Contents
1Randomnumbers2
2Pseudorandomgenerators3
3
2.1De?nition..........................
2.2Propertiesrequiredforagoodpseudo-randomnumbers
generator.......................... 4
2.3Constructingpseudo-randomnumbergenerators...... 6
3Low-discrepancysequences8
3.1De?nition.......................... 8
3.2Generalremarksonlowdiscrepancysequences...... 10
3.3Sobolsequences....................... 11
4Simulationofnon-uniformrandomvariablesorvectors14
4.1Inversemethod.......................
14
4.2SimulationofGaussianstandardvariables......... 14
4.3SimulationofGaussianvectors............... 17
5PrincipleoftheMonteCarloSimulation19
5.1Limittheorems.......................
19
5.2Estimationprinciple..................... 20
5.3Properties.......................... 22
6VarianceReductionTechniques24
6.1AntitheticVariables..................... 24
6.2ControlVariables...................... 26
6.3ImportanceSampling.................... 27
6.4Ef?ciencyoftheMonteCarlomethods........... 29
7QuasiMonteCarloSimulation31
7.1Generalprinciple...................... 31
7.2Koksma-Hlawkainequality................. 31
7.3Remarks........................... 33
8ComputingtheGreeksby(Quasi)MonteCarlo35
8.1FiniteDifferences......................
35
8.2Derivationofthepayoff................... 36
8.3Payoffregularization.................... 37
9(Quasi)MonteCarloalgorithmsforvanillaoptions40
9.1(Q)MCBS1D........................ 40
9.2(Q)MCBS2D........................ 57
10Simulationofprocesses80
10.1BrownianMotion...................... 80
10.2Black¨CScholesmodel................... 83
10.3Generaldiffusions:EulerandMilshteinscheme...... 85
10.4Hestonmodel........................ 87
10.5MonteCarloSimulationforProcesses........... 89
11(Quasi)MonteCarlomethodsforExoticOptions91
91
11.1Lookbackoptions......................
11.2AndersenandBrotherton-RatcliffeAlgorithmforLook-
backOptions........................ 94
11.3Barrieroptions....................... 104
11.4Asianoptions........................ 107
12Treesforvanillaoptions110
12.1Cox-Ross-RubinsteinasanapproximationtoBlack-Scholes 110
12.2Algorithm(CRR)...................... 116
12.3VariantsoftheCRRtree.................. 123
12.4Trinomialtrees....................... 126
12.5Algorithm(Kamrad-Ritchken)............... 131
12.6Miscellaneousremarks................... 138
13Treesforexoticoptions144
13.1Inaccuracyofthedirectmethodforbarrieroptions.... 144
13.2TheRitchkenalgorithmforbarrieroptions......... 145
13.3Customizationoftrees................... 145
14FiniteDifferencesforEuropeanVanillaOptions149
14.1LocalizationandDiscretization............... 149
14.2The!Π-scheme!±...................... 158
14.3ExplicitMethod....................... 160
14.4ImplicitMethods...................... 161
15FiniteDifferencesforAmericanVanillaOptions167
15.1Variationalinequalityin?nitedimension..........
167
15.2Linearcomplementarityproblem.............. 168
15.3Splittingmethods...................... 173
16FiniteDifferenceΠ-schemeAlgorithmforVanillaOptions176
17FiniteDifferencesfor2D VanillaOptions
17.1NumericalintegrationbyanA.D.I.Method........ 192
17.2AmericanOptions...................... 193
17.3Algorithm(A.D.I.BS2D).................. 194
FiniteDifferencesforExoticOptions208
18.1LookbackOptions...................... 208
18.2BarrierOptions....................... 209
18.3AsianOptions........................ 211
Dynamictests215
19.1Delta-hedging........................ 215
19.2DynamictestsusingBrownianBridgeintheBlack¨CSchole
model............................ 218
Conclusion224 | 