很好的金融计算与随机过程入门书籍，需要微积分的基础。

 

 

 

Contents
1IntroductiontoProbabilityTheory11
1.1TheBinomialAssetPricingModel..........................11
1.2FiniteProbabilitySpaces ...............................16
1.3LebesgueMeasureandtheLebesgueIntegral....................22
1.4GeneralProbabilitySpaces ..............................30
1.5Independence.....................................40
1.5.1Independenceofsets.............................40
 -algebras.........................41
1.5.2Independenceof
1.5.3Independenceofrandomvariables ......................42
1.5.4Correlationandindependence........................44
1.5.5Independenceandconditionalexpectation..................45
1.5.6LawofLargeNumbers............................46
1.5.7CentralLimitTheorem............................47
2ConditionalExpectation49
2.1ABinomialModelforStockPriceDynamics....................49
2.2Information......................................50
2.3ConditionalExpectation...............................52
2.3.1Anexample..................................52
2.3.2De?nitionofConditionalExpectation ....................53
2.3.3FurtherdiscussionofPartialAveraging...................54
2.3.4PropertiesofConditionalExpectation....................55
2.3.5ExamplesfromtheBinomialModel.....................57
2.4Martingales......................................58

3ArbitragePricing59
3.1BinomialPricing...................................59
3.2Generalone-stepAPT.................................60
3.3Risk-NeutralProbabilityMeasure ..........................61
3.3.1PortfolioProcess...............................62

................62
3.3.2Self-?nancingValueofaPortfolioProcess
3.4SimpleEuropeanDerivativeSecurities ........................63
3.5TheBinomialModelisComplete...........................64
4TheMarkovProperty67
4.1BinomialModelPricingandHedging........................67
4.2ComputationalIssues.................................69
4.3MarkovProcesses...................................70
4.3.1DifferentwaystowritetheMarkovproperty................70
4.4ShowingthataprocessisMarkov..........................73
4.5ApplicationtoExoticOptions............................74
5StoppingTimesandAmericanOptions77
5.1AmericanPricing...................................77
5.2ValueofPortfolioHedginganAmericanOption...................79
5.3InformationuptoaStoppingTime ..........................81
6PropertiesofAmericanDerivativeSecurities85
6.1Theproperties.....................................85
6.2ProofsoftheProperties................................86
6.3CompoundEuropeanDerivativeSecurities ......................88
6.4OptimalExerciseofAmericanDerivativeSecurity..................89
7Jensen!ˉsInequality9
7.1Jensen!ˉsInequalityforConditionalExpectation...................91
7.2OptimalExerciseofanAmericanCall........................92
7.3StoppedMartingales .................................94
8RandomWalks97
8.1FirstPassageTime..................................97

8.2
 isalmostsurely?nite................................97
 ........................99
8.3Themomentgeneratingfunctionfor
 ....................................100
8.4Expectationof
8.5TheStrongMarkovProperty.............................101
8.6GeneralFirstPassageTimes.............................101
8.7Example:PerpetualAmericanPut..........................102
8.8DifferenceEquation..................................106
8.9DistributionofFirstPassageTimes..........................107
8.10TheRe?ectionPrinciple...............................109
PricingintermsofMarketProbabilities:TheRadon-NikodymTheorem.111
9.1Radon-NikodymTheorem ..............................111
9.2Radon-NikodymMartingales... ..........................112
9.3TheStatePriceDensityProcess...........................113
9.4StochasticVolatilityBinomialModel .........................116
9.5AnotherApplicatonoftheRadon-NikodymTheorem. ...............118
CapitalAssetPricing119
10.1AnOptimizationProblem...............................119
GeneralRandomVariables123
11.1LawofaRandomVariable ..............................123
11.2DensityofaRandomVariable.. ..........................123
11.3Expectation......................................124
11.4Tworandomvariables .................................125
11.5MarginalDensity...................................126
11.6ConditionalExpectation...............................126
11.7ConditionalDensity..................................127
11.8MultivariateNormalDistribution...........................129
11.9Bivariatenormaldistribution.............................130
11.10MGFofjointlynormalrandomvariables .......................130
Semi-ContinuousModels131
12.1Discrete-timeBrownianMotion...........................131

12.2TheStockPriceProcess................................132
12.3RemainderoftheMarket...............................133
12.4Risk-NeutralMeasure.................................133
12.5Risk-NeutralPricing.................................134
12.6Arbitrage.......................................134
12.7StalkingtheRisk-NeutralMeasure..........................135
12.8PricingaEuropeanCall................................138
13BrownianMotion139
13.1SymmetricRandomWalk ...............................139
13.2TheLawofLargeNumbers..............................139
13.3CentralLimitTheorem................................140
13.4BrownianMotionasaLimitofRandomWalks ...................141
13.5BrownianMotion...................................142
13.6CovarianceofBrownianMotion...........................143
13.7Finite-DimensionalDistributionsofBrownianMotion................144
13.8FiltrationgeneratedbyaBrownianMotion ......................144
13.9MartingaleProperty..................................145
13.10TheLimitofaBinomialModel............................145
13.11StartingatPointsOtherThan0............................147
13.12MarkovPropertyforBrownianMotion........................147
13.13TransitionDensity...................................149
13.14FirstPassageTime..................................149
?
14TheItoIntegral153
14.1BrownianMotion...................................153
14.2FirstVariation.....................................153
14.3QuadraticVariation..................................155
14.4QuadraticVariationasAbsoluteVolatility ......................157
14.5ConstructionoftheIt?oIntegral............................158
14.6It?ointegralofanelementaryintegrand........................158
14.7PropertiesoftheIt?ointegralofanelementaryprocess................159
14.8It?ointegralofageneralintegrand...........................162

14.9Propertiesofthe(general)It?ointegral........................163
14.10QuadraticvariationofanIt?ointegral.........................165
?
Ito!ˉsFormula16
15.1It?o!ˉsformulaforoneBrownianmotion........................16
15.2DerivationofIt?o!ˉsformula..............................16
15.3GeometricBrownianmotion.............................169
15.4QuadraticvariationofgeometricBrownianmotion.................170
15.5VolatilityofGeometricBrownianmotion ......................170
15.6FirstderivationoftheBlack-Scholesformula....................170
15.7MeanandvarianceoftheCox-Ingersoll-Rossprocess................172
15.8MultidimensionalBrownianMotion .........................173
15.9Cross-variationsofBrownianmotions........................174
15.10Multi-dimensionalIt?oformula............................175
MarkovprocessesandtheKolmogorovequations177
16.1StochasticDifferentialEquations...........................177
16.2MarkovProperty...................................178
16.3Transitiondensity ...................................179
16.4TheKolmogorovBackwardEquation ........................180
16.5ConnectionbetweenstochasticcalculusandKBE..................181
16.6Black-Scholes.....................................183
16.7Black-Scholeswithprice-dependentvolatility ....................186
Girsanov!ˉstheoremandtherisk-neutralmeasure18
f
IP ..........................191
17.1Conditionalexpectationsunder
17.2Risk-neutralmeasure.................................193
MartingaleRepresentationTheorem197
18.1MartingaleRepresentationTheorem.........................197
18.2Ahedgingapplication.................................197
d-dimensionalGirsanovTheorem..........................199
18.3
d-dimensionalMartingaleRepresentationTheorem.................200
18.4
18.5Multi-dimensionalmarketmodel. ..........................200

19Atwo-dimensionalmarketmodel203
19.1Hedgingwhen1 << 1 ..............................204
 =1 .................................205
19.2Hedgingwhen
20PricingExoticOptions209
20.1Re?ectionprincipleforBrownianmotion......................209
20.2UpandoutEuropeancall...............................212
20.3Apracticalissue....................................218
21AsianOptions219
21.1Feynman-KacTheorem................................220
21.2Constructingthehedge................................220
21.3PartialaveragepayoffAsianoption..........................221
22SummaryofArbitragePricingTheory223
22.1Binomialmodel,HedgingPortfolio.........................223
22.2Settingupthecontinuousmodel. ..........................225
22.3Risk-neutralpricingandhedging...........................227
22.4Implementationofrisk-neutralpricingandhedging.................229
23RecognizingaBrownianMotion233
23.1Identifyingvolatilityandcorrelation .........................235
23.2Reversingtheprocess.................................236
24Anoutsidebarrieroption239
24.1Computingtheoptionvalue..............................242
24.2ThePDEfortheoutsidebarrieroption........................243
24.3Thehedge.......................................245
25AmericanOptions247
25.1PreviewofperpetualAmericanput..........................247
25.2FirstpassagetimesforBrownianmotion:?rstmethod................247
25.3Driftadjustment....................................249
25.4Drift-adjustedLaplacetransform...........................250
25.5Firstpassagetimes:Secondmethod.........................251