Chapter23

EstimatingVolatilitiesandCorrelationsOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull20141StandardApproachtoEstimatingVolatility

(page521)Definesnasthevolatilityperdaybetweendayn-1anddayn,asestimatedatendofday

n-1DefineSiasthevalueofmarketvariableatendofdayiDefineui=ln(Si/Si-1)Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull20142SimplificationsUsuallyMade

inRiskManagement(page522)Set

ui=(Si?Si-1)/Si-1AssumethatthemeanvalueofuiiszeroReplacem?1bymThisgivesOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull20143WeightingScheme InsteadofassigningequalweightstotheobservationswecansetOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull20144ARCH(m)Model

(page523)InanARCH(m)modelwealsoassignsomeweighttothelong-runvariancerate,VL:Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull20145EWMAModelInanexponentiallyweightedmovingaveragemodel,theweightsassignedtotheu2declineexponentiallyaswemovebackthroughtimeThisleadstoOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull20146ToShowthatWeightsDeclineExponentiallyOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull20147AttractionsofEWMARelativelylittledataneedstobestoredWeneedonlyrememberthecurrentestimateofthevariancerateandthemostrecentobservationonthemarketvariableTracksvolatilitychanges0.94isapopularchoiceforlOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull20148GARCH(1,1)

page525 InGARCH(1,1)weassignsomeweighttothelong-runaveragevariancerate Sinceweightsmustsumto1g+a+b=1Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull20149GARCH(1,1)continued Settingw=gVtheGARCH(1,1)modelis andOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201410Example

(Example23.2,page525)SupposeThelong-runvariancerateis0.0002sothatthelong-runvolatilityperdayis1.4%Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201411ExamplecontinuedSupposethatthecurrentestimateofthevolatilityis1.6%perdayandthemostrecentpercentagechangeinthemarketvariableis1%.Thenewvariancerateis Thenewvolatilityis1.53%perdayOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201412GARCH(p,q)

Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201413MaximumLikelihoodMethodsInmaximumlikelihoodmethodswechooseparametersthatmaximizethelikelihoodoftheobservationsoccurringOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201414Example1Weobservethatacertaineventhappensonetimeintentrials.Whatisourestimateoftheproportionofthetime,p,thatithappens?TheprobabilityoftheeventhappeningononeparticulartrialandnotontheothersisWemaximizethistoobtainamaximumlikelihoodestimate.Result:p=0.1(asexpected)Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201415Example2

EstimatethevarianceofobservationsfromanormaldistributionwithmeanzeroOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201416ApplicationtoGARCH WechooseparametersthatmaximizeOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201417S&P500ExcelApplicationStartwithtrialvaluesofw,a,andb

UpdatevariancesCalculateUsesolvertosearchforvaluesofw,a,andbthatmaximizethisobjectivefunctionForefficientoperationofSolver:setupspreadsheetsothatthreenumbersthatarethesameorderofmagnitudearebeingsearchedforOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201418S&P500ExcelApplication(Table23.1)Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201419DateDaySiui=(Si?Si-1)/Si-1vi=si2?ln(vi)

?ui2/vi18-Jul-200511221.1319-Jul-200521229.350.00673120-Jul-200531235.200.0047590.000045319.502221-Jul-200541227.04?0.0066060.000044479.0393…….…..…….………..………….…………13-Aug-201012791079.25?0.0040240.000163278.6209Total10,228.2349TheResults(Figure23.2,page530)Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201420VarianceTargetingOnewayofimplementingGARCH(1,1)thatincreasesstabilityisbyusingvariancetargetingWesetthelong-runaveragevolatilityequaltothesamplevarianceOnlytwootherparametersthenhavetobeestimatedOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201421HowGoodistheModel?TheLjung-BoxstatistictestsforautocorrelationWecomparetheautocorrelationofthe ui2withtheautocorrelationoftheui2/si2Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201422ForecastingFutureVolatility

(equation23.13,page532)

Afewlinesofalgebrashowsthat ThevariancerateforanoptionexpiringondaymisOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201423ForecastingFutureVolatilitycontinued(equation23.14,page534)Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201424S&PExamplew=0.0000013465,a=0.083394,b=0.910116Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201425OptionLife(days)103050100500Volatility(%perannum)27.3627.1026.8726.3524.32VolatilityTermStructuresGARCH(1,1)suggeststhat,whencalculatingvega,weshouldshiftthelongmaturityvolatilitieslessthantheshortmaturityvolatilitiesWheninstantaneousvolatilitychangesbyDs(0),volatilityforT-dayoptionchangesbyOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201426ResultsforS&P500(Table23.4)Wheninstantaneousvolatilitychangesby1%Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201427OptionLife(days)103050100500Volatilityincrease(%)0.970.920.870.770.33CorrelationsandCovariances(page535-537)Definexi=(Xi?Xi-1)/Xi-1andyi=(Yi?Yi-1)/Yi-1Alsosx,n:dailyvolofXcalculatedondayn?1sy,n:dailyvolofYcalculatedondayn?1covn:covariancecalculatedondayn?1Thecorrelationiscovn/(sx,nsy,n)Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201428UpdatingCorrelationsWecanusesimilarmodelstothoseforvolatilitiesUnderEWMA covn

=lcovn-1+(1-l)xn-1yn-1Options,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201429PositiveFiniteDefiniteCondition Avariance-covariancematrix,W,isinternallyconsistentifthepositivesemi-definitecondition

forallvectorswOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201430Example Thevariance-covariancematrix isnotinternallyconsistentOptions,Futures,andOtherDerivatives,9thEdition,Copyright?JohnC.Hull201431VolatilitiesandCorrelationsforFour-Inde