Multi-Period Portfolio Selection with No-Shorting Constraints: Duality Analysis

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1 Joura of Maheaca Face hp://wwwscrporg/oura/f ISSN Oe: 6-44 ISSN Pr: Mu-Perod Porfoo Seeco wh No-Shorg Cosras: Duay Aayss Ju Q La Y Maagee Schoo Ja Uversy Guagzhou Cha How o ce hs paper: Q J ad Y L (7 Mu-Perod Porfoo Seeco wh No-Shorg Cosras: Duay Aayss Joura of Maheaca Face hps://doorg/436/f7734 Receved: February 7 Acceped: Jue 7 Pubshed: Augus 3 7 Copyrgh 7 by auhors ad Scefc Research Pubshg Ic hs work s cesed uder he Creave Coos Arbuo Ieraoa Lcese (CC BY 4 hp://creavecoosorg/ceses/by/4/ Ope Access Absrac hs paper cosders a u-perod ea-varace porfoo seeco probe wh o shorg cosra We assue ha he sape space s fe ad he possbe secures prce vecor rasos s equvae o he uber of secures By akg use of he ebeddg echque of L ad Ng ( he orga oseparabe probe ca be soved by roducg a auxary probe Afer he rsk eura probaby s cacuaed he auxary probe ca be soved by usg he argae ehod of Pska (986 Fay we derve a cosed for of he opa souo o he orga cosraed probe Keywords Mu-Perod Mea-Varace Foruao Auxary Marke Margae Mehod Rsk Neura Probaby Duay Opa radg Sraegy Iroduco Porfoo heory deas wh he queso of how o fd a opa dsrbuo of he weah aog varous asses Mea-varace aayss ad expeced uy foruao are wo dffere oos for deag wh porfoo seecos A fudaea bass for porfoo seeco a sge perod was provded by Markowz Uder he assupo ha shor-seg of socks s o aowed aayca expresso of he ea-varace effce froer sge-perod porfoo seeco was derved by sovg a quadrac prograg probe Markowz (95 [] Laer a aayca souo o he sge-perod eavarace probe wh assupo ha shor-seg s aowed s derved Mero (97 [] Recey a u-perod porfoo seeco probe has bee suded hs probe s ore eresg as vesors aways ves her weah u DOI: 436/f7734 Aug 3 7

2 J Q L Y perods sead of oy oe perod Work of L ad Ng ( [3] cosders he u-perod porfoo seeco probe a ea-varace fraework whe shor-seg of socks s aowed L ad Ng have derved he aayca foru- ao of he froer of he u-perod porfoo seeco by ebeddg he asses-oy u-perod ea-varace probe o a arge racabe probe Whe shor-seg s o aowed he u-perod porfoo seeco probe s uch ore dffcu o dea wh For couous-e ea-varace por- foo seecos Xu Xuyu ad Adrew ( [4] use sochasc opa ear-quadrac ehod For u-perod seg he porfoo seeco pro- be wh o-shorg cosra has bee suded Xu ad Shreve (99 [5] [6] hese papers vesgaed a uy axzao probe wh a o shor- seg cosra usg a duay aayss he obecve of hs paper s o vesgae dyac ea-varace porfoo seeco whe shor-seg s o aowed Isead of usg opzao e- hod hs paper used a argae approach whch was orgay proposed by Pska (986 [7] o our kowedge o aayca uerca ehod usg argae easure for fdg he opa porfoo pocy wh o-shor sheg cosra for he uperod ea-varace foruao has bee repored he eraure I hs sese hs paper exeds exsg eraure by uzg a argae approach o sove a opa porfoo seeco probe wh o-shorg cosra hs approach aso showed ha a uque equvae argae easure exs he o-arbrage copee arke ode A effec- ve agorh s derved for fdg he axu quadrac uy fuco wh o-shor seg cosra o oue of hs paper I Seco we bud up he secury arke ode I Seco 3 we cosder he opa porfoo seeco probe wh o shorseg cosra By rasforg he orga arke o soe auxary arkes he opa vaue of orga cosraed probe ca be derved by he opa vaued of he ucosraed probe he auxary arkes I Seco 4 we use argae approach o sove he ucosraed probe he auxary arkes I Seco 5 he opa era weah was derved by sovg a dua probe I Seco 6 he derve he opa radg sraegy based o he opa era weah A uerca exape s aso gve he Seco 7 Fay we cocude he paper Secury Marke Mode We cosder a u-perod secury arke ode wh radg daes (dexed by ad he e horzo s fe here are rsky secures ad oe bod he arke Le ( Ω P be he probaby space Suppose here are fe saes of he word ad e K { N } be he sae space of he ecooy a e he sape space Ω of he ecooy has a fe uber of eee ω { e e e } wh e K he frao F { } where s geeraed by { e e e } reveas Ω ad We he forao o he ecooy Specfcay { } 75

3 ca ha he process { k K ; } A \ J Q L Y K s -adaped For ay P A > he secures are raded he arke whou rasaco cos Deoe he sochasc process of he secury prce as S { S; } S ( S( S( s a rado vecor ad he bod prce process as B { B ; } where B s cosa Le { R ; } secury reur process defed by R ( R( R( ad R S S for S > ad { r; } where R be he rsky r be he bod reur process defed by r B B wh r r for a Assupo he sae space a e has N eees; For ay ω { e e e } f e K he e K ( { } K( ; 3 Deoe a arx D of he secures prces D rak D ( ( B S S B( S( S( ( for ad K he above assupo akes he secury arke a copee oe We ca easy verfy ha he sape space of he arke Ω has ( eees uder assupo We cosder a vesor he faca arke wh a weah v She or he foows a sef-facg radg sraeges H { H; } where H ( H( H( ad H s he uber of us of he h rsky secury hed bewee e ad he uber of oey vesed he bod s h Assupo he vesor vess her or hs weah he copee arke wh o shor-seg cosra ha s H for ( Le V be he vaue of porfoo a e sasfes V hb HS V hb r H S R ( where S S ( S For our coveece we roduce he dscoued prce process S S S S S S S B So he { } wh ad ; dscoued vaue of porfoo s V h HS he chage of he dscoued δ δ δ wh prces of rsky secury s defed as ( δ S S 753

4 J Q L Y We ca aso defed he sef-facg radg sraeges as π { π ; } where π π ( π ad π s he fraco of oey vesed h rsky secury a e Sary he o shor-seg requres ha wre as π π s o-egave for ay herefore he vaue of porfoo ca be re- V V ( r π R I s easy o verfy ha HS V π 3 Pra ad Auxary Probe he u-perod porfoo opzao probe uder ea-varace frae- work hs paper ca be foruaed as foows: E ( V Var ( V ax ω s & for ω Varyg he vaue of ω yeds he se of effce souos As dcaed L ad Ng ( above probe s dffcu o be soved drecy because of he o-separaby he sese of dyac prograg I L ad Ng ( he reao bewee he u-perod ea-varace porfoo seeco probe wh a fxed vese horzo ad a separabe porfoo seeco probe wh a quadrac uy fuco s vesgaed ad he aayca souo s derved by usg a ebeddg schee Foruaey heores ad L ad Ng ( ca be aso apped he curre subec wh a ucera vese horzo We ow cosder he foowg auxary probe: ( λ ω ax E V V s ( & ( he obecve fuco of he auxary probe s equvae o he quadrac λ uy fuco U( x β x x β > I s cocave ad wce co- ω uousy dffereabe fuco Proposo 3 he frs dervave of U( x s U ( x β x he verse fuco of U ( x s I( y : β y he opa porfoo probe s o axze he expecao of U( V uder he o shor seg cosra π { K π R ; π } So he cosraed opa porfoo probe s: ax EU V s π V v where deoe he se of a adssbe radg sraeges beog K Sce here s a o shor-seg cosra he opa porfoo seeco probe s dffcu o be soved drecy by dyac prograg We w ry o sove he probe by roducg ucosra auxary probes Deoe he suppor fuco σ ( x of K by σ ( x ( π x sup π K 754

5 J Q L Y I order o eae he suao σ ( x doa of σ s he covex coe σ ad ( x { ; } wh ( we defed ha he effecve K { x R : x < } { x R ; x } σ for x K We roduce he predcabe process ( K for a Le deoe he se of a such process Defe a auxary arke M for each by odfyg he reur processes for he bod ad he rsky secures as: r r R R B Specay he arke M wh s he orga arke We cosder he ucosra opa porfoo probe he arke Le J ( v ax s EU V V v M : deoe he correspodg opa obecve vaue he arke M heore 3 Suppose probe ad where J ( v J s he opa souo of he pra cosraed J s he opa souo of he dua probe ( v J s he opa obecve vaue he ucosraed arke M assocaed wh he opa souo If he opa radg sraegy π for he ucosraed arke M sasfes a π b π for a he π s he opa sraegy for he orga cosraed arke ad J J Proof For he arke M ad opa radg sraegy π whch sasfes (a ad (b he vaue of porfoo a e s ( π π ( v r π R r B v r π R r V V v r R B r π ( π As π s a feasbe souo of he orga cosraed probe he expeced uy of V ( π s saer ha or equa o he opa vaue of he orga cosraed probe So we have J EU ( V EU ( V J O he overhads for a arbrary arke M ad he opa radg sraegy π of he orga cosraed porfoo probe we have ( V π v r π R B r v r π R r Bπ r π R r V ( π 755

6 J Q L Y Sce ( π ( π ( π ( π EU V EU V J v J v for ay herefore J EU V EU V J v for ay Hece Pug ogeher he above wo equaes we have 4 Margae Mehod Now we ry o sove he auxary probes ax s EU V V v Deoe he rsk eura probaby he arke ( ω ( ω ( ω L Q P be he sae prce desy J J M as Q Le J J v J Proposo 4 Uder he o-arbrage cosderao he expeced ds- coued era weah based o he rsk eura probaby s equa o he a weah e E V B v So he probe s equvae o ax s Q EU V ( E V B v Q heore 4 For he above opa probe wh quadrac uy fuco he opa aaabe weah s: v E L B β V β L B E( L ( B ad he opa obecve vaue s Proof EU V ( β ( λ Q ( ( λe ( V L B EU V E V EU V ( ( v β E L B E ( L ( B P ω U V ω λv ω L ω B he ecessary codos o axze hs expresso us be: U V ω λl ω B ω Ω ( for a hs s equvae o V ω I λl ω B β λl ω B ω Ω ( for a he vaue of he paraeer λ s he oe ha akes EQ V v Hece EQ (( β λlb B E( ( β λlb LB βe( LB λe( L B v V sasfes 756

7 J Q L Y herefore λ βe( LB v E( L B ad he opa obecve vaue Hece we have v β ELB [ ] V β LB a ω Ω EL B J ( v β 5 Opa era Weah Now we coe o he dua probe: ( v ( v β ELB [ ] E L B ( v β ELB [ ] J β EL B Sce E[ L B] E[ L] B B ad s equvae o E L B E L B he probe ( ( ω E L Q Defo 5 For arbrary { ω e e} he prce chage a e as D e where ( he arke s a sae K( Defo 5 Deoe D e ( Ω we defed he arx of ( e ( e ( e δ δ δ δ( e δ( e δ( e δ s he chage of dscoued prce of h secury a e whe a e as a arx whch coes fro e D by deeg cou Defo 53 Deoe D e ( arx whch coes fro e D by repacg he row wh heore 5 Uder he assupo he arke exss a uque rsk eura probaby Q ( ω e e e ( D ( e D ( e S ( e ( e e D S ( he sae K Proof We ca see ha ( s he prce of he h rsky secury a e whe he arke s a Q ω > for ay ( Frs we ry o Q ω s equa o oe prove ha he su of For we have Q ( ω e ( D ( e D ( e S ( ( D ( 757

8 J Q L Y ad e Because where hece Q ( ω ( D ( D( S( ( D ( ( D ( S( ( ( D ( S( ( D D D ( S( ( A S A ( δ ( δ ( δ ( δ ( δ ( δ ( δ ( δ ( δ ( δ δ δ ( ( ( ( ( Q ( ω D D ( Suppose for k k k s oefor k he sape space s Ω k Uder he assupo Ω k ca be dvded o ( k k ( subspace Ω k Ωk Ωk Ωk where Ω k cudes a he ω whch has he sae sae he frs k perod (e ( k e ek ω ad e e ( { k K k ek e k e k } Ωk Ω k for ay So for arbrary subspace Ω k Q ω ω Ωk k ω Ω k he su of he Q ( ω e e e ( D ( e D ( e S ( e ( e k D ( e e e e ( D ( e D ( e S ( e ( e k ω Ωk D ( e e k ek ek ( Dk ( ek D ( ek Sk ( e k k ( e k k D ( e k e e e ( D ( e D ( e S ( e ( e k D ( e 758

9 J Q L Y we ca verfy ha ω Ωk ( ω Q ( ω Q k k herefore k k k ( ( ( Qk ( ω Qk ( ω Qk ( ω ω Ω k So we cocude ha for ay he su of Q( ω s oe ad hs defes a probaby Now we ry o prove ha he probaby s a rsk eura probaby he arke Cosder a arbrary e ad arbrary eve A correspodg o (e for ω A ( e e ω have bee kow Suppose e we kow ha ω A e K { } We deoe he subse of A as A A where a eee A has e ( A A A A rsky secury we have For h E S E σ S Q( ω σ ( ω S ( ω ( ω Q Q ω A Q w ( D D ( ( ( S D ( S ( ( ( σ ( D σ ( D ( S ( ( σ ( Q( w D ( D D ( ( ( S S ( ( D ( D σ ( D ( D ( ( ( ( S σ D ( D σ D S ( ( D ( D ( ( ( S S ( ( D ( Q w here are four par For par ( D σ C D where C s he h row of D 759

10 J Q L Y For par ( because for where For par (3 For par (4 Hece ( D ( S ( ( σ( C S ( ( D ( C s he h row of D Hece D ( S ( ( σ ( D ( σ ( D C ( ( D ( D C C ( ( D ( D ( ( D ( S ( ( ( ( A S E Q σ S So hs probaby s a rsk eura probaby of he arke he dua probe s equvae o e e D ( e D ( e S ( e ( e o spfy our probe we gve he foowg oaos: So he probe ca be rewre as: a D ( b D S b b b e a e b e e e here have soe speca properes of he obecve fuco whch akes he cacuus uch easy Deoe 76

11 J Q L Y {( e e ; e K( } e ϕ a e b e e Proposo 5 Uder he assupo pery: ϕ e e e a e b e e {( e e e ; e K( } So we have {( e e e ; e K( e K } ϕ ϕ have he foowg pro- a e b e e a b e ( f a e b e e e : e e a b( a b K K a b ( 3 K( 3 K he probe s separabe ad ca be soved by dyac prograg b b has fu rak for ay ad he he opa souo of he dua probe s heore 5 If ϕ K ( ( ( for ad wh ( where ( ( ( ( s: Proof Le ( ( ( f f < ( b b b a ϕ ϕ ( a b ( ϕ ϕ : K K ( f a b K a b ( K a b f 76

12 J Q L Y for K ( { ( } A s ( K f a b We separaey sove he foowg probe for each : ax s a b ( ( We sove he ucosraed probe ad he opa s: ( ( b b b a he opa souo of he cosraed probe s where ( ( ( ( ( So f ϕ a b f f < ad f ( a b ϕ K Now we coe o s We separaby o sove he foowg probe for : ax a b ( ϕ ( s ( We sove he ucosraed probe ad he opa s: ( ( b b ϕ b a ϕ he opa souo of he cosraed probe s where ( ( ( ( ( f f < So f( ϕ( a b ( ϕ ad 3 3 3( ϕ K 3 f a b Geeray a sage s we suppose s ( ϕ K ( f a b 76

13 J Q L Y We separaby o sove he foowg probe for each K( ( ϕ ax a b s We sove he ucosraed probe ad he opa s: ( : ( ϕ b b ϕ b a he opa souo of he cosraed probe s where ( ( ( ( f ( f < Hece he opa souo of he dua probe s ( ( ( for ad wh ( where ( ( ( ( s: ( ( ( f f < ( b b b a ϕ ϕ ( a b ( ϕ ϕ Hece he rsk eura probaby wh opa Q ( ω s e e e ( D ( e D ( e S ( e ( e e D ad he opa era weah s V L B L E ( L ( B E ( L vb β β Q( ω ( EQ( ω v β E L B vb β ( ω β β Hece he expecao ad varace of era weah are: β E V a bv 763

14 J Q L Y where c ( Var V c vb β ( ω ( ω EQ a ( EQ b ( ( EQ EQ ( ω ( ω B ( ω ( ω ( E Q( ω EQ EQ he opa β us sasfy he opay codo of 6 Opa radg Sraegy Le α vb β ( EQ( ω a ωcvb β ωc so V ( ω β αq( ω Proposo 6 For ay { } ω Ω {( e e ( e e ω } where We deoe du dβ ha s V ω are equvae for ; { } ω s beog o he perod sape space ad ( We have he reaoshp bewee V ( ω V ( ω V ad V for ay : ( ω π ( ( ω V r R B r V So we ca eravey derve V ad correspodg radg sraegy π for each For we cosder ω Ω {( e } e ; e e ω ( { } ad ΩΩ Ω ( For each here are eee he se Ω For a arbrary se Ω we oae he eee as ω ω So ( ( ( ( ( ( ( V ω r π R ω B r V ω V ω r π R ω B r V ω V ω r π R ω B r V ω Rewre he above equaos as he arx for as G X b where 764

15 J Q L Y ( V ( ω R ( ω B r G V ω ( R ω B r V ( ω X π b r If G s fu rak X G b Geeray for we cosder ω Ω {( ; ( e e e e } ω ( { ( } ad Ω Ω Ω For each here are ( eee he se Ω For a arbrary se Ω we oae he eee as ω ω So V ( ω r ( R ( B r V ( π ω ω V ( ω r ( R ( B r V ( π ω ω V ( ω r ( R ( B r V ( π ω ω Rewre he above equaos as he arx for as G X b where V ( ω ( R ( ω B r G V ( ω ( R ( ω B r If G s fu rak X G b 7 Nuerca Exape ( ω V X π b r We cosder a arke wh oe rsky secury ad oe bod ad he vese horzo s 3 Suppose he bod prce s cosa he prces of he rsky secury are: ω S S S S 3 ω ω ω ω ω ω ω ω

16 J Q L Y he rsk eura probaby arke M are: ω Q ω ω ω ω 3 ω 4 ω 5 ω 6 ω 7 ω 8 ( ( 8 ( 9 48 ( ( 8 ( ( ( 8 ( 6 36 ( ( 8 ( 6 36 ( 3 ( 4 ( 6 36 ( 3 ( 4 ( 6 36 ( 3 ( 4 ( 3( 3 36 ( 3 ( 4 ( 3( 3 36 We sove he dua probe: ax 8 Q ( ω s for 3 he opa souo of hs s easy foud o be: he opa vaue s 3 5 ( 68 ( ω V ω ω ω ω ω ω ω ω ω

17 J Q L Y Fay we ge he opa radg sraegy of he orga probe by sovg soe ear equaos he opa sraegy are ω π π 87 π π 46 ω 3 π 87 π π 46 ω 3 π 87 π π 84 ω 3 3 π 87 π π 84 ω 4 3 π 87 π 44 π 687 ω 5 3 π 87 π 44 π 687 ω 6 3 π 87 π 44 π 569 ω 7 3 π 87 π 44 π 569 ω Cocuso Opa ea-varace uperod porfoo seeco wh o shorg cosras probe s suded he paper We coec he orga ea-varace probe o a auxary probe by usg a ebeddg echque Sce he auxary probe s dffcu o sove drecy we exed he eraure by usg duay heory ad argae approach o do he aayss Fay he derved aayca opa uperod porfoo sraegy provdes vesors wh he bes sraegy o foow a o-shor seg dyac vese evroe he ao hs paper s ha we derve he opa porfoo pocy by axzg he quadrac uy fuco A fuure research subec s vesgao of a opa souo usg dffere uy obecve fuco Ackowedgees Ja Uversy scefc research cuvao ad Iovao Fud Nuber: 7JNQN5 Refereces [] Markowz H (983 Porfoo Seeco he Joura of Face [] Mero RC (97 A Aayc Dervao of he Effce Porfoo he Joura of Faca ad Quaave Aayss hps://doorg/37/396 [3] Dua L ad Ng WL ( Opa Dyac Porfoo Seeco: Muperod Mea-Varace Foruao Maheaca Face hps://doorg// [4] L X Zhou XY ad L AEB ( Dyac Mea-Varace Porfoo Seeco wh No-Shorg Cosras Joura o Coro ad Opzao hps://doorg/37/s [5] Xu GL ad Shreve SE (99 A Duay Mehod for Opa Cosupo ad Ivese uder Shor-Seg Prohbo Geera Marke Coeffces Aas of Apped Probaby 87- hps://doorg/4/aoap/

18 J Q L Y [6] Xu GL ad Shreve SE (99 A Duay Mehod for Opa Cosupo ad Ivese uder Shor-Seg Prohbo Cosa Marke Coeffces Aas of Apped Probaby hps://doorg/4/aoap/77576 [7] Pska SR (986 A Sochasc Cacuus Mode of Couous radg: Opa Porfoos Maheacs of Operaos Research hps://doorg/87/oor37 Sub or recoed ex auscrp o SCIRP ad we w provde bes servce for you: Accepg pre-subsso qures hrough Ea Facebook LkedI wer ec A wde seeco of ouras (cusve of 9 subecs ore ha ouras Provdg 4-hour hgh-quay servce User-fredy oe subsso syse Far ad swf peer-revew syse Effce ypeseg ad proofreadg procedure Dspay of he resu of dowoads ad vss as we as he uber of ced arces Maxu dsseao of your research work Sub your auscrp a: hp://papersubssoscrporg/ Or coac f@scrporg 768