Financial Modelling with Jump Processes
Chapman & Hall/CRC Financial Mathematics Series
© 2004 by CRC Press LLC
527页
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Contents
1 Financial modellingbeyondBrownianmotion
1.1 Modelsinthelightof empirical facts
1.2 Evidencefromoptionmarkets
1.2.1 Impliedvolatilitysmilesandskews
1.2.2 Short-termoptions
1.3 Hedgingandriskmanagement
1.4 Objectives
I Mathematicaltools
2 Basictools
2.1 Measuretheory
2.1.1 σ-algebrasandmeasures
2.1.2 Measuresmeetfunctions: integration
2.1.3 Absolutecontinuityanddensities
2.2 Randomvariables
2.2.1 Randomvariablesandprobabilityspaces
2.2.2 Whatis(Ω,F,P)anyway?
2.2.3 Characteristicfunctions
2.2.4 Momentgeneratingfunction
2.2.5 Cumulantgeneratingfunction
2.3 Convergenceof randomvariables
2.3.1 Almost-sureconvergence
2.3.2 Convergenceinprobability
2.3.3 Convergenceindistribution
2.4 Stochasticprocesses
2.4.1 Stochasticprocessesasrandomfunctions
2.4.2 Filtrationsandhistories
2.4.3 Randomtimes
2.4.4 Martingales
2.4.5 Predictableprocesses(*)
2.5 ThePoissonprocess
2.5.1 Exponential randomvariables
2.5.2 ThePoissondistribution
2.5.3 ThePoissonprocess: definitionandproperties
2.5.4 CompensatedPoissonprocesses
2.5.5 Countingprocesses
2.6 Randommeasuresandpointprocesses
2.6.1 Poissonrandommeasures
2.6.2 CompensatedPoissonrandommeasure
2.6.3 BuildingjumpprocessesfromPoissonrandommeasures
2.6.4 Markedpointprocesses(*)
3L´evyprocesses: definitionsandproperties
3.1 FromrandomwalkstoL´evyprocesses
3.2 CompoundPoissonprocesses
3.3 Jumpmeasuresof compoundPoissonprocesses
3.4 InfiniteactivityL´evyprocesses
3.5 Pathwisepropertiesof L´evyprocesses
3.6 Distributional properties
3.7 Stablelawsandprocesses
3.8 L´evyprocessesasMarkovprocesses
3.9 L´evyprocessesandmartingales
4 BuildingL´evyprocesses
4.1 Model buildingwithL´evyprocesses
4.1.1 “Jump-diffusions”vs. infiniteactivityL´evyprocesses
4.2 BuildingnewL´evyprocessesfromknownones
4.2.1 Lineartransformations
4.2.2 Subordination
4.2.3 TiltingandtemperingtheL´evymeasure
4.3 Modelsof jump-diffusiontype
4.4 BuildingL´evyprocessesbyBrowniansubordination
4.4.1 General results
4.4.2 Subordinatingprocesses
4.4.3 ModelsbasedonsubordinatedBrownianmotion
4.5 Temperedstableprocess
4.6 Generalizedhyperbolicmodel
5 Multidimensional modelswithjumps
5.1 MultivariatemodellingviaBrowniansubordination
5.2 BuildingmultivariatemodelsfromcommonPoissonshocks
5.3 Copulasforrandomvariables
5.4 DependenceconceptsforL´evyprocesses
5.5 CopulasforL´evyprocesseswithpositivejumps
5.6 Copulasforgeneral L´evyprocesses
5.7 BuildingmultivariatemodelsusingL´evycopulas
5.8 Summary
II Simulationandestimation
6 SimulatingL´evyprocesses
6.1 Simulationof compoundPoissonprocesses
6.2 Exactsimulationof increments
6.3 Approximation of an infinite activity L´evy process by a compoundPoissonprocess
6.4 Approximationof small jumpsbyBrownianmotion
6.5 Seriesrepresentationsof L´evyprocesses(*)
6.6 Simulationof multidimensional L´evyprocesses
7 Modellingfinancial timeserieswithL´evyprocesses
7.1 Empirical propertiesof assetreturns
7.2 Statistical estimationmethodsandtheirpitfalls
7.2.1 Maximumlikelihoodestimation
7.2.2 Generalizedmethodof moments
7.2.3 Discussion
7.3 Thedistributionof returns: ataleof heavytails
7.3.1 Howheavytailedisthedistributionof returns?
7.4 Timeaggregationandscaling
7.4.1 Self-similarity
7.4.2 Arefinancial returnsself-similar?
7.5 Realizedvarianceand“stochasticvolatility”
7.6 Pathwisepropertiesof pricetrajectories(*)
7.6.1 H¨ olderregularityandsingularityspectra
7.6.2 Estimatingsingularityspectra
7.7 Summary: advantagesandshortcomingsof L´evyprocesses
III Optionpricingin modelswithjumps
8 Stochasticcalculusforjumpprocesses
8.1 Tradingstrategiesandstochasticintegrals
8.1.1 Semimartingales
8.1.2 Stochasticintegralsforcagladprocesses
8.1.3 StochasticintegralswithrespecttoBrownianmotion
8.1.4 Stochastic integrals with respect to Poisson random measures
8.2 Quadraticvariation
8.2.1 Realizedvolatilityandquadraticvariation
8.2.2 Quadraticcovariation
8.3 TheItˆoformula
8.3.1 Pathwisecalculusforfiniteactivityjumpprocesses
8.3.2 Itˆ oformulafordiffusionswithjumps
8.3.3 Itˆ oformulaforL´evyprocesses
8.3.4 Itˆ oformulaforsemimartingales
8.4 Stochasticexponentialsvs. ordinaryexponentials
8.4.1 Exponential of aL´evyprocess
.......
9 MeasuretransformationsforL´evyprocesses
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10Pricingandhedginginincompletemarkets
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