Ambiguity, Risk and Portfolio Choice under Incomplete Information

Transcription

1 ANNALS OF ECONOMICS AND FINANCE 1-2, (29) Ambiguiy, Risk and Porfolio Choice under Incomplee Informaion Jianjun Miao * Deparmen of Economics, Boson Universiy, 27 Bay Sae Road, Boson, MA 2215, and Zhongnan Universiy of Economics and Law miaoj@bu.edu This paper sudies opimal consumpion and porfolio choice in a Meronsyle model wih incomplee informaion when here is a disincion beween ambiguiy and risk. The laer disincion is afforded by adopion of recursive muliple-priors uiliy. The fundamenal issues are: (i) How does he agen opimally esimae he unobservable processes as new informaion arrives over ime? (ii) Wha are he effecs of ambiguiy and incomplee informaion on behavior? This paper shows ha i is opimal o firs use any prior o perform Bayesian esimaion and hen o maximize expeced uiliy wih ha prior based on he resuling esimaes. Finally, he paper shows ha a hedging demand arises ha is affeced by boh ambiguiy and esimaion risk. Key Words: Ambiguiy; Recursive muliple-priors uiliy; Incomplee informaion; Porfolio choice; Hedging; Esimaion risk. JEL Classificaion Numbers: D81, G INTRODUCTION In economic analysis, i is ypically assumed ha a decision maker s beliefs are represened by a single probabiliy measure. Frank Knigh (1921) emphasizes he disincion beween risk where here are probabiliies o guide choice, and ambiguiy where likelihoods of evens are oo imprecise o be adequaely summarized by probabiliies. The Ellsberg Paradox (1961) ells us ha his disincion is behaviorally significan. This suggess ha here are wo dimensions of he decision maker s beliefs abou he likeli- * I would like o hank Jerome Deemple, Larry Epsein, and Ali Lazrak for helpful commens. This paper was originally wrien in 21. I have made no subsanial changes excep ha I have updaed references /29 All righs of reproducion in any form reserved.

2 258 JIANJUN MIAO hoods of evens: risk and ambiguiy. In sandard models, ambiguiy is negleced or i is assumed ha he decision maker is indifferen o i. Added moivaion for he analysis o follow comes from he finance lieraure on incomplee informaion. In he real world, invesors make consumpion and invesmen decisions based on informaion available from sources such as newspapers, financial repors, and marke daa. I is unrealisic o assume ha hey observe he driving uncerainy processes underlying prices and reurns. These unobservable processes (or parameers) mus be learned as new informaion arrives over ime. In he sandard Bayesian analysis, he decision maker has a unique prior over he unobservable processes. The prior hen is updaed by Bayes Rule as new informaion arrives. Moreover, esimaes of hese processes are obained by Bayesian esimaion. This Bayesian approach emphasizes he effec of esimaion risk on opimal behavior. To incorporae ambiguiy, his paper asks he following quesion: How does a decision maker choose when he is averse o ambiguiy and when his informaion is incomplee? The firs sep in addressing his quesion is o formulae a uiliy funcion ha permis he disincion beween risk and ambiguiy under incomplee informaion. Chen and Epsein (22) provide such a disincion under complee informaion by generalizing Gilboa and Schmeidler s (1989) saic model o a dynamic seing; hey call heir model recursive muliple-priors uiliy. 1 This paper adaps heir formulaion o an environmen wih incomplee informaion. The resuling model of uiliy is applied o sudy a single agen s consumpion and invesmen decisions in a coninuous-ime Meron-syle model wih incomplee informaion. The issues hen become: (i) How does he agen opimally esimae he unobservable uncerainy processes underlying asse prices as informaion arrives over ime. (ii) Wha are he effecs of ambiguiy and incompleeness of informaion on behavior? If one views he agen s planning problem as a conrol problem, for he sandard expeced uiliy model, here is a well-known separaion principle in he conrol lieraure (e.g., Flemming and Rishel (1975)). This principle saes ha conrol under incomplee informaion can be solved separaely by he wo independen problems of filering (or esimaion) and conrol under complee informaion. I is naural o conjecure ha his principle is also rue for recursive muliple-priors uiliy. Because here is a unique prior under expeced uiliy or risk-based uiliy such as sochasic differenial uiliy proposed by Duffie and Epsein (1992), esimaion is no a problem because sandard Bayesian analysis applies. 1 Epsein and Schneider (23) develop an aximomaic foundaion for recursive mupiple-priors in a discree-ime framework wih complee informaion.

3 AMBIGUITY, RISK AND PORTFOLIO CHOICE 259 However, when he agen has muliple priors, i is no clear a priori how o perform esimaion: Can one perform Bayesian esimaion using one of he priors in he se? If so, which ones are suiable? This paper shows ha he separaion principle sill holds for recursive muliple-priors uiliy. In paricular, opimaliy is consisen wih he use of any measure in he se of priors o perform Bayesian esimaion. In an incomplee informaion environmen, he key o he above resuls is ha (i) he se of priors is updaed by applying Bayes Rule o each prior in he se, and his leads o dynamic consisency; (ii) all measures in he se of priors and heir resricions on he observaion filraion are muually absoluely coninuous. Thus given ha he rue probabiliy measure is one of he priors, one can obain Bayesian esimaes of unobservable processes using any measure in he se of priors and equivalenly rewrie he agen s budge consrain in erms of hese esimaes under he corresponding measure. Accordingly, he agen s opimizaion problem is ransformed ino an environmen wih complee informaion and he preceding esimaion procedure is opimal. Wih regard o he characerizaion of opimal consumpion and porfolio choice, I find ha consisen wih he separaion principle, a wo-sep procedure consising of ordinary filering and ordinary maringale mehods can be used o solve he agen s problem. Finally, I provide examples wih logarihmic and power feliciy funcions ha deliver closed form soluions. I show ha under complee informaion here is no hedging demand even when ambiguiy is presen. The effec of ambiguiy is ha he agen myopically holds a mean-variance efficien porfolio bu wih disored mean values of asse reurns. In conras, here is a hedging demand under incomplee informaion. This demand is affeced by boh ambiguiy and esimaion risk Relaed Lieraure Chen and Epsein (22) formulae recursive muliple-priors uiliy in coninuous ime. They also apply his uiliy o a Lucas-syle represenaive agen model o sudy asse pricing implicaions. Epsein and Miao (2) apply recursive muliple-priors uiliy o sudy a heerogeneous agen model o address he consumpion home bias and equiy home bias puzzles. Boh of hese papers assume complee informaion. There is a large lieraure sudying consumpion and porfolio choice wih incomplee informaion in he expeced uiliy framework (see Bawa, Brown and Klein (1979), Deemple (1986), Gennoe (1986), Karazas and Xue (1991), Feldman (1992), Lakner (1995), Brennan (1998), Lakner (1998), Barberis (2), Karazas and Zhao (1998), and Xia (2) and he references cied herein). Cvianic e al. (2) sudy he corresponding prob-

4 26 JIANJUN MIAO lem for sochasic differenial uiliy. This paper adds o his lieraure using recursive muliple-priors uiliy. My model is relaed o a series of papers by Hansen and Sargen and heir coauhors (see Anderson e al. (23), Cagei e al. (22), Hansen e al. (26) and Hansen and Sargen (21)). These papers sudy models of robus conrol where he decision maker fears model uncerainy and seeks robus decision-making, which are also moivaed in par by he Ellsberg Paradox Ouline The paper proceeds as follows. Secion 2 defines recursive muliple-priors uiliy under incomplee informaion. Secion 3 applies his uiliy o sudy opimal consumpion and porfolio choice in a Meron-syle model wih incomplee informaion. Secion 4 provides examples ha deliver explici soluions. Proofs are relegaed o an appendix. 2. RECURSIVE MULTIPLE-PRIORS UTILITY This secion adaps Chen and Epsein (21) and defines recursive muliplepriors uiliy under incomplee informaion ha also conforms wih he axiomaizaion in Epsein and Schneider (23) Informaion Srucure Time is coninuous in he finie horizon [, T ]. There is a complee filered probabiliy space (Ω, F T, {F } T =, P ) on which a d -dimensional sandard Brownian moion on R d W = (W 1,..., W d ) is defined. 3 The filraion {F } T = or simply {F } represens complee informaion. The probabiliy measure P is a reference measure. The decision maker s available informaion is represened by a sub-filraion {G } where each G F. Assume ha {G } is generaed by some R d -valued observable diffusion process (y ). The following assumpion is crucial and common in he lieraure on incomplee informaion. Assumpion 1. There is a d-dimensional sandard Brownian moion Ŵ defined on he filered probabiliy space (Ω, G T, {G }, P ) such ha he augmened naural filraion generaed by Ŵ is idenical o {G }. 2 See Hansen e al. (26) and Hansen and Sargen (21) for surveys of he robus conrol model and Epsein and Schneider (21) for deailed comparison wih he recursive muliple-priors model. 3 All processes o appear in he sequel are progressively measurable and all equaliies and inequaliies involving random variables (processes) are undersood o hold dp a.s. (d dp a.s.). Denoe by E Q [ ] and E Q [ ] he expecaion and condiional expecaion aken wih respec o he measure Q. When Q is suppressed i is undersood ha Q = P. Finally, denoe by he Euclidean norm.

5 AMBIGUITY, RISK AND PORTFOLIO CHOICE 261 Because he Brownian moion W is unobservable, I use he observable Brownian moion Ŵ o define uiliy under incomplee informaion in he sequel. The Brownian moion Ŵ is ofen referred o as an innovaion process. I can be exraced from he decision maker s observaion process (y ) by filering heory. 4 For example, suppose ha d = 2, d = 1 and ha he decision maker observes (y ) bu no (W 1, W 2 ) and (x ) where dx = x d + dw 1 and dy = x d + σ y dw 2. Assume σ Y is a nonzero consan. Then Ŵ is delivered by dŵ = (σ y ) 1 (dy E [x G ] d). Noe ha d migh no be equal o d because he decision maker may observe an arbirary dimensional process (y ). However, I assume d = d in he laer applicaions Consumpion Space There is a single perishable consumpion good. A consumpion process c is nonnegaive, real-valued, progressively measurable [ wih respec o he ] T filraion {G } and square inegrable (i.e. E c2 d < ). Denoe by C he se of all consumpion processes Uiliy A recursive muliple-priors uiliy process (V (c)) for each c C is defined by five primiives: informaion srucure ((Ω, G T, {G }, P )), he Brownian moion Ŵ, he se of priors (probabiliy measures) P on (Ω, G T ), he discoun rae β >, and he feliciy funcion u : R + R. The consrucion of he se of priors P is key. 5 Take all measures in P o be equivalen o P. They can be defined via heir densiies by use of densiy generaors and Girsanov s Theorem. Specifically, define a densiy generaor θ = (θ ) as an R d -valued {G }-adaped process saisfying sup θ i () κ i, i = 1,..., d, where κ = (κ 1,..., κ d ). Denoe by Θ he se of all such densiy generaors. This specificaion of Θ is referred o as κ-ignorance in Chen and Epsein (22). 6 4 See Lipser and Shiryayev (1977) for an inroducion o filering heory. 5 Noe ha he se of priors is delivered as par of he uiliy represenaion from behavior (see Epsein and Schneider (23)). In applicaions, one mus specify his se so ha i is consisen wih behavior, e.g., some axiomaic foundaion. 6 See Chen and Epsein (22) for more general specificaions of Θ.

6 262 JIANJUN MIAO Then each densiy generaor θ generaes a (P, {G })-maringale (z θ ) : z θ { = exp 1 2 θ s 2 ds } θ s dŵs, T, (1) which deermines a probabiliy measure Q θ on (Ω, G T ) via dq θ dp The se of priors is defined by = zθ T,, and dqθ dp = z θ. (2) G P = {Q θ : θ Θ and Q θ is given by (2)}. (3) Because P expands as κ increases, one can inerpre κ as an ambiguiy aversion parameer. Finally, define he recursive muliple-priors uiliy process (V (c)) for each c C as: [ ] T V (c) = min E Q e β(s ) u(c s ) ds Q P G, T. (4) Abbreviae V ( ) by V ( ) and refer o i as recursive muliple-priors uiliy. The recursive muliple-priors uiliy model under complee informaion sudied in Chen and Epsein (22) corresponds o he case where {G } = {F } and Ŵ = W. Finally, he sandard expeced uiliy model is obained when κ = in which case P = {P }. Wih regard o he properies of uiliy, firs he uiliy process (V (c)) is dynamically consisen because he following recursive relaion holds: [ τ ] V = min E Q e β(s ) u(c s ) ds + e β(τ ) V τ G, < τ T. Q P This propery follows from he fac ha he uiliy process (V (c)) is he unique soluion o he following backward sochasic differenial equaion (BSDE), 7 dv = [ u(c ) + βv + max θ Θ θ σ V ] d + σ V dŵ, V T =. (5) 7 Sufficien condiions h are ha u be Borel measurable and ha i saisfy a growh R i condiion ensuring E T u2 (c ) d < for all c in C. See El Karoui e al. (1997) for an excellen survey of he heory and applicaions of he backward sochasic differenial equaions.

7 AMBIGUITY, RISK AND PORTFOLIO CHOICE 263 Noe ha he volailiy (σ V ) a c, denoed more fully by (σ V (c)), is deermined as par of he soluion o he BSDE; i plays a key role in he sequel. 8 Because 9 max θ σ V = θ σ V, for θ = κ sgn(σ V (c)), (6) θ Θ BSDE (5) can be wrien as dv = [ u(c ) + βv + θ σ V ] d + σ V dŵ, V T =. (7) Noe ha he measure delivered by he densiy generaor θ achieves he minimum in (4). Finally, assume ha u >, u <. Then by Chen and Epsein (22), each V ( ) is coninuous, increasing and sricly concave. Also assume ha he following Inada condiion holds: lim x + u (x) = and lim x u (x) =. 3. OPTIMAL CONSUMPTION AND PORTFOLIO CHOICE This secion applies recursive muliple-priors uiliy o sudy he opimal consumpion and porfolio choice problem wih incomplee informaion The Environmen Financial markes. Uncerainy is represened by a complee filered probabiliy space (Ω, F T, {F } T =, P ) on which is defined a d-dimensional sandard Brownian moion W = (W 1,..., W d ). There are d + 1 securiies consising of one riskless bond and d non-dividend-paying socks. The price of he riskless bond is given by S = e r, [, T ], where he riskless rae r is a posiive consan. Denoe by S i he price of he i h sock and by R i = ds i /S i is reurn, i = 1,..., d. Assume ha he iniial price S i is a given posiive consan and ha he vecor of reurns R = (R 1,..., R d ) saisfies dr = µ R d + σ R dw, (8) 8 Boh (V ) and (σ V ) are progressively measurable wih respec o {G} and sequare inegrable. 9 For any d-dimensional vecor x, sgn(x) is he d-dimensional vecor wih i h componen equal o sgn(x i ) = x i / x i if x i and = if x i =. For any y R d, y sgn(x) denoes he vecor in R d wih i h componen y i sgn(x i ).

8 264 JIANJUN MIAO where he volailiy σ R is a d d marix of real-valued consans. On he oher hand, he vecor of mean reurns µ R = (µ R 1,..., µ R d ) : Ω R d is an F -measurable random variable wih disribuion ν(a) = P (µ R A) for any Borel se A in R d ha saisfies: R d b ν(db) <. Thus µ R is independen of W. In he sandard Bayesian analysis, ν is he prior disribuion of µ R. Assume ha he volailiy marix σ R saisfies he following assumpion which ensures ha financial markes are complee (e.g., Duffie (1996)). Assumpion 2. σ R is inverible. Define he marke price of uncerainy process (η ) by 1 η = (σ R ) 1 (µ R r1), T, (9) where 1 is he vecor in R d wih each componen equal o 1. Then he following Lemma holds (see Lakner (1995)). Lemma 1. The process Z defined by { Z = exp η s dw s 1 2 } η s 2 ds is a (P, {F })-maringale. Informaion srucure. Assume ha he bond price S and sock prices S are given exogenously. Denoe by {F S } he augmened filraion generaed by he price processes. Complee informaion is represened by {F }, he augmened filraion generaed by µ R and W. However, he agen does no observe he Brownian moion W and he mean reurns µ R. Raher, his informaion is represened by he filraion {F S } where each F S F. Thus we are in he se-up of secion 2.1 wih {G } = {F S }. Budge consrain. There is a single consumpion good aken as he numeraire. Consumpion processes lie in he consumpion space C defined in secion 2.2. Denoe he wealh process by (X ). A porfolio (share) ψ is an R d -valued {F S }-adaped progressively measurable process such ha 1 Following Chen and Epsein (22) and Epsein and Miao (23), he deviaion from he usual erminology of marke price of risk is o emphasize ha uncerainy includes boh risk and ambiguiy in he model.

9 AMBIGUITY, RISK AND PORTFOLIO CHOICE 265 T ψ s 2 ds <. The componen ψ i () represens he proporion of wealh invesed in he i h socks a ime. Thus 1 ψ 1 is he proporion invesed in he bond. Denoe he se of all porfolios by Ψ. Endowed wih iniial wealh X >, he agen makes consumpion and invesmen decisions based on informaion represened by {F S }. His budge consrain is given by dx = {[ r + (ψ ) (µ R r1) ] X c } d + X (ψ ) σ R dw. (1) Preferences. The above environmen is sandard. The deparure from he sandard model is ha preferences are represened by he recursive muliple-priors uiliy funcion V corresponding o he se of priors defined in (3). In order o ensure ha V is well defined, inroduce he process (Ŵ): Ŵ = (σ R ) 1 [dr τ µ R τ dτ], (11) where µ R () E [ ] µ R F S is a measurable version of he condiional expecaion of µ R wih respec o he price filraion {F S }. The following lemma implies ha Ŵ defined in (11) saisfies Assumpion 1. Thus recursive muliple-priors uiliy V is well defined. The proof of his lemma is sandard (see, e.g., Lipser and Shiryayev (1977)). Lemma 2. Ŵ is a (P, {F S })-Brownian moion. Moreover, he augmened filraion generaed by he Brownian moion Ŵ coincides wih {F S } The Decision Problem and Separaion Principle Decision problem. The agen makes consumpion and invesmen plans for he enire horizon a ime zero by solving: subjec o (1) and sup V (c) (12) (c,ψ) C Ψ X, [, T ], X > given. (13) The credi consrain (13) rules ou doubling sraegies (e.g., Dybvig and Huang (1988)). Noe ha he consumpion and porfolio processes are required o be adaped o he price filraion {F S }. Finally, because he

10 266 JIANJUN MIAO uiliy process is dynamically consisen, he opimal plan will be carried ou as ime proceeds. Separaion principle. I solve his problem by he separaion principle. In order o undersand how his principle works for recursive muliple-priors uiliy, consider firs he sandard seing where κ = and V is an expeced uiliy funcion. In his case, he agen s unique prior is represened by P. By (8) and (9), (Ŵ) defined in (11) saisfies Ŵ = W + (η s η s )ds, (14) where η E [ ] η F S is a measurable version of he condiional expecaion of η wih respec o {F S }. Denoe µ R () = ( µ R 1 (),..., µ R d ()). Then η = (σ R ) 1 ( µ R r1). (15) By (8), (11) and (15), under prior P he agen s perceived reurns dynamics is and he budge consrain (1) becomes dr = µ R d + σ R dŵ (16) dx = (rx c )d + X (ψ ) σ R [dŵ + η d]. (17) Because Ŵ and η are adaped o {F S }, all processes in (17) are adaped o {F S }. Thus he agen s problem has been ransformed ino one wih complee informaion and filraion {F S } where he Bayesian esimae η ( µ R ) is reaed as he rue marke price of uncerainy (mean reurns). Afer using sandard filering heory (see Lipser and Shiryayev (1977)) o deermine he condiional disribuion of η (or µ R ), he usual opimizaion ools under complee informaion can be applied. Wha happens when he agen has a se of priors? Noe ha he above ransformaion (16) is performed using he single prior P. When he agen has a se of priors P, his ransformaion can ake many forms depending on which prior in he se P is used. Formally, consider any Q P and denoe by θ he corresponding densiy generaor. By Girsanov s Theorem, he process Ŵ Q defined by dŵ Q = dŵ + θ d is a Q-Brownian moion and he naural filraion generaed by Ŵ Q coincides wih {F S }. Then he budge consrain can be wrien as [ ] dx = (rx c )d + X (ψ ) σ R dŵ Q + ( η θ )d. (18)

11 AMBIGUITY, RISK AND PORTFOLIO CHOICE 267 Because ( η θ) and Ŵ Q are {F S }-adaped, when he agen reas ( η θ ) as he observable esimae of he marke price of uncerainy using measure Q, he problem is ransformed ino he complee informaion world. Consequenly, he usual opimizaion ools under complee informaion can be applied. In sum, because all measures in he se of priors are equivalen all corresponding ransformed budge consrains are equivalen o he original one (1). Hence using any measure in he se of priors o perform esimaion leads o he same opimum Two-sep Procedure Consisen wih he separaion principle, I firs use he reference measure P o perform esimaion and ransform he budge consrain (1) ino (17) as in he preceding subsecion. Then he problem is reformulaed as a saic Arrow-Debreu problem. Finally, from his problem, I derive opimal consumpion and porfolio choice (e.g., Duffie and Skiadas (1994)). Filering. By Lemma 1 and Girsanov s Theorem, one can define a probabiliy measure P equivalen o P on (Ω, F T ) via d P /dp = Z T such ha he d-dimensional process W defined by W = W + is a ( P, {F })-Brownian moion. Then, by (8) and (9), dr = rd + σ R d W. η s ds (19) Thus, P is an equivalen maringale measure because he vecor of discouned prices, (e r S ), is a P -maringale. The following facs are imporan for he characerizaion of opima. By Lakner (1998), he (P, {F S })-maringale (Ẑ) defined by Ẑ E [ Z F S ], T, is an indisinguishable version of he process { exp η s dŵs 1 } η s 2 ds, T. (2) 2 Therefore, by (14) and Girsanov s Theorem, he process W defined by (19) saisfies W = W + η s ds = Ŵ + η s ds (21)

12 268 JIANJUN MIAO and i is a ( P, {F S })-Brownian moion. Moreover, he augmened naural filraion of W coincides wih he price filraion {F S } (see Lakner (1995) Proposiion 4.1). Noe ha P is also a probabiliy measure on (Ω, F S T ) defined by d P /dp = ẐT. Saic Arrow-Debreu problem. The exisence of an equivalen maringale measure P and he credi consrain (13) rule ou arbirage opporuniies (see Duffie (1996)). Because here is no arbirage and markes are complee, a unique sae price densiy process (p ) relaive o measure P is delivered by p = e r Ẑ. The following heorem is sandard (see Karazas and Xue (1991) or Lakner (1995)). Theorem 1. (i) For any consumpion process c C, here exis a porfolio process ψ and a wealh process X such ha (c, ψ, X) saisfies he dynamic budge consrain (17) and he credi consrain (13) if and only if [ ] T E p c d X. (22) (ii) If he above inequaliy holds wih equaliy, he porfolio process ψ is unique up o equivalence and given by ψ = e r ((σ R ) ) 1 φ /X, (23) where [ ] T e r X = E ep e r c d F S = X + φ s d W s. The corresponding wealh process is given by [ ] X = 1 T E p s c s ds p F S. (24) Thus he consumpion process c can be found by solving he saic Arrow-Debreu problem: [ T ] sup V (c) subjec o E p c d X. (25) c C

13 AMBIGUITY, RISK AND PORTFOLIO CHOICE 269 The opimal porfolio process ψ is hen delivered by (23). Uiliy supergradien. In order o solve problem (25), i is useful o find he supergradiens for V. A supergradien for V a he consumpion process c C is a process (π ) saisfying [ ] T V (c ) V (c) E π (c c ) d, for all c in C. By Chen and Epsein (22), for each θ saisfying (6), he process is a supergradien for V a c. π (c) = e β u (c ) z θ, T, (26) Opimal plan. Denoe by J he value funcion of problem (25). Assume ha J(X ) <. Then i is easy o show ha he value funcion for problem (12) is also finie and equal o J(X ). The following heorem characerizes an opimum for problem (12). Theorem 2. (i) The opimal consumpion process c is given by e β z θ u (c ) = λp, (27) where λ > is such ha [ ] T E p c d = X, (28) and (θ ) saisfies θ = κ sgn(σ V (c )). (29) Here (V (c ), σ V (c )) is he unique soluion o BSDE (7) for c = c. (ii) The opimal wealh process X is given by (24) where c = c. (iii) The opimal porfolio ψ is given by where (φ ) saisfies e r X + ψ = e r ((σ R ) ) 1 φ X, [ ] T e r c ds = E ep e r c d F S = X + φ s d W s. (3)

14 27 JIANJUN MIAO As is well known, he opimal porfolio is relaed o he inegrand of he maringale represenaion in (3). Secion 4 will give wo examples o clarify he naure of he opimal porfolio. If an opimum exiss i mus be unique. This is because sric concaviy of V ( ) implies he opimal consumpion process is unique. Then by (27) he opimal densiy generaor is also unique. In sum, he following wo-sep procedure can be used o solve opimal consumpion and porfolio choice described above. Sep 1. (Ordinary filering) Firs, use he sandard filering echnique (e.g., Karazas and Zhao (1998)) o solve for he condiional disribuion of µ R and he condiional expecaion µ R = E[µ R F S ] for each. Nex use (15) and (2) o solve for Ẑ. Finally, le p = e r Ẑ. Sep 2. (Ordinary maringale mehod) Given any θ Θ, solve he following sysem of wo equaions: e β z θ u (c ) = λp, [ ] T E p c d = X, for c and λ o yield c = g(θ) where g maps Θ ino C. Second, solve BSDE (7) for he volailiy of (V (c)) when c = g(θ) o obain σ(g(θ)). The opimal densiy generaor θ is given by he following fixed poin problem: θ = κ sgn ( σ V (g(θ)) ), T. (31) Finally, if here exiss a soluion θ o (31), he opimal consumpion process c and porfolio ψ are given by Theorem Hedging Moives In order o undersand he effecs of ambiguiy on opimal choice, i is useful o consider firs a limied observaional equivalence poined ou in Chen and Epsein (22) and Epsein and Miao (23). Noice ha equaion (27) is idenical o ha for an expeced uiliy maximizer who uses he single prior Q corresponding o he densiy generaor θ : dq /dp = exp { 1 2 T θ s 2 ds T θ s dŵs }. (32) Thus he opimum characerized in Theorem 2 can be generaed in a sandard model wihou ambiguiy where he agen uses a disored belief Q.

15 AMBIGUITY, RISK AND PORTFOLIO CHOICE 271 Because Q is endogenously delivered by ambiguiy, following Epsein and Miao (23), i is naural o refer o Q as ambiguiy adjused probabiliy beliefs. By (16) and Girsanov s Theorem, under Q he agen s perceived reurns dynamics is dr = ( µ R σ R θ ) d + σ R dŵ, (33) where he (Q, {F S })-Brownian moion (Ŵ ) is defined by dŵ = dŵ + θ d. (34) From (33), here are wo facors influencing he deviaions of he agen s perceived mean reurns from heir rue values: µ R ( µ R σ R θ ) = (µ R E[µ R F S ]) + σ R θ. The firs erm represens esimaion risk and he second erm reflecs ambiguiy. Because hese erms are ime-varying, invesmen opporuniies change over ime and wo separae hedging moives arise. 4. EXAMPLES Consider he power feliciy funcion: u(x) = x γ /γ, x R +, γ < 1, where 1 γ is he coefficien of relaive risk aversion. The following heorem characerizes an opimum. Theorem 3. where λ = (i) The opimal consumpion process is given by ( c = ( e β ) z 1 θ 1 γ, (35) λp ( θ = κ σ H /γ + 1 ) 1 γ ( η θ ), (36) E [ T ] ) 1 γ (p ) γ 1 γ (e β z θ ) 1 1 γ d /X, and (H, σ H ) is given below. The dynamics of c is given by dc /c = µ c d + σ c dŵ,

16 272 JIANJUN MIAO where (µ c ) and (σ c ) saisfy σ c = 1 1 γ ( η θ ) and (37) µ c = 1 1 γ (r β) (2 γ)σc σ c + σ c θ. (38) (ii) The uiliy process a c is given by where (H, σ H ) is he unique soluion o he BSDE: where µ H = γ 1 γ V = (c ) γ γ H, (39) dh /H = µ H d + σ H dŵ, H T =, (4) [ β/γ r ( η θ ) ( η θ ] ) H 1 +(θ ασ c ) σ H. (41) 2(1 γ) (iii) The opimal wealh process is given by [ ] X = 1 T E p s c p sds F S = c H. (42) (iv) The opimal porfolio is given by ψ = 1 1 γ (σr (σ R ) ) 1 ( µ R r1) 1 1 γ ((σr ) ) 1 θ + ((σ R ) ) 1 σ H. (43) I focus discussions on he opimal porfolio as he behavior of opimal consumpion can be deduced from (35), (37) and (38). Firs, i is useful o rewrie (linear) BSDE (4) in inegral form: H = E Q [ T { γ exp 1 γ s } [r β/γ + (1 γ)στ c στ c /2] dτ ds F S (44) where dq/p = zt θ and (zθ ) is deermined by he densiy generaor θ = θ ασ c. Thus, H >. From (44) and Io s Lemma, H σ H is he inegrand ],

17 AMBIGUITY, RISK AND PORTFOLIO CHOICE 273 of he maringale represenaion of he maringale: [ T { γ s } ] E Q exp [r β/γ + (1 γ)στ c στ c /2] dτ ds 1 γ F S, T. Thus, subsiuing (37) ino he above reveals ha boh ambiguiy (represened by θ ) and esimaion risk (represened by µ R ) affec σ H which deermines hedging demands represened by he hird erm in (43). As shown in secion 3.4, ambiguiy disors mean reurns a ime by an amoun of σ R θ under he ambiguiy adjused belief Q. The second erm in (43) represens his saic effec due o ambiguiy. As γ, he firs-order condiions converge o hose for he logarihmic case. Accordingly, he opimal consumpion and porfolio processes converge o he plans ha are opimal in he logarihmic case. In paricular, when γ =, H = β 1 [ 1 e β(t )] and σ H =. This reflecs he well known fac ha wih logarihmic feliciy he agen behaves myopically so ha here is no hedging demand agains fuure changes of invesmen opporuniies. As a resul, he opimal porfolio rule is idenical o ha in a model wih complee informaion and mean reurns ( µ R ). Nex, he above heorem subsumes soluions for he sandard model wih expeced uiliy, obained by seing κ = (e.g., Brennan (1998)). 11 In he absence of ambiguiy, esimaion risk is he only source of hedging demand. 12 Brennan (1998) inerpres his demand as being induced by he agen s learning abou he rue mean reurns. Theorem 3 can also deliver soluions for he case of complee informaion where he agen observes {F } so ha {F S } = {F }. For example, under expeced uiliy, i is easy o show ha σ H =. Thus he opimal porfolio is given by he mean-variance efficien demand: ψ = 1 1 γ (σr (σ R ) ) 1 (µ R r1). Under ambiguiy, he opimal porfolio is characerized by he following corollary: Corollary 1. In he case of complee informaion, if κ < η, hen θ = κ and he opimal porfolio is given by ψ = 1 1 γ (σr (σ R ) ) 1 (µ R r1) 1 1 γ ((σr ) ) 1 κ. 11 Brennan (1998) assumes ha he disribuion of µ R is normal and ha he agen maximizes expeced uiliy from erminal weah. 12 Explici expression for he hedging demand can be derived from Corollary 2.

18 274 JIANJUN MIAO Thus under complee informaion, ambiguiy as modeled using κ-ignorance, does no induce any hedging demand even when mean reurns are random. In conras, under incomplee informaion hedging demands arise as revealed by he hird componen of opimal porfolio given in (43). Hedging demands naurally arise in sandard models wih incomplee informaion due o esimaion risk. In my model ambiguiy affecs hese hedging demands even when (θ ) is consan as will be shown laer. Therefore, ambiguiy has an ineremporal hedging effec. Finally, in general i is difficul o solve for (θ ) and (ψ ) explicily because (θ ) is endogenously deermined by a fixed-poin problem (36) and (σ H ) can no be characerized explicily. However, he following corollary provides a condiion o ensure ha θ = κ and characerizes he opimal porfolio (ψ ) explicily in erms of Malliavin derivaives and sochasic inegrals. 13 Corollary 2. If he following condiion hold: [ T ( s ) ] ( η κ) + er X ((σ R ) ) 1 E ep e rs c s D η τ d W τ ds F S >, (45) where for θ = κ, (c ) is given by (35), (X ) is given by (42) and Q is deermined by (32), hen θ = κ is opimal and he opimal porfolio is given by + γ 1 γ ψ = 1 1 γ (σr (σ R ) ) 1 ( µ S r1) 1 1 γ ((σr ) ) 1 κ (46) e r X [ T ( s ) ] ((σ R ) ) 1 E ep e rs c s D η τ d W τ ds F S. Even hough θ = κ is consan, he opimal wealh and consumpion processes (X ) and (c ) depend on κ. Thus he hird erm in (46) is affeced by κ so ha ambiguiy sill affecs he hedging demand. 13 The Malliavin derivaive operaor D is defined on D 1,1, he space of smooh funcionals of { W c ; T }. For he exac definiion of D 1,1 and an inroducion o Malliavin calculus, he reader is referred o Ocone and Karazas (1991) and Nualar (1995).

19 AMBIGUITY, RISK AND PORTFOLIO CHOICE 275 APPENDIX Proof of Theorem 2: By (26), he uiliy supergradien a c is given by π = e β z θ u (c ), where θ = κ sgn(σ V (c )) and (V (c ), σ V (c )) is he unique soluion o BSDE (7) for c = c. From (27), he firs-order condiion for he problem (25) is saisfied since λ is he Lagrange muliplier associaed wih he consrain (22). Since V is concave, his condiion is also sufficien for c o be an opimum for problem (25) and hence o problem (12) subjec o (17) and (13). The remaining sep is o find he opimal porfolio ψ. This follows immediaely from Theorem 1. Proof of Theorem 3: Equaions (35), (37) and (38) follow from he firs-order condiion e β z θ (c ) γ 1 = λp, (A.1) and Io s Lemma. The Lagrange muliplier λ is deermined by (28). Defer he proof of (36) for he momen. For par (ii), i suffices o show ha for V in (39) he process V + ((c s) γ /γ βv s θ σ V )ds, T, is a (P, F S )-maringale so ha (V ) solves BSDE (7). Apply Io s Lemma o (39) o derive where dv + ((c ) γ /γ βv θ σ V )d = B d + ( σ H + ασ c ) d Ŵ, (A.2) V B = µ H + γ(µ c σ c θ ) 1 2 γ(1 γ)σc σ c β + H 1 (θ ασ c ) σ H. By (37), (38) and (41), one obains ha B = as desired. By (A.2), he volailiy of uiliy process is given by σ V (c ) = V (σ H + ασ c ). Then, equaion (36) follows from (29). Turn o he proof of par (iii). By Io s Lemma and eliminaing he resuling maringale erm afer aking expecaions, [ ] ] T V e β z θ E [V T e βt zt θ F S = E e βs zs θ (c s) γ /αds F S.

20 276 JIANJUN MIAO Use V T =, (A.1) and (24) o derive X = αv e β z θ (λp ) 1 = γ(c ) 1 γ V. Equaion (42) follows from he above ideniy and (39). Finally, apply Io s Lemma o (42) and mach he resuling volailiy wih ha in (1) o obain ψ = (σ R ) 1 (σ c + σ H ). Insering (37) yields he opimal porfolio (43). Proof of Corollary 1: 1 Guess θ = κ. Then using he same compuaion as above one can show ha σ c = 1 1 γ (η κ) and H = E Q»Z T j exp γ 1 γ» r β/γ + 1 (η κ) (η κ)/2 1 γ ff (s ) ds F where dq/p = z θ T and (zθ ) is deermined by he densiy generaor θ = κ ασ c. Thus H >. Because η = (σ R ) 1 (µ R r1) and µ R is a random variable independen of he Brownian moion W, one can show ha σ H =. Apply Io s Lemma o (39) o derive Because (c ) γ H >, σ V (c ) = (c ) γ H σ c., Thus if sgn(σ V (c )) = sgn(σ c ). κ i < η i for all i, hen σ c >. By (39), σ V (c ) >. Thus θ = κ saisfies (36) and he expression in he corollary gives he opimal porfolio. Proof of Corollary 2: Firs I guess θ = κ. The key sep is o compue σ H. Then one verifies ha he guess is consisen wih (36) so ha θ = κ is indeed opimal. 1 I can also be proved from Corollary 2.

21 AMBIGUITY, RISK AND PORTFOLIO CHOICE 277 Le F T e rs c sds. Then by (3) and Ocone and Karazas (1991) Theorem 2.5, [ ] [ φ = E ep D F F S ] T E ep F D η τ d W τ F S. (A.3) Apply he following seps o compue his expression. Sep 1. Compue he firs erm in (A.3). Use (35) and he definiions of F and p o derive DF = Z T e rs D e βs z θ s λp s! 1 1 γ Z T «1 ds = e rs e βs 1 γ D z θ λe rs s / Z b 1 1 γ s ds. (A.4) By (1), (2), (21) and he chain rule of Malliavin derivaive, D zs θ / Z b 1 1 γ s j» γ 2 = D exp = = 1 1 γ zs θ / Z b 1 1 γ s θ ( )1 [,s] ( ) + Z s 1 zs θ / Z 1 γ b 1 1 γ s Z s j θ τ 2 dτ Z s Z s θ τ d c W τ Z s Z s bη τ 2 dτ + (Dθτ )θτ dτ Dθτ dw c τ Z s ff (Dbη τ )bη τ dτ + Dbη τ dw c τ + bη( )1 [,s] ( ) j bη( )1 [,s] ( ) κ1 [,s] ( ) + where 1 [,s] ( ) is an indicaor funcion. Subsiuing his expression ino (A.4) yields [ [ E ep D F F S ] T = E ep = 1» Z T 1 γ (bη θ )E ep e rs c sds F S Z s Dbη τ d f W τ ff, ( ) 1 e rs D e βs zs θ 1 γ /(λp s ) + 1» Z T 1 γ E P e e rs c s Z s ] ds F S Z s η τ d c W τ ff D bη τ dw f τ ds F S Sep 2. Compue he second erm in (A.3). By he definiions of F and Malliavin derivaive, ] [ T T ( s ) ] E ep [F D η τ d W τ F S = E ep e rs c s D η τ d W τ ds F S..

22 278 JIANJUN MIAO Sep 3. Compue σ H. By (43), σ H = γ 1 γ e r X [ T ( s ) ] E ep e rs c s D η τ d W τ ds F S. Thus if condiion (45) holds, hen θ = κ saisfies (36) so ha i is indeed opimal. Sep 4. Compue he opimal porfolio. By Theorem 2, + γ 1 γ ψ = e r ((σ S ) ) 1 φ /X = 1 1 γ (σr (σ R ) ) 1 ( µ S r1) 1 1 γ ((σr ) ) 1 κ e r X [ T ( s ) ] ((σ R ) ) 1 E ep e rs c s D η τ d W τ ds F S. REFERENCES Anderson E., L. P. Hansen and T.J. Sargen, 23. A quare of semigroups for model specificaion, robusness, prices of risk, and model deecion. Journal of he European Economic Associaion 1, Barberis N., 2. Invesing for he long run when reurns are predicable. Journal of Finance 1, Bawa V., S. J. Brown and R. W. Klein, Esimaion Risk and Opimal Porfolio Choice. Norh-Holland Publishing Company. Brennan M. J., The role of learning in dynamic porfolio decisions. Eur. Fin. Rev. 1, Cagei M., L. P. Hansen, T. J. Sargen, and N. Williams, 22. Robusness and pricing wih uncerain growh. Review of Financial Sudies 15, Chen Z. and L. Epsein, 22. Ambiguiy, risk and asse reurns in coninuous ime. Economerica 7, Cvianic J., A. Lazrak, M.C. Quenez and F. Zapaero, 2. Incomplee informaion wih recursive preferences. Working paper, Universiy of Souhern California. Deemple J., Asse pricing in a producion economy wih incomplee informaion. Journal of Finance 41, Duffie D., Dynamic Asse Pricing Theory, 2nd Ed., Princeon Universiy Press, Princeon, NJ. Duffie D. and L. Epsein, Sochasic differenial uiliy. Economerica 6, (Appendix C wih C. Skiadas).

23 AMBIGUITY, RISK AND PORTFOLIO CHOICE 279 Duffie D. and C. Skiadas, Coninuous-ime securiy pricing: a uiliy gradien approach. Journal of Mahemaical Economics 23, Dybvig P. and C. Huang, Nonnegaive wealh, absence of arbirage and feasible consumpion plans. Review of Financial Sudies 1, El Karoui, N., S. Peng and M. C. Quenez, Backward sochasic differenial equaions in finance. Mahemaical Finance 7, Ellsberg D., Risk, ambiguiy, and he Savage axioms. Quarerly Journal of Economics 75, Epsein L. and J. Miao, 23. A wo person dynamic equilibrium under ambiguiy. Journal of Economics Dynamics and Conrol 27, Epsein L. and M. Schneider, 23. Recursive muliple-priors uiliy. Journal of Economic Theory 113, Epsein L. and T. Wang, Ineremporal asse pricing under Knighian uncerainy. Economerica 62, Feldman D., Logarihmic preferences, myopic decisions, and incomplee informaion. J. of Fin and Quan. Analysis 27, Flemming W. H. and R. W. Rishel, Deerminisic and Sochasic Opimal Conrol. Springer-Verlag. Gennoe G., Opimal porfolio choice under incomplee informaion. Journal of Finance 41, Gilboa I. and D. Schmeidler, Maximin expeced uiliy wih nonunique prior. Journal Mahemaical Economics 18, Hansen L. P. and T. J. Sargen, 21. Robus conrol and model uncerainy. American Economic Review 91, Hansen L. P., T. J. Sargen, G. A. Turmuhambeova and N. Williams, 21. Robus conrol and model misspecificaion. Journal of Economic Theory 128, Karazas I. and S. E. Shreve, Brownian Moion and Sochasic Calculus, 2nd Ed., Springer-Verlag, New York. Karazas I. and X. Xue, A noe on uiliy maximizaion under parial observaions. Mahemaical Finance 2, Karazas I. and X. Zhao, Bayesian adapive porfolio opimizaion. Working paper, Columbia Universiy. Knigh F. H., Risk, Uncerainy and Profi. Houghon Mifflin. Lakner P., Uiliy maximizaion wih parial informaion. Sochasic Processes and Their Applicaions 56, Lakner P., Opimal rading sraegy for an invesor: he case of parial informaion. Sochasic Processes and Their Applicaions 76, Lipser R. S. and A. N. Shiryayev, Saisics of Random Processes I. Berlin, Springer-Verlag. Nualar D., The Malliavin Calculus and Relaed Topics. Springer Verlag, New York. Ocone D. L. and I. Karazas, A generalized Clark represenaion formula, wih applicaion o opimal porfolios. Soch. and Soch. Repors 34, Xia Y., 2. Learning abou predicabiliy: he effecs of parameer uncerainy on dynamic asse allocaion. Journal of Finance 56,