Stochastic Processes and their Applications in Financial Pricing by Andrew Shi

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StochasticProcessesandtheirApplicationsinFinancialPricingAndrewShiJune3,2010Contents1Introduction22Terminology22.1Financial.............................................22.2Stochastics............................................23MathematicalStochastics33.1BrownianMotion.........................................33.2TheItoIntegralandtheItoDierential............................33.3Ito'sLemma............................................44FinancialApplications54.1BlackScholesEquation.....................................55Conclusion6AAppendix71 1IntroductionInvestorspurchasestocksandbondsinthenancialmarket,puttingtheirfundsatriskfortheopportunitytoearnareturn.SincethetimeofPhoenicians,theyhavesoughttominimizethisriskvalueforeachlevelofexpectedreturn.Inordertodoso,awholerangenancialinstrumentshavebeendeveloped,knownasderivatives,assetswhoderiveassetsfromanothernancialasset.ThenatureofderivativeassetsprovidesaninterestingconduitfortheanalysisandapplicationofBrown-ianmotionandsolvingpartialderivativeequations,whilemaintainingitsrealworldapplications.Numer-ousarticleshavebeenwrittenonmodelingmovementsinnancialmarketswithstochasticcalculus.Per-hapsthemostfamousofthesedescribedtheNobelPrizewinningBlack-Scholesoptionpricingmodel[2].Inseveralarticles,mathematicians,specicallyRobertAlmgren's[1]andAnastasiosMalliaris[5],haveattemptedtomorerigorouslybridgethegapbetweenrandommotionandoptionpricing.2Terminology2.1FinancialAsset:Anobjectthatprovidesaclaimtofuturecash ows.EcientMarketHypothesis:ThereisnoopportunityforarbitrageinthemarketDerivative:Anancialassetthatderivesitsvaluefromanotherasset.Option:Aderivativethatprovidestheopportunity,butnotobligationtobuyorsellanassetatapredeterminedpriceinthefuture.StrikePrice:Thepredeterminedpriceforexecutinganoption.Foracalloption,ifthemarketpricerisesabovethestrikeprice,theinvestorwillbewillingtobuy.Foraputoption,ifthemarketpricefallsbelowthestrikeprice,theinvestorwillwanttoselltheunderlyingasset.2.2StochasticsProbabilitySpace:Aconstructofthreecomponents,(;F;P),where1.isthesetofallpossibleoutcomes.2.Fisthesetofallevents,whereeacheventhaszeroormoreoutcomes.3.PistheassignmentofprobabilitiestoeacheventWithProbability1:Alsoknownasalmostsurely.Theprobabilityofaneventoccuringtendsto1givensomelimit.Notethatthisdiersfromsurelyinthatsurelyindicatesthatnoothereventispossible,whilealmostsurelyindicatesthatothereventsbecomelessandlesslikely.S1AcollectionofsetsFiscalleda-algebraifforasequenceofsetsAk2F,1Ak2Fandisclosedundercomplementation.ThesetsA2FareF-measurable.M[0;T]denotesthesetoffunctionsf(t)suchthatf(t)isdenedon[0;T],measurablewithrespectRTtothe-algebraFforallt,andjf(t)j2dtisnitewithprobability1.t02 3MathematicalStochastics3.1BrownianMotionTherealmofnancialassetpricingborrowsheavilyfromtheeldofstochasticcalculus.ThepriceofastocktendstofollowaBrownianmotion.DenitionAstochasticprocessw(t)issaidtofollowaBrownianmotionon[0;T]ifitsatisesthefollowing:1.w(0)=0.2.w(t)isalmostsurelycontinuous.3.Forarbitraryt1;t2;:::;tn,where0