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DerivativePricingProf.JieLiatCAFDFall,2010LectureITheParableofaBookmakerAbookmakeristakingbetsonatwo-horserace.Inascienti¯cway,hecorrectlyestimatedthechanceofwinningforeachhorse(25%,75%),andhesettheoddstobe3-1againstand3-1onrespectively.Supposethetotalbetsaddupto$5,000and10,000.Let'scalculatethepro¯t/lossforthebookmaker.¡¡¡¡¡µ@@@@@R25%75%(¡3;5000)(¡1=3;10000)(¡3)£5000+10000=¡5000(¡1=3)£10000+5000=1667Isitfairtohim?Let'sgettheexpectedvalueofthepayo®s:25%£(¡5000)+75%£1667=0.TheanswerisYES.ButitisYESinthelongrun;hemaybearagreatlossintheshortrun.Togetridoftherisks,hechangestheoddsinto2-1againstand2-1on.Thepro¯t/losspatternwouldbe:¡¡¡¡¡µ@@@@@R25%75%(¡2;5000)(¡1=2;10000)(¡2)£5000+10000=0(¡1=2)£10000+5000=0Ifhewantstoguaranteeapro¯tat1000,healterstheoddsinto®1;®2.TheP/Lpatternbecomes:1¡¡¡¡¡µ@@@@@R25%75%(¡®1;5000)(¡®2;10000)(¡®1)£5000+10000=1000(¡®2)£10000+5000=1000®1=9=5;®2=2=5,thus,9-5againstand5-2on.ExpectationPricingConsiderthegameofcointossing.$1forheadsand$0fortails.Whatshouldbetheentryfeeforthegame?Entryfee=1=2£1+1=2£0=0:5)expectationofthegamespayo®s.KomogrovStrongLawofLargeNumbers:Letxi;i=1;2;:::beiidrandomnumberswithmean¹,thenSn=1=nPni=1xi!¹withprob.1.AccordingtoKomogrovStrongLaw,weknowtheaverageP/Lpergameis0.5inthelongrunforsure.So,0.5isthefairpriceofthegame.Anypricedeviatingfrom0.5willendupextraPorLforsureinthelongrun.ReturnofMonetaryMarketAccount(TimeValueofMoney)Supposeweborrow$1foroneyearatanominalinterestrateofrperyear,compoundedcontinuously.Howmuchisitowedattheendoftheyear?CompoundingTimes$Once(1+r)Twice(1+r=2)2ThreeTimes(1+r=3)3......ntimes(1+r=n)nWhenn!1,limn!1(1+r=n)n=er.Thus,iftheprincipalisp,thenperisowedinoneyear.2Iftime=tinsteadof1,thentherearentcompoundingbytimet.Sotheamountattimetwouldbep(1+r=n)ntifcompoundedntimesayear.Thus,undercontinuouscompounding,limn!1p(1+r=n)nt=plimn!1(1+r=n)nt=p(limn!1(1+r=n)n)t=pert.UseD0todenotethetimevalueofthemoneynowandDt,thetimevalueofmoneyattimet.ThenDt=D0ertandD0=Dtexp(¡rt).IfDt=1,everyonedollarattimetisvaluedatexp(-rt)dollarsattime0.StockModelDe¯nearandomvariableXasthechange(di®erence)inthelogarithmofthestockpriceoversometimeperiodT.ThenX=logST¡logS0orST=S0exp(X).ForwardContractandForwardPriceAforwardcontractisanagreementtobuyandsellanassetatcertainfuturedateforcertainprice.ThepricethatmakesthecontractvalueatZeroiscalledforwardprice.LetStbetheassetpriceattimet,Kbetheforwardprice.AttimeT:E(ST¡K)=0,andE(ST)=K.So,K=E(S0exp(X)).3LetX»N(¹;¾),wehavethepdfofX,f(X)=1¾p2¼e¡12(X¡¹)2¾2.E(S0eX)=Z1¡11¾p2¼e¡12(x¡¹)2¾2S0exdx=S0¾p2¼Z1¡1e¡x2¡2x¹+¹2¡2¾2x2¾2dx=S0¾p2¼Z1¡1e¡x2¡2x(¹+¾2)+¹22¾2dx=S0¾p2¼Z1¡1e¡[x2¡2x(¹+¾2)+z}|{¹2+2¹¾2+(¾2)2]¡2¹¾2¡(¾2)22¾2dx=S0¾p2¼Z1¡1e¡[x2¡2x(¹+¾2)+(¹+¾2)2]¡2¹¾2¡(¾2)22¾2dx=S0¾p2¼Z1¡1e¡[x¡(¹+¾2)]22¾2+¹+¾2=2dx=S0e¹+¾2=2Z1¡11¾p2¼e¡[x¡(¹+¾2)2]2¾2dx|{z}=S0e¹+¾2=2Therefore,theexpectedassetpriceattimeTisE(ST)=E(S0eX)=S0e¹+¾2=2=KButthispriceisnotenforceableintheshortrun.TheenforceablepriceshouldbeS0erTbyarbitrage.Why?1.IfKS0erT,Icanshorttheforwardcontract.Thestrategytoguaranteepro¯tisto:BorrowS0amountofmoneyandbuyoneunitofthestockattime0.AttimeT,mypayo®isK¡S0erT0.2.IfKS0erT,Icanlongtheforwardcontract.Thestrategytoguaranteepro¯tisto:Shortoneunitofthestockattime0anddepositS0amountofmoneyand.AttimeT,mypayo®isS0erT¡K0.BranchStockModelWeneedtwothingsinourmodeltosimulatethereal¯nancialworld:(a)randomnessofstockprices;(b)thetimevalueofmoney.4¡¡¡¡¡µ@@@@@RS1p1¡p1n3S32S2nnTime0B0Time1B0er±tThetimedi®erenceb/w0and1is±twithinterestrater.Letfbethepayo®function,then(1¡p)f(2)+pf(3)istheexpectedvalueoffattime1.Canwepricethederivative(forwardinparticular)ate¡r±t[(1¡p)f(2)+pf(3)]?NO,again,itisnotenforceable!TheenforceablepriceisS1er±tfromthepreviousarbitragediscussion.Canwereplicatethepayo®functionbyusingboththestockandthebond?Ifso,wecanpricethederivativesbythereplicationcost.Consideraportfolio(Á;')withÁunitsofstockand'unitsofbonds.Thecostofsuchportfolioattime0is:ÁS1+'B0.¡¡¡¡¡µ@@@@@RS1p1¡p1n3ÁS3+'B0er±tS32ÁS2+'B0er±tS2nnIfwewanttoreplicatethepayo®sbythisportfolio,weneedtoset8:ÁS3+'B0er±t=f(3)ÁS2+'B0er±t=f(2)5Therefore,8:Á=f(3)¡f(2)S3¡S2'=B¡10e¡r±t[f(3)¡(f(3)¡f(2))S3S3¡S2]Thus,thepriceofthederivativewhichisthereplicationcostisV=ÁS1+'B0=S1f(3)¡f(2)S3¡S2+e¡r±t[f(3)¡(f(3)¡f(2))S3S3¡S2]Thisisthepricingformulanotonlyforforward,butalsoforanyotherderivativeswhicharewrittenonthefundamentalasset,stock.Example:¡¡¡¡¡µ@@@@@RS1p1¡p1n3S3=2S32S2=0:5S2nnAndwealsohaveS1=1;B0=1;B1=1,whichimpliesnointerestforbonds.Howtopriceabetwhichpays$1ifthestockgoesup?So,thepayo®functionsf(3)=1andf(2)=0.Weuse(Á;')toreplicateÁ=f(3)¡f(2)S3¡S2=1¡02¡0:5=2=3'=B¡10e¡r±t[f(3)¡(f(3)¡f(2))S3S3¡S2]=1£1£(1¡(1¡0)£22¡0:5)=¡1=3So,thereplicationcostisÁS1+'B0=2=3£1¡1=3=1=3,whichisthepriceofthebet.Howtocheckportfolio(Á;')doesreplicatethebet'spayo®s?Afteranup-move,thevalueoftheportfoliois2=3£2¡1=3£1=1)exactlymatchesthe6payo®ofthebetafteranup-move(f(3)).Afteradown-move,thevalueis2=3£0:5¡1=3£1=0)f(2).NoticethatThepricingformuladoesn'tincludeanythingabouttheprobabilitymeasurep.TobetterunderstandV,wemanipulateitabita