Variance reduction techniques for estimating Value

RC21577(97324)October22,1999ComputerScience/MathematicsResearchReportVarianceReductionTechniquesforEstimatingValue-at-RiskPaulGlassermanColumbiaBusinessSchoolColumbiaUniversityNewYork,NewYork10027PhilipHeidelbergerIBMT.J.WatsonResearchCenterP.O.Box218YorktownHeights,NY10598PerwezShahabuddinIEORDepartmentColumbiaUniversityNewYork,NewYork10027IBMResearchDivisionAlmaden-Austin-Beijing-Haifa-T.J.Watson-Tokyo-ZurichLIMITEDDISTRIBUTIONNOTICE:ThisreporthasbeensubmittedforpublicationoutsideofIBMandwillprobablybecopyrightedifacceptedforpublication.IthasbeenissuedasaResearchReportforearlydisseminationofitscontents.Inviewofthetransferofcopyrighttotheoutsidepublisher,itsdistributionoutsideofIBMpriortopublicationshouldbelimitedtopeercommunicationsandspecicrequests.Afteroutsidepublication,requestsshouldbelledonlybyreprintsorlegallyobtainedcopiesofthearticle(e.g.,paymentofroyalties).CopiesmayberequestedfromIBMT.J.WatsonResearchCenter[Publications16-220ykt]P.O.Box218,YorktownHeights,NY10598.email:[e-mail]leontheinternetat(\delta-gamma)approximationtothechangeinportfoliovaluetoguidetheselectionofeectivevariancereductiontechniques;specicallyimportancesamplingandstratiedsampling.Iftheapproximationisexact,thentheimportancesamplingisshowntobeasymptoticallyoptimal.Numericalresultsindicatethatanappropriatecombinationofimpor-tancesamplingandstratiedsamplingcanresultinlargevariancereductionswhenestimatingtheprobabilityoflargeportfoliolosses.1IntroductionAnimportantconceptforquantifyingandmanagingportfolioriskisvalue-at-risk(VAR)[17,19].VARisdenedasaquantileofthelossinportfoliovalueduringaholdingperiodofspeciedduration.IfthevalueoftheportfolioattimetisV(t),theholdingperiodist,andthevalueoftheportfolioattimet+tisV(t+t),thenthelossinportfoliovalueduringtheholdingperiodisL=V(t)V(t+t).Foragivenprobabilityp,theVAR,xp,isdenedtobethe(1p)’thquantileofthelossdistribution:PfLxpg=p:(1)Typically,theintervaltisonedayortwoweeksandpisclosetozero,oftenp0:01:MonteCarlosimulationisfrequentlyusedtoestimatetheVAR.Insuchasimulation,changesintheportfolio’s\riskfactors(e.g.,interestrates,currencyexchangerates,stockprices,etc.)duringtheholdingperiodaregeneratedandtheportfolioisre-evaluatedusingthesenewvaluesfortheriskfactors.Thisisrepeatedmanytimessothatthelossdistributionmaybeestimated.However,thecomputationalcostrequiredtoobtainaccurateVARestimatesisoftenenormous.Thisisduetotwofactors.First,theportfoliomayconsistofaverylargenumberofnancialinstruments.1Furthermore,computingthevalueofanindividualinstrumentmayitselfrequirerepeatedMonteCarlotrials.Thuseachportfolioevaluationmaybecostly.Second,alargenumberofruns(portfolioevaluations)arerequiredinordertoobtainaccurateestimatesofthelossdistributionintheregionofinterest.Wefocusonthissecondissue:thedevelopmentofvariancereductiontechniquesdesignedtodramaticallyreducethenumberofrunsrequiredtoachieveaccurateestimatesoflowprobabilities.Ageneraldiscussiononvariancereductiontechniquesmaybefoundin[13].Thetechniquedescribedinthispaperbuildsonthemethodsof[8,10],whichweredevelopedtoreducethevariancewhenpricingasingleinstrument.Preliminarynumericalresultsforthetechniquedescribedinthispaper,andforrelatedtechniques,werereportedin[9].Ourapproachistoapproximatetheportfoliolossbyaquadraticfunctionoftheunderlyingriskfactorsandtousethisapproximationtodesignvariancereductiontechniques.Quadraticapproxi-mationsarewidelyusedwithoutsimulation;indeedthesecondorderTaylorseriesapproximationiscommonlycalledthe\delta-gammaapproximation[17,18,19].Whileourapproachcouldbecombinedwithotherquadraticapproximations,manyoftherstandsecondderivativesneededforthedelta-gammaapproximationareroutinelycomputedforotherpurposesquiteapartfromthecalculationofVAR.ApremiseofthispaperisthatthesederivativesarethusreadilyavailableasinputstobeusedinaVARsimulationanddonotrepresentanadditionalcomputationalburden.Whenthechangeinriskfactorshasamultivariatenormaldistribution,asiscommonlyassumed(andaswewillassume),thenthedistributionofthedelta-gammaapproximationcanbecomputednumerically[16,18].WhilethisapproximationisnotalwaysaccurateenoughtoprovidepreciseVARestimates,wedescribehowitmaybeusedtoguideinselectinganimportancesampling(IS)changeofmeasureforsamplingthechangesinriskfactors.ISisaparticularlyappropriatetechniquefor\rareeventsimulations,whichcorrespondstotheVARproblemwithasmallvalueofp.See[1,3,8,10,14]andtherefere