Zongxia Liang and Ming Ma. Department of Mathematical Sciences, Tsinghua University, Beijing , China

Transcription

1 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE IN INCOMPLETE MARKETS Zongxia Liang and Ming Ma Deparmen of Mahemaical Science, Tinghua Univeriy, Beijing 184, China Abrac. In hi paper, we udy he imple invemen problem of an incomplee marke, where he expeced uiliy of erminal wealh i maximized under a ochaic inere rae and a ock invemen. Unle iuing a rolling bond o hedge he inere rae rik, he incompleene caue he following difficul: he variey of equivalen maringale meaure confue he radiional maringale approach. Thu we aim a clarifying he maringale approach in he incomplee marke, whoe eence i finding he opimal neural meaure over all admiible meaure, which i exacly a minimax meaure. The five-ep maringale approach i ummarized by he coniency heorem and he ufficien condiion. Indeed, he coniency heorem allow u o uing maringale approach dynamically. Baed on hee udie, he opimal raegie are derived explicily under CRRA and CARA uiliie. Finally, numerical illuraion reveal he behavior of opimal premium, paricularly, he ime and capial effec on he behavior. MSC(1): 91G1, 91G3, 93E. Keyword: Maringale approach; Incomplee marke; Opimal neural meaure; Minimax meaure; Premium ad curve; Sochaic inere rae. 1. Inroducion Wih he variey of rik in financial marke, more and more derivaive are iuing o hedge rik. Bu here are only everal developed derivaive due o he complexiy of rik or he policy uperviion. Thu he compleene of marke doe no hold and ome incomplee marke have o be conidered. We inveigae he imple opimizaion ha he expeced uiliy of erminal wealh i uppoed o be maximized under a ochaic inere rae and a ock invemen. The only conrollable hing i he fracion of he oal wealh inveed in ock. I i well-known ha if a rolling bond i iued o hedge he inere rae rik, hen he marke urn o be complee and here are wo developed Correponding auhor. zliang@mah.inghua.edu.cn(z.liang), maming9@gmail.com(m.ma) 1

2 ZONGXIA LIANG AND MING MA mehod o ue. One i he dynamic programming, which i propoed by Meron[16](1971) and uppored by he heory of Hamilon-Jacobi- Bellman equaion. The oher one i he maringale approach developed by Cox and Huang (1989)[3] baed on he heory of Lagrange muliplier, which uccefully olve an opimal conumpion-porfolio problem under hyperbolic abolue rik averion uiliie. Thee wo mehod boh have developed heorie for ime-conien problem in complee marke. Bu in incomplee marke, he HJB equaion derived from dynamic programming come o a higher-dimenional parially differenial equaion, whoe explici oluion can hardly be found. The concep of vicoiy oluion may be ued o olve a HJB equaion in an incomplee marke(cf. Duffie e al.(1997)[4]). The core of maringale approach i rewriing he problem o be adaped o he heory of Lagrange muliplier by an equivalen maringale meaure. Bu here are infinie equivalen maringale meaure in an incomplee marke and i i no clear which one hould be choen. The main arge of hi paper i o clarify he maringale approach in incomplee marke, by which we pick ou an equivalen maringale meaure called opimal neural meaure. The udy in Secion 3 ell u ha uing maringale approach in incomplee marke i equal o finding he opimal neural meaure from a lo of admiible meaure over Q defined by (3.1) below. If he inere rae i conan, hi kind of opic ha been widely dicued a he opimal crierion for he e of all maringale meaure in an incomplee marke. According o prior reearche, people uually udy hi opic baed on he following wo perpecive o find he appropriae candidae meaure. The fir perpecive i o define a diance from maringale meaure o he original meaure, which i uppoed o be minimized. For inance, Cizár(1975)[] ue he I-divergence or Kullback-Leibler informaion a a diance o find he I-projecion of he original meaure on ome meaure e. Delbaen and Schachermayer(1996)[5] inroduce he variance-opimal maringale meaure which minimize he L norm of deniy dq. The relaive enropy of Q wih repec o P i defined by dp Frielli()[6] and lead u o he minimal enropy maringale meaure. Along hee hough, he minimax maringale meaure i preened

3 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 3 by Goll & Rüchendorf(1)[8] and Bellini & Frielli()[1], which exacly minimize he f-divergence diance. Grandi()[7] generalize he minimal maringale meaure for quai-lef-coninue proce wih bounded jump via he definiion of minimal Hellinger maringale meaure. The econd perpecive i o conider a family of pricing or opimizaion problem wih a given uiliy, which correpond o a family of admiible meaure. For inveor wih differen uiliie, i i reaonable ha he choen meaure are differen. Karaza e al.(1991)[11] firly dicu he uiliy maximizaion and i dual problem for geing an opimal invemen raegy in an incomplee marke, where hey elec an opimal meaure by vanihing hoe ficiiou ock. Kallen(1999)[1] give a crieria of meaure o hedge rik in incomplee marke by maximizing expeced local uiliy. Indeed, hee kind of choen meaure have cloed relaionhip wih above minimax meaure, which ha been revealed by Goll & Rchendorf(1)[8]. Under he framework of opimal crierion, he opimal neural meaure in hi paper i a differen minimax meaure, which maximize he expeced uiliy of wealh bu no he deflaed wealh. If he inere rae i conan, our problem i imilar o he reearch of Karaza e al.(1991)[11]. If in addiion he uiliy i CARA, he expeced uiliy of wealh i proporional o he expeced uiliy of deflaed wealh, hu he udy of Goll & Rüchendorf(1)[8] operae. Bu conidering a ochaic inere rae, he problem become complex and non-linear and hu ome inereing effec appear. More preciely, he rik premium correponding o he opimal neural meaure may be ochaic and relaed o he inananeou ae a well a mauriie. Becaue of hi, he ime and capial boh have ignifican impac on he behavior of rik premium. The re of hi paper i organized a follow. Secion formulae he original opimizaion problem, where he dynamic of ae and he admiible conrol e are rigorouly defined. In Secion 3, we eablih he five-ep maringale approach in incomplee marke uppored by ome heorem. The expanded pace and he opimal neural meaure are preened for olving he original opimizaion. Baed on he five-ep

4 4 ZONGXIA LIANG AND MING MA approach, we calculae wo cae in Secion 4 where CRRA and CARA uiliie are conidered. The numerical illuraion of wo cae are boh diplayed in Secion 5, according o which we reveal he behavior of opimal premium. Paricularly, we ummarize he ime and capial effec on he behavior.. Problem decripion The invemen problem we conidered in hi paper i maximizing he uiliy of erminal wealh conrolled by invemen raegy, i.e., max π V[,T] E[U(X T)], (.1) where he uiliy funcion U( ) i uually concave and ricly increaing due o he rik averion. Firly, we decribe he wealh proce X } in a financial marke. The financial marke i compoed of wo ae: a ock andacah wih a ochaic inere rae. The probabiliy pace (Ω,F,F,P) conruced in he financial marke i generaed by wo independen Brownian moion W } and B }. The equipped filraion F = F } i defined by F = σ(w u,b u ) : u } and F = σ( F ). Now we characerize he dynamic of ae. The inere rae i ochaic and decribed by Vaicek model: dr = a( r r )d+σ r db, (.) where he parameer a, r and σ r are all poiive. The ock proce S } i a geomeric Brownian moion aifying he following SDE: ds S = (r +ν)d+σ 1 dw +σ db, (.3) where he inrinic volailiy σ 1 i poiive and ν repreen he oal rik premium. The correlaed volailiy σ i uually negaive becaue people rend o depoi when inere rae become higher. Under he aumpion of elf-finance, he wealh proce can be uniquely deermined by he adaped raegy proce π }, where π i he fracion of he oal wealh inveed in ock a ime. Thu, he dynamic

5 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 5 of wealh proce X } i ds dx =(1 π )X r d+π X S =(r +π ν)x d+π X σ 1 dw +π X σ db, X =x. (.4) Secondly, we inroduce he admiible raegy e V[, T]. A raegy π i called an admiible raegy if (i) π L F (,T;R), i.e., π i a F } -adapedr-valued proce and E π() d} < ; (ii) here exi an unique oluion of equaion (.4), denoed by X( ); (iii) X( ) aifie he wealh conrain, i.e., X( ). Propoiion.1. Le L F (Ω;C([,T];R)) be he e of all coninuou procee in L F (,T;R). Then we have L F(Ω;C([,T];R)) V[,T] L F(,T;R). Proof. IiobviouhaV[,T] L F (,T;R)byhedefiniionofV[,T]. So we ju need o how, for any π L F (Ω;C([,T];R)), here exi an unique oluion of equaion (.4) and i i non-negaive. Becaue π i coninuou on [,T], i i bounded and he coefficien in SDE (.4) are all Lipchiz coninuou. By Theorem 6.3 in chaper 1 of Yong and Zhou(1999)[18], he uniquene and exience of rong oluion X( ) boh hold. Finally, X( ) i non-negaive due o he propery of geomeric Brownian moion. Corollary.. Suppoe XT i he opimal erminal wealh o he opimizaion problem: max E[U(X T)], X T V (.5) where V = X T : π L F (,T;R).. X T = x + (r +π ν)x d+ π X ( σ1 dw +σ db ) }, (.6) and π i he correponding opimal raegy of X in (.6) and coninuou. Then he π maximize he original problem (.1). We omi he proof becaue i can be eaily verified uing Propoiion.1. Thi corollary ell u ha if he opimal invemen raegy of he

6 6 ZONGXIA LIANG AND MING MA erminal wealh problem (.5) i coninuou, hen he oluion of (.5) lead u o he oluion of he original problem (.1). Forunaely, he condiion of coninuiy in corollary. uually hold hank o he feedback conrol. Furhermore, he ranformaion form (.1) o (.5) i crucial becaue maringale approach ju work for hi kind of erminal wealh opimizaion. 3. Five-ep maringale approach Becaue he number of ochaic ae i le han he number of Brownian moion, he maringale meaure in he finical marke i no unique. We uually call hi kind of marke a incomplee marke. I i well-known he maringale meaure i crucial in maringale approach and influence he budge conrain direcly. In complee marke, he uniquene of equivalen maringale meaure guaranee he efficiency of he maringale approach. Once he equivalen maringale meaure i divere, here exi differen o-called opimal raegie under differen meaure. Hereo, a meaure leading u o he real opimal raegy i demanded Neural meaure and expanded pace. According o he hiorical daa of ock, premium ν can be eimaed by he aiical propery of deflaed log-reurn. Thu, any meaure accorded wih he oal premium can be choen a a maringale meaure. Significanly, he premium correponding o he equivalen maringale meaure can be ochaic. The e Q conained all admiible meaure i decribed accuraely a follow: Q= Q Λ : dqλ dp =exp( 1 [λ 1 (u) +λ (u) ]du λ 1 (u)dw u } ) λ (u)db u, Λ=(λ1,λ ) L F (,T;R ).. σ 1 λ 1 +σ λ ν, (3.1) where Λ i he ochaic premium correponding o he meaure Q Λ. For any equivalen maringale meaure in Q, he correponding expanded pace of erminal wealh i V(Λ) = X T : E QΛ [exp( r u du)x T ] = x }. (3.)

7 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 7 The budge conrain in (3.) mean he erminal wealh can generae a dicouned wealh proce e rudu X E QΛ [e rudu X T F ], which i a Q Λ -maringale wih mean x. The following heorem ell he relaionhip beween he elf-finance pace and he expanded pace. Theorem 3.1. V = Λ:Q Λ Q} V(Λ). Proof. Firly, we how V V(Λ) for any Q Λ Q. For X T V we have he following: X T =x + =x + =x + (r +π ν)x d+ r X d+ r X d+ π X (σ 1 dw +σ db ) π X (σ 1 (dw +λ 1 ()d)+σ (db +λ ()d)) π X (σ 1 d W +σ d B ), where W and B are Brownian moion under meaure Q Λ. Uing Iô formula, he dicouned erminal wealh can alo be expreed a a ochaic inegral. Thu V = X T : π L F(,T;R).. e rudu X T = x + } π e rudu X (σ 1 d W +σ d B ), i.e., X T aifie he budge conrain and o X T V(Λ). (3.3) Secondly, how V Λ:Q Λ Q} V(Λ). By maringale repreenaion heorem(cf. Chaper 3 of Karaza and Shreve(1)[13]), we can rewrie V(Λ) a he follow: V(Λ) = X T : I 1,I L F (,T;R).. e } T rudu X T =x + I 1 ()σ 1 d W +I ()σ d B = X T : I 1,I L F(,T;R).. e rudu X T =x + [I ()ν+(i 1 () I ())σ 1 λ 1 ()]d+i 1 ()σ 1 dw +I ()σ db }, (3.4) wherei i (i = 1,)ianadapedochaicproceobainedbymaringale decompoiion and we call i maringale coefficien in hi paper. Thu for any X T Λ:Q Λ Q}, Λa and Λ b L F (,T;R ) here exi I a 1,Ia,Ib 1,Ib

8 8 ZONGXIA LIANG AND MING MA L F (,T;R) aifying = = e rudu X T x [I a ()ν +(I a 1() I a ())σ 1 λ a 1()]d+I a 1()σ 1 dw +I a ()σ db [I b ()ν +(I b 1() I b ())σ 1 λ b 1()]d+I b 1()σ 1 dw +I b ()σ db. Comparing he Iô inegral and he Lebegue inegral in he la wo equaliie we have I a 1 = I b 1, I a = I b and I a 1 = I a, repecively. Thu, if we denoe I I a 1 = Ib 1 = Ia = Ib and π So X T V. = = e rudu X T x e I() rudu X, hen I()νd+I()σ 1 dw +I()σ db π e rudu X (νd+σ 1 dw +σ db ). 3.. Opimizaion in expanded pace. Before propoing he opimal neural meaure, we pecify he radiional four-ep maringale approach wih a given meaure. Fixing an equivalen maringale meaure, maringale approach in incomplee marke ha no difference from complee marke, which bae on he heory of Lagrange muliplier developed by Cox and Huang[3](1989). Reader can alo refer o Karaza and Shreve [1](1998) for he deail and we ummarize hee reul by Propoiion 3.. For a fixed equivalen maringale meaure Q Λ and any iniial ime, denoe he Lagrangian funcion by [ ] L(,x,r,y,X T ) = E [U(X T ) F ]+y x E(H Λ (,T)X T F ),(3.5) where H Λ (,T) = exp ( r udu )( ) dq Λ dp exp ( r udu )( dq Λ dp = exp r u du 1 λ 1 (u) +λ (u) du ) T λ 1 (u)dw u +λ (u)db u }. If here exi X T V(Λ) and F -meaurable random variable y aifying (i) X T maximize L(,x,r,y,X T) over V(Λ), (ii) budge conrain hold, i.e., y [x E(H Λ (,T)X T F )] =,

9 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 9 hen XT i he unique opimal oluion o he problem in expanded pace V(Λ), i.e., E[U(XT ) F ] = max E[U(X T ) F ]. X T V(Λ) The Propoiion 3. enure he efficiency of maringale approach, which exacly i a rong dualiy heorem in he ochaic conrol. The proof abou opimaliy i imilar o he deermined programming, and inereed reader can refer o Luenberger and Ye(8)[15]. The proof abou uniquene i an applicaion of Jenen inequaliy hank o he well propery of he uiliy funcion. Now we can li he radiional maringale approach a a four-ep approach. Given an equivalen maringale meaure Q Λ, maringale approach o opimal wealh wih iniial x and r can be realized by ep(i) ge he opimal wealh X T (y ) a a funcion of y, which maximize he Lagrangian funcion L(,x,r,y,X T ); ep(ii) ubiue X T (y ) ino budge conrain and ge he opimal Lagrange muliplier y ; ep(iii) for any [,T], calculae he explici form of dicouned wealh a ime : e rudu X E QΛ [e rudu X T F ]; ep(iv) differeniaee rudu X andhengei1(;,x,r ),I(;,x,r ) a ime [,T]. Indeed, he fir wo ep calculae he opimal erminal wealh and Lagrange muliplier while he la wo ep realize he maringale decompoiion o ge maringale coefficien in (3.4). Bu in pracice, i i complexoexprei i (i = 1,)aeveryime. Sohefollowingconiency heorem devoe o improving he la wo ep. Theorem 3.3. (Coniency heorem) Regarding ep(iii) and ep(iv) a he definiion of maringale coefficien I i (;,x,r )( ) for i = 1,, we have where I i(;,x,r ) = e rudu I i(;,x (;,x,r ),r ), r = r + e rudu X (;,x,r )=x + a( r r )d+ σ r db, I 1(u;,x,r )σ 1 d W u +I (u;,x,r )σ d B u.

10 1 ZONGXIA LIANG AND MING MA Proof. We firly prove X (T;,x,r ) = X (T;,X (;,x,r ),r ), (3.6) which mean he global opimum i he local opimum. By he definiion of opimal wealh, we know E[U(X (T;,x,r ))] E[U(Y 1 )], Y 1 V(Λ), E[U(X (T;,X (;,x,r ),r )) F ] E[U(Y ) F ] fory V(Λ;,X (;,x,r ),r ),wherev(λ;,x (;,x,r ),r )iimilar o V(Λ) in (3.4), and defined by V(Λ;,X (;,x,r ),r ) X T : I 1,I L F (,T;R).. e T r udu X T = X (;,x,r )+ I 1 (u)σ 1 d W u +I (u)σ d B u }. Chooing Y 1 = X (T;,X (;,x,r ),r ) and Y = X (T;,x,r ), he equaliy (3.6) can be proved by he uniquene of X (T;,x,r ) and he following inequaliie: E[U(X (T;,x,r ))] E[U(X (T;,X (;,x,r ),r ))] =E[E[U(X (T;,X (;,x,r ),r )) F ]] E[E[U(X (T;,x,r )) F ]] =E[U(X (T;,x,r ))]. Secondly, reropecing (3.4), he relaionhip beween X T and I i (i = 1, ) provide e rudu X (T;,x,r )=x + and e rudu X (T;,X (;,x,r ),r )) =X (;,x,r )+ I 1 (u;,x,r )σ 1 d W u +I (u;,x,r )σ d B u, e rudu I 1 (u;,x (;,x,r ),r ))σ 1 d W u +e rudu I (u;,x (;,x,r ),r ))σ d B u. Thu he inegrand a ime aifie I i (;,x,r ) = e rudu I i (;,X (;,x,r ),r ) for i = 1,. The proof hen i complee.

11 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 11 Remark 3.4. Coniency heorem provide anoher approach o I i (;,x,r )(i = 1,, > ), which can replace he la wo ep of radiional maringale approach. For any [, T], differeniaing E QΛ [e rudu X T F ] w.r. ime can ge I i (;,X,r )(i = 1,), furhermore, I i(;,x,r )(i = 1,) i obained by he relaionhip in Theorem 3.3. I i no obviou ha hi new approach become impler, bu i really doe ince I i(;,x,r )(i = 1,) ha complex form for >. Wih a higher perpecive, he maringale approach can be realized dynamically according o he coniency heorem Opimal neural meaure. Afer inroducing he expanded pace wih repec o differen equivalen maringale meaure, we are inereed in he relaionhip beween opimizaion on he original pace and he expanded pace. By Theorem 3.1, he following inequaliy hold: max E[U(X T)] max E[U(X T)], Λ Λ : Q Λ Q}. X T V X T V(Λ) A conequen key queion i wheher he equaliy can hold a ome Λ, and wheher hi Λ make XT in V. If hee wo queion have poiive anwer, we can olve he original opimizaion by he expanded pace V(Λ ). The following definiion i propoed o realize hi idea. Definiion 3.1. A equivalen maringale meaure Q Λ Q i called an opimal neural meaure in Q, if here exi an opimal erminal wealh X T and an opimal premium Λ L F (,T;R ) aifying (i) X T V; (ii) X T maximize E[U(X T)] over V(Λ ). Propoiion 3.5. Le V(Λ) max XT V(Λ)E[U(X T )] be a he value funcion on Q. Then he opimal neural meaure Λ reache he minimum of V( ). Proof. Suppoe X T i he opimal erminal wealh correponding o Λ. For Λ Q, by Theorem 3.1 and Definiion 3.1, we have V(Λ) max X T V E[U(X T)] E[U(X T )] = V(Λ ).

12 1 ZONGXIA LIANG AND MING MA Theorem 3.6. A ufficien and neceary condiion for Λ being he opimal neural meaure i I 1 (;,x,r ) = I (;,x,r ), [,T], where I 1 ( ) and I ( ) are maringale coefficien correponding o X T in (3.4) and X T maximize E[U(X T)] over V(Λ ). Proof. By coniency heorem 3.3, we ubiue an equivalen condiion: I 1 (;,x,r ) = I (;,x,r ), [,T]. Neceiy i obviou by (3.3) and Definiion 3.1. Sufficiency i no complicaed ye. We ju need o verify X T V under he aumpion I 1 I. Chooing π I 1(;,x,r ) in (3.3), he proof i hen complee. e rudu X The concluion above formulae he maringale approach in incomplee marke. For an opimizaion problem in an incomplee marke, we need o olve a family of problem in expanded pace parameerized by Λ, by which we can find he opimal neural meaure. Combining he radiional four ep, he Remark 3.4 and Theorem 3.6, we ummarize he maringale approach in incomplee marke a he following five-ep approach: ep(i) for any Λ and any iniial ime, ge he opimal wealh X T (y ) a a funcion of y, which maximize he Lagrangian funcion L(,x,r,y,X T ); ep(ii) ubiue X T (y ) ino budge conrain and ge opimal Lagrange muliplier y ; ep(iii) for any [,T], calculae he explici form of dicouned wealh a ime : e rudu X E QΛ [e rudu X T F ]; ep(iv) differeniae e rudu X wih repec o, hen ge he maringale coefficien: I 1(;,x,r,Λ) and I (;,x,r,λ); ep(v) findheopimalneuralmeaureλ aifyingi 1 (;,x,r,λ ) = I (;,x,r,λ ) and he opimal invemen amoun π X i I 1 (;,X,r,Λ ). Moreover, in he view of opimal crierion, he opimal neural meaure i indeed a minimax meaure over Q. Thi i accuraely explained by Corollary 3.7. If he opimal neural meaure Λ exi, hen max E[U(X T)] = max E[U(X T)] = min X T V X T V(Λ ) Λ:Q Λ Q} max E[U(X T)]. X T V(Λ)

13 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 13 Proof. The concluion can be happily achieved via combining Propoiion 3.5 wih he exience of opimal neural meaure. Anoher key poin we hould expound i he opimal neural meaure doe no exi if here i a rik oally independen of he invemen. For example, conidering an ae liabiliie managemen(alm), he deah in inurance marke are oally independen of inveable ae. Hainau and Devolder(6)[9] provide an approximaion by minimizing he econd momen, while conidering he ALM in an incomplee marke. Liang and Ma (15)[14] udy he non-hedging moraliy and alary rik in penion fund by hi approximae maringale approach. To review hee reearche in our framework, he meaure hey chooe ju minimize he difference beween I 1 and I, becaue he condiion I 1 = I in Theorem 3.6 can no hold a all. Forunaely, hough he marke i incomplee in our paper, he whole rik are embodied in he ock invemen, o we have an opporuniy o ge real opimal raegy by he opimal neural meaure. 4. Explici oluion abou raegy and meaure In hi ecion, CRRA and CARA uiliie are choen and he explici oluion are obained by a lo of complicaed calculaion. To pecific he main idea and implify he noaion, we pu ome calculaion in appendixe and omi ome Λ a upercrip if i doe no caue ambiguiy CRRA. The CRRA uiliy repreen a conan relaive rik averionwihuiliyfuncionu(x) = 1 γ xγ andwenowrealizefive-epmaringale approach under hi uiliy. For a fixed meaure Q Λ, he Lagrangian funcion defined by (3.5) a ime i [ 1 [ ] L(,x,r,y,X T ) = E ]+y γ Xγ x E(H(,T)X T F ). T F The local opimal wealh X T (y ) expreed by y i X T (y ) = [y H(,T)] 1 γ 1. Subiuing i ino he budge conrain (3.), we have: (y ) 1 γ 1 = x E[H(,T) γ γ 1 F ].

14 14 ZONGXIA LIANG AND MING MA Thu we ge he opimal wealh: X T (y ) = x E[H(,T) γ γ 1 F ] [H(,T)] 1 γ 1. So far he fir wo ep have been finihed, bu he la wo ep will be more complicaed. e rudu X E Q [exp r u du } XT F ] =E[H(,T)exp r u du } XT F ] =E[H(,T)exp r u du } = E[H(,T) γ γ 1 F ] [H(,T)] 1 x E[H(,T) γ γ 1 F ] [H(,)] 1 γ 1 exp r u du } E x = E[H(,T) γ γ 1 F ] exp x exp γ γ 1 A(T )r +C(T ; r u du } [H(,)] 1 γ 1 γ γ 1 )}, γ 1 F ] [[H(,T)] γ γ 1 F ] where A( ) i a deermined funcion and C( ) i a funcion relaed o Λ and γ γ 1. Their expreion are boh in appendix A. Uing Iô formula, he differenial of e rudu X a ime = i: ( X 1 γ 1 λ 1()d W 1 γ 1 λ ()d B + γ γ 1 A(T )σ rd B where B and W are boh Brownian moion under Q Λ. So we ge he following explici expreion: Leing I 1 (;,x,r ) = X λ 1 () σ 1 (γ 1), λ ()+γa(t )σ r I (;,x,r ) = X. σ (γ 1) λ 1 () = νσ 1 σ 1 σ σ r γa(t ), σ1 +σ λ () = νσ +σ 1 σ 1 σ r γa(t ), σ1 +σ ), (4.1) i i eay o verify Q (λ 1,λ ) Q and Λ = (λ 1,λ ) i he opimal neural meaure by Theorem 3.6. By ep(v), he correponding opimal raegy

15 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 15 in elf-financial pace i π = ν σ σ r γa(t ) (1 γ)(σ 1 +σ ). (4.) I i noeworhy Λ become conan once σ r =, which repreen he inere rae i no ochaic. So, while conidering a ochaic inere rae, he rik premium behave exraordinarily and really deerve a deep udy. 4.. CARA. The CARA uiliy repreen a conan abolue rik averion wih uiliy funcion U(x) = 1 α e αx. For a fixed meaure Q Λ, he Lagrangian funcion a ime i L(,x,r,y,X T ) = E [ 1 [ ] α exp( αx T) F ]+y x E(H(,T)X T F ). (4.3) The local opimal wealh X T (y ) expreed by y i X T(y ) = 1 α ln[y H(,T)]. By he budge conrain (3.), he opimal Lagrange muliplier aifie Thu he opimal wealh i ln(y ) = αx +E[H(,T)ln(H(,T)) F ]. E[H(,T) F ] X T (y ) = x + 1 α E[H(,T)ln(H(,T)) F ] E[H(,T) F ] 1 α ln(h(,t)). We ill omi ome ubcrip Λ in nex wo ep. Following he reul in appendix A and B, we have e rudu X E Q[ exp r u du } ] XT F =E [ H(,T)exp r u du } 1 α ln(y ) 1 α ln(h(,)) 1 α ln(h(,t))} ] F =exp r u du } 1 α ln(y ) 1 α ln(h(,))} E [ ] H(,T) F 1 α ( r u du)e [ ] H(,T)ln(H(,T)) F

16 16 ZONGXIA LIANG AND MING MA =exp r u du } 1 α ln(y ) 1 α ln(h(,))} exp A(T )r +C(T ) } 1 α exp r u du } exp A(T )r +C(T ) } (A(T )r +F(T )) = 1 α exp r u du } exp A(T )r +C(T ) } ln(y ) +ln(h(,))+a(t )r +F(T ) }. Uing Iô formula, we ge he differenial of e rudu X a ime = : exp [ A(T )r +C(T ) ] ( λ1 () α [ λ () + αa(t )σ r ) d W +exp [ A(T )r +C(T ) ] x exp(a(t )r +C(T )) 1 ] A(T )σr d B. α Acually, ince e rudu X i a Q Λ maringale, i i ufficien o focu on he differenial of Brownian moion in above calculaion, which ake a huge implificaion. Then he maringale coefficien are I 1 (;,x,r ) = λ 1()exp[A(T )r +C(T )], σ 1 α I (;,x,r ) = exp[a(t )r +C(T )]A(T )σ r σ [ λ () + αa(t )σ r x exp(a(t )r +C(T )) 1 α ]. To make he ufficien condiion hold, we wan I 1 = I, which equal o λ () = νσ A(T )σ r σ1 [αx exp( A(T )r C Λ (T )) 1]. σ1 +σ (4.4) I mu be emphaized ha (4.4) i no an explici formulaion becaue C Λ (T ) i dependen on he informaion of Λ afer ime. Thu, he opimal pair (X,Λ ) hould aify he following equaion:

17 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 17 X T =x + (r +π ν)x d+ π X (σ 1 dw +σ db ), ( )( ) ν X π =exp A(T )r +C Λ σ λ () (T ), ασ 1 λ () = νσ A(T )σ r σ1 [αx exp( A(T )r C Λ (T )) 1], σ1 +σ C Λ (T ) = (A(T )+T ) r + 1 [σ r A(T u)] du [ ( ) ]) F +ln (E Q(λ 1,(λ σra(t ))) exp σ r A(T u)λ (u)du. (4.5) Thi family of forward-backward equaion expree a complex relaionhip beween he opimal wealh and he opimal neural meaure. Though i i hard o ge an explici oluion, (4.5) ill decribe he opimal neural meaure compleely. In appendix C, we provide an algorihm o approach he opimal premium which i ued in he numerical illuraion. 5. Numerical illuraion Thi ecion how he numerical illuraion of he opimal neural meaure a well a he opimal raegie. An opimal neural meaure canbeoallydeerminedbyanopimalpremiumproceλ = (λ 1,λ ),o we diplay he opimal premium proce inead. The aniheic raegy i derived from a conan premium, which i exacly he opimal premium under a conan inere rae. The fir reul in hi ecion i ha he opimal premium proce ecure a beer wealh proce in differen ene under differen uiliie. We conider a peron wih uni wealh a iniial ime = and he invemen horizon i year, i.e., T =. For he ochaic inere rae, we uppoe he iniial rae r i %, he peed of adjumen a i 1.7%, he revering mean r i 3.88%, and he volailiy σ r i 1.75%. For he ock proce, we uppoe he oal rik premium ν i 5.35%, he inrinic volailiy σ 1 i 15.4% and he correlaed volailiy σ i 1% CRRA. We chooe he parameer of CRRA uiliy: γ = 5% and denoe X1 a he correponding wealh proce deermined by opimal

18 18 ZONGXIA LIANG AND MING MA raegy (4.). X repreen he wealh proce deermined by following aniheic raegy ˆπ wih repec o a conan premium: ˆπ = λ 1(), [,T], σ 1 (γ 1) λ 1 λ 1(T) = νσ 1. σ1 +σ I i remarkable he opimal invemen proporion under CRRA uiliy i uually a proce independen o wealh, which mean he inveor hould keep he invemen proporion no maer how he marke behave. Thi appearance i exacly caued by he relaive averion. Moreover, he relaive averion i ill refleced in he behavior of opimal wealh. The Figure 1 compare he differen behavior of opimal wealh X1 and aniheic wealh X, where he relaive exce reurn X1 X and X he abolue exce reurn X1 X are depiced repecively in he lef and righ graph. I evidence X1 gain a anding relaive exce reurn regarding X a a benchmark, bu X1 behave wore in he ene of abolue exce reurn. Thi phenomenon i reaonable once noing he uiliy i conan relaive rik avere o ha we ill realize an inveor wih CRRA uiliy ju concenrae on he relaive exce reurn. Afer 1.5 x 1 4 average of relaive exce wealh 1.5 (X1 X)./X ime(year) average of abolue exce wealh X1 X ime(year) Figure 1. Comparion beween opimal wealh and anihei under CRRA hi comparion, we udy he opimal premium under CRRA. By (4.1), Λ i a deermined funcion o we draw i direcly in he Figure. There are hree ignifican appearance in hi figure. Fir one i he ock premium λ 1 keep poiive while inere rae premium λ i negaive. Thi reflec a pychology ha people wih CRRA like inere rae rik and hae ock rik. Acually, ock rik reduce he

19 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 19 abiliy of growh bu he inere rae rik promoe he growh due o he convexiy of exponenial funcion. Second one i λ 1 i an order of magniude larger han λ, which ell u he influence of ock rik i greaer han inere rae rik. Thi coincide wih he relaionhip beween he inrinic volailiy and he correlaed volailiy. The la one i λ 1 and λ are boh decreaing in he coure of ime. Thi how a rik-avere inveor urn o be relaively fearle while approaching he erminal ime, becaue he foregone concluion can be hardly changed. We call hi phenomenon a ime effec. premium of ock rik λ 1.3 λ premium of inere rae rik ime(year) ime(year) Figure. The opimal premium under CRRA 5.. CARA. We chooe he parameer α = 5%. Our ulimae aim i o imulae he opimal pair (premium, raegy and wealh) in (4.5), bu he expecaion under opimal neural meaure in FBSDE i hardly o handle. So an approximae pair i ued in hi imulaion, which i deailed calculaed in appendix C. In hi par, X1 i he approximae opimal wealh inroduced in he appendix and X i he aniheic wealh proce deermined by a imilar-formed raegy ˆπ wih a conan premium. ˆπ X = ν σ rσ A(T ) α(σ1 +σ ) exp [ A(T )r +C(T ) ] + σ rσ A(T ) X σ }}, 1 +σ }} conan correcion C(T ) = A(T u)a( r σ rλ (u) )+.5A(T u) σ a rdu, λ λ (T) = νσ. σ1 +σ I i worhwhile o noe ha he opimal invemen amoun under CRRA uiliy i he um of wo componen. The conan erm dominae when

20 ZONGXIA LIANG AND MING MA wealh amoun i le or he erminal i upcoming. The correcion erm dominae ju a he beginning of invemen wih a bigger amoun of wealh. I ell u a CARA inveor i ap o keep a conan invemen amoun no maer how he marke behave. Furhermore, hi abolue rik averion i ill refleced in he opimaliy of X1. The Figure 3 compare he differen behavior of he opimal wealh X1 and he aniheic wealh X, where he relaive exce reurn X1 X X and he abolue exce reurn X1 X are depiced. I evidence X1 gain a anding abolue exce reurn regarding X a a benchmark, bu X1 canno win X in he ene of relaive exce reurn. So we aer an inveor wih CARA uiliy ju concenrae on he abolue exce reurn. average of relaive exce wealh (X1 X)./X ime(year) average of abolue exce wealh X1 X ime(year) Figure 3. Comparion beween opimal wealh and anihei under CARA Now we udy he opimal premium. By (4.4), Λ i a ochaic proce o we can only draw i average behavior in he Figure 4 o udy i. The paen of opimal premium under CARA ha ome difference from CRRA condiion, hough ome explicaion in CRRA condiion i ill valid. We ju emphaize he premium ad curve now. The ad curve i a ynheical appearance caued by ime effec and capial effec ogeher. The ime effec ha already menioned in CRRA condiion a inveor urn o be fearle while approaching he erminal ime. The capial effec do no appear in CRRA condiion becaue inveor wih relaive averion conider he relaive growh bu no he abolue amoun. When inveor have abundan wealh, he abolue exce reurn can be grealy gained or lo. So, a an abolue rik-avere inveor

21 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 1 wih more abundan wealh, i rik premium become higher. Thi exacly explain a phenomenon ha ome rich men refue he invemen in ock by he pychology of abolue rik averion. Following hi logic, he average premium urn o be higher a he beginning of invemen ince he expeced wealh gain. The decreaing of premium ad curve a erminal ime i dominaed by ime effec of coure average premium of ock rik λ 1 average premium of inere rae rik λ ime(year) ime(year) Figure 4. The premium ad curve under CARA 6. Concluion In hi paper, we conider he imple invemen problem in an incomplee marke. The invemen proporion in ock i he only hing ha an inveor can decide, which mean he number of invemen i le han he rik and hu he marke i incomplee. The invemen arge i maximizing he expeced uiliy of erminal wealh. The uual wo mehod, HJB equaion by dynamic programming and maringale approach, boh mee difficulie in incomplee marke, repecively. The five ep of maringale approach in incomplee marke are diplayed in our paper by defining an opimal neural meaure. The coniency heorem i proved o improve he ep(iii) and ep(iv), which admi u uing maringale approach dynamically. Baed on above udy, we implemen he maringale approach under CRRA and CARA uiliie. The opimal premium under CRRA uiliy i deerminiic indeed, and an explici opimal raegy i obained. Under CARA uiliy, he opimal premium i really ochaic and we can only ue a family of equaion o decribe i. In order o gain a deep knowledge of opimal neural meaure, we make numerical illuraion

22 ZONGXIA LIANG AND MING MA a he epilog of reearch. In fac, an equivalen maringale meaure repreen a rik premium, which i he minimum amoun of money by which an inveor i willing o ake a rik. The udy of opimal premium curve embodie he pychological change during he invemen period. Under CRRA uiliy, he opimal premium i decreaing which mean a rik-avere inveor urn o be fearle while approaching he erminal ime. We call hi phenomenon he ime effec in behavioral finance. There appear a capial effec a well a he ime effec in he CARA uiliy condiion. The capial effec decribe an abolue rikavere inveor become more pruden when he own abundan wealh. Combining he ime and capial effec, he average opimal premium preen a a concave curve, and we call i premium ad curve. 7. Appendix Appendix A: Calculaion of E[H(,T) γ F ] Firly, we give an explici expreion of r u du relaed o he SDE (.). Thank o he propery of Ornein-Uhlenbeck proce, r u = e a(u ) r + r(1 e a(u ) )+ Inegraing by par we have u σ r e a(u v) db v, u >. r u du = A(T )r (A(T )+T ) r +σ r A(T u)db u, u >, where A(x) = 1 a (e ax 1). Now we ar he following calculaion baed on hee preparaion: E[H(,T) γ F ] = E exp [ γ( r u )du γ } ] γ λ 1 (u)dw u +λ (u)db F u =E exp γ λ 1 (u) +λ (u) du [γa(t )r γ(a(t )+T ) r +γσ r A(T u)db u ] } F λ 1 (u) +λ (u) du γ λ 1 (u)dw u +λ (u)db u

23 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 3 =E Q(γλ 1,γ(λ σra(t ))) exp γ λ 1 (u) +λ (u) du+ 1 ] } F +[γλ 1 (u)] du [ γa(t )r γ(a(t )+T ) r [ =exp γa(t )r γ(a(t )+T ) r + 1 [ E Q(γλ 1,γ(λ exp σra(t ))) ] } F γ σ r A(T u)λ (u)du } exp γa(t )r +C Λ (T ;γ). [γσ r A(T u) γλ (u)] γ γ (λ 1 (u) +λ (u) ) ] [γσ r A(T u)] du We emphaize C( ) i relaed o he equivalen maringale meaure Λ and parameer γ a C Λ (T ;γ) = γ(a(t )+T ) r + 1 [γσ r A(T u)] du [ ( +ln (E Q(γλ 1,γ(λ γ γ σra(t ))) exp (λ 1 (u) +λ (u) ) ) ]) F γ σ r A(T u)λ (u)du. And when γ = 1, we uually omi γ in C Λ (T ;γ) for implificaion. Appendix B: Calculaion of E[H(,T)ln(H(,T)) F ] Referring o Lemma 8.6. in Okendal(13)[17] we have E[H(,T)ln(H(,T)) F ] = exp A(T )r +C Λ (T ;1) } E R [ln(h(,t)) F ], where R i a new probabiliy meaure uch ha dr dp o appendix A, we have = H(,T). Similar E R[ ] ln(h(,t)) F } H(,T) =E E[(H(,T)) F ] ln(h(,t)) F

24 4 ZONGXIA LIANG AND MING MA ( =E 1 exp λ 1 (u)dw u + [σ r A(T u) λ (u)]db u 1 ) ( [λ (u) σ r A(T u)] du exp σ r A(T u)λ (u)du (E Q(λ 1,(λ σra(t ))) [ exp [ ( = (E Q(λ 1,(λ σra(t ))) exp ( E Q(λ 1,(λ [exp σra(t ))) λ 1 (u) du ( ) ]) 1 F σ r A(T u)λ (u)du ln(h(,t)) } F σ r A(T u)λ (u)du ) F ] ) 1 ) σ r A(T u)λ (u)du ln(h(,t)) ] F [ ( =A(T )r + (E Q(λ 1,(λ σra(t ))) exp σ r A(T u)λ (u)du ) ] ) 1 F ( E Q(λ 1,(λ exp σra(t ))) 1 + )( σ r A(T u)λ (u)du (A(T )+T ) r λ 1 (u) +λ (u) du+ λ 1 (u) +[λ (u) σ r A(T u)] du ) } F σ r A(T u) λ (u)dbu Q λ 1 (u)dwu Q [ ( =A(T )r + (E Q(λ 1,(λ σra(t ))) exp σ r A(T u)λ (u)du ) ] ) 1 F ( E Q(λ 1,(λ exp σra(t ))) + 1 λ 1 (u) du+ A(T )r +F Λ (T ). )( σ r A(T u)λ (u)du (A(T )+T ) r ) } F [λ (u) σ r A(T u)].5λ (u) du We emphaize F( ) i relaed o he equivalen maringale meaure Λ oo and i doe no have a clear form unle Λ i a deermined funcion. [ ( F Λ (T ) = (E Q(λ 1,(λ σra(t ))) exp σ r A(T u)λ (u)du ) ] ) 1 F ( E Q(λ 1,(λ exp σra(t ))) + 1 λ 1 (u) du+ )( σ r A(T u)λ (u)du (A(T )+T ) r ) } F [λ (u) σ r A(T u)].5λ (u) du. ) Appendix C: Approximae opimal pair under CARA

25 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 5 Propoiion 7.1. Suppoe (Y 1,Y ) are wo dimenional normal diribuion wih mean (µ 1,µ ) and covariance marix ( ) σ Σ = 1 ρσ 1 σ ρσ 1 σ σ. Then E[exp(Y 1 )Y ] = exp(µ 1 + σ 1 )(µ ρσ 1 σ ). Proof. E[exp(Y 1 )Y ] = E E[Y Y 1 ]exp(y 1 ) } =E µ exp(y 1 )+ρ σ σ 1 (Y 1 µ 1 )exp(y 1 ) } =(µ ρ σ σ 1 µ 1 )E[exp(Y 1 )]+ρ σ σ 1 E[Y 1 exp(y 1 )] =(µ ρ σ σ 1 µ 1 )exp(µ 1 + σ 1 )+ρσ σ 1 exp(µ 1 + σ 1 )(µ 1 σ 1) =exp(µ 1 + σ 1 )(µ ρσ 1 σ ). Equaion (4.5) are forward-backward equaion wih a ochaic riple pair (X,Λ,π). We ry o implify hem by he following approximaion: [ ( ) ]} F ln E Q(λ 1,(λ σra(t ))) exp σ r A(T u)λ (u)du σ r A(T u)e[λ (u) F ]du. Acually, above approximaion ignore he meaure ranformaion form P o Q (λ 1,(λ σ ra(t ))), and he volailiy of ochaic λ. Thi can be rue when Λ i minucule wih mall volailiy. We ue a new noaion λ (u;) o replace E[λ (u) F ]. Baing on hi ubiuion, he equaion (4.5) change ino X T = x + X π = exp (r +π ν)x d+ A(T )r +C Λ (T ) π X (σ 1 dw +σ db ), }( ν σ λ (;) λ (u;) = νσ A(T u)σ r σ1(αex u exp( A(T u)r u C Λ (T u)) F } 1), σ1+σ 1 C Λ (T )= (A(T )+T ) r+ [σ ra(t u)] σ r A(T u)λ (u;) } du. ασ 1 ),

26 6 ZONGXIA LIANG AND MING MA Thi family of equaion deermine he X1 in numerical illuraion and we ju need o calculae E[X exp( A(T )r C(T )) F ] now. Subiuing he opimal invemen amoun X π ino he dynamic of wealh, we ge dx =X r d+x π (σ 1 d W +σ d B ) ν σr σ A(T ) =X r d+ exp(a(t )r α(σ1 +σ) +C(T )) + σ } rσ A(T )X (σ σ1 1 d W +σ d B ). +σ By he heory of general one-dimenional linear equaion(cf.chaper 5 of Karaza and Shreve(1)[13]), he explici oluion of X can be expreed by where Thu, X = Z Y, [ Y = X + Z 1σ r σ A(T u)(ν σ r σ A(T u)) ] u α(σ1 +σ) expa(t u)r u +C(T u)}du + Zu 1 ν σ r σ A(T u) α(σ1 +σ ) expa(t u)r u + C(T u)}(σ 1 d W u +σ d B u ), Z = exp (r u.5 [σ rσ A(T u)] )du σ1 +σ } σ r σ A(T u) + (σ σ1 +σ 1 d W u +σ d B u ). } E X exp( A(T )r C(T )) F } =E exp( A(T )r C(T )+ln(z ))Y F E[exp(U )Y F ] =exp E[U F ]+Var[U F ]/ } E[Y F ] Cov[U,Y F ] }, where he la-econd equaliy come from a definiion and he la equaliy i guaraneed by Propoiion 7.1. To calculae he expecaion and

27 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 7 covariance of (U,Y ), we decompoe hem repecively ino he following: U = A(T )r C(T )+ r u.5 [σ } rσ A(T u)] du σ1 +σ σ r σ A(T u) + (σ σ1 +σ 1 d W u +σ d B u ) ) = A(T ) (e a( ) r +(1 e a( ) ) r+σ r e a( u) db u C(T )+ r u.5 [σ } rσ A(T u)] du σ1 +σ σ r σ A(T u) + (σ σ1 1 d W u +σ d B u ) +σ = A(T )(e a( ) r +(1 e a( ) ) r) A(T )σ r e a( u) db u C(T ) + r u + σ rσ A(T u)ν σ1 +σ + σ r σ A(T u) (σ σ1 1 dw u +σ db u ) +σ.5 [σ rσ A(T u)] σ 1 +σ } du = A(T )(e a( ) r +(1 e a( ) ) r) A(T )σ r e a( u) db u C(T )+ r(a( )+ ) A( )r σ r A( u)db u σr σ A(T u)ν +.5 [σ } rσ A(T u)] du σ1 +σ σ1 +σ σ r σ A(T u) + (σ σ1 +σ 1 dw u +σ db u ) =(A(T ) A(T )+ ) r A(T )r C(T ) σr σ A(T u)ν +.5 [σ } rσ A(T u)] du σ1 +σ σ1 +σ σ r σ A(T u) + σ σ1 1 dw u +σ + A(T )σ r e a( u) σ r A( u)+ σ rσ A(T u) σ σ1 +σ }db u, Y = X + Zu 1 σ r σ A(T u)(ν σ r σ A(T u)) α(σ1 +σ) exp A(T u)r u +C(T u) }} du + Zu 1 ν σ r σ A(T u) α(σ1 +σ ) exp A(T u)r u +C(T u) } (σ 1 d W u +σ d B u )

28 8 ZONGXIA LIANG AND MING MA = X + Z 1 (ν σ r σ A(T u)) u α(σ1 +σ ) expa(t u)r u +C(T u)}du+ Z 1 ν σ r σ A(T u) u α(σ1 +σ ) exp A(T u)r u +C(T u) } (σ 1 dw u +σ db u ). By he properie of Gauian diribuion and maringale momen equaliie we ge E[U F ] =(A(T ) A(T )+ ) r A(T )r σr σ A(T u)ν +.5 [σ rσ A(T u)] σ1 +σ σ1 +σ Var[U F ] = E[Y F ] =X + =X + = = Cov[U,Y F ] } du A(T u)a( r σ rλ (u;) )+.5A(T u) σr a } du, [A(T )σ r e a( u) +σ r A( u) σ rσ A(T u) σ σ1 +σ ] +[ σ rσ A(T u) σ σ1 1 ] }du, +σ (ν σ r σ A(T u)) α(σ 1 +σ ) E Z 1 u exp[a(t u)r u +C(T u)] F } du (ν σ r σ A(T u)) α(σ1 +σ ) exp E[U u F ]+Var[U u F ]/ } du, [ σ rσ A(T u) σ1 +σ ν σ rσ A(T u) α(σ 1 +σ ) σ1 +A(T )σ r σ e a( u) σ r σ A( u)+ σ rσ A(T u) σ σ1 +σ ] E[Z 1 u exp(a(t u)r u +C(T u)) F ]du σ r σ [A(T u) A( u)] ν σ rσ A(T u) α(σ 1 +σ ) exp E[U u F ]+Var[U u F ]/ } du, where we have ued he following equaliy in above calculaion. E[exp( U u ) F ] =E[Z 1 u exp(a(t u)r u +C(T u)) F ] =exp E[U u F ]+Var[U u F ]/}. Reference [1] Bellini, F., & Frielli, M. (). On he exience of minimax maringale meaure. Mahemaical Finance, 1(1): 1-1.

29 MARTINGALE APPROACH AND OPTIMAL NEUTRAL MEASURE 9 [] Cizár, I. (1975). I-divergence geomery of probabiliy diribuion and minimizaion problem. The Annal of Probabiliy, 3: [3] Cox, J. C., Huang, C. F., Opimal conumpion and porfolio policie when ae price follow a diffuion proce. Journal of economic heory 49(1): [4] Duffie, D., Fleming, W., Soner, H. M., & Zariphopoulou, T. (1997). Hedging in incomplee marke wih HARA uiliy. Journal of Economic Dynamic and Conrol, 1(4): [5] Delbaen, F., & Schachermayer, W.(1997). The variance-opimal maringale meaure for coninuou procee. Bernoulli, 1: [6] Frielli, M. (). The minimal enropy maringale meaure and he valuaion problem in incomplee marke. Mahemaical Finance, 1(1): [7] Grandi, P. (). On maringale meaure for ochaic procee wih independen incremen. Theory of Probabiliy & I Applicaion, 44(1): [8] Goll, T., & Rchendorf, L. (1). Minimax and minimal diance maringale meaure and heir relaionhip o porfolio opimizaion. Finance and Sochaic, 5(4): [9] Hainau, D., Devolder, P., 6. A maringale approach applied o he managemen of life inurance. Iniue of Acuarial Science, Univeri Caholique de Louvain, WP6, 1. [1] Kallen, J. (1999). A uiliy maximizaion approach o hedging in incomplee marke. Mahemaical Mehod of Operaion Reearch, 5(): [11] Karaza, I., Lehoczky, J. P., Shreve, S. E., & Xu, G. L. (1991). Maringale and dualiy mehod for uiliy maximizaion in an incomplee marke. SIAM Journal on Conrol and Opimizaion, 9(3):7-73. [1] Karaza, I., Shreve, S. E., Mehod of Mahemaical Finance, Vol. 39, Springer. [13] Karaza, I., & Shreve, S. (1). Brownian moion and ochaic calculu (Vol. 113). Springer Science & Buine Media. [14] Liang, Z., & Ma, M. (15). Opimal dynamic ae allocaion of penion fund in moraliy and alary rik framework. Inurance: Mahemaic and Economic,64:151C161. [15] Luenberger, D. G., & Ye, Y. (8). Linear and nonlinear programming (Vol. 116). Springer Science & Buine Media. [16] Meron,R.C., Opimum conumpion and porfolio rule in a coninuouime model. Journal of Economic Theory, 3(4): [17] Okendal, B. (13). Sochaic differenial equaion: an inroducion wih applicaion. Springer Science & Buine Media.

30 3 ZONGXIA LIANG AND MING MA [18] Yong, J., & Zhou, X. Y. (1999). Sochaic conrol: Hamilonian yem and HJB equaion (Vol. 43). Springer Science & Buine Media.