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TypesofStochasticProcesses
Discretetime;discretevariable
Discretetime;continuousvariable
Continuoustime;discretevariable
Continuoustime;continuousvariable
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ModelingStockPrices
Wecanuseanyofthefourtypesofstochasticprocessestomodelstockprices
Thecontinuoustime,continuousvariableprocessprovestobethemostusefulforthepurposesofvaluingderivatives
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MarkovProcesses(Seepages259-60)
InaMarkovprocessfuturemovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegotwhereweare
WeassumethatstockpricesfollowMarkovprocesses
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Weak-FormMarketEfficiency
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AMarkovprocessforstockpricesisconsistentwithweak-formmarketefficiency
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ExampleofaDiscreteTimeContinuousVariableModel
Astockpriceiscurrentlyat$40
Attheendof1yearitisconsideredthatitwillhaveanormalprobabilitydistributionofwithmean$40andstandarddeviation$10
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Questions
Whatistheprobabilitydistributionofthestockpriceattheendof2years?
½years?
¼years?
Dtyears?
Takinglimitswehavedefinedacontinuousvariable,continuoustimeprocess
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Variances&StandardDeviations
InMarkovprocesseschangesinsuccessiveperiodsoftimeareindependent
Thismeansthatvariancesareadditive
Standarddeviationsarenotadditive
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Variances&StandardDeviations(continued)
Inourexampleitiscorrecttosaythatthevarianceis100peryear.
Itisstrictlyspeakingnotcorrecttosaythatthestandarddeviationis10peryear.
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GeneralizedWienerProcesses (continued)
MeanchangeinxintimeTisaT
VarianceofchangeinxintimeTisb2T
StandarddeviationofchangeinxintimeTis
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