rewrite515hh

UniversityofConnecticutOpenCommons@UConnHonorsScholarThesesHonorsScholarProgramSpring5-1-2017MathematicalModelingofFinancialDerivativePricingKellyL.
CosgroveUniversityofConnecticut,cosgrove.
kelly@gmail.
comFollowthisandadditionalworksat:https://opencommons.
uconn.
edu/srhonors_thesesPartoftheFinanceandFinancialManagementCommons,andtheOtherAppliedMathematicsCommonsRecommendedCitationCosgrove,KellyL.
,"MathematicalModelingofFinancialDerivativePricing"(2017).
HonorsScholarTheses.
515.
https://opencommons.
uconn.
edu/srhonors_theses/515MathematicalModelingofFinancialDerivativePricingKellyCosgroveMay6,2017AbstractThebinomialasset-pricingmodelisusedtopricenancialderivativesecurities.
Thistextwillbeginbygoingovertheprobabilityconceptsnecessarytounderstandthisdiscrete-timemodel.
Itthendevelopsthetheorybehindthebinomialmodelanddierentpropertiesthatarise.
Itshowshowtousethebinomialmodeltopredictfuturestockprices,andthenusesthisinformationtopricederivativesecurities.
ItinitiallyfocusesontheEuropeancalloption,butgoesontoprovideapricingmethodfortheAmericanputoption.
However,manyofthetheoremsdevelopedareapplicabletoallderivativesecurities.
Thetextwrapsupbyconsideringadierentmethodusedinpricingderivativesecurities,theBlack-Scholesmodel,whichisbasedoncontinuous-timeconcepts.
Contents1ProbabilityTheory31.
1FiniteProbabilitySpaces31.
2RandomVariables41.
3MartingalesandMarkovProcesses62TheBinomialModel72.
1Structure72.
2PricingDerivatives82.
3Properties123ApplicationtoAmericanDerivativeSecurities153.
1Introduction153.
2Path-Independent153.
3StoppingTimes193.
4Path-Dependent214TheBlack-ScholesModel264.
1RandomWalk264.
2TheBlack-ScholesModel281IntroductionThebinomialasset-pricingmodelisusedtopricenancialderivativesecurities.
Inthistext,wewillmostlyusetheexampleoftheEuropeancalloptiontoillustratethefunctionthebinomialmodelserves.
Thistypeofderivativeisonethatallowsitsownertheright(butnottheobligation)tobuyastockataspeciedstrikepriceonaspeciedexpirationdate.
AsimilarderivativeistheEuropeanputoptionwhichgivestheownertherighttosellstockataspecicpriceonaspecicdate.
Webeginbyreviewinggeneralprobabilityconceptsneededtodevelopthebinomialasset-pricingmodelinChapter1.
Topicsincludeniteprobabilityspaces,randomvariables,andpropertiesofconditionalexpectations.
Chapter1thengoesontodevelopthepropertiesofcertainadaptedstochasticprocesses.
Namely,weshalllookatmartingalesandMarkovprocesses.
Wedeveloptheideaofthebinomialasset-pricingmodelandhowtouseittopriceEuropeanderivativesecuritiesinChapter2.
Inordertoaccomplishthis,wewillneedtogureouthowtoreplicatethederivativesecurityinthestockandmoneymarkets.
Onceweareabletodothis,wecandetermineafairpriceforthederivativesecuritybysettingitsvalueattimenequaltothatofourreplicatedportfolio.
InChapter3wewillexploreAmericanderivativesecuritieswhicharedenedsimilarlytotheirEuropeancounterpartsexceptforthefactthatitsownerhastherighttoexercisetheoptionatanypointuptoorontheexpirationdate.
Thiscomplicatestheprocessofbuildingareplicatingportfoliobecausetherewillexistanoptimalexercisedateonwhichtheownerofthederivativesecurityshouldexerciseit(whichwemustnd).
TheconceptofstoppingtimeswillbedevelopedinthischapterandwillbecrucialtoguringouttheoptimalexercisedateofanAmericanderivativesecurity.
Chapter4wrapsupthepaperbyintroducinganalternatemethodforpricingderivativesecuritiesknownastheBlack-Scholesmodel.
Itbeginsbydescribingtherandomwalk,whosecontinuous-timecounterpart,Brownianmotion,isanunderlyingassumptionoftheBlack-Scholesmodel.
ItthenprovidesthegeneralideabehindBlack-Scholesandhowitisrelatedtothebinomialmodel.
Severalsourceswerereferencedinordertocompletethispaper,butStevenE.
Shreve'sStochasticCalculusforFinanceITheBinomialAssetPricingModel[Shreve,2004]wasthemostprominentlyreferenced,asthesourceofthematerialinChapters1,2,and3,andSection4.
1.
Further,RossitsaYalamova'sSimpleheuristicapproachtointroductionoftheBlack-Scholesmodel[Yalamova,2010]wasreferencedtocompleteSection4.
2,andNogaAlonandJoelH.
Spencer'sTheProbabilisticMethod[AlonandSpencer,2015]wasreferencedforTheorem4.
2.
1(Azuma'sInequality).
2Chapter1ProbabilityTheory1.
1FiniteProbabilitySpacesBecausethebinomialasset-pricingmodelisanapplicationofprobabilitytheory,itiscrucialtohaveagoodunderstandingofgeneralprobabilityconceptsinordertounderstandthemodelitself.
Inthissectionwewillbuildupourunderstandingofaniteprobabilityspace.
Wewillusetheexampleofacointosstoillustratethecomponentsofaniteprobabilityspace.
Saywetossacoin3times.
LetHdenoteaheadsipandTdenoteatailsip.
Thesamplespace,,isthenitesetofallpossibleoutcomesthatcouldresultfromtossingthatcoin,or={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}(1.
1)Sequencesofoutcomesaredenotedby!
=!
1!
2!
3.
Wecallsubsetsofevents.
Forexample,theeventAthatwegetatleasttwotails:A={HTT,THT,TTH,TTT}Ifwelet#T(!
1!
2!
3)bedenedasthenumberoftailsthatappearinoursequenceof3cointosses,thenwecanrewriteAas:A={!
2;#T(!
1!
2!
3)2}(1.
2)Finally,wemustdeterminethelikelihoodofeachsequenceoccurring.
Wecandothisbylettingpbetheprobabilityofthecoinlandingonheadsandq=1–pbetheprobabilityoftails.
Then,theprobabilitiesofeach!
occurringareP(HHH)=p3,P(HHT)=p2q,P(HTH)=p2q,P(THH)=p2q,P(HTT)=pq2,P(THT)=pq2,P(TTH)=pq2,P(TTT)=q3(1.
3)Further,wecangureouttheprobabilityofaneventbyaddinguptheindividualprobabilitiesofeachpossiblesequenceintheevent:P(A)=P(HTT)+P(THT)+P(TTH)+P(TTT)=pq2+pq2+pq2+q3(1.
4)NotethatP!
2P(!
)=1,orP()=1.
Denition1.
1.
1.
AniteprobabilityspaceconsistsofasamplespaceandaprobabilitymeasurePthattakeseachelement!
2andassignsitavalueintheinterval[0,1].
Wedenoteitby(,P).
31.
2RandomVariablesTheniteprobabilityspacemodelsasituationinwhicharandomexperimentisconducted.
Theseexperimentstypicallyproducenumericaldatawhichcanbewrittenasrandomvariables.
Denition1.
2.
1.
Let(,P)beaniteprobabilityspace.
ArandomvariableisafunctionthatmapsontoR.
ItisimportanttonotethatrandomvariablesthemselvesdonotdependontheprobabilitymeasureP.
ThedistributionofarandomvariableusesPtodeterminetheprobabilitiesoftherandomvariabletakingdierentvalues.
Wecankeepoursamplespaceandrandomvariablefunctionconsistentbetweentwoniteprobabilityspaces,butiftheirprobabilitymeasuresdier,thesamerandomvariablecanhavetwodierentdistributions.
Thisisillustratedintheexamplebelow.
Example1.
2.
2.
Supposewetossacoin3timessuchthat={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}.
NowdenetherandomvariableXtobethetotalnumberofheadsandYtobethetotalnumberoftails.
Therefore,X(HHH)=3X(HHT)=X(HTH)=X(THH)=2X(HTT)=X(THT)=X(TTH)=1X(TTT)=0(1.
5)Y(HHH)=0Y(HHT)=Y(HTH)=Y(THH)=1Y(HTT)=Y(THT)=Y(TTH)=2Y(TTT)=3(1.
6)WehavenotyetsettheprobabilitymeasurePforourniteprobabilityspaceandwewerestillabletodeterminethevariousvaluesforourrandomvariablesXandY.
Now,letusspecifyP1suchthattheprobabilityofgettingheadsis12(andthereforetheprobabilityofgettingtailsis1–12=12).
Then,P1(!
)=121212=18forall!
2.
LetP{X=i}:=P{!
2;X(!
)=i}.
Thedistributionwouldbeasfollows:P1{X=3}=P1{HHH}=18P1{X=2}=P1{HHT,HTH,THH}=38P1{X=1}=P1{HTT,THT,TTH}=38P1{X=0}=P1{TTT}=18(1.
7)NowletususeadierentP,sayP2,suchthattheprobabilityofgettingheadsis23andtheprobabilityofgettingtailsis13.
Bysimilarcalculations,wegetthefollowingdistribution:P2{X=3}=P2{HHH}=827P2{X=2}=P2{HHT,HTH,THH}=1227P2{X=1}=P2{HTT,THT,TTH}=627P2{X=0}=P2{TTT}=127(1.
8)4Clearlythesedistributionsdier,eventhoughwearemeasuringthesamerandomvariable.
Knowingtheprobabilitydistributionofarandomvariable,wecancalculateasinglevalueofwhatweexpecttheresultofourrandomexperimenttobeifweweretoactuallyconductit.
Denition1.
2.
3.
LetXbearandomvariabledenedonaniteprobabilityspace(,P).
Theex-pectedvalue,E,ofXisdenedasEX=X!
2X(!
)P(!
)(1.
9)ThevarianceofXisVar(X)=E(XEX2)(1.
10)Usingoursamplespace(,P2),wecanexpectthenumberofheadsthatshowupafterthreecointossestobeEX=X!
2X(!
)P2(!
)=X(HHH)P2{HHH}+X(HHT)P2{HHT}+X(HTH)P2{HTH}+X(THH)P2{THH}+X(HTT)P2{HTT}+X(THT)P2{THT}+X(TTH)P2{TTH}+X(TTT)P2{TTT}=3827+2427+2427+2427+1227+1227+1227+0127=2(1.
11)Thisagreeswithourintuitionthatifwehavea23probabilityofgettingheads,thenevery3tosses,weshouldexpecttosee2heads.
Thisisthebestestimateifweareprovidedwithnoadditionalinformation.
However,saywechangethescenariosothatwehavealreadyippedthecointwotimes,andbothtimeshaveturneduptails!
Surely,gettingtwoheadsbyourthirdtosswouldbeimpossible.
Wecanadjustourex-pectedvalueusingtheinformationaboutthersttwotossesthatwenowhave.
ThisiscalledtheconditionalexpectationofXbasedontheinformationwehave.
Denition1.
2.
4.
Givenncointossessuchthat1nN,thereare2Nnpossiblecontinua-tions!
n+1.
.
.
!
Nofthesequence!
1.
.
.
!
n.
Letpbetheprobabilityofgettingheadsandq=1-pbetheprobabilityofgettingtails.
Let#H(!
n+1.
.
.
!
N)bethenumberofheadsin!
n+1.
.
.
!
Nand#T(!
n+1.
.
.
!
N)bethenumberoftailsin!
n+1.
.
.
!
N.
Then,ourconditionalexpectationofXbasedontheinformationattimenisEn[X](!
1.
.
.
!
n)=X!
n+1.
.
.
!
Np#H(!
n+1.
.
.
!
N)q#T(!
n+1.
.
.
!
N)X(!
1.
.
.
!
n!
n+1.
.
.
!
N)(1.
12)Theorem1.
2.
5(Fundamentalpropertiesofconditionalexpectations).
LetNbeapositiveintegerandletX,YberandomvariablesdependingontherstNcointosses.
Let0nNbegiven.
Then,thefollowingpropertieshold.
(i)Linearityofconditionalexpectations.
Forallconstantsc1andc2,En[c1X+c2Y]=c1En[X]+c2En[Y](1.
13)(ii)Takingoutwhatisknown.
IfXactuallydependsontherstncointosses,thenEn[XY]=X·En[Y](1.
14)5(iii)Iteratedconditioning.
If0nmN,thenEn[Em[X]]=En[X](1.
15)(iv)Independence.
IfXdependsonlyontossesn+1throughN,thenEn[X]=E[X](1.
16)Theproofofthistheoremresultslargelyfromthedenitionofconditionalexpectationandwillbelefttothereadertoworkthroughifdesired.
1.
3MartingalesandMarkovProcessesDenition1.
3.
1.
Considerthecointossscenario.
LetM0,M1,MNbeasequenceofrandomvariablessuchthateachMn,0nN,dependsonlyontherstncointosses.
Then,wecallthissequenceanadaptedstochasticprocess.
Wecanfurtherclassifyadaptedstochasticprocessesbyhowweexpectthemtochangefromonecointosstothenext.
GivenMn,andcalculatingMn+1usingourdenitionofconditionalexpectation,wewillndthatcertainadaptedstochasticprocessescanbeexpectedtorise,otherscanbeexpectedtofall,andotherscanbeexpectedtoremainconstant.
Suchclassicationsaredenedformallybelow.
Denition1.
3.
2.
LetthesequenceofrandomvariablesM0,M1,MNbeanadaptedstochasticprocess.
IfMn=En[Mn+1],n=0,1,N1,thisprocessisamartingale.
IfMnEn[Mn+1],n=0,1,N1,thisprocessisasubmartingale.
IfMnEn[Mn+1],n=0,1,N1,thisprocessisasubmartingale.
Thoughthemartingalepropertyis"one-step-ahead"byonlyspecifyingtheexpectationoftheimmediatenextvariableinthesequence,wecanextendthepropertyforanyvariableappearingafterMnusingtheiteratedconditioningpropertyofconditionalexpectation.
Forexample,ifthesequenceM0,M1,MNisamartingaleandnN2,weknowthatMn+1=En+1[Mn+2].
(1.
17)Applyingtheiteratedconditioningproperty,weseethatEn[Mn+1]=En[En+1[Mn+2]]=En[Mn+2].
(1.
18)AndsinceMn=En[Mn+1],Mn=En[Mn+2].
(1.
19)Thenotionofconditionalexpectationsgiverisetomanyalgorithmsthatserveaspowerfulpredictivetoolsindierentscenarios.
But,ifanadaptedstochasticprocessispathindependent;namely,ifavariableinthesequenceonlydependsontheimmediateformervariable,oftentimesthesealgorithmscanbegreatlysimplied.
ThistypeofprocessisknownasaMarkovprocess.
Denition1.
3.
3.
LetX0,X1,XNbeanadaptedstochasticprocess.
Iffor0nN1thereexistsafunctiong(x)foreveryf(x)dependingonnandfsuchthatEn[f(Xn+1)]=g(Xn)(1.
20)WecallX0,X1,XNaMarkovprocess.
Themartingaleisaspecialcaseof(1.
20)withf(x)=xandg(x)=x.
Butsincetheremustexistag(x)foreveryf(x)suchthatequation(1.
20)holdsinorderforaprocesstobeMarkov,noteverymartingaleisaMarkovprocess.
Ofcourse,sinceg(x)doesnothavetoequalx,noteveryMarkovprocessismartingale.
6Chapter2TheBinomialModel2.
1StructureS3(HHH)=u3S0S2(HH)=u2S0S1(H)=uS0S3(HHT)=S3(HTH)=S3(THH)=u2dS0S0S2(HT)=S2(TH)=udS0S1(T)=dS0S3(HTT)=S3(THT)=S3(TTH)=ud2S0S2(TT)=d2S0S3(TTT)=d3S0Athree-periodbinomialmodel.
Thebinomialmodelabovedisplaysthepossiblepathsofastockprice,withS0>0beingtheinitialpricepershareattimet=0.
Themodelisbrokenupintoperiods,withtimet=1beingtheendoftherstperiod,t=2theendofthesecond,andt=3theendofthethird.
Weextendourcointossscenarioontothismodelandletheadsrepresentthestockpriceincreasingandtailsrepresentthestockpricedecreasing.
Attheendofeachperiodthestockcantakeontwopossiblevalues,oneofwhichisgreaterthanitspreviousvalueandoneofwhichisless.
Herewecanintroducetwonewvariables,uandd,thatserveasthetwopossibleratiosofthenewstockpricetotheformerstockprice.
IfwehaveatotalofNperiods,thenforeachn,11and0common,butnotnecessary,tohavedandusuchthatd=1u.
2.
2PricingDerivativesInordertoensurenoarbitragewhenpricingourderivatives,thereexistsanarbitragepricingtheorybyreplicatingitthroughtradinginthestockandmoneymarkets.
Firstly,oneshouldassignapricetothesecuritytopreventthepossibilityofarbitrage.
Secondly,foranexpirationtimeN,atanytimencomeoftherstcointoss,wereallyhavetwoequations:X1(H)=0S1(H)+(1+r)(V00S0)(2.
4)X1(T)=0S1(T)+(1+r)(V00S0)(2.
5)Theagentcanadjustherportfoliobasedontheoutcomeoftherstcointoss.
Shenowdecidestohold1sharesofstockandinvestsX11S1inthemoneymarket.
Becausewearereplicatingtheoption,wewanttheportfoliovaluedatV2.
Inotherwords,wewantV2=1S2+(1+r)(X11S1)(2.
6)But,S2andV2dependonthersttwocointossessowereallyhavefourequations:V2(HH)=1(H)S2(HH)+(1+r)(X1(H)1(H)S1(HH))(2.
7)V2(HT)=1(H)S2(HT)+(1+r)(X1(H)1(H)S1(HT))(2.
8)V2(TH)=1(T)S2(TH)+(1+r)(X1(T)1(T)S1(TH))(2.
9)V2(TT)=1(T)S2(TT)+(1+r)(X1(T)1(T)S1(TT))(2.
10)Now,wehavesixequationsandsixunknowns:V0,0,1(H),1(T),X1(H),X1(T).
Let'srstndX1(T)and1(T)bylookingatequations(2.
9)and(2.
10).
Wecaneasilysolvefor1(T)bysubtracting(2.
10)from(2.
9)andsolvingforit.
Thesolutionweobtainiscalledthedelta-hedgingformula.
1(T)=V2(TH)V2(TT)S2(TH)S2(TT)(2.
11)NextwesolveforX1(T).
SinceS2(TH),S2(TT),S1(TH),andS1(TT)arerandomvariables,weknowthatifourrstcointossistails,wehavesomeprobabilitypandq=1pprobabilityofgettingtails.
Inotherwords,wehavepprobabilityofequation(2.
9)andqprobabilityofequation(2.
10).
InordertondtheexpectedvalueofX1(T),wemultiply(2.
9)bypand(2.
10)qandaddthemtogether.
Wearealsogoingtodividealltermsby(1+r).
11+rpV2(TH)+qV2(TT)=1(T)11+r[pS2(TH)+qS2(TT)](pq)S1(T)+(pq)X1(T)(2.
12)Since(p+q)=1,wecansimplify(2.
12)toget11+rpV2(TH)+qV2(TT)=1(T)11+r[pS2(TH)+qS2(TT)]S1(T)+X1(T)(2.
13)IfwechoosepsuchthatS1(T)=11+r[pS2(TH)+qS2(TT)](2.
14)Wecangreatlysimplify(2.
13).
UseS1(T)=dS0,S2(TH)=duS0,andS2(TT)=ddS0,andq=1p.
Then,(2.
14)becomesdS0=11+r[pduS0+(1p)ddS0](2.
15)DividebothsidesbydS0toget1=11+r[pu+(1p)d](2.
16)9Andfromtherewecaneasilysolveforp.
Similarly,wecansolve(2.
14)forq.
p=1+rdud,q=u1rudTheseprobabilitiespandqarecalledriskneutralprobabilities.
Normally,theaveragegrowthrateofastockexceedsthatofthemoneymarket;otherwise,itwouldnotmakesensetoriskinvestinginstock.
So,theactualprobabilitiespandqofagivenstockshouldsatisfy(1+r)S1(T)comeupwiththefollowingtwoequationsfor1(H)andV1(H)=X1(H):1(H)=V2(HH)V2(HT)S2(HH)S2(HT)(2.
19)V1(H)=11+rpV2(HH)+qV2(HT)(2.
20)Thelatterequationisthepriceoftheoptionattimet=1ifthersttossresultsinhead.
Finally,wecansolveforV0and0bypluggingourvaluesforX1(H)=V1(H)andX1(T)=V1(T)intoequations(2.
4)and(2.
5).
Clearly,wehavethreestochasticprocesses:(0,1),(X0,X1,X2),and(V0,V1,V2).
Werecursivelydenedourportfolio,andifwespecifyvaluesforX0,0,1(H),and1(T),wecanrecursivelydeneareplicatingportfoliowithanynumberofperiodsbythewealthequationXn+1=nSn+1+(1+r)(XnnSn)(2.
21)andwecanpriceourstockoptioninasimilarmanneraswedidforthetwo-periodmodel.
Theorem2.
2.
1(Replicationinthemultiperiodbinomialmodel).
ConsideranN-periodbinomialasset-pricingmodelwith0comesXn+1(H)=nuSn+(1+r)(XnnSn)(2.
27)Fromequation(2.
24),andusingoursimpliednotation,weknowthatn=Vn+1(H)Vn+1(T)Sn+1(H)Sn+1(T)(2.
28)Wemayrewritethisasn=Vn+1(H)Vn+1(T)(ud)Sn(2.
29)Now,wecanrewriteequation(2.
27)andsubstitutethisinforn,alongwithusingourhypothesisthatVn=Xn:Xn+1(H)=(1+r)Xn+nSn(u(1+r))=(1+r)Vn+(Vn+1(H)Vn+1(T))(u(1+r)udWecanalsouseequation(2.
22)tosubstituteqintotheequation:Xn+1(H)=(1+r)Vn+qVn+1(H)qVn+1(T)=pVn+1(H)+qVn+1(T)+qVn+1(H)qVn+1(T)=Vn+1(H)AsimilarargumentshowsthatXn+1(T)=Vn+1(T).
So,nomatterwhat!
n+1is,wehaveXn+1(!
1!
2.
.
.
!
n+1)=Vn+1(!
1!
2.
.
.
!
n+1)Theinductionstepiscomplete.
ThistheoremwasprefacedwiththeexampleofaEuropeancalloptionforwhichthepayoonlydependsonthenalstockprice.
Thetheoremalsoappliestopathdependentoptionswhosepayodependsonthedierentvaluesthestocktakesonbetweenitsinitialvalueanditsnalvalue.
WewillexplorethepathdependentexampleofAmericanderivativesinChapter3.
112.
3PropertiesIntheprevioussectionwecameupwithrisk-neutralprobabilitiespandq.
WecandenetheseprobabilitiesastheprobabilitymeasureP.
Similarly,ourrisk-neutralexpectedvalueforrandomvariableXundertheserisk-neutralprobabilitiesisEX.
Recallthatp=1+rdud,q=u1rudItiseasytocheckthatpu+qd1+r=1So,forallnandforeverycointosssequence!
1.
.
.
!
n,wecanwriteSn(!
1.
.
.
!
n)=11+rpSn+1(!
1.
.
.
!
nH)+qSn+1(!
1.
.
.
!
nT)(2.
30)But,usingourknowledgeofconditionalexpectationfromDenition1.
2.
4,wecanrewritethetheaboveequationasfollows:Sn=11+rEn[Sn+1](2.
31)Ifwedividebothsidesof(2.
31)by(1+r)n,wegettheequationSn(1+r)n=EnhSn+1(1+r)nn+1i(2.
32)whereSn(1+r)nisthediscountedstockpriceattimen.
Thisequationexpressesakeyfactthatundertherisk-neutralprobabilitymeasureP,thebestestimateofthediscountedstockpriceattimen+1isthediscountedstockpriceattimen.
Inotherwords,thediscountedstockpriceprocessisamartingale.
Theorem2.
3.
1.
Considerthegeneralbinomialmodelwith0computethestockpriceattimen+1usingthefollowingformulaSn+1(!
1.
.
.
!
n!
n+1)=uSn(!
1.
.
.
!
n)if!
n+1=HdSn(!
1.
.
.
!
n)if!
n+1=T(2.
37)13Therefore,En[f(Sn+1)](!
1.
.
.
!
n)=pf[uSn(!
1.
.
.
!
n)]+qf[dSn(!
1.
.
.
!
n)]=g[Sn(!
1.
.
.
!
n)](2.
38)Whereg(x)isdenedbyg(x)=pf(ux)+qf(dx).
Hence,wehaveanalgorithmforndingg(x)foreveryf(x)andthestockpriceprocessisMarkov.
And,itisMarkovundertherisk-neutralprobabilitymeasureortheactualprobabilitymeasure.
WeknowthatthepayoofaderivativesecurityattimeNisafunctionvNofthestockpriceattimeN.
Inotherwords,VN=vN(SN).
Ifwetakeequation(2.
36)anddividebothsidesby(1+r)n,wegetVn=11+rEn[Vn+1],n=0,1,N1(2.
39)SincethestockpriceprocessisMarkov,wecandeneVN1asVN1=11+rEN1[vN(SN)]=vN1(SN1)(2.
40)forsomefunctionvN1.
Similarly,wecandeneVN2:VN2=11+rEN2[vN1(SN1)]=vN2(SN2)(2.
41)Ingeneral,wecandenerecursivelybackwardsthefunctionvnsuchthatVn=vn(Sn)bythealgorithmvn(s)=11+rhpvn+1(us)+qvn+1(ds)i,n=N1,N2,0(2.
42)ThisshowsthatthepriceprocessofanyderivativesecurityisMarkovundertherisk-neutralprobabilitymeasure.
Wecanrestateourndingsasatheorem.
Theorem2.
3.
5.
LetX0,X1,XNbeaMarkovprocessundertherisk-neutralprobabilitymeasurePinthebinomialmodel.
LetvN(x)beafunctionofthedummyvariablex,andconsideraderivativesecuritywhosepayoattimeNisvN(XN).
Then,foreachnbetween0andN,thepriceofVnofthisderivativesecurityissomefunctionvnofXn,i.
e.
Vn=vn(Xn),n=0,1,N(2.
43)ThereisarecursivealgorithmforcomputingvnwhoseexactformuladependsontheunderlyingMarkovprocessX0,X1,XN.
14Chapter3ApplicationtoAmericanDerivativeSecurities3.
1IntroductionSofar,wehaveonlylookedatEuropeanderivativesecuritiesofwhichtheownercanonlychoosetoexerciseonagivenexpirationdate.
ThischapterdiscussesAmericanderivativesecuritiesthatcanbeexercisedatanypointonorbeforetheexpirationdate.
Thismeansthattherewillexistanoptimalexercisedate,afterwhichthederivativesecuritywilltendtolosevalue.
Asaresult,thediscountedpriceprocessofAmericanderivativesecuritiesisasupermartingale(unliketheEuropeancase,forwhichthediscountedpriceprocessisamartingale).
WewillstartbylookingatthoseAmericanderivativesecuritiesthatarenotpathdependentandthenmoveontothosethatarepathdependent.
WewillndthatwecanstillcomeupwithapricingalgorithmbyreplicatingthederivativeusingthestockandmoneymarketsmuchlikewedidwithEuropeanderivativesecurities.
3.
2Path-IndependentRecallthepricingalgorithmforaEuropeanderivativesecurity.
WeuseanN-periodbinomialmodelwithupfactoru,downfactord,andinterestratersuchthat0come,wewillalwaysholdontoouroptionuntilatleasttimet=1.
2^willdependonthecointossing:Y2^(HH)=Y2(HH)=0,Y2^(HT)=Y2(HT)=0.
64,Y2^(TH)=Y1(T)=2.
40,Y2^(TT)=Y1(T)=2.
40Notethatevenifthestoppingtimeislessthan2,theprocesscontinuesuntilt=2.
Thevalueoftheprocessmerelyfreezesuponreachingthestoppingtime.
Therefore,wemayillustrateastoppedprocessusingabinomialmodel,albeitslightlymodiedfromwhatweareusedtosinceY2^(TH)doesnotequalY2^(HT).
Y2^(HH)=0Y1^(H)=0.
32Y2^(HT)=0.
64Y0^=1.
36Y2^(TH)=2.
40Y1^(T)=2.
40Y2^(TT)=2.
40Astoppedprocess.
Thisprocessisamartingale(whichisalsoasupermartingalebydenition).
ThisfactistrueforallstoppedpriceprocessesofanAmericanputoptionundertherisk-neutralprobabilitymeasureP.
Additionally,EYn^EYn.
Theorem3.
3.
2(OptionalSamplingPartI).
Amartingalestoppedatastoppingtimeisamartingale.
Asupermartingale(orsubmartingale)stoppedatastoppingtimeisasupermartingale(orsubmartin-gale,respectively).
Theorem3.
3.
3(OptionalSamplingPartII).
LetXn,n=0,1,Nbeasupermartingaleandletbeastoppingtime.
ThenEXn^EXn.
IfXnisasubmartingale,thenEXn^EXn.
IfXnisamartingale,thenEXn^=EXn.
Thisdoesnotholdforallexercisetimes,suchas.
203.
4Path-DependentUsingournewfoundknowledgeofstoppingtimes,wemaynowworkouthowtopriceAmericanderivativesecuritiesthatarepermittedtobepath-dependent.
First,wemustredenethepriceprocessVntoincludestoppingtimes.
WestillhaveanN-periodbinomialmodelwithuanddastheup-factoranddown-factorrespectively,alongwithinterestratersuchthat0Combiningthesetwofacts,wegetVn(!
1.
.
.
!
n)maxnGn(!
1.
.
.
!
n),11+rhpVn+1(!
1.
.
.
!
nH)+qVn+1(!
1.
.
.
!
nT)io(3.
17)But,equation(3.
16)tellsusthatbothsidesofthisinequalityareequal.
So,Vn(!
1.
.
.
!
n)isassmallaspossible.
Finally,wewillprovetheprocessforreplicatingpath-dependentAmericanderivativesinthestockandmoneymarkets.
Theproofisanotherinductiononn,sowewillskipit.
Theorem3.
4.
4.
ConsideranN-periodbinomialasset-pricingmodelwith0Gnforalln,includingforn=N.
But,becauseofequation(3.
15),itmustbetruethatGNcometosimilarconclusionsfor(2j3)cointosses.
Inaddition,itisclearthatforj2,P(1=2j1)=P(12j1)P(12j3)=[1P(M2j1=1)][1P(M2j3=1)]=P(M2j3=1)P(M2j1=1)=122j3(2j3)!
(j1)!
(j2)!
122j1(2j1)!
j!
(j1)!
=122j1(2j3)!
j!
(j1)!
h4j(j1)2(j1)(2j2)i=122j1(2j3)!
j!
(j1)!
h2j(2j2)2(j1)(2j2)i=122j1(2j3)!
j!
(j1)!
(2j2)=122j1(2j2)!
j!
(j1)!
Wecanstatethisasatheorem.
Theorem4.
1.
1.
Let1betherstpassagetimetolevel1ofasymmetricrandomwalk.
Then,P(1=2j1)=122j1(2j2)!
j!
(j1)!
j=1,2,4.
2)274.
2TheBlack-ScholesModelAnotherpopularmethodforpricingderivativesecuritiesisBlack-Scholes.
ThestockpricemodelofBlack-Scholescanbethoughtofasanaturalextensionofthebinomialmodel,sinceitisbasedonacontinuous-timemodelandthebinomialmodelisdiscrete.
Itisasifwedividetheperiodsofthebinomialmodelintosmallerandsmallerparts,approachingaperiodlengthofzero.
Thiswillgiveusamodelwithcontinuousprices.
ThefollowingtheoremhelpstoestimatethedistributionofstockpricesasanassumptionoftheBlack-Scholesmodel.
Theorem4.
2.
1(Azuma'sInequality).
Let0=X0,Xmbeamartingalewith|Xi+1Xi|1forall0im1.
Let>0bearbitrary.
Then,theprobabilitythatXm>pmislessthane22.
ByTheorem2.
3.
1,weknowthatthediscountedstockpriceisamartingale.
Black-Scholesinsteadtakesintoaccountthelogarithmofstockprices,andthisprocessisalsoamartingale.
Azuma'sInequalitystatesthat,aftermanyperiods,thestockpricesinthemiddleofourbinomialmodelhaveamuchhigherprobabilitythanthoseattheverytopandverybottom.
And,infact,thelogarithmofthestockpriceprocessfollowsanormaldistribution.
Thismeansourstockpriceshavealognormaldistributionwithmeanandstandarddeviation.
ThisalsoallowsfortheBlack-ScholesmodeltofollowBrownianmotioninestimatingstockprices,whichassumesthat(1)thereturnsofastockarenormallydistributedand(2)thestandarddeviationofthesereturnscanbedeterminedfromhistoricaldata.
Inthisscenario,weequateastock'sstandarddeviationtoitsvolatility.
Thelessvolatileastockis,thegreaterourabilitywillbetopredictfutureprices.
TheBlack-Scholesformulausesz-scoresofastockprice'slognormaldistribution.
Az-scorecanbedenedasfollows:Denition4.
2.
2.
Considerastandardnormaldistribution,denetobethemean,tobethestandarddeviation,andxtobeanobservedvalue.
Then,theprobabilityofxcanbedenedintermsofitsz-score:z=x(4.
3)whichmeasureshowfarawayxisfromthemeanintermsofstandarddeviations.
Whenpricingacalloption,Black-Scholeslooksforprobabilitiesthataresmallerthanthestock'sz-score.
28AcknowledgementsResearchingforandwritingthisthesishastakenupagoodpartofmysenioryearasanundergraduateattheUniversityofConnecticut,andIdenitelycouldnothavecompleteditwithoutthehelpofseveralpeople.
Firstandforemost,Iwouldliketoexpressmygratitudetomythesisadvisor,ProfessorAlexanderRussell,forguidingmetotheresearchtopicofnancialderivativepricingandhelpingmeeverystepofthewayafterthat.
WithouthiscontinualinvolvementandassistanceIsurelywouldnothavebeenabletocompletethisthesisinthetimeframethatIdid.
Iwouldalsoliketothankmyhonorsadvisor,ProfessorLukeRogers.
HehelpedmenavigatetherequirementsofthehonorsprogramforthelastseveralyearsandwasagreathelpinmakingsureIstayedontracktograduatewithhonors.
Lastly,Iwouldliketothankallofmyfriendsinthehonorsprogram.
Wemutuallysupportedeachotherincompletingourseniortheses,especiallyattimeswherethetaskseemedparticularlydauntingandoutofreach.
Wedidit!
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