Ch14_ValueatRis(金融工程学,华东师大)k

Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.1ValueatRiskChapter14Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.2TheQuestionBeingAskedinValueatRisk(VaR)“WhatlosslevelissuchthatweareX%confidentitwillnotbeexceededinNbusinessdays?”Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.3MeaningisProbability2(0,)Pr()()*YNYppN(1-)%%ZOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.4VaRandRegulatoryCapitalRegulatorsrequirebankstokeepcapitalformarketriskequaltotheaverageofVaRestimatesforpast60tradingdaysusingX=99andN=10,timesamultiplicationfactor.(Usuallythemultiplicationfactorequals3)Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.5AdvantagesofVaR•Itcapturesanimportantaspectofriskinasinglenumber•Itiseasytounderstand•Itasksthesimplequestion:“Howbadcanthingsget?”Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.6DailyVolatilities•Inoptionpricingweexpressvolatilityasvolatilityperyear•InVaRcalculationsweexpressvolatilityasvolatilityperdaydayyear252Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.7DailyVolatility(continued)•Strictlyspeakingweshoulddefinedayasthestandarddeviationofthecontinuouslycompoundedreturninoneday•InpracticeweassumethatitisthestandarddeviationoftheproportionalchangeinonedayOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.8IBMExample(p.343)•Wehaveapositionworth$10millioninIBMshares•ThevolatilityofIBMis2%perday(about32%peryear)•WeuseN=10andX=99Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.9IBMExample(continued)•Thestandarddeviationofthechangeintheportfolioin1dayis$200,000•Thestandarddeviationofthechangein10daysis20000010456,$632,Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.10IBMExample(continued)•Weassumethattheexpectedchangeinthevalueoftheportfolioiszero(ThisisOKforshorttimeperiods)•Weassumethatthechangeinthevalueoftheportfolioisnormallydistributed•SinceN(0.01)=-2.33,theVaRis233632456473621.,$1,,Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.11AT&TExample•Considerapositionof$5millioninAT&T•ThedailyvolatilityofAT&Tis1%(approx16%peryear)•TheSTDper10daysis•TheVaRis5000010144,$158,158114233405,.$368,Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.12Portfolio(p.344)•NowconsideraportfolioconsistingofbothIBMandAT&T•Supposethatthecorrelationbetweenthereturnsis0.7Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.13STDofPortfolio•Astandardresultinstatisticsstatesthat•Inthiscasex=632,456andY=158,114andr=0.7.Thestandarddeviationofthechangeintheportfoliovalueistherefore751,665rXYXYXY222Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.14VaRforPortfolio•TheVaRfortheportfoliois•Thebenefitsofdiversificationare(1,473,621+368,405)-1,751,379=$90,647•WhatistheincrementaleffectoftheAT&TholdingonVaR?751665233751379,.$1,,Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.15TheLinearModelWeassumeThechangeinthevalueofaportfolioislinearlyrelatedtothechangeinthevalueofmarketvariablesThechangesinthevaluesofthemarketvariablesarenormallydistributedOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.16TheGeneralLinearModelcontinued(Equation14.5)PxiiiinPijijijjninPiiijijijijiniPrr121122212whereisthevolatilityofvariableandistheportfolio'sstandarddeviationOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.17HandlingInterestRates•Wedonotwanttodefineeveryinterestrateasadifferentmarketvariable•AnapproachistousethedurationrelationshipP=-DPysothatP=DPyy,whereyisthevolatilityofyieldchangesandPisasbeforethestandarddeviationofthechangeintheportfoliovalueOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.18Alternative:CashFlowMapping(p.347)•Wechooseasmarketvariableszero-couponbondpriceswithstandardmaturities(1mm,3mm,6mm,1yr,2yr,5yr,7yr,10yr,30yr)•Supposethatthe5yrrateis6%andthe7yrrateis7%andwewillreceiveacashflowof$10,000in6.5years.•Thevolatilitiesperdayofthe5yrand7yrbondsare0.50%and0.58%respectivelyOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity14.19CashFlowMapping(continued)•Weinterpola