《随机微积分和金融》(1)

StevenShreve:Stochas[e-mail][e-mail]:PLEASEDONOTDISTRIBUTEcCopyright;StevenE.Shreve,1996July25,1997Contents1IntroductiontoProbabilityTheory111.1TheBinomialAssetPricingModel..........................111.2FiniteProbabilitySpaces...............................161.3LebesgueMeasureandtheLebesgueIntegral....................221.4GeneralProbabilitySpaces..............................301.5Independence.....................................401.5.1Independenceofsets.............................401.5.2Independenceof-algebras.........................411.5.3Independenceofrandomvariables......................421.5.4Correlationandindependence........................441.5.5Independenceandconditionalexpectation..................451.5.6LawofLargeNumbers............................461.5.7CentralLimitTheorem............................472ConditionalExpectation492.1ABinomialModelforStockPriceDynamics....................492.2Information......................................502.3ConditionalExpectation...............................522.3.1Anexample..................................522.3.2DefinitionofConditionalExpectation....................532.3.3FurtherdiscussionofPartialAveraging...................542.3.4PropertiesofConditionalExpectation....................552.3.5ExamplesfromtheBinomialModel.....................572.4Martingales......................................58123ArbitragePricing593.1BinomialPricing...................................593.2Generalone-stepAPT.................................603.3Risk-NeutralProbabilityMeasure..........................613.3.1PortfolioProcess...............................623.3.2Self-financingValueofaPortfolioProcess................623.4SimpleEuropeanDerivativeSecurities........................633.5TheBinomialModelisComplete...........................644TheMarkovProperty674.1BinomialModelPricingandHedging........................674.2ComputationalIssues.................................694.3MarkovProcesses...................................704.3.1DifferentwaystowritetheMarkovproperty................704.4ShowingthataprocessisMarkov..........................734.5ApplicationtoExoticOptions............................745StoppingTimesandAmericanOptions775.1AmericanPricing...................................775.2ValueofPortfolioHedginganAmericanOption...................795.3InformationuptoaStoppingTime..........................816PropertiesofAmericanDerivativeSecurities856.1Theproperties.....................................856.2ProofsoftheProperties................................866.3CompoundEuropeanDerivativeSecurities......................886.4OptimalExerciseofAmericanDerivativeSecurity..................897Jensen’sInequality917.1Jensen’sInequalityforConditionalExpectations...................917.2OptimalExerciseofanAmericanCall........................927.3StoppedMartingales.................................948RandomWalks978.1FirstPassageTime..................................9738.2isalmostsurelyfinite................................978.3Themomentgeneratingfunctionfor........................998.4Expectationof....................................1008.5TheStrongMarkovProperty.............................1018.6GeneralFirstPassageTimes.............................1018.7Example:PerpetualAmericanPut..........................1028.8DifferenceEquation..................................1068.9DistributionofFirstPassageTimes..........................1078.10TheReflectionPrinciple...............................1099PricingintermsofMarketProbabilities:TheRadon-NikodymTheorem.1119.1Radon-NikodymTheorem..............................1119.2Radon-NikodymMartingales.............................1129.3TheStatePriceDensityProcess...........................1139.4StochasticVolatilityBinomialModel.........................1169.5AnotherApplicatonoftheRadon-NikodymTheorem................11810CapitalAssetPricing11910.1AnOptimizationProblem...............................11911GeneralRandomVariables12311.1LawofaRandomVariable..............................12311.2DensityofaRandomVariable............................12311.3Expectation......................................12411.4Tworandomvariables.................................12511.5MarginalDensity...................................12611.6ConditionalExpectation...............................12611.7ConditionalDensity..................................12711.8MultivariateNormalDistribution...........................12911.9Bivariatenormaldistribution.............................13011.10MGFofjointlynormalrandomvariables.......................13012Semi-ContinuousModels13112.1Discrete-timeBrownianMotion...........................131412.2TheStockPriceProcess................................13212.3RemainderoftheMarket...............................13312.4Risk-NeutralMeasure.................................13312.5Risk-NeutralPricing.................................13412.6Arbitrage......