Moral Hazard in Dynamic Risk Management

Transcription

1 Moral Hazard in Dynamic Rik Managemen Jakša Cvianić, Dylan Poamaï and Nizar ouzi arxiv: v2 [q-fin.pm] 16 Mar 215 March 17, 215 Abrac. We conider a conracing problem in which a principal hire an agen o manage a riky projec. When he agen chooe volailiy componen of he oupu proce and he principal oberve he oupu coninuouly, he principal can compue he quadraic variaion of he oupu, bu no he individual componen. hi lead o moral hazard wih repec o he rik choice of he agen. We idenify a family of admiible conrac for which he opimal agen acion i explicily characerized, and, uing he recen heory of ingular change of meaure for Iô procee, we udy how rericive hi family i. In paricular, in he pecial cae of he andard Homlrom-Milgrom model wih fixed volailiy, he family include all poible conrac. We olve he principal-agen problem in he cae of CARA preference, and how ha he opimal conrac i linear in hee facor: he conracible ource of rik, including he oupu, he quadraic variaion of he oupu and he cro-variaion beween he oupu and he conracible rik ource. hu, like ample Sharpe raio ued in pracice, pah-dependen conrac naurally arie when here i moral hazard wih repec o rik managemen. In a numerical example, we how ha he lo of efficiency can be ignifican if he principal doe no ue he quadraic variaion componen of he opimal conrac. Keyword: principal agen problem, moral hazard, rik-managemen, volailiy/porfolio elecion. 2 Mahemaic Subjec Claificaion: 91B4, 93E2 JEL claificaion: C61, C73, D82, J33, M52 We would like o expre our graiude o Paolo Guaoni, Yuliy Sannikov, Julio Backhoff, and paricipan a: he Calech brown bag eminar, he Bachelier eminar, he 214 Rik and Sochaic conference a LSE, he 214 Arbirage and Porfolio Opimizaion conference a BIRS, Banff, he 214 Oberwolfach conference on Sochaic Analyi in Finance and Inurance, he 214 CREE Conference on Reearch on Economic heory & Economeric, he 5h SIAM Conference on Financial Mahemaic & Engineering, and he 214 conference on Opimizaion mee general equilibrium heory, dynamic conracing and finance a Univeriy of Chile. Calech, Humaniie and Social Science, M/C , 12 E. California Blvd. Paadena, CA 91125, USA; cvianic@h.calech.edu. Reearch uppored in par by NSF gran DMS CEREMADE, Univeriy Pari-Dauphine, place du Maréchal de Lare de aigny, Pari, France; poamai@ceremade.dauphine.fr CMAP, Ecole Polyechnique, Roue de Saclay, Palaieau, France; nizar.ouzi@polyechnique.edu 1

2 1 Inroducion In many cae manager are in charge of managing expoure o many differen ype of rik, and hey do ha dynamically. A well-known example i he managemen of a porfolio of many riky ae. Neverhele, virually all exiing coninuou-ime principal-agen model wih moral hazard and coninuou oupu value proce uppoe ha he agen conrol he drif of he oupu proce, and no i volailiy componen. he drif i wha he agen conrol in he eminal model of Holmrom and Milgrom 1987, henceforh HM 1987, in which uiliy i drawn from erminal payoff, and of Sannikov 28, in which uiliy i drawn from iner-emporal paymen. In fac, in hoe paper he moral hazard canno arie from volailiy choice anyway when here i only one ource of rik one Brownian moion, becaue if he principal oberve he oupu proce coninuouly, here i no moral hazard wih repec o volailiy choice: he volailiy can be deduced from he oupu quadraic variaion proce. However, when he agen manage many non-conracible ource of rik, hi choice of expoure o he individual rik ource canno be deduced from he oupu obervaion, even coninuou. One reaon hi problem ha no been udied i he previou lack of a workable mahemaical mehodology o ackle i. When he drif of an Iô proce i picked by he agen, hi can be formulaed a a Giranov change of he underlying probabiliy meaure o an equivalen probabiliy meaure, and here i an exenive mahemaical heory behind i. However, changing volailiy componen require ingular change of meaure, a problem ha, unil recenly, ha no been uccefully udied. We ake advanage of recen progre in hi regard, and ue he new heory o analyze our principal-agen problem. However, we depar from he uual modeling aumpion ha he agen effor coni in changing he diribuion of he oupu i.e., he underlying probabiliy meaure, and we, inead, apply he andard ochaic conrol approach in which he agen acually change he value of he conrolled proce, while he probabiliy meaure ay he ame. he reaon why all he exiing lieraure ue he former, o-called weak formulaion, i ha he agen problem become racable for any given conrac. Inead, we make he agen problem racable by rericing he family of admiible conrac o a naural e of conrac ha lead o a racable characerizaion of he agen problem. Eenially, we reric he admiible conrac o hoe for which he agen problem i olvable. I could be argued, ha, from a pracical poin of view, hee are he only relevan conrac he oher, for which he principal doe no know wha incenive hey will provide o he agen, will likely no be offered. Moreover, we how ha in he claical Holmrom-Milgrom model he rericed family i no acually rericed a all. hu, in addiion o olving he new agency problem wih volailiy conrol, we offer an alernaive new way o udy claical problem wih drif conrol only. Wha we have ju decribed i our main conribuion on he mehodological ide. In principle, our approach can be applied o any uiliy funcion. However, wih erminal paymen only, a in HM 2

3 1987, he only racable cae i he one wih CARA exponenial uiliy funcion. Our main economic inigh of he paper i a follow. he opimal conrac i linear in he CARA cae, bu no only in he oupu proce a in HM 1987, raher, alo in hee facor: he oupu, i quadraic variaion, he conracible ource of rik if any, and he cro-variaion beween he oupu and he conracible rik ource. hu, he ue of pah dependen conrac naurally arie when here i moral hazard wih repec o rik managemen. In paricular, our model i conien wih he ue of he ample Sharpe raio when compenaing porfolio manager. However, unlike he ypical ue of Sharpe raio, here are parameer value for which he principal reward he agen for higher value of quadraic variaion, hu, for aking higher rik. In cae here are wo ource of rik, and a lea one i obervable and conracible, he fir be i aained, becaue here are wo rik facor and a lea wo conracible variable, he oupu and a lea one rik ource; however, o aain he fir be, he opimal conrac make ue of he quadraic and cro-variaion facor. In cae of wo non-conracible ource of rik, we olve numerically a CARA example wih a quadraic co funcion. In hi cae, fir be i generically no aainable. Numerical compuaion how ha he lo in expeced uiliy can be ignifican if he principal doe no ue he pah-dependen componen of he opimal conrac. Lieraure review. An early coninuou-ime paper on volailiy moral hazard i Sung However, in ha paper moral hazard i a reul of he oupu being obervable only a he erminal ime, and no becaue of muliple ource of rik. Conequenly, he opimal conrac i ill a linear funcion of he erminal oupu value only. he paper Ou-Yang 23 how ha he opimal conrac depend on he final value of he oupu and a benchmark porfolio, in an economy in which all he ource of rik all he riky ae available for invemen are obervable, bu he oupu i obervable a final ime only. Some of hi reul are exended in Cadenilla, Cvianić and Zapaero 27, who how ha, if he marke i complee, fir-be i aainable by conrac ha depend only on he final value of he oupu. hu, econd be may be differen from fir be only if he marke i no complee. In our model, he principal oberve he whole pah of he oupu, bu no all he exogenou ource of rik, and alo here i a non-zero co of effor, which make he marke incomplee. hu, fir be and econd be are indeed differen for generic parameer configuraion. More recenly, Wong 213 conider he moral hazard of rik-aking in a model differen from our: he horizon i infinie, a in Sannikov 28, and, while he volailiy i fixed, he agen effor influence he arrival rae of Poion hock o he oupu proce. Lioui and Ponce 213, like u, conider a principal-agen problem in which he volailiy i choen by he agen. heir i he fir-be framework; however, unlike he above menioned paper, hey aume ha he agen ha enough bargaining power o require ha he conrac be linear in he oupu and in a benchmark facor. Working paper Leung 214 propoe a model in which volailiy moral hazard arie becaue here i an exogenou facor muliplying he one-dimenional volailiy 3

4 choice of he agen, and ha facor i no oberved by he principal. In erm of mehodological echnique, a number of paper in mahemaic lieraure ha been developing ool for comparing ochaic differenial yem correponding o differing volailiy rucure. We cie hem in he main body of he paper, a we ue a lo of heir reul. Here, we only menion wo paper ha no only conribue o he developmen of hoe ool, bu alo apply hem o problem in financial economic: Epein and Ji 213 and 214. heir i no he principal-agen problem, bu he ambiguiy problem, ha i, a model in which he deciion-maker ha muliple prior on he drif and he volailiy of marke facor. here i no ambiguiy in our model, i i he agen who conrol he drif and he volailiy of he oupu proce. We ar by Secion 2 preening he imple poible example in our conex, wih CARA uiliy funcion and quadraic penaly, we decribe he general model in Secion 3, in Secion 4 we preen he conracing problem and our approach o olving i, we conider he cae wih no exogenou conracible facor in Secion 5, and conclude wih Secion 6. he longer proof are provided in Appendix. 2 Example: Porfolio Managemen wih CARA Uiliie and quadraic co A an illuraive racable example, we preen here he Meron porfolio elecion problem wih wo riky ae, S 1 and S 2, and a rik-free ae wih he coninuouly compounded rae e equal o zero. Holding amoun v i in ae i, he porfolio value proce X follow he dynamic dx = v 1 S 1 ds 1 + v 2 S 2 ds 2. Suppoe ds i, /S i, = b i d + db i, where B i are independen Brownian moion and b i are conan. We have hen dx = [v 1, b 1 + v 2, b 2 ]d + v 1, db 1 + v 2, db 2. he principal hire an agen o manage he porfolio, ha i, o chooe he value of v = v 1,,v 2,, coninuouly in ime. We aume ha he agen i paid only a he final ime in he amoun ξ. he uiliy of he principal i U P X ξ and he uiliy of he agen i U A ξ K v where K v := kv d and k : R 2 R i a non-negaive convex co funcion. If he principal only oberve he pah of X, hen he can deduce he value of v 2 1, + v2 2,, bu no he value of v 1, and v 2, eparaely, which lead o moral hazard. On he oher hand, if he alo oberve he price pah of one of he ae, ay S 1, hen he can deduce he abolue value of 4

5 v 1, and of v 2,. o diinguih beween hee cae, denoe by 1 O he indicaor funcion ha i equal o one if he pah of S 1 i obervable and conracible, and zero oherwie. We aume ha he principal and he agen have exponenial uiliy funcion, U P x = e RPx, U A x = e RAx. and ha he agen running co of porfolio v 1,v 2 i of he form kv 1,v 2 = 1 2 β 1v 1 α β 2v 2 α 2 2. hu, i i coly o move he volailiy v i away from α i, and he co ineniy i β i. An inerpreaion i ha α i are he iniial rik expoure of he firm a he ime he manager ar hi conrac. 2.1 Fir-be conrac Given a bargaining-power parameer ρ >, he principal fir-be problem i defined a up v he fir order condiion for ξ i hen Wih CARA uiliie, we obain up ξ E[U P X ξ + ρu A ξ K v ]. 1 ξ = R A + R P U PX ξ = ρu A ξ K v. R P X + R A K v + log ρra R P. hu, he opimal fir be conrac i linear in he final value X of he oupu. Plugging back ino he opimizaion problem, we ge ha i i equivalen o [ C ρ inf E exp R ] AR P X K v v R A + R P [ = C ρ inf E E R AR P v dx exp R AR P v R A + R P R A + R P ] f v d, where x y denoe he inner produc, C ρ i a conan, E denoe he Doléan-Dade ochaic exponenial, and f v := b v kv 1 R A R P v 2. 2 R A + R P Sochaic exponenial i defined by u E X db = e 1 u 2 X 2 d+ u X db. 5

6 Under echnical condiion, Giranov heorem can be applied and he above can be wrien a [ C ρ inf E P exp R AR ] P f v d. v R A + R P for an appropriae probabiliy meaure P. hu, fir be opimal v = v FB i deerminiic, found by he poinwie maximizaion of he funcion f, annd given by v FB i 2.2 Second be conrac := b i + β i α i β i + R, where 1 R := 1 R A + 1 R P. 2.1 We now ake ino accoun ha i i no he principal who conrol v, bu he agen. We conider linear conrac baed on he pah of he obervable porfolio value X, he obervable quadraic variaion of X, and, poibly, on S 1 via B 1, and he co-variaion of X and B 1. More preciely, le [ ξ = ξ + Z X dx +Y X d X + 1 O Z 1 db 1 +Y 1 d X,B 1 + H d ], 2.2 for ome conan ξ, and ome adaped procee Z X,Z 1,Y X,Y 1 and H. o be conien wih he noaion in he general heory ha follow laer, for fuure noaional convenience we work inead wih arbirary adaped procee Z X,Z 1,Γ X,Γ 1 and G uch ha Y X = 1 2 Γ X + R A Z X 2, Y 1 = Γ 1 + R A Z X Z 1, H = G R AZ 1 2. Solving he agen problem if hoe procee were deerminiic would be eay, bu no for arbirary choice of hoe procee. We will allow hem o be ochaic, bu we will reric he choice of proce G, moivaed by a ochaic conrol analyi of he agen problem, dicued in a laer ecion. We will ee in ha ecion ha he naural choice for G i G := GZ X,Z 1,Γ X,Γ 1, where GZ X,Z 1,Γ X,Γ 1 := up gv 1,v 2,Z X,Z 1,Γ X,Γ 1 v 1,v 2 := up v 1,v 2 { kv 1,v ΓX v 21 + v 22 + Z X b v + 1 O Γ 1 v 1 }. 2.3 One of our main reul, heorem 4.1, characerize he conrac ha have a repreenaion of he above form, wih G a defined here. In paricular, he heorem will how ha applying he ame mehod o he claical Holmrom-Milgrom 1987 problem lead o no lo of generaliy in he In he cae he co β i i zero, he fir be volailiy v FB i aggregae prudence 1/ R. 6 i imply a produc of he rik premium b i and he

7 choice of poible conrac including nonlinear conrac, and we will argue ha, in general, hi choice of conrac include all pracically relevan conrac. he agen i maximizing E[ exp R A ξ K v ] wih ξ a in 2.2. hi urn ou o be an eay opimizaion problem: by Giranov heorem auming appropriae echnical condiion, imilarly a in he fir be problem, he agen objecive can be wrien a [ E P e R ] A [g G ]d, for an appropriae probabiliy P. Wih our definiion of G, we ee ha hi i never larger han minu one, and i i equal o minu one for any pair v 1,v 2 if i exi ha maximize g := gz X,Z 1,Γ X,Γ 1, by, and ω by ω. hu, he agen would chooe one of uch pair. Noice ha hi implie ha he principal can alway make he agen indifferen abou one of he porfolio poiion. For example, if b 2 i no zero, he can e Z X = α 2 β 2 /b 2, and Γ X β 2, o make g independen of v 2, hence he agen indifferen wih repec o v 2. hi i alo poible if here i only one ock, ay S 2, o ha in ha cae he fir be i aained wih uch a conrac, if we aume ha he agen will chooe wha i be for he principal, when indifferen Conracible S 1 : fir be i aained Wih wo riky ae, if S 1 i oberved, hen alo he covariaion beween S 1 and X i oberved, which mean ha v 1 i oberved. Since alo he quadraic variaion i oberved, hen v 2 i oberved, and, if he oberved procee are conracible, we would expec he fir be o be aainable. Indeed, v 1,v 2 i obained by maximizing g = 1 2 β 1v 1 α β 2v 2 α Z X b v + Γ 1 v ΓX v 2 + Z 1 b 1 + Z Z 1 Z X v 1. Aume, for example, ha b 2, β 2 β 1. Suppoe he principal e hen, Γ X β 2, Z X α 2 β 2 /b 2, Γ 1 = α 1 β 1 Z X b 1 + β 1 β 2 v FB 1, Z 1. [ ] 1 g = β 2 β 1 2 v2 1 v 1 v FB 1 + con. We ee ha he agen i indifferen wih repec o which v 2 he applie, and he would chooe v 1 = vfb 1. hu, if, when indifferen, he agen will chooe wha i be for he principal, he will chooe he fir be acion. I can alo be verified ha he principal will aain he fir be expeced uiliy wih hi conrac. 7

8 2.2.2 Non-conracible S 1 If S 1 i no conracible, he opimal v 1,v 2 i obained by maximizing gv 1,v 2 = 1 2 β 1v 1 α β 2v 2 α Z X b v ΓX v 2. Aume, for example, β 2 β 1. If Γ X > β 2, hen he agen will opimally chooe v 2 =. I i raighforward o verify ha hi canno be opimal for he principal. he ame i rue if Γ X = β 2 and Z X i no equal o α 2 β 2. If Γ X = β 2 and Z X = α 2 β 2, hen he agen i indifferen wih repec o which v 2 o chooe. If Γ X < β 2 β 1, he opimal poiion are v i = ZX b i + α i β i β i Γ X. he principal uiliy i proporional o [ E e R P [1 Z X dx 2 1 ΓX d X +G d] ]. Wih he above choice of v i, by a imilar Giranov change of probabiliy meaure a above, i i raighforward o verify ha maximizing hi i he ame a maximizing, over Z = Z X and Γ = Γ X, b v Z,Γ 1 2 [R AZ 2 + R P 1 Z 2 ] v 2 kv Z,Γ. hi i a problem ha can be olved numerically. We now preen a numerical example ha will how u fir, ha fir be i no aained, econd, ha he opimal conrac conain a non-zero quadraic variaion componen and ha ignoring i can lead o ubanial lo in expeced uiliy, and hird, ha here are parameer value for which he principal reward he agen for aking high rik unlike he ypical ue of porfolio Sharpe raio in pracice. In Figure 1 we plo he percenage lo in he principal econd be uiliy cerainy equivalen relaive o he fir be, when varying he parameer α 2, and keeping everyhing ele fixed. he lo can be ignifican for exreme value of iniial expoure α 2. ha i, when he iniial rik expoure i far from deirable, he moral hazard co of providing incenive o he agen o modify he expoure i high. In Appendix, we provide ufficien condiion for exience of opimizer. Numerically, we found opimizer for all parameric choice we ried. 8

9 In Figure 2, we compare he principal econd be cerainy equivalen o he one he would obain if offering he conrac ha i opimal among hoe ha are linear in he oupu, bu do no depend on i quadraic variaion. Again we ee ha he correponding relaive percenage lo can be large. Figure 3 plo he value of he coefficien he eniiviy muliplying he quadraic variaion in he opimal conrac. We ee ha he principal ue quadraic variaion a an incenive ool: for low value of he iniial rik expoure α 2 he wan o increae he rik expoure by rewarding higher variaion he eniiviy i poiive, and for i high value he wan o decreae i by penalizing 9

10 high variaion he eniiviy i negaive. hi i becaue when he iniial rik expoure α 2 i no a he deired value v 2, incenive are needed o make he agen apply coly effor o modify he expoure. In he re of he paper, we generalize hi example and we aim o characerize he conrac payoff ha can be repreened a in 2.2, wih G defined analogouly o he General Model We conider he following general model for he oupu proce X v,a : X v,a = σ v b a d + db, 3.1 where v,a repreen he conrol pair of he agen, allowing alo for eparae conrol a of he drif, where v and a are adaped procee aking value in ome ube V A of R m R n, for ome m,n N N. Moreover, b : [, ] A R d and σ : V R d are given deerminiic funcion uch ha ba + σv C 1 + v + a, 3.2 for ome conan C >, and B = B 1,...,B d i a d-dimenional Brownian moion, and he produc are inner produc of vecor, or a marix acing on a vecor. he example of delegaed porfolio managemen correpond o he cae in which m = n = d, and σ v := v σ, b a := b, 3.3 for ome fixed b R d and ome inverible d d marix σ. In ha cae, he inerpreaion of he proce X v,a i ha of he porfolio value proce dependen on he agen choice of he vecor 1

11 v of porfolio dollar-holding in d riky ae wih volailiy marix σ, and he vecor b of rik premia. Anoher pecial cae, when d = 1, v i fixed and he agen conrol a only, i he original coninuou-ime principal-agen model of Holmrom and Milgrom In addiion o he oupu proce X v,a, we may wan o allow conrac baed on addiional obervable and conracible rik facor B 1,...,B d, for ome 1 d < d. For example, S = S + µ S + σ S B 1 migh be a model for a conracible ock index. Uually in conrac heory, for ake of racabiliy, he model i conidered in i o-called weak formulaion, in which he agen change he oupu proce no by changing direcly he conrol v, a, bu by changing he probabiliy meaure over he underlying probabiliy pace. When, a in andard coninuou-ime conrac model, he effor i preen only in he drif, changing meaure i done by he mean of he Giranov heorem. Unil recenly, hough, uch a ool had no been available for ingular change of meaure ha are needed when changing volailiy, a i he cae here. he mahemaic o formulae rigorouly he weak formulaion of our problem i now available. However, we ake a differen approach: inead of auming he weak formulaion, we will adop he andard rong formulaion of ochaic conrol no change of meaure. We will ill be able o explicily characerize he oluion o he agen problem, bu only in a rericed family of admiible conrac. We will provide an aumpion under which a conrac belong o he rericed family, we will argue ha he family include all conrac relevan in pracice, and we will how he aumpion i no rericive if v i fixed. hu, we alo provide a new alernaive way o olve he Holmrom-Milgrom 1987 problem. We fir need o inroduce ome noaion and he framework. We work on he canonical pace Ω of coninuou funcion on [, ], wih i Borel σ-algebra F. he d dimenional canonical proce i denoed B, and F := F i i naural filraion. Le P denoe he d dimenional Wiener meaure on Ω. hu, B i a d dimenional Brownian moion under P. We denoe by E he expecaion operaor under P. A pair v,a of F-predicable procee aking value in V A i aid o be admiible if [ ] σ v b a < +, P a.., E exp p σ v 2 d < +, for all p >, 3.4 and E b a db i a P -maringale in L 1+η, for ome η >, 3.5 We ued he weak formulaion in an earlier verion of hi paper. Inereingly, o derive hee reul, even hough we work wih he rong formulaion, we need o ue he weak formulaion in he proof. In fac, for he admiible conrac in our rericed family, he weak and he rong formulaion for he agen problem are equivalen. 11

12 where σ v 2 d i=1 σ v i Moreover, we aume ha he fir d enrie of vecor b do no depend on he conrol proce a becaue hey will correpond o exogenou conracible facor. Remark 3.1. he condiion 3.6 i acually no needed o prove our reul; i i only needed in our compuaion ha moivae he definiion of he admiible conrac. A in he example in he previou ecion, we aume ha he agen i paid only a he final ime in he amoun ξ. When he agen chooe he conrol v,a U, he uiliy of he principal i U P X v,a ξ, and he uiliy of he agen i e R Aξ K v,a, where K v,a, = kv,a d. and k : R m R n R i a non-negaive convex co funcion. 4 Second Be wih Conracible Rik In hi ecion, we aume here i exacly one exogenou conracible rik facor, ha i, we e d = 1. hu, we inerpre B 1 a he obervable and conracible yemic rik. 4.1 Seup We allow he conrac payoff o depend boh on he oupu X v,a and B 1. ha i, given a pair u := v, a choen by he agen, he principal can offer conrac payoff meaurable wih repec o F ob, a σ-field conained in he filraion Fob := F Xv,a F B1 generaed by X v,a,b 1, where F Xv,a := {F Xu } i he compleed filraion generaed by he oupu proce X u. Recall ha our aumpion imply ha b 1 doe no depend on a, and inroduce he following noaion for he conracible and non-conracible facor: B ob,v,a := X v,a,b 1 = µ v,a d + Σ v db, he above inegrabiliy condiion are uiable for he cae of CARA uiliie,and hey may have o be modified for oher uiliy funcion. We dicu briefly he general cae laer below. Noice alo ha condiion 3.6 in he cae of porfolio managemen problem wih a = and σ = I d, mean ha he inveor ha o inve in a lea one of he non-conracible ource of rik. When here are no conracible ource of rik ha i when d =, hi condiion reduce o σ v 1,d. We could equally have any ube of {B 1,...,B d } conracible, and he re non-conracible. We aume ha here i only one conracible rik ource for impliciy of noaion, and becaue i i alo conien wih he yemic rik ock index inerpreaion. We conider he cae in which none of he rik ource i conracible in he following ecion. 12

13 B ob,v,a := Σ v db, where for any v,a V A and any [, ], he R 2 vecor µ v,a and he 2 d marix Σ v are defined by σ vb a σ v µ v,a :=, Σ v :=,wih I 1,d := i a d 2 d marix aify- and where p,q denoe he p q marix of zero. Furhermore, Σ ing, for any v V and any [, ], I 1,d Σ vσ v = d 2,2 and Σ v Σ v = Id ,d 1, In oher word, we ake a he non-conracible ource of rik everyhing which i orhogonal o he conracible ource of rik. Noe ha he vecor B ob,v,a B ob,v,a generae he ame filraion F a B if and only if he deniy of i quadraic variaion i inverible. Uing he definiion of Σ and Σ, we can readily compue ha d B ob,v,a B ob,u d = Σ v Σ v 2,d 2 d 2,2 I d 2 which i inverible if and only if Σ v Σ v i ielf inverible. Simple calculaion lead u o he following neceary and ufficien condiion σ v 2 σ 1 v 2, which i exacly 3.6 when d = 1, and hu explain why we aumed 3.6., 4.2 Admiible conrac In hi ubecion we will define he e of admiible conrac payoff. o moivae our definiion, conider he agen value funcion a ime, for a given choice of conrol v,a U [ V A,v,a := eup E U A ξ K v,a, ], 4.7 v,a U v,a, where he e U v,a, denoe he ube of elemen of U which coincide wih v,a, d P a.e. on [,]. Noe ha we have he following explici relaionhip beween he payoff and he erminal value of he value funcion: ξ = U 1 A A,v,a V

14 he idea i o conider he mo general repreenaion of he value funcion V A ha we can reaonably expec o have, and hen define he admiible conrac payoff via 4.8. When U A i a CARA uiliy funcion, we may gue from 4.8 ha conrac ξ are uch ha ξ can be wrien a a linear combinaion of variou inegral. For example, in he pecial cae d = wih no ouide conracible facor, we migh expec he conrac o aify [ ] U A ξ = C exp R A Z dx v,a + Y d X v,a + H d, for ome conan C < and ome adaped procee H,Y,Z. Wih U A x = e R Ax, hi would give ξ = C + Z dx v,a + Y d X v,a + H d, 4.9 for ome conan C. We will admi exacly he conrac of hi form, bu, aking ino accoun 4.8, we will impoe ome rericion on proce H. We preen hi reaoning nex, in an informal way, o moivae he definiion which will follow hereafer. Noice ha V A,v,a i an F meaurable funcion and can hu be wrien in a funcional form a V A,v,a = V A,B ob,v,a.,b. ob,v,a = V A,B ob,v,a,b ob,v,a. If he value funcion i mooh in he ene of he Dupire 29 funcional differeniaion ee alo Con and Fournié 213 for more deail, hen, he Dupire ime derivaive V A,v,a exi, and one can find predicable procee Z Z v,a ob,v,a = Z ob,v,a Γ and Γ v,a ob,v,a = Γ ob,v,a, wih Z ob,v,a, Z ob,v,a aking value in R 2 and R d 2, repecively, Γ ob,v,a, Γ ob,v,a aking value in he pace S 2 and S d 2 of ymmeric marice, repecively, uch ha we have he following generalizaion of Iô rule ee heorem 1 in Dupire 29 or heorem 4.1 in Con and Fournié 213, dv A,v,a = V A,v,a + 1 [ ] 2 r Γ ob,v,a d B ob,v,a + 1 [ 2 r Γ ob,v,a + Z ob = + V A,v,a db ob,v,a r[ Γ ob,v,a Σ v Z ob,v,a + Σ,v,a d B ob,v,a ] + Z ob,v,a db ob,v,a Σ v Σ v ] r[ Γ ob,v,a ] + Z ob,v,a µv,a d v Z ob db, P a.. 4.1,v,a For inance, if we happen o be in he Markov cae in which V A,v,a = f,b ob,v,a,b ob,v,a for ome mooh funcion f,x,y, hen, i follow from Iô rule ha he procee Z ob,v,a, Z ob,v,a 14

15 and Γ ob,v,a, Γ ob,v,a are given by Z ob,v,a Γ ob,v,a = x f,b ob,v,a = xx f,b ob,v,a,b ob,v,a,v,a, Z ob,v,a, Γ ob,b ob,v,a = y f,b ob,v,a = yy f,b ob,v,a,b ob,u,,b ob,v,a, wih x, xx, y, yy denoing parial derivaive wih repec o he correponding variable. Nex, from he maringale opimaliy principle of he claical ochaic conrol heory, he dynamic programming principle ugge ha he proce V A,v,a e R AK v,a, hould be a upermaringale for all admiible conrol v,a, and ha i hould be a maringale for any opimal conrol v,a, provided uch exi. Wriing formally ha he drif coefficien of a upermaringale i non-poiive, and ha of a maringale mu vanih, we obain he following pah-dependen HJB Hamilon-Jacobi-Bellman equaion: V A,v,a and where + R A V A,v,a G 1,Z v,a,γ v,a =, where Z v,a,γ v,a := 1 R A V A,v,a Z v,a, Γ v,a, 4.11 wih G 1,z,γ := up v,a V A g 1,z,γ,v,a, 4.12 g 1,z,γ,v,a := kv,a + z ob µ v,a r[ γ ob Σ vσ v ] r[ γ ob ]. Subiuing he above ino 4.1, i follow ha, P a.., dv A,v,a e R AK v,a, = R A V A,v,a e R AK v,a, g1,z v,a,γ v,a R A V A e R AK v,a, Σ v Z ob,v,a +,v,a G 1,Z v,a,γ v,a d Σ v Z ob,v,a db We hen ee by direcly olving he laer ochaic differenial equaion ha [ V A,v,a = V A exp R A G 1,Z v,a,γ v,a 1 2 r[ Γ ob,v,a d B ob,v,a ] 1 2 r[ Γ ob [ exp R A Z ob,v,a db ob,v,a + R ] A Z ob,v,a d B ob,v,a Z ob,v,a 2 [ exp R A Z ob,v,a db ob,v,a + R ] A Z ob,v,a 2 d, P a.. 2,v,a ] ] d Nex, we recall ha he principal mu offer a conrac baed on he informaion e F ob only. From he definiion of G 1 we can check ha he expreion for ξ doe no depend on Γ ob,v,a, and we expec i alo no o depend on Z ob,v,a, ha i, o have Z ob,v,a. In ha cae, from 4.8, denoing G ob 1,zob,γ ob := up v,a V A g ob 1,zob,γ ob,v,a,

16 and g ob 1,zob,γ ob,v,a := kv,a + z ob µ v,a r[ γ ob Σ vσ v ]. he conrac payoff ξ would be a in he following definiion: Definiion 4.1. An admiible conrac payoff ξ = ξ Z,Γ i a F ob meaurable random variable ha aifie U A ξ Z,Γ = Ce R A {Z db ob,v,a G ob 1,Z,Γ d+ 1r[ ]} 2 Γ +R A Z Z d B ob,v,a for ome conan C <, and ome pair Z,Γ of bounded F ob predicable procee wih value in R 2 and S 2, repecively, ha are uch ha here i a maximizer v Z,Γ,a Z,Γ U of g ob 1,Z,Γ, d dp -a.e.. We denoe by C he e of all admiible conrac, and by U he e of he correponding Z,Γ. he aumpion of boundedne i echnical, aumed o implify he proof, and i can be relaxed. If, a in he example ecion, he opimal Z and Γ are conan procee, hen he aumpion i aified. he aumpion ha g ob 1,Z,Γ ha a maximizer i needed o prove he incenive compaibiliy of conrac ξ, and o olve he principal problem. Remark 4.2. In addiion o a conan erm and he d inegral erm, wih U A a CARA uiliy funcion, an admiible conrac i i linear in he inegraion ene in he following facor: he conracible variable, ha i, he oupu and he conracible ource of rik; and he quadraic variaion and cro-variaion procee of he conracible variable. A een in he numerical example, he opimal conrac generally make ue of all of hee componen. hi i o be conraed wih he fir be conrac, and wih he cae of conrolling he drif only a in Holmrom- Milgrom 1987, in which only he oupu i ued,. Remark 4.3. We argue now ha, under echnical condiion, an opion-like conrac of he form ξ = FX v,a,b1, for a given funcion F i an admiible conrac. For noaional impliciy, aume b = and a =, and conider he PDE, wih ubcrip denoing parial derivaive, and wih G 1 = G 1 z 1,z 2,γ 1,γ 2,γ 3 where γ 1 and γ 2 are he diagonal enrie of a ymmeric marix γ and γ 3 i he value of he off-diagonal enrie, u + G 1 u x,u y,u xx R A u 2 x,u yy R A u 2 y,u xy R A u x u y =,,x [, R, u,x,y = Fx,y. hen, auming ha he PDE ha a mooh oluion, i follow from Iô formula applied o u,x v,,b 1 ha FX v, +,B1 = u,, + 1 uxx,x v, 2 G ob 1 u x,x v,,b 1 dx v, +,B 1 d X v, + u yy,x v,,b 1 d B 1 + u y,x v,,b 1 db 1 ux,u y,u xx R A u 2 x,u yy R A u 2 y,u xy R A u x u y,x v,,b 1 d. 16 u xy,x v,,b 1 d X v,,b 1

17 hu, FX v,,b1 i of he form ξ Z,Γ, where he vecor Z ha enrie given by u x,x v,,b 1 and u y,x v,,b 1, and where he marix Γ ha diagonal enrie given by u xx R A u 2 x,x v,,b 1, u yy R A u 2 y,x v,,b 1, and off-diagonal enrie given by u xy R A u x u y,x v,,b 1. A he end of hi ecion, we how ha, a deired, under he above definiion of admiible conrac, one can characerize he agen opimal acion. Inroduce he e of he conrol ha are opimal for maximizing g ob 1, given Z,Γ: V 1 Z,Γ = {v Z,Γ,a Z,Γ, uch ha he condiion of Definiion 4.1 are aified}. he nex propoiion ae ha for a given conrac ξ Z,Γ C, any conrol v Z,Γ,a Z,Γ V 1 Z,Γ i opimal for he agen. Propoiion 4.1. An admiible conrac ξ Z,Γ C a defined in 4.15 i incenive compaible wih V 1 Z,Γ. ha i, given he conrac ξ Z,Γ, any conrol v Z,Γ,a Z,Γ V 1 Z,Γ i opimal for he agen. Moreover, he correponding agen value funcion aifie equaion 4.13 wih Z ob,z ob = Z,, and Γ ob,γ ob = Γ,. Proof: Le Z,Γ be an arbirary pair proce in U, and conider he agen problem wih conrac ξ Z,Γ: V A,v,a ξ Z,Γ [ := eup E U A ξ Z,Γ K v,a, ], P a.. v,a U,v,a We fir compue, for all v,a U,v,a, U A ξ Z,Γe R AK v,a., = U A ξ Z,ΓE R A Z r Σ r v rdb r [ exp R A G ob 1 r,z r,γ r g ob 1 r,z r,γ r,v r,a r ] dr, P a.., where U A ξ Z,Γ ha he ame form 4.15 a U A ξ Z,Γ, when we ubiue for. Since Z i bounded by definiion and σ aifie he linear growh condiion 3.2, we have by definiion of U ee 3.4 in paricular ha [ R 2 ] E exp A Σ 2 r v rz r 2 dr < +. Hence, by Novikov crierion, we may define a probabiliy meaure P equivalen o P via he deniy hen, d P. F = E R A Z r Σ r v dp rdb r [ E U A ξ Z,Γ ] e R AK v,a, = U A ξ Z,Γ [ E P e R ] A G ob 1 r,z r,γ r g ob 1 r,z r,γ r,v r,a rdr.. 17

18 Since G ob 1 gob 1, we ee ha [ E U A ξ Z,Γ ] e R AK v,a, U A ξ Z,Γ, and by he arbirarine of v,a U,v,a, i follow ha V A,v,a ξ Z,Γ U A ξ Z,Γ. hu, any conrol v,a for which G ob 1 r,z r,γ r = g ob 1 Z r,γ r,v r,a r, aain he upper bound. Hence, V A,v,a ξ Z,Γ = U A ξ Z,Γ, and he dynamic of V A,v,a are a aed. Remark 4.4. When U A i no a CARA uiliy funcion, he ame approach would work if he agen draw uiliy/diuiliy of he form U A ξ K v,a, ha i, if he co of effor i eparable from he agen uiliy funcion. For example, in he cae in which only X = X v,a i conracible and σ = I d, we would define admiible conrac ξ = ξ Z,Γ o be hoe ha aify 1 U A ξ Z,Γ = C + Z u dx u + 2 Γ ud X u G Z u,γ u du, for ome conan C, ome F Xv,a predicable procee Z and Γ wih value in R, and a proce proce G Z,Γ defined imilarly a above. hu, U A ξ, raher han ξ, would be required o be linear in he inegraion ene. However, while he agen problem would be racable, he difficuly i ha, in general, i would be hard o olve he principal maximizaion problem How general i cla C? hi queion i relaed o he poibiliy of HJB characerizaion of he value funcion in he non- Markovian cae, which ha been approached by inroducing and udying o-called econd-order BSDE; ee, e.g., Soner, ouzi and Zhang 212. However, idenificaion or even exience of he opimal conrol i a very hard ak, which may require rong regulariy aumpion. Uing reul of ha recen heory, we prove in Appendix ha, in he cae of drif conrol only a in Hormrom and Milgrom 1987 any feaible conrac payoff ha he form 4.15, while, wih volailiy conrol, hi i rue under addiional regulariy aumpion. More preciely, we have he following heorem. heorem 4.1. If v i unconrolled and fixed o v V, and if here i a conan C > and ε [1, + uch ha kv,a lim = +, D a kv,a C 1 + a ε, a + a A poible inerpreaion i ha he funcion k repreen, in a ylized way, join effec of he rik averion o he choice of v and a and of heir co. 18

19 a condiion aified in he cae of quadraic co, hen any F ob -meaurable random variable ξ for which he agen value funcion i well-defined can be repreened a in More generally, 4.15 i implied by moohne of he non-maringale componen of he agen value funcion in he ene of exience of a econd-order eniiviy proce Γ a in Aumpion 7.1 in Appendix. Baically, by impoing he form 4.15 wih an appropriaely choen G ob 1, we enure ha he correponding conrac payoff i ufficienly mooh o make he agen value funcion proce alo mooh. In oher word, wih uch a conrac, he value funcion of he agen i a oluion o he pah-dependen HJB equaion ha we derived heuriically in In he exiing lieraure in which only drif conrol i preen hi i alway he cae, wheher he model i wih finie or infinie horizon. Indeed, he maringale repreenaion and he comparion heorem ype reul i alway ued, which i equivalen o he pah dependen HJB equaion. I eem unlikely ha one could olve he agen problem for conrac for which he value funcion would be o degenerae ha i would no aify hi weak form of moohne. In pracice, hi mean ha he principal alo wouldn know wha he agen would do given uch conrac, and he principal would likely decide no o conider hem. 4.3 Aainabiliy of fir be. In hi ecion we aume ha drif effor a i unconrolled by he agen and fixed o he value of zero. Moreover, we adop he eing of 3.3, wih V = R d and CARA uiliy funcion. We how ha fir be i alway aainable when he co funcion i zero. In oher word, wih no conrain on he choice of v and no co of chooing i, here i no agency fricion. For a pair Z,Γ R 2 S 2, we inroduce he noaion z 1 γ 1 γ 2 Z := and Γ :=. z 2 γ 2 γ 3 If we chooe γ 1 < and aume ha he co kv, i, hen i i eay o ee ha he opimal v R d in he definiion of G ob 1 Z,Γ i he unique oluion of he following equaion z 1 σb + γ 2 σ.1 + γ 1 d i=1σ.i v σ.i =, where for any 1 i d, σ.i i he i-h column of marix σ. hi lead direcly o he principal may chooe he value v = 1 γ 1 σσ 1 z 1 σb + γ 2 σ z 1 := R P, γ 1 := R AR 2 P R A + R P R A + R P 2, γ 2 :=, 19

20 and arbirary value for z 2 and γ 3. hen, hi conrac i clearly admiible, ince Z and Γ are bounded and conan, and i follow from 4.16 ha when σ = I d, i implemen he following opimal volailiy v = R A + R P R A R P b. When k =, hi i exacly equal o he fir be volailiy v FB obained in 2.1 for d = 2, bu i hold for any d, a can eaily be verified, and he ame hold for general σ. hi reul i in agreemen wih Cadenilla, Cvianić and Zapaero 27, who how ha in a fricionle and a complee marke i.e., zero co of volailiy effor and he number of Brownian moion equal o he number of obervable ae, wih arbirary uiliy funcion, fir be i aainable uing a conrac ha depend only on he final value of he oupu. We ee here ha wih exponenial uiliy funcion compleene i no neceary, and a linear conrac i opimal. Indeed, wih he above z 1 and Γ, we have, eing z 2 =, ξ Z,Γ = con. + which i he ame a he fir be conrac when kv =. R P R A + R P X, 4.4 he principal problem wih CARA uiliy In hi ecion, we aume ha he uiliie are exponenial for boh he principal and he agen, ha i we have U I x = e RIx, I = R A,R P. Fix an admiible ξ Z,Γ C and inroduce he noaion Z X Γ X Γ XB1 Z = and Γ =. Z B1 he principal maximize he expeced uiliy of her erminal payoff X v,a ξ Z,Γ. Since he conrac ξ Z,Γ i incenive compaible in he ene of Propoiion 4.1, he opimal volailiy and drif choice by he agen correpond o any v Z,Γ,a Z,Γ V 1 Z,Γ. hen, auming ha he agen le he principal chooe among he conrol choice ha are opimal for he agen, he principal problem i: Γ XB1 Γ B1 up E [ U P X v,a ξ Z,Γ ]. Z,Γ U, v Z,Γ,a Z,Γ V 1 Z,Γ Denoing v,a := v Z,Γ,a Z,Γ, and ubiuing he expreion for ξ Z,Γ, we ge X v,a ξ Z,Γ = C + σ v b a 1 Z X 1 σ v 2 2 Γ X + R A Z X 2 d + G ob 1,Z,Γ 1 Γ B 1 + R A Z B 1 2 d 2 Γ XB 1 + R A Z X Z B1 d X,B 1 Z B1 db ob,v,a Z X v σdb v,a, P a..

21 Inroduce a vecor θ := σ v and denoe i fir enry θ1, and denoe by θ 1 he d 1- dimenional vecor wihou he fir enry. Arguing exacly a in he proof of Propoiion 4.1, in paricular, by iolaing he appropriae ochaic exponenial, i follow ha he principal problem reduce o maximizing θ Z,Γ b a Z,Γ 1 Z X 1 2 θ Z,Γ 2 Γ X + R A Z X 2 + G ob 1,Z,Γ 1 Γ B R A Z B 1 2 Γ XB1 + R A Z X Z B1 θ1,z,γ R [ P θ 2 1,Z,Γ 2 1 Z X 2 + θ 1,Z,Γ 1 Z X ] Z B Since he upremum in he definiion of G ob 1,Z,Γ i aained a v,a, hi i equivalen o he minimizaion problem { inf θ,z,γ b a Z,Γ + 1 Z,Γ,v Z,Γ,a Z,Γ 2 θ Z,Γ 2 R A Z X 2 + kv,a + R A Z B R A Z X Z B1 θ 1,Z,Γ + R P 2 [ θ Z,Γ 2 1 Z X 2 2θ 1,Z,Γ 1 Z X Z B1 + Z B1 2 ]} Noe ha if minimizer Z, Γ, v and a exi, hey are hen necearily deerminiic, ince b, σ and k are non-random. By Propoiion 4.1, he conrac ξ Z,Γ i incenive compaible for v Z,Γ,a Z,Γ, if ξ Z,Γ C. Moreover, a we have ju hown, i i alo opimal for he principal problem, ha i, we have proved he following. heorem 4.2. Conider he e of admiible conrac ξ Z,Γ C. hen, he conrac ha i opimal in ha e and provide he agen wih expeced uiliy V A < i ξ Z,Γ correponding o Z,Γ,v,a which are he minimizer in 4.18, provided uch minimizer exi, wih v. Moreover, he conrac cah conan C i given by C := 1 R A log V A. Proof. he only hing o check here i he admiibiliy of he conrac ξ Z,Γ, bu hi i ju a conequence of he opimizer being deerminiic, noing ha condiion 3.6 i aified when v. Nex, we provide here ufficien condiion for exience of a lea one minimizer of 4.18 when, for impliciy, here i no opimizaion wih repec o a, and when he co funcion k i uperquadraic. he cae of a quadraic k i acually harder and i reaed in Appendix in Propoiion 7.1. Propoiion 4.2. Aume ha he agen doe no conrol he drif, i.e. a =, and conider he eing of 3.3 wih V = R d. Aume moreover ha he co funcion kv := kv, i a lea C 1 and aifie for ome conan C > kv C 1 + v 1+ε, for ome ε >, and hen, he infimum in 4.18 i aained. 21 lim v + kv v 2 = +.

22 5 Second-be wih non-conracible rik Conider now he cae in which he only conracible proce i X v,a. In ha cae, we need o modify our approach by adoping he following change, a can be verified uing imilar argumen. Fir of all, he principal can now only offer conrac payoff meaurable wih repec o F Xv,a, a igma-field conained in he filraion F Xv,a := {F Xv,a } generaed by he oupu proce X v,a. Following exacly he ame inuiion from he ochaic conrol heory a in he previou ecion, we inroduce he funcion G ob, he counerpar of he funcion Gob 1 above, defined for any,z,γ [, ] R R by G ob,z,γ := up g ob,z,γ,v,a v,a V A := up v,a V A { σ vb az + 1 } 2 σ v 2 γ kv,a, z,γ R. Once again, if a maximizer exi, we denoe i by v z,γ,a z,γ. We now inroduce he e of admiible conrac in hi cae. Definiion 5.2. An admiible conrac payoff ξ = ξ Z,Γ i an F Xv,a meaurable random variable ha aifie U A ξ Z,Γ := Ce R A {Z dx v,a G ob ]},Z,Γ d+ 1 2 Γ +R A Z d X 2 v,a for ome conan C, and ome pair Z,Γ of bounded F Xv,a predicable procee wih value in R, and uch ha here i a maximizer v Z,Γ,a Z,Γ U of g ob,z,γ, d dp -a.e.. We denoe by C he e of all admiible conrac, and by U he e of he correponding Z,Γ. Similarly a before, we inroduce he e of conrol ha are opimal for maximizing g ob, given Z,Γ: V Z,Γ = {v Z,Γ,a Z,Γ, uch ha he condiion of Definiion 5.2 are aified}. he following propoiion i he analogue of Propoiion 4.1 in hi eing, and can be proved by he ame argumen. Propoiion 5.3. An admiible conrac ξ Z,Γ, a defined in 5.19, i incenive compaible wih V Z,Γ. ha i, given he conrac ξ Z,Γ, any conrol in V Z,Γ i opimal for he agen problem. Accordingly, he principal problem i modified a follow, denoing again for noaional impliciy v,a := v Z,Γ,a Z,Γ, and auming U P x = e RPx : [ { inf E exp R P σ Z,Γ,v,a v b a Z U V Z,Γ 2 σ v 2 Γ + R A Z 2 d G ob u,z u,γ u du + 22 Z 1dX v,a }].

23 Similarly, a above, denoe by θ he vecor σ v. he principal problem hen become { inf θ Z,Γ,v,a Z,Γb a + R A 2 θ Z,Γ 2 Z 2 + k v,a + R P 2 θ Z,Γ 2 1 Z 2}. 5.2 We hen have, imilarly a before, heorem 5.3. Conider he e of admiible conrac ξ Z,Γ C. hen, he opimal conrac ha provide he agen wih opimal expeced uiliy V A i he one correponding o Z,Γ,v,a which are he minimizer in 5.2, provided uch minimizer exi wih v. Moreover, he conrac cah erm in he conrac i given by R 1 A log V A. he analogue of Propoiion 4.2 ill hold in hi conex, wih he ame aemen and he ame proof. 6 Concluion We build a framework for udying moral hazard in dynamic rik managemen, uing recenly developed mahemaical echnique. While hoe allow u o olve he problem in which uiliy i drawn olely from erminal payoff, we leave for a fuure paper he imilar problem on infinie horizon wih iner-emporal paymen, a la Sannikov 28. In he cae of erminal payoff, we find ha he opimal conrac i implemened by compenaion baed on he oupu, i quadraic variaion correponding in pracice o he ample variance ued when compuing Sharpe raio, he conracible ource of rik, and he cro-variaion beween he oupu and he rik ource. Or, i could be implemened by derivaive ha provide uch payoff. I i an open queion how much, if any, mahemaical generaliy we loe in rericing he family of admiible conrac he way we do. We argue ha from a pracical perpecive, we do include all he conrac ha are likely o be conidered by a principal in he real world. In our framework i i aumed ha he principal know he model parameer for example, he mean reurn rae and he variancecovariance marix of he reurn of he ae he hedge fund manager i inveing in. In pracice, he principal may no have ha informaion, and i would be of inere o exend he model o include hi advere elecion problem. hi migh be approached by combining our model wih he one of Leung 214. Reference [1] Bicheler, K Sochaic inegraion and L p -heory of emimaringale, Ann. Prob., 91: [2] Briand, P., Hu, Y. 28. Quadraic BSDE wih convex generaor and unbounded erminal condiion, Prob. heory and Relaed Field, :

24 [3] Cadenilla, A., Cvianić, J., Zapaero, F. 27. Opimal rik-haring wih effor and projec choice, Journal of Economic heory, 133: [4] Con, R., Fournié, D Funcional Iô calculu and ochaic inegral repreenaion of maringale, Annal of Probabiliy, 411: [5] Cvianić, J., Wan, X., Yang, H Dynamic of conrac deign wih creening, Managemen Science, 59: [6] Dupire, B. 29. Funcional Iô calculu, preprin. [7] El Karoui, N., Peng, S., Quenez, M.-C Backward ochaic differenial equaion in finance, Mahemaical Finance, 7:1 71. [8] El Karoui, N., Quenez, M.-C Dynamic programming and pricing of coningen claim in an incomplee marke, SIAM J. Conrol Opim., 331: [9] Epein, L., Ji, S Ambiguou volailiy and ae pricing in coninuou ime, Rev. Financ. Sud., 26: [1] Epein, L., Ji, S Ambiguou volailiy, poibiliy and uiliy in coninuou ime, Journal of Mahemaical Economic, 5: [11] Holmrom B., Milgrom, P Aggregaion and lineariy in he proviion of ineremporal incenive, Economerica, 552: [12] Kazi-ani, N., Poamaï, D., Zhou, C Second-order BSDEJ wih jump: formulaion and uniquene, Ann. of App. Probabiliy, o appear. [13] Leung, R.C.W Coninuou-ime principal-agen problem wih drif and ochaic volailiy conrol: wih applicaion o delegaed porfolio managemen, working paper, hp://rn.com/abrac= [14] Lioui, A., Ponce, P 213. Opimal benchmarking for acive porfolio manager, European Journal of Operaional Reearch, 2262: [15] Kobylanky, M. 2. Backward ochaic differenial equaion and parial differenial equaion wih quadraic growh, Ann. Prob., 2: [16] Neufeld, A., Nuz, M Meaurabiliy of emimaringale characeriic wih repec o he probabiliy law, Soc. Proc. App., 12411: [17] Nuz, M Pahwie conrucion of ochaic inegral, Elec. Com. Prob., 1724:

25 [18] Nuz, M., Soner, H.M Superhedging and dynamic rik meaure under volailiy uncerainy, SIAM Journal on Conrol and Opimizaion, 54: [19] Nuz, M., van Handel, R Conrucing ublinear expecaion on pah pace, Sochaic Procee and heir Applicaion, 1238: [2] Pagè, H., Poamaï, D A mahemaical reamen of bank monioring incenive, Finance and Sochaic, 181: [21] Peng, S., Song, Y., Zhang, J A complee repreenaion heorem for G-maringale, Sochaic, 86: [22] Poamaï, D., Royer, G., ouzi, N On he robu uper hedging of meaurable claim, Elec. Comm. Prob., 1895:1 13. [23] Ou-Yang, H. 23. Opimal conrac in a coninuou-ime delegaed porfolio managemen problem, he Review of Financial Sudie, 16: [24] Sannikov, Y. 28. A coninuou-ime verion of he Principal-Agen problem, Review of Economic Sudie, 75: [25] Soner, H.M., ouzi, N., Zhang, J Quai-ure ochaic analyi hrough aggregaion, Elecronic Journal of Probabiliy, 1667: [26] Soner, H.M., ouzi, N., Zhang, J Dual formulaion of econd order arge problem, Annal of Applied Probabiliy, 231: [27] Soner, H.M., ouzi, N., Zhang, J. 212 Wellpoedne of econd-order backward SDE, Prob. h. and Rel. Field, : [28] Sung, J Lineariy wih projec elecion and conrollable diffuion rae in coninuou-ime principal-agen problem, RAND J. Econ. 26, [29] Wong,.-Y Dynamic agency and exogenou rik aking, working paper. 7 Appendix Proof of heorem 4.1. We adap argumen of Soner, ouzi & Zhang 211, 212, 213. In Sep 1 we ranform our problem o he weak formulaion ued in hoe paper. he agen problem i hen analyzed in Sep 2. Finally, Sep 3 pecialize o he cae of unconrolled volailiy. Sep 1: An alernaive formulaion of he agen problem Le u conider he following family of procee, indexed by admiible procee v: 25

26 M v := σ v db B 1. = B d Σ v db Σ v db, P a.., 7.1 Σ v db where, imilarly a in Secion 4.1, for any [, ] and any v V he d + 1 d marix Σv i defined by σ v Σ v :=,wih I d,d := I d,d I d d,d d. Furhermore, Σ i now a d d 1 d marix aifying, for any [, ] and any v V, Σ vσ v = d d 1,d +1 and Σ vσ v = I d d 1. We hen e P m o be he e of probabiliy meaure P v on Ω,F of he form P v := P M ṿ 1, for any admiible v. 7.2 We recall ha by Bicheler 1981, we can alo define a pahwie verion of he quadraic variaion proce B and of i deniy proce wih repec o he Lebegue meaure, a poiive ymmeric marix α: α := d B d he elemen of family correpond o poible choice of he volailiy vecor σ v by he agen. We remark ha by our definiion, we have eh following weak formulaion: he law of B, α under P v = he law of M v, Σ vσ v d d 1,d +1 I d d 1. d +1,d d 1 under P. Moreover, exacly a in Secion 4.1, he deniy of he quadraic variaion of B i inverible under P v, if and only if Σ vσ v i inverible, which can be hown o be equivalen o which i he ame a 3.6. σ v 2 d i=1 σ v i 2, hi family could alo be characerized by conidering all he choice of conrol u for which here exi a lea one rong oluion o he SDE for M ee Soner, ouzi and Zhang 211, or, equivalennly, o a cerain maringale problem ee Kazi-ani, Poamaï and Zhou 213 and Neufeld and Nuz

27 hen, according o Lemma 2.2 in Soner, ouzi, Zhang 213, for every admiible v, here exi a F-progreively meaurable mapping β v : [, ] Ω R d uch ha B = β v M v Σ β v BΣ, P a.., α β v B B = d d 1,d +1 I d d 1 hi implie in paricular, ha he proce W := d +1,d d 1, P v a.. α 1/2 db, P a., 7.3 i a R d -valued, P-Brownian moion, for every P P m. In paricular, hi implie ha he canonical proce B admi he following dynamic, for every admiible v, B d +2. B d B 1 = B 1 + σ v W dw, P v a.., B j = B j =. +W j 1 B d +2 B d, P v a.., j = 2...d + 1, if d >, + Σ v W dw, P v a hu, he fir coordinae of he canonical proce i he deired oupu proce, oberved by boh he principal and he agen, he following d coordinae repreen he conracible ource of rik, while he remaining one repreen he facor ha are no conracible. he inroducion of he conrolled drif b a can now be done by uing Giranov ranformaion. We define for any v,a U and any P v P m, he equivalen probabiliy meaure P v,a by dp v,a dp v and we denoe by P := P v,a v,a U. := E b a dw Nex, by Giranov heorem, he following proce W a i a P v,a -Brownian moion W a := W b a d, P v,a a.. hu alo P v a.. Noice ha we hould acually have conidered a family W P P Pm, ince he ochaic inegral in 7.3 i, a priori, only defined P-a.. However, we can ue reul of Nuz 212 o provide an aggregaed verion of hi family, which i he proce we denoe by W. ha reul hold under a good choice of e heoreic axiom ha we do no pecify here. Noe ha by aumpion ha he fir d enrie of vecor b do no depend on a, he choice of conrol a doe no modify he diribuion of he exogenou ource of rik B 2,...,B d +1., 27