A New Simple Proof of the No-arbitrage Theorem for Multi-period Binomial Model

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Everett Baldwin
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1 Inernaional Jornal of Bsiness an Social Science ol No ; Jl A New Simple Proof of e No-arbirage eorem for Mli-perio Binomial Moel Liang Hong ASA PD Assisan Professor of Maemaics an Acarial science Deparmen of Maemaics Brale Universi 5 Wes Brale Avene Peoria IL 665 USA aress: long@bralee Ame Elsaa CFM PD Assisan Professor of Finance Deparmen of Finance an Qaniaive Meos Foser College of Bsiness Brale Universi 5 Wes Brale Avene Peoria IL 665 USA aress: aelsaa@bralee Absrac Binomial opion pricing moel is one of e wiel se moels o price opion conracs wic are commonl emploe o ege agains risks in e insrance fiel One of e main nerling assmpions of e binomial moel is e no-arbirage coniion is coniion simpl provies a necessar an sfficien coniion for e moel o be free of arbirage opporniies Previos aemps were mae o assess e viabili of is assmpion B e were qie complicae an leng wic en o obscre e nerling meaning an ma appear o be aning an inaccessible o e general aience In is paper we sppl a clear clean an simple proof sing onl pre-rigonomeric algebra Inrocion Insrance companies ofen rel on invesmen opporniies osie e raiional insrance secor o manage eir financial risk In fac e paoffs of man poplar eqi-linke insrance conracs sc as variable anni A) conracs segregae fn conracs an eqi-inexe anni conracs can be expresse in erms of opions cf Har 3)) Man opion pricing moels ave been propose an sccessfll se in qaniaive finance over e pas ecaes cf Hll 9)) an Karaas an Sreve 5)) e binomial moel is a well-known asse pricing moel firs inroce b Cox Ross an Rbinsein 979) I is wiel se for opion pricing cf Hll 9) or Sreve 5)) Is simplici makes i eas o appl in man cases For Eropean opions e moel sall provies a close-form solion cf Breale Mers an Allen 5) Copelan Weson an Sasri 5) an Hll 9)) For American opions a close form solion is sall no available b one can work backwar an vale i analicall wi e ai of a comper program cf Geske an Jonson 984)) Moreover e moel is imporan in is own rig as e famos Black- Scoles-Meron formla can be erive as a limi of e binomial moel cf Cox Ross an Rbinsein 979) an Sreve 5)) In aiion e binomial ree moel can be exene o a rinomial ree moel for pricing exoic opions wi regime-swicing cf Ye an Yanc )) Besies is applicaion in financial opion e binomial moel as also been applie o evalae real opions cf Copelan Weson an Sasri 5) an rigeorgis 996)) s a orog nersaning of e binomial moel is esseniall for eoriss empiriciss an praciioners in economics finance an acarial science In e following we mainl aop e noaions in Bjork 4) e mli-perio binomial moel can be escribe as follows Assme ere exiss a bon B a ime wi e price B were We also assme ere exiss a sock wi e price S wic is a ranom variable We consier ime as e beginning of e perio s B an S are e iniial prices of e bon an e sock respecivel

Bsiness Brale Universi 5 Wes Brale Avene Peoria IL 665 USA Email aress: aelsaa@bralee Absrac Binomial opion pricing moel is one of e wiel se moels o price opion conracs wic are commonl emploe o ege

2 Cenre for Promoing Ieas USA wwwijbssnecom We assme S is a posiive consan) A e en of e firs perio e sock price will eier move from S p o S wi p probabili p or go from S own o S wi own probabili p were an p p s we ave S wi probabili p S S wi probabili p Similarl a e en of e secon perio e sock price will eier move from S p o S wi p probabili p or go from S own o S wi own probabili p Hence S wi probabili p S S wi probabili p is process conines wi e p probabili be expresse as S S p an e own probabili p ncange erefore S can an are wo ii binomial ranom variables Following is line of reasoning we ma simpl wrie e price of e sock a e en of - perio ) as S S were S is a consan an are ii binomial ranom variables sc a k wi probabili p wi probabili p Definiion A porfolio x ) represens e nmber of bons an sares a ime We assme a is a measrable fncion of S S S S S S wi e convenion a From avance probabili cf Billingsle 995) Cow an eicer 997) or Siraev 984)) we know iself is a ranom variable Definiion e vale of e porfolio a ime is enoe b Assming compon ineres wi consan effecive rae r we en ave x B S x S ) o sae e no-arbirage coniion eorem we nee o inroce e following concep: Definiion 3 A porfolio is calle self-financing if e following ieni ols for all : x S x S 3) Remark Recall B Eqaion 3) means no exernal financing is neee a an ime e porfolio once forme can fn iself If we le x 4) en eqaion 3) can be wrien as S S 5)

probabili p S S wi probabili p is process conines wi e p probabili be expresse as S S p an e own probabili p ncange erefore S can an are wo ii binomial ranom variables Following is line of reasoning

3 Inernaional Jornal of Bsiness an Social Science ol No ; Jl Noe a Bjork 4) ses 5) o efine self-financing Definiion 4 An arbirage possibili is a self-financing porfolio sc a P P ) ) Wi all ese conceps we now can sae e no-arbirage coniion eorem eorem No-arbirage Coniion eorem) e mli-perio binomial moel is arbirage-free if an onl if r 6) Sreve 5) sows e necessi in e one-perio moel sing raing sraeg Bjork 4) gives a leng proof for e mli-perio moel sing maringale probabili an binomial algorim In e nex secion we provie a new an simple proof sing onl elemenar algebra A New an Simple Proof of No-arbirage Coniion eorem Since eqaion 5) implies x S ) From 5) we also ave a S S ) Firs le's consier e case In is case x S wi probabili p x S wi probabili p Now b 4) an ) we can wrie r r as S wi probabili p S wi probabili p Mlipling e above eqaliies ogeer we see rs or rs if an onl if S r r 3) However 3) ols if an onl if r Since we assme p an p is sows ere exis no arbirage opporniies if an onl if r Now le's consier e case B ) an ) we ave S S S S 4) Sbsiing 4) ino ) we obain S S S 3

Bjork 4) gives a leng proof for e mli-perio moel sing maringale probabili an binomial algorim In e nex secion we provie a new an simple proof sing onl elemenar algebra A New an Simple Proof of

4 Cenre for Promoing Ieas USA wwwijbssnecom In view of ) if ere exiss an arbirage opporni en e following ineqaliies ol: P U an D en e above ineqaliies can be wrien as U U D D U D U D Noe a e lef-an sies of ese ineqaliies are e irec proc of D an U wi r ) an r ) Sppose 6) ols en D U an ence e irec proc of D an U wi r ) an r ) canno all be posiive In oer wors 5) canno ol is conraics or assmpion a is an arbirage opporni s sfficienc is prove On e oer an if 6) fails o ol sa ie D U 5) en an porfolio wi is clearl an arbirage opporni Similarl if ie D U en an porfolio wi will be an arbirage opporni s e necessi is esablise oo is complees e proof for e case e general case can be sown b following exacl e same line of reasoning se for e case b in e space of iger imensions 3 Smmar e necessar an sfficien coniion for e mli-perio binomial moel o be arbirage-free can be clearl sae in erms of e p facor e own facor an e ineres rae e formla is simple However mos proofs in e exising lierare involve avance maemaics sc as maringale eor Here we provie a proof wic onl reqires elemenar algebra Hence or argmen is compleel accessible o e general aience e proof assmes a consan force of ineres Relaxing is assmpion can lea o frer researc In a case o caracerie e coniion for e moel o be arbirage-free can be callenging One probabl as o resor o eav probabilisic macineries sc as maringales However e proof in is paper will offer some insig Acknowlegmens: e firs aor sincerel anks Dr Pavel Bleer for a frifl an enligening iscssion abo e problem His eep insig leas o some resls of e paper 4

is an arbirage opporni s sfficienc is prove On e oer an if 6) fails o ol sa ie D U 5) en an porfolio wi is clearl an arbirage opporni Similarl if ie D U en an porfolio wi will be an arbirage opporni

5 Inernaional Jornal of Bsiness an Social Science ol No ; Jl REFERENCES Billingsle P 995 Probabili an Measre 3r eiion Wile Bjork 4 Arbirage eor in Coninos ime n eiion Wile Breale RA Mers SC an Allen F 5 Principles of Corporae Finance 8 eiion Wile Cow YS eicer H 997 Probabili eor: Inepenence Inercangeabili Maringales 3r eiion Springer Copelan E Weson JF an Sasri K 5 Financial eor an Corporae Polic 4 eiion Aison Wesle Cox JC Ross S Rbinsein M 979 Opion Pricing: A Simplifie Approac Jornal of Financial Economics Geske R an Jonson HE 984e American P Opion ale Analicall e Jornal of Finance XXXIX 5) 5-54 Har M 3 Invesmen Garanees: Moeling an Risk Managemen for Eqi-Linke Life Insrance Wile Hll J 9 Opions Fres an Oer Derivaives 7 eiion Prenice Hall Siraev AN 984 Probabili n eiion Springer Sreve S 5 Socasic Calcls for Finance I: e Binomial Asse Pricing Moel Springer Sreve S 5 Socasic Calcls for Finance II: Coninos-ime Moel Springer rigeorgis L 996 Real Opions; Managerial Flexibili an Sraeg in Resorce Allocaion MI Press Ye FL an Yan L Opion Pricing wi Regime Swicing b rinomial ree Meo Jornal of Compaional an Applie Maemaics

4 eiion Aison Wesle Cox JC Ross S Rbinsein M 979 Opion Pricing: A Simplifie Approac Jornal of Financial Economics 3 45-66 Geske R an Jonson HE 984e American P Opion ale Analicall e Jornal of Finance

Commun Nonlinear Sci Numer Simulat

Commun Nonlinear Sci Numer Simulat Commn Nonlinear Sci Nmer Simla 4 (2009) 3463 347 Conens liss available a ScienceDirec Commn Nonlinear Sci Nmer Simla jornal omepage: www.elsevier.com/locae/cnsns Oscillaion of second-order srongly sperlinear