《隨機微積分和金融》(pdf 348)內容簡介
Contents
1 Introduction to Proba bil ITy Theory 11
1.1 The binomialAssetPri cingModel . . 11
1.2 Fin ITe Proba bil ITy Spaces . 16
1.3 LebesgueMeasure andtheLebesgue Integral . . 22
1.4 General Proba bil ITy Spaces 30
1.5 Independence . 40
1.5.1 Independenceof sets . . 40
1.5.2 Independence of -algebras . 41
1.5.3 Independence of random variables . 42
1.5.4 Correlationandindependence 44
1.5.5 Independence andcond ITionalexpectation45
1.5.6 LawofLargeNumbers . 46
1.5.7 CentralLim ITTheorem . 47
2 Cond ITional Expectation 49
2.1 A binomialModel forS TOCkPriceDynamics . . 49
2.2 Information . . 50
2.3 Cond ITionalExpectation . 52
2.3.1 Anexample . 52
2.3.2 Definition of Cond ITional Expectation . . 53
2.3.3 FurtherdiscussionofPartialAveraging . 54
2.3.4 Propert IEsofCond ITionalExpectation . . 55
2.3.5 Examples fromthe binomialModel 57
2.4 Martingales . . 58
1
2
3 Ar bitrage Pri cing 59
3.1 binomialPri cing . . 59
3.2 Generalone-stepAPT60
3.3 Risk-Neutral Proba bil ITyMeasure . . 61
3.3.1 PortfolioProcess . 62
3.3.2 Self-finan cing Value of a Portfolio Process . 62
3.4 Simple European Derivative Secur IT IEs 63
3.5 The binomialModel isComplete 64
4 The Markov Property 67
4.1 binomialModelPri cingandHedging 67
4.2 ComputationalIssues 69
4.3 MarkovProcesses . . 70
4.3.1 Differentways towr ITe theMarkovproperty . 70
4.4 Showingthata process isMarkov . . 73
4.5 ApplicationtoExoticOptions . 74
5 Stopping Times and American Options 77
5.1 AmericanPri cing . . 77
5.2 ValueofPortfolioHedginganAmericanOption . 79
5.3 Information up to a Stopping Time . . 81
6 Properties of American Derivative Secur IT IEs 85
6.1 Thepropert IEs . 85
6.2 Proofsof thePropert IEs . . 86
6.3 Compound European Derivative Secur IT IEs . 88
6.4 OptimalExer ciseofAmericanDerivativeSecur ITy89
7 Jensen’s In EQual ITy 91
7.1 Jensen’s In EQuality for Cond ITional Expectations . 91
7.2 OptimalExer ciseof anAmericanCall 92
7.3 StoppedMartingales 94
8 RandomWalks 97
8.1 FirstPassageTime . 97
3
8.2 is almost surelyfin ITe . . 97
8.3 The moment generating function for 99
8.4 Expectation of 100
8.5 TheStrongMarkovProperty . . 101
8.6 GeneralFirstPassageTimes . . 101
8.7 Example: P ERPetualAmerican Put . . 102
8.8 Difference EQuation . 106
8.9 DistributionofFirstPassageTimes . . 107
8.10 TheReflectionPrin ciple . 109
9 Pri cing in terms ofMarket Proba bil IT IEs: The Radon-Nikodym Theorem. 111
9.1 Radon-Nikodym Theorem 111
9.2 Radon-NikodymMartingales . . 112
9.3 TheStatePriceDens ITyProcess 113
9.4 S TOChastic Volatil ITy binomial Model . 116
9.5 Another Applicaton of the Radon-Nikodym Theorem . 118
10 Cap ITal Asset Pri cing 119
10.1 AnOptimizationProblem. 119
11 General Random Variables 123
11.1 Law of a Random Variable 123
11.2 Dens ITy of a Random Variable . 123
11.3 Expectation . . 124
11.4 Two random variables 125
11.5 MarginalDens ITy . . 126
11.6 Cond ITionalExpectation . 126
11.7 ConditionalDens ITy . 127
11.8 MultivariateNormalDistribution 129
11.9 bivariatenormal distribution . . 130
11.10MGF of jointly normal random variables . . 130
12 Semi-Continuous Models 131
12.1 Discrete-timeBrownianMotion 131
4
12.2 TheS TOCkPriceProcess . . 132
12.3 Remainder of theMarket . 133
12.4 Risk-NeutralMeasure 133
12.5 Risk-NeutralPri cing 134
12.6 Ar bitrage 134
12.7 StalkingtheRisk-NeutralMeasure . . 135
12.8 Pri cingaEuropeanCall . . 138
13 BrownianMotion 139
13.1 Symmetric RandomWalk . 139
13.2 TheLawofLargeNumbers 139
13.3 CentralLim ITTheorem . . 140
13.4 BrownianMotion as a Lim ITof RandomWalks . 141
13.5 BrownianMotion . . 142
13.6 CovarianceofBrownianMotion 143
13.7 Fin ITe-DimensionalDistributionsofBrownianMotion. 144
13.8 Filtration generated by a BrownianMotion . 144
13.9 MartingaleProperty . 145
13.10TheLim ITof a binomialModel . 145
13.11StartingatPointsOtherThan0 . 147
13.12MarkovPropertyforBrownianMotion 147
13.13Transition Dens ITy . . 149
13.14FirstPassageTime . 149
14 The IT?o Integral 153
14.1 BrownianMotion . . 153
14.2 FirstVariation . 153
14.3 QuadraticVariation . 155
14.4 Quadratic Variation as Absolute Volatil ITy . 157
14.5 Construction of the IT?oIntegral . 158
14.6 IT?ointegralof an elementaryintegrand 158
14.7 Propert IEs of the IT?ointegralof anelementary process . 159
14.8 IT?ointegralof a general integrand 162
5
14.9 Propert IEs of the (general) IT?ointegral 163
14.10Quadratic variation of an IT?ointegral . 165
15 IT?o’s Formula 167
15.1 IT?o’s formula for oneBrownianmotion 167
15.2 Derivation of IT?o’s formula 168
15.3 GeometricBrownianmotion . . 169
15.4 QuadraticvariationofgeometricBrownianmotion . . 170
15.5 Volatil ITy of Geometric Brownian motion . 170
15.6 Firstderivationof theBlack-Scholes formula . . 170
15.7 Mean andvarianceof theCox-Ingersoll-Rossprocess . 172
15.8 Multidimensional Brownian Motion . 173
15.9 Cross-variationsofBrownianmotions 174
15.10Multi-dimensional IT?oformula . 175
16 Markov processes and the Kolmogorov EQuations 177
16.1 S TOChasticDifferential EQuations 177
16.2 MarkovProperty . . 178
16.3 Transition dens ITy . . 179
16.4 The Kolmogorov Backward EQuation 180
16.5 Connectionbetweens TOChasticcalculusandKBE 181
16.6 Black-Scholes . 183
16.7 Black-Scholes with price-dependent volatil ITy . . 186
17 Girsanov’s theorem and the risk-neutral measure 189
17.1 Cond ITional expectations under
fIP . . 191
17.2 Risk-neutralmeasure 193
18 Martingale Representation Theorem 197
18.1 MartingaleRepresentationTheorem . 197
18.2 Ahedgingapplication 197
18.3 d-dimensionalGirsanovTheorem . . 199
18.4 d-dimensionalMartingaleRepresentationTheorem . . 200
18.5 Multi-dimensionalmarket model 200
6
19 A two-dimensional market model 203
19.1 Hedging when 1 < < 1 204
19.2 Hedging when = 1 205
20 Pri cing Exotic Options 209
20.1 Reflectionprin cipleforBrownianmotion . 209
20.2 UpandoutEuropeancall. 212
20.3 Apractical issue 218
21 Asian Options 219
21.1 Feynman-KacTheorem . . 220
21.2 Constructingthehedge . . 220
21.3 Partial average payoffAsianoption . . 221
22 Summary of Ar bitrage Pri cing Theory 223
22.1 binomialmodel,HedgingPortfolio . 223
22.2 Setting up the continuousmodel 225
22.3 Risk-neutralpri cingandhedging 227
22.4 Implementationof risk-neutralpri cingandhedging . . 229
23 Recognizing a BrownianMotion 233
23.1 Identifying volatil ITy and correlation . 235
23.2 Reversingtheprocess 236
24 An outside barr IEr option 239
24.1 Computingtheoptionvalue 242
24.2 ThePDEfor theoutsidebarr IEroption 243
24.3 Thehedge 245
25 American Options 247
25.1 Prev IEwofp ERPetualAmerican put . . 247
25.2 Firstpassage times forBrownianmotion: firstmethod. 247
25.3 Driftadjustment 249
25.4 Drift-adjustedLaplace transform 250
25.5 Firstpassage times: Secondmethod . 251
7
25.6 P ERPetualAmericanput . . 252
25.7 Valueof thep ERPetualAmerican put . 256
25.8 Hedgingtheput 257
25.9 P ERPetualAmericancontingentclaim. 259
25.10P ERPetualAmericancall . . 259
25.11Putw IThexpiration . 260
25.12Americancontingentclaimw IThexpiration 261
26 Options on dividend-paying s TOCks 263
26.1 Americanoptionw IThconvexpayoff function . . 263
26.2 Dividendpayings TOCk . . 264
26.3 Hedging at time t1 . 266
27 Bonds, forward contracts and futures 267
27.1 Forwardcontracts . . 269
27.2 Hedginga forwardcontract 269
27.3 Future contracts 270
27.4 Cashflowfroma future contract 272
27.5 Forward-future spread272
27.6 Backwardationandcontango . . 273
28 Term-structure models 275
28.1 Computing ar bitrage-free bond prices: first method . . 276
28.2 Some interest-ratedependentassets . 276
28.3 Terminology. . 277
28.4 Forwardrate agreement . . 277
28.5 Recovering the interest r(t) fromthe forwardrate 278
28.6 Computing ar bitrage-free bond prices: Heath-Jarrow-Morton method . . 279
28.7 Checkingfor absenceof ar bitrage . . 280
28.8 Implementationof theHeath-Jarrow-Mortonmodel . . 281
29 Gaussian processes 285
29.1 Anexample: BrownianMotion . 286
30 Hull and Wh ITe model 293
8
30.1 Fiddlingw IThthe formulas 295
30.2 Dynamics of the bond price 296
30.3 Calibrationof theHull&Wh ITemodel 297
30.4 Option on a bond . . 299
31 Cox-Ingersoll-Ross model 303
31.1 EQuilibrium distribution of r(t) . 306
31.2 Kolmogorov forward EQuation . 306
31.3 Cox-Ingersoll-Ross EQuilibrium dens ITy . . 309
31.4 Bondprices inthe ciRmodel . 310
31.5 Option on a bond . . 313
31.6 Deterministictime changeof ciRmodel . . 313
31.7 Calibration . . 315
31.8 Tracking down '0(0) inthe time change of the ciRmodel . 316
32 A two-factor model (Duff IE& Kan) 319
32.1 Non-negativ ITy of Y . 320
32.2 Zero-coupon bond prices . 321
32.3 Calibration . . 323
33 Change of num?eraire 325
33.1 Bond price as num?eraire . 327
33.2 S TOCk price as num?eraire . 328
33.3 Mertonoptionpri cingformula . 329
34 Brace-Gatarek-Mus IEla model 335
34.1 Rev IEw of HJM under risk-neutral IP . 335
34.2 Brace-Gatarek-Mus IEla model . 336
34.3 LIBOR . 337
34.4 ForwardLIBOR 338
34.5 The dynamics of L(t; ) . 338
34.6 ImplementationofBGM . 340
34.7 Bondprices . . 342
34.8 Forward LIBOR under more forward measure . . 343
9
34.9 Pri cingan interest rate caplet . . 343
34.10Pri cingan interest rate cap 345
34.11CalibrationofBGM. 345
34.12Longrates 346
34.13Pri cinga swap . 346
..............................
1 Introduction to Proba bil ITy Theory 11
1.1 The binomialAssetPri cingModel . . 11
1.2 Fin ITe Proba bil ITy Spaces . 16
1.3 LebesgueMeasure andtheLebesgue Integral . . 22
1.4 General Proba bil ITy Spaces 30
1.5 Independence . 40
1.5.1 Independenceof sets . . 40
1.5.2 Independence of -algebras . 41
1.5.3 Independence of random variables . 42
1.5.4 Correlationandindependence 44
1.5.5 Independence andcond ITionalexpectation45
1.5.6 LawofLargeNumbers . 46
1.5.7 CentralLim ITTheorem . 47
2 Cond ITional Expectation 49
2.1 A binomialModel forS TOCkPriceDynamics . . 49
2.2 Information . . 50
2.3 Cond ITionalExpectation . 52
2.3.1 Anexample . 52
2.3.2 Definition of Cond ITional Expectation . . 53
2.3.3 FurtherdiscussionofPartialAveraging . 54
2.3.4 Propert IEsofCond ITionalExpectation . . 55
2.3.5 Examples fromthe binomialModel 57
2.4 Martingales . . 58
1
2
3 Ar bitrage Pri cing 59
3.1 binomialPri cing . . 59
3.2 Generalone-stepAPT60
3.3 Risk-Neutral Proba bil ITyMeasure . . 61
3.3.1 PortfolioProcess . 62
3.3.2 Self-finan cing Value of a Portfolio Process . 62
3.4 Simple European Derivative Secur IT IEs 63
3.5 The binomialModel isComplete 64
4 The Markov Property 67
4.1 binomialModelPri cingandHedging 67
4.2 ComputationalIssues 69
4.3 MarkovProcesses . . 70
4.3.1 Differentways towr ITe theMarkovproperty . 70
4.4 Showingthata process isMarkov . . 73
4.5 ApplicationtoExoticOptions . 74
5 Stopping Times and American Options 77
5.1 AmericanPri cing . . 77
5.2 ValueofPortfolioHedginganAmericanOption . 79
5.3 Information up to a Stopping Time . . 81
6 Properties of American Derivative Secur IT IEs 85
6.1 Thepropert IEs . 85
6.2 Proofsof thePropert IEs . . 86
6.3 Compound European Derivative Secur IT IEs . 88
6.4 OptimalExer ciseofAmericanDerivativeSecur ITy89
7 Jensen’s In EQual ITy 91
7.1 Jensen’s In EQuality for Cond ITional Expectations . 91
7.2 OptimalExer ciseof anAmericanCall 92
7.3 StoppedMartingales 94
8 RandomWalks 97
8.1 FirstPassageTime . 97
3
8.2 is almost surelyfin ITe . . 97
8.3 The moment generating function for 99
8.4 Expectation of 100
8.5 TheStrongMarkovProperty . . 101
8.6 GeneralFirstPassageTimes . . 101
8.7 Example: P ERPetualAmerican Put . . 102
8.8 Difference EQuation . 106
8.9 DistributionofFirstPassageTimes . . 107
8.10 TheReflectionPrin ciple . 109
9 Pri cing in terms ofMarket Proba bil IT IEs: The Radon-Nikodym Theorem. 111
9.1 Radon-Nikodym Theorem 111
9.2 Radon-NikodymMartingales . . 112
9.3 TheStatePriceDens ITyProcess 113
9.4 S TOChastic Volatil ITy binomial Model . 116
9.5 Another Applicaton of the Radon-Nikodym Theorem . 118
10 Cap ITal Asset Pri cing 119
10.1 AnOptimizationProblem. 119
11 General Random Variables 123
11.1 Law of a Random Variable 123
11.2 Dens ITy of a Random Variable . 123
11.3 Expectation . . 124
11.4 Two random variables 125
11.5 MarginalDens ITy . . 126
11.6 Cond ITionalExpectation . 126
11.7 ConditionalDens ITy . 127
11.8 MultivariateNormalDistribution 129
11.9 bivariatenormal distribution . . 130
11.10MGF of jointly normal random variables . . 130
12 Semi-Continuous Models 131
12.1 Discrete-timeBrownianMotion 131
4
12.2 TheS TOCkPriceProcess . . 132
12.3 Remainder of theMarket . 133
12.4 Risk-NeutralMeasure 133
12.5 Risk-NeutralPri cing 134
12.6 Ar bitrage 134
12.7 StalkingtheRisk-NeutralMeasure . . 135
12.8 Pri cingaEuropeanCall . . 138
13 BrownianMotion 139
13.1 Symmetric RandomWalk . 139
13.2 TheLawofLargeNumbers 139
13.3 CentralLim ITTheorem . . 140
13.4 BrownianMotion as a Lim ITof RandomWalks . 141
13.5 BrownianMotion . . 142
13.6 CovarianceofBrownianMotion 143
13.7 Fin ITe-DimensionalDistributionsofBrownianMotion. 144
13.8 Filtration generated by a BrownianMotion . 144
13.9 MartingaleProperty . 145
13.10TheLim ITof a binomialModel . 145
13.11StartingatPointsOtherThan0 . 147
13.12MarkovPropertyforBrownianMotion 147
13.13Transition Dens ITy . . 149
13.14FirstPassageTime . 149
14 The IT?o Integral 153
14.1 BrownianMotion . . 153
14.2 FirstVariation . 153
14.3 QuadraticVariation . 155
14.4 Quadratic Variation as Absolute Volatil ITy . 157
14.5 Construction of the IT?oIntegral . 158
14.6 IT?ointegralof an elementaryintegrand 158
14.7 Propert IEs of the IT?ointegralof anelementary process . 159
14.8 IT?ointegralof a general integrand 162
5
14.9 Propert IEs of the (general) IT?ointegral 163
14.10Quadratic variation of an IT?ointegral . 165
15 IT?o’s Formula 167
15.1 IT?o’s formula for oneBrownianmotion 167
15.2 Derivation of IT?o’s formula 168
15.3 GeometricBrownianmotion . . 169
15.4 QuadraticvariationofgeometricBrownianmotion . . 170
15.5 Volatil ITy of Geometric Brownian motion . 170
15.6 Firstderivationof theBlack-Scholes formula . . 170
15.7 Mean andvarianceof theCox-Ingersoll-Rossprocess . 172
15.8 Multidimensional Brownian Motion . 173
15.9 Cross-variationsofBrownianmotions 174
15.10Multi-dimensional IT?oformula . 175
16 Markov processes and the Kolmogorov EQuations 177
16.1 S TOChasticDifferential EQuations 177
16.2 MarkovProperty . . 178
16.3 Transition dens ITy . . 179
16.4 The Kolmogorov Backward EQuation 180
16.5 Connectionbetweens TOChasticcalculusandKBE 181
16.6 Black-Scholes . 183
16.7 Black-Scholes with price-dependent volatil ITy . . 186
17 Girsanov’s theorem and the risk-neutral measure 189
17.1 Cond ITional expectations under
fIP . . 191
17.2 Risk-neutralmeasure 193
18 Martingale Representation Theorem 197
18.1 MartingaleRepresentationTheorem . 197
18.2 Ahedgingapplication 197
18.3 d-dimensionalGirsanovTheorem . . 199
18.4 d-dimensionalMartingaleRepresentationTheorem . . 200
18.5 Multi-dimensionalmarket model 200
6
19 A two-dimensional market model 203
19.1 Hedging when 1 < < 1 204
19.2 Hedging when = 1 205
20 Pri cing Exotic Options 209
20.1 Reflectionprin cipleforBrownianmotion . 209
20.2 UpandoutEuropeancall. 212
20.3 Apractical issue 218
21 Asian Options 219
21.1 Feynman-KacTheorem . . 220
21.2 Constructingthehedge . . 220
21.3 Partial average payoffAsianoption . . 221
22 Summary of Ar bitrage Pri cing Theory 223
22.1 binomialmodel,HedgingPortfolio . 223
22.2 Setting up the continuousmodel 225
22.3 Risk-neutralpri cingandhedging 227
22.4 Implementationof risk-neutralpri cingandhedging . . 229
23 Recognizing a BrownianMotion 233
23.1 Identifying volatil ITy and correlation . 235
23.2 Reversingtheprocess 236
24 An outside barr IEr option 239
24.1 Computingtheoptionvalue 242
24.2 ThePDEfor theoutsidebarr IEroption 243
24.3 Thehedge 245
25 American Options 247
25.1 Prev IEwofp ERPetualAmerican put . . 247
25.2 Firstpassage times forBrownianmotion: firstmethod. 247
25.3 Driftadjustment 249
25.4 Drift-adjustedLaplace transform 250
25.5 Firstpassage times: Secondmethod . 251
7
25.6 P ERPetualAmericanput . . 252
25.7 Valueof thep ERPetualAmerican put . 256
25.8 Hedgingtheput 257
25.9 P ERPetualAmericancontingentclaim. 259
25.10P ERPetualAmericancall . . 259
25.11Putw IThexpiration . 260
25.12Americancontingentclaimw IThexpiration 261
26 Options on dividend-paying s TOCks 263
26.1 Americanoptionw IThconvexpayoff function . . 263
26.2 Dividendpayings TOCk . . 264
26.3 Hedging at time t1 . 266
27 Bonds, forward contracts and futures 267
27.1 Forwardcontracts . . 269
27.2 Hedginga forwardcontract 269
27.3 Future contracts 270
27.4 Cashflowfroma future contract 272
27.5 Forward-future spread272
27.6 Backwardationandcontango . . 273
28 Term-structure models 275
28.1 Computing ar bitrage-free bond prices: first method . . 276
28.2 Some interest-ratedependentassets . 276
28.3 Terminology. . 277
28.4 Forwardrate agreement . . 277
28.5 Recovering the interest r(t) fromthe forwardrate 278
28.6 Computing ar bitrage-free bond prices: Heath-Jarrow-Morton method . . 279
28.7 Checkingfor absenceof ar bitrage . . 280
28.8 Implementationof theHeath-Jarrow-Mortonmodel . . 281
29 Gaussian processes 285
29.1 Anexample: BrownianMotion . 286
30 Hull and Wh ITe model 293
8
30.1 Fiddlingw IThthe formulas 295
30.2 Dynamics of the bond price 296
30.3 Calibrationof theHull&Wh ITemodel 297
30.4 Option on a bond . . 299
31 Cox-Ingersoll-Ross model 303
31.1 EQuilibrium distribution of r(t) . 306
31.2 Kolmogorov forward EQuation . 306
31.3 Cox-Ingersoll-Ross EQuilibrium dens ITy . . 309
31.4 Bondprices inthe ciRmodel . 310
31.5 Option on a bond . . 313
31.6 Deterministictime changeof ciRmodel . . 313
31.7 Calibration . . 315
31.8 Tracking down '0(0) inthe time change of the ciRmodel . 316
32 A two-factor model (Duff IE& Kan) 319
32.1 Non-negativ ITy of Y . 320
32.2 Zero-coupon bond prices . 321
32.3 Calibration . . 323
33 Change of num?eraire 325
33.1 Bond price as num?eraire . 327
33.2 S TOCk price as num?eraire . 328
33.3 Mertonoptionpri cingformula . 329
34 Brace-Gatarek-Mus IEla model 335
34.1 Rev IEw of HJM under risk-neutral IP . 335
34.2 Brace-Gatarek-Mus IEla model . 336
34.3 LIBOR . 337
34.4 ForwardLIBOR 338
34.5 The dynamics of L(t; ) . 338
34.6 ImplementationofBGM . 340
34.7 Bondprices . . 342
34.8 Forward LIBOR under more forward measure . . 343
9
34.9 Pri cingan interest rate caplet . . 343
34.10Pri cingan interest rate cap 345
34.11CalibrationofBGM. 345
34.12Longrates 346
34.13Pri cinga swap . 346
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《隨機微積分和金融》(pdf 348)簡介結束