An approximate approach to the exponential utility indifference valuation

Transcription

1 A approximae approach o he expoeial uiliy idifferece valuaio akuji Arai Faculy of Ecoomics, Keio Uiversiy, Mia, Miao-ku, okyo, , Japa arai@ecokeioacjp) Absrac We propose, i his paper, a ew valuaio mehod for a coige claim, which approximaes o he expoeial uiliy idifferece valuaio I paricular, we rea boh ask ad bid valuaios I he defiiio of he expoeial uiliy idifferece valuaio, we require a srog codiio relaed o he uderlyig coige claim he ew valuaio i his paper succeeds i reducig his codiio by usig a kid of power fucios isead of he expoeial fucio Furhermore, we shall ivesigae some basic properies ad a asympoic behavior of our ew valuaio Keywords: Icomplee markes, Idifferece value, p-opimal marigale measure, Reverse Hölder iequaliy 1 Iroducio Our aim of his paper is o obai a approximae approach o he expoeial uiliy idifferece valuaio EUIV, for shor) by usig a kid of power fucios I mahemaical fiace, he problem of valuaio for a coige claim i a icomplee marke is very impora Recely, may researchers have sudied he uiliy idifferece valuaio mehod, of which he defiiio is give by as follows: We sar wih a icomplee marke wih he mauriy >, whose asse flucuaio is described by a semimarigale X Moreover, we cosider a ivesor havig iiial capial x a ime, ad who ieds o sell a coige claim Le U be his/her uiliy fucio I oher words, U is a R-valued coiuous icreasig cocave fucio defied o R We defie a adaped process C ) by esssup ϑ Θ E Ux + G, ϑ)) F = esssup ϑ Θ E Ux + C )+G, ϑ) ) F, 11) 1

2 where G, ϑ) := ϑ s dx s ad Θ a suiable se of predicable processes, represes he se of all self-fiacig sraegies he, we call C ) he uiliy idifferece valuaio, which is oe of cadidaes for he askig price of he coige claim I addiio, he valuaio C ) srogly depeds o he preferece of he ivesor who ieds o sell he lef had side of 11) is he expeced uiliy maximizaio problem whe he/she does o sell he coige claim O he oher had, he righ had side is he case where he/she sells for he price C ) a ime ad agrees o pay a he mauriy I paricular, here has bee much lieraure o he expoeial uiliy case, ha is, he case where U is give by Ux) = e x, for > See echerer 24), Frielli 2), Rouge ad El Karoui 2), Musiela ad Zariphopoulou 24a, 24b), Youg 24), ad so o esides, Maia ad Schweizer 25) MS, for shor) have provided he dyamics for he case where he asse price process is give by a coiuous semimarigale Remark ha we call C ) he expoeial uiliy idifferece valuaio EUIV), if U is he expoeial uiliy fucio O he oher had, whe we defie he EUIV, we eed o assume he followig srog codiio wih respec o he uderlyig coige claim: E e < 12) For example, i he case where is a Europea call opio ad X is give by a geomeric rowia moio, 12) does o hold, because, roughly speakig, he disribuio of is ear o oe of e Y, where Y is a ormal radom variable Hece, models saisfyig he codiio 12) do o iclude some ypical impora oes as he above example A his, we ry o reduce he codiio 12) o, for a sufficie large N, E <, 13) equivalely Ee Y < Now, we recall he defiiio of e as follows: e x = lim x, ) he, for ay sufficie large, we ca say ha x is ear o e ) x If we deoe, for a sufficie large umber, Ux) = x ) or 1 x he we ca approximae he EUIV uder he codiio 13) Remark ha his fucio U is o a uiliy fucio exacly, sice o cocave Alhough, for x < /, U is cocave, so ha we ca say ha U is almos cocave Isead of he expoeial uiliy, if we adop he fucio U as he uderlyig uiliy fucio, he we may obai a approximae approach o he EUIV O he oher had, i is difficul for us o rea U direcly herefore, we ry o decompose he ), 2

3 value x + C ) +G, ϑ) io he F -measurable par x + C ), he gai process par G, ϑ) ad he coige claim par hus, isead of U, we cosider, for > ad N, U, x, y, z) := x ) 1 y ) +1 z Noe ha, if is sufficie large, he U, x, y, z) is very ear o ) x + y z) or 1 ) x + y z) O he oher had, if we deoe U,exp x, y, z) := exp x + y z)), he he EUIV, deoed by C,exp ), saisfies he followig: ) esssup ϑ Θ E U,exp x,g, ϑ), ) F = esssup ϑ Θ E U,exp x + C,exp ),G, ϑ),) F Remark ha C,exp ) does o deped o he iiial capial x hus, by he same way as he EUIV, we defie a adaped process C, ) as a process saisfyig esssup ϑ Θ E U, x,g, ϑ), ) F = esssup ϑ Θ E U, x + C, ),G, ϑ),) F his process C, ) may be a srog cadidae of approximaios o he EUIV Hece, we shall ivesigae some properies of C, ) i his paper Remark ha C, ) depeds o x Heceforh, we fix x = he srucure of his paper is as follows: I Secio 2, we sae he sadig assumpios ad he exac defiiio of our ew valuaio C, ) I paricular, we eed he closedess of he se of all self-fiacig sraegies i he L +1 sese his closedess is close relaed o he opimal marigale measure hus, some sadig assumpios are cocered i i Moreover, remark ha i is close relaed o he projecio of 1 oo a suiable space of he sochasic iegraios I addiio, we iroduce, i Secio 3, a example saisfyig he all sadig assumpios I order o make sure ha our ew valuaio is useful as a approximae approach o he EUIV, we ivesigae is basic properies ad he asympoic behavior as eds o I Secio 4, we prove ha our ew valuaio has same basic properies as he EUIV approximaely I paricular, we show ha here exiss a dualiy relaioship bewee a porfolio opimizaio problem relaed o our ew valuaio ad a opimizaio problem amog equivale marigale measures, which is relaed o he opimal marigale measure Furhermore, we asser i Secio 5 ha C, ) coverges o he EUIV as eds o i probabiliy o see his, i is worh while o oice ha he p-opimal marigale measure coverges o he miimal marigale measure as 3

4 p eds o 1, which has bee proved by Gradis ad Rheiläder 22) GR, for shor) O he oher had, we ca say ha he defiiio of he uiliy idifferece valuaio is a ask-pricig mehod, which is oe from a seller s view hus, i Secio 6, we exed our ew valuaio o oe from a buyer s view ha is, we rea a bid-pricig mehod I addiio, we iroduce a dualiy relaio as i Secio 4, ad ivesigae some basic properies of he valuaio from a buyer s view 2 Prelimiaries I his secio, we iroduce he hree sadig assumpios ad some oaios Moreover, we formulae he exac defiiio of our ew valuaio C, ) uder he sadig assumpios I oher words, we give he defiiio of he se of all self-fiacig sraegies hroughou his paper, we cosider a icomplee fiacial marke composed of oe riskless asse whose price is 1 a all ime, ad d risky asses described by a R d -valued coiuous semimarigale X Suppose ha he mauriy is > Le Ω, F,P; F = {F }, ) be a compleed filered probabiliy space wih a righ-coiuous filraio F such ha F is rivial ad coais all ull ses of F, ad F = F Furhermore, i his paper, we rea a suiable se of R d -valued predicable X-iegrable processes ϑ as he se of all self-fiacig sraegies, deoed by Θ Le be a F -measurable radom variable hroughou his paper, we regard as a coige claim, ha is, a pay-off a he mauriy We fix a posiive real umber ad a large odd umber o simplify oaios, we resric wihi odd umbers For all uexplaied oaios, we refer o Dellacherie ad Meyer 1982) ad GR hroughou his paper, C deoes a cosa i, ) which may vary from lie o lie Firsly, we give oe of he sadig assumpios relaed o he uderlyig coige claim Assumpio 21 We assume ha ad L P ) I he defiiio of he EUIV, we do o assume he posiiviy of However, sice he erm ) 1 appears i he sequel, we resric o posiive i his paper Nex, we prepare some oaios i order o iroduce he oher sadig assumpios Le P be a probabiliy measure which is equivale o P, ad p>1 Defiiio 22 1) Le S be a soppig ime We deoe by S VP ) he liear subspace of L P ) spaed by he simple sochasic iegrals of he form h r X 2 X 1 ), where S 1 2 are soppig imes such ha he sopped process X 2 is bouded, ad h is a bouded R d -valued F 1 -measurable 4

5 radom variable Se VP )= VP ) 2) A siged marigale measure uder P is a siged measure Q P wih dq dq E P = 1 ad E dp P dp f = for all f VP ) 3) M s P ) is he space of all siged marigale measures uder P, ad M e P ) is he subse of M s P ) cosisig of probabiliy measures beig equivale o P Moreover, we se M x pp ):=M x P ) L p P ) for x {e, s} 4) he p-opimal marigale measure wih respec o P is defied as he eleme of M s pp ) which miimizes L p P )-orm 5) Le Y be a uiformly iegrable P -marigale wih Y = 1 ad Y > We say ha Y saisfies he reverse Hölder iequaliy R p P ), if here is a cosa C such ha for every soppig ime S, we have ) p Y FS E P C Y S he opimal marigale measure will play a impora role, so ha he followig assumpio is esseial Assumpio 23 We assume ha he opimal marigale measure Q) exiss i M e P ), ad is desiy process Z ) saisfies he reverse Hölder 1 iequaliy R 1 P ) Sice X is a coiuous semimarigale, i is special uder P, ad is caoical decomposiio is give by X = X + M + A wih M a local marigale, A a predicable process, ad M = A = Moreover, if P is equivale o P, he X is also a special semimarigale uder P Le us deoe is caoical decomposiio uder P as follows: X = X + M + A Defiiio 24 1) We deoe by S K p P ) he closure i L p P )of S VP ) for a soppig ime S I paricular, le K p P ):= K p P ) 2) Le L p M ) be he space of all R d -valued predicable processes ϑ such ha ) p/2 ϑ Lp M ) := E 1/p P ϑ r dm ϑ < 3) Le L p A ) be he space of all R d -valued predicable processes ϑ such ha ) p ϑ Lp A ) := E 1/p P ϑ r da < 4) A posiive process Y saisfies R LLogL P ) if here exiss a cosa C> such ha sup E Y P log + Y FS C, S Y S Y S 5

6 where he supremum is ake over all soppig imes S We defie ad Θ +1 P ):=L +1 A ) L +1 M ) { } G, Θ) := ϑ s dx s ϑ Θ, for a suiable se Θ of R d -valued X-iegrable predicable processes I paricular, we deoe G Θ) := G, Θ) Remark ha we ca rearrage he defiiio of Θ +1 P )as Θ +1 P ):={ϑ Gϑ) S +1 P )} y heorem 41 of Gradis ad Krawczyk 1998), G Θ +1 P )) is L +1 P )- closed uder Assumpio 23 Proposiio 47 of GR yields ha here exiss he miimal eropy marigale measure he desiy process of which saisfies R LLogL P ) I addiio, Lemma 21 of GR implies G, Θ +1 P )) = K +1 P ) Moreover, sice is odd, Proposiios 42 ad 44 of GR imply, by passig o a versio if ecessary, ) Z ) ), := Z) /Z = C ) f ), where C ) Z ) dq ) := E F, dp is a F -measurable posiive radom variable, ad f ) K +1 P ) I paricular, f ) / is he projecio of 1 oo K +1 P )il +1 P ) hirdly, we defie a probabiliy measure P, as dp, dp := C, ), where C, R + Furhermore, we deoe Z,, Z, := Z, = C, ) ad Z, := E dp, F, dp where C, is a F -measurable posiive radom variable Remark ha X is also a semimarigale uder P, 6

7 Assumpio 25 We assume ha he 1 -opimal marigale measure Q), wih respec o P, exiss i M e P, ), ad is desiy process Z ), wih 1 respec o P, saisfies R 1 P, ), where Z ), := E P, dq ), F dp, We have Z ),, := Z ), /Z ), = C ), ) f ),, where C ), is a F -measurable posiive radom variable, ad f ), K +1 P, ) I paricular, f ), / is he projecio of 1 oo K +1 P, ) i L +1 P, ) I order o defie he process C, ) exacly, we have o deermie he se of all self-fiacig sraegies Noe ha we have, for a R d -valued predicable process ϑ, E 1 ) +1 G, ϑ) ) F = 1 E P, 1 ) +1 G F, ϑ) C, Moreover, by he same sor of argume as he above, G Θ +1 P, )) is L +1 P, )-closed uder Assumpio 25 hus, here exiss a soluio o he followig miimizaio problem: max E ϑ Θ +1 P, ) 1 ) +1 G, ϑ) ) F Hece, Θ +1 P, ) should be he se of all self-fiacig sraegies Now, we defie { } Θ +1 P, ):= ϑgϑ) isaq ), -marigale ad G ϑ) L +1 P, ) I addiio, we eed oe more preparaio Defiiio 26 A P -marigale Y is i bmo p P ) if here exiss a cosa C such ha, for ay,, E P Y Y ) p/2 F C he, we have he followig relaioship uder Assumpios 21, 23 ad 25: Lemma 27 We have Θ +1 P, )= Θ +1 P, ) 7

8 Proof Firsly, we prove ha, for ay ϑ Θ +1 P, ), Gϑ) is a local Q ), -marigale Le he caoical decomposiio of X uder P, be give by X = X + M, + A, y Lemma 46 of Gradis ad Krawczyk 1998), he desiy process Z ), of Q ), wih respec o P, is deoed by Z ), = EN ), ), where N ), is a P, -marigale ad i bmo 1 P, ) O he oher had, sice Gϑ) S +1 P, ), ϑdm, is i L +1)/2 P, ) hus, we have E P, ϑdm,,n ), 1 ϑdm, 2 N ), 1 2 L L +1 P, 1 P, ) ) <, ha is, ϑdm,,n ), is P, -iegrable Sice he produc Z ), X is a local P, -marigale, we obai A, = N ),,M, hus, we have, for ay ϑ Θ +1 P, ), ϑ r da, = ϑ r d N ),,M, As a resul, by Corollary 316 of Choulli, Krawczyk ad Sricker 1998), Gϑ) is a local Q ), -marigale I addiio, by heorem V2 of Proer 199), here exiss a C>such ha G ϑ) L 1 P, ) sup G ϑ) L 1 P, ) C Gϑ) S +1 P, ) < Sice Gϑ) is a local Q ), -marigale, Gϑ) isaq ), -marigale hus, we obai Suppose ha ϑ Θ +1 P, ) y heorem 412 of Choulli, Krawczyk ad Sricker 1999), here exiss a cosa C> such ha Gϑ) S +1 P, ) C G ϑ) L +1 P, ) < Hece, ϑ Θ +1 P, ), from which holds y Lemma 27, we ca cosider ha Θ +1 P, ) is appropriae as he se of all self-fiacig sraegies Now, we deoe Θ ) := Θ+1 P, )= Θ +1 P, ) 8

9 Agai, we defie he process C, ) as follows: esssup ) ϑ Θ E U,,G, ϑ), ) F = esssup ) ϑ Θ E U, C, ),G, ϑ),) F, where U, is give by U, x, y, z) = x ) 1 y ) +1 z ) O he oher had, as for he expoeial uiliy case, we defie { } Θ exp := ϑ LX) Gϑ) isaq-marigale for all Q M e e P ), ad dq where HQ P ) = E dp } M e e {Q P ):= M e P ) HQ P ) <,, if Q P, =, oherwise he, he dq log dp expoeial uiliy idifferece valuaio EUIV) C,exp F-adaped process saisfyig ) is defied as a esssup ϑ Θ exp E U,exp,G, ϑ), ) F = esssup ϑ Θ exp E U,exp C,exp ),G, ϑ),) F, where U,exp x, y, z) = exp x + y z)) 3 Examples I his secio, we iroduce a example saisfyig Assumpios 21, 23 ad 25 Alhough Assumpios 21 ad 23 are aural comparaively, Assumpio 25 is arifical oe he, we have o reveal ha models saisfyig hese assumpios iclude some impora ad ypical cases o be simpliciy, we cosider oly he case where d = 1 Le W := W 1,W 2,,W l )beal-dimesioal rowia moio, where l 2 Suppose ha he filraio F is he P -augmeaio of he filraio geeraed by W, ad here exiss a predicable process λ such ha he caoical decomposiio of X is give by X = X + M + A = X + M + λ s d M s 31) I oher words, we assume wha is called he srucure codiio SC) for he process X Moreover, we deoe K := λ 2 sd M s, 9

10 which is called he mea-variace rade-off process he, we assume ha K is bouded Furhermore, i view of he marigale represeaio heorem, we ca represe M as l M = σsdw i s, i for some R l -valued predicable process σ hus, X is represeed as X = X + l i=1 i=1 σ i sdw i s + l i=1 λ s σ i s) 2 ds Now, we assume ha X is i L P ) ad e X is o iegrable For example, he case where σ = X η ad λ = ζ /X for some R l ad R-valued bouded predicable processes η ad ζ, respecively I addiio, we assume ha each η i ad ζ are posiive far away from Nex, suppose ha he uderlyig coige claim saisfies Assumpio 21 For example, he Europea call opio X K) +, where K is is srike price Sice L P ), we have, by Corollary 4 of heorem IV42 of Proer 199), ) = E ) l ) E νsdw i s i, i=1 ) where ν i is a predicable process such ha ν i 2 s ds < Le P be he miimal marigale measure see Föllmer ad Schweizer 1991)), so ha is desiy process Ẑ is give by Ẑ = E λ s dm s ) = exp λ s dm s 1 2 λ 2 sd M s ) Remark ha P is i M e P ), so ha Assumpio 23 is saisfied by heorem 41 of Gradis ad Krawczyk 1998) As for Ẑ, we have he followig 1 represeaio: l Ẑ = exp λ s σsdw i s i 1 l ) ) λs σs i 2 ds 2 i=1 i=1 Hece, if we deoe Ẑ, d P F := E P,, dp, he here exiss a cosa C>such ha Ẑ, ) 1 F E P, Ẑ, 1

11 = E Ẑ = E Ẑ E 1, ) 1 = E exp exp 1 = E E, C, exp Z, Z, l i=1 i= ) 1 F l ) λ s dm s E 1, i=1 1 ) λ s dm s 1 2 ν i sdw i s l i=1 ν i sdw i s) l ) ) +1 λ sσs i + νi s dws i l λs σs i + νs i ) 2 ) ds F i=1 F 1 ) ) ν i 2 s ds F ) λ 2 s d M s where E, := E /E, ha is, P saisfies R 1 P, ) Hece, we ca coclude ha Assumpio 25 is saisfied I summary, he models of which he asse price is expressed by 31) saisfy he all sadig assumpios uder he followig codiios: 1) X is i L P ), 2) he uderlyig filraio F is give by he P -augmeaio of he rowia moio, 3) he mea-variace rade-off process is uiformly bouded 4 Dualiy ad some properies We focus o some basic properies of our ew valuaio C, I paricular, we are ieresed wheher or o C, properies as he EUIV o see his, we eed some preparaios For a F -measurable radom variable x, we defie ) ) i his secio ) saisfies he same basic V,, x ) := esssup ) ϑ Θ EU, x,g, ϑ),) F he, we ca rewrie he defiiio of C, ) as V,, ) = V,, C, )) We have V,, ) V,, ) = V V,,,, ) C, )) = C, )), 11

12 amely, C, ) = V,, ) V,, ) ) 1 1 Remark ha, by Proposiio 44 of GR ad Assumpio 23, we have V,, ) = esssup ) ϑ Θ E U,,G, ϑ), ) F = esssup ϑ Θ +1 P ) E 1 ) +1 G F, ϑ) ) = E f ) +1 F Recall Q ) M e P ), so ha, f ) 1 > holds Hece, V,, ) < For ay Q M e P, ), we deoe 1 Z Q, := ZQ Z Q Moreover, we defie Ṽ,, Remark ha we have Ṽ,,, ad Z Q := E dq dp F ) := essif Q M e 1 P, ) E Q Z Q 1, ) 1 F = essif Q M e 1 P ) E Q Z Q, ) 1 F = C ) I order o ivesigae basic properies, we eed o show a dualiy relaioship bewee a porfolio opimizaio problem ad a opimizaio problem wih respec o equivale marigale measures heorem 41 We have he followig dualiy relaioship: E 1 ) +1 G, ϑ) ) F Proof esssup ϑ Θ ) = { essif Q M e 1 P, ) E Q Z Q, ) 1 ) 1 ) 1 } F 41) Firsly, we calculae he lef had side of 41) as follows: LHS of 41) = 1 C, essif ) ϑ Θ E P, 1 ) +1 G F, ϑ) 12

13 = 1 C, = 1 C, = 1 C, E P, 1 C ), 1 C ), E Q ), ) +1 F f ), ) f ), F he secod equaliy owes o Proposiio 44 of GR O he oher had, we have RHS of 41) = { essif Q M e 1 P, ) E Z Q,,, = 1 { C, essif Q M e 1 P, ) E P, = 1 C, = 1 C, where Z Q,,, { E Q ), 1 C ),, := ZQ,, Z Q,, ) Z ), 1 }, F ad his complees he proof of heorem 41 Z Q,, ) Z, 1, Z,, ) 1 C, ) Z Q,, 1 }, F := E P, dq F dp, heorem 41 provides he followig represeaio of C, ): Corollary 42 y he resul of heorem 41, we obai ad V,, C, ) = ) = Ṽ,, ) { } Ṽ,, Ṽ,, 1 ) 1 F } Nex, we sudy basic properies of C, ) by usig he above dualiy relaio Firs of all, we iroduce he basic properies of he EUIV, which have bee proved i MS Proposiio 43 Proposiio 4 of MS) We assume ha ad are bouded o ecessarily posiive) For fixed, ad >, C,exp ) has he followig properies: 13

14 1) C,exp ), 2) if, he C,exp ) C,exp ), 3) C, λ +1 λ) ) λc, )+1 λ)c, ), for ay λ, 1, 4) C, + x )=C, )+x, for ay x L F ) MS called C,exp ) a covex moeary uiliy fucioal Furhermore, hey remarked ha C,exp ) is close relaed o a covex moeary risk measure see Cheridio, Delbae ad Kupper 24)) I order o see ha our ew valuaio C, ) is available as oe of approximae approaches o he EUIV, we wish o prove ha C, ) saisfies Proposiio 43 Heceforh, we shall prove ha his fac holds approximaely Firsly, we obai he followig resul beig relaed o 1) ad 2) of Proposiio 43 Proposiio 44 For ay,, we have he followig: 1) for L + P ), C, ), 2) uder Assumpios 21 ad 25 for, = C, ) C, ) Proof 1) V,, ) esssup ) ϑ Θ E 1 G, ϑ) ) +1 ) F V,, = ) esssup ) ϑ Θ E 1 G, ϑ) ) +1 F ) y he same way, implies V,, )/V,, ) 1 Hece, we have C, ) 2) Firsly, we obai Θ ) Θ), sice ϑ Θ ) G ϑ) L +1 P, ) E G +1 ϑ) ) < E ϑ) ) G +1 < G ϑ) L +1 P, ) ϑ Θ ) Remark ha he las iclusio is derived from heorem 412 of of Choulli, Krawczyk ad Sricker 1999) hus, we have V,, ) esssup ) ϑ Θ E 1 G, ϑ) ) +1 ) F V,, = ) esssup ) ϑ Θ E 1 G, ϑ) ) +1 F 1 essif ) ϑ Θ E G, ϑ) ) +1 ) F 1 essif ) ϑ Θ E G, ϑ) ) +1 F 14

15 essif ϑ Θ ) essif ) ϑ Θ = V,, ) V,, ) his complees he proof of 2) of Proposiio 44 1 E G, ϑ) ) +1 ) F E 1 G, ϑ) ) +1 F Secod, we deal wih 3) of Proposiio 43 We defie V,, ) := essif Q M e 1 P, ) E Q Z Q 1, 1 ) F Se := / ) Sice we have we obai Now, we deoe 1 = ) 1, Ṽ,, = V,, C, ) := V,, { } Ṽ,, V,, 1 We prove, o accou, ha C, ) saisfies 3) of Proposiio 43 Proposiio 45 For fixed,, ay λ, 1 ad, L + P ) such ha, /, we have Proof C, )λ +1 λ),, ) λ C )+1 λ) C ) Remark ha, for L + P ), M e P, )=M e P ) holds We 1 1 have oly o prove ha V,,λ+1 λ) ) 1 λ y he covexiy of 1/x, we have ) 1 V,,λ+1 λ) V,, ) 1 +1 λ) V,, ) 1 = { ) essif Q M e 1 P ) E Q Z Q 1, λ 1 ) +1 λ) 1 } 1 )) F { λv,, λ V,, +1 λ)v,, + 1 λ V,, } 1 15

16 hus, Proposiio 45 follows Now, i order o see ha C, ) saisfies 3) of Proposiio 43 approximaely, we prove ha, for ay sufficie large, C, ) is very ear o C, ) Proposiio 46 For ay sufficie large, here exiss a C>depedig o such ha ) C, ) C Proof sup C, Remark ha we have C, ) C, ) = Ṽ,, V,, Firsly, we have he followig esimaio: Ṽ,, V,, Ṽ,, 42) Ṽ,, V,, = Ṽ,, essif Q M e 1 P ) E Q Z Q, Ṽ,, ) 1 1 ) F 1 ) essifq M e 1 P ) E Q Z Q, ) 1 F 1 ) 1 43) Remark ha, for ay sufficie large, 1 > holds Moreover, we have Ṽ,, V,, = essif Q M e 1 P ) E Q Z Q, essif Q M e 1 P ) E Q Z Q, = essif Q M e 1 P ) E Q Z Q, ) 1 ) 1 F ) 1 essif Q M e 1 P ) E Q Z Q, Q E eq Z e ) 1, Q E eq Z e ) 1, 1 ) F ) 1 1 ) F 1 + ) 2 ) 1 ) F ) 1 1 ) F 1 + ) 2 ) 1 ) F ) 2 E Q eq Z e ) 1, ) 1 F, 44) 16

17 where Q is give by ) essif Q M e 1 P ) E Q Z Q 1, 1 ) F Q = E eq Z e ) 1, 1 ) F O he oher had, we have Ṽ,, = essif Q M e 1 P ) E Q Z Q, y 44) ad 45), Ṽ,, V,, Ṽ,, essif Q M e 1 P ) E Q Z Q, ) 1 ) 1 F ) 1 1 ) F Q = E eq Z e ) 1, 1 ) F 45) 2 ) E Q eq Z e ) 1, ) 1 F Q E eq Z e ) 1, 1 ) F 2 ) 1 46) As a resul, by 42),43) ad 46), we obai, ogeher wih C, ),, sup C ) C, ) 2 ) 1 ) 2 C, C ) y usig Proposiio 46 ogeher wih Proposiio 45, we obai ha C, ) is F -measurably covex i approximaely Corollary 47 Suppose ha λ, 1 ad, L + P ) For fixed, ad ay sufficie large, here exiss a cosa C> depedig o ad such ha C, λ +1 λ) ) λc, )+1 λ)c, )+C Proof Remark ha, for L + P ), Ṽ,, V,, y Proposiios 45 ad 46, we have C, λ +1 λ) ) C, λ +1 λ) ),, λ C ) +1 λ) C ) λ C, )+C ) +1 λ) C, )+C ) = λc, ) +1 λ)c, )+C 17

18 hirdly, we cocerae o 4) of Proposiio 43 y a similar way wih he above, we prove he followig: Proposiio 48 Le x be a bouded F -measurable radom variable For fixed, ad ay sufficie large, here exiss a cosa C> depedig o x ad such ha C, + x ) C, ) x C 47) Proof We prove firsly he case where C, + x ) C, ) x Remark ha C, Firsly, we have Ṽ,, V,,+x + x ) C, ) Ṽ,, = essif Q M e 1 P ) E Q Z Q, essif Q M e 1 P ) E Q Z Q, Ṽ,, ) 1 ) 1 F V,,+x Ṽ,, V,,+x ) 1 1 ) + x F ) Q E eq Z e ) 1, ) 1 F Q E eq Z e, ) ) 2 + x where Q M e 1 P ) is give by 48) ) 1 1 ) + x F ) Q E eq Z e ) 1, F, 49) essif Q M e 1 P ) E Q Z Q, = E eq Z e Q, ) 1 1 ) + x F ) ) 1 1 ) + x F ) he las iequaliy of 49) is give from ) 1 ) 2 = ) 1 ) 2 Nex, we have Ṽ,, V,,+x = essif Q M e 1 P ) E Q Z Q, ) 1 Ṽ,, ) 1 ) 1 F Q E eq Z e ) 1, 1 ) + x F ) 1 ) Q + x ) E eq Z e ) 1, F 18

19 hus, we ca coclude RHS of 48) = Ṽ,, ) ) 2 + x Q E eq Z e ) 1, F ) x ) ) Q E eq Z e ) 1, F Ṽ,, ) ) 2 + x 1 + x ) O he oher had, 1 + x ) = 2 + x 1 + x ) C 411) 41) ad 411) imply 47) for he case where C, +x ) C, ) x Nex, we rea he reverse case Wihou loss of geeraliy, we assume ha x is posiive Remark ha ) 1 + x ) 1 + x ) x ) 2 Sice we ca prove V,, Ṽ,,+x where Q M e 1 P ) is give by x x 2 ) ) Q E eq Z e ) 1, F, essif Q M e 1 P ) E Q Z Q, = E eq Z e Q, ) 1 1 ) F, ) 1 1 ) F 41) ad V,, Ṽ,,+x Q E eq Z e ) 1, F Ṽ,,, here exiss a cosa C> such ha C, + x ) C, ) Ṽ,, V,, V,, Ṽ,,+x Ṽ,,+x x x 2 ) x C Hece, Proposiio 48 follows 19

20 5 Asympoic behavior We have o sudy he asympoic behavior of our ew valuaio C, ) as eds o so as o make sure ha C, ) is jusified as a approximae approach o he EUIV I his secio, we prove ha C, ) coverges o he EUIV i probabiliy Remark ha GR have proved ha he p-opimal marigale measure coverges o he miimal eropy marigale measure as p eds o 1 I he proof of he followig heorem, his asympoic behavior will play a vial role heorem 51 Suppose ha L + P ) For fixed,, C, ) coverges o C,exp ) i probabiliy as Proof Sep 1 We shall prove ha V,, V,, coverges o esssup ϑ Θ exp E U,exp,G, ϑ),) F i probabiliy esssup ϑ Θ exp E U,exp,G, ϑ), ) F For small ε>, here exiss a sufficie large odd umber such ha, for ay odd umber, 1 ε)e e ε ) e P -as Sice + 1 is eve, we have esssup ) ϑ Θ E 1 ε) esssup ϑ Θ ) 1 ) +1 G, ϑ) e F E 1 G, ϑ) V,, ) Now, we defie a probabiliy measure P exp, equivale o P as dp exp, = C exp, e, dp where C exp, R + We deoe Z exp, dp exp, := E F, dp ad where C exp, LHS of 51) = Z exp,, C exp, := Z exp, /Z exp, = C exp, e, esssup ϑ Θ ) ) +1 e F 51) is a F -measurable posiive radom variable he, we have 1 E P exp, 1 ) +1 G F, ϑ) 2

21 = 1 C exp, essif ϑ Θ +1 P exp, ) E P exp, 1 = 1 C exp, E P exp, ) f ),exp, +1 F, ) +1 G F, ϑ) where f ),exp, is he projecio of 1 oo K +1 P exp, )il +1 P exp, ) Remark ha, L + P ) yields Θ ) =Θ+1 P, )=Θ +1 P exp, )=Θ +1 P ) y Lemma 413 of GR, f ),exp, coverges i probabiliy o f exp,, which is i K +1 P exp, ) hus, for every sequece k, we ca exrac a subsequece sill deoed by k ) such ha f k),exp, f exp, P -as Remark ha here exiss he miimal eropy marigale measure P exp, for P exp,, he desiy process of which saisfies R LLogL P exp, ) We deoe dp Z exp, exp, F := E P exp, dp exp,, ad, by he proof of Proposiio 415 of GR, we ca represe Z exp,, := Z exp, /Z exp, = C exp, exp f exp, ), where C exp, is a F -measurable posiive radom variable I addiio, he proof of Proposiio 415 of GR implies ha lim k E P exp, = lim k E P exp, = E P exp, exp f k),exp, k f exp, ) k+1 F f k),exp, k ) F ) k F P -as Hece, we obai E P exp, ) f ),exp, +1 F E P exp, exp ) f exp, F 52) i probabiliy Moreover, as for he sequece f ), Lemma 413 of GR yields ha f ) f exp i probabiliy esides, f exp is icluded i K +1 P ) 21

22 Remark ha he miimal eropy marigale measure P for P is give by dp dp = C exp f exp ), where C R + Hece, he same sor of argume as he above shows V,, V,, V,, E exp f exp ) F Sice ε is arbirary, 51), 52) ad 53) yield exp f exp, )e F E Eexp f exp ) F i probabiliy 53) i probabiliy 54) Now, we shall prove ha exp f exp, ) F = essif ϑ Θ exp E P exp, exp G, ϑ)) F 55) E P exp, y Proposiio 1 of MS, here exiss a η Θ exp such ha G, η) = f exp, For ay ϑ Θ exp, we deoe ϑ = ϑ + η he, we have essif ϑ Θ exp E P exp, exp G, ϑ)) F = essif ϑ Θ exp E P exp, e G, ϑ) e G, η) F 1 = essif ϑ Θ exp E exp, P e G, ϑ) F C exp, Jese s iequaliy yields E exp, P e G, ϑ) F exp{e P exp,g, ϑ) F } =1 O he oher had, if we se ϑ, he e G, ϑ) F =1 E P exp, Hece, we obai 55) Le us go back o 54) We ca coclude ha, ogeher wih 55), V,, esssup ϑ Θ exp E U,exp,G, ϑ),) F V,, i probabiliy esssup ϑ Θ exp E U,exp,G, ϑ), ) F Sep 2 We prove he followig: Lemma 52 Le A be a compac se o R ad {X } 1 a sequece of A-valued radom variables such ha X coverges o a radom variable X i probabiliy Moreover, suppose ha a sequece f of coiuous fucios coverges o a coiuous fucio f o A he, we have f X ) fx) i probabiliy 22

23 Proof of Lemma 52 We fix a ε> arbirarily We have oly o show lim P { f X ) fx) <ε}) =1 Now, we calculae he lower boud of he lef had side P { f X ) fx) <ε}) = P { f X ) fx )+fx ) fx) <ε}) P { f X ) fx ) + fx ) fx) <ε}) P { f X ) fx ) <ε/2} { fx ) fx) <ε/2}) = P { f X ) fx ) <ε/2})+p { fx ) fx) <ε/2}) P { f X ) fx ) <ε/2} { fx ) fx) <ε/2}) P { f X ) fx ) <ε/2})+p { fx ) fx) <ε/2}) 1 Sice X A for ay 1 P -as ad f f o A, we have, for ay sufficie large, P { f X ) fx ) <ε/2}) =1 Moreover, here exiss a δ>such ha x y <δ fx) fy) <ε/2 hus, P { fx ) fx) <ε/2}) P { X X <δ}) 1 his complees he proof of Lemma 52 Sep 3 We deoe he, we ca represe U := V,, ) V,, ) C, ) = { } U 1 1 Furhermore, U saisfies 1 U ) e y Sep 1, U coverges o esssup ϑ Θ exp E U,exp,G, ϑ),) F =: U) esssup ϑ Θ exp E U,exp,G, ϑ), ) F i probabiliy Icideally, MS have proved C,exp ) = 1 log U 23

24 Moreover, f x) := { x 1/ 1 } coverges o fx) := 1 log x o he compac se 1, exp{ } y Lemma 52, we ca coclude ha C, ) C,exp ) i probabiliy his complees he proof of heorem 51 6 Exesio o bid valuaio he defiiio of he uiliy idifferece valuaio is give from view of a seller I oher words, i is a proposiio of a askig-price for a coige claim hus, whe we ry o sugges a bid-price, i is aural ha we improve he uiliy idifferece valuaio o a adaped process C ) saisfyig esssup ϑ Θ E Ux + G, ϑ)) F = esssup ϑ Θ E Ux C ) +G, ϑ)+) F I paricular, as for he EUIV, is bid valuaio is give by C,exp ) Remark ha he EUIV is defied for bouded coige claims which may value i egaive umbers O he oher had, our valuaio C, ) is available for oly posiive coige claims A leas, we have o resric ha has a lower boud i order ha C, ) is well-defied Hece, we should defie a bid valuaio oher ha he ask valuaio C, ) Firsly, we defie obediely a bid valuaio C, ) correspodig o our ew valuaio as follows: esssup ) ϑ Θ E 1 ) +1 G F, ϑ) = esssup ) ϑ Θ E 1 ) C, ) 1 ) +1 G, ϑ) 1 ) F, 61) where Θ ) ad Θ ) are suiable spaces of R d -valued predicable processes However, C, ) is o coveie, because if we deoe dp, dp := C, 1 ), where C, is a posiive cosa, he P, is o equivale o P i geeral Noe ha we should se Θ ) =Θ +1 P, ) for 61) hus, we sugges, i his secio, aoher defiiio of a bid valuaio, deoed by Č, ), for C, ) by a approximae way as follows: esssup ϑ ˇΘ E 1 ) +1 ) G F, ϑ) = esssup ϑ ˇΘ E ) ) Č, ) 1 ) +1 G, ϑ) ) F, 24

25 where ˇΘ ) := Θ+1 ˇP, ) ad ˇP, is defied as, d ˇP dp := Č, ) Noe ha Č, is a posiive cosa y he defiiio of ˇP,, here exiss he 1 -opimal marigale measure ˇQ ), wih respec o ˇP, i M e ˇP, ), 1 ad is desiy process Ž ), wih respec o ˇP, saisfies R 1 ˇP, ) Moreover, if we deoe ˇV,, x ) := esssup ϑ ˇΘ E ) ) x 1 ) +1 G, ϑ) ) F he, we have, by he defiiio of Č, ), his implies ha ˇV,, ) ˇV,, ) = = = ˇV,, Hece, Č, ˇV,, ) Č, esssup ϑ ˇΘ ) 1 )) ˇV,, esssup ϑ ˇΘ ) E Č, ) ) ) = ˇV,, Č, )) E 1 G, ϑ) ) +1 ) F Č, ) is represeed as Č, ) = ˇV,, ) ˇV,, ) ) ) 1 G, ϑ) ) +1 ) F ) 1 1 Jus as i he ask valuaio, we give a dualiy relaioship wih respec o ) Č, heorem 61 We have he followig dualiy relaioship wih respec o Č, esssup ϑ ˇΘ ) = E 1 ) +1 G, ϑ) ) F { essif Q M e 1 ˇP, ) E Q ): Z Q, ) 1 ) } F 62) 25

26 Proof Ž,, where Č, Ž ),, We deoe Ž, := Ž, := Ž), Ž ), Č ), = Č, ) ad Ž, := E d, ˇP F, dp is a F -measurable posiive radom variable Moreover, we deoe ) ), ˇf = Č), d ad Ž), := E ˇQ), F ˇP, d ˇP,, ˇf ), where is a F -measurable posiive radom variable, ad K +1 ˇP, ), ) I paricular, ˇf / is he projecio of 1 oo K +1 ˇP, ) i L +1 ˇP, ) Now, we compue he lef had side of 62) LHS of 62) = 1 Č, essif ϑ ˇΘ E ) ˇP 1 ) +1, G F, ϑ) = 1 Č, = 1 Č, = 1 Č, O he oher had, we have RHS of 62) = { essif Q M e 1 E ˇP, 1 Č ), 1 Č ), ˇP, ) E ŽQ,,, ) ), +1 ˇf F E ˇQ ), ) Ž, 1, = 1 { ŽQ,, Č, essif Q M e 1 ˇP, ) E ˇP,, = 1 { Ž), ) 1 } Č, E ˇQ ),, F = 1 Č, where Ž Q,,, 1 Č ),, := ŽQ,, Ž Q,, ad his complees he proof of heorem 61 ) ), ˇf F Ž,, ) 1 Č, ) 1 F } Ž Q,, dq F := E ˇP, d ˇP, y usig heorem 61, we obai aoher represeaio of Č, ) ) 1 F } 26

27 Corollary 62 Deoe ˇV,, := essif Q M e 1 ˇP, ) E Q Z Q, ) 1 ) F he, we have ) ˇV,, ) = ˇV,,, ad Č, ) = ˇV,, ˇV,, 1 As i Secios 4 ad 5, we have o ivesigae wheher or o Č, ) saisfies Proposiio 43 approximaely ad coverges o he EUIV Sice C, ) defied i 61) is equivale o C, ) as log as is bouded ad is a sufficie large, we ca prove easily ha C, ) approximaely saisfies 1), 2) ad 4) of Proposiio 43, ad he reverse iequaliy of 3), ha is, we ca say ha C, ) is a approximae cocave moeary uiliy fucioal hus, we have oly o sudy he relaioship bewee Č, ) ad C, ) i order o cofirm ha Č, ) is meaigful as a bid valuaio of our ew approximae approach o he EUIV heorem 63 For L + P ) ad ay sufficie large, here exiss a cosa C> depedig o such ha sup Č, ) C, ) C Proof Remark ha, sice is bouded ad is a sufficie large, P, is equivale o P ad Θ ) ) = ˇΘ = Θ) holds Now, we defie Č, ) asa adaped process saisfyig E 1 ) +1 G F, ϑ) esssup ϑ Θ ) = esssup ϑ Θ ) he, we have E Č, ) = 1 Č, ) ) 1 ) +1 G, ϑ) ) F Č, ) Č, ) = ˇV,, ˇV,, 1 We prove ha here exiss a C>such ha C, ) Č, ) C Noe ha we have C, ) Č, ) = C, ) Č, ) 27

28 = { } Ṽ,, Ṽ,, 1 + = Ṽ,, Ṽ,, ˇV,, Ṽ,, We esimae he righ had side of he above Firsly, we have Ṽ,, ˇV,, = Nex, we compue Ṽ,, Ṽ,, ˇV,, Ṽ,, essif Q M e 1 ˇP, ) E Q = essif Q M e 1 ˇP, ) E Q essif Q M e 1 ˇP, ) E Q = essif Q M e 1 ˇP, ) E Q ˇV,, Z Q, ) 1 as follows: ˇV,, Z Q, ) 1 1 ) 1 F ) F Z Q, ) 1 Z Q, ) 1 ˇV,, ˇV,, 1 ) F 1 + ) 2 1 ) 1 ) F ) F essif Q M e 1 ˇP, ) E Q Z Q, ) 1 E Q Z Q, ) 1 ) ) 1 ) ) F 2 ) 1 1 ) EQ Z Q F,, ) 1 where essif Q M e 1 ˇP, ) E Q Z Q, ) 1 ) F = E Q Z Q, ) 1 ) F Remark ha M e P, )=M e ˇP, ) Moreover, we have 1 1 Ṽ,, = essif Q M e 1 ˇP, ) E Q Z Q, ) 1 1 ) 1 F As a resul, we obai essif Q M e 1 ˇP, ) E Q Z Q, ) 1 = E Q Z Q, ) 1 ) F E Q Z Q F Ṽ,,, ) 1 ˇV,, Ṽ,, ) F ) ) 28

29 Hece, we ca coclude ha here exiss a C>such ha C, ) Č, ) ) 2 1 ) 1 C Č, O he oher had, sice Ṽ,, ) Moreover, sice Č, Č, hus, heorem 63 follows ˇV,, ) Č, ) Č, ), we obai C, ) ), here exiss a C>such ha Č, ) Č, ) C Cosequely, he bid valuaio Č, ) saisfies approximaely he same basic properies as a cocave moeary uiliy fucioal, ad coverges o he bid valuaio of he EUIV i probabiliy 7 Cocludig Remarks he resuls of Secios 4, 5 ad 6 mea ha, uder some assumpios relaed o he opimal marigale measure, our ew valuaios C, ) ad Č, ) are available as approximae approaches o he EUIV for ask ad bid, respecively I oher words, we succeed i relaxig he codiio 12) o 13) for a sufficie large by usig a kid of power fucios I addiio, Secio 3 shows ha here exis may impora examples saisfyig he all sadig assumpios O he oher had, i order ha we calculae our valuaio C, ) cocreely, we eed o obai he desiy of P, ad he projecio of 1 oo K +1 P, ) hese are big difficulies for us o realize our ew valuaio However, here exis several cases where we ca compue cocreely he desiy of he p-opimal marigale measure, for example, see Hobso 24) Fially, we elimiae some fuure problems We have o researched for he dyamics of our ew valuaio i his paper Hece, very lile is kow abou properies of our valuaio as processes For example, he ime-cosise or he locally Lipschiz coiuiy i see Secio 5 of MS) Furhermore, we are ieresed i he asympoic behavior of C, ) as eds o or For example, MS have proved ha he EUIV coverges o he codiioal expecaio uder he miimal eropy marigale measure as eds o, ad o he superreplicaio price process as eds o Ackowledgmes he auhor would like o express his graiude o Mari Schweizer ad Shigeo Kusuoka for heir much valuable advice he fiacial suppor of he auhor has bee parially graed by Gra-i-Aid for Youg Scieiss ) No from he Miisry of Educaio, Culure, Spors, Sciece ad echology of Japa 29

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