invesor under parial max E x,i [U(X (h) h T )], subjec o he iniial value of he wealh X (h) = x and iniial sae Y = i. 3 Opimizaion problem under complee informaion In his secion, we assume ha he invesor s filraion is given by G = (G ), ha is, he invesor is assumed o observe he rue sae of he marke and he sock price. Accordingly, a porfolio sraegy is admissible if h G, for all and A1, A hold. We denoe he se of admissible sraegies by H. In he following we solve he invesmen problems for he case of logarihmic and power uiliy preferences. The value funcion of he invesor is denoed by [ ] 1 V (, x, i) = sup E x,i (h) (X T h H θ )θ. (4) 3.1 Logarihmic uiliy We firs consider he porfolio opimizaion problem in he case of logarihmic uiliy. In his case i is possible o solve he opimizaion problem for a general impac funcion ha is regular enough. Proposiion 1 Suppose ha g is coninuously differeniable and U(x) = log(x). Then he opimal sraegy h = arg max E x,i [U(X (h) T )] h H exiss. Moreover, for all (, i) [, T ] E,h (, i) H log i, where H log i is defined by H log i { ( ) } g(l) ={ L, L} l : M i r + g(l) + l σ =. (5) l Proof Given he dynamics in () we apply Iô s formula for U(x) = log(x) and obain T U(X (h) T ) = log(x) + T + h s σ dw s. (h s (μ(y s ) + g(h s )) + (1 h s )r h s σ ) ds

8 Z. Eksi, H. Ku For any h H, he sochasic inegral T h s σ dw s is well defined and has an expeced value zero. Thus, we have [ T E x,i [U(X (h) T (h )]=log(x) + E s (μ(y s ) + g(h s )) + (1 h s )r h s σ ) ] ds. (6) Now we denoe he inegrand in (6)by f (s, l) = l (μ(y s ) + g(l)) + (1 l)r l σ. I follows from he coninuiy of g ha for any s [, T ], f (s, ) is a coninuous funcion defined on he compac se [ L, L]. Hence, maximizer of f (s, ) exiss. Moreover, he maximizer is an elemen of eiher {l : f (s,l) l = } or { L, L}. Tha is, h s Hlog i. Remark 1 I is possible o exend he resul of Proposiion 1 o he case where he funcion g( ) is differeniable excep for finiely many poins of he domain [ L, L]. Le H denoes he se( of he poins where g( ) is no differeniable. Then, Proposiion 1 holds wih h (, i) H log i H ). As a specific case, we now consider he opimizaion problem under complee informaion wih linear impac funcion and logarihmic uiliy. We se g(h) = βh, β>. The following corollary is an immediae resul of Proposiion 1. Corollary 1 Suppose U(x) = log(x) and g(h) = βh, β>. Then { } H log i = L, L, Mi r σ. β In paricular, depending on he given se of model parameers we have he following cases: i) if β σ <, hen, for all (, i) [, T ] E, he opimal sraegy is given by h (, i) = Mi r σ β, and he value funcion has he following sochasic represenaion: [ T V (, x, i) = log(x) + r(t ) + E x,i (μ(y s ) r) ] (σ β) ds. ii) If β σ, hen, for all (, i) [, T ] E, he opimal sraegy is given by h (, i) = L ( 1 {M i r } 1 {M i r<}),

9 Porfolio opimizaion for large invesor under parial and he value funcion in his case has he following sochasic represenaion: V (, x, i)=log(x)+(t ) ((β σ ) ) [ T ] L +r + E x,i L μ(y s ) r ds. Remark In Corollary 1 he case wih parameer condiion i) implies ha if M i r <, which can be inerpreed as an indicaion of an unfavorable marke environmen, opimal porfolio sraegy is o pu negaive weigh in he risky asse (shor selling) and posiive in he bank accoun and vice versa. When he parameer condiion ii) holds and if M i r, hen i is opimal o borrow as much as possible from he bank accoun and inves he proceeds in he risky asse. If, on he oher hand, M i r, hen he opimal is o sell he risky asse shor as much as possible and inves he proceeds in he bank accoun. Remark 3 In he model wihou price impac he opimal porfolio sraegy is given by (see, Bäuerle and Rieder 4) h (, i) = Mi r, σ and he corresponding value funcion is V (, x, i) = log(x) + r(t ) + E x,i [ T (μ(y s ) r) σ ] ds. (7) Corollary 1 suggess ha he value funcion of he invesor in he presence of he price impac dominaes he value funcion given in (7). Tha is, he invesor benefis from he presence of he price impac. 3. Power uiliy Nex we assume ha he uiliy funcion is U(x) = 1 θ xθ,<θ<1. This gives a consan relaive risk aversion (CRRA) ype preferences wih risk aversion (1 θ)/x. In conras o he case of logarihmic uiliy i is no possible o solve he opimizaion problem direcly. Insead, we address his problem by dynamic programing approach. To his, for any funcion v C 1, and (, x, i) [, T ] R + E, h H, we define he differenial operaor A h v(, x, i) v(, x, i) v(, x, i) = + x(h(m i + g(h)) + (1 h)r) x + 1 v(, x, i) x x h σ + (v(, x, j) v(, x, i))q ij. j Hypoheically, he following Hamilon Jacobi Bellman (HJB) equaion has o be solved

10 Z. Eksi, H. Ku sup A h v(, x, i) = h v(t, x, i) = 1 θ xθ for all (x, i) R + E. (8) Here noe ha due o he general form of he impac funcion g, i is no possible o characerize he soluion for he curren opimizaion problem. In he following we insead consider he case wih a linear impac funcion and obain explici resuls. Proposiion Suppose U(x) = 1 θ xθ and g(h) = βh, β> and i) β (1 θ)σ <, hen he opimal sraegy h is given by h (, i) = M i r (1 θ)σ β, (9) and V (, x, i) = 1 θ xθ u(, i), for all (, x, i) [, T ] R + E, where u(, i) >, wih u(t, i) = 1, i E, is he unique soluion of he following sysem of linear differenial equaions u(, i) + a(i)u(, i) + j (u(, j) u(, i))q ij =, (1) wih a(i) = θr + θ(mi r) sochasic represenaion ((1 θ)σ β). Moreover, he value funcion has he following V (, x, i) = xθ θ exp (rθ(t ))Ex,i [ ( T exp θ(μ(y s ) r) )] ((1 θ)σ β) ds. ii) β (1 θ)σ, hen he opimal sraegy h is given by h (, i) = L ( 1 {M i r } 1 {M i r<}), and V (, x, i) = 1 θ xθ u(, i), for all (, x, i) [, T ] R + E, where u(, i) >, wih u(t, i) = 1, i E, is he unique soluion of he following sysem of linear differenial equaions u(, i) + a(i)u(, i) + j (u(, j) u(, i))q ij =, (11) ( ) wih a(i) = θr + θ L M i r +θl β + (θ 1)σ. Moreover, he value funcion has he following sochasic represenaion

11 Porfolio opimizaion for large invesor under parial [ ( V (, x, i) = xθ θ Ex,i exp θ(t ) (L (β + θ L T μ(y s ) r ds )]. ) ) (1 θ)σ + r Proof I follows from he form of he uiliy funcion and he linear srucure of he dynamics of he wealh process ha for all i {1,...,K } he value funcion can be rewrien as v(, x, i) = 1 θ xθ u(, i), for some funcion u wih u(t, i) = 1. This gives following parial derivaives v(, x, i) v(, x, i) x = 1 θ u(, i) xθ, = (θ 1)x θ u(, i). v(, x, i) x = x θ 1 u(, i), Subsiuing hese and g(h) = βh in (8), we obain ru(, i) = { sup h [ L,L] + 1 u(, i) θ h(m i r)u(, i) + h (β + θ Q ij (u(, j) u(, i)), θ j } σ )u(, i) u(t, i) = 1 for all i {1,...,K }. (1) We have he following necessary condiion for he maximizer h(β + σ (θ 1) )u(, i) + (M i r)u(, i) =. Suppose β < (1 θ)σ. These ogeher wih u(, i) > (for he posiiviy, see Remark 4 below) imply ha he necessary condiion is also sufficien. Tha is, he maximizer is given by (9). Insering his maximum and afer some simple algebra, we obain (1). This differenial equaion has a unique soluion u and we have he following Feynman Kac ype represenaion of u(, i) (see Bäuerle and Rieder 4, Lemma ) ( T u(, x, i) = exp (rθ(t ))E [exp x,i θ(μ(y s ) r) )] ((1 θ)σ β) ds. In fac, his funcion v(, x, i) = 1 θ xθ u(, i) is a soluion of he HJB equaion (8), v C 1,, and saisfies v(, x, i) K (1+ x ) for a suiable consan K. By applying a Verificaion Theorem (see, e.g., Bäuerle and Rieder 4, Theorem 1), we obain v(, x, i) is indeed an opimal value funcion V (, x, i). Nex suppose β (1 θ)σ. Then, he maximum is aained in one of he end poins of he inerval [ L, L]. I is clear by inspecion ha he maximizer depends

12 Z. Eksi, H. Ku on he value of (M i r). Namely, h (, i) = L for M i r > and h (, i) = L oherwise. Hence, insering hese in (1) and afer some simple algebra we obain he sysem of differenial equaions in (11). By he same argumen as in case i), he proof is compleed. Remark 4 Noe ha he above represenaions of u imply ha he funcion u(, i) says posiive provided ha he given parameer resricions are saisfied. Remark 5 Proposiion suggess ha for any parameer condiion he curren value funcion dominaes he value funcion given in Bäuerle and Rieder (4), Theorem 3. This means ha he invesor benefis from he presence of he price impac also in he case of power uiliy preferences. 4 Opimal conrol under parial informaion Throughou his secion we assume ha he sae process Y is no direcly observable by he large invesor. Insead, she observes he price process S and knows he model parameers, ha is, Π, Q, M and he funcion g( ). Hence, informaion available o he large invesor is carried by he filraion F = (F ), F = σ {S u, u }. We noe ha F G. Recall ha he opimizaion problem of he large invesor is o find invesmen sraegies ha maximize he expeced uiliy from erminal wealh. We assume ha his decision depends only on he informaion available o he invesor a ime. Tha is, we consider he self-financing invesmen sraegies h such ha h is F -adaped. Accordingly, an F-adaped self-financing invesmen sraegy which saisfy A1 and A is called an admissible invesmen sraegy. We denoe he se of admissible invesmen sraegies by H F. Suppose we are given a concave, increasing and wice coninuously differeniable uiliy funcion U : R + R. The opimizaion problem of he large invesor is given by [ ] sup E x U(X (h) T ), h H F where E x denoes he condiional expecaion given X = x. Considering F-adaped invesmen sraegies, we naurally end up wih an opimal conrol problem under parial informaion. In he nex par, in order o solve his problem we will derive an equivalen conrol problem under complee informaion via he so-called reducion approach (see, e.g., Fleming and Pardoux 198). 4.1 Reducion of he opimal conrol problem The reducion approach requires he derivaion of he filering equaion for he unobservable sae of he underlying sae variable. In wha follows we will denoe he filer for he unobserved sae of he Markov chain by p = (p 1,, pk ) wih

13 Porfolio opimizaion for large invesor under parial p k = P(Y = e k F ). I follows from he finie-sae propery of he Markov chain ha We now inroduce he process E[μ(Y ) F ]= K μ(e k )p k = Mp. k=1 Ŵ := W + μ(y s ) Mp s ds. (13) σ The following lemma shows ha Ŵ is an F-Brownian moion and gives he F - dynamics of he filer process p. Lemma 1 The process given in (13) is an F -Brownian moion. Moreover, he filer process p k, 1 k K is he unique soluion of dp k = j Q jk p j d + ( M k ) Mp p k σ dŵ, (14) wih he iniial condiion p k = Π k. Proof In order o prove ha Ŵ is an F-Brownian moion we will follow he similar argumens given in he proof of Ellio (198, Lemma 1.4). Firs, i follows from definiion (13) and he Fubini heorem ha we have ] [ E [Ŵ Ŵ Fs s = E s = a.s.. ] μ(y u ) Mp u du + W W Fs s σ Hence, Ŵ is an F-maringale. Second, Ŵ is a coninuous G-semimaringale wih he quadraic variaion Ŵ, Ŵ = W, W =. As his quadraic variaion process is deerminisic, i says same under a change of filraion. Tha is, he F-quadraic variaion of Ŵ a ime is also equal o. Finally, i follows from he Levy s characerizaion of he Brownian moion ha Ŵ is an F-Brownian moion. Showing ha Ŵ is a Brownian moion wih respec o he observaion filraion F, we can make use of he well-known maringale represenaion resuls. This brings us o a siuaion where we can apply he sandard resuls given in, for example, Wonham (1964) and Ellio e al. (1994, Chaper 8), and obain (14). Now i follows from () and Lemma 1 ha F semimaringale decomposiion of X is given by dx (h) X (h) = (h (Mp + g(h )) + (1 h )r) d + h σ dŵ, X = x. (15)

14 Z. Eksi, H. Ku Then, i follows from (14) and (15) ha he (K + 1)-dimensional process (X, p) [, T ] R + Δ K, where Δ K is he K -dimensional simplex, is an F-Markov process. Considering his (K + 1)-dimensional process as he sae process, we inroduce he equivalen opimal conrol problem under complee informaion wih he following: max E x,p [U(X (h) h T )], where E x,p denoes he condiional expecaion given X = x and p = p. Accordingly, he value funcion of he invesor in he reduced model is denoed by he following V (, x, p) = sup E x,p [U(X (h) T )]. (16) h H F 4. Logarihmic uiliy As he firs case we consider he porfolio opimizaion problem for a large invesor wih logarihmic uiliy preferences, ha is, we assume ha U(x) = log(x). In his case, he opimal conrol problem can be solved direcly. Proposiion 3 Suppose g is coninuously differeniable and U(x) = log(x). Then he opimal sraegy h = arg h H F max E x,p [U(X (h) T )] exiss. Moreover, for all [, T ], h H log, where H log is defined by H log { ={ L, L} l : Mp r + g(l) + l( g(l) } σ ) =. (17) l Proof Given he dynamics in (15) we apply Iô s formula for U(X ) = log(x ) and ge T U(X (h) T ) = log(x) + (h s (Mp s + g(h s )) +(1 h s )r h s σ ) T ds + h s σ dŵ s. For any h H F, he sochasic inegral T h s σ dŵ s is well defined and has an expeced value zero. Thus, we have [ T E x,p [U(X (h) T (h )]=log(x)+ex,p s (Mp s +g(h s )) + (1 h s )r h s σ ) ] ds. (18)

15 Porfolio opimizaion for large invesor under parial Now we denoe he inegrand in (18) by f (s, l) = l (Mp s + g(l)) + (1 l)r l σ. I follows from he coninuiy of g ha for any s [, T ], f (s, ) is a coninuous funcion defined on he compac se [ L, L]. Hence, { a maximizer } of f (s, ) exiss. Moreover, he maximizer is an elemen of eiher l : f (s,l) l = or { L, L}. Tha is, h s Hlog s for all s [, T ]. Remark 6 I is possible o exend he resul of Proposiion 3 o he case where he funcions g( ) is coninuously differeniable excep for finiely many poins of he domain [ L, L].LeH denoes he ( se of he poins where g( ) is no differeniable. Then, Proposiion 3 holds wih h H log H ). Nex, as an example we consider he opimizaion problem under parial-informaion wih linear impac funcion and logarihmic uiliy. Formally, we assume g(h) = βh, β>. The following corollary follows from Proposiion 3. Corollary Suppose U(x) = log(x) and g(h) = βh,β >. Then, for all [, T ], { H log = L, L, Mp } r σ. β In paricular, depending on he given se of model parameers we have he following cases: i) If β σ <, hen, for all [, T ], he opimal porfolio sraegy is given by h = Mp r σ β, and he value funcion V (, x, p), for all (, x, p) [, T ] R + Δ K, has he following sochasic represenaion: [ T V (, x, p) = log(x) + r(t ) + E x,p (Mp s r) ] (σ β) ds. ii) If β σ, hen, for all [, T ], he opimal porfolio sraegy is given by h = L ( 1 {Mp r } 1 {Mp r<}), and he value funcion V (, x, p), for all (, x, p) [, T ] R + Δ K, has he following sochasic represenaion: V (, x, p) = log(x)+(t ) ((β σ ) [ T ] )L +r + E x,i L Mp s r ds.

16 Z. Eksi, H. Ku Remark 7 Here, if Mp r < (ha is, if he curren marke environmen is inferred o be unfavorable) he opimal sraegy is o pu negaive weigh in he risky asse (shor selling) and o inves proceeds in he bank accoun. On he oher hand, if Mp r, hen i is opimal o borrow as much as possible from he riskless rae and inves he amoun in he risky asse. In all of he above cases, when compared o he case wih complee informaion he resuling opimal sraegy is obained by replacing he unknown drif by is filer esimae. Tha is, he cerainy equivalence principle holds. 4.3 Power uiliy In wha follows we assume ha U(x) = 1 θ xθ,<θ<1. Here we mainly follow he procedure given in he previous secion and solve he problem by using dynamic programming mehods. To begin wih, for any funcion v C 1, and (, x, p) [, T ] R + Δ K, h H F we define he differenial operaor A h v(, x, p) v(, x, p) v(, x, p) = + x(h(mp + g(h)) + (1 h)r) x + 1 v(, x, p) x x h σ + v(, x, p) Q jk p j p k, j k + 1 σ + xh k k, j v(, x, p) (M k Mp)(M j Mp)p k p j p k p j v(, x, p) x p k (M k Mp )p k. (19) Hypoheically, he value funcion solves he following HJB equaion sup A h v(, x, p) = h v(t, x, p) = 1 θ xθ for all (x, p) R + Δ K. () Due o he general form of he impac funcion g, i is no possible o show he exisence of a soluion or o characerize i as he soluion of equaion () forhe curren opimizaion problem. Insead we consider he opimizaion problem under parial informaion wih power uiliy preferences and linear impac funcion. This case allows o derive he opimal conrol in he feedback form and yields a probabilisic represenaion for he corresponding value funcion. Proposiion 4 Suppose U(x) = θ 1 xθ,g(h) = βh, β>and β (1 θ)σ <. Define γ = (1 θ)σ β. The value funcion is given by V (, x, p) = 1 σ β θ xθ u(, p) γ,for all (, x, p) [, T ] R + Δ K, where u >, wih u(t, p) = 1, for all p Δ K,is he soluion of he parabolic parial differenial equaion

17 Porfolio opimizaion for large invesor under parial ( θ u(, p) (r + γ + k u(, p) p k (Mp r) γ(σ β) Q jk p j + θ γ j )) + 1 σ k, j u(, p) (M k Mp)(M j Mp)p k p j p k p j (Mp r)(m k Mp) σ β p k + u(, p) Moreover, he opimal porfolio sraegy h is given in feedback form as k =. (1) h Mp r (, p) = (1 θ)σ β + γ u(,p) p k (M k Mp)p k u(, p)((1 θ)σ β). () Proof Due o he form of he uiliy funcion and he linear srucure of he dynamics of he wealh process, we use he ansaz v(, x, p) = 1 θ xθ u(, p) γ, for some funcion u γ > wih u(t, p) = 1, for all p Δ K. This gives he following parial derivaives v(, x, p) = 1 θ xθ γ 1 u(, p) γ u(, p), v(, x, p) x = (θ 1)x θ u(, p) γ, v(, x, p) p k p j v(, x, p) p k x = 1 θ xθ v(, x, p) = x θ 1 u(, p) γ, x v(, x, p) = 1 p k θ xθ γ 1 u(, p) γ u(, p), p k ( γ u u γ(γ 1)u(, p) + γ u(, p) γ 1 u(, p) p k p j p k p j = x θ 1 γ 1 u(, p) γ u(, p). p k Subsiuing hese and g(h) = βh in (), we obain he following equaion for funcion u for some γ ha will be deermined below. { sup u(, p)(h(mp + βh) + (1 h)r) + θ 1 u(, p)h σ h [ L,L] + hγ u(, p) } (M k Mp)p k + 1 γ p k k σ (M k Mp)(M j Mp)p k p j θ k, j ( ) u(, p) 1 u(, p) u(, p) + (γ 1)u(, p) + γ u(, p) p k p j p k p j θ + γ u(, p) Q jk p j =, (3) θ p k k, j wih u(t, p) = 1. The necessary condiion for he opimizer of he above equaion is given by ), h(β (1 θ)σ )u(, p) + (Mp r)u(, p) + γ k u(, p) p k (M k Mp)p k =.

18 Z. Eksi, H. Ku Provided ha β (1 θ)σ < and u(, p) > (see Proposiion 5 below) he necessary condiion is also sufficien. This suggess ha he maximizer is (). Now we muliply (3) byθ/γ and inser h. This gives u(, p) = + u(, p) Q jk p j + θ u(, p)r p k, j k γ + h (, p) u(, p) θ ( (1 θ)σ ) β + 1 γ σ (M k Mp)(M j Mp)p k p j k, j ( ) u(, p) 1 u(, p) u(, p) + (γ 1)u(, p). (4) p k p j p k p j This is a non-linear equaion. In wha follows, in order o eliminae he non-lineariy in (4) we follow he idea given in Zariphopoulou (1) and choose γ o saisfy Wih his choice of γ we have = γ = (1 θ)σ β σ. β ( γ k p u k (M k Mp)p k u(, p)((1 θ)σ β) ( + γ k u p k (M k Mp)p k ) u(, p) θ ( (1 θ)σ γ ) 1 γ 1 u(, p) σ, ) β and hence we ge he linear parabolic differenial equaion given in (1). Here noe ha he coefficiens are coninuous and bounded funcions and hence here exiss a soluion o his parabolic differenial equaion, and also he soluion o his Cauchy problem is unique (see, for example, Friedman 1983). We have proved ha he funcion v(, x, p) = 1 θ xθ u(, p) γ is a soluion of he HJB equaion (). Nex we will show v(, x, p) is indeed he opimal value funcion V (, x, p).leh H F be an arbirary invesmen sraegy. By applying Iô s formula o v(, x, p), wehave T v(t, X (h) T, p T ) = v(, x, p) + T + + T v(s, X (h) k A h v(s, X s (h), p s ) ds s, p s ) x v(s, X (h) s, p s ) p k X s (h) h s σ dŵ s ( M k ) Mp s ps k σ dŵ s

19 Porfolio opimizaion for large invesor under parial T v(, x, p) + (X s (h) ) θ u(s, p s ) γ h s σ dŵ s T 1 (h) + (X s ) θ γ u(s, p s ) γ 1 u(s, p ( s) M k ) Mp s ps k θ p k k σ dŵ s (5) Noe ha he inequaliy (5) follows from he HJB equaion. Under A, he above sochasic inegrals wih respec o Ŵ s are local maringales. Considering hey are bounded below due o he fac v, hey are supermaringales. Taking condiional expecaions, [ ] 1 E,x,p (h) (X T θ )θ v(, x, p). Therefore, we have V (, x, p) v(, x, p). If we have a porfolio sraegy h (, p) given by (), we have equaliy in (5) wih h (, p). Moreover, h (, p) is bounded, u(s, p s ) and u(s,p s) p k are coninuous, hus he sochasic inegrals are in fac maringales. Taking condiional expecions, we have The proof is now compleed. E,x,p [ 1 θ (X (h ) T ) θ ] = v(, x, p). Remark 8 If β (1 θ)σ >, he opimal sraegy will be eiher L or L. For he complee informaion case, his decision only depends he relaion beween he Markov modulaed par of he drif and he risk-free rae. On he conrary, for he parial informaion case he decision is raher complicaed and depends on he values of he variables in (3). Here, comparing he resuling opimal conrol wih he conrol corresponding o he complee informaion case, we conclude ha he cerainy equivalence principle does no hold for he case of power uiliy. In paricular, in he curren case here is an addiional erm arising due o he uncerainy abou he sae of he marke. Proposiion 5 Suppose β (1 θ)σ < holds. Then, for all (, p) [, T ] Δ K, he funcion u has he following sochasic represenaion, { θr θ u(, p) = E [exp P (T ) + γ γ (σ β) T p ] (Mp s r) ds} = p, where he kh componen of process p has he following dynamics under measure P : dp k = j Q jk p j + θ γ (6) (Mp r)(m k Mp ) σ p k d + Mk Mp p k β σ dwp.

20 Z. Eksi, H. Ku Opimal Trading Sraegy (wih β=) h *,p =. 1 h *,p =.5 1 h *,p =.9 1 h *, h *, Time.5 Opimal Trading Sraegy (wih β=.) Time Fig. 1 Opimal porfolio sraegy for power uiliy preferences wih and wihou price impac for full (dashed) and parial (solid) informaion seings. h,1 (h, ): opimal sraegy when he Markov chain is in he good (bad) sae, h,p 1=a : opimal sraegy when he filer is in sae p = (a, 1 a) Proof Resul immediaely follows from he applicaion of he Feynman-Kac heorem (see, e.g., Pham (9, Thm ) for he pde given in (1). 5 Numerical sudy In he numerical sudy, we firs compare he opimal rading sraegy for he full and parial informaion seings wih and wihou price impac. To his, we se he following se of parameers: T =, β =., σ =.8, M = (.6,.), r =.1, θ =.3, (Q 1, Q 1 ) = (,.1). Opimal rading sraegies for he fullinformaion case obained in a sraighforward way, ha is, by insering he parameers on he formulas given in Proposiion. On he oher hand, o obain he corresponding opimal sraegies in he case of parial informaion, we use an explici finie-difference

21 Porfolio opimizaion for large invesor under parial h *,p =. 1 h *,p =.5 1 h *,p =.9 1 h *, h *,1 Opimal Trading Sraegy θ Fig. Opimal porfolio sraegy wih full (dashed) and wih parial (solid) informaion for power uiliy preferences wih differen levels of risk aversion. h,1 (h, ): opimal sraegy when he Markov chain is in he good (bad) sae, h,p 1=a : opimal sraegy when he filer is in sae p = (a, 1 a) 1 Value Funcion V β=. (,.) V β= (,.) V β=. (,.9) V β= (,.9) Time Fig. 3 Value funcion for parial informaion case wih (dashed) and wihou (solid) price impac for power uiliy preferences when he filer is in sae p = (.,.98) and p = (.9,.1) mehod and solve he pde given in (1) numerically. Figure 1 shows ha he presence of he price impac yields an upward shif on he opimal rading sraegies, wih an amoun depending on he ime and on he sae of he filer. In paricular, he higher he value of p 1, he larger he shif. Noe also ha he amoun of he shif decreases as he ime ges closer o he mauriy ime T. Nex, we analyze he behavior of opimal rading sraegies wih respec o he risk aversion level, represened by θ. Figure suggess ha all ypes of opimal rading sraegies are increasing wih a decreasing level of risk aversion.

22 Z. Eksi, H. Ku We finally compare he value funcions of he opimal conrol problem under parial informaion wih and wihou price impac. We conclude ha he value funcion wih price impac dominaes he one wihou he price impac for any sae of he filer. Resuls for he case when he filer is in sae p = (.,.98) and p = (.9,.1) are given in Fig Conclusion We invesigae he problem of maximizing he expeced uiliy from erminal wealh of a large invesor whose acion has some lasing impac on he price process under he changing marke environmen. We solve he opimal invesmen problem explicily wih he linear price impac under complee informaion. The decision of opimal sraegies depends on he relaionship beween he Markov modulaed par of he drif and he risk-free rae, and he relaionship beween he price impac par of he drif and he volailiy. We hen sudy he invesmen problem furher for he large invesor under parial informaion, and obain he opimal invesmen sraegies by he reducion approach. For power uiliy, he opimal sraegy is given, unlike he complee informaion case, wih an addiional erm due o he uncerainy abou he marke condiion. We observe, for logarihmic and power uiliy funcions reaed in he paper, he large invesor would gain benefis from he price impac by choosing opimal sraegies under parial informaion as well as complee informaion. The quesions on opimal invesmen sraegies for he non-consan volailiy σ( ) case remain unanswered. This may be an ineresing and exciing case since acions of he invesor creae a rade-off beween he increase in he conrolled par and he decrease in he precision of he esimaes of he unobserved par of he drif. We leave his challenging issue for fuure sudy. Acknowledgemens The auhors hank Rüdiger Frey for helpful discussions. The auhors also hank he paricipans of 14h Europ Workshop on Advances in Coninuous Opimizaion and 9h Bachelier Finance Congress for discussions. References Almgren R, Chriss N (1) Opimal execuion of porfolio ransacions. J Risk 3:5 4 Bäuerle N, Rieder U (4) Porfolio opimizaion wih markov-modulaed sock prices and ineres raes. IEEE Trans Auom Conrol 49(3): Bäuerle N, Rieder U (5) Porfolio opimizaion wih unobservable markov-modulaed drif process. J Appl Probab 4: Björk T, Davis MHA, Landén C (1) Opimal invesmen under parial informaion. Mah Mehods Oper Res 71(): Busch M, Korn R, Seifried FT (13) Opimal consumpion and invesmen for a large invesor: an inensiybased conrol framework. Mah Financ 3(4): Cuoco D, Cvianić J (1998) Opimal consumpion choices for a largeinvesor. J Econ Dyn Conrol (3): Cvianić J, Ma J e al (1996) Hedging opions for a large invesor and forward-backward sde s. Ann Appl Prob 6(): Ellio Rober J (198) The non-linear filering equaions. In: Kohlmann M, Chrisopei N (eds) Sochasic Differenial Sysems. Springer, Berlin, pp

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