Stochastic Processes and their Applications in Financial Pricing by Andrew Shi

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1、StochasticProcessesandtheirApplicationsinFinancialPricingAndrewShiJune3,2010Contents1Introduction22Terminology22.1Financial.............................................22.2Stochastics............................................23MathematicalStochastics33

2、.1BrownianMotion.........................................33.2TheItoIntegralandtheItoDierential............................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