金融随机微积分培训知识英文版(pdf 348)

1 IntroductiontoProbabilITyTheory11
1.1 ThebinomialAssetPricingModel......1 1
1.2 FinITeProbabilITySpaces.....1 6
1.3 LebesgueMeasureandtheLebesgueIntegral......2 2
1.4 GeneralProbabilITySpaces......3 0
1.5 Independence.....4 0
1.5.1 Independenceofsets.....4 0
1.5.2 Independenceof-algebras.....4 1
1.5.3 Independenceofrandomvariables......4 2
1.5.4 Correlationandindependence......4 4
1.5.5 IndependenceandcondITionalexpectation......4 5
1.5.6 LawofLargeNumbers......4 6
1.5.7 CentralLimITTheorem......4 7
2 CondITionalExpectation49
2.1 AbinomialModelforSTOCkPriceDynamics......4 9
2.2 Information......5 0
2.3 CondITionalExpectation.....5 2
2.3.1 Anexample......5 2
2.3.2 DefinitionofCondITionalExpectation......5 3
2.3.3 FurtherdiscussionofPartialAveraging.....5 4
2.3.4 PropertIEsofCondITionalExpectation......5 5
2.3.5 ExamplesfromthebinomialModel.....5 7
2.4 Martingales......5 8
3 ArbitragePricing59
3.1 binomialPricing.....5 9
3.2 Generalone-stepAPT.....6 0
3.3 Risk-NeutralProbabilITyMeasure......6 1
3.3 .1 PortfolioProcess.....6 2
3.3 .2 Self-financingValueofaPortfolioProcess
......6 2
3.4 SimpleEuropeanDerivativeSecurITIEs......6 3
3.5 ThebinomialModelisComplete.....6 4
4 TheMarkovProperty67
4.1 binomialModelPricingandHedging......6 7
4.2 ComputationalIssues.....6 9
4.3 MarkovProcesses.....7 0
4.3 .1 DifferentwaystowrITetheMarkovproperty......7 0
4.4 ShowingthataprocessisMarkov......7 3
4.5 ApplicationtoExoticOptions......7 4
5 StoppingTimesandAmericanOptions77
5.1 AmericanPricing.....7 7
5.2 ValueofPortfolioHedginganAmericanOption.....7 9
5.3 InformationuptOAStoppingTime......8 1
6 PropertiesofAmericanDerivativeSecurITIEs85
6.1 ThepropertIEs.....8 5
6.2 ProofsofthePropertIEs......8 6
6.3 CompoundEuropeanDerivativeSecurITIEs......8 8
6.4 OptimalExerciseofAmericanDerivativeSecurITy......8 9
7 Jensen’sInEQualITy91
7.1 Jensen’sInEQualityforCondITionalExpectations.....9 1
7.2 OptimalExerciseofanAmericanCall......9 2
7.3 StoppedMartingales.....9 4
8 RandomWalks97
8.1 FirstPassageTime......9 7
8.2  isalmostsurelyfinITe......9 7
8.3 Themomentgeneratingfunctionfor......9 9
8.4 Expectationof......1 00
8.5 TheStrongMarkovProperty.....1 01
8.6 GeneralFirstPassageTimes.....1 01
8.7 Example:PERPetualAmericanPut......1 02
8.8 DifferenceEQuation......1 06
8.9 DistributionofFirstPassageTimes......1 07
8.1 0TheReflectionPrinciple.....1 09
9 PricingintermsofMarketProbabilITIEs:TheRadon-NikodymTheorem.1 11
9.1 Radon-NikodymTheorem......1 11
9.2 Radon-NikodymMartingales.....1 12
9.3 TheStatePriceDensITyProcess.....1 13
9.4 STOChasticVolatilITybinomialModel.....1 16
9.5 AnotherApplicatonoftheRadon-NikodymTheorem......1 18
10 CapITalAssetPricing119
10.1 AnOptimizationProblem.....1 19
11 GeneralRandomVariables123
11.1 LawofaRandomVariable......1 23
11.2 DensITyofaRandomVariable......1 23
11.3 Expectation......1 24
11.4 Tworandomvariables.....1 25
11.5 MarginalDensITy.....1 26
11.6 CondITionalExpectation.....1 26
11.7 ConditionalDensITy......1 27
11.8 MultivariateNormalDistribution.....1 29
11.9 bivariatenormaldistribution.....1 30
11.1 0MGFofjointlynormalrandomvariables.....1 30
12 Semi-ContinuousModels131
12.1 Discrete-timeBrownianMotion.....1 31
12.2 TheSTOCkPriceProcess......1 32
12.3 RemainderoftheMarket.....1 33
12.4 Risk-NeutralMeasure.....1 33
12.5 Risk-NeutralPricing.....1 34
12.6 Arbitrage.....1 34
12.7 StalkingtheRisk-NeutralMeasure......1 35
12.8 PricingaEuropeanCall......1 38
13 BrownianMotion139
13.1 SymmetricRandomWalk.....1 39
13.2 TheLawofLargeNumbers......1 39
13.3 CentralLimITTheorem......1 40
13.4 BrownianMotionasaLimITofRandomWalks.....1 41
13.5 BrownianMotion.....1 42
13.6 CovarianceofBrownianMotion.....1 43
13.7 FinITe-DimensionalDistributionsofBrownianMotion......1 44
13.8 FiltrationgeneratedbyaBrownianMotion......1 44
13.9 MartingaleProperty......1 45
13.1 0TheLimITofabinomialModel......1 45
13.1 1StartingatPointsOtherThan0......1 47
13.1 2MarkovPropertyforBrownianMotion......1 47
13.1 3TransitionDensITy.....1 49
13.1 4FirstPassageTime......1 49
14 TheITˆoIntegral153
14.1 BrownianMotion.....1 53
14.2 FirstVariation.....1 53
14.3 QuadraticVariation......155
14.4 QuadraticVariationasAbsoluteVolatilITy......157
14.5 ConstructionoftheITˆoIntegral......158
14.6 ITˆointegralofanelementaryintegrand......158
14.7 PropertIEsoftheITˆointegralofanelementaryprocess......159
14.8 ITˆointegralofageneralintegrand.....1 62
14.9 PropertIEsofthe(general)ITˆointegral......1 63
14.1 0QuadraticvariationofanITˆointegral.....1 65
15 ITˆo’sFormula167
15.1 ITˆo’sformulaforoneBrownianmotion......1 67
15.2 DerivationofITˆo’sformula......1 68
15.3 GeometricBrownianmotion.....1 69
15.4 QuadraticvariationofgeometricBrownianmotion.....1 70
15.5 VolatilITyofGeometricBrownianmotion......1 70
15.6 FirstderivationoftheBlack-Scholesformula......1 70
15.7 MeanandvarianceoftheCox-Ingersoll-Rossprocess......1 72
15.8 MultidimensionalBrownianMotion.....1 73
15.9 Cross-variationsofBrownianmotions......1 74
15.1 0Multi-dimensionalITˆoformula......1 75
16 MarkovprocessesandtheKolmogorovEQuations177
16.1 STOChasticDifferentialEQuations.....177
16.2 MarkovProperty.....178
16.3 TransitiondensITy.....179
16.4 TheKolmogorovBackwardEQuation......180
16.5 ConnectionbetweensTOChasticcalculusandKBE......181
16.6 Black-Scholes.....1 83
16.7 Black-Scholeswithprice-dependentvolatilITy......186
17Girsanov’stheoremandtherisk-neutralmeasure189
17.1 CondITionalexpectationsunder
fIP......1 91
17.2 Risk-neutralmeasure.....1 93
18 MartingaleRepresentationTheorem197
18.1 MartingaleRepresentationTheorem.....1 97
18.2 Ahedgingapplication.....1 97
18.3 d-dimensionalGirsanovTheorem......1 99
18.4 d-dimensionalMartingaleRepresentationTheorem.....2 00
18.5 Multi-dimensionalmarketmodel.....2 00
19 Atwo-dimensionalmarketmodel203
19.1 Hedgingwhen􀀀1<<1......2 04
19.2 Hedgingwhen=1.....2 05
20 PricingExoticOptions209
20.1 ReflectionprincipleforBrownianmotion......2 09
20.2 UpandoutEuropeancall.....2 12
20.3 Apracticalissue......2 18
21 AsianOptions219
21.1 Feynman-KacTheorem......2 20
21.2 Constructingthehedge......2 20
21.3 PartialaveragepayoffAsianoption......2 21
22 SummaryofArbitragePricingTheory223
22.1 binomialmodel,HedgingPortfolio.....2 23
22.2 Settingupthecontinuousmodel.....2 25
22.3 Risk-neutralpricingandhedging.....2 27
22.4 Implementationofrisk-neutralpricingandhedging.....2 29
23 RecognizingaBrownianMotion233
23.1 IdentifyingvolatilITyandcorrelation.....2 35
23.2 Reversingtheprocess.....2 36
24 AnoutsidebarrIEroption239
24.1 Computingtheoptionvalue......2 42
24.2 ThePDEfortheoutsidebarrIEroption......2 43
24.3 Thehedge.....2 45
25 AmericanOptions247
25.1 PrevIEwofpERPetualAmericanput......2 47
25.2 FirstpassagetimesforBrownianmotion:firstmethod......2 47
25.3 Driftadjustment......2 49
25.4 Drift-adjustedLaplacetransform.....2 50
25.5 Firstpassagetimes:Secondmethod.....2 51
25.6 PERPetualAmericanput......2 52
25.7 ValueofthepERPetualAmericanput.....2 56
25.8 Hedgingtheput......2 57
25.9 PERPetualAmericancontingentclaim.....2 59
25.1 0PERPetualAmericancall......2 59
25.1 1PutwIThexpiration......2 60
25.1 2AmericancontingentclaimwIThexpiration.....2 61
26 Optionsondividend-payingsTOCks263
26.1 AmericanoptionwIThconvexpayofffunction......2 63
26.2 DividendpayingsTOCk......2 64
26.3 Hedgingattimet1......2 66
27 Bonds,forwardcontractsandfutures267
27.1 Forwardcontracts.....2 69
27.2 Hedgingaforwardcontract......2 69
27.3 Futurecontracts......2 70
27.4 Cashflowfromafuturecontract.....2 72
27.5 Forward-futurespread.....2 72
27.6 Backwardationandcontango.....2 73
28 Term-structuremodels275
28.1 Computingarbitrage-freebondprices:firstmethod.....2 76
28.2 Someinterest-ratedependentassets.....2 76
28.3 Terminology......2 77
28.4 Forwardrateagreement......2 77
28.5 Recoveringtheinterestr(t)fromtheforwardrate......2 78
28.6 Computingarbitrage-freebondprices:Heath-Jarrow-Mortonmethod......2 79
28.7 Checkingforabsenceofarbitrage......2 80
28.8 ImplementationoftheHeath-Jarrow-Mortonmodel.....2 81
29 Gaussianprocesses285
29.1 Anexample:BrownianMotion......2 86
30 HullandWhITemodel293
30.1 FiddlingwIThtheformulas......2 95
30.2 Dynamicsofthebondprice......2 96
30.3 CalibrationoftheHull&WhITemodel......2 97
30.4 Optiononabond.....2 99
31 Cox-Ingersoll-Rossmodel303
31.1 EQuilibriumdistributionofr(t)......3 06
31.2 KolmogorovforwardEQuation......3 06
31.3 Cox-Ingersoll-RossEQuilibriumdensITy.....3 09
31.4 BondpricesintheciRmodel......3 10
31.5 Optiononabond.....3 13
31.6 DeterministictimechangeofciRmodel.....3 13
31.7 Calibration......3 15
31.8 Trackingdown'0(0)inthetimechangeoftheciRmodel.....3 16
32 Atwo-factormodel(DuffIE&Kan)319
32.1 Non-negativITyofY......3 20
32.2 Zero-couponbondprices.....3 21
32.3 Calibration......3 23
33 Changeofnum´eraire325
33.1 Bondpriceasnum´eraire.....3 27
33.2 STOCkpriceasnum´eraire.....3 28
33.3 Mertonoptionpricingformula......3 29
34 Brace-Gatarek-MusIElamodel335
34.1 RevIEwofHJMunderrisk-neutralIP.....3 35
34.2 Brace-Gatarek-MusIElamodel......3 36
34.3 LIBOR......3 37
34.4 ForwardLIBOR......3 38
34.5 ThedynamicsofL(t;).....3 38
34.6 ImplementationofBGM.....3 40
34.7 Bondprices......3 42
34.8 ForwardLIBORundermoreforwardmeasure......3 43
34.9 Pricinganinterestratecaplet.....3 43
34.1 0Pricinganinterestratecap......3 45
34.1 1CalibrationofBGM......3 45
34.1 2Longrates.....3 46
34.1 3Pricingaswap.....3 46

The binomial asset pricing model provides a powerful tool to understand arbitrage pricing theory
and probability theory. In this course, we shall use IT for both these purposes.
In the binomial asset pricing model, we model sTOCk prices in discrete time, assuming that at each
step, the sTOCk price will change to one of two possible values. Let us begin with an initial posITive
sTOCk price S0. There are two positive numbers, d and u, wITh

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