NORMAL VARIANCE-MEAN MIXTURES (III) OPTION PRICING THROUGH STATE-PRICE DEFLATORS

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1 Joural of Mahmaial i: Adva ad Appliaio Volum 5 4 Pag 3-4 NORMAL VARIANCE-MEAN MIURE (III) OPION PRICIN HROUH AE-PRICE DEFLAOR WERNER HÜRLIMANN Wolr Klur Fiaial rvi fldra 69 CH-88 Zürih izrlad -mail: rr.hurlima@olrlur.om Ara A alraiv o h Bla-hol-Vai dflaor i propod. I i ad o a impl mulivaria poial ormal varia-ma miur Lév pro. Clod-form aalial igral formula for priig h Europa gomri a opio ih a dflad ormal varia-ma pri pro ar oaid. Appliaio o h varia-gamma h ormal ivr auia ad h ormal mprd al pro ar iludd. A dd Bla-hol formula ha a io aou h orrlaio ruur of h mar i alo drivd.. Iroduio h pr oriuio follo up h auhor ivigaio aroud h ormal varia-ma (NVM) miur modl irodud i Bardorff-Nil al. [6]. I i moivad h arh for o- auia mulivaria opio priig modl a a alraiv o h Mahmai u Claifiaio: 6P5 6P 9B4 9B5 9. Kord ad phra: ormal varia-ma miur Lév pro varia-gamma ormal ivr auia ormal mprd al a-pri dflaor mar pri of ri gomri a opio dd Bla-hol formula. Rivd Dmr iifi Adva Pulihr

2 4 WERNER HÜRLIMANN mulivaria Bla-hol modl. Our mphai i o rul ha a formulad for a poil larg la of NVM miur modl i h piri of Korholm [3]; Bigham ad Kil [7]; ad p ad a [5] amog ohr. h op of a-pri dflaor or ohai diou faor hih ha irodud Duffi [7] pp. 3 ad 97 i a ovi igrdi of gral fiaial priig rul. I oai iformaio aou h valuaio of pam i diffr a a diffr poi i im. h a-pri dflaor i a aural io of h oio of a pri ha r irodud arlir ad udid Arro [ 3 4 5]; Dru [4]; Ngihi [44]; ad Ro [48] a milo i h hior of a priig ( Dimo ad Muavia [6]). hough gral framor for drivig a-pri dflaor i (.g. Milr ad Pro [4]; Jala al. [3]; Mu [43]) hr ar o ma papr hih propo plii prio for hm ad hir orrpodig diriuio fuio. A hor aou of h o follo. io rall h mulivaria NVM miur modl i i impl Lév pro form. I graliz h oruio of h mulivaria poial varia-gamma pro oidrd i Co ad aov []; Luiao ad hou [35]; ad fir udid i Hürlima [4 5]. horial ad aiial uifiaio for i u ar rifl miod. I io 3 driv h mulivaria NVM dflaor a alraiv o h mulivaria Bla-hol-Vai (BV) dflaor irodud i Hürlima [] ( alo Hürlima [3 8]). Aalial igral formula for priig h Europa gomri a all opio ihi h mulivaria NVM mar ih NVM dflaor ar drivd ad diud i io 4. h fial io 5 irodu a mulivaria uordiad a pri modl ho uivaria vrio oiid ih h modl Hur al. []. A a pial a i ilud a dd Bla-hol formula ha a io aou h orrlaio ruur of h mar. h pr ovl approah i gral ad impl. From a aiial prpiv paramr imaio a do ih a mulivaria mom mhod ( Hürlima [7]). Morovr h drivaio of h dd Bla-hol formula ha a lmar proailii flavour.

3 NORMAL VARIANCE-MEAN MIURE (III) 5. A lio of Mulivaria Normal Varia-Ma Miur Pro h uivaria ormal varia-ma (NVM) miur pro i dfid a a drifd Broia moio im hagd a idpd miig pro. Vid from h iiial im i i dfid hr θ W > > < θ < (.) W i a adard Wir pro ad i a idpd uordiaor ha i a iraig poiiv Lév pro. i i a Lév pro h dami of h pro i drmid i diriuio a ui im. h radom varial follo a diriuio ho umula graig fuio (gf) C ( u) l E [ p( u )] i aumd o i ovr om op irval. B oidrig i paralll diffr uordiaor a uifid approah o vral impora NVM miur modl i poil. Our fou i rrid o hr of hm (for furhr poil hoi oul Hürlima [6] io 4). (a) Varia-gamma (V) pro h uordiaor ~ ( Γ ) i a gamma pro ih ui ma ra ad varia ra. h radom varial ~ V( θ ) follo a hr paramr V diriuio ih gf: C ( u) l{ ( θu u )} > < θ <. C (.) hi formula i oaid from h gf ( u) l( u) of h gamma radom varial odiioig uig ha ~ N( θ ) i ormall diriud. h irm of h pro follo a V < diriuio aml ~ V( θ ).

4 6 WERNER HÜRLIMANN () Normal ivr auia (NI) pro h gf of h ivr auia uordiaor i ( u) C { u } >. h radom varial ~ NI( θ ) follo a hr paramr NI diriuio ih gf: C ( u) { ( θu u )} > < θ <. h irm of h pro ar NI( θ ) diriud. () Normal mprd al (N) pro (.3) h uordiaor follo a laial mprd al pro ih α α gf ( u) α { ( u ) } > α ( ). h radom C varial ~ N ( θ α ) follo a four paramr N diriuio ih gf: C α α α α ( u) α { ( ( θu u )) } > α ( ) < θ <. (.4) α h irm of h pro ar N( θ α ) diriud. h pial a α i h NI pro. imilarl o ho mulivaria vrio miod i Hürlima [4] for h V pro diffr mulivaria modl for ah NVM miur pro a dfid. For implii oidr hr ol mulivaria Lév pro ih NVM ompo of h p hr h E θ W (.5) W ar orrlad adard Wir pro uh ha () i ( [ dw dw ) ] d. Dpi all i horomig h u of h modl i

5 NORMAL VARIANCE-MEAN MIURE (III) 7 (.5) i uifid horiall looig a h varia of i NVM margi a ui im aml Var [ ( ) ] Var[ ] θ. (.6) Eah varia dompo io a idiorai ompo ha i ariud o h Broia moio ad a ogou ompo Var[ ] θ ha i du o h im hag of h Broia moio. h paramr θ govr h pour of h margi o h gloal mar urai maurd h ommo varia Var[ ]. imilarl o o ha h ad uroi ar alo affd h igl margial ig ad h ommo uordiad paramr() ( Hürlima [7] horm.). O h ohr had h aiial mom mhod dvlopd i Hürlima [7] ad i uful appliaio o ral-orld daa uifi i u i praial or. For h rao i i lgiima o fou fir o h modl (.5). O o ha h gf of h NVM miur modl i giv C ( u) C ( θ u u u) θ ( θ θ ) ( ii ) u ( u u ). (.7) h oi gf of h oidrd mulivaria NVM pro a prd i lod-form. Propoiio.. h oi gf of h mulivaria NVM miur () ( pro ( ) ) ih paramr θ ( θ θ ) ( ) ad uordiaor i drmid a follo: Mulivaria V pro i i gf h mulivaria V radom vor ~ V( θ ) ha oi C ( u) l{ ( θ u u u)} u ( u u ). (.8)

6 8 WERNER HÜRLIMANN Mulivaria NI pro gf h mulivaria NI radom vor ~ NI( θ ) ha oi C ( u) { ( θ u u u)} u ( u u ). (.9) Mulivaria N pro α h mulivaria N radom vor ~ N( θ α ) ha oi gf C α α α ( u) α { ( ( θ u u u)) } u ( u u ). (.) Proof. hi follo from (.7) ad h plii form for h gf of h uordiaor. 3. Uifid a-pri Dflaor Rpraio for h Epoial NVM Miur Pro I h folloig implif ad graliz h produr i Hürlima [4] io 4. Coidr h folloig la of a priig modl. iv h urr pri of ri a a iiial im hir fuur pri a im > ar drid poial NVM miur pro ( ( ) p µ ω ) (3.) hr µ rpr h ma logarihmi ra of rur of h -h ( ri a pr im ui ad h radom vor ( ) ) follo a mulivaria NVM miur pro. Uig h dfiig rlaiohip [ ( ) ] E p( µ ) a ui im o ha ω C ( )() < hr o aum ha h gf of

7 NORMAL VARIANCE-MEAN MIURE (III) 9 i ovr om op irval hih oai o. uppo ha h mulivaria NVM dflaor of dimio ha h am form a h pri pro i (3.). For om paramr α ad vor ( ) (oh o drmid) o for i (a Ehr raformd maur) D p ( α ) >. (3.) A impl gf alulaio ho ha h a-pri dflaor marigal odiio E r [ ] [ ( ) ] D E D > (3.3) ar quival ih h m of quaio i h uo α ω (u ha i a Lév pro h C ( u) C ( u)): ( ( r α C µ ω α ) C ) ( ( ) ( ) ) δ. (3.4) Irig h fir quaio io h od o ild h ar rlaiohip ( ( µ r ω C ) ) C ( ). (3.5) i h m (3.4) ha dgr of frdom h uo ho ariraril a ω a ω µ r C (3.6) hih i irprd a h (im-idpd) NVM mar pri of h -h ri a. Wih h mad rriio o h gf hi valu i ala fii. Ird io (3.5) ho ha h paramr vor i drmid h quaio C ( ( ) ) C ( ). (3.7)

8 WERNER HÜRLIMANN W ar rad o ho h folloig uifid NVM dflaor rpraio: horm 3. (Mulivaria NVM dflaor). iv i a ri-fr a ih oa rur r ad ri a ih ral-orld pri (3.) hr o aum ha h gf of i ovr om op irval hih oai o. h h mulivaria NVM dflaor of h poial NVM miur pro i drmid D p( α ( ) ) α r C ( ) θ ( γ ) γ. (3.8) Morovr i h uivaria a o ha γ. Proof. h fir quaio i (3.4) ild α. i ar Lév pro o ha C ( u) C ( u) C C ( u) h (.7) i quival ih h quaio C ( u) C ( θ u u u). I follo ha h odiio (3.7) ar quival ih h quaio: θ θ. A raighforard alulaio ho ha h lar i quival ih h ad odiio for hr i a o ha γ (mp um). 4. Priig omri Ba Opio for h Epoial NVM Miur Pro I h liraur o diiguih o p of a opio. h arihmi a opio i dfid o h ighd arihmi avrag of a pri uh ha

9 NORMAL VARIANCE-MEAN MIURE (III) (4.) hr h igh ( ) a gaiv ad i hi iuaio i ilud prad opio. h gomri a opio i dfid o h ighd gomri avrag of a pri ( [ ) ] >. (4.) i diriuio fuio of ighd um of orrlad a pri a uuall o ri i plii lod form h priig of arihmi a opio i rahr hallgig. Diffr ad mol approima mhod o pri hm hav dvlopd o far ma auhor iludig urull ad Wama [5]; Milv ad Por [4]; Krl al. [34]; Carmoa ad Durrlma []; Borova al. [8]; Wu al. [53]; Varamaa ad Aladr [5]; Aladr ad Varamaa []; ad Brigo al. [9]. h priig of h gomri a opio i mor raighforard. o illura h uful i opio priig of h mulivaria NVM dflaor rri h aio o h priig of gomri a opio. W oai a gral aalial NVM priig formula hih a vid a a gralizaio of h Bla-hol formula. Morovr plii aalial formula ar diplad for h poial V ad NI pri pro. I pariular a implr alraiv o h NI lod-form formula Wu al. [53] i drivd. Coidr a Europa gomri a all opio ih mauri da ad ri pri K i h mulivaria NVM mar ih ri a ha follo h pri pro (3.) ad i u o h NVM dflaor (3.8). I pri a iiial im i giv C E[ D ( K ) ]. (4.3)

10 WERNER HÜRLIMANN A raighforard alulaio hih a io aou h ormalizig odiio r µ ω ho ha { } p C D { }. p r C K K D (4.4) I h folloig uppo ha h di fuio f of h miig radom varial i. hrough odiioig o a rri (4.3) a d f C C C ih [( { ( )} W E C θ p p{ ( )}) ]. W K r θ (4.5) Eah of h o odiioal orrlad ormall diriud um i (4.5) i ormall diriud ad hir oi diriuio i ivaria ormal. hrfor h diriuio of h odiioal radom oupl ( ) i drmid h odiioal ma [ ] m m E θ [ ] m m E θ (4.6) h odiioal varia

11 NORMAL VARIANCE-MEAN MIURE (III) 3 [ ] Var i i i i i [ ] Var i i i i (4.7) ad h odiioal ovaria [ ] Cov. i i i i i (4.8) No l Φ do h adard ormal diriuio Φ Φ i urvival fuio ad Φ ϕ i di. h ivaria adard ormal di i dfid ad dod. p ; π ϕ From (4.5) ad h dfiiio (4.6)-(4.8) o oai. ; dd K C m r m ϕ (4.9) h prio i h ra of (4.9) i o-gaiv providd ih. l m m r K i ( ) ; ϕ ϕ ϕ a paraio of h doul igral ild d J C ϕ ih h ir igral

12 WERNER HÜRLIMANN 4 { } m r m K J ( ). d ϕ (4.) A raighforard appliaio of Lmma A i h Appdi ild ( ) J m Φ ( ). Φ K m r o implif oaio rri h argum ihi h ormal diriuio fuio a ih a ( ) l m m K r a. l m m K r Furhrmor o ha. ϕ ϕ ϕ ϕ No uig i h Lmma A of h Appdi o oai. K a C m r m Φ Φ Bad o h aov prio for h offii ad a o oai furhr

13 NORMAL VARIANCE-MEAN MIURE (III) 5 C ( m ) Φ r ( l( K ) m Ω m Ω Ω ) r ( m ) K Φ r ( l( K ) m Ω m Ω Ω ) Ω Ω Ω Ω. (4.) ummarizig hav ho h folloig mai rul: horm 4. (omri a all opio formula i h mulivaria NVM mar). iv i h mulivaria poial NVM miur pro (3.) u o h NVM dflaor (3.8). h i h aov oaio o ha C C E[ D ( K ) ] r { Ψ ( a d ) K Ψ ( a )} d (4.) Ψ a ( a d ) Φ( d ) f d a m m m ( ) Ω Ω r l( K ) d. (4.3) Ω A pariular ia for hih h formula (4.) implifi oidral i h pial a d ha i h ri pri i qual o K. I hi iuaio (4.) rri a r C r C E[ D ( ) ] a { Φ( ) a Φ( )} f d. (4.4)

14 6 WERNER HÜRLIMANN A hi poi a impora oio ih h adard o-arirag framor of mahmaial fia mu miod (.g. Wührih al. [54] uio.5; ad Wührih ad Mrz [55] Chapr ). B h fudamal horm of a priig h aumpio of o-arirag (a form of ffii mar hpohi) i quival ih h i of a quival marigal maur for dflad pri pro. I ompl mar h quival marigal maur i uiqu prf rpliaio of oig laim hold ad raighforard priig appli. I iompl mar a oomi modl i rquird o did upo hih quival marigal maur i appropria. No l P do h ral-orld maur ad P do a quival marigal maur. h o a ihr or udr P hr h pri pro ar dflad ih a a-pri dflaor. Alraivl o a or udr P diouig h pri pro ih h a aou umrair. Worig ih fiaial irum ol o of or udr P. Bu if addiioall iura liailii ar oidrd o or udr P ( Wührih al. [54] Rmar.3). A r o-rivial ampl i priig of h guarad maimum iflaio dah fi (MIDB) opio (Hürlima []; Equaio (5.4) i Hürlima [3]). horm 4. dmora h praiaili of h a-pri dflaor approah for poial NVM pri pro a applid o h Europa gomri a all opio. I a hr h di fuio of h miig radom varial i o h formula (4.)-(4.4) ild a lod-form aalial priig m for h gomri a opio. h odiio udr hih h mulivaria NVM mar i ompl ad arirag-fr ha i hr i a uiqu quival marigal maur ad pri ar uiqul dfid (hhr udr P or udr P ih a-pri dflaor) rmai o foud. hi i a o-rivial prolm ha ha ald o far oll for h mulivaria Bla-hol modl ( Dha al. [5]).

15 NORMAL VARIANCE-MEAN MIURE (III) 7 Eampl 4.. Mulivaria poial V a priig modl. h uordiaor ~ Γ( γ γ) γ i gamma diriud ih di γ γ f γ ( γ) Γ( γ). (4.5) Maig h hag of varial z ( a) h Ψ -fuio i (4.3) i of h form ad i Koz al. [33] p. 96 (Europa ri-ural all opio pri for a poial V pri pro) hih i h origial lod form formula Mada al. [37] horm ad Appdi ( alo Mada [36] uio 6.3. Equaio (4)). h advaag of h alraiv priig formula (4.) i h impl paramr ruur (4.3). Alo h pial a (4.4) implifi oidral. A imilar aalial priig m for h Margra opio ha oidrd prvioul i Hürlima [4 5]. I pariular for a Erlag uordiaor ih igr paramr γ formula of h p (4.4) rdu o fii um prio ( Hürlima [4] Appdi ). A a rul driv a lod-form prio for h Ψ -fuio a a um of a iompl a fuio ad a igrad Madoald fuio. I i muh implr ha h origial formula i rm of h Madoald fuio (modifid Bl fuio of h od id) ad h dgra hpr-gomri fuio of o varial. I h folloig l a ( γ) ~ Γ( γ ) a adardizd gamma γ radom varial ih di fa ( γ ) Γ( γ) B( γ ) a a ih diriuio fuio γ F ( ) d (4.6) B γ B( γ )

16 8 WERNER HÜRLIMANN ad V ( γ α ) a varia-gamma ih di (.g. Hürlima [9] Equaio (A4.)) f V γ γ ( γ α ) ( α) πγ( γ) α p α K ( ( α ) ) γ. (4.7) horm 4. (Clod-form V Ψ -fuio rpraio). h ormal gamma mid igral Ψ( a γ) Φ( a ) f d aifi h folloig proailii rpraio: Ψ( a γ) { g( ) F a B ( )} γ a λ g ( a) f ( ) d. (4.8) V γ g a g a Proof. hi i ho i h Appdi. I pariular h V formula (4.4) a rri i rm of h iompl a fuio. Eampl 4.. Mulivaria poial NI a priig modl. h uordiaor ~ I( ) i ivr auia diriud ih di f p{ π 3 ( ) }. (4.9) Priig of h gomri a opio udr a quival marigal maur ha prvioul oidrd i Wu al. [53]. Hovr hir fa Fourir raform priig formula i mor ompl ha h lodform prio (4.) lo.

17 NORMAL VARIANCE-MEAN MIURE (III) 9 I h folloig l NRI ( α δ) a mmri NVM miur radom varial ih riproal ivr auia miig di αδ f NRI ( α δ) α K ( α δ ). (4.) π I i oaid from h gralizd hproli diriuio a NRI ( α δ) H( α δ). h di of h ormal ivr auia NI ( α δ) H( α δ) i dod δ α K ( α δ ) fni α δ αδ (4.) π δ horm 4.3 (Clod-form NI Ψ -fuio rpraio). h ormal gamma mid igral Ψ( a γ) Φ( a ) f d aifi h folloig proailii rpraio: ( a Ψ γ) g( ) F NRI ( ) g ( a) F ( ) ( a ). (4.) NI g a A io of h proailii rpraio (4.8) ad (4.) o arirar gralizd ivr auia miig dii i foud i Hürlima [3]. Eampl 4.3. Mulivaria poial N a priig modl. h laial mprd al uordiaor ha o a raal di fuio u a rlaivl impl hararii fuio. I hi iuaio o approima h igral (4.3)-(4.4) u of h fa Fourir raform (FF) of f (.g. Hürlima [7] Appdi ).

18 3 WERNER HÜRLIMANN 5. A impl Mulivaria uordiad A Pri Modl h pial hoi of h mulivaria NVM dflaor (3.8) impli ha C ( ) α r θ ( γ ). I follo ha h NVM dflaor dgra o h ri-fr diou faor D. I hi mulivaria uordiad mar h ri a follo h pri pro (ir (3.6) io (3.)) ( ( ) p r γ W ) (5.) hr h [ () i ( dw ) dw ] i r W ar orrlad adard Wir pro uh ha E ad i a idpd uordiaor. h priig of h Europa gomri a all opio ih mauri da ad ri pri K implifi omha. horm 5. (omri a all opio formula i h mulivaria uordiad mar). iv i h mulivaria uordiad a pri modl (5.) u o h ri-fr diou faor D. h o ha r C r E[ D ( K ) ] Ψ ( m d) K ( m ) (5.) Ψ d Ψ ( ) m ( m d) Φ d ( m ) ( ) f d Ψ d m ( m d) Φ f d (5.3)

19 NORMAL VARIANCE-MEAN MIURE (III) 3 l. r K d m i i i i Proof. hi rul i a pial ia of horm 4. ad a imilarl drivd i a implr a (o-dimioal igraio ol). h pializaio o a gomri a ih a igl ri a i iruiv. Corollar 5. (Europa a-pri dflad all opio pri for a igl uordiad a). iv i h mulivaria uordiad a pri modl (5.) u o h ri-fr a-pri dflaor. r D h o ha [ ( () ) ] () ( { } ) ( { } ) d K d K D E C r Ψ Ψ (5.4) ( { } ) ( ) d f d d Φ Ψ ( { } ) ( ) d f d d Φ Ψ. l r K d (5.5) Proof. o ha m ad h rul follo immdial hrough irio io h formula (5.). Corollar 5. i rlad o om rmaral prviou rul. uppo ha i h igl a i uorrlad ih h

20 3 WERNER HÜRLIMANN ohr ri a h. h o rovr h opio formula Hur al. [] for h uivaria uordiad a pri modl ( alo Rahv ad Mii [47]; ad Rahv al. [46] uio 7.6 ih orrio of h mipri i h o pri modl hovr). If ih proaili o (Bla-hol-Mro modl) h (5.4) ild a dd Bla-hol formula ha a io aou h orrlaio ruur of h mar. Bid h mulivaria poial V NI ad N pro h propod mulivaria uordiaor mar oai a rih la of fail modl. If i h al pro propod Madlro [39 4] ad Fama [8] o ha a logal modl. If i h CIR pro Co al. [3] o oai a mulivaria vrio of h modl Ho [9] ( Rahv al. [46] uio 7.6). h CMY ad Mir pifiaio Carr al. [] rp. hou ad ugl [49]; ad Pima ad Yor [45] ar alo uordiad modl a ho Mada ad Yor [38]. om diffr ih h approah Hur al. [] a od. Whil h auhor or udr a quival marigal maur P i h uivaria a ol or i h mulivaria a udr P uig a a-pri dflaor. Hur al. argu i io 5 ha a quival marigal maur i o pd o i ul o aum ha µ r. I hi a hir (quival marigal maur) mar pri of ri rad (Equaio (5.)) ω ( θ ). (5.6) h aumpio µ r orrpod o h hoi µ ω r i h dflaor approah. B viru of (3.6) h (a-pri dflaor) mar pri of ri qual () C ( θ ) C ( ( )). ω µ r C (5.7)

21 NORMAL VARIANCE-MEAN MIURE (III) 33 Udr h mad aumpio oh mar pri of ri ar qual ad vaih if ad ol if o ha ad i hi iuaio h formula (5.4) oiid ih h Hur-Pla-Rahv formula. Our mulivaria uordiad a pri modl i a gui io au i ovr a ih o-vaihig mar pri of ri for hih µ r. For hi h odiio i fiaial mar i rahr h rul ha h pio. hrfor h approah i of primordial impora. i h quival marigal approah ad h a-pri dflaor approah ar quival (omm afr horm 4.) hr i o d o dipla h maur P i h mulivaria a. Rfr [] C. Aladr ad A. Varamaa Aalial approimaio for muli-a opio priig Mahmaial Fia (4) () [] K. Arro A Eio of h Bai horm of Claial Wlfar Eoomi I: Prodig of h d Brl mpoium o Mahmaial aii ad Proaili pp Uivri of Califoria Pr 95. [3] K. Arro L rol d valur ourièr pour la répariio la millur d riqu Eoomria 4 (953) 4-47 (Eglih ralaio Arro (964)). [4] K. Arro h rol of urii i h opimal alloaio of ri-arig Rvi of Eoomi udi 3 (964) [5] K. Arro Ea i h hor of Ri-Barig Norh-Hollad 97. [6] O. E. Bardorff-Nil J. K ad M. or Normal varia-ma miur ad z diriuio I. ai. Rvi 5 (98) [7] N. H. Bigham ad R. Kil mi-paramri modllig i fia: horial foudaio Quaiaiv Fia () -. [8]. Borova F. J. Prmaa ad H. V. D. Wid A lod form approah o h valuaio ad hdgig of a ad prad opio h Joural of Drivaiv 4(4) (7) 8-4. [9] D. Brigo F. Rapiarda ad A. ridi h arirag-fr mulivaria miur dami modl: Coi igl-a ad id volaili mil ariv:3.7v [q-fi.pr] (3).

22 34 WERNER HÜRLIMANN [] R. Carmoa ad V. Durrlma ralizig h Bla-hol formula o mulivaria oig laim h Joural of Compuaioal Fia 9() (6) [] P. Carr H. ma D. B. Mada ad M. Yor h fi ruur of a rur: A mpirial ivigaio J. of Bui 75() () [] R. Co ad P. aov Fiaial Modllig ih Jump Pro CRC Fiaial Mahmai ri Chapma ad Hall 4. [3] J. C. Co J. E. Igroll ad. A. Ro A hor of h rm ruur of ir ra Eoomria 53 (985) [4]. Dru Valuaio quilirium ad Paro opimum Prodig of h Naioal Aadm of i 4 (954) [5] J. Dha A. Kuuh ad D. Lidr h mulivaria Bla & hol mar: Codiio for ompl ad o-arirag hor Proa. Mah. ai. 88 (3) -4. [6] E. Dimo ad M. Muavia hr uri of a priig J. of Baig ad Fia 3 (999) [7] D. Duffi Dami A Priig hor Prio Uivri Pr N Jr 99. [8] E. Fama Madlro ad h al Paria hpohi J. of Bui 36(4) (963) [9]. L. Ho A lod form oluio for opio ih ohai volaili ih appliaio o od ad urr opio Rvi of Fiaial udi 6 (993) []. R. Hur E. Pla ad.. Rahv Opio priig for a log-al a pri modl Mahmaial ad Compur Modllig 9 (999) 5-9. [] W. Hürlima A io of h Bla-hol ad Margra formula o a mulipl ri oom Applid Mahmai (4) (a) [] W. Hürlima Aalial priig of a iura mddd opio: Alraiv formula ad auia approimaio J. of Iformai ad Mah. i 3() () [3] W. Hürlima h algra of opio priig: hor ad appliaio I. J. of Algra ad aii () () [4] W. Hürlima Margra formula for a impl ivaria poial variagamma pri pro (I): hor I. J. iifi Iovaiv Mah. Rarh () (3a) -6. [5] W. Hürlima Margra formula for a impl ivaria poial variagamma pri pro (II): aiial imaio ad appliaio I. J. iifi Iovaiv Mah. Rarh () (3)

23 NORMAL VARIANCE-MEAN MIURE (III) 35 [6] W. Hürlima Normal varia-ma miur (I): A iquali ad uroi Adva i Iqualii ad Appliaio (i pr) 4 (3). URL:hp://i.org/id.php/aia/aril/vi/7 [7] W. Hürlima Normal varia-ma miur (II): A mulivaria mom mhod J. of Mahmaial ad Compuaioal i (i pr) (3d). [8] W. Hürlima Opio priig i h mulivaria Bla-hol mar ih Vai ir ra Mahmaial Fia Lr () (3) -8. [9] W. Hürlima Porfolio raig ffii (I): Normal varia gamma rur Iraioal J. of Mah. Arhiv 4(5) (3f) 9-8. [3] W. Hürlima Proailii rpraio of a ormal gralizd ivr auia igral: Appliaio o opio priig J. of Adva i Mahmai 4() (3g) [3] M. Jala M. Yor ad M. Ch Mahmaial Mhod for Fiaial Mar prigr Fia prigr-vrlag Lodo Limid 9. [3] L. Korholm h miparamri ormal varia-ma miur modl adiavia Joural of aii 7 () -67. [33]. Koz. J. Kozuoi ad K. Podgori h Lapla Diriuio ad ralizaio: A Rvii ih Appliaio Birhäur Boo. [34] M. Krl J. d Ko R. Kor ad.-k. Ma A aali of priig mhod for a opio Willmo Magazi (4) [35] E. Luiao ad W. hou A mulivaria ump-driv fiaial a modl Quaiaiv Fia 6(5) (6) [36] D. Mada Purl Dioiuou A Priig Pro I: E. Jouii J. Cviai ad M. Muila (Ed.) Opio Priig Ir Ra ad Ri Maagm 5-53 Camridg Uivri Pr Camridg. [37] D. Mada P. Carr ad E. Chag h varia gamma pro ad opio priig Europa Fia Rvi (998) [38] D. Mada ad M. Yor Rprig h CMY ad Mir Lév pro a im hagd Broia moio h Joural of Compuaioal Fia () (8) [39] B. Madlro h variaio of rai pulaiv pri J. of Bui 36 (963) [4] B. Madlro h variaio of om ohr pulaiv pri J. of Bui 4 (967) [4] M. A. Milv ad. E. Por A lod-form approimaio for valuig a opio h Joural of Drivaiv 5(4) (998) [4] K. R. Milr ad. A. Pro Priig ra of rur guara i a Hah- Jarro-Moro framor Iura: Mahmai ad Eoomi 5(3) (999)

24 36 WERNER HÜRLIMANN [43] C. Mu Fiaial A Priig hor Oford Uivri Pr Oford 3. [44]. Ngihi Wlfar oomi ad i of a quilirium for a ompiiv oom Mrooomria (96) [45] J. Pima ad M. Yor Ifiil diviil la aoiad ih hproli fuio Caadia Joural of Mahmai 55() (3) [46].. Rahv Y.. Kim M. L. Biahi ad F. J. Faozzi Fiaial Modl ih Lév Pro ad Volaili Clurig J. Wil & o I. Hoo N Jr. [47].. Rahv ad. Mii al Paria Modl i Fia J. Wil & o NY. [48]. A. Ro A impl approah o h valuaio of ri ram J. of Bui 5 (978) [49] W. hou ad J. L. ugl Lév pro polomial ad marigal Commuiaio i aii-ohai Modl 4 (998) [5] A. p ad E. a d ormal varia-ma modl for a priig ad h mhod of mom Iraioal aiial Rvi 74() (6) 9-6. [5]. M. urull ad L. M. Wama A qui algorihm for priig Europa avrag opio h Joural of Fiaial ad Quaiaiv Aali 6(3) (99) [5] A. Varamaa ad C. Aladr Clod form approimaio for prad opio Applid Mahmaial Fia ifir () -6. [53] Y.-C. Wu.-L. Liao ad.-d. hu Clod-form valuaio of a opio uig a mulivaria ormal ivr auia modl Iura: Mah. Eoom. 44() (9) 95-. [54] M. V. Wührih H. Bühlma ad H. Furrr Mar-Coi Auarial Valuaio EAA Lur No Vol. (d Rvid ad Elargd Ediio) prigr- Vrlag. [55] M. V. Wührih ad M. Mrz Fiaial Modlig Auarial Valuaio ad olv i Iura prigr Fia 3. Appdi: Igral Idii of Normal p ad Proailii Ψ -Fuio Rpraio g h ruial idii ud i h drivaio of horm 4. ar ad ad provd paral.

25 NORMAL VARIANCE-MEAN MIURE (III) 37 Lmma A. For a ral umr µ ad σ > o ha h idi σ µ σ ( ) µ ϕ µ σ d Φ σ. σ (A.) Proof. Coidr fir h a µ σ. From h rlaio ϕ ϕ( ) o g ϕ d ϕ() d Φ ( ). Uig hi o oai a hag of varial σ ϕ( ( µ ) σ) d µ σ ϕ ( µ ) σ µ σ () µ d Φ σ. σ Lmma A. For a ral umr a µ ad σ > o ha h idi a σ ( ) a µ Φ a ϕ µ σ d Φ. (A.) σ Proof. Coidr h fuio F ( z) Φ( z ) ϕ d ( z) Φ( a z) ϕ d. ad F ( z) ϕ( z ) ϕ d ϕ( z ) a O o ha F ( ) Φ ϕ d from hih i follo ha a F ( a) F F ( z) dz Φ. O h ohr had o ha a a z () F ( a) Φ ad a ( z) z ( a z) d ϕ ϕ ( z ) 3

26 38 WERNER HÜRLIMANN a a ϕ h a ( ) a ( z) dz. Φ I follo z ha ( ) a µ σ Φ a ϕ µ σ d µ σ Φ. σ a Proof of horm 4.. A rpad appliaio of h mhod ud i Lmma A ill do. I h pial a a J ( z) Ψ( z γ) λ Φ( z ) fa d z. O ha J J( ) J J ( z) dz. A alulaio ho ha J ( z ) ϕ( z ) f ( ) d a λ ( ) ( γ z ) d πγ( γ) B( γ ) ( z ) γ hr h la prio follo oig ha h igrad i rlad o a gamma di. Wih h hag of varial z u o ha du J( ) B( γ ) ( u ) γ. A furhr hag of varial u ( ) ho ha du γ ( u ) ( ) γ ( ) d. Wih (4.6) hi impli (4.8) for a. h formula for a < follo oig ha Φ ( ) Φ( ) Φ( ) h

27 NORMAL VARIANCE-MEAN MIURE (III) 39 J( ) J( ). I gral if a I z Φ( z ) fa( λ) ( ) d z. O ha I J( ) I( a) J( ) I ( z) dz ad I ( z) ϕ( z ) fa( λ)d a z πγ( γ) ( γ ) ( ) z d. h igrad i rlad o a gralizd ivr auia di of h form f I ( α p) p ( α ) K ( α ) p p ( α ) p γ α ( ) z. ( ) o ha I ( z) π z γ z K Γ( γ) γ γ ( z) γ α A ompario ih (4.7) ho ha I ( z) fv ( z) α α. h formula (4.8) for a follo. If a < o o ha Φ( a ) Φ( a ) Φ( a ) h Ψ( a γ) Ψ( a γ).

28 4 WERNER HÜRLIMANN Proof of horm 4.3. If a J( z) Ψ( z γ) Φ( z ) d z. O ha J( ) J ( z) dz. A alulaio ho ha f J z ϕ z f d p( {( z ) }) d. π h igrad i rlad o a harmoi la H ( α ) I( α ) ih di ( α ) f ( ) H α α z K( α ) uh ha J ( z) K( z ). π Comparig ih (4.) ho (4.) for a. h formula for a < follo from h rlaio J( ) J( ). If a h o a ha I( a) J( ) I ( z)dz ih I( z) Φ( z ) f d z. hrough alulaio o oai I ( z) ϕ( z ) f d z p( {( ) ( z ) }) d π

29 NORMAL VARIANCE-MEAN MIURE (III) 4 ho igrad i rlad o h di of a gralizd ivr auia p I α ih paramr. z p α I follo ha. z z K z I z π Comparig ih (4.) ho ha z f z I NI ad (4.) for a i ho. h formula for < a i oaid from h rlaiohip. γ Ψ γ Ψ a a