DHC Iniuional Day Pari November nd 006 4:00-5:30 he heory and (Be) Pracice of iabiliy-driven Invemen (DI) ionel Marellini DHC Rik and Ae Managemen Reearch Cenre lionel.marellini@edhec.edu.edhec-rik.com DHC Iniuional Day & Summi 006
Inroducion A Brief Hiory of AM An Academic Perpecive on DI Soluion Implemening iabiliy-maching Porfolio Concluion Mahemaical Appendix Ouline
Inroducion A Brief Hiory of AM An Academic Perpecive on DI Soluion Implemening iabiliy-maching Porfolio Concluion Mahemaical Appendix
Inroducion Penion und Crii S&P 500 DB Penion Plan Ne urplu over $00 billion Ne defici over $00 billion in abou 3 year $400 000 40% (in million $300 000 $00 000 $00 000 $0 -$00 000 99 99 993 994 995 996 997 998 999 000 00 00 003 004 0% 00% 80% -$00 000 -$300 000 penion funding au 60% 99 99 993 994 995 996 997 998 999 000 00 00 003 004 Penion funding raio 350 300 50 00 50 00 50 0 000 00 00 003 004 Overfunded fund Underfunded fund Source: S&P 500 Penion Sau Repor 005
Inroducion Penion und Crii he iuaion ha improved recenly bu remain problemaic. By ome eimae () he aggregae penion defici decreaed by $4.8 billion during 005 leaving an aggregae penion defici of $96.0 billion for he 00 companie involved in he 006 Millimun penion fund urvey. During he pa hree year he funding defici for he 00 companie ha been reduced by $67.5 billion. ou of he 00 companie ere in a urplu poiion a he end of 005 up from 9 in 004 7 in 003 and only in 00 alhough ill ignificanly le han he 90 ha had urplu ae a he end of 999. () Millimun penion fund urvey 006: hp://.milliman.com/penion_fund_urvey/.
o primary caue Inroducion Origin of he Crii Marke Condiion Perfec orm of advere marke condiion Decline in equiy marke decreae in penion plan ae Decreae in inere rae increae in penion liabiliie Weakne of rik managemen and ae allocaion pracice Could he crii have been avoided by beer ae allocaion deciion?
mai-05 Inroducion Perfec Sorm of Advere Marke Condiion Bae rae in US urope and UK 700 voluion of S&P 500 index 600 500 400 300 00 Rae in % 00 000 janv-05 janv-00 mai-00 ep-00 janv-0 mai-0 ep-0 janv-0 mai-0 ep-0 janv-03 mai-03 ep-03 janv-04 mai-04 ep-04 UK USA U
Inroducion Ae Allocaion & Rik Managemen (Mi)Pracice Invemen in equiy By 99 holding in equiie by penion fund ere 75% in he UK 47% in he US 8% in he Neherland and 3% in Sizerland. A a reul of he dominaion of equiie he increae in liabiliy value ha folloed he decreae in inere rae a only parially offe by he parallel increae in he value of he bond porfolio. Wa exceively high invemen in equiy before he ar of he bear marke he real problem? In 00 miday hrough he bear marke penion fund had 64% of heir oal ae in equiie in he UK 60% in he US 50% in he Neherland and 39% in Sizerland. Beer ae allocaion and rik managemen pracice ould have avoided he penion fund crii and migh help conribue o finding a oluion o he problem.
Inroducion A Brief Hiory of AM An Academic Perpecive on DI Soluion Implemening iabiliy-maching Porfolio Concluion Mahemaical Appendix
Cah-flo maching & immunizaion A (Very) Brief Hiory of AM Differen Approache o AM Cah-flo maching: hi involve a perfec mach beeen he cah flo from he porfolio of ae and he commimen in he liabiliie; inflaionlinked inrumen are ofen ued in hi perpecive. Immunizaion: o he exen ha perfec maching i no poible hi echnique allo he reidual inere rae rik creaed by he imperfec mach beeen he ae and liabiliie o be managed in an opimal ay. AM counerpar: inveing in rik-free ae. Surplu opimizaion In a concern o improve he profiabiliy of he ae and herefore o reduce he level of conribuion i i neceary o inroduce ae clae (ock nominal bond) ha are no perfecly correlaed ih he liabiliie ino he raegic allocaion. AM counerpar: inveing in opimal riky porfolio.
Miing ingredien? A (Very) Brief Hiory of AM ooking orard o-fund eparaion heorem: in AM he opimal porfolio involve a combinaion of he rik-free ae and he riky porfolio ih he be rikreurn rade-off. Porfolio inurance: hee raegie generae a limied expoure o he riky porfolio hen i perform ell and a donide rik proecion hen i perform poorly. Quid of DI oluion?
Inroducion A Brief Hiory of AM An Academic Perpecive on DI Soluion Implemening iabiliy-maching Porfolio Mahemaical Appendix
An Academic Perpecive on DI Soluion We conider n riky ae he price of hich are given by n i i j dp P id ijdw μ i... n j Ae Price A rik-free ae he 0h ae i alo raded in he economy; he reurn on ha ae ypically a defaul free bond i given by dp 0 P 0 rd (W) (W W n ) i a andard n-dimenional Bronian moion.
Inroduce a eparae independen proce for pecific liabiliy rik Becaue of he independence beeen yemaic rik expoure and pecific liabiliy rik e obain ih An Academic Perpecive on DI Soluion iabiliy Proce ε ε μ j n j j dw dw d d η η () () () () () dw d dw d ε ε ε η μ η exp exp
An Academic Perpecive on DI Soluion Objecive Objecive: here he invemen policy i a (column) predicable proce vecor repreening allocaion o riky ae ih he reminder inveed in he rik-free ae. Differen from andard AM problem (Meron (969 97)): In AM ha maer i no ae value per e bu ae value relaive o liabiliy value (a.k.a. funding raio) he dynamic of hich i given by: A Max A Max ε ε ε μ μ dw dw d r d r A A d d
An Academic Perpecive on DI Soluion Opimal porfolio raegy Soluion ( ) ( μ r ) ( ) We hu obain a o- (hree-)fund eparaion heorem he fir porfolio i he andard log-opimal efficien porfolio. he amoun inveed i inverely proporional o he inveor Arro-Pra coefficien of rik-averion. he econd porfolio i a liabiliy-hedging porfolio: i can be hon o have he highe correlaion ih he liabiliie; alernaively i i a porfolio ha minimize he local volailiy of he funding raio. ( )( ) ε
An Academic Perpecive on DI Soluion iabiliy-hedging Porfolio 05 0 Relaive MSD ronier 05 xcee reu 0 iabiliy Hedging Porfolio 005 0 0 005 0 05 0 05 03 Relaive Rik
An Academic Perpecive on DI Soluion iabiliy-maching Porfolio 03 05 0 xp. Reur 05 Relaive MSD ronier 0 005 iabiliy Maching Porfolio 0 0 005 0 05 0 05 03 035 Relaive Rik
An Academic Perpecive on DI Soluion Building block # DI Soluion hi i cah-flo or duraion maching: ue derivaive along ih radiional fixed-income inrumen o manage he inere rae rik (immunizaion/dedicaion). everage can be ued in hi leg. hi par i cuomized ih repec o he liabiliy rucure of he clien. Building block # here i a need for rik managemen and no only ae managemen in he performance-eeking porfolio. In principle hi par i independen of he clien. Allocaion o hee o building block i aic and depend on rik-averion.
An Academic Perpecive on DI Soluion unding Raio Conrain In mo counrie implici or explici funding raio conrain are in effec In he Unied Sae he Penion Benefi Guarany Corporaion (PBGC) hich provide a parial inurance of penion charge a higher premium o fund reporing a funding level of under 90% of "curren" liabiliie. In he UK here a a Minimum unding Requiremen (MR) ha came ino effec in 995; he 004 Penion Bill ha replaced he MR ih a chemepecific auory funding objecive o be deermined by he ponoring firm and fund ruee. In Germany Penionkaen and Penionfond mu be fully funded a all ime o he exen of he guaranee hey have given. In Sizerland he minimum funding level i 00% ih an incenive o conervaive managemen (invemen in equiie for example i limied o 30% of oal ae for fund ih le han 0% coverage raio). In he Neherland he minimum funding level i 05% plu addiional buffer for invemen rik. A k Max We capure hi in he folloing ay:
An Academic Perpecive on DI Soluion Opimal porfolio raegy MR Conrain Dynamic DI Soluion k ( ) ( ) ( μ r ) ( ) We no have a ae-dependen a oppoed o aic allocaion o he o fund. he fir porfolio i he andard mean-variance efficien porfolio. Conider he fracion of ealh A allocaed o he opimal groh porfolio I i given by m ( ) ( μ r) ( ) ( μ r) ( ) ( μ r) A k ( ) ( μ r) A ( A k ) k
An Academic Perpecive on DI Soluion MR Conrain Inerpreaion I appear ha he fracion of ealh A allocaed o he opimal groh porfolio i equal o a conan muliple m of he cuhion i.e. he difference beeen he ae value and he floor defined a A -k. ( ) ( μ r) m hi i reminicen of CPPI raegie hich he preen eup exend o a relaive rik managemen conex. While CPPI raegie are deigned o preven final erminal ealh from falling belo a pecific hrehold exended CPPI raegie (a.k.a. coningen immunizaion raegie) are deigned o proec ae value from falling belo a pre-pecified fracion of ome benchmark value here liabiliy value.
Inroducion A Brief Hiory of AM An Academic Perpecive on DI Soluion Implemening iabiliy-maching Porfolio Concluion Mahemaical Appendix
Implemening iabiliy-maching Porfolio Cah veru Derivaive Inrumen We have argued ha an opimal AM policy involve inveing in o fund A liabiliy hedging porfolio for rik managemen purpoe he andard groh opimal porfolio for ae managemen purpoe here are variou poible non-excluive ay o implemen he liabiliy-maching porfolio Cah inrumen: nominal and real (i.e. inflaion-proeced) bond Derivaive: fuure ap opion
Implemening iabiliy-maching Porfolio ocuing on Inere Rae Rik he mo dominan facor in liabiliy reurn i inere rae rik. A a reul bond are dominan ingredien in he liabiliy maching porfolio. A recalled earlier everal facor make he replicaion of liabiliie ih bond omeha problemaic. In paricular perfec replicaion of he liabiliy i never poible becaue of he lack of available mauriie in he bond marke.
Implemening iabiliy-maching Porfolio Inere Rae Rik Managemen An iniuional inveor i expoed o ignifican inere rae rik in cae of a duraion mimach beeen ae and liabiliie. hi expoure may repreen a large unacknoledged raegic be on inere rae and he mimach in duraion expoe he company o uncompenaed rik. A poiive duraion gap indicae ha ae are more inere rae eniive han liabiliie on average. hu hen inere rae rie (fall) ae ill fall proporionaely more (le) in value han liabiliie and he marke value of equiy ill fall (rie) accordingly. On he oher hand a negaive duraion gap indicae ha eighed liabiliie are more inere rae eniive han ae. hu hen inere rae rie (fall) ae ill fall proporionaely le (more) in value han liabiliie and he marke value of equiy ill rie (fall). A given decreae in he level of inere rae ill have a more dramaic impac on he value of he liabiliie han on he value of he ae ih a dramaic decreae of urplu ize a a conequence. uure can help reduce he gap.
Implemening iabiliy-maching Porfolio iabiliy Hedging ih uure Surplu-Defici-Diribuion Bund 0Y ihou hie noie Surplu-Defici-Diribuion Bund 0Y ih hie noie -0.7% -0.5% -0.3% -0.% 0.% 0.3% 0.5% 0.7% -0.7% -0.5% -0.3% -0.% 0.% 0.3% 0.5% 0.7% 0.9% Surplu-Defici-Diribuion Buxl 0Y ihou hie noie Surplu-Defici-Diribuion Buxl 0Y ih hie noie -0.7% -0.5% -0.3% -0.% 0.% 0.3% 0.5% 0.7% -0.7% -0.5% -0.3% -0.% 0.% 0.3% 0.5% 0.7% 0.9% Surplu-Defici-Diribuion Duraion mached P 0Y ihou hie noie Surplu-Defici-Diribuion Duraion mached 0Y ih hie noie -0.7% -0.5% -0.3% -0.% 0.% 0.3% 0.5% 0.7% -0.7% -0.5% -0.3% -0.% 0.% 0.3% 0.5% 0.7% 0.9% Surplu/defici beeen ae and liabiliie afer a hree monh period. 0 year duraion i aumed for liabiliie. Source: Golz Marellini Ziemann (006). he Bund (Buxl) conrac i a medium (long) mauriy conrac.
Implemening iabiliy-maching Porfolio Inroducing Inflaion Rik xample: Reail Price Inflaion (RPI)-linked annuiie Mach ih inflaion-linked ae bu only an imprecie cahflo mach poible ih index-linked bond Again perfec replicaion of he liabiliy i never poible becaue of he lack of available mauriie in he inflaion linked marke. Coupon paid on hee bond do no mach liabiliy paymen and have o be reinveed (replicaion ould imply being hor poiion in loer mauriie). 300000000 xample iabiliy lo v IG Cahflo (marke eighed) Marke implied inflaion 50000000 00000000 50000000 00000000 50000000 0 -Apr-5 -Apr-48 -Apr-44 -Apr-40 -Apr-36 -Apr-3 -Apr-8 -Apr-4 -Apr-0 -Apr-6 -Apr- -Apr-08 -Apr-04 iabiliy lo Ae lo
Implemening iabiliy-maching Porfolio Uing Sap for everaged Poiion in AM Conider a liabiliy ih benefi ha are predominanly inflaion-linked. herefore inflaion-linked ae ould be an appropriae mach. Inflaion-linked bond are cah inrumen ih he loe liabiliy rik. An alernaive ae and liabiliy mach ould be o: Buy a riky porfolio of raigh bond or inve in an abolue reurn performance porfolio; Ue he ap marke o conver he bond cahflo o he precie inflaionlinked cahflo required o pay he cheme projeced benefi paymen. Penion cheme pay fixed flo (exraced from he porfolio) Scheme receive inflaion-linked cah flo ailored o mee he projeced liabiliie. Relaive o index-linked gil hi oluion ould: Provide more precie managemen of inflaion rik; Inroduce leverage and hence generae a higher expeced reurn due o addiional performance of riky porfolio bu addiional rik (in cae he porfolio underperform he fixed ap rae).
Implemening iabiliy-maching Porfolio rom iabiliy-hedging o iabiliy-maching An inflaion ap can be ued o exchange cah-flo generaed by a bond porfolio for RPI-linked cah-flo o mach he precie naure and iming of annuiy paymen. hi give a more precie inflaion mach han ih index-linked gil and provide he freedom o inve in a ide range of underlying ae. RPI/PI linked cahflo o mach benefi paymen RPI/PI linked benefi paymen Bank und Cahflo generaed by performance porfolio 300000000 xample iabiliy lo v IG Cahflo (marke eighed) Marke implied inflaion 300000000 Inflaion-inked Benefi Paymen v Inflaion-inked Sap lo 50000000 50000000 00000000 00000000 50000000 50000000 00000000 00000000 50000000 50000000 0 0 004 008 0 06 00 04 08 03 036 004 008 0 06 iabiliy lo 00 04 Ae lo 08 03 036 iabiliy lo Ae lo
Implemening iabiliy-maching Porfolio Illuraion Con. I hould be noed ha he fund i hedged again inflaion and inere rae rik bu no again moraliy rik. high impac on he penion fund cah flo profile Projeced Benefi Pa ymen 00 90 80 70 60 50 40 30 0 0 0 year le age raing Sandard Moraliy 009 04 09 04 09 034 039 044 049 054 059 064 069 074 079 084
Inroducion A Brief Hiory of AM An Academic Perpecive on DI Soluion Implemening iabiliy-maching Porfolio Concluion Mahemaical Appendix
Concluion Regulaory Conrain and Accouning Impac We have preened a brief overvie of he echnique ued in ae-liabiliy managemen. Modern porfolio heory allo one o: i) raionalize he DI approach a a pecific example of a eparaion heorem hich ae ha rik managemen and performance are be managed hen handled by eparae porfolio; ii) exend he DI approach o allo for dynamic rik managemen benefi. In pracice he queion remain a o ho o implemen hee oluion hile aking ino accoun regulaory conrain he and accouning impac. In he nex eion Prof. oulquier ill ouline ome of he challenge involved in erm of he implemenaion of hee oluion in he conex of inurance companie.
Inroducion A Brief Hiory of AM An Academic Perpecive on DI Soluion Implemening iabiliy-maching Porfolio Concluion Mahemaical Appendix
Mahemaical Appendix Reference Proof and Mahemaical Preciion In ha follo e preen more deailed reference and mahemaical reul. hi maerial illurae he poer of he change of numeraire echnique ofen ued in derivaive pricing (quano opion exchange opion fixed-income derivaive ec.) in porfolio opimizaion problem. By recognizing ha he liabiliy porfolio i he naural numeraire in hi economy and by conidering porfolio value dynamic under he aociaed equivalen maringale meaure e can ue convex dualiy echnique o olve he opimal allocaion problem.
he Model Ae Price We conider n riky ae he price of hich are given by n i i j dp P id ijdw μ i... n j A rik-free ae he 0h ae i alo raded in he economy; he reurn on ha ae ypically a defaul free bond i given by dp 0 P 0 rd We aume ha he calar r he (xn) (column) vecor μ (μ ι ) i n and he (nxn) marix ( ij ) ij n are progreively-meaurable and uniformly bounded procee and ha i a non ingular marix ha i alo progreively-meaurable and bounded uniformly. (W) (W W n ) i a andard n-dimenional Bronian moion.
Under hee aumpion he marke i complee and arbiragefree and here exi a unique equivalen maringale meaure Q. Define Z ( 0 ) exp θ dw θ θ d 0 0 here θ i he rik premium proce θ μ he Model Rik Premium Proce hen Z i a maringale and Q i he meaure ih a Radon- Nikodym deniy Z ih repec o he hiorical probabiliy meaure P. According o he Giranov heorem he folloing proce i a Q- maringale 0 Q W W θ d 0 ( r ) 0
Inroduce a eparae independen proce for pecific liabiliy rik Becaue of he independence beeen yemaic rik expoure and pecific liabiliy rik e obain: ih he Model iabiliy Proce ε ε μ j n j j dw dw d d η η () () () () () dw d dw d ε ε ε η μ η exp exp
In he preence of liabiliy rik ha i no panned by exiing ecuriie he e of all meaure under hich dicouned price are maringale here he rik-free ae i ued a numeraire i given by: dq A Q; θ.. Z Z 0 0 dp ih: Z Z ( 0 ) exp ε ( 0 ) exp θ () dw θ d 0 0 θ dw 0 0 he Model MM in he P 0 -Marke θ θ d here he rik premium for pure liabiliy rik i (for ε 0) ( μ ( θ ) θ r ε
iabiliie a a Numeraire Porfolio he liabiliy porfolio i he naural numeraire porfolio in hi economy; he value of ae relaive o liabiliie i he funding raio he key variable in AM. he dynamic of relaive price i dpˆ Pˆ d P hich can alo be rien a dpˆ Pˆ Q Q ε dw dw MM in he -Marke i ( P ) ε ( )( dw ( θ ) d) ( dw ( θ ) d) i here e have defined he folloing procee (independen Bronian moion under Q - ee nex lide) dw dw Q Q ε ε ( θ ) ε dw d ε dw ( θ )d ε ε
iabiliie a a Numeraire Porfolio In he -marke he e of all meaure under hich dicouned price are maringale here he liabiliy porfolio i ued a numeraire i given by dq A Q; θ.. 0 0 dp ih ( 0 ) exp κ dw κ ()() κ d 0 0 In he -marke he rik premium are or marke price: or pure liabiliy rik (for ε 0): MM in he -Marke (Con.) ε ( 0 ) exp κ dw κ () 0 κ θ 0 d () θ κ ε
Objecive here he invemen policy i a (column) predicable proce vecor repreening allocaion o riky ae ih he remainder inveed in he rik-free ae Define by he funding raio proce i.e. ae a ime relaive o he liabiliie for an inveor folloing he raegy aring ih iniial ealh A 0 and given iniial liabiliy value 0 he Unconrained Problem Objecive A Max ε ε ε μ μ dw dw d r d r A A d d 0... n
he Unconrained Problem Soluion Opimal erminal funding raio value Value funcion ih [ ] η η A A g A J exp ε ε ε θ θ θ g
Opimal porfolio raegy he Unconrained Problem Soluion Con ( ) ( μ r ) ( ) We hu obain a o- (hree-)fund eparaion heorem he fir porfolio i he andard log-opimal efficien porfolio. he amoun inveed i inverely proporional o he inveor Arro-Pra coefficien of rik-averion. he econd porfolio i a liabiliy-hedging porfolio: i can be hon o have he highe correlaion ih he liabiliie; alernaively i i a porfolio ha minimize he local volailiy of he funding raio. ( )( ) ε
he Unconrained Problem Skech of he Proof Maringale Approach he opimizaion program read uch ha agrangian OC: A Max Q A A A A A η η λ η η A η λ η η A A η η () ()
he Unconrained Problem Skech of he Proof Con. rom () e obain Subiuing (A-3) ino (A-) e olve for λ hich e plug back ino (3) o ge We hen obain he indirec uiliy funcion and andard calculaion of expecaion of an exponenial of a Gauian variable give he announced reul η η λ η A (3) [ ] η η η η A A [ ] η η J
he Unconrained Problem Skech of he Proof Con. o obain opimal porfolio eigh conider Ue Io lemma o find he dynamic of hi funding raio proce and idenify he volailiy erm [ ] [ ] G Q η η η ε ε dw dw d d... ε ε κ dw dw d G dg...
he Unconrained Problem Dynamic Programming Uing he dynamic programming approach e check ha ih and φ a oluion o he non-linear Cauchy problem hich i eparable in and can be rien a: ih g a before. r ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ g 0 ϕ μ ϕ ϕ
he Unconrained Problem Welfare Gain from Marke Compleion In he pa fe year invemen bank have ared iuing dedicaed liabiliy-maching OC derivaive. hee dedicaed derivaive oluion allo for beer hedging of inveor liabiliie In pracice hey allo for quai-perfec hedging of financial rik bu no of oher rik (acuarial rik) unle a pecific re-inurance oluion i pu ino place We can eimae he increae in inveor elfare ha ould emanae from compleing he marke a J J ( )( ) κ exp ε ε ε complee incomplee hi meaure can be ued in an empirical exercie. ( )
he Conrained Soluion Minimum unding Requiremen (MR) We no add conrain on he funding raio in a complee marke eup. here can be o ype of conrain explici or implici. In a program ih explici conrain marginal indirec uiliy from ealh diconinuouly jump o infiniy: uch ha In a program ih implici conrain marginal uiliy goe moohly o infiniy a he MR: A Max a.. k A k A Max
he Conrained Soluion Implici Conrain Soluion (Complee Marke) Conider he folloing AM conrain problem Opimal erminal funding raio value Value funcion k k A J k k k A Max
Opimal porfolio raegy he Conrained Soluion Implici Conrain Soluion Con. k ( ) ( ) ( μ r ) ( ) We no have a ae-dependen a oppoed o aic allocaion o he o fund. he fir porfolio i he andard mean-variance efficien porfolio. Conider he fracion of ealh A allocaed o he opimal groh porfolio I i given by m ( ) ( μ r) ( ) ( μ r) ( ) ( μ r) A k ( ) ( μ r) A ( A k ) k
he Conrained Soluion Implici Conrain Inerpreaion I appear ha he fracion of ealh A allocaed o he opimal groh porfolio i equal o a conan muliple m of he cuhion i.e. he difference beeen he ae value and he floor defined a A -k. ( ) ( μ r) m hi i reminicen of CPPI raegie hich he preen eup exend o a relaive rik managemen conex. While CPPI raegie are deigned o preven final erminal ealh o fall belo a pecific hrehold exended CPPI raegie are deigned o proec he ae value from falling belo a pre-pecified fracion of ome benchmark value.
he Conrained Problem Skech of he Proof he agrangian read OC: ih λ from he budge conrain [ ] { } k λ k k λ λ λ k
he Conrained Problem o obain opimal porfolio eigh conider Skech of he Proof Con. G Q ( ) ( ( )) k ( ) ( )( ( )) k Ue Io lemma o find he dynamic of hi funding raio proce dg k dg G G κ and idenify he volailiy erm ih ( G k) κ dw (...) d dw (...)d d (... d ) dw
he Conrained Souion xplici Conrain Soluion (Complee Marke) Conider he folloing conrain AM problem uch ha Opimal erminal funding raio value Value funcion A Max a.. k A max k J J 0 max max k k k u
he Conrained Soluion xplici Conrain Soluion Con. Opimal porfolio raegy: ih: he fracion of ealh A allocaed o he opimal groh porfolio d kn r d kn μ d k N A r μ k d ) ln( κ κ κ κ
he Conrained Soluion xplici Conrain Inerpreaion Noe ha: A k max ( u 0) max( u k k A k 0) he replicaing porfolio for he payoff k coni in inveing in he liabiliy hedging porfolio hich i perfecly correlaed o he liabiliy porfolio in he complee marke eing; i iniial value i k 0. herefore he opimal raegy coni in allocaing he iniial ealh A 0 o a o inve k 0 in he liabiliy-replicaing porfolio and inve he remaining ealh A 0 k 0 in an opion ha ill deliver he urplu if any of he value of he unconrained payoff i.e. he payoff reuling from folloing he opimal raegy from he unconrained oluion. hi i reminicen of OBPI raegie hich he preen eup exend o a relaive rik managemen conex (exchange opion).
he Conrained Problem Skech of he Proof he agrangian read OC: hen [ ] { } { } k c λ λ c λ λ 0 max λ λ k c k k max max λ
he Conrained Problem o obain opimal porfolio eigh conider G Q ( ) Q k max ( ) Skech of he Proof Con. Price he opion componen and ue Io lemma o find he dynamic of hi funding raio proce and idenify he volailiy erm ih ( d ) ( ) dg kn dw (...)d G κ d (... d ) dw k0