FM-StockSimulation

SimulatingStockPricesSimulatingStockPricesFiilMdliFinancialModelingProf.DougBlackburnWhatdoweknowaboutstockprices?1.Futurepricesareunknown.2.Stockpricesarecontinuousandcanchangeatanypointintime.atanypointintime.Stkilllth3.Stockpricesaregenerallylargerthanzero.Whatdoweknowaboutstockprices?4.Expectedreturnsfromholdingastocktendstoincreaseovertime.WeexpecthigherreturnsthelongerweholdthestockWeexpecthigherreturnsthelongerweholdthestock.5Thepriceuncertaintyincreaseswithtime5.Thepriceuncertaintyincreaseswithtime.Varianceoftomorrow’sstockpricesissmallerthanthevarianceofthestockpriceinonemonth.6.Stockpricesappeartowiggleastheymovethroughtime.AfirstmodelofstockpricesAgoodmodelwillincorporateallofthesefeatures.Stochasticcalculusisneededforthis…wewillfocusonintuition.Thefundamentalmodel:Thefundamentalmodel:S=stockpriceμ=meanreturnr=stockreturnσ=standarddeviationofreturnrstockreturnσstandarddeviationofreturndtdtSdSr~~CASE1Supposenorisk.Whatisthepriceattime=T?~dtSdSrTST0)ln()ln(000TSSdtSdSTTSSTWithoutrisk,pricesgrow]ep[0)ln()ln(0TSSTSSTatexponentialrate.Thisisthefuturevalueflithti]exp[0TSSTformulawithcontinuouscompounding.CASE2Supposethemeanrateofreturniszeroandstandarddeviationispositive.dtSSdrN~~~)1,0(~~DefineStockpriceisexpectedtoEdtdtErE0~~]~[ppstaythesame.dtdtErErrErVar]~[]~[])~[(~22222Vitiit(dtki)dddtrDevSt]~[.Variationinreturns(andstockprices)dependontime.Largetimeincrementleadstohighvolatility.ThemodelrevisitedThereturngeneratingfunctionismadeupoftwoparts.Thfittidthlitd1.Thefirstpartprovidesthegeneralgrowingtrend.2.Thesecondpartprovidesvolatility.dtdtNrdtdtSdSr2~~~Aswelookfartherintothefuture,theexpectedreturndtdtNr,~andvolatilitybothincrease.FutureValueRelationPricesarecontinuous.Therefore,thefuturevalueofthepriceis:]~exp[0TrSSTThisimplies:timeofunitoneisTassuming~ln~0SSrTLogNormalDistributionSupposereturnsarenormallydistributedand~0ln~SSrTThenSTislognormallydistributedDefinition:Ifacontinuousrandomvariableislognormallydistributed,thenthelogofthevariableisnormallydistributednormallydistributed.LogNormalDistributionReturnsarenormall0.80.9Returnsarenormallydistributed…cantakepositiveandnegativevalues0.50.60.7utionvaluesPilllditibtd020.30.4DistribuPricesarelognormallydistributed…cantakeonlypositivevalues.00.10.20123456789StockPriceLogNormalDistributionLetln(x)~N(μ,σ)Thenxislognormallydistributedwithstatistics:22/exp:Mean222exp1exp:Variance2/exp:Mean2exp1exp:VarianceModelforstockprice…aasmodeledandddistributenormallyarereturnsSupposeThenmotion.browniangeometricaasmodeledandddistributenormallyarereturnsSuppose~~dtdtSdSr:pricestockfutureforthesolvewecalculus,stochasticUsing2timecontinuous~2exp)(22dtdtSdttSttimediscrete~2exp)(2ttSttStStatisticsforlogstockprices:thatseecanwemodelpricestocktheFromttNSSttt,2~ln2orttSNSll2ttSNSttt,2ln~lnRecursiveModelAiitiltkiiSAssumeinitialstockpriceisS0.ChooseatimeincrementCooseateceetConsider1-day=1/250ofayear.Note:timeisgenerallyinyears.Usingpastdatafromaparticularstock,estimatetheappropriatemeanandstandarddeviationofprices.EstimateS1–thestockpriceatday1(dependsonS0)EstimateS–thestockpriceatday2(dependsonS)EstimateS2–thestockpriceatday2(dependsonS1)Etc…ExampleofSimulatedPricesSimulatedStockPrice$25$10$15$20$0$5$10050100150200250Price($10)($5)DaysArestockpricescontinuous?Somearguethatpricesarenotcontinuous.Whenreallybad(good)newsoccurs,thepricecanlose(gain)alotofvalueallatonepricecanlose(gain)alotofvalueallatonetime…adiscontinuity.WecallthesequickchangesinpricesJumpsModelingajumpThiilThisrequirestwovalues:1.HowmanyjumpsdoweexpecteachyearHbihj2.HowbigarethejumpsonaverageHence,wenowhavethreesourcesofrisk1.Thegeneralvolatilityofprices2.Theuncertainriskofajump3.TheuncertainsizeofthejumpArrivalProcessesAnoccurrenceofaneventmyphonerings,someonewalksinthedoor,astockpricejumps.WedefineNt(w)asthenumberofarrivalsoccurringinthetimeinterval[0,t].g[,]N={Nt(w),t≥0}iscalledanarrivalprocess.PoissonArrivalProcessThreeaxioms1.Ntisanon-negativeinteger.2.Fort,s0,Nt+s–Ntisindependentofthehistoryuptotimet,{Nu,ut}.Thenumberofarrivalsinatime-perioddoesnotdependonthepast.3Forts0NNisindependentoft3.Fort,s0,Nt+s–Ntisindependentoft.Thenumberofarrivalsinthefutureisidenticallydistributed(stationary).(y)PoissonArrivalProcessLemma1:IfNisaPoissonprocess,thenforallt≥0andsomeλ0hλ≥0,thenLemma2:tNPtexp)0(Lemma2:IfNisaPoissonprocessthenthenumberofarrivalsinanintervaloflengththasaPoissondistributioninanintervaloflengththasaPoissondistributionwithparameterλt:tktttektkNP!)(Whatisλ?tketkkNPkNEktkktt00!)(][kkk00!λisthearrivalrate…theexpectednumberofarrivalsduringeachintervaloftime(intervaloflth1)length1).AsimplificationWillidh(TlWewillconsiderthreestates(useTaylorexpansion):]1[)(1]0[tNNP