Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model

Transcription

1 Invemen and valuaion under backward and forward dynamic exponenial uiliie in a ochaic facor model Marek Muiela and Thaleia Zariphopoulou BNP Pariba, London and The Univeriy of Texa a uin Fir verion: January 004 Thi verion: June 006 brac We inroduce a new cla of dynamic uiliie ha are generaed forward in ime. We dicu he aociaed value funcion, opimal invemen and indifference price and we compare hem wih heir radiional counerpar, implied by backward dynamic uilie. Inroducion Thi paper i a conribuion o inegraed porfolio managemen in incomplee marke. Incompleene em from a correlaed ochaic facor affecing he dynamic of he raded riky ecuriy ock. The inveor rade beween a rikle bond and he ock, and may incorporae in hi porfolio derivaive and liabiliie. The opimal invemen problem i embedded ino a parial equilibrium one ha can be olved by he o called uiliy-baed pricing approach. The opimal porfolio can be, in urn, conruced a he um of he policy of he plain invemen problem and he indifference hedging raegy of he aociaed claim. BNP Pariba, London. marek.muiela@bnppariba.com. The Univeriy of Texa a uin. zariphop@mah.uexa.edu. The auhor acknowledge parial uppor from gran NSF DMS and DMS-FRG Thi work wa preened a he Workhop in Semimaringale heory and pracice in Finance, Banff June 004, Imperial College June 004, Cornell Univeriy November 004, he Workhop in Mahemaical Finance a Carnegie Mellon Univeriy January 005, he Second Bachelier Colloquium, Meabief January 005, London School of Economic pril 005, he Conference on Sochaic Procee and heir applicaion, cona June 005, he SIM Conference on Conrol and Opimizaion, New Orlean July 005, he Sochaic Modeling Workhop on PDE and heir applicaion o Mahemaical Finance a KTH, Sockholm ugu 005, he Bachelier Seminar May 006 and he Conference in Honor of S. Sehi a he Univeriy of Texa a Dalla May 006. The auhor would like o hank he paricipan for fruiful commen and uggeion.

2 In a variey of applicaion, he invemen horizon and he mauriie of he claim do no coincide. Thi mialignmen migh caue price dicrepancie, if he curren opimal expeced uiliy i no correcly pecified. The focu herein i in exploring which clae of uiliie preclude uch pahological iuaion. In he radiional framework of expeced uiliy from erminal wealh, he correc dynamic uiliy i eaily idenified, namely, i i given by he implied value funcion. Such a uiliy i, hen, called elf-generaing in ha i i indiinguihable from he value funcion i produce. Thi i an inuiively clear conequence of he Dynamic Programming Principle. There are, however, wo imporan underlying ingredien. Firly, he rik preference are a priori pecified a a fuure ime, ay T, and, econdly, he uiliy, denoed by U B x; T, i generaed a previou ime 0 T. Herein, T denoe he end of he invemen horizon and x repreen he wealh argumen. Due o he backward in ime generaion, T i called he backward normalizaion poin and U B x; T he backward dynamic uiliy. lbei heir populariy, he radiional backward dynamic uiliie coniderably conrain he e of claim ha can be priced o he one ha expire before he normalizaion poin T. Moreover, even in he abence of payoff and liabiliie, uiliie of erminal wealh do no eem o capure very accuraely change in he rik aiude a he marke environmen evolve. In many apec, in he familiar uiliy framework uiliie move backward in ime while marke hock are revealed forward in ime. Moivaed by uch conideraion, he auhor recenly inroduced he noion of forward dynamic uiliie ee Muiela and Zariphopoulou 005a and 005b in a imple muli-period incomplee binomial model. Thee uiliie, like heir backward counerpar, are creaed via an expeced crierion bu, in conra, hey evolve forward in ime. Specifically, hey are deermined oday, ay a, and hey are generaed for fuure ime, via a elf-generaing crierion. In oher word, he forward dynamic uiliy, U F x;, i normalized a preen ime and no a he end of he generic invemen horizon. In hi paper, we exend he noion of forward dynamic uiliie in a diffuion model wih a correlaed ochaic facor. For impliciy, we aume ha he uiliy daa, a boh backward and forward normalizaion poin, are aken o be of exponenial ype wih conan rik averion. Thi aumpion can be relaxed wihou loing he fundamenal properie of he dynamic uiliie. We, alo, concenrae our analyi o European-ype liabiliie o ha cloed form variaional reul can be obained. The wo clae of dynamic uiliie, a well a he emerging price and invemen raegie, have imilariie bu, alo, riking difference. menioned above, boh uiliie are elf-generaing and, herefore, price dicrepancie are precluded in he aociaed backward and forward indifference pricing yem. conequence of elf-generaion i ha an inveor endowed wih backward and forward uiliie receive he ame dynamic uiliy acro differen invemen horizon. I i worh noing ha while he backward dynamic uiliy i unique, he forward one migh no be. The aociaed indifference price have very diinc characeriic. Back-

3 ward indifference price depend on he backward normalizaion poin, and, hu, implicily on he rading horizon even if he claim maure before T. However, forward indifference price are no affeced by he choice of he forward normalizaion poin. For he cla of European claim examined herein, backward price are repreened a nonlinear expecaion aociaed wih he minimal relaive enropy meaure. On he oher hand, forward price are alo repreened a nonlinear expecaion bu wih repec o he minimal maringale meaure. The wo price do no coincide unle he marke i complee. Thi i a direc conequence of he fac ha he inernal marke incompleene - coming form he ochaic facor - i proceed by he backward and forward dynamic uiliie in a very diinc manner. The porfolio raegie relaed o he backward and forward uiliie have alo very differen characeriic. The opimal backward invemen coni of he myopic porfolio, he backward indifference dela and he exce riky demand. The laer policy reflec, in conra o he myopic porfolio, he incremenal change in he opimal behavior due o he movemen of he ochaic facor. The forward opimal invemen have he ame rucure a heir backward counerpar bu do no include he exce riky demand. The paper i organized a follow. In Secion, we inroduce he invemen model and i dynamic uiliie. In Secion 3, we provide ome auxiliary echnical reul relaed o he minimal maringale and minimal enropy meaure. In Secion 4 and 5, repecively, we conruc he backward and forward dynamic uiliie, he aociaed price and he opimal invemen. We conclude in Secion 6, where we provide a comparaive udy for inegraed porfolio problem under he wo clae of dynamic rik preference. The model and i dynamic uiliie Two ecuriie are available for rading, a rikle bond and a riky ock whoe price olve ds = µ Y S d + σ Y S dw for 0 and S 0 = S > 0. The bond offer zero inere rae. The cae of deerminiic non-zero inere rae may be handled by raighforward caling argumen and i no dicued. The proce Y, o be referred o a he ochaic facor, i aumed o aify dy = b Y d + a Y dw for 0 and Y 0 = y R. The procee W and W are andard Brownian moion defined on a probabiliy pace Ω, F, F,P wih F being he augmened σ-algebra. We aume ha he correlaion coefficien ρ, and, hu, we may wrie dw = ρdw + ρ dw, 3 3

4 wih W, being a andard Brownian moion on Ω, F, F, P orhogonal o W. For impliciy, we aume ha he dynamic in and are auonomou. We denoe he ock Sharpe raio proce by λ = λy = µ Y σ Y. 4 The following aumpion will be anding hroughou. umpion : The marke coefficien µ, σ, a and b are aumed o be C R funcion ha aify, f y C + y for f = µ, σ, a and b, and are uch ha and have a unique rong oluion aifying S > 0 a.e. for 0. There alo exi ε > 0 uch ha σ y > ε, for y R. Nex, we conider an arbirary rading horizon [0, T], and an inveor who ar, a ime 0 [0, T], wih iniial wealh x R and rade beween he wo ecuriie. Hi/her curren wealh X, 0 T, aifie he budge conrain X = π 0 +π where π 0 and π are elf-financing raegie repreening he amoun inveed in he bond and he ock accoun. Direc calculaion, in he abence of inermediae conumpion, yield he evoluion of he wealh proce dx = µ Y π d + σ Y π dw 5 wih X 0 = x R. The e of admiible raegie i defined a = } T0 {π : π i F meaurable, elf-financing and E P σ Y π d <. Furher conrain migh be binding due o he pecific applicaion and/or he form of he involved uiliy payoff. In order, however, o keep he expoiion imple and o concenrae on he new noion and inigh, we chooe o abrac from uch conrain. We denoe by D he generic paial olvency domain for x, y. We ar wih an informal moivaional dicuion for he upcoming noion of backward and forward dynamic uiliie. In he radiional economic model of expeced uiliy from erminal wealh, a uiliy daum i aigned a a given ime, repreening he end of he invemen horizon. We denoe he uiliy daum by u x and he ime a which i i aigned by T. 0 0, he inveor ar rading beween he available ecuriie ill T. inermediae ime [ 0, T], he aociaed value funcion v : D [0, T] R i defined a he maximal expeced condiional on F uiliy ha he agen achieve from invemen. For he model a hand, v ake he form v x, y, = up E P u X T X = x, Y = y, [ 0, T] 6 wih he wealh and ochaic facor procee X, Y olving 5 and. The cope i o pecify v and o conruc he opimal conrol policie. The dualiy approach can be applied o general marke model and provide characerizaion reul for he value funcion, bu limied reul for he opimal porfolio. The laer can be conruced via variaional mehod for cerain clae of diffuion model. To dae, while here i a rich body of work for he value funcion, very lile i underood abou how inveor adju heir 4

5 porfolio in erm of heir rik preference, rading horizon and he marke environmen. i Inegraed model of porfolio choice In a more realiic eing, he inveor migh be inereed in incorporaing in hi porfolio derivaive ecuriie, liabiliie, proceed from addiional ae, labor income ec. Given ha uch iuaion arie frequenly in pracice, i i imporan o develop an approach ha accommodae inegraed invemen problem and yield quaniaive and qualiaive reul for he opimal porfolio. Thi i he aim of he udy below. To implify he preenaion, we aume, for he momen, ha he inveor face a liabiliy a T, repreened by a random variable C T F T. We recall ha F T i generaed by boh he raded ock and he ochaic facor and ha he inveor ue only elf-financing raegie. In a complee marke e-up e.g. when he procee S and Y are perfecly correlaed he opimal raegy for hi generalized porfolio choice model i a follow: a iniiaion 0, he inveor pli he iniial wealh, ay x, ino he amoun E Q C T F 0 and x = x E Q C T F 0, wih E Q C T F 0 being he arbirage-free price of C T. The reidual amoun x i ued for invemen a if here wa no liabiliy. The dynamic opimal raegy i, hen, he um of he opimal porfolio, correponding o iniial endowmen x and he hedging raegy, denoed by δ C T, of a European-ype coningen claim wrien on he raded ock, mauring a T and yielding C T. Uing o denoe opimal policie, we may wrie π x, = π x, + δ C T wih x = x E Q C T F 0 7 Thi can be eablihed eiher by variaional mehod or dualiy. Thi remarkable addiive rucure, ariing in he highly nonlinear uiliy eing, i a direc conequence of he abiliy o replicae he liabiliy. Noe ha for a fixed choice of wealh uni, he econd porfolio componen i no affeced by he rik preference. When he marke i incomplee, imilar argumenaion can be developed by formulaing he problem a a parial equilibrium one and, in urn, uing reul from he uiliy-baed valuaion approach. The liabiliy may be, hen, viewed a a derivaive ecuriy and he opimal porfolio choice problem i embedded o an indifference valuaion one. Uing payoff decompoiion reul ee, for example, Muiela and Zariphopoulou 00 and 004a, Soikov and Zariphopoulou 004 and Monoyio 006, we aociae o he liabiliy an indifference hedging raegy, ay C T, ha i he incomplee marke counerpar of i arbirage-free replicaing porfolio. Denoing he relevan indifference price by ν C T, we obain an analogou o 7 decompoiion of he opimal invemen raegy in he ock accoun, namely, π x, = π x, + C T wih x = x ν 0 C T 8 5

6 Becaue he indifference valuaion approach incorporae he inveor rik preference, he choice of uiliy will influence - in conra o he complee marke cae - boh componen of he emerging opimal invemen raegy. Noe, however, ha due o he dynamic naure of he problem, uiliy effec evolve boh wih ime and marke informaion. In order o correcly quanify hee effec, i i imperaive o be able o pecify he dynamic value of our invemen raegie acro horizon, mauriie and uni. he analyi below indicae, he cornerone of hi endeavor i he pecificaion of a dynamic uiliy rucure ha yield conien valuaion reul and invemen behavior acro opimally choen elf-financing raegie. Before we inroduce he dynamic uiliie, we fir recall he auxiliary concep of indifference value. To preerve impliciy, we conider he aforeinroduced ingle liabiliy C T. To calculae i indifference price, ν C T, for [0, T], we look a he inveor modified uiliy, v CT x, y, = up E P u X T C T X = x, Y = y 9 and, ubequenly, impoe he equilibrium condiion v x ν C T,y, = v CT x, y,. 0 The opimal policy i given by 8 and can be rerieved in cloed form for pecial cae. For example, when he uiliy i exponenial, u x = e γx wih γ > 0, he ock Sharpe raio i conan and C T = GY T, for ome bounded funcion G, variaional argumen yield he opimal invemen repreenaion π x, = π x, + ρ a Y σ Y g y, y y=y,= wih x = x g y, 0, and g : R [0, T] R olving a quailinear pde, of quadraic gradien nonlineariie., wih erminal condiion g y, T = Gy. ii Liabiliie and payoff of horer mauriie Conider a liabiliy o be paid before he fixed horizon T, ay a T 0 < T. There are wo way o proceed. The fir alernaive i o work wih porfolio choice in he iniial invemen horizon, [ 0, T]. In hi cae, he uiliy 9 become v CT 0 x, y, = up E P u X T C T0 X = x, Y = y, where we ook ino conideraion ha he rikle inere rae i zero. The indifference value i, hen, calculaed by he pricing condiion 0, for [ 0, T 0 ]. However, uch argumen migh no be eaily implemened, if a all, a i i he cae of liabiliie and payoff of random mauriy and/or variou exoic characeriic. 6

7 The econd alernaive i o derive he indifference value by conidering he invemen opporuniie, wih and wihou he liabiliy, up o he claim mauriy T 0. For hi, we fir need o correcly pecify he value funcion, denoed, repecively, by v and v CT 0, ha correpond o opimaliy of invemen in he horer invemen horizon, [ 0, T 0 ]. Working along he line heir long-horizon counerpar, v and v CT, were defined, le u, hypoheically, aume ha we are given a uiliy daum for he poin T 0. We denoe hi daum by ūx, y, T 0. We will, henceforh, ue he noaion for all quaniie, i.e. uiliie, invemen and indifference value, aociaed wih he horer horizon. For [ 0, T 0 ], and v x, y, = up E P ūx T0, Y T0, T 0 X = x, Y = y v CT 0 x, y, = up E P ūx T0 C T0, Y T0, T 0 X = x, Y = y. The aociaed indifference value, ν C T0, will be, hen, given by v x ν C T0,y, = v CT 0 x, y,. Clearly, in order o have a well pecified valuaion yem, we mu have, for all C T0 F T0 and [ 0, T 0 ], ν C T0 = ν C T0, which rongly ugge ha he uiliy daum ūx, y, T 0 canno be exogenouly aigned in an arbirary manner. Such iue, relaed o he correc pecificaion and alignmen of inermediae uiliie, and heir value funcion, wih he claim poibly differen and/or random mauriie, were fir dicued in Davi and Zariphopoulou 995 in he conex of uiliy-baed valuaion of merican claim in marke wih ranacion co. Recall ha when early exercie i allowed, he fir alernaive compuaional ep, cf., canno be implemened becaue T 0 i no a priori known. For he ame cla of early exercie claim, bu when incompleene come excluively from a non-raded ae which doe no affec he dynamic of he ock, and he claim i wrien on boh he raded and nonraded ae, furher analyi on he pecificaion of preference acro exercie ime, wa provided in Kallen and Kuehn 004, Oberman and Zariphopoulou 003 and Muiela and Zariphopoulou 004b. In he laer paper, he relaed inermediae uiliie, and valuaion condiion ook, repecively, he form ūx, S, = v x, S, and v x, S, = ūx, S,, v Cτ x, S, z, = up E P v X τ C S τ, Z τ,s τ, τ X = x, S = S, Z = z, T where T i he e of opping ime in [ 0, T]. The procee S and Z repreen he raded and nonraded ae and X he wealh proce. The early exercie indifference price of C τ i, hen, given by v x, S, = v Cτ x + ν C S τ, Z τ,s, z,. 7

8 While he above calculaion migh look pedanic when a ingle exogenou cah flow liabiliy or payoff i incorporaed, he argumen ge much more involved when a family of claim i conidered and arbirary, or ochaic, mauriie are allowed. Naurally, he relaed difficulie diappear when he marke i complee. However, when perfec replicaion i no viable and a uiliy-baed approach i implemened for valuaion, dicrepancie leading o arbirage migh arie if we fail o properly incorporae in our model dynamic rik preference ha proce and price he marke incompleene in a conien manner. Thi iue wa expoed by he auhor in Muiela and Zariphopoulou 005a and 005b, who iniiaed he conrucion of indifference pricing yem baed on he o-called backward and forward dynamic exponenial uiliie. In hee paper, indifference valuaion of arbirary claim and pecificaion of inegraed opimal policie were udied in an incomplee binomial cae. Even hough hi model e-up wa raher imple, i offered a aring poin in exploring he effec of he evoluion of rik preference o price and invemen. Wha follow i, o a grea exen, a generalizaion of he heory developed herein. iii Uiliy meauremen acro invemen ime Le u now ee how a dynamic uiliy can be inroduced and incorporaed in he ochaic facor model we are inereed in. We recall ha he anding aumpion are: i he rading horizon [0, T] i preaigned, ii a uiliy daum i given for T and iii T dominae he mauriie of all claim and liabiliie in conideraion. We nex aume ha inead of having he ingle aic meauremen of uiliy, u, a expiraion, he inveor i endowed wih a dynamic uiliy, u x, y; T, [0, T]. Being vague, for he momen, we view hi uiliy a a funcional, a each inermediae ime, of her curren wealh and he level of he ochaic facor. Obviouly, we mu have u T x, y; T = u x, in which cae, we ay ha u x, y; T i normalized a T. a conequence, we will refer o T a he normalizaion poin. For reaon ha will be apparen in he equel, we chooe o carry T in our noaion. If a maximal expeced crierion i involved, he aociaed value funcion, denoed wih a ligh abue of noaion by v, will naurally ake he form v x, y, T = up E P u T X T,Y T;T X = x, Y = y, in an arbirary ub-horizon [, T ] [ 0, T] and wih X, Y olving 5 and. Le u now ee how he generic liabiliy C T0 F T0 would be valued under uch a uiliy rucure. For [ 0, T 0 ], he relevan maximal expeced dynamic uiliy will be v CT 0 x, y, T 0 = up E P u T0 X T0 C T0, Y T0 ; T X = x, Y = y. 8

9 Repecively, he indifference value, ν C T0 ; T, mu aify, for [ 0, T 0 ], v x ν C T0 ; T,y, T 0 = v CT 0 x, y, T 0. Oberve ha becaue u i normalized a T, he aociaed value funcion will depend on he normalizaion poin. The laer will alo affec he indifference price ν C T0 ; T, even hough he claim maure a an earlier ime. So far, he above formulaion eem convenien, and flexible enough, for he valuaion of claim wih arbirary mauriie, a long a hee mauriie are horer han he ime a which rik preference are normalized. However, a he nex wo example how, i i wrong o aume ha a dynamic uiliy can be inroduced in an ad hoc way. In boh example, i i aumed ha he erminal uiliy daum i of exponenial ype, and independen of he level of he ochaic facor, u T x, y; T = e γx wih x, y D and γ being a given poiive conan. I i alo aumed ha here i a ingle claim o be priced. I payoff i aken o be of he form C T0 = GY T0, for ome bounded funcion G : R R +. lbei he fac ha in he model conidered herein, uch a payoff i, o a cerain exen, arificial, we, neverhele, chooe o work wih i becaue explici formulae can be obained and he expoiion i, hu, coniderably faciliaed. Example : Conider a dynamic uiliy of he form e γx T < T u x, y; T = e γx 0 < T, wih T > T 0, γ a in and γ γ. Le u now ee how C T0 will be valued under he above choice of dynamic uiliy. If he inveor chooe o rade in he original horizon [ 0, T], he aociaed inermediae uiliie are v 0,CT 0 x, y, T = up E P u T X T C T0, Y T X = x, Y = y = up E P e γxt CT 0 X = x, Y = y. Obviouly, he diconinuiy, wih regard o he rik averion coefficien of u will no aler he above value funcion. Following he reul of Sircar and Zariphopoulou 005 yield ν C T0 = γ ρ ln E Q me e γ ρ GY T0 Y = y, 3 wih Q me being he minimal relaive enropy maringale meaure ee nex ecion for he relevan echnical argumen. 9

10 If, however, he inveor chooe o rade olely in he horer horizon [ 0, T ], analogou argumenaion yield v 0,CT 0 x, y, T = up E P e γx T GY T0 X = x, Y = y, where we ued he noaion o denoe he horer horizon choice. The aociaed indifference price i ν C T0 = and we eaily deduce ha, in general, γ ρ ln E Q me e γ ρ GY T0 Y = y, ν C T0 ν C T0, an obviouly wrong reul. Noe ha even if we naively allow γ = γ price dicrepancie will ill emerge. Example : Conider he dynamic uiliy u x, y; T = e γx Fy,;T wih γ a in and F y, ; T = E P T λ Y d Y = y, where P i he hiorical meaure. If he agen chooe o inve in he longer horizon, [ 0, T], he indifference value remain he ame a in 3. However, if he chooe o inve excluively ill he liabiliy i me, we have, for [ 0, T 0 ], Seing v 0,CT 0 x, y, T 0 = up E P u T0 X T0 GY T0 ;T X = x, Y = y = upe P e γxt 0 GYT 0 FY T0,T 0;T X = x, Y = y Z T0 = γ F Y T 0, T 0 ; T we deduce ha, in he abence of he liabiliy, he curren uiliy i v 0 x, y, T 0 = up E P e γxt 0 +ZT 0 X = x, Y = y. Noe ha, by definiion, Z T0 F T0. Therefore, we may inerpre v 0 a a buyer value funcion for he claim Z T0, in a radiional non-dynamic exponenial uiliy eing of conan rik averion γ and invemen horizon [0, T 0 ]. Then,. 0

11 v 0 x, y, T 0 = e γx+ µzt 0 Hy,;T 0 wih H y,, T 0 being he aggregae enropy funcion ee equaion 4 in nex Secion and µ Z T0 = γ ρ lne Q me e γ ρ Z T0 Y = y. Proceeding imilarly, we deduce, v CT 0 x, y, T 0 = up E P e γxt 0 +ZT 0 GYT 0 X = x, Y = y = e γx+ µzt 0 GYT 0 Hy,;T 0 wih µ Z T0 GY T0 = γ ρ lne Q me e γ ρ Z T0 GY T0 Y = y. pplying he definiion of he indifference value, we deduce ha, wih regard o he horer horizon, = ν GY T0 = µ Z T0 µ Z T0 GY T0 E e γ ρ Z T0 GY T0 γ ρ ln Qme Y = y e γ ρ Z T0 Y = y E Q me which, in general, doe no coincide wih ν GY T0 given in 3. iv Backward and forward dynamic exponenial uiliie The above example expoe ha an ad hoc choice of dynamic uiliy migh lead o price dicrepancie. Thi, in view of he rucural form of he opimal policy for he inegraed model cf. 8 would, in urn, yield wrongly pecified invemen policie. I i hu imporan o inveigae which clae of dynamic uiliie preclude uch pahological iuaion. For he imple example above, he correc choice of he dynamic uiliy i eenially obviou, namely, u x for = T u x, y; T = v x, y, for [ 0, T], wih v a in 6. Thi imple obervaion indicae he following: fir, oberve ha if u i he candidae dynamic uiliy, hen, in all rading ub-horizon, ay [, ], he aociaed dynamic value funcion v will be v x, y; T = up E P u X, Y ; T X = x, Y = y.

12 Dicrepancie in price will be, hen, precluded if a all inermediae ime he dynamic uiliy coincide wih he dynamic value funcion i generae, u x, y; T = v x, y; T. We hen ay ha he dynamic uiliy i elf-generaing. Building on hi concep, we are led o wo clae of dynamic uiliie, he backward and forward one. Their definiion are given below. Becaue he applicaion herein are concenraed on exponenial preference, we work wih uch uiliy daa. Throughou, we ake he rik averion coefficien o be a poiive conan γ. While he backward dynamic uiliy i eenially he radiional value funcion, he concep of forward uiliy i, o he be of our knowledge, new. menioned earlier, i wa recenly inroduced by he auhor in an incomplee binomial eing ee Muiela and Zariphopoulou 005a and 005b and i i herein exended o he diffuion cae. We coninue wih he definiion of he backward dynamic uiliy. Thi uiliy ake he name backward becaue i i fir pecified a he normalizaion poin T and i hen generaed a previou ime. Definiion Le T > 0. n F -meaurable ochaic proce U B x; T i called a backward dynamic uiliy BDU, normalized a T, if for all, T i aifie he ochaic opimaliy crierion e γx, = T U B x; T = up E P U X B T T;T 4 F, 0 T T, wih X given by 5 and X = x R. The above equaion provide he coniuive law for he backward dynamic uiliy. Noe ha even hough hi dynamic uiliy coincide wih he familiar value funcion, i noion wa creaed from a very differen poin of view and cope. Under mild regulariy aumpion on he coefficien of he ae procee, i i eay o deduce ha he above problem ha a oluion ha i unique. There i ample lieraure on he value funcion and, hu, on he backward dynamic uiliy ee, for example, Kramkov and Schachermayer 999, Rouge and El Karoui 000, Delbaen e al. 00 and Kabanov and Sricker 00. The fac ha U B i elf-generaing, i immediae. Indeed, in an arbirary ub-horizon [, T ], he aociaed value funcion V B, given by V B x, T; T = up E P U B T X T;T F, coincide wih i aociaed dynamic uiliy, U B x; T = V B x, T; T by Definiion.

13 conequence of elf-generaion i ha he inveor receive he ame dynamic uiliy acro differen invemen horizon. Thi i een by he fac ha, for T T, elf-generaion yield and and he horizon invariance, V B V B V B x, T; T = U B x; T x, T ; T = U B x; T x, T; T = V B x, T ; T follow. In mo of he exiing uiliy model, he dynamic uiliy - or, equivalenly, i aociaed value funcion - i generaed backward in ime. The form of he uiliy migh be more complex, a i i he cae of recurive uiliie where dynamic rik preference are generaed by an aggregaor. Neverhele, he feaure of uiliy prepecificaion a a fuure fixed poin in ime and generaion a previou ime are ill prevailing. One migh argue ha an ad hoc pecificaion of uiliy a a fuure ime i, o a cerain exen, non inuiive, given ha our rik aiude migh change wih he way he marke environmen enfold from one ime period o he nex. Noe ha change in he invemen opporuniie and loe/gain are revealed forward in ime while he radiional value funcion appear o proce hi informaion backward in ime. Such iue have been conidered in propec heory where, however, uiliy normalizaion a a given fuure poin i ill preen. From he valuaion perpecive, working wih uiliie normalized in he fuure everely conrain he cla of claim ha can be priced. Indeed, heir mauriie mu be alway dominaed by he ime a which he backward uiliy i normalized. Thi preclude opporuniie relaed o claim arriving a a laer ime and mauring beyond he normalizaion poin. In order o be able o accommodae claim of arbirary mauriie, one migh propoe o work in an infinie horizon framework and o employ eiher dicouned a opimal growh uiliy funcional or uiliie allowing for inermediae conumpion. The perpeual naure of hee problem, however, migh no be appropriae for a variey of applicaion in which he agen face defaul, conrain due o reporing period and oher real-ime iue. Moivaed by hee conideraion, he auhor recenly inroduced he concep of forward dynamic uiliie. Their main characeriic i ha hey are deermined a preen ime and, a heir name indicae, are generaed, via heir coniuive equaion, forward in ime. Definiion Le 0. n F -meaurable ochaic proce U F x; i called a forward dynamic exponenial uiliy FDU, normalized a, if, for all, T, 3

14 wih T, i aifie he ochaic opimizaion crierion e γx, = U F x; = up E P U F T X T ; 5 F,. Oberve ha by conrucion, here i no conrain on he lengh of he rading horizon. Like i backward dynamic counerpar, he forward dynamic uiliy i elfgeneraing and make he inveor indifferen acro diinc invemen horizon. Indeed, elf-generaion, i.e., wih V F U F x; = V F x, T; x, T; = up E P U F T X T ; F i an immediae conequence he above definiion. For he horizon invariance, i i enough o oberve ha in differen ub-horizon, ay [, T] and [, T ], and and, herefore, V F V F x, T; = up E P U F T X T ; F, x, T; = up E P U F T X T; F, V F x, T; = V F x, T;. We re ha, in conra o heir backward dynamic counerpar, forward dynamic uiliie migh no be unique. In general, he problem of exience and uniquene i an open one. Thi iue i dicued in Secion 5. Deermining a naural cla of forward uiliie in which uniquene i eablihed i a challenging and, in our view, inereing queion. 3 uxiliary echnical reul In he upcoming ecion, wo equivalen maringale meaure will be ued, namely, he minimal maringale and he minimal enropy one. They are denoed, repecively, by Q mm and Q me and are defined a he minimizer of he enropic funcional H 0 Q mm P = min E P ln dq Q Q e dp and dq H Q me dq P = min E P ln, Q Q e dp dp 4

15 where Q e and for he e of equivalen maringale meaure. There i ample lieraure on hee meaure and on heir role in valuaion and opimal porfolio choice in he radiional framework of exponenial uiliy; ee, repecively, Foellmer and Schweizer 99, Schweizer 995 and 999, Bellini and Frielli 00 and Frielli 000, Rouge and El Karoui 000, rai 00, Delbaen e al. 00, Kabanov and Sricker 00. For arbirary T > 0, he rericion of Q mm and Q me on he σ-algebra F T = σ { Wu, W } u : 0 u T, can be explicily conruced a i i dicued nex. We remark ha, wih a ligh abue of noaion, he rericion of he wo meaure are denoed a heir original counerpar. The deniy of he minimal maringale meaure i given by dq mm T T dp = exp λ dw λ d 6 0 wih λ being he Sharpe raio proce 4. Calculaing he deniy of he minimal relaive enropy meaure i more involved and we refer he reader o Rheinlander 003 ee, alo, Grandi and Rheinlander 00 for a concie reamen. For he diffuion cae conidered herein, he deniy can be found hrough variaional argumen and i repreened by dq me dp = exp T 0 λ dw T 0 ˆλ dw, wih W, a in 3. The proce ˆλ i given by wih Y olving and ˆλ : R [0, T] R + defined a 0 T λ 0 + ˆλ d 7 ˆλ = ˆλY, ; T 8 ˆλ y, ; T = a y f y y, ; T ρ f y, ; T, 9 where f : R [0, T] R + i he unique C, R [0, T] oluion of he erminal value problem f + a yf yy + b y ρλyayf y = ρ λ yf 0 fy, T =. The proof can be found in Benh and Karlen 005 ee, alo, Soikov and Zariphopoulou 004 and Monoyio 006. The dependence of f and ˆλ on he end of he horizon, T, i highlighed due o he role ha i will play in he upcoming dynamic uiliie. 5

16 I eaily follow ha he aggregae, relaive o he hiorical meaure, enropie of Q mm and Q me are, repecively, dq H Q mm mm T dqmm P = E P ln = E Q mm dp dp λ d 0 and H Q me P = E P dq me dp T dqme ln = E Q me λ + dp 0 ˆλ d. When he marke become complee, he wo meaure, Q mm and Q me, coincide wih he unique rik neural meaure. In general, hey differ and heir repecive relaive enropie are relaed in a nonlinear manner. Thi wa explored in Soikov and Zariphopoulou 005, Corollary 3., where i wa hown ha T H Q me P = E Q mm λ d F 0. The condiional nonlinear expecaion E Q of a generic random variable Z F T and meaure Q on Ω, F T i defined, for [0, T] and γ R +, by E Q Z F ; γ = γ ρ lne Q 0 e γ ρ Z F, γ R +. The aggregae enropy H Q me P i hen he nonlinear expecaion of he random variable Z T = T 0 λ d, for Q = Qme and γ =. Nex we inroduce wo quaniie ha will faciliae our analyi. Namely, for 0 T T, we define he aggregae relaive enropy proce H, T T = E Q me λ Y + ˆλY, ; T d F 3 and he funcion H [ : R 0, T ] R +, H y, ; T T = E Q me λ Y + ˆλY, ; T d Y = y, 4 for λ, ˆλ defined in 4 and 8. We, alo, inroduce he linear operaor L Y = a y y + b y y, 5 L Y,mm = a y y + b y ρλyay y 6 6

17 and L Y,me = a y y + b y ρλyay y +a y f y y, ; T f y, ; T y 7 wih f olving 0. The following reul follow direcly from he definiion of H, and 9 and 0. Lemma 3 For T T, he funcion H : R [0, T] R +, olve he quailinear equaion H + L Y,mm H ρ a y H y + λ y = 0, or, equivalenly, he linear equaion H + L Y,me λ H y + ˆλY, ; T + = 0 wih H y, T; T = 0, and L Y,mm, L Y,me a in 6 and 7. 4 Invemen and valuaion under backward dynamic exponenial uiliie In hi ecion, we provide an analyic repreenaion of he backward dynamic exponenial uiliy cf. Definiion and conruc he agen opimal invemen in an inegraed porfolio choice problem. We recall ha he invemen horizon i fixed, he uiliy i normalized a i end and ha no liabiliie, or cah flow, are allowed beyond he normalizaion poin. For convenience, we occaionally rewrie ome of he quaniie inroduced in earlier ecion. Propoiion 4 Le Q me be he minimal relaive enropy maringale meaure and H, T he aggregae relaive enropy proce cf. 3, H, T T = E Q me λ Y + ˆλ Y, ; T d F wih λ and ˆλ a in 4 and 8. Then, for x R, [0, T], he proce U B F, given by U B x; T = e γx H,T 8 i he backward dynamic exponenial uiliy. The proof i, eenially, a direc conequence of he Dynamic Programming Principle and he reul of Rouge and El Karoui 000. For he pecific echnical argumen, relaed o he ochaic facor model we examine herein, we refer he reader o Soikov and Zariphopoulou 004. We eaily deduce he following reul. 7

18 Corollary 5 The backward dynamic uiliy i given by U B x; T = u x, Y, ; T wih u : R R + [0, T] R defined a γx Hy,;T u x, y, ; T = e wih H y, ; T T = E Q me λ Y + ˆλY, ; T d Y = y. i Backward indifference value Nex, we revii he claical definiion of indifference value bu in he framework of backward dynamic uiliy. Thi framework allow for a concie valuaion of claim and liabiliie of arbirary mauriie, provided ha hee mauriie occur before he normalizaion poin. Due o elf-generaion, he noion of dynamic value funcion become redundan. Herein we concenrae on he indifference reamen of a liabiliy, or, equivalenly, on he opimal porfolio choice of he wrier of a claim, yielding payoff equal o he liabiliy a hand. Definiion 6 Le T be he backward normalizaion poin and conider [ a claim C T F T, wrien a 0 0 and mauring a T T. For 0, T ], he backward indifference value proce BIV ν B C T;T i defined a he amoun ha aifie he pricing condiion U B x ν B C T;T;T = up E P U B T X T C T;T F, 9 for all x R, X = x. We noe ha he backward indifference value coincide wih he claical one, bu i i conruced from a quie differen poin of view and cope. The focu herein i no on rederiving previouly known quaniie bu, raher, in exploring how he backward indifference value are affeced by he normalizaion poin and he change in he marke environmen, a well, a how hey differ from heir forward dynamic counerpar. We addre hee queion for he cla of bounded European claim and liabiliie, for which we can deduce cloed form variaional expreion. Propoiion 7 Le T be he backward normalizaion poin and conider a European claim wrien a 0 0 and mauring a T T, yielding payoff C T = C S T,Y T. For [ 0, T ], i backward indifference value proce ν B C T;T i given by ν B C T;T = p B S, Y, 8

19 where S and Y olve and, and p F : R + R [0, T] R aifie Herein, p B + L S,Y,me p B + γ ρ a y p B y = 0 p B S, y, T = C S, y. 30 L S,Y,me = σ ys S + ρσ ysa y S y + a y y 3 + b y ρλyay + a y f y y, ; T f y, ; T y, and f olve cf. 0 f + a yf yy + b y ρλyayf y = ρ λ yf fy, T =. Proof. For convenience, we recall he enropic quaniie H, = E Q me λ Y + ˆλ Y, ; T d F and H y, ; = E Q me λy u + ˆλY u, u; T du Y = y for 0 T T. We fir calculae he righ hand ide of 9, which, in view of Propoiion 6, become up E P e γx T C T H T;T F = up E P e γx T G T F wih G T = C S T,Y T γ H T; T. One may, hen, view hi problem a a radiional indifference valuaion one in which he rading horizon i [, T ] and he uiliy i he exponenial funcion a T. For he ochaic facor model we conider herein, we obain ee Sircar and Zariphopoulou 005 and Graelli and Hurd 004 up E P e γx T G T F 9 γx hs,y, H; = e T

20 wih h : R + R [0, T] R olving h + L S,Y,me h + γ ρ a yh y = 0 h S, y, T = C S, y γ H y, T; T. Nex, inroduce he funcion p B : R + R [0, T] R p B S, y, = h S, y, + γ H y, ; T H y, ; T. Uing he equaion aified by h, we deduce ha p B olve 30. On he oher hand, Corollary 5 and he above equaliie yield up E P e γx T G T X = x, S = S, Y = y = e γx pb S,y, Hy,; T+ Hy, T;T = e γx p B S,y, Hy,;T and he aerion follow from Definiion 6 and Propoiion 5. ii Opimal porfolio under backward dynamic uiliy Nex, we conruc he opimal porfolio raegie in he inegraed porfolio problem. We ar wih he agen opimal behavior in he abence of he liabiliy/payoff. We concenrae our aenion o opimal behavior in a horer horizon. For impliciy, i end i aken o coincide wih T, he poin a which he liabiliy i me. Propoiion 8 Le T be he backward normalizaion poin and [, T] [, T] be he rading horizon of an inveor endowed wih he backward exponenial dynamic uiliy U B. The procee, π B, and π B,0,, repreening he opimal invemen in he riky and rikle ae, are given, for [, T], by and π B, Herein, X B, = π B, X B,, Y, = µ Y γσ Y ρay σ Y H y Y, ; T 3 π B,0, olve 5 wih π B, = π B,0, X, Y, = X B, π B,. being ued, and H : R [0, T] R + aifie H + L Y,me λ H y + ˆλy, ; T + = 0 wih erminal condiion H y, T; T T = E Q me λy + ˆλY, ; T d Y T = y. 33 T 0

21 Given he diffuion naure of he model, he form of he uiliy daa and he regulariy aumpion on he marke coefficien, opimaliy follow from claical verificaion reul ee, among oher, Duffie and Zariphopoulou 993, Zariphopoulou 00, Pham 00, Touzi 00. Due o he ochaiciy of he invemen opporuniy e, he opimal invemen raegy in he ock accoun coni of wo componen, namely, he myopic porfolio and he o-called exce riky demand, given, repecively, by µy γσ Y and ρ ay σy H y Y, ; T. The myopic componen i wha he inveor would follow if he coefficien of he riky ecuriy remained conan acro rading period. The exce riky demand i he required invemen ha emerge from he local in ime change in he Sharpe raio ee, among oher, Kim and Omberg 996, Liu 999, Campbell and Viceira 999, Chacko and Viceira 999, Wacher 00 and Campell e al Noe ha even hough he rading horizon [, T] i horer han he original one, [, T], he opimal policie depend on he longer horizon becaue he dynamic rik preference are normalized a T and no a T. Remark: The reader familiar wih he repreenaion of indifference price, migh ry o inerpre he exce riky demand a he indifference hedging raegy of an appropriaely choen claim. Such queion were udied in Soikov and Zariphopoulou 004 where he relevan claim wa idenified and priced. We coninue wih he opimal raegie in he preence of a European-ype liabiliy C T, which, we recall, i aken o be bounded. Propoiion 9 Le T be he backward normalizaion poin and conider an inveor endowed wih he backward dynamic exponenial uiliy U B and facing a liabiliy C T = C S T,Y T. The procee, π B, and π B,0,, repreening he opimal invemen in he riky and rikle ae, are given, for [, T], by and π B, Herein, X B, olve 30. = π B, X B,, S, Y, = µ Y γσ Y ρay σ Y H y Y, ; T 34 +S p B S S, Y, + ρ a Y σ Y pb y S, Y, π B,0, = π B,0, X, S, Y, = X B, olve 5 wih π B, Proof. In he preence of he liabiliy, we oberve π B,. being ued, H a in Propoiion 8 and p B up E P U B T X T C T;T F = u C x, S, Y,, where u C : R R + R [ 0, T ] R olve he Hamilon-Jacobi-Bellman equaion u C + max π σ yπ u C xx + π σ ysu C xs + ρayσ yu C xy + µ yu C x

22 wih and +L S,Y u C = 0, u C x, S, y, T = e γx CS,y Hy, T;T, L S,Y = σ ys S + ρσ ysa y S y + a y y 35 +µ y S + b y y. Verificaion reul yield ha he opimal policy π B, form wih π B, = π B, X B,, S, Y, i given in he feedback π B, x, S, y, = σ ysu C xs + ρayσ yuc xy + µ yuc x σ yu C xx On he oher hand, from Propoiion 8, u C x, S, y, = e γx pb S,y, Hy,:T. Combining he above and he feedback form of π B, x, S, y,, we conclude. 5 Invemen and valuaion under forward dynamic exponenial uiliie We now rever our aenion o porfolio choice and pricing under he newly inroduced cla of forward dynamic uiliie. We ar wih he analyic conrucion of uch a uiliy. menioned in Secion, general exience and uniquene reul for forward dynamic uiliie are lacking. a maer of fac, an alernaive oluion o 5 i Example 3. Propoiion 0 Le 0 be he forward normalizaion poin. Define, for, he proce h, = λy u du 36 wih λ being he Sharpe raio 4. Then, he proce U F x;, given, for x R and, by U F x; = e γx+h, 37 i a forward dynamic exponenial uiliy, normalized a. Proof. The fac ha U F x; i F -meaurable and normalized a i immediae. I remain o how 5, namely, ha for arbirary T, e γx+h, = up E P e γxt+h,t F..

23 Uing 36, he above reduce o e γx = up E P e γxt+h,t F. 38 Nex, we inroduce he funcion u : R R [0, T] R, u x, y, = up E P e γxt+r T λ Y d X = x, Y = y. Claical argumen imply ha u olve he Hamilon-Jacobi-Bellman equaion wih u + L Y u + λ y u + max π σ yπ u xx + π ρayσ yu xy + µ yu x u x, y, T = e γx and L Y a in 5. We deduce ee, for example, Duffie and Zariphopoulou 993 and Pham 00 ha he above equaion ha a unique oluion in he cla of funcion ha are concave and increaing in x, and are uniformly bounded in y. We hen ee ha he funcion ǔx, y, = e γx i uch a oluion and, by uniquene, i coincide wih u. The re of he proof follow eaily. wih We nex preen an alernaive forward dynamic uiliy. Example 3: Conider, for x R and, he proce U F x; = e γx Z, Z, = λ d + = 0 λ dw. 39 Oberve ha, for X = x, he forward ochaic crierion cf. 5, e γx Z, = up E P e γxt Z,T F will hold if we eablih e γx = up E P e γxt Z,T F or, equivalenly, e γx = up E P e γxt Z,T X = x, Y = y, Z = 0 3

24 wih Z a in 39. Defining v : R R R [0, T] R by v x, y, z, = up E P e γxt Z,T X = x, Y = y, Z = z, we ee ha i olve he Hamilon-Jacobi-Bellman equaion v + max π σ yπ u xx + π λyσ yu xz + ρayσ yu xy + µ yu x wih + λ yv zz + ρλyayv zy + a yv yy + b yv y + λ yv z v x, y, z, T = e γx z. Subiuing above he funcion ˆv x, y, z, = e γx z, and afer ome calculaion, yield λyσ y ˆv xz + µ y ˆv x σ y ˆv xx + λ yˆv zz + ˆv z = 0. We eaily conclude ha ˆv v and he aerion follow. i Forward indifference value We nex inroduce he concep of forward indifference value. Like i backward counerpar, i i defined a he amoun ha generae he ame level of dynamic uiliy wih and wihou incorporaing he liabiliy. Noe, alo, ha in he Definiion below, i i only he forward dynamic uiliy ha ener, eliminaing he need o incorporae in he definiion he forward dynamic value funcion. Thi allow for a concie reamen of payoff and liabiliie of arbirary mauriie. Finally, we remark, ha he nomenclaure forward doe no refer o he erminology ued in derivaive valuaion perinen o wealh expreed in forward uni. Raher, i refer o he forward in ime manner ha he dynamic uiliy evolve. While he concep of forward indifference value appear o be a raighforward exenion of he backward one, i i imporan o oberve ha he mauriie of he claim in conideraion need no be bounded by any prepecified horizon. Thi i one of he riking difference beween he clae of claim ha can be priced by he wo diinc dynamic uiliie we conider herein. Finally, we remark, ha he nomenclaure forward doe no refer o he erminology ued in derivaive valuaion perinen o wealh expreed in forward uni. Raher, i refer o he forward in ime manner ha he dynamic uiliy evolve. Definiion Le 0 be he forward normalizaion poin and conider a claim C T F T, wrien a 0 and mauring a T. For [ 0, T], he 4

25 forward indifference value proce FIP ν F C T; i defined a he amoun ha aifie he pricing condiion U F x ν F C T ; ; = up E P U F T X T C T ; F 40 for all x R, X = x. We coninue wih he valuaion of a bounded European-ype liabiliy and we examine how i forward indifference value i affeced by he choice of he normalizaion poin. We how ha even hough boh forward dynamic uiliie, enering in 40 above, depend on he normalizaion poin, he emerging forward price doe no. Thi i anoher imporan difference beween he backward and he forward indifference value. Propoiion Le 0 be he forward normalizaion poin and conider a European claim wrien a 0 and mauring a T yielding payoff C T = C S T, Y T. For [ 0, T], i forward indifference value ν F C T ; i given by ν F C T ; = p F S, Y, where S and Y olve and, and p F : R + R [0, T] R aifie p F + L S,Y,mm p F + γ ρ a y p F y = 0 p F S, y, T = C S, y, 4 wih L S,Y,mm = σ ys S + ρσ ysa y S y + a y y Proof. We fir noe ha = e R + b y ρλyay y. up E P U F T X T C T ; F = up E P e γxt CT+R T λ Y udu up λ Y udu E P e γxt CT +R T F λ Y udu F where we ued Propoiion 0 and he meaurabiliy of he proce h cf. 36. Define u C : R R + R [0, T] R, u C x, S, y, = up E P e γxt CT +R T λ Y udu X = x, S = S, Y = y 5

26 and oberve ha i olve he Hamilon-Jacobi-Bellman equaion u C + max π σ yπ u C yy + π σ ysu C xs + ρayσ yu C xy + µ yu C x wih +L S,Y u C + λ y u C = 0 u C x, S, y, T = e γx CS,y and L S,Y a in 35. Uing he ranformaion u C x, S, y, = e γx pf S,y, we deduce, afer ediou bu raighforward calculaion, ha he coefficien p F S, y, olve 4. We, hen, eaily, ee ha up E P U F T X T C T ; F = e R λ Y udu u C x, S, Y,, and, uing Propoiion and Definiion we conclude. ii Opimal porfolio under forward dynamic uiliie We coninue wih he opimal invemen policie under he forward dynamic rik preference. Propoiion 3 Le 0 be he forward normalizaion poin and [, T] he rading horizon, wih. The procee, π F, and π F,0,, repreening he opimal invemen in he riky and rikle ae, are given, repecively, for u [, T], by and wih X F, u π F, u π F,0, u olving 5 wih π F, u = π F, X F, u, Y u, u = µ Y u γσ Y u = π F,0, X u, Y u, u = X F, u being ued. π F, u, 4 Two imporan fac emerge. Firly, boh opimal invemen policie π F, and π F,0, are independen of he po normalizaion poin. Secondly, he invemen in he riky ae coni enirely of he myopic componen. Indeed, he exce hedging demand, which emerge due o he preence of he ochaic facor, ha vanihed. The inveor ha proceed he ochaiciy of he marke environmen ino her preference, ha are dynamically updaed, following, forward in ime, he marke movemen. 6

27 Propoiion 4 Le 0 be he forward normalizaion poin and conider an inveor endowed wih he forward exponenial dynamic uiliy U F and facing a liabiliy C T = C S T, Y T. The procee, π F, and π F,0,, repreening he opimal invemen in he riky and rikle ae in he inegraed porfolio choice problem, are given, for u [, T], by and Herein, X F, π F, u = π F, X F, u, S u, Y u, u = µ Y u γσ Y u +S u p F S S u, Y u, u + ρ a Y u σ Y u pf y S u, Y u, u 43 π F,0, u = π F,0, X u, S u, Y u, u = X F, u π F, u. olve 5 wih π F, being ued, and p F aifie 4. 6 Concluding remark: Forward veru backward uiliie and heir aociaed indifference price In he previou wo Secion, we analyzed he invemen and pricing problem of inveor endowed wih backward BDU and forward FDU dynamic exponenial uiliie. Thee uiliie have imilariie bu, alo, riking difference. Thee feaure are, in urn, inheried o he aociaed opimal policie, indifference price and rik monioring raegie. Below, we provide a dicuion on hee iue. We fir oberve ha he backward and forward uiliie are produced via a condiional expeced crierion. They are boh elf-generaing, in ha hey coincide wih heir implied value funcion. Moreover, in he abence of exogenou cah flow, inveor endowed wih uch uiliie are indifferen o he invemen horizon. Backward and forward dynamic uiliie are conruced in enirely differen way. Backward uiliie are fir pecified a a given fuure ime, T, and, hey are, ubequenly, generaed a previou o T ime. Forward uiliie are defined a preen,, and are, in urn, generaed forward in ime. The ime T and, a which he backward and forward uiliy daa are deermined, are he backward and forward normalizaion poin. We recall, from equaion 8 and 37, ha he BDU and FDU procee, U B and U F, are F adaped and given, repecively, by U B x; T = e γx H,T and U F x; = e γx h,, 7

28 wih and T H, T = E Q me λ u + ˆλ u du F h, = λ udu. Herein, λ and ˆλ are given in 4 and 8, and Q me i he minimal relaive enropy meaure. Boh BDU and FDU have an exponenial, affine in wealh, rucure. However, he backward uiliy compile change in he marke environmen in an aggregae manner, while he forward uiliy doe o in a much finer way. Thi i een by he naure of he procee H, T and h,. I i worh oberving ha H, T E Q me h, T F and ha U F i no affeced by ˆλ cf. 8 and 9 ha repreen he orhogonal componen of he marke price of rik. Backward and forward uiliie generae differen opimal invemen raegie ee, repecively, Propoiion 9 and 4. Under backward dynamic preference, he inveor inve in he riky ae an amoun equal o he um of he myopic porfolio and he exce riky demand. The former invemen raegy depend on he rik averion coefficien γ, bu no on he backward normalizaion poin T. The exce riky demand, however, i no affeced by γ bu depend on he choice of he normalizaion poin, even if invemen ake place in a horer horizon. Under forward preference, he inveor inve in he rik ae olely he myopic porfolio. The myopic raegy doe no depend on he forward normalizaion poin or he invemen horizon. a conequence of he above difference, he emerging backward BIV and forward FIV indifference value, ν B C T;T and ν F C T;, 0 T T, have very diinc characeriic. Concenraing on he cla of bounded European claim, we ee ha ν B C T;T and ν F C T; are conruced via oluion of imilar quailinear pde. While he nonlineariie. in he pricing pde are of he ame ype, he aociaed linear operaor, L S,y,me and L S,y,mm differ ee, repecively, 30 and 4. The former, appearing in he BIV equaion, correpond o he minimal relaive enropy meaure while he laer, appearing in he FIV equaion, o he minimal maringale meaure. Denoing he oluion of hee pde a nonlinear expecaion, we may formally repreen - wih a ligh abue of noaion - he wo indifference value a ν B C T;T = E Q me C T;T and ν F C T; = E Q mm C T;. 8

29 The FIV i independen on he forward normalizaion poin. The BIV depend, however, on he backward normalizaion poin, even if he claim maure in a horer horizon. and he inveor become rik neural, γ 0, we obain lim γ 0 νb C T;T = E Q me C T F lim γ 0 νf C T; = E Q mm C T F. However, a he inveor become infiniely rik avere, γ, boh BIV and FIV converge o he ame limi given by he uper replicaion value, lim γ νb C T;T = lim γ νf C T; = C T L {. F }. In he preence of he liabiliy and under backward dynamic uiliy, he invemen in he riky ae coni of he myopic porfolio, he exce riky demand and he backward indifference rik monioring raegie ee Propoiion 0. Wih he excepion of he myopic porfolio, all oher hree porfolio componen depend on he normalizaion poin T. When, however, he inveor ue forward dynamic uiliy, hi opimal inegraed policy doe no include he exce riky demand. The enire policy i independen of he forward normalizaion poin, and depend excluively on he mauriy of he claim and he change in he marke environmen. When he marke become complee, he backward and forward pricing meaure, Q me and Q mm, coincide wih he unique rik neural meaure, Q, and BIV and FIV reduce o he arbirage free price. In general, he backward and forward indifference value do no coincide. The underlying reaon i ha hey are defined via he backward and forward dynamic uiliie ha proce he inernal model incompleene, generaed by he ochaic facor Y, in a very differen manner. Characerizing he marke environmen a well a he claim for which he wo price coincide i an open queion. 7 Bibliography [] rai, T: The relaion beween he minimal maringale meaure and he minimal enropy maringale meaure, ia-pacific Financial Marke, 8, 67-77, 00. [] Bellini, F. and M. Frielli: On he exience of he minmax maringale meaure, Mahemaical Finance,, -. [3] Benh, F. E. and K. H. Karlen: PDE repreenaion of he deniy of he minimal maringale meaure in ochaic volailiy marke, Sochaic and Sochaic repor, 77,