Basic Fin Math.pdf

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TheBasicsofFinancialMathematics
Spring2003
RichardF.Bass
DepartmentofMathematics
UniversityofConnecticut
Thesenotesarec2003byRichardBass.Theymaybeusedforpersonaluseor
classuse,butnotforcommercialpurposes.Ifyoufindanyerrors,Iwouldappreciate
hearingfromyou:bass@math.uconn.edu
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1.Introduction.
Inthiscoursewewillstudymathematicalfinance.Mathematicalfinanceisnot
aboutpredictingthepriceofastock.Whatitisaboutisfiguringoutthepriceofoptions
andderivatives.
Themostfamiliartypeofoptionistheoptiontobuyastockatagivenpriceat
agiventime.Forexample,supposeMicrosoftiscurrentlysellingtodayat$40pershare.
AEuropeancalloptionissomethingIcanbuythatgivesmetherighttobuyashareof
Microsoftatsomefuturedate.Tomakeupanexample,supposeIhaveanoptionthat
allowsmetobuyashareofMicrosoftfor$50inthreemonthstime,butdoesnotcompel
metodoso.IfMicrosofthappenstobesellingat$45inthreemonthstime,theoptionis
worthless.Iwouldbesillytobuyasharefor$50whenIcouldcallmybrokerandbuyit
for$45.SoIwouldchoosenottoexercisetheoption.Ontheotherhand,ifMicrosoftis
sellingfor$60threemonthsfromnow,theoptionwouldbequitevaluable.Icouldexercise
theoptionandbuyasharefor$50.Icouldthenturnaroundandselltheshareonthe
openmarketfor$60andmakeaprofitof$10pershare.ThereforethisstockoptionI
possesshassomevalue.Thereissomechanceitisworthlessandsomechancethatitwill
leadmetoaprofit.Thebasicquestionis:howmuchistheoptionworthtoday?
ThehugeimpetusinfinancialderivativeswastheseminalpaperofBlackandScholes
in1973.Althoughmanyresearchershadstudiedthisquestion,BlackandScholesgavea
definitiveanswer,andagreatdealofresearchhasbeendonesince.Thesearenotjust
academicquestions;todaythemarketinfinancialderivativesislargerthanthemarket
instocksecurities.Inotherwords,moremoneyisinvestedinoptionsonstocksthanin
stocksthemselves.
Optionshavebeenaroundforalongtime.Theearliestoneswereusedbymanu-
facturersandfoodproducerstohedgetheirrisk.Afarmermightagreetosellabushelof
wheatatafixedpricesixmonthsfromnowratherthantakeachanceonthevagariesof
marketprices.Similarlyasteelrefinerymightwanttolockinthepriceofironoreata
fixedprice.
Thesectionsofthesenotescanbegroupedintofivecategories.Thefirstiselemen-
taryprobability.Althoughsomeonewhohashadacourseinundergraduateprobability
willbefamiliarwithsomeofthis,wewilltalkaboutanumberoftopicsthatarenotusu-
allycoveredinsuchacourse:-fields,conditionalexpectations,martingales.Thesecond
categoryisthebinomialassetpricingmodel.Thisisjustaboutthesimplestmodelofa
stockthatonecanimagine,andthiswillprovideacasewherewecanseemostofthemajor
ideasofmathematicalfinance,butinaverysimplesetting.Thenwewillturntoadvanced
probability,thatis,ideassuchasBrownianmotion,stochasticintegrals,stochasticdier-
entialequations,Girsanovtransformation.Althoughtodothisrigorouslyrequiresmeasure
theory,wecanstilllearnenoughtounderstandandworkwiththeseconcepts.Wethen
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returntofinanceandworkwiththecontinuousmodel.WewillderivetheBlack-Scholes
formula,seetheFundamentalTheoremofAssetPricing,workwithequivalentmartingale
measures,andthelike.Thefifthmaincategoryistermstructuremodels,whichmeans
modelsofinterestratebehavior.
IfoundsomeunpublishednotesofSteveShreveextremelyusefulinpreparingthese
notes.Ihopethathehasturnedthemintoabookandthatthisbookisnowavailable.
Thestochasticcalculuspartofthesenotesisfrommyownbook:ProbabilisticTechniques
inAnalysis,Springer,NewYork,1995.
IwouldalsoliketothankEvaristGin´ewhopointedoutanumberoferrors.
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2.Reviewofelementaryprobability.
Let’sbeginbyrecallingsomeofthedefinitionsandbasicconceptsofelementary
probability.Wewillonlyworkwithdiscretemodelsatfirst.
Westartwithanarbitraryset,calledtheprobabilityspace,whichwewilldenote
by,thecapitalGreekletter“omega.”WearegivenaclassFofsubsetsof.Theseare
calledevents.WerequireFtobea-field.
Definition2.1.AcollectionFofsubsetsofiscalleda-fieldif
(1);2F,
(2)2F,
(3)A2FimpliesA c 2F,and
(4)A 1 ,A 2 ,...2Fimpliesboth[ 1 i=1 A i 2Fand\ 1 i=1 A i 2F.
HereA c ={!2:! /2A}denotesthecomplementofA.;denotestheemptyset,that
is,thesetwithnoelements.Wewillusewithoutspecialcommenttheusualnotationsof
[(union),\(intersection),(containedin),2(isanelementof).
Typically,inanelementaryprobabilitycourse,Fwillconsistofallsubsetsof
,butwewilllaterneedtodistinguishbetweenvarious-fields.Hereisanexam-
ple.Supposeonetossesacointwotimesandletsdenoteallpossibleoutcomes.So
={HH,HT,TH,TT}.Atypical-fieldFwouldbethecollectionofallsubsetsof.
InthiscaseitistrivialtoshowthatFisa-field,sinceeverysubsetisinF.Butif
weletG={;,,{HH,HT},{TH,TT}},thenGisalsoa-field.Onehastocheckthe
definition,buttoillustrate,theevent{HH,HT}isinG,sowerequirethecomplementof
thatsettobeinGaswell.Butthecomplementis{TH,TT}andthateventisindeedin
G.
Onepointofviewwhichwewillexploremuchmorefullylateronisthatthe-field
tellsyouwhateventsyou“know.”Inthisexample,Fisthe-fieldwhereyou“know”
everything,whileGisthe-fieldwhereyou“know”onlytheresultofthefirsttossbutnot
thesecond.Wewon’ttrytobeprecisehere,buttotrytoaddtotheintuition,suppose
oneknowswhetheraneventinFhashappenedornotforaparticularoutcome.We
wouldthenknowwhichoftheevents{HH},{HT},{TH},or{TT}hashappenedandso
wouldknowwhatthetwotossesofthecoinshowed.Ontheotherhand,ifweknowwhich
eventsinGhappened,wewouldonlyknowwhethertheevent{HH,HT}happened,which
meanswewouldknowthatthefirsttosswasaheads,orwewouldknowwhethertheevent
{TH,TT}happened,inwhichcasewewouldknowthatthefirsttosswasatails.But
thereisnowaytotellwhathappenedonthesecondtossfromknowingwhicheventsinG
happened.Muchmoreonthislater.
Thethirdbasicingredientisaprobability.
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Definition2.2.AfunctionPonFisaprobabilityifitsatisfies
(1)ifA2F,then0P(A)1,
(2)P()=1,and
(3)P(;)=0,and
(4)ifA 1 ,A 2 ,...2Farepairwisedisjoint,thenP([ 1 i=1 A i )= P 1 i=1 P(A i ).
AcollectionofsetsA i ispairwisedisjointifA i \A j =;unlessi=j.
Thereareanumberofconclusionsonecandrawfromthisdefinition.Asone
example,ifAB,thenP(A)P(B)andP(A c )=1−P(A).SeeNote1attheendof
thissectionforaproof.
Someonewhohashadmeasuretheorywillrealizethata-fieldisthesamething
asa-algebraandaprobabilityisameasureoftotalmassone.
Arandomvariable(abbreviatedr.v.)isafunctionXfromtoR,thereals.To
bemoreprecise,tobear.v. Xmustalsobemeasurable,whichmeansthat{!:X(!)
a}2Fforallrealsa.
Thenotionofmeasurabilityhasasimpledefinitionbutisabitsubtle.Ifwetake
thepointofviewthatweknowalltheeventsinG,thenifY isG-measurable,thenwe
knowY.Phrasedanotherway,supposeweknowwhetherornottheeventhasoccurred
foreacheventinG.ThenifY isG-measurable,wecancomputethevalueofY.
Hereisanexample.Intheexampleabovewherewetossedacointwotimes,letX
bethenumberofheadsinthetwotosses.ThenXisFmeasurablebutnotGmeasurable.
Toseethis,letusconsiderA a ={!2:X(!)a}.Thiseventwillequal
8 > <
ifa0;
{HH,HT,TH}if0< a1;
{HH} if1< a2;
; if2< a.
> :
Forexample,ifa= 3 2 ,thentheeventwherethenumberofheadsis 3 2 orgreateristhe
eventwherewehadtwoheads,namely,{HH}.NowobservethatforeachatheeventA a
isinFbecauseFcontainsallsubsetsof.ThereforeXismeasurablewithrespecttoF.
HoweveritisnottruethatA a isinGforeveryvalueofa–takea= 3 2 asjustoneexample
–thesubset{HH}isnotinG.SoXisnotmeasurablewithrespecttothe-fieldG.
Adiscreter.v.isonewhereP(!:X(!)=a)=0forallbutcountablymanya’s,
say,a 1 ,a 2 ,...,and P i P(!:X(!)=a i )=1.Indefiningsetsoneusuallyomitsthe!;
thus(X=x)meansthesameas{!:X(!)=x}.
Inthediscretecase,tocheckmeasurabilitywithrespecttoa-fieldF,itisenough
that(X=a)2Fforallrealsa.Thereasonforthisisthatifx 1 ,x 2 ,...arethevaluesof
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