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1、CHAPTER13WienerProcessesIt’sLemmaPracticeQuestionsProblem13.1.WhatwoulditmeantoassertthatthetemperatureatacertainplacefollowsaMarkovprocessDoyouthinkthattemperaturesdoinfactfollowaMarkovprocessImaginethatyouhavetofecastt
2、hefuturetemperaturefroma)thecurrenttemperatureb)thehistyofthetemperatureinthelastweekc)aknowledgeofseasonalaveragesseasonaltrends.IftemperaturefollowedaMarkovprocessthehistyofthetemperatureinthelastweekwouldbeirrelevant.
3、ToanswerthesecondpartofthequestionyoumightliketoconsiderthefollowingscenariofthefirstweekinMay:(i)MondaytoThursdayarewarmdaystodayFridayisaverycoldday.(ii)MondaytoFridayareallverycolddays.WhatisyourfecastftheweekendIfyou
4、aremepessimisticinthecaseofthesecondscenariotemperaturesdonotfollowaMarkovprocess.Problem13.2.Canatradingrulebasedonthepasthistyofastock’spriceeverproducereturnsthatareconsistentlyaboveaverageDiscuss.Thefirstpointtomakei
5、sthatanytradingstrategycanjustbecauseofgoodluckproduceaboveaveragereturns.Thekeyquestioniswhetheratradingstrategyconsistentlyoutperfmsthemarketwhenadjustmentsaremadefrisk.Itiscertainlypossiblethatatradingstrategycoulddot
6、his.Howeverwhenenoughinvestsknowaboutthestrategytradeonthebasisofthestrategytheprofitwilldisappear.Asanillustrationofthisconsideraphenomenonknownasthesmallfirmeffect.Ptfoliosofstocksinsmallfirmsappeartohaveoutperfmedptfo
7、liosofstocksinlargefirmswhenappropriateadjustmentsaremadefrisk.Researchwaspublishedaboutthisintheearly1980smutualfundsweresetuptotakeadvantageofthephenomenon.Thereissomeevidencethatthishasresultedinthephenomenondisappear
8、ing.Problem13.3.Acompany’scashpositionmeasuredinmillionsofdollarsfollowsageneralizedWienerprocesswithadriftrateof0.5perquarteravariancerateof4.0perquarter.Howhighdoesthecompany’sinitialcashpositionhavetobefthecompanytoha
9、vealessthan5%chanceofanegativecashpositionbytheendofoneyearSupposethatthecompany’sinitialcashpositionis.Theprobabilitydistributionofxthecashpositionattheendofoneyearis(40544)(2016)xx???????????whereisanmalprobabilitydist
10、ributionwithmeanvariance.The()mv??mvprobabilityofanegativecashpositionattheendofoneyearisThevariabletherefefollowsageneralizedWienerprocesswithdriftrate12XX?variancerate.12???2212122???????Problem13.5.Consideravariableth
11、atfollowstheprocessSdSdtdz????Fthefirstthreeyearsfthenextthreeyears.Ifthe2??3??3??4??initialvalueofthevariableis5whatistheprobabilitydistributionofthevalueofthevariableattheendofyearsixThechangeinduringthefirstthreeyears
12、hastheprobabilitydistributionS(2393)(627)???????Thechangeduringthenextthreeyearshastheprobabilitydistribution(33163)(948)???????Thechangeduringthesixyearsisthesumofavariablewithprobabilitydistribution(627)??avariablewith
13、probabilitydistribution.Theprobabilitydistributionofthe(948)??changeistherefe(692748)????(1575)???Sincetheinitialvalueofthevariableis5theprobabilitydistributionofthevalueofthevariableattheendofyearsixis(2075)??Problem13.
14、6.Supposethatisafunctionofastockpricetime.SupposethatGSS?G?arethevolatilitiesof.ShowthatwhentheexpectedreturnofincreasesbySGSS??thegrowthrateofincreasesbywhereisaconstant.GG???FromIt’slemmaGSGGSS?????AlsothedriftofisG222
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