Market Conditions under Frictions and without Dynamic Spanning

Transcription

1 Mar Codiio udr Friio ad wihou Dyai Spaig Jui Kppo Hlii Uivriy of hology Sy Aalyi aboraory O Bo FIN-5 HU Filad hp://wwwhufi/ui/syaalyi ISBN Hlii Uivriy of hology ISSN 78-3 Sy Aalyi aboraory iblla Rarh Rpor Oaii 998 A7 Fbruary 998

2 MARK CONDIIONS UNDR FRICIONS AND WIHOU DYNAMIC SANNING * Jui Kppo Sy Aalyi aboraory Hlii Uivriy of hology Oaaari 5 poo Filad -ail: JuiKppo@hufi ABSRAC I hi papr w udy h odiio for h ab of arbirag igl ag opialiy ad quilibriu i a ar udr friio ad wih or our of uraiy ha radabl a I ohr word w aalyz a apri dflaor i a dflaor wih h propry ha h dflad pri of radabl a ar arigal i h pr of friio ad iopl A uiqu a-pri dflaor ay i oly o projd ar I i how ha if h ar odiio hold for h projd ar hy hold alo for h iiial ar KYWORDS: Arbirag quilibriu iopl ar opiizaio raaio o INRODUCION Mar ar iopl if hr i or ha o a-pri dflaor i a dflaor wih h propry ha h dflad pri of a radabl a i a arigal If h ar ar iopl h pri of a oig lai ay dpd o h a-pri dflaor wih rp o whih i i prid I hi papr w d h frawor of ab of arbirag opial porfolio ad oupio hoi ad uriy ar quilibriu i opl ar o ovr ar wih friio ad wihou dyai paig * h auhor i graful o oa Björ a Joivuoll Sau ura Sapa Saila o Valila ad uoo Vuolaho for hlpful uggio ad o

3 h gral frawor for arbirag fr odiio i opl ar i drivd i Harrio ad Krp 979 Harrio ad lia 98 Krp 98 ad Co ad Huag 986 Mro Co ad Huag 989 ad Karaza hozy ad Shrv 987 hav olvd opial porfolio ad oupio hoi probl i opl ar Suriy ar quilibriu i opl ar i udid g i Brd 979 Duffi ad Za 989 ad Huag 987 Jouii ad Kallal 995 driv h arbirag fr odiio udr raaio o hy how ha i h pr of raaio o h ab of arbirag odiio i quival o h i of a quival probabiliy aur ha rafor o pro bw h bid ad a pri pro io a arigal Hdgig ad porfolio opiizaio udr raaio o ar aalyzd i Cviai ad Karaza 996 Cviai ad Karaza 993 oidr hdgig i h pr of gral lod ov orai of porfolio pro ad hir hodologi a alo b applid o h a of diffr ir ra for borrowig ad ldig h igl ag opialiy i a iopl ar i udid g i Cviai ad Karaza 99 Duffi ad Su 99 ad lad 985 Groa ad Shillr 98 ad Ba 99 driv ar quilibriu wihou dyai paig odiio h odl ar h aalyi fro h a-pri dflaor ipliily giv by igl ag ad h driv h pd ra of rur o all urii fro h ovaria of rur wih aggrga oupio ir ad h ar-ri-avrio oa I hi papr w oidr h a id of probl ha ar udid i Jouii ad Kallal 995 ad Cviai ad Karaza ad w uiliz hir frawor I ora o Joui ad Kallal w oidr alo ar wihou dyai paig ad w driv h ar odiio by uig h quoi pa of pri of radabl a h iiial ohai variabl ar projd io a w pa o whih ar ar opl ad friiol ad w a ploy h frawor of opl ar h projd ar ar h a id of fiiiou ar ha ar ud i Cviai ad Karaza I addiio o Cviai ad Karaza w l h volailiy pro of radabl a diffr bw variou fiiiou ar ad w alo oidr gral ar friio for ia friio i gig iforaio ad i-varyig raaio o W how ha h ar odiio hold i h iiial ooy if hy hold i h projd ar h r of h papr i orgaizd a follow: Sio dfi h frawor ud i h papr ad Sio 3 driv h projd ar Sio 4 driv h ar quilibriu odiio ad Sio 5 olud MOD W plor a ooy whr iru ar radd oiuouly wihi a i horizo [ τ hr i a fii of radabl a H whr H ad a of ag dod by M A ag M i dfid by a ozro oupio dow pro ad a rily iraig uiliy fuio U I dribig h probabilii ruur of h ooy orrpodig o a ag M w rfr o a udrlyig probabiliy pa F F : τ < Hr i a alog wih h adard filraio [ { } F i a σ-algbra of ub of grad by a -diioal Browia oio B K B ad i a probabiliy aur o F h probabiliy pa ay diffr bw variou ag g bau hr ay b friio i gig ar iforaio ad bau h porfolio of diffr ag ay parly dpd o variou our of uraiy W do by h la of fuio f : [ τ R i a Hilbr pa uh ha i ω a f ω i B ii f ω i F -adapd F -aurabl whr B do h Borl σ-algbra o [ τ iii ah oordia f i of f i R aifi < f i d paio wih rp o Hrafr h id of i oid h followig aupio harariz or our ooy ω [ τ whr do h 3

4 ASSUMION A: h poibl radabl a pri ha ar dioud by ri-fr ir ra for a ag M ar giv by h lod pa o h orrpodig a pri X h pri ar uh ha h bid pri ar alway lowr or qual Aupio A a ha hr do o hav o b a uiqu pri vor rprig h radabl a hi i du o friio i h ar ad/or bid-a prad for ir ra I our odl h bid pri i h llig pri afr raaio o ad ohr friio ad h a pri i h buyig pri afr h ar friio h pri pro dpd o ag bau ar friio ay diffr bw variou ag For ia du o h friio i h diribuio of iforaio variou ag ay hav diffr pro ia h pri pro ar alo fuio of h porfolio pro bau raaio o uually dpd o h porfolio ir h pri ay dpd alo o h ag aio hi happ wh h ivor i a igifia playr i h ar h pro i X ar h pri pro of our fiiiou ar orrpodig o h ag M h followig aupio i ud i h alulaio of opial oupio ad radig ragi ASSUMION A: For all M hr i a pri pro i X uh ha giv h pri pro h ag i o willig o rad i h ar Aupio A ipli ha all h ag hav a opio o rad i h ar ad hy a op h radig for a i if hy wa ASSUMION A3: h ohai variabl of radabl a ha ar dioud by ri-fr ir ra orrpodig o a ag M follow a Iô ohai diffrial quaio whr [ τ d α d + σ db X X i a rado variabl whih i idpd of F ad [ < α σ B i a -diioal Browia oio o h probabiliy pa F ad X i h rag of pro i X a i orrpodig o h ag h opraor i dfid a follow α ' [ α K α σ M σ σ O K K σ σ ad ' i h rapo of W will rfr α ad σ a h drif ad volailiy pro of Aupio A3 a ha hr ay i or our of uraiy ha hr ar radabl a h oordia of a b dpd a hy uually ar Cobiig aupio A ad A3 w ha ah pro i X i giv by quaio For ia quaio hold for h pro ha fir qual h bid pri of h radabl a ad h h a pri hi aupio i ad i ordr o iplify our aalyi Coparig A ad A3 wih Cviai ad Karaza w ha i our odl h volailiy pro of diffr fiiiou ar ay diffr ad variou ag ay hav dii pro ia hi i g du o friio i gig iforaio uraiy i raaio o ag pifi ri ad uraiy i illiquid a M ASSUMION A4: For ah ag M hr i a appig ψ : uh ha ψ σ ω σ ω ϑ ω [ τ X a whr ϑ ϑ ω : R R i oo bu o ary o o o ϑ aifi Noviov odiio 3 p ϑ d < [ τ 4

5 ad ψ σ α ω [ τ h iuaio of Aupio A4 rg bau alo urly σ ay b igular W will rfr o ϑ a h ar pri of ri A4 i a ral aupio of h papr bau a w will h ar odiio hold if i i ru If σ i igular for o [ τ h ag M hoo o ar pri of ri pro for ah pro i X aordig o hi or hr a abou ri aog h pro i h ar ha aify A4 If σ i ivribl [ τ ar ar opl ad ϑ i uiqu ASSUMION A5: h uiliy fuio of ivor ar ooh-addiiv A5 a ha h uiliy fuio U: R + R of h ag M i dfid by 4 U u d [ τ whr i a adapd o-gaiv oupio i [ u : R + R i ooh o R + ad for ah u o R + u : R + R i iraig rily oav wih a uboudd parial drivaiv u aifyig h Iada odiio: if u ad up [ 3 UOIN SACS I hi io w driv h quoi pa of Klly 955 ad [for h diuio of quoi pa g σ u dfi h quival rlaio R by ig σ R σ whr σ σ ω ad ω ϑ σ ω ϑ alo urly whr ϑ i giv by Aupio A4 [ τ if X M h rag of pro i a b dividd io quival la { σ ; σ R σ } { σ σ } σ [ τ R ad h of quival la i dod by ad i alld h quoi pa of wih rp o R h quoi pa i a pariio of R ha i vry l of blog o o ad oly o la i h oiuou appig : R i alld projio ad h iag of σ i h quival la io whih σ blog h appig i a liar urjio whih oidu a σ- algbra o R Now w a a h followig la ψ 5 MMA : ψ dfid i Aupio A4 idu a oiuou liar bijiv appig : whr R σ ϑ uh ha σ ϑ ϑ [ τ X M a ϑ ω : R R i a bijio ha aifi h Noviov odiio ad σ R σ ad aoial faorizaio of ψ i ψ ψ o Figur illura h iuaio ha i ψ σ ψ σ ad h ψ 5

6 ψ R Figur h aoial faorizaio of y ROOF: Fir w o ha ψ i wll dfid ha i σ σ ψ σ ψ σ alo urly whr σ ω σ [ τ X ad M Bau ψ : i a oiuou liar urjio ad i a oiuou liar projio whih oidu h σ-algbra of R R ψ i oiuou ad liar I i alo a bijio i ψ i ijio by h dfiiio of ad hrfor hi a ha hr ould b o ohr ϑ ha aifi α ω σ ω ϑ alo urly Bau ϑ aifi ω h Noviov odiio ad i boudd alo R o a probabiliy pa F whr ϑ aifi h odiio Now F : F [ i a probabiliy aur o h aur pa liiaig [ τ F B i a pro i i h σ-algbra grad by F B h a l ha wr liiad fro ψ i ordr o g B ad i obaid fro B by ψ D a a ha w hri h diio of h Browia oio i ordr o g h bijio alhough w ar o abl o fid a uiqu ar pri of ri fuio i ASSUMION A6: σ i piwi oa w a fid i i ϑ ha i Aupio A6 ur ha w a igra wih rp o B -diioal Browia oio i o h piwi irval B i a W do by X h quoi pa of h la of X ar X ad {y} whr y blog o h opl of X i y \ X X i obaid fro by projig X io o poi : X ha i l o pro io whih all h pro i X ar projd hi projio i a urjio whih oidu h σ-algbra of 6 Now w a dfi h pro of d X by α d + σ X R db ; whr M aifi h odiio of A3 α ad σ ad h wll-dfid a- pri dflaor by [ τ 7 p ϑ ω ' db ϑ ω d [ τ 6

7 quaio 6 ipli ha hr i a uiqu pri vor ad pro for radabl a i h quoi pa X ad quaio 7 a ha hr i a uiqu a-pri dflaor o F h pri pro of a fiiiou ar ad i h a-pri dflaor of h ar W illura our frawor wih a apl h pro of b dfid a follow for i 8 d d d + d + db db [ [ db [ τ whr B ad B ar idpd Browia oio pro τ > ad M h ar pri of ri vor i dfid a 9 ϑ ϑ [ [ τ I hi a h pro of i d d d + d + db db [ [ τ 4 ARBIRAG OIMAIY AND UIIBRIUM I hi io w oidr h ab of arbirag odiio igl ag opialiy ad uriy ar quilibriu Fro h prviou io w g h followig la MMA : hr i X ad uh ha i a arigal o F whr i a quival arigal aur wih Rado-Niody drivaiv d d o F M ad p ϑ db ϑ ω d if ad oly if F whr i a quival arigal aur wih Rado-Niody drivaiv i a arigal o d d o F ad i giv by quaio 7 ROOF: If ω whih pro i 3 d [ α σ ϑ [ σ ϑ ' db i a arigal h alo whih pro i 4 d d + [ α σ ϑ [ σ ϑ ' db d + 7

8 i a arigal hi i bau 5 σ ϑ σ ϑ [ τ a ad 6 proj all h pro i X io ha i α σ ϑ α Covrly if ω i a arigal h hr i aordig o Aupio A4 σ 7 α σ ϑ [ τ a hi giv ω wih rp o ϑ X ad i alo a arigal ad [ [ [ τ a a ha if hr i alo [ τ a uh ha 6 hold ad whr a h paio D X ad i a arigal o F h ab of arbirag o F giv by 7 uh ha i a arigal o F h ad vi vra hi yild h fa ha h uffii odiio for i ha hr i a uiqu a-pri dflaor o F ha i M Giv Aupio A4 ah pro of X i a arigal wih rp o i ow arigal aur hrfor i our aalyi a oly fi h arigal aur odiio i provd i hor h arbirag-fr HORM : hr i o arbirag i h iiial ar if ad oly if hr i a a-pri dflaor o F M ROOF: hr i o arbirag o F Rado-Niody drivaiv if ad oly if hr i a quival arigal aur wih [ Harrio ad Krp 979 Harrio ad lia 98 Krp 98 ad Clar 993 W a wri h dflad pro of a dyai porfolio whr ' i a radig ragy pro o F a follow 8 d[ d[ + + [ τ whr M ad p ϑ ω db ϑ d Bau i h a pri dflaor of X ad X w g fro Aupio A h followig odiio 9 d[ [ τ X ha i if h ad if h hi i du o raaio o Cobiig 8 ad 9 w ha i a uprarigal If hi hold M h hr i o arbirag i h iiial ar Covrly if w au ha hr i o arbirag i h iiial ar h hr i X uh ha i a arigal M quaio 8 ad 9 ur ha hi ipli h ab of arbirag odiio rojig all h pro i X io F bau w ha hr i o arbirag o 8

9 i a arigal M D hor ipli h a ha i alo provd i Jouii ad Kallal 995 ha i h ab of arbirag odiio i quival o h i of a quival probabiliy aur ha rafor o pro bw h bid ad a pri pro io a arigal I our aalyi all h pro of h iiial ooy ar arigal wih rp o hir ow arigal aur Howvr i h proof of hor w oly d o au ha M hr i o pro i X uh ha i i a arigal udr h quival arigal aur h probl i fidig h opial hdgig oupio ad porfolio pro udr raaio o ad ohr friio i h fa ha radig a b a zro-uiliy oupio A iod arlir h pro X i a fuio of h radig ragy ad hrfor alo h arigal aur of i a fuio of h radig ragy W wri pliily ad W dfi h -auriy uppr- ad lowr- hdgig pri of a oig lai C i our iiial ooy a up C if if { C a } ad M low C up up{ C a } M low up whr h porfolio ar lf-fiaig i h udrlyig pri pro ad C C Bau i i poibl ha h ar i iopl hr ay b uorollabl ri i h ag porfolio pro h uppr hdgig pri i fii if i i idpd of h uorollabl ri ad h a pri ar idpd of h ag aio Uig h frawor of Cviai ad Karaza 993 w g h followig propoiio ROOSIION : If h uppr hdgig pri i fii w g { [ if { up up C C F } ad M low 3 { { [ up if C C F } whr [ ad [ τ M ROOF: If i h arigal aur of h C [ C F friiol ar h pri o F 4 [ C C F i ow giv by aig upru ovr all arigal aur w g up 5 up{ [ } C C F whr i h pri i a opl up C i h uppr pri orrpodig o ag I h a way w g for h lowr pri { [ } C F lowr C if Now w a ay lf-fiaig porfolio pro ha aifi C Bau [ F [ C F up 6 up{ [ } C C F h projd arigal aur w g fro 9 9

10 Siilarly wih h lowr hdgig pri w g [ { } if lowr F C C by oruig ay lffiaig porfolio ha aifi C aig h ifiu ovr all h uppr pri w g h ar uppr pri ad aig h upru ovr all h lowr pri w g h ar lowr pri D ropoiio ipli ha if h friio ar zro i h bid ad a pri ar qual h ag i M h h uppr- ad lowr hdgig pri ar qual if h uppr pri ar fii Now w ar o oidr h opial oupio ad porfolio pro Giv h adapd oupio dow pro o F hr i a dyai porfolio whr ' i a radig ragy pro whih fia a adapd oupio pro o F if 7 [ [ [ d d * + ad i rial oupio i zro whr M ad [ τ ar h orrpodig pro of ad o F quaio 7 giv h followig la MMA 3: Giv h dow pro ad ay adapd hr i a pro ' fiaig if ad oly if 8 [ d ad 9 [ + d d d * whr [ τ ad alo urly ROOF: S g Co ad Huag 989 for h proof ha a 3 hold for ad o F Now proj all h pro i X io ad w g by uig 8 ad 7 h followig fiaig odiio 3 [ [ [ [ [ d d d ad [ + whr Bau fia w g fro 3 quaio 8 ad 3 [ [ [ [ d d * + aig io aou h rial oupio odiio w g 9

11 Covrly if fia h 3 hold Bau ad ar idpd pro w g quaio 7 ad 3 fro 3 Uig agai h rial oupio odiio ad h frawor of Co ad Huag 989 w g 8 ad 9 D a 3 ipli ha i a iopl ar hr a b uorollabl ri i h ag dow ad porfolio pro Giv a 3 a igl ag M fa h followig probl o F 3 U up ubj o 33 [ d Now w a a h followig hor HORM : hr i a opial oupio ad radig ragy o F for ag M if ad oly if hr i a opial oupio ad radig ragy o F h opial oupio hoi o F for ag M o i priod [ i 34 [ [ I γ whr [ I ivr u aig ha u I ad ad > γ i a agrag uliplir aifyig [ { } d I γ h opial porfolio olv 35 [ [ ' ' a ϑ σ ϕ whr [ { } ' db F d I ϕ γ Furhr h opial oupio ad porfolio pro aify 36 [ [ d I u d I u γ γ ad [ I γ ha aify quaio 34 ad 35 W alo u hav 37 < U ROOF: By h addl poi hor [ g ubrgr 969 ad ri oooiiy of U h opial oupio pro olv h uoraid probl 38 [ d U up γ

12 Fro Aupio A5 ad 38 w g 39 I [ γ [ hr i agrag uliplir uh ha { [ } I γ d oiuou ad rily draig ad ap hold i I [ I io ilf wih I [ + ad [ Fro 4 7 a 3 ad { I[ γ } d F ϕ db ' w g 4 ϕ ' [ σ ϑ ' h opial pro hav o alo aify quaio 36 ad 37 ad h oluio do o hav o b uiqu h opial oupio ad porfolio pro o F ar quaio 9 h ar opial pro bau h drif pro of pro of + ad ri ha i h ag ad + whr i olv i qual o zro ad h volailiy blog o h a quival la wih rp o h ag ar pri of M i ri ural wih rp o Corrpodigly if hr i opial ragi o F hollowig rpraio quaio 9 If + whr h h opial oupio ha h i a oupio ragy o F ad olv ha olv h hr i a oupio ragy ha giv or uiliy or hr do o i a opial oupio ragy ad a radig ragy ha fia h oupio pro hi i a oradiio i + i a opial oupio pro ad w g ha hr i opial oupio ad porfolio pro o F D hor ipli ha h for of h opial oupio radig ragy i h a a h orrpodig ragi i opl friiol ar hi opl ooy i dfid by h a-pri dflaor ipliily giv by quaio 34 ad 35 Diffr ag ay hav dii a-pri dflaor bau hir friio uiliy fuio ad dow pro ay diffr h diffiuly i olvig quaio i ha w rquir ha h orrpodig a-pri dflaor i ud wih a opial ragy ha i w a o ju fi h a-pri dflaor ad olv h opial oluio bau uually hi lad o ragi ha ry o a advaag fro friio If a ag ri o a advaag fro h friio h Aupio A i applid ad w h a pri qual o a pri bw h bid ad a pri uh ha h opial porfolio ir i zro Uig Aupio A hor ad h frawor of Cviai ad Karaza 996 w g h followig propoiio ROOSIION : If h opial oupio ad radig ragi of a ag M i h hy aify 4 U if u I [ γ d whr h ifiu i a ovr all a-pri dflaor [ τ i h opial oupio pro + { I[ γ } d i h opial radig ragy olv quaio 9 ad γ olv

13 ROOF: Fro hor w g U if u I [ γ d Now w rla h odiio ha h orrpodig a-pri dflaor ad pri pro ar ud wih a radig ragy u fi h pro ad driv h opial oupio ad radig ragi h h dflad walh pro i giv by 43 W d [ τ whr i h radig ragy whih fia h opial oupio i h fiiiou ar h walh pro ha a io aou h friio i giv by 44 W d[ + [ τ Fro 9 ad 3 w g W W hi giv [ { [ } if W F W F h ifiu i a ovr all arigal aur aig io aou hor w g 45 U if u I [ γ d whr D ropoiio ipli ha if hr i a oupio pro ha giv or uiliy ha h opial oupio pro h hr do o i a porfolio pro ha fia h oupio Auig Aupio A w g ha h radig i zro if h ag i ryig o a advaag fro h raaio o i if h or h i ryig o ll a h a pri or buyig a bid pri Suriy po-ar quilibriu i a ollio 46 { X M } uh ha giv h uriy-pri pro i 3 ad 33 ad ar lar ad h followig hor M X ad h a-pri pro for ah ag olv M [ HORM 3: h quilibriu odiio lad o h followig a qualiy 47 I [ γ [ M M Uig hor w g whr I ivr pro of u i h a-pri dflaor of ag M ad i h opial porfolio ROOF: Fro 47 w dirly ha oodiy ar lar ad ah ag ha opial oupio ad radig ragy hor giv i a ar lar bau ϕ D M Fro hor ad hor 3 w ha h uffii odiio for h i of a quilibriu o h iopl ar wih friio i ha hr i opial ragi for ah ag o hi or hr opl friiol ar ha i iid h iiial ar If h iiial ar i opl ad friiol h ad ar idiy appig ha i h rul of hi papr a b a a io o h orrpodig hor i opl ar M 3

14 5 SUMMARY hi papr udi h a-pri dflaor i h pr of friio ad wihou dyai paig A uiqu a-pri dflaor ay oly i o projd ar h i of h dflaor i quival o h i of a urjio ha ap h volailiy fuio oo h drif fuio hr i o arbirag if hr i a uiqu a-pri dflaor o a projd ar If hr i a opial oupio pro ad a porfolio ha fia h oupio o a projd ar h hr i a opial ragi alo o h iiial ar Giv h opial oupio pro ad radig ragy for igl ag hr i a quilibriu for radabl a 4

15 Rfr Ba K 99: A riig for Gral ro Joural of Mahaial ooi Brd D 979: A Irporal A riig Modl wih Sohai Coupio ad Iv Opporuii Joural of Fiaial ooi Clar S 993: h Valuaio robl i Arbirag ri hory Joural of Mahaial ooi Co J ad C F Huag 986: Mulipriod uriy ar wih diffrial iforaio Joural of Mahaial ooi Co J ad C F Huag 989: Opial Coupio ad orfolio olii Wh A ri Follow a Diffuio ro Joural of ooi hory Cviai J ad Karaza 99: Cov dualiy i Coraid orfolio Opiizaio Aal of Applid robabiliy Cviai J ad Karaza 993: Hdgig Coig Clai wih Coraid orfolio Aal of Applid robabiliy Cviai J ad Karaza 996: Hdgig ad orfolio Opiizaio udr raaio o: a Marigal Approah Mahaial Fia Duffi D ad C Siada 99: Coiuou-i Suriy riig: A Uiliy Gradi Approah Joural of Mahaial ooi Duffi D ad Sug 99: raaio Co ad orfolio Choi i a Dir-Coiuou i Sig Joural of Dyai ad Corol Duffi D ad W Za 989: h Coupio-Bad Capial A riig Modl ooria Groa S ad R Shillr 98: Coupio Corrlad ad Ri Maur i ooi wih No- radd A ad Hrogou Iforaio Joural of Fiaial ooi 95- Harrio J M ad S Krp 979: Marigal ad Arbirag i Mulipriod Surii Mar Joural of ooi hory Harrio J M ad S lia 98: Marigal ad Sohai Igral i h hory of Coiuou radig Sohai ro ad hir Appliaio 5-6 Huag C -F 987: A Irporal Gral quilibriu A riig Modl: h Ca of Diffuio Iforaio ooria Jouii ad H Kallal 995: Marigal ad Arbirag i Suriy Mar wih raaio Co Joural of ooi hory Karaza I J hozy ad S Shrv 987: Opial orfolio ad Coupio Diio for a Sall Ivor o a Fii Horizo SIAM Joural of Corol ad Opiizaio Klly J 955: Gral opology Nw Yor Sprigr-Vrlag Krp D 98: Arbirag ad quilibriu i ooi wih Ifiily May Coodii Joural of Mahaial ooi lad H 985: Opio riig ad Rpliaio wih raaio o Joural of Fia ubrgr D 969: Opiizaio by Vor Spa Mhod Nw Yor Wily Mro R 969: ifi orfolio Slio udr Uraiy: h Coiuou i Ca Rviw of ooi ad Saii Mro R 97: Opiu Coupio ad orfolio Rul i a Coiuou i Modl Joural of ooi hory