DECISION SCIENCES INSTITUTE Portfolio Management Determined by Initial Endowment or Terminal Wealth in A Consumer Finance Market With Jumps

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1 Porfolo Managemen n A Marke Wh Jumps DECISION SCIENCES INSIUE Porfolo Managemen Deermned by Inal Endowmen or ermnal Wealh n A Consumer Fnance Marke Wh Jumps (Full Paper Submsson) Gan Jn Lngnan (Unversy) College & Lngnan Insue for Economcs Sun Ya-sen Unversy Guangzhou 575, Chna lnsjg@mal.sysu.edu.cn. ABSRAC In he feld of sochasc consumer fnance marke wh jumps, we sudy porfolo and wealh processes decded by nal endowmen or by nal and ermnal wealh, as well as conngen clams and opon valuaon decded by ermnal wealh. Comparng deflaor wh dscoun facor, we gve wo knds of proofs for each mporan heorem by sochasc analyss mehod. And we gve he necessary suffcen condons for he hedgng sraeges o duplcae compleely he dervave secures conngen clams n markes, under he condons of deermnsc coeffcens, usng sochasc analyss and paral dfferenal-dfference equaons. KEYWORDS: Porfolo managemen, Inal endowmen, ermnal wealh, Conngen clams, Valuaon INRODUCION Accordng o Meron s pon of vew [33, 34], he oal change n he sock prce s posed o be he composon of wo ypes of changes: () he normal vbraons n prce, for example, due o a emporary mbalance beween supply and demand, changes n capalzaon raes, changes n he economc oulook, or oher new nformaon ha causes margnal changes n he sock s value. In essence, he mpac of such nformaon per un me on he sock prce s o produce a margnal change n he prce (almos ceranly). hs componen s modeled by a sandard geomerc Brownan moon wh a consan varance per un me and has a connuous sample pah. In hs suaon, here s ample scope for he famous Black-Scholes formula. () he abnormal vbraons n prce are due o he arrval of mporan new nformaon abou he sock ha has more han a margnal effec on prce, for example, due o he Sep. aacks, whch caused dsasers n he world as well as n he fnancal markes. Usually, such nformaon wll be specfc o he frm or possbly s ndusry. I s reasonable o expec ha here wll be acve mes n he sock when such nformaon arrves and que mes when dose no arrve alhough he acve and que mes are random. Accordng o s very naure, mporan nformaon arrves only a dscree pons of me. hs componen s modeled by a jump process reflecng he non-margnal mpac of he nformaon (hs jump process can also be consdered as a model reflecng he sudden breaks). And on hs occason of abnormal vbraons, he Black-Scholes formula s no vald, even n he connuous lm, because he sock prce dynamcs canno be represened by a sochasc process wh a connuous sample pah. We mus use sochasc dfferenal equaons wh jumps o sudy hose problems on porfolo and wealh processes decded by nal endowmen or by nal and ermnal wealh, as well as problems on valuaon of conngen clams and opon decded by ermnal wealh. In hs

2 Porfolo Managemen n A Marke Wh Jumps paper, comparng deflaor wh dscoun facor, we show wo knds of proofs for some mporan heorems on nvesmen decsons when choosng porfolo and wealh processes, usng sochasc analyss mehodology. And we gve he necessary suffcen condons for he hedgng sraeges o duplcae compleely he dervave secures conngen clams n he markes, under he condons of deermnsc coeffcens, usng sochasc analyss and paral dfferenal-dfference equaons, cf. [,,, 3, 4, 5, 33, 34, and 4].. he fnancal marke model Le us frs nroduce a pon process { n : n }, where n s he me of he n h jump. We denoe N () = sup { n: n } = #{ n : n } (.) as he number of random jumps o he marke by me. N s a counng process assocaed wh { : n }, wll represen boh he jump process, as well as he erm n (.). n Le s consder a fnancal marke subjec o boh dffusve uncerany and jump uncerany. Uncerany comes from an R d -valued Brownan moon W () = ( W(),..., Wd ()), and a onedmensonal jump process N () denoed above. W () s defned on a probably space ( Ω W, F W, P W ) and () N N N N on ( Ω, F, P ). W N W N Le ( Ω, F, P) be he produc probably space,.e., Ω=Ω Ω, F = F F, and W N P = P P. here are d + sources of uncerany presen alogeher. We assume ha here s a fxed me horzon <. Assumpon. he jump process N () have a ( P, F )-sochasc nensy λ (), n oher words, λ () s he rae of he jump process a me. he process λ () s { F }-predcable, posve and unformly bounded away from on [, ]. For more deals on pon process, see Bremaud [7]. Because of ha N () Ns (), s, we have ha EN ( ( ) F) ENs ( ( ) F ) = Ns ( ), hs means ha N s a submarngale. he pon process { : n } s also a sequence of soppng mes. Indeed, n s { ω : N ( ) < n} = { ω : n > }, and N () s F -adaped, hus we have ha { ω : N ( ) < n} F, hs mples s.e., n s a soppng me. { ω : n > } F,

3 Porfolo Managemen n A Marke Wh Jumps In he marke we consder, here are d + secures (asses) beng raded connuously. he frs one s rskless, called a bond, wh prce P( ) gven by dp() P()() r d, P( ). = = (.) he oher d + secures are rsky asses, call socks, subjec o he uncerany n he marke. he prces are modeled by he lnear sochasc equaons d dp() = P( ) b() d σj() dwj() ρ() dq() + + j = = P( )[ b () d + σ () dw() + ρ () dq(), ] P() = p, for =,, d +, where Q () = N () λ() sds, (.4) s a P -marngales and represens he conrbuon of he jumps o he secury reurns. r () s he nsananeous rae of neres. b () = ( b(),, b () ) s he vecor of he nsananeous d + apprecaon raes on he socks. σ() ( σ ) s a ( d + ) d volaly marx process and ρ ( ρ ) j s a ( d + ) one. hen σ() [ σ(), ρ()] s a ( d + ) ( d + ) volaly marx process. σ denoes he h row vecor of σ. Assumpon. r (), b (), σ () are predcable wh respec o { F }, and are bounded unformly n (, ω) [, ] Ω. Furhermore, + ρ ( ) > for all and [, ], o ensure lmed lably of he sock. And he covarance marx process a () σ() σ () s assumed o be srongly nondegenerae [5,]. Remark. he assumpon + ρ ( ) > s also essenal n mahemacs, we wll see hs laer. Noe ha he szes of he jumps n he secury reurns are random, wh he randomness comng from he process ρ. And he jump process s no predcable (he jump process s rgh connuous and lef lmed, abbrevaed RCLL). However, he effec of a jump s predcable, snce ρ s predcable. hs means ha, a me, he effec of a possble jump a s known. (.3) From marngale heory, he processes W and Q of (.3) are acually P -marngales, and he prce process P of sock, s a semmarngale wh drf rae b (), =,, d +. Defne he dscoun facor as { r s ds} () exp (). (.5) P () β = Now we envson a small nvesor who sars wh some nal endowmen x and nvess

4 Porfolo Managemen n A Marke Wh Jumps n he d + secures descrbed prevously. Le η () denoe he number of shares of secury held by he nvesor a me. hen me s d + X() x = η () p, p =, and he nvesor s wealh a = d + = X () = η P() (.6) If ransacon of shares akes place a dscree me pons, for example, a τ and τ + h, and τ, τ + h (.e., ransacon never akes place a me of jump), and here s no nfuson or n n whdrawal of funds (.e., akng a self-fnancng sraegy), hen s obvous ha η ( ) = consan for ( ττ, + h). Assume ha η ( ) ( ) τ + = η τ, hen η s rgh-connuous and wh lef-lm a ransacon me pon. We should emphasze he dynamc propery of X a me τ and We clam ha X( τ ) = X( τ + ). Indeed, d + X( τ ) = η ( ) ( ) τ P τ = P τ = P τ + = P τ snce and ( ) ( ) ( ) d +, X( τ + ) = ( ) ( ) η τ + P τ, = P s connuous a τ (for τ + h. τ n ). Suppose ha a me τ, he nvesor sells (shors) a se I {,,, d + } of secures, and buys (longs) smulaneously I {,,, d + } I,.e., he res secures raded. Snce he nvesor akes a self-fnancng sraegy, we have he followng equaon, and hen [ η ( ) ( ) ] ( ) [ ( ) ( ) ] ( ) τ η τ + P τ = η τ + η τ P τ I I d+ d+ η ( ) ( ) ( ) ( ) τ P τ = η τ + P τ = =,.e., X( τ ) X( τ ) hus, defne X( τ) X( τ ) = X( τ + ). And smlarly, X( τ + h) X( τ + h ) = X( τ + h + ). So = η ( τ) P( τ + h) η ( τ) P( τ) = η ( τ) [ P( τ + h) P( τ). ] = +. d+ d+ + = + + = = = d+ d+ d+ = = = X( τ h) X( τ) X( τ h ) X( τ ) η ( τ h ) P( τ h) η ( τ ) P( τ) (.7) On he oher hand, f he nvesor chooses a me reduce he wealh accordngly, hen (.7) should be replaced by d + = τ + h o consume an amoun hc( τ + h) and X( τ + h) X( τ) = η( τ)[ P( τ + h) P( τ)] hc( τ + h). (.8)

5 Porfolo Managemen n A Marke Wh Jumps he connuous-me analogue of (.8) s d dx() = η() dp() C() d. = akng (.), (.3), (.6) no accoun and denong he amoun nvesed n he h sock by π () η () P(), =,, d +, we oban dx() = (() r X() C()) d + π ()[() b r()] d + π () σ() dw() + π () ρ() dq(), where s a vecor whose every componen s. And hs s he Meron s model. Defnon.. A porfolo process π() ( π(),..., π d ()) d + = + s an F -predcable, R -valued process on ( Ω, F,P ) for whch d π () d= π() d <, a.s. (.9) = I represens he nvesmen ha he nvesor manans n he d + socks. A consumpon process C (), s non-negave predcable w.r.. { F }, and sasfes Cd< (), a.s. (.) Remark. We allow π () ake negave values correspondng o shor sellng he h sock. In addon, f we suppose ha each sock pays ou a connuous sream of dvdends deermned by a dvdend rae process δ (),, =...,, d +,.e., he dvdend pad ou for each un of money nvesed n he sock, hen he nvesor s wealh process X sasfes dx() = (() r X() C()) d + π ()[() b + δ() r()] d + π () σ() dw() + π () ρ() dq(), X() = x (.) where he process δ () s assumed o be predcable and bounded, smlar o he processes b and r. And he nsananeous expeced reurn from nvesmen n sock s b () + δ (), he rsk premum s b () + δ () r(). d + Defne he R -valued process of relave rsk as () W + = θq (), θ() ( σ()) [() b δ() r()] d where θ W () s an R -valued process and θ () s an R -valued process. he process θ () s Q bounded, measurable and predcable w.r.. { F }, by he assumpons on b, δ, r and σ. I represens he relave rsk-premum as mpled by sock reurns and sock volales. Defne he followng processes θ

6 Porfolo Managemen n A Marke Wh Jumps θ W Q W () W () + () sds, Q () Q () + θ () sds, (.) hen (.3) can be cas as dp() s = P( s ) rsds () δ() sds+ σ() sdws () + ρ() sdqs (), P() = p. (.3) We now nroduce he boundedness condon on he relave rsk process of he jumps. Assumpon 3. We requre he followng nequales θ () λ() nf θ () λ() a >, a.s. (.4) Q < and { Q } Remark. hs means ha he relave rsk-premum process s bounded from above. hs condon wll be requred o consruc he rsk-neural measure. Inuvely, he rsk-neural measure changes he drf n he sock prces o r δ. If he rsk-premum s posve, hen he drf b s greaer han r δ and mus be brough down by he new measure. In hs case, f he relave rsk-premum s hgher han he rae a whch jumps are conrbung o he upward drf, we canno ge a measure o brng he sock prce drf down o he level we wan. Assumpon 3 s also necessary n mahemacs. Consder he followng equaons where dz () = Z () θ () dw(), Z () =, (.5) W W W W dz () = Z ( ) μ() dq(), Z () =, (.6) Q Q Q μ() = θ () λ(). (.7) Q Noe μ () s well defned snce λ () >. Furhermore, μ () > snce (.4). he soluons of (.5) and (.6) are gven as follow ZW() = exp θw() s dw() s θw() s ds { }, (.8) { } Z () = ( μ( ) + )exp μ()() s λ s ds Q N () n. (.9) n = Here we can see n mahemacal sense why we should have + μ( ) >. Oherwse, (.9) would be nvald. Le us denoe Z () Z () Z (), and from Iô s formula we oban W Q dz() = Z( ) θw () dw() μ() dq(), Z() =, (.) In oher words, Z () sasfes equaon (.). he nex lemma descrbes he rsk-neural measure.

7 Porfolo Managemen n A Marke Wh Jumps Lemma.[3, 4]. he process Z s a P -marngale wh E[ Z( )] =. Defne an auxlary probably measure on ( Ω, F ) as PA ( ) EZ [ ( ) A], A F. hen W and Q are marngales under P. In parcular, he jump process N adms ( P, F ) -sochasc nensy λ () = ( μ() + ) λ(). We can oban he soluon of (.3) n explc form N () { } { } n = { σ sdws σ s ds} β() P() = p exp δ () sds ( + ρ ( )) exp ρ sλ() sds n exp ( ) ( ) ( ). Also, he soluon of (.3) can be expressed as N () { b s ds } P = p + ρ n { } ρ s λ s ds n = exp { σ( sdws ) ( ) σ( s) ds}. exp ( ) ( ) ( ( )) exp ( ) ( ) (.) (.) From (.), we can see ha he expeced apprecaon rae s exacly r δ for he h sock, and he oal expeced reurn from nvesmen n any sock s equal o he neres rae r, under he rsk-neural measure P. Bu under P, he nsananeous apprecaon rae of h sock s b, and he nsananeous expeced reurn from nvesmen n sock s b() + δ(). In pracce, b s very dffcul o esmae, whereas, b can be observed drecly. hs shows ha he rsk-neural measure s no only helpful n heorecal research, bu also useful n praccal handlng. Under P, n erms of W and Q, (.) can be rewren as dx() = (() r X() C()) d + π () σ() dw () + π () ρ() dq (), X() = x. (.3) A soluon o hs dfferenal equaon wh X() = x s β() X() + β() s C() s ds = x + β() s π () s σ() s dw () s + β() s π ()() s ρ s dq () s, (.4) so M () β() X () + β() scsds () (.5) s a local marngale under P, by Lemma.. hs s because process β and σ are bounded unformly. π () ds < a.s., and he Defnon.. A porfolo and consumpon processes par ( π,c ) s admssble w.r.. nal wealh x f he correspondng wealh process X sasfes

8 Porfolo Managemen n A Marke Wh Jumps X (),, a.s., and equaon (.)(or (.3) or (.4)). Denoe he class of admssble par w.r.. nal wealh x as A () x. For any ( π, C) A ( x), he nonnegave local P -marngale M of (.5) s bounded from below and s hence a nonnegave P -supermarngale (cf. [4, -5, 56]). Accordng o he Lemma below, β () X () s also a supermarngale. Lemma.3. Assume ha Y () s a supermarngale (or marngale), and A () s a monoonc ncreasng process, hen Y () A () s a supermarngale. Proof. EY [ () A () F] (or =) Y() s E[ A () F] = Y() s E[ A () As () + As () F ] s s s = Ys () As () EA [ () As () F ] Ys () As (), for any s. Indeed, s Le τ = nf { [, ] : X( ) = }, hen X( τ ) =, τ τ holds on { τ < } a.s. we have ha β() X () = M () β() scsds () s a supermarngale by Lemma.3, hen for any τ τ, Eβτ ( ) X( τ) I Eβτ ( ) X( τ) I =. { τ< } { τ< } And comes ou ha Eβτ ( ) X( τ ) I{ τ < } =. For βτ ( ) X( τ) a.s., hen we ge X( τ ) =, a.s. on { τ < }. If τ <, we nerpre ha bankrupcy occurred. Applyng Iô s rule o he produc of he processes Z () and β () X () of (.) and (.4) gves us dz()[ β() X()] = Z() [ β() C() d + β() π () σ() ( dw() + θw() d ) + β() π () ρ() θq() d ] β() XZ () () θ () dw () β() π σ() θ () Zd () W + {[ β() X ( ) + β() π ρ() ][ Z ( )( + μ()) ] β() X ( ) Z ( ) } dq () + [ β( )( X( ) + π ( ) ρ())][ Z ( )( + μ())] β() XZ () () { } [ W ] [ ] Z () β() π ρ() β() XZ () () μ() λ() d W = Z () β() Cd () + Z () β() π σ() X () θ () dw () + β() Z ( ) ( X ( ) + π ρ() )( + μ() ) X ( ) dq (). (.6) Denoe ζ() β() Z(). We can assume ha pahs of β () X () are RCLL (cf. P33, [9]). hen from (.6) and he properes of sochasc negral, we have ha M () ζ() X () + ζ() scsds () (.7) s bounded a.s. on [, ], hus ζ () X () s also bounded a.s. on [, ]. Consequenly, M () s a nonnegave local P -marngale, and s a nonnegave P -supermarngale.

9 Porfolo Managemen n A Marke Wh Jumps ζ acs as a deflaor, has he meanng of ha mulplcaon by ζ () convers wealh held a me o he equvalen amoun of wealh held a me zero. Applyng (.5), (.) and Iô s formula leads o he lnear sochasc dfferenal equaon dζ() = dβ() Z() = ζ( )[ r() d + θ () dw() μ() dq() ]. (.8) Lemma.4. Suppose ha g () s a measurable, hen = W F- -adaped process, and E g() s ds E Z()() s g s ds and E Z()() s g s ds <,. Proof. Accordng o Fubn heorem, E g() s ds <, > E g() s ds = E g() s ds = E Z()() s g s ds = E Z()() s g s ds. Usng Fubn agan gves E gsds () = Egsds () = E[ Zsgs ()()] ds= E [ Zsgs ()()] ds, where [, ]. In he proof of Lemma.4, we have used he followng fac Lemma.5. If Y s an negrable, measurable and F -adaped process, hen we have EY [ ( ) : A] = E [ Y ( ) : A] for any [, ] and any A F. Here, we denoe E s( ξ : A) E[ Z( s) ξ : A] for any negrable random varable ξ, any s [, ] and any A F s. Proof. For any [, ], E [ Y (): A] = EZY [ ( ) (): A] = E[ EZY [ ( ) () F]: A] = EYEZ [ () [ ( ) F ]: A] = EZY [ ( ) ( ) : A] = E [ Y( ) : A], where A F s arbrary. Lemma.5 shows he conssency of P (or E ). Lemma.6. Under he condons of Lemma.4, Proof. For any ( ()() ) ( ( ) () ) () ( = = () ) E Z s g s ds F E Z g s ds F Z E g s ds F,. A F, E Z( ) g() s ds : A = E g() s ds : A = E [(): g s Ads ] = E[ Z()(): s g s Ads ] = E Z()() s g s ds : A, hus we have shown he frs equaly of hs lemma. he second equaly can be derved from Bayes formula. Remark. Bayes formula s ha: for s < and any F -measurable, P -negrable random varable, Z() s EY [ F s] = EYZ [ () F s]. Indeed, for any A F s, he Bayes formula can follow from

10 Porfolo Managemen n A Marke Wh Jumps ( ) ( ) E Z() s EY [ F]: A = E EY [ F ]: A = EY [ : A] = E [ Y : A] = EYZ [ (): A]. s s From Lemma.4 and he supermarngale propery of M and M, follows ha E ζ() X() ζ() s C() s ds + = E ( β() X() + β() s C() s ds) x. (.9) hen we ge necessary condon for ( π, C) A ( x) (admssbly) ζ E () scsds () = E β() scsds () x. (.3) E[ ζ() X() ] = E ( β() X() ) x. (.3) Wh he nerpreaon of he process ζ as a deflaor, he nequaly (.9) has he sgnfcance of a budge consran; mandaes ha he expeced oal value of curren wealh and consumpon-o-dae, boh deflaed down o =, does no exceed he nal capal. Defnon.7. For every gven x, le () C ( x) (respecvely, D () x ) denoe he class of consumpon processes C whch sasfy (.3) (respecvely, (.3) as an equaly); () L () x (respecvely, M ( x) ) denoe he class of nonnegave random varables B on ( Ω, F, P ) whch sasfy E( Bζ( ) ) = E ( Bβ( ) ) x (.3) (respecvely, (.3) as an equaly); () P () x denoe he class of porfolo processes π such ha ( π,) A ( x) and he correspondng ermnal wealh X ( ) M ( x). From he nequaly (.9), we deduce ( π, C) A( ( x) C C ( x) and X ( ) L ( x) (. We should show ha C ( x) consss of exacly hose reasonable consumpon processes, for whch an nvesor, sarng ou wh wealh x a me =, s able o consruc a porfolo ha wll avod deb (.e., negave wealh) on [, ], almos surely. From (.9), we have shown ha he condons (.3) and (.3) are necessary for admssbly, nex we wll see hey are also suffcen n a sense for admssbly. heorem.8. (Porfolo deermned by nal endowmen and consumpon). For any gven C C ( x), here exss a porfolo process π such ha ( π, C) A ( x) ;. Le ξ be a nonnegave, F -measurable random varable, C s a consumpon process such ha

11 Porfolo Managemen n A Marke Wh Jumps ( ) E ζ() ξ ζ() s C() s ds + = E β() ξ + β() s C() s ds x, hen here exss a porfolo π such ha ( π, C ) A ( x) wh he ermnal wealh X( ) E( ζ( ) ξ) ζ( ) ( β( X ) ( ) E( β( ) ξ) ). Proof.. Proof I. Le D C()() s ζ s ds, defne he nonnegave process as Ω() E( D F ) E[ D] + x C()() s ζ s ds, (.33) and assume ha ED ( F ) s RCLL, hen he ( P, F )-marngale u () ED [ F ] ED [ ], d can be represened as a sochasc negral w.r.. ( WQ, ),.e., here exs { F} }-predcable R - valued process η W and { F }-predcable R -valued process η Q, wh ( ηw() ηq() ) s + s ds <, a.s., such ha ηw Q u () = () sdws () + η () sdqs (). (.34) hs s because ha he famly of marngales { W : j d} and Q has he predcable represenaon propery on he produc space [4, 7, 44]. Le [ ] [ ( ) ] ζ( ) π () σ() X( ) θ () = η (), ζ( ) π () + μ() ρ() + X( ) μ() = η (), j W W Q hen we ge from he above wo equaons ( ηw( ) + ζ( ) X( ) θw( ) ) ζ( ) π() = [ σ () ] ( ) ( ηq () ζ( ) X( ) μ() ζ( ) μ( ) ), (.35) + so ha (.33) becomes (.6) when we make he denfcaon Ω() ζ() X(). he process defned n (.35) s { F }-predcable and sasfes wealh process X s gven by π () d <, a.s. he correspondng ζ() X () = E( Cs ()() ζ sds F ) ED [ ] + x. Furhermore, X ( ) = ( x ED) ζ( ). (.36) Proof II. Le D C() s β() s ds, and defne a nonnegave process by Ω() E [ D F ] ED + x C() s β() s ds. (.33 ) From Bayes Formula, we oban E[ DZ( ) ] m () ED ( ) ED F F = EDZ ( ( )). Z ()

12 Porfolo Managemen n A Marke Wh Jumps Assume ha pahs of marngale N () EDZ ( ( ) F ) are RCLL, a.s., hen here exs { F }- predcable d R -valued process Y W and { F }-predcable R -valued process Y Q, and ( W() Q() ) Y s + Y s ds <, a.s., such ha N () = EDZ ( ( )) + Y () sdws () + Y () sdqs () W Q. Applyng Iô s formula on N () Z (), wh (.) akng no accoun yelds dm() = d ( N() Z() ) = YW() dw() N() ( Z() ) [ Z () θw() dw() ] Z () 3 ( )( ( )) ( ) ( ) ( ( ) ) ( ( ) + N Z ZθW d Z ZθW ( Y ) W ( ) ) d N( ) + YQ() N ( ) N() + YQ() N ( ) YQ() N () + dq() + + Z () μ() λ() d Z ( )( + μ()) Z ( ) Z ()( + μ() ) Z () Z() Z () = ( YW() + N() θw() ) dw() + ( YW() + N() θw() ) θw() d Z () Z () YQ () N( ) μ() μ() ( YQ () N( ) μ() ) + dq () + λ() d Z ( )( + μ()) Z ( )( + μ() ) YQ () = ( YW() + N() θw() ) dw N ( ) μ( ) () + dq (). (.34 ) Z () Z ( )( + μ( )) YQ () N( ) μ() Le β() π () σ() = ( YW() + N( ) θw() ), β() π () ρ() =. Z ( ) Z ( )( + μ( )) hen oban ha ( ) ( ) Y W() + N( ) θw() ζ( ) π() = σ () ( YQ () N( ) μ() ) [ ζ( )( + μ() )]. (.35 ) When we se Ω() β() X(), (.33 ) wll become o (.4). π () defned n (.35 ) s { F }- predcable and sasfes by π () <, a.s. And he correspondng wealh process s decded β() X () = E ( Cs () β() sds F ) ED [ ] + x.. Clearly, C( ) C ( x) by he assumpon. And from he prevous proof of, here exss a porfolo process π such ha ( π, C) A ( x), and X( ) ζ( ) = x E ζ( s) C( s) ds E( ξ ζ( ) ). And for smlar reasons, we have β( X ) ( ) E ( ξ β( ) ). (.36)

13 Porfolo Managemen n A Marke Wh Jumps Remark. In he prevous heorem, wheher nroducng he deflaor or dscoun facor n he proofs, we can always oban he same conclusons. Bu n he Proof II, we should represen he marngale as he sum of wo sochasc negrals wh respec o W and Q [3, 4]. hs represenaon canno be obaned from a drec applcaon of marngale represenaon heorem o he P -marngale, snce he flraon { F } s he augmenaon (under P or P ) of W N { } W N F F, no of { } F F (cf. P375, [5]). We say ha wo measurable sochasc processes AB, on [, ] are equvalen, f A (, ω) = B (, ω) holds for λ P -a.e. (, ω) [, ] Ω. Here λ s Lebesgue measure on [, ]. Proposon.9. For each C D ( x) defned n Defnon.7, we have he followng: () he correspondng wealh process X sasfes X ( ) =, a.s.; () he process M of (.7) (or respecvely, M of (.5)) s a P (or P )-marngale. Furhermore, = ( ζ F ) (or. β() X () = E( β() scsds () ) ζ() X () E () scsds () F ), ; () he porfolo π s unque up o equvalence, and he wealh process X s also unque. Proof. For each gven C D( x) C ( x), here exss a porfolo process π such ha ( π, C) A ( x) by heorem.8. We have from (.9) he nequaly E( ζ( ) X( ) ) x E ζ() s C() s ds =, whch proves (), as well as EM( ) = E ζ() s C() s ds = x = EM(). And EM( ) EM( ) EM(),, snce M s a supermarngale, hus EM() x, and hs leads o M () beng a P -marngale. Smlarly, M () s also a P - marngale. As for unqueness, le π, π be wo arbrary porfolo processes such ha boh ( π, C ) and ( π, C) are n A () x, and le X, X be he correspondng wealh processes and M, M he correspondng marngales from (.5). hen we oban by (.4) ha ( M M)() = β() ( X () X () ) = β() s ( π () s π () s ) σ() s dw () s + β() s ( π () s π () s ) dq (). s Moreover, M( ) = M( ) = β( s) C( s) ds and M M s a P -marngale, hen ( M M )( ) = E ( M M )( ) = E ( ) ( F ) F, and hen

14 Porfolo Managemen n A Marke Wh Jumps. M M ( ) = β ( s) ( π ( s) π ( s) ) σ( s) ds + β ( s) ( π ( s) π ( s) ) ρ( s) λ( s) ds hus, ( π () π () ) σ() =, so π() π() =, a.s., a.e., snce σ s nondegenerae. hen follows ha π, π are equvalen. Furhermore, follows from he prevous ha ( M M)( ) = β( ) ( X ( ) X ( ) ),.e., he wealh process X and marngale M correspondng o C D () x are all unque. In heorem.8, we show ha one of he wealh process correspondng o C C () x can be expressed as () () ( ζ ζ() () F ) ( ζ() () ) X = E sc sds E sc sds + x, and now we have shown ha he wealh process correspondng o C D () x s unque, hus ceranly can be expressed as ( ζ ) ζ() X () = E () scsds () F,. Remark. heorem.8 shows ha: for any C C ( x), we can consruc a porfolo π, such ha he correspondng M (or M ) s P (or P )-marngale. Bu hs s vald only under some specal consrucon; Now Proposon.9 shows ha: for any C D () x, X, π, M (or M ) can be unquely deermned, and he M (or M ) s marngale, no dependng on he consrucons. heorem.. (Porfolo deermned by nal endowmen and ermnal wealh) For each B L () x, here exss a par ( π, C) A ( x) such ha he correspondng wealh process X sasfes X ( ) = B, a.s. Proof. Proof I. Le Q Bζ( ), and defne ha ( ) ( ) ζ() X () E Q F + ( x EQ) x+ m () ρ, (.37) here m ( ) = EQ ( F ) EQ, ρ = ( x EQ), X() = xx, ( ) = B, a.s. By he smlar argumen n he Proof I of heorem.8., we oban a sochasc negral represenaon of P -marngale m () n he form (.34), and hen (.37) can be cas n he form (.6) f we ake π as n (.35) and C () = ρζ(). Proof II. Le Q Bβ( ), and defne β() X () E Q F + ( x EQ) x+ m () ρ, (.37 ) ( ) ( )

15 Porfolo Managemen n A Marke Wh Jumps where m ( ) = EQ ( F ) EQ, ρ = ( x EQ ), X() = x, X ( ) = B, a.s. By analogy wh Proof II of heorem.8., a sochasc negral represenaon of P -marngale m () s obaned n he form (.34'), and (.37') can be cas n he form (.4) f we ake π as n (.35 ) and C () = ρβ(). Proposon.. For any B M () x, we have he followng () he par ( π, C) A ( x) s unquely deermned up o equvalence, and C (), a.s., a.e. And he correspondng wealh process X s also unque wh X ( ) = B. Here π P () x ; () he process M () (or M ()) s P (or P )-marngale. In parcular, he correspondng wealh process X s gven by ζ() X () = E( Bζ( ) F ) (or β() X () = E ( Bβ( ) F )),. Proof. From (.9), we have E ζ() s C() s ds x E( Bζ( ) ) =, whch verfes C (), a.s., a.e., and unqueness of Cs (), as well as EM( ) = E ( Bζ( ) ) = x = EM(), whch shows ha M s a marngale snce supermarngale M has a consan expecaon value. Oher conclusons such as unqueness of π and X, ec can be shown n smlar ways as Proposon.9. Remark. heorem. and he relaon (.3) show ha L () x consss of precsely hose levels of ermnal wealh whch are aanable from he nal endowmen x, by usng some porfolo-consumpon par ha avods deb. In oher words, L () x s an aanable se of X wh X() = x. he ermnology here s due o Plska [43]. Proposon. shows ha he exreme elemens of L () x are aanable by sraeges ha mandae zero consumpon. Defnon.. Defne an arbrage opporuny as a porfolo π such ha ()( π,) A (), and ()he wealh process X correspondng o ( π,) and he nal capal X() = x =, sasfes P( X( ) > ) >. In oher words, an arbrage opporuny s an admssble nvesmen sraegy wh zero nal capal and zero consumpon, whose ermnal wealh s posve wh posve probably. I s also called a free lunch somemes. Our model excludes arbrage opporunes. Indeed, he necessary condons (.3) and (.3) wh nal capal x = yeld C () a.e., a.s. and X ( ) = a.s.. Valuaon of European conngen clams A European conngen clam (Abrvaed, ECC) s a fnancal nsrumen conssng of a

16 Porfolo Managemen n A Marke Wh Jumps dvdend payoff rae g (), [, ], and a ermnal lqudaon payoff value f. Here, g s a nonnegave, bounded, measurable and F -adaped process, and f s a nonnegave, F - measurable random varable. In addon, g and f are assumed o sasfy ( ( ) β β()() ) ( ζ( ) ζ()() ) E f + sgsds = E f + sgsds <. (.) Le x be gven, a par ( π, C) A ( x) s called a hedgng sraegy agans he conngen clam (, gf ), f C () = g (), ; and X ( ) = f hold a.s., where X s he wealh process assocaed wh he par ( π, C ) and wh he nal condon X() = x. Denoe H( x) = {( π, C) : ( π, C) s a hedgng sraegy agans he ECC ( g, f) wh nal wealh X() = x}. In words, a hedgng sraegy ( π, C) H ( x) sars ou wh nal wealh x and duplcaes he payoff from he ECC. Suppose ha a me =, we sgn a conrac ha gves us he opon o buy, a he specfed me (.e., maury or expraon dae), one share of he sock = a a specfed prce q (he conracual exercse prce ). A me, f he prce P ( ) of he share s below he exercse prce, hen he conrac s worhless o us. On he oher hand, f P ( ) > q, we can exercse our opon a hs me by buyng one share of he sock a he pre-assgned prce q, and hen sellng he share mmedaely n he marke for P ( ), earnng a prof of P ( ) q. hus, hs conrac s equvalen o a paymen of ( P ( ) q ) + a maury, s called a European call opon. A European call opon s a specal case of an ECC wh g and f = ( P ( ) q ) +. How do we decde he far prce o pay a = for he ECC? If here exss a hedgng sraegy ( π, C) H ( x) whch s admssble for some x >, hen an agen who no only can buy he ECC ( gf, ) a me = bu also can nsead nves he wealh x n he fnancal marke accordng o he porfolo π and consume a he rae C. A maury, he agen can sll duplcae he payoff from buyng he ECC. Consequenly, he prce he should be prepared o pay a = for he ECC canno possbly be greaer han hs amoun x. hen s naural o defne he far prce of as he smalles value of he nal wealh, whch allows consrucng a hedgng sraegy. ECC. Le us defne he number v nf { x > : ( π, C) H ( x) } as he far prce a = for he Denoe Q ζ( ) f + ζ()() s g s ds, Q β( ) f + β()() s g s ds.

17 Porfolo Managemen n A Marke Wh Jumps heorem... he far prce of he ECC ( f, g ) s gven by v EQ = EQ. here exss a unque (up o equvalence) par ( π, C) H ( x), correspondng o a wealh process X whch s also unque and gven by = ( + F ) or β ( β β ) ζ() X () E ζ( f ) ζ()() sgsds () X () = E ( f ) + ()() sgsds F, [, ]. Furhermore, he process M n (.7) (or M n (.5)) s a P (or P )-marngale; If g, hen = F,.e., X ( ) = E( Xu ( ) β( u) β( ) ) β() X () E( β( uxu ) ( ) ). For any nonnegave number x (, ) ( ) π C A x, and he correspondng wealh process F for < u. v, here exss a porfolo process π such ha he par X sasfes he ermnal condons ζ( ) X ( ) E( ζ( ) f ) or β( ) X ( ) E ( β( ) f ). Proof.. Le C () = g (), and consder he P -marngale u () EQ ( F ) EQ,. We can assume ha pahs of hs P -marngale are RCLL, a.s (cf. P33, [9]). By smlar argumen n he Proof I of heorem.8., u () has a sochasc negral represenaon n he form (.34), and ζ() X () E( Q F ) ζ()() sgsds (.) can be cas n he form (.6) f we ake π as n (.35), here X() = EQ. Now we show EQ s he far prce of he ECC ( gf, ). By (.9), any oher hedgng sraegy agans he conngen clam ( gf, ) should cos a leas as much as EQ, and EQ s a lower bound on he far prce. In he above proof, f we replace he deflaor ζ by he dscoun facor β, and mae he analogous argumen under P nsead of P, we wll ge an anoher proof smlar wh he proof II of heorem.8.. Snce EM ( ( )) = EQ= X() = EM ( ()), hen he mahemacal expecaon of supermarngale M s a consan, and follows ha M s a marngale. he proof of he unqueness s smlar o he correspondng par n Proposon.9. And comes ou of (.) ha ( ) ζ() X () E ζ( f ) ζ()() sgsds If g, hen for any < u, = + F, ( ) ( ). β() X () = E β( X ) ( ) F = E E ( β( X ) ( ) F ) F = E ( β( uxu ) ( ) F ). We may prove by analogy of heorem.8.. u

18 Porfolo Managemen n A Marke Wh Jumps 3. he valuaon equaon for dervave secures of ECC For he hedgng ECC ( gf, ) problem, we wll consder dervave secures, e., ECC whose payoffs and values depend on he prces of he underlyng asses P = ( P,..., P d + ). In hese cases, he prce of he clam can be expressed as a funcon of me and he curren sock prces vecor P = ( P,, P n + ). he dvdend sream and he ermnal payoff boh depend on he d + values of he sock prces beng raded, e., g s a measurable funcon [, ] R R, and + + d + f s a measurable funcon R R. Noe ha g and f depend only on he curren prces + + on he sock markes, no he pah of he prce process. he prevous resuls ndcae ha here exss a replcang porfolo for he dervave secury. Now we would lke o relae he prce (value) of he dervave secury and he composon of replcang porfolo o he prces of he underlyng socks. We wll use hedgng argumens o derve a valuaon equaon ha mus be sasfed by he ECC s prce, and ndcae some knd of he equvalen relaons beween he soluons of he wealh process equaon and of he valuaon equaon. Assume ha all marke coeffcens are deermnsc, and he value of he dervave secury s a funcon of me and he sock prces, e., here exss a funcon, d + V C ([, ) R R ), such ha VP (, ( )), [, ) s he value (prce) of he dervave + + secury a me. We wll use a rple ( VP (, ( )), gp (, ( )), fp ( ( ))) o express he ECC dervave secury, where VP (, ()) sands for he value of he dervave secury, P () are he curren sock prces, as well as gp (, ()) and f ( P ( )) sand respecvely for he dvdend rae and he ermnal payoff of he ECC dervave secury. Le x, ( π, C) A ( x) be he hedgng sraegy agans he above-menoned dervave secury assocaed wh he wealh process X (). If X () VP (, ()), C( ) = g(, P( ) ), a.s.,, hen we say ha he hedgng sraegy ( π, C ) duplcaes (or replcaes) compleely he above-menoned dervave secury. Here X () s a wealh process of he hedgng sraegy, sasfyng dx () = rxd () () + π σ() dw () + π ρ() dq () gp (, ()) dx, ( ) = fp ( ( )), <. (3.) he sock prces sasfy he equaon (.3). use. d + For any vecor p = ( p,, p ) R, denoe p [ p ( + ρ ( )),, p ( + ρ ( )) ] for laer d + + d+ d+

19 Porfolo Managemen n A Marke Wh Jumps heorem 3.. If ( π, C ) duplcaes compleely he ECC dervave secury ( VP (, ( )), gp (, ( )), fp ( ( ))), hen he value VP (, ()) of he dervave secury mus sasfy he followng dfferenal-dfference equaon d+ d+ δ =, j= j j σ σj d + VP (, ()) ρ λ = P VP (, ()) VP (, ()) VP (, ()) + P()[ r() ()] + P() P () () () P P P + V( P, ()) VP (, ()) P () () () = rvp () (, ()) gp (, ()), VP (, ( )) = fp ( ( )), [, ). (3.) he replcang porfolo s gven by d + VP (, ( )) π( P, ( )) = P( ) σ( ), V( P, ( )) VP (, ( )) σ ( ) = P. (3.3) Furhermore, f d + VP (, ()) P () ρ() = VP (, ()) VP (, ()), (3.4) P = hen π s gven by VP (, ( )) π(, P ()) = P ( ). (3.5) P Proof. Usng Iô s formula o VP (, ()) leads o d + VP (, ()) VP (, ()) dv(, P()) = d + P() ( r() δ() ) d + σ() dw () = P d + VP (, ()) + P() Pj() σ() σj () d+ { V( P, ( ) ) VP (, ( )) } dq () j, = P Pj d + VP (, ()) + V( P, ()) VP (, ()) P() ρ() λ () d. = P Comparng hs equaon wh he wealh process equaon (3.) of he hedgng sraegy agans he conngen clam ( VP (, ( )), gp (, ( )), fp ( ( ))), follows ha (3.) and (3.3). And wh condon (3.4) n force, we oban (3.5) by comparng he coeffcens before dw () and dq () n (3.6) and (3.). Conversely, we can derve a hedgng sraegy o duplcae compleely he ECC ( VP (, ( )), gp (, ( )), fp ( ( ))) by solvng a paral dfferenal-dfference equaon, we show hs n he followng heorem. Noe ha σ () s bounded unformly snce a () = σ() σ () s assumed o be srongly nondegenerae. (3.6)

20 Porfolo Managemen n A Marke Wh Jumps heorem 3.. Suppose ha, d + Vp (, ) C ([, ) R R ) and sasfes he followng dfferenal-dfference equaon + + d+ d+ δ =, j= j jσ σj d + Vp (, ) ρ λ = p Vp (, ) Vp (, ) Vp (, ) + p [ r() ()] + p p () () p p p + Vp (, ) Vp (, ) p () () = rvp () (, ) gp (, ), [, ), Vp (, ) = fp ( ), hen here exss a hedgng sraegy ( π, C ) o duplcae compleely he ECC (3.7) ( VP (, ( )), gp (, ( )), fp ( ( ))), and he replcang porfolo s gven by π( P, ( )) = P( ) σ ( ), V P, ( ) VP (, ( )) σ ( ) d + VP (, ( )) ( ) = p. Furhermore, he replcang porfolo can be gven by ha Vp (, ) sasfes d + Vp (, ) p ρ() = V (, p ) V (, p ). p = VP (, ( )) π(, P ()) = P ( ) provded p Proof. I gves us (3.5) when applyng Iô s formula on VP (, ( ). Wh (3.7) ake no consderaon, we have { ( ) } d + VP (, ()) dv(, P()) = r() V(, P()) d + P() σ() dw () p = + V P, ( ) VP (, ( )) dq ( ) gp (, ( )) d, VP (, ( )) = fp ( ( )). (3.8) d + VP (, ( )) π( P, ( )) P( ) σ ( ), V P, ( ) VP (, ( )) σ ( ) = p, hen (3.8) becomes Le ( ) (3.), and π (, P ()) d<, a.s. If we le C () gp (, ()) = more, hen he hedgng sraegy ( π, C ) duplcaes compleely he ECC dervave secury ( VP (, ( )), gp (, ( )), fp ( ( ))). he remans can be proved smlarly wh heorem 3.. In combnaon of he above wo heorems, we derve he necessary suffcen condon for he hedgng sraegy ( π, C ) o duplcae compleely he ECC dervave secury ( VP (, ( )), gp (, ( )), fp ( ( )))., d + heorem 3.3. Suppose ha Vp (, ) C ([, ) R R ). (I) he necessary suffcen + + condon for he hedgng sraegy ( π, C ) o duplcae compleely he dervave secury

21 Porfolo Managemen n A Marke Wh Jumps ( VP (, ( )), gp (, ( )), fp ( ( ))) s ha VP (, ()) sasfes he followng dfferenal-dfference equaon d+ d+ δ =, j= j j σ σj d + VP (, ()) ρ λ = P VP (, ()) VP (, ()) VP (, ()) + P()[ r() ()] + P() P () () () P P P + V( P, ()) VP (, ()) P() () () = rvp () (, ()) gp (, ()), [, ), subjec o he boundary condon VP (, ( )) = fp ( ( )). he replcang porfolo s gven by π(, P ()) = P( ) σ (), V P, ( ) VP (, ( )) σ () d + VP (, ( )) ( ) = P ; (II) Furhermore, f VP (, ()) sasfes he followng equaon d + VP (, ()) P () ρ() = VP (, ()) VP (, ()), P = hen he necessary suffcen condon for he hedgng sraegy ( π, C ) o duplcae compleely he dervave secury ( VP (, ( )), gp (, ( )), fp ( ( ))) s ha VP (, ()) sasfes d+ d+ δ j j σ σ =, j= j VP (, ()) VP (, ()) VP (, ()) + P()[() r ()] + P() P () () () P P P = rvp () (, ()) gp (, ()), [, ), subjec o he boundary condon VP (, ( )) = fp ( ( )). And π s gven by VP (, ( )) π(, P ()) = P ( ). P Remark.. heorem. ensures ha here exss a hedgng sraegy o duplcae he ECC, whereas heorems here ell us when here exss a hedgng sraegy o duplcae compleely he ECC dervave secury. And he unqueness leads o ha hese wo hedgng sraeges are he same for a gven ECC dervave secury.. Alhough he jumps are pure non-sysemac rsk, hs doesn mean ha he jump componen wll no affec he equlbrum opon prce. hus, we can no ac as f he jump componen was no here and compue he correc opon prce. hese are emphaszed many mes by Meron [33, 34]. Appendx. o solve a hea equaon he paral dfferenal equaon (hea equaon)

22 Porfolo Managemen n A Marke Wh Jumps ϕ(, y) + βϕ(, y) αyϕy(, y) θ yϕyy(, y) = gy (, ), () ϕ( y, ) = fy ( ), [, ), y R, β, α, θ are all consan, s ofen used n sochasc fnancal marke whou jump, and he famous Black-Scholes formula s derved from hs knd of equaon. Now le us solve hs equaon. x x Le y = e, hen x = lny (f y <, hen le y = e, all he conclusons are he same), and hen ϕ x x ϕy = = ϕx = e ϕx, x y y x x x x x x x ϕyy = ( ϕy ) x = e ( e ϕx ) x = e ( e ϕx + e ϕxx ) = e ( ϕxx ϕx ). y Subsue no (), we oban x where hx (, ) = ge (, ). Make a ransformaon of Subsue no (), we arrve o θ x xx hx (, ) + βϕ ϕ ( α θ ) ϕ = ϕ, () θ ϕ = ϕξ ξ + ϕτ τ = α ϕξ + ϕτ, θ ξ = x α, τ =, hen ϕ = ϕ ξ + ϕ τ = ϕ, ϕ = ( ϕ ) ξ + ( ϕ ) τ = ϕ. x ξ x τ x ξ xx x ξ x x τ x ξξ where θ K(, τξ) = h τξ, + α τ θ K(, τξ) + βϕ ϕτ = ϕξξ(, τξ), βτ. Mulplyng wo sde of he prevous equaon by e leads o Le ψ = e βτ ϕ, he equaon becomes βτ βτ θ βτ τ K(, τξ) e ( e ϕ) = [ e ϕ] ξξ. βτ θ K(, τξ) e ψτ = ψξξ, ξ R, τ [, ), β ξ+ ( α θ ) ψ(, ξ) = e f( e ). I follows from [4] ha he exsence, unqueness and sably for he soluon of hs hea equaon are vald. And here s an explc expresson n [4] (P84) for he soluon. Afer dervng he soluon of (3), make nverses of above-menon ransformaons (Noe: Fnally, we should subsue back x = ln y no he equaon, because here s he possbly of y < ), and he soluon of he orgnal equaon comes ou. (3)

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