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1、StochasticProcessesandtheirApplicationsinFinancialPricingAndrewShiJune3,2010Contents1Introduction22Terminology22.1Financial.............................................22.2Stochastics............................................23MathematicalStochastics33
2、.1BrownianMotion.........................................33.2TheItoIntegralandtheItoDi erential............................33.3Ito'sLemma............................................44FinancialApplications54.1BlackScholesEquation..........................
3、...........55Conclusion6AAppendix711IntroductionInvestorspurchasestocksandbondsinthe nancialmarket,puttingtheirfundsatriskfortheopportunitytoearnareturn.SincethetimeofPhoenicians,theyhavesoughttominimizethisriskvalueforeachlevelofexpectedreturn.Inorderto
4、doso,awholerange nancialinstrumentshavebeendeveloped,knownasderivatives,assetswhoderiveassetsfromanother nancialasset.ThenatureofderivativeassetsprovidesaninterestingconduitfortheanalysisandapplicationofBrown-ianmotionandsolvingpartialderivativeequations
5、,whilemaintainingitsrealworldapplications.Numer-ousarticleshavebeenwrittenonmodelingmovementsin nancialmarketswithstochasticcalculus.Per-hapsthemostfamousofthesedescribedtheNobelPrizewinningBlack-Scholesoptionpricingmodel[2].Inseveralarticles,mathematici
6、ans,speci callyRobertAlmgren's[1]andAnastasiosMalliaris[5],haveattemptedtomorerigorouslybridgethegapbetweenrandommotionandoptionpricing.2Terminology2.1FinancialAsset:Anobjectthatprovidesaclaimtofuturecash
ows.EcientMarketHypothesis:Thereisnoopportunit
7、yforarbitrageinthemarketDerivative:A nancialassetthatderivesitsvaluefromanotherasset.Option:Aderivativethatprovidestheopportunity,butnotobligationtobuyorsellanassetatapredeterminedpriceinthefuture.StrikePrice:Thepredeterminedpriceforexecutinganoption.
8、Foracalloption,ifthemarketpricerisesabovethestrikeprice,theinvestorwillbewillingtobuy.Foraputoption,ifthemarketpricefallsbelowthestrikeprice,theinvestorwillwanttoselltheunderlyingasset.2.2StochasticsProbabilitySpace:Acons
