: Cox,RossandRubinstein,,7(1979),229 263,andCoxandRubinstein(1985),OptionsMarkets,(PricingaCallOption)Supposeu d r (interestrate),S .(Inthisandallexamples,theinterestratequotedisperunittime,andthestockpricesS S areindexedbythesametimeperiods).WeknowthatS if Hif TFindthevalueattimezeroofacalloptiontobuyoneshareofstockattime1for$50($50).Thevalueofthecallattime1isV S if Hif TSupposetheoptionsellsfor$:$
$$$ 59
isportfolio,thecashoutlayattime1is: H TPayoffoption Sellstock Payoffdebt Thearbitragepricingtheory(APT)valueoftheoptionattime0isV .AssumptionsunderlyingAPT: Unlimitedshortsellingofstock. Unlimitedborrowing. smallinvestor ,.,his/:-stepAPTSupposeaderivativesecuritypaysofftheamountV attime1,whereV isanF -measurablerandomvariable.(Thismeasurabilityconditionisimportant;thisiswhyitdoesnotmakesensetousesomestockunrelatedtothederivativesecurityinvaluingit,atleastinthestraightforwardmethoddescribedbelow). SellthesecurityforV attime0.(V istobedeterminedlater). Buy sharesofstockattime0.( isalsotobedeterminedlater)InvestV S inthemoneymarket,atrisk-freeinterestrater.(V S mightbe negative).Thenwealthattime1isX S r V r V S S r S WewanttochooseV and sothatX V regardlessofwhetherthestockgoesupordown.
(whichisfortunatesincetherearetwounknowns): r V S H r S V H () r V S T r S V T ()NotethatthisiswhereweusethefactthatthederivativesecurityvalueVkisafunctionofSk,.,whenSkisknownforagiven ,Vkisknown(andthereforenon-random)atthat fromthe rstgives SV HH VS TT ()Plugtheformula()for into(): r V VV H S H r S H V Hu V T d S u r S u d u d V H V H V T u r u r ddV H u u drV T Wehavealreadyassumedu d .Wenowalsoassumed r u(otherwisetherewouldbeanarbitrageopportunity).De nep
u r dd q
u u dr Thenp
andq
.Sincep
q
,wehave p
andq
p
.Thus,p
q
. Thusthepriceofthecallattime0isgivenbyV r p
V H q
V T ()-NeutralProbabilityMeasureLet bfytheformulaIfP n p
fj j Hgq
ffj j IfE rV
,thediscountedstockpriceprocessf r kSk Fkgnk :IfE r r k Sk jFk k p
u q
d Sk r k u u rd d d u r u d Sk r k u ur uud ddu d dr Sk r k u ud d r Sk r kSk n ,where kisthenumberofsharesofstockheldbetweentimeskandk .Each kisFk-measurable.(Noinsidertrading).- nancingValueofaPortfolioProcess StartwithnonrandominitialwealthX , nerecursivelyXk k Skr X r Xk kSk ()k k Sk r Sk () ,thediscountedself- nancingportfolioprocessvaluef r kXk Fkgnk :Wehave r k Xk r kXk k r k Sk r kSk
,IfE IfEr k Xk jFk r kXkjF IIffEEk r kk kSk jFk r k Xr kSkjFk k(requirement(b)ofco nditionalexp.) k IfEr k rk S k kSk jFk (takingoutwhatisknown)r kX(property(b))k() ()AsimpleEuropeanderivativesecuritywithexpirationtimemisanFm-measurablerandomvariableVm.(Here,mislessthanorequalton,thenumberofperiods/coin-tossesinthemodel).De ()AsimpleEuropeanderivativesecurityVmissaidtobehedgeableifthereexistsaconstantX andaportfolioprocess m suchthattheself- nancingvalueprocessX X XmgivenbyX()satis es Vm Inthiscase,fork m,()IfasimpleEuropeansecurityVmishedgeable,thenforeachk m,theAVPT va lueattimekofVmisk r kIfE r mVmjFk ()Proof:We rstobservethatiffMk Fk k mgisamartingale,.,satis esthemartingalepropertyIfE Mk jFk Mkforeachk m ,thenwealsohaveIfE MmjFk Mk k m ()Whenk m ,theequation() m
,weusethetowerpropertyftowriteIE MmjFm IIMffEE IfE MmjFm jFm m M m jFm
64Wecancontinuebyinductiontoobtain().IfthesimpleEuropeansecurityVmishedgeable,thenthereisaportfolioprocesswhoseself- nancingvalueprocessX X Xmsatis esXm nition, r X r mXmisamartingale,andsoforeachk, r kXk IfE r mXmjFk IfE r mVmjFk Therefore,Xk r kIfE r mVmjFk yes , no ,,letVmbeasimpleEuropeanderiva-tivesecurity,andVsetk k r kIfE r mVmjFk k () k k SVkk k H Vk k T k H S k k T ()Sta rtingwithinitialwealthV IfE r mVm ,theself- nancingvalueoftheportfolioprocess m istheprocessV V :LetV Vm Xand m bede nedby()and().SetX V andde netheself- nancingvalueoftheportfolioprocess m :k kSk r Xk kSk WeneedtoshowthatXk Vk k f mg () ,()holdsbyde nitionofX .Assumethat()holdsforsomevalueofk,.,foreach xed k ,wehavek k Vk k
k H Vk k H fk k T Vk k T Weprovethe rstequality; rstthatIE r k Vk jFk IIffEE IfE rm mVmjFk jFk r f rk VVkmjFk Inotherwords,f r kVkgnVk ,k k IfE r Vk jFk k r p
Vk k H q
Vk k T Since k willbe xedfortherestoftheproof,,wewritethelastequationasVk r p
Vk H q
Vk T WecomputeXk H kSk Sk k H r Xk kSk Vk H r Sk r VkSk Vk p
V HH SVkk TT Sk H r Sk k H q
Vk T u HS Vk T p
Vk dSk uSk r Sk k H q
Vk T Vk H Vk T u u dr p
Vk H q
Vk T VVkk HH Vk T q
p
Vk H q
Vk T
