Optimal Portfolio Strategy with Discounted Stochastic Cash Inflows When the Stock Price Is a Semimartingale

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1 Journal of Mahemaial Finane, 6, 6, hp:// SSN Online: 6-44 SSN Prin: Opimal Porfolio Sraegy wih iouned Sohai Cah nflow When he Sok Prie a Semimaringale Onhuie Baraedi, Elia Offen eparmen of Mahemai Univeriy of Bowana, Gaborone, Bowana How o ie hi paper: Baraedi, O. and Offen, E. (6 Opimal Porfolio Sraegy wih iouned Sohai Cah nflow When he Sok Prie a Semimaringale. Journal of Mahemaial Finane, 6, hp://dx.doi.org/.436/jmf Reeived: Augu 9, 6 Aeped: November 6, 6 Publihed: November 9, 6 Copyrigh 6 by auhor and Sienifi Reearh Publihing n. hi work i liened under he Creaive Common Aribuion nernaional Liene (CC BY 4.. hp://reaiveommon.org/liene/by/4./ Open Ae Abra hi paper diue opimal porfolio wih diouned ohai ah inflow (SC. he ah inflow are inveed ino a marke ha i haraeried by a ok and a ah aoun. i aumed ha he ok and he ah inflow are ohai and he ok i modeled by a emi-maringale. he nflaion linked bond and he ah inflow are Geomeri. he ah aoun i deerminii. We do ome ienifi analye o ee how he diouned ohai ah inflow i affeed by ome of he parameer. Under hi eing, we develop an opimal porfolio formula and laer give ome numerial reul. Keyword Sohai Cah nflow, Porfolio, nflaion-linked Bond, Semimaringale. nroduion For example in finanial mahemai, he laial model for a ok prie i ha of a geomeri Brownian moion. However, i i argued ha hi model fail o apure properly he jump in prie hange. A more realii model hould ake jump ino aoun. n he Jump diffuion model, he underlying ae prie ha jump uperimpoed upon a geomeri Brownian moion. he model herefore oni of a noie omponen generaed by he Wiener proe, and a jump omponen. involve modelling opion prie and finding he repliaing porfolio. Reearher have inreaingly been udying model from eonomi and from he naural iene where he underlying randomne onain jump. Aording o Nkeki [], he war, deiion of he Federal Reerve, oher enral bank, and oher new an aue he O:.436/jmf November 9, 6

2 ok prie o make a udden hif. o model hi, one would like o repreen he ok prie by a proe ha ha jump (Ba []. Liu e al. (3 [3] olved for he opimal porfolio in a model wih ohai volailiy and jump when he inveor an rade he ok and a rik-free ae only. hey alo found ha Liu and Pan (3 [4] exended hi paper o he ae of a omplee marke. Arai [5] onidered an inomplee finanial marke ompoed of d riky ae and one rikle ae. Branger and Laren [6] olved he porfolio planning problem of an ambiguiy avere inveor. hey onidered boh an inomplee marke where he inveor an rade he ok and he bond only, and a omplee marke, where he alo ha ae o derivaive. n Guo and Xu (4 [7], reearher applied he mean-variane analyi approah o model he porfolio eleion problem. hey onidered a finanial marke onaining d + ae: d riky ok and one bond. he euriy reurn are aumed o follow a jump-diffuion proe. Unerainy i inrodued by Brown moion proee and Poion proee he general mehod o olve mean-variane model i he dynami programming. ynami programming ehnique wa firly inrodued by Rihard Bellman in he 95 o deal wih alulu of variaion and opimal onrol prob-lem (Weber e al. [8]. Furher developmen have been obained ine hen by a number of holar inluding Florenin (96, 96 and Kuhner (6, among oher. n Jin and Zhang [9], reearher olved he opimal dynami porfolio hoie problem in a jump-diffuion model wih ome realii onrain on porfolio weigh, uh a he no-hor-elling onrain and he no-borrowing onrain. Beginning wih work of Nkeki [] whih involve opimiaion of he porfolio raegy uing diouned ohai ah inflow, hi work explore opimal porfolio raegy uing jump diffuion model. n Nkeki [], he ok prie i modelled by oninuou proe whih i geomeri and bu in hi work we aume ha he ok prie proe i driven by a emimaringale; defined in Shiryaev e al. []. he jump diffuion model ombine he uual geomeri Brownian moion for he diffuion and he general jump proe uh ha he jump ampliude are normally diribued. Semimaringale a a ool of modelling ok prie proee ha a number of advanage. For example hi la onain diree-ime proee, diffuion proee, diffuion proee wih jump, poin proee wih independen inremen and many oher proee (Shiryaev []. he la of emimaringale i able wih repe o many ranformaion: aboluely oninuou hange of meaure, ime hange, loaliaion, hange of filraion and o on a aed in (Sharyaev []. Sohai inegraion wih repe o emimaringale deribe he growh of apial in elf-finaning raegie. n hi reearh, a uffiien maximum priniple for he opimal onrol of jump diffuion i ued howing dynami programming and going appliaion o finanial opimiaion problem in a marke deribed by uh proe. For jump diffuion wih jump, a neeary maximum priniple wa given by ang and Li, ee alo Kabanov and Kohlmann ( kendal and Sulem []. f ohai onrol aifie he maximum priniple ondiion, hen he onrol i indeed opimal 66

3 for he ohai onrol problem. i believed ha uh reul involve a ueful ompliaed inegro-differenial equaion (he Hamilon-Jaobi-Bellmann equaion in he jump diffuion ae. he inveor ohai Cah inflow (CS ino he ah aoun, on inflaion-linked bond and ok were onidered. Mo alulaion and mehod ued were influened by he work of Nkeki [], Nkeki [3] kendal [4], kendal and Sulem [], Klebaner [5] and Con and ankov [6].. Model Formulaion denoe he flow of informaion a diued in he definiion. Mahemaially he laer mean ha oni of σ-algebra, i.e. for all,. he Brownian moion S W ( W, W i a -dimenional proe on a given filered probabiliy pae ( Ω,, F(,, [, ], where i he real world probabiliy meaure, i he ime period, i he erminal ime period, W ( i he Brownian moion wih W i he Brownian moion Le ( Ω,, be a probabiliy pae where repe o he noie ariing from he inflaion and wih repe o he noie ariing from he ok marke. he dynami of he ah aoun wih he prie Q( i given by: d Q(, Q rd ( Q ( where r i he hor erm inere a defined in Nkeki []. he prie of he inflaion-linked bond B (, ( i given by he dynami: d B (, B (, ( r+ σφ d+ σb dw ( where σ ( σ, (, ( B b B i he volailiy of inflaion-linked bond, φ i he marke prie of inflaion rik, i he inflaion index a ime and ha he dynami: + σ d qd dw where q i he expeed rae of inflaion, whih i he differene beween nominal inere rae, r real inere r and σ i he volailiy of inflaion index. Suppoe he finanial proe ( ok reurn S ( S i given on a filered S i of exponenial form. probabiliy pae. Aume ha S S e, H, (3 H where H ( H i a emi-maringale wih repe o and. Uing ô formula for emimaringale (ee Appendix and hen differeniaing he proe we have ds S dhˆ (4 where ( H H ˆ H + H + e H (5 < 66

4 Uing random meaure if jump (ee [] ˆ x H H + H + ( e x µ (6 hene d H ˆ d ( ϕ + d H + d d ( e d d ( e d H + µ ν + ν (7 Subiuing on Equaion (7 ino Equaion (5 we have ( ϕ µ ν ν d S S d H e d e d H (8 We know ha differenial of our ok prie an wrien a where σ ( ρσ S, ρ σ S ( α σ ds S d + dw + jump (9 and W defined a before. Now omparing Equaion (8 wih Equaion (9, we an now ee ha when we equae he prediable par we have S d d d, ϕ S α + d d, ϕ + α ϕ + α ( Equaing he oninuou par we ge and he jump par give and hene we le From ( i follow ha dh σ d W, H σ W ( ( ( µ ν ( ν d e d + e d jump ( e d µ ν + e dν J ( H σ and hene i follow ha H ( ϕ α α σ Subiuing Equaion ( ino Equaion (9 we have ( α σ µ ν ν (3 d S S d + d W + e d + e d and furher imply i o ( ( α + σ + ϕ ( µ ν + ϕ ν (4 d S S d d W d d where ϕ ϕ + e Uing ó formula for jump diffuion 663

5 S Sexp α σ + σw + ln ϕ + ϕ ν d, d ( ( ln ( ϕ ϕ ν ( d, d ( ln ( ϕ ( µ ν( d, d (ee Appendix. Hene we define he following σφ σ B, Σ, α r σ (5 θ λ θ, λ θs he marke prie of he marke rik i given by where, φ φ Σ α r ρσφ φ S σ ρ φ (6 φ i he marke prie of ok marke rik. We aume he proe P( whih i geomeri and wih he no arbirage ondiion applied o i obain he following ohai differenial equaion, ( φ ψν dp P dw + d,d (7 Uing ó formula for jump diffuion equaion on 7 we have P P ( exp φ W φ (, ( d,d ψ ν + (8 where ψ (, ln ( + ψ (, ψ (, (ee Appendix. P( i a maringale ha i alway poiive and aifie Z( Now we have he prie deniy given by. P Λ (9 Q where P Λ P( exp φw r+ φ + ψ (, ν ( d,d Q ( 3. he ynami of Sohai Cah nflow he dynami of he ohai ah inflow wih proe, ( i given by d ( kd σ d W, ( where (, + ( σ σ σ i he volailiy of he ah inflow and k i he expeed growh rae of he ah inflow. σ i he volailiy ariing from inflaion and σ i he volailiy ariing from he ok marke. Solving for ( we ue ò formula for oninuou proee. Le f ln and 664

6 f df d+ d + d ( ( kd+ σ dw σd k σd+ σ dw d ln k d dw σ + σ ln ln k d dw σ + σ ln + k σ σ W ( expk σ + σ W ( 4. he ynami of he Wealh Proe f X ( i he wealh proe and θ ( θ θ θ where θ i number of uni in he ah aoun, inflaion bond and,, S i he admiible porfolio θ i he number of uni in he θ S i he number of uni in he ok. n an inomplee marke θ θ θ. he dynami of he wealh proe i given wih no arbirage we have S by where ( ( + θ + + ( Σθ + Xθλ ( ϕ ( µ ν ( d,d + ϕν ( d,d dx X r d X dw (3 ( W δ ( δ ( (4 λ exp αδ + σ δ + ϕ µ ν d,d + ϕ ν d,d (ee Appendix. For µ ν we have he dynami of he wealh proe a ( ( θ ( θ dx X r+ + d+ X Σ dw (5 For he Poion jump meaure we have he dynami of he wealh proe a ( ( + θ + + ( Σθ + Xθλ ϕ( N ν ( d,d + ϕν ( d,d dx X r d X dw ( where N i he Poion meaure and ν i he ompenaor on he Poion meaure N. 5. he iouned Value of SC n hi Seion, we inrodue efiniion. he diouned value of he expeed fuure SC i defined a (6 665

7 where ( F Filraion { F } and P exp ( r ( u Λ ( udu Λ (7 i he ondiional expeaion wih repe o he Brownian Λ i he ohai dioun faor whih adju for nominal inere rae and marke prie of rik for ok and inflaion-linked bond (Nkeki []. Propoiion. f i he diouned value of he expeed fuure SC, hen { } ( exp ( k r σφ( ψν ( d,d + ( η + k r σ φ Proof. By definiion, we have ha Applying hange of variable on 3, we have aring wih ( τ ( τ ( u ( u d (8 Λ u Λ (9 ( τ ( τ ( ( Λ dτ Λ (3 Λ exp φwτ r + φ τ+ ψν ( d,d Λ we have and laly exp k σ τ + σwτ exp d,d ( σ φ Wτ k r σ φ τ ψν {( σ φ Wτ ( k r σ φ τ ψν ( } exp d,d Λ Λ ( τ ( τ ( ( {( k r σ φ τ ψν ( } exp + + d,d { } exp ( k r σ φ dτ ψν ( d,d + + We furher ake noe ha for ν we have he diouned value of he SC a exp{ ( σφ( } k r η + k r σ φ he differenial form of i given by ( ( σ φ σ ψ (3 d r + d + d W + d d d (3 Equaion (3 i obained by differeniaing a hown in he proof below 666

8 { } ( exp ( k r σφ( ψν ( d,d + ( η + k r σ φ differeniaing boh ide, { } d exp β( ψν ( d,d + d ( η+ β { } d( exp β( ψν ( d,d + + η+ β + + η+ β { } ( η β exp β( ψν ( d,d ( σ β ψν d { } k d + dw exp + d,d + η+ β { ψν d,d } O. Baraedi, E. Offen ( η+ β ( exp β( + β + η d d η+ β β + η { } ( σ β ψν k d + dw exp + d,d + η+ β ( exp{ + d,d } β ψν η+ β ( ( η β σ (( β η σ + d+ k d + dw d k d+ dw d ( σφ σ ψν r+ d+ dw + d d d he urren diouned ah inflow an be obained by puing ino Equaion (8, f { } ( exp ( k r σφ( ψν ( d,d + ( η + k r σ φ r + σφ > k and ψν ( d,d < we an hange he horion by allowing our o go up o i.e. { } ( exp ( k r σφ( ψν ( d,d + lim ( lim η + k r σφ ( η + k r σ φ n ae of deerminii ae, we have φ and ψ, o (33 667

9 and for r > k, and we have ( {( } exp k r ( k r ( {( } exp k r lim ( lim k r r k Sine ( (,, krσ,, kr σ we need o ake ( kr σ. hen we an look a he eniiviy analyi of ( (34 i a onan, if we are inereed o ee how i behave wih repe o,,, and,,, and Finding parial derivaive of we obain he followng iffereniaing wih repe o, we have + ( β ψν( dξ a a funion of, exp (35 iffereniaing wih repe o, we have iffereniaing wih repe o k, we have (36 ( ( η + β + + k + η + β ( η β (37 iffereniaing wih repe o r, we have ( ( η + β + r + η + β ( η β (38 iffereniaing wih repe o where ( η β σ, we have ( ( η + β + σ + η + β β k r σ φ and η ψν( d and ψ ln + ψ ψ he following alulaion how how we differeniaed ( wih repe o ( exp β + ψν ( d,d differeniaing wih repe o ( ( β ψν ( β + η exp d,d + β + η β + η ( β ψν( ( β η ( β ψν d exp + d,d + d β + η exp + d,d We repeaed he following proedure for all oher variable. When we have a deerminii ae, differeniaing parially we have he fol- (39 668

10 lowing iffereniaing wih repe o, we have β exp iffereniaing wih repe o, we have β iffereniaing wih repe o r, we have ( β r β iffereniaing wih repe o σ, we have φ ( β σ β (4 (4 (4 (43 iffereniaing wih repe o k, we have ( ( β + k β able how he eniiviy of variable. Seniiviy analyi an be inorporaed ino diouned ah inflow analyi by examining how he diouned ah inflow of eah proje hange wih hange in he inpu ued. hee ould inlude hange in revenue aumpion, o aumpion, ax rae aumpion, and dioun rae. alo enable managemen o have oningeny plan in plae if aumpion are no me. alo how ha he reurn on he proje i eniive o hange in he projeed revenue and o. Looking a able, one an ee ha hanging a variable an make able. Simulaion of he eniiviy analyi. (44 σ σ r θ θ S k

11 an impa on he SC. An inveor mu do he eniiviy analyi in order o know hange an be made on he marke o improve he reul of an invemen. 6. he ynami of he Value Proe Propoiion. f V ( i he value proe and V + X where i he diouned value of he expeed fuure SC hen he differenial form of V ( i given by ( ( + + θ + σφ + ( Σ θ + σ +( ψν ( d d + λθ X ( ϕν ( d d + λθ X ( ϕ ( µ ν ( d d dv r X X d X dw Proof. iffereniaing V ( and ubiuing Equaion (3 and (6 on he differenial obained we have dv d + dx (( r+ σφ d+ σ dw +( ψν ( d d d ( θ θ λθ ϕν + λθ X ( ψ ( µ ν ( d d+ d ( r( X ( σφ θx d ( σ X θ dw +( ψν ( d d+ λθ X ( ϕν ( d d + λθ X ( ψ ( µ ν ( d d + X r+σ d+ Σ dw d+ X d d Σ For µ ν, he jump par beome ero and we obain dv r( X + + θx + σ φ d + Σ θx + σ dw 7. Finding Opimal Porfolio ( ( heorem 3. Le X ( be he worh proe whoe dynami i defined by Equaion (3, he diouned value of expeed fuure ohai ah inflow a defined in proporion (, V ( he value proe a defined in proporion ( and γ v U( v he uiliy funion and if we aume ha ν, he opimal porfolio i γ given by θ ( θ,, θ θs where φv σ σγ X σx θ ( ( θs σ α r + σφρ σ ρ σ V ( σσ σσ σσ ( ρ γx σσ ρ X and (45 67

12 O. Baraedi, E. Offen θ θ θ ( ( φ V ( σ ( σ (α r + σ φ ρ σ ρ σ V ( + σ γ X σ X σ σ ( ρ γ X ( + (σ σ (46 σ σ ( σ σ ρ X ( he proof i given in Appendix. From Equaion (7, (σ (α r + σ φ ρ (σ ρ σ σ σ ( ρ γ X ( laial porfolio raegy a ime and (σ σ V σ σ ( σ σ ρ X ( repreen he repreen he iner- emporal hedging erm ha offe hok from he SC a ime. Some Numerial Value Figure wa obained by eing, φ.8, r.4, k.99, α.9, σ d [.5;.36], σ.4, σ.4,, ρ.6 and γ.5 in Equaion (7. hi figure how ha when, he porfolio value i.5 whih i equivalen o 5.% when he value of he wealh i 4, and he porfolio value i.59 whih i equivalen o 5.9% when he value of he wealh i,,. When, he porfolio value i.6 whih i equivalen o 6% when he value of he wealh i 4, Figure. Porfolio value in inflaion-linked bond. 67

13 and he porfolio value i.64 whih i equivalen o 6.4% when he value of he wealh i,,. hi how ha here i a huge inreae on he porfolio value from o when he value of he wealh i mall and here in le hange when he value of he wealh i large. Figure wa obained by eing, φ.8, r.4, k.99, α.9, σ d [.5;.36], σ.4, σ.4,, ρ.6 and γ.5 in Equaion (7. hi figure how ha when, he porfolio value i.97 whih i equivalen o 9.7% when he value of he wealh i 4, and he porfolio value i.99 whih i equivalen o 9.9% when he value of he wealh i,,. When, he porfolio value i.97 whih i equivalen o 9.7% when he value of he wealh i 4, and he porfolio value i.97 whih i equivalen o 9.7% when he value of he wealh i,,. hi how ha here i a huge dereae on he porfolio value from o when he value of he wealh i mall and here in le hange when he value of he wealh i large. Figure 3 wa obained by eing, φ.8, r.4, k.99, α.9, σ d [.5;.36], σ.4, σ.4,, ρ.6 and γ.5 in Equaion (7. hi figure how ha when, he porfolio value i.57 whih i equivalen o 5.7% when he value of he wealh i 4, and he porfolio value i.63 whih i equivalen o 6.3% when he value of he wealh i,,. When, he porfolio value i.65 whih i equivalen o 6.5% when he value of he wealh i 4, and he porfolio value i.63 whih i equivalen o 6.3% when he value of he wealh i,,. hi how ha here i a huge dereae on he porfolio value from o when he value of he wealh i mall and Figure. Porfolio value in ok. 67

14 Figure 3. Porfolio value in ah aoun. here in le hange when he value of he wealh i large. For ν, we have a problem beaue we anno olve he equaion expliily. we need o ome up wih a ompuer program. 8. Conluion Semimaringale eem o model finanial proee beer ine he aer for he jump ha our in he yem. he oninuou proee may be onvenien beaue one an eaily produe reul. For example, in our iuaion we managed o find he porfolio for oninuou proee bu we ouldn for he one wih jump. hi work an be exended deigning a MALAB program ha will olve he equaion for porfolio θ. Aknowledgemen We hank he edior and he referee for heir ommen. We alo hank Profeor E. Lungu for he guidane he gave u on ahieving hi. Laly, we hank he Univeriy of Bowana for he reoure we ued o ome up wih hi paper. No forgeing he almighy God, he reaor. Referene [] Nkeki, C.. (3 Opimal Porfolio Sraegy wih iouned Sohai Cah nflow, Journal of Mahemaial Finane, 3, hp://dx.doi.org/.436/jmf.3.3 [] Ba, B.F. (4 Sohai ifferenial Equaion wih Jump. Probabiliy Survey,, -9. hp://dx.doi.org/.4/ [3] Liu, J., Pan, J. and Wang,. ( An Equilibrium Model of Rare Even Premia. Working 673

15 Paper, UCLA, M Sloan, and UBC. [4] Liu, J. and Pan, J. (3 ynami erivaive Sraegie. Journal of Finanial Eonomi, 69, hp://dx.doi.org/.6/s34-45x(38- [5] Arai,. (4 Minimal Maringale Meaure for Jump iffuion Proee. Journal of Applied Probabiliy, 4, hp://dx.doi.org/.7/s9494 [6] Branger, N. and Laren, L.S. (3 Robu Porfolio Choie wih Unerainy abou Jump and iffuion Rik. Journal of Banking and Finane, 37, hp://dx.doi.org/.6/j.jbankfin [7] Guo, W. and Xu, C. (4 Opimal Porfolio Seleion When Sok Prie Follow an Jump- iffuion Proe. Mahemaial Mehod of Operaion Reearh, 6, hp://dx.doi.org/.7/ [8] Aevedo, N., Pinheiro,. and Weber, G.W. (4 ynami Programming for a Markov- Swihing Jump iffuion. Journal of Compuaional and Applied Mahemai, 67, -9. hp://dx.doi.org/.6/j.am.4.. [9] Jin, X. and Zhang, K. (3 ynami Opimal Porfolio Choie in a Jump-iffuion Model wih nvemen Conrain. Journal of Banking and Finane, 37, hp://dx.doi.org/.6/j.jbankfin.3..7 [] Shiryaev, A.N., Buhlmann, H., elbaen, F. and Embreh, P. (995 No-Arbirage, Change of Meaure and Condiional Eher ranformaion. hp://people.mah.eh.h/~delbaen/fp/preprin/bes-cw.pdf [] Shiryaev, A. (3 Eenial of Sohai Finane: Fa, Model and heory. World Sienifi, Singapore, 3. [] Okendal, B. and Sulem, A. (9 Applied Sohai Conrol of Jump iffuion. 3rd Ediion, Springer, Berlin. [3] Nkeki, C.. (3 Opimal nvemen under nflaion Proeion and Opimal Porfolio wih iouned Cah Flow Sraegy. Journal of Mahemaial Finane, 3, hp://dx.doi.org/.436/jmf.3.3 [4] Okendal, B. ( Sohai ifferenial Equaion: An nroduion wih Appliaion. 6h Ediion, Springer, Berlin. [5] Klebaner, F.C. (998 nroduion o Sohai Calulu wih Appliaion. mperial College Pre, London. hp://dx.doi.org/.4/p [6] Con, R. and ankov, P. (4 Finanial Modelling wih Jump Proee. CRC Pre, Boa Raon. 674

16 Appendix Appendix A H Aume ha f e and f. Uing ô formula for ememaringale (ee Jaod [?], Proer [?], Shiryaev [], Shiryaev [] f, one obain f H f H f H H f H d H ( + d + + f ( H f ( H f ( H H < x o find our SE, aume ha f ( x e and ubiue on Equaion (47. Simplifying will give he following reul H H H H H d d H H e e + e H + e H + e e e H < H e + e H + e H + e e e H H H H H H d d < iffereniaing will give; de d e dh d e d H d e e e H H H H H H H < (47 (48 H H d d H H H e H + e H + e e e H H H H H H H d d + e H + e H + ( e e e H, H H H H H H H e dh + e d H + e ( e H H H e dh + d H + ( e H H H e dh + d H + d ( e H < H H e d H + H + ( e H < H H d e e dhˆ (49 Now he differenial of he ok proe i given by ds S dhˆ (5 where ( H H ˆ H + H + e H (5 < hen, uing o formula for emimaringale (Proer [?], we have d d X X X X X X Y e + e X + e X e e e X + e + e X + e X + e e e X ( X X X d d X X X (5 675

17 and in differenial form, hi an be expreed a X X X X X X X X+ X X X dy e dx + e d X + e e e X e dx + e d X + e e e X X X X X d d e X + e X + e ( e X X X e d X ( d ˆ + X + e X Y X, < (53 Appendix B Auming Y ln S and ubiuing i on he formula we ge Y Y Y Y dy d+ ds + ds ln ln d + S S S + S S S S [ ] ( ϕ ϕν Y + ln ln + d + S ln ln d αd+ σdw σd+ ( ln ( ϕ ϕ ν ( d + + ln + d + ln ϕ µ ν d ( S Sϕ S Sϕν ( ( ( S Sϕ S( µ ν( ( ( ϕ ϕ ν ( ( ln S ln S α σ + σw + ln ϕ + ϕ ν d, d ( ( ln ( ϕ ϕ ν ( d, d ( ln ( ϕ ( µ ν( d, d S Sexp α σ + σw + ln ϕ + ϕ ν d, d ( (54 ( ln ( ϕ ϕ ν ( d, d ( ln ( ϕ ( µ ν( d, d Appendix C Le f( P, Y ln P and Y Y Y dy d+ dp + dp P P [ ] (, ξ(, (, ξ(, ν ( d,d P P ( P( φdw P( φdw Y + Y P Y P P + ln ( P( P( ψ (, ln P( P ψ, ν d,d x + P( P ( ( + ψ (, ( d P( φ φ + ψ ν dw d ln, d, φdw φ d+ ln ψ, ψ, ν d,d + ( ( 676

18 d ln P φdw φ d + ln ( ψ (, ψ (, ν d, d + ln P ln P ( φ W φ ln ( (, (, d, d + + ψ ψ ν P W P ( ( ln φ φ + ln + ψ, ψ, ν d, d P exp φw φ + ( ln ( ψ (, ψ (, ν ( d, d P + P P( exp φw φ + ( ln ( (, (, ( d, d + ψ ψ ν P( exp φw φ + ψ (, ν ( d,d (55 Appendix ( ( ds d B, dq dx X θs + X θ + X θ + d S B Q ( S S X + W + + S S θs ( αd σ d ϕ ( µ ν( d,d ϕν ( d,d ( ( d B, dq + X θ + X θ + d B Q ( S X + W + + S θs ( αd σ d ϕ ( µ ν( d,d ϕν ( d,d θ (( σφ d σbd + X r+ + W ( θ θ + X rd + d θs ( αd σ d λ ϕ ( µ ν( d,d ϕν ( d,d θ (( σφ σb ( θ θ X + W X r+ d+ dw + X rd + d θs ( αd σ d θs λ ϕ ( µ ν( d,d X + W + X X ϕ ( ν( d,d X θ ( r σφ d σbdw X ( θ θ ( rd d θ ( α r d+ X θσφd+ rx d+ X θsσdw S ( θσ θsλ ϕ ( µ ν ϕν + X BdW + X d,d + d,d + d α r σ X S rx X + + S W σφ σb ( θ, θ d d ( θ, θ d, λ ( d,d d,d d θ d θ { d θλ ( ϕ ( µ ν + ϕν} + ( θs θ ϕ ( µ ν ϕν + X + + X r+ + Σ W + d,d d,d d 677

19 hen ( ( + θ + + ( Σθ + Xθλ ( ϕ ( µ ν ( d,d + ϕν ( d,d dx X r d X dw (56 where ( W δ ( δ λ exp αδ + σ δ + ϕ µ ν d,d + ϕ ν d,d wih δ and δw W W λ wa found by imply dividing S by S i.e. ( α + σ + ϕ ( µ ν + ϕ ( ν + + ( + Sexp W d,d d,d λ S exp W d,d d,d ( α σ ϕ µ ν ϕν ( α( σ( W W ϕ ( µ ν( ϕν ( δ δ ( αδ σδw ϕ ( µ ν ( ϕ ν (,d exp + + d,d + d,d exp + + d,d + d Appendix E Le f, C and define hai proe wih jump and (, X, d (, (,,. hen Y f V f X d Y (, X, d +, X, dx X + f + X (, X, dx f + (, X, d f + ( (, X, d dx X f( V, ( κ (, ( Y i a o- + f( V, ( + κ(,, ω f( V, ( V, κ,, ω ν d X + + { f( V, ( κ (,, ω f( V, ( }( µ ν( d (, ω f V, V, κ,, ω ν d X + + ( + f( V, ( + κ3(,, ω f( V, ( V, κ3,, ω ν d ( 678

20 ake Y f( V, and ubiuing on 58 o have d Y (, X, d +, X, X r+ d+σθdw X ( ( σ φ ( ( ( θ ( ( ( θ ( ( ( θ +, X, r+ d+ σdw f +, X, X r+ d+σθdw X f +, X, r+ d+σθdw f +, X, X r+ θ d+σθdw r+ σφ d+ σdw ( X ( + f( V, ( + κ(,, ω f( V, ( V, κ,, ω ν d X { f( V, ( κ (,, ω f( V, ( }( µ ν( d ( + f( V, ( + κ(,, ω f( V, ( V, κ,, ω ν d X + + ( + f( V, ( + κ3(,, ω f( V, ( V, κ3,, ω ν d ( ( σ φ ( (, X, ( X θ d (, X, ( σ σd ( ( θ (, X, ( X σ θd X ( d Y (, X, d +, X, X r+ d+σθdw X +, X, r+ d+ σdw f + Σ X f + f + Σ + + f( V, ( κ(,, ω f( V, ( ( V, ( κ (,, ω ν ( d X + f( V, ( + κ(,, ω f( V, ( V, κ,, ω ν d X { f( V, ( κ (,, ω f( V, ( }( µ ν( d + + ( + f( V, ( + κ3(,, ω f( V, ( V, κ 3,, ω ν d Chooing f( V, J( V, ( uh ha for a given porfolio raegy θ (no 679

21 neearily opimum, we inrodue he aoiaed uiliy θ ( J x,,, θ x,, UV (57 Subiuing κ xθλϕ, κ xθλϕ and κ3 ψ we now have ( ( θ d J V, Jd+ Jx x r+ d+σθdw + J r+ + W + J x Σ ( σϕ d σd xx ( θ d + J + J x Σ ( σ σd x ( σ θd { J( v xθλϕ J( v Jx xθλϕ} ν ( d + + { J( v xθλϕ J( v Jx xθλϕ} ν ( d + + { J( v xθλϕ J( v }( µ ν ( d + + { J( v ψ J( v J ψ} ν ( d + + J + x r+ negraing boh ide we ge ( θ J + ( r+ σ ϕ J + x Σθ J x xx + σ J + xσ Σ θj d+ ( Σ θj + σ J dw x x { J( v xθλϕ J( v Jx xθλϕ} ν ( d + + { J( v xθλϕ J( v Jx xθλϕ} ν ( d + + { J( v xθλϕ J( v }( µ ν ( d + + { J( v ψ J( v J ψ} ν ( d + + (, (, + ( + ( + θ x + ( + σφ J V J V J x r J r J x J J x J d + Σ θ xx + σ + σσθ x ( θ σ + xσ J + J dw { J( v xθλϕ J( v Jx xθλϕ} ν ( d,d { J( v xθλϕ J( v Jx xθλϕ} ν ( d,d { J( v xθλϕ J( v }( µ ν ( d,d { J( v ψ J( v J ψ } ν ( d,d x aking he expeaion on boh ide we have 68

22 x,, (, (, x,, ( ( θ x ( σφ x J J x J d + Σ θ xx + σ + σσθ x J V J V + J + x r+ J + r+ J { J( v xθλ ( ϕ ϕ J( v Jx xθλ ( ϕ ϕ } ν ( d,d { J( v ψ J( v J ψ} ν ( d,d + + For impliiy we have x,, (, (,, x,, ( ( θ x ( σφ x J J x J d + Σ θ xx + σ + σσθ x J V J x + J + xr+ J + r+ J { J( v xθλψ J( v Jx xθλψ } ν ( d,d + + { J( v ψ J( v J ψ} ν ( d,d and ( ( Where ψ ϕ ϕ J, V J, x,. Sine we know ha θ θ J, V U V, we now have Whih give u θ U( V ( x,, (,, x,, ( ( θ x ( σφ J x + J + xr+ J + r+ J x J J x J d + Σ θ xx + σ + σσθ x { J( v xθλψ J( v Jx xθλψ } ν ( d,d + + { J( v ψ J( v J ψ} ν ( d,d + + θ (,, O. Baraedi, E. Offen ( ( θ ( σ φ J x UV J + xr+ J + r+ J x,, x,, x x J J x J d + Σ θ xx + σ + σσθ x { J( v xθλψ J( v Jx xθλψ } ν ( d,d + + { J( v ψ J( v J ψ} ν ( d,d + + By Equaion (57, we have he inegral on he righ hand ide being equal o ero. ha i 68

23 ( x,, + + x+ + ( θ ( σ φ x θ J σ J x σ θj d + Σ xx + + Σ x J x r J r J { J( v xθλψ J( v Jx xθλψ } ν ( d,d + + { J( v ψ J( v J ψ} ν ( + d,d + iffereniaing boh ide we obain following parial differenial equaion wih jump. Conider he value funion ( θ ( σ φ J + x r+ J + r+ J x + x Σ θ J + σ J + xσ ΣθJ xx x { J( v xθλψ J( v Jx xθλψ } ν ( d + + { J( v ψ J( v J ψ} ν ( + + d (,, up J( x,,, θ V x (58 θ where J i a in Equaion (57 Under ehnial ondiion, he value funion V aifie U + up x Σ θ U + θu + xσ ΣθU xx x x θ + { U( v xθλψ Ux } ( d x θλψ ν + + σ U ( r σφ U rxu x { U ( v ψ J ( v U ψ} ν ( d (59 hi ake u o he HJB equaion; θ U v, + max Φ x,, (6 θ π where θ i he eond linear operaor for jump diffuion. Hene θ U x( r+ θ U + ( r+ σ φ U + x Σθ U x xx + σ U + xσ ΣθU x { U( v xθλψ U( v Ux xθλψ } ν ( d + + { U( v ψ U( v U ψ} ν ( d + + (6 aking our uiliy funion a γ v U( v (6 γ We onider he funion of θ whih i 68

24 Γ ( θ Σ θ + θ + σ Σθ + + x Uxx Ux x Ux { U( v xθλψ Ux x θλψ } ν ( d iffereniaing U( v and ubiue on (63, we ge Γ ( θ x γσ θ v + θv γxσ Σθv + + γ γ γ γ γ γ ( v xθλψ v xθλψ ν ( d Sine Γ ( θ i a onave funion of θ, o find i maximum we differeniae (64 wih repe o θ o obain Γ ( θ x γ Σ θv θ + v γx σ Σ v + + γ γ γ γ γ { xλψ ( v xθλψ v xλψ } ν ( d For ν we an olve for θ beaue we have a linear equaion below γ γ γ (63 (64 (65 x v γx Σ θv γxσ Σ v (66 γ γ x v γxσσv θ γ γ γx Σ v γx Σ v v σ Σ v Σ σ γ xσ xσ γ x x and θ will be given by Σ σ Σ V θ (67 γ X X ubiuing Σ, and where and σ a defined, we obain he following φv σ γσ X σ X θ ( σ α r + σσρσ S S ρ σ V ( σ σ ρσ Sσ γσ Sσ ( ρ X σσ S ρ X φv σ σγ X σx θ ( ( θs σ α r + σφρ σ ρ σ V ( σ σ ρσ σ σσ ( ρ γx σσ ρ X V σ X σx (68 (69 φ θ (7 σγ 683

25 ( ( ( ( σ α r + σφρ σ ρ σ V σσ σσ θs σσ ρ γx σσ ρ X We an now ee ha θ θ θ ( σσ σσ ( σ ( α r + σφρ ( σ ρ σ V φ σ + σγx σx σσ ρ γx + V σσ ρ X ( (7 (7 Submi or reommend nex manurip o SCRP and we will provide be ervie for you: Aeping pre-ubmiion inquirie hrough , Faebook, Linkedn, wier, e. A wide eleion of journal (inluive of 9 ubje, more han journal Providing 4-hour high-qualiy ervie Uer-friendly online ubmiion yem Fair and wif peer-review yem Effiien ypeeing and proofreading proedure iplay of he reul of download and vii, a well a he number of ied arile Maximum dieminaion of your reearh work Submi your manurip a: hp://paperubmiion.irp.org/ Or ona jmf@irp.org 684