风险管理与金融机构(第4版)第8章答案

Chapter8:HowTradersManageTheirRisks8.1交易组合价值减少10500美元。8.2当波动率变化2%时，交易组合价格增长200×2＝400美元。8.3两种情形均为0.5*30*4=60美元8.41000份期权短头寸的Delta等于-700，可以通过买入700份股票的形式使交易组合达到Delta中性。8.5Theta为-100的含义是指在股价与波动率没有变化的情况下，期权价格每天下降100美元。假如交易员认为股价及隐含波动率在将来不会改变，交易员可以卖出期权，并且Theta值越高越好。8.6当一个期权承约人的Gamma绝对值较大，Gamma本身为负，并且Delta等于0，在市场变化率较大的情况下，期权承约人会有较大损失。8.8看涨及看跌期权的多头头寸都具备正的Gamma，由图6.9可以看出，当Gamma为正时，对冲人在股票价格变化较大时会有收益,而在股票价格变化较小时会有损失，因此对冲人在（b）情形收益更好，当交易组合包含期权的空头头寸时，对冲人在（a）情形收益会更好。8.9Delta的数值说明当欧元汇率增长0.01时，银行交易价格会增加0.01*30000=300美元，Gamma的数值说明，当欧元价格增长0.01时，银行交易组合的Delta会下降0.01*80000=800美元；为了做到Delta中性，我们应该卖出30000欧元；当汇率增长到0.93时，我们期望交易组合的Delta下降为（0.93-0.9）*80000=24000，组合价值变为27600。为了维持Delta中性，银行应该对2400数量欧元空头头寸进行平仓，这样可以保证欧元净空头头寸为27600。当一个交易组合的Delta为中性，同时Gamma为负时资产价格有一个较大变动时会引起损失。因此银行可能会蒙受损失。8.15.Thegammaandvegaofadelta-neutralportfolioare50per$per$and25per%,respectively.Estimatewhathappenstothevalueoftheportfoliowhenthereisashocktothemarketcausingtheunderlyingassetpricetodecreaseby$3anditsvolatilitytoincreaseby4%.Withthenotationofthetext,theincreaseinthevalueoftheportfoliois0.5×gamma×(ΔS)2vega×ΔσThisis0.5×50×3225×4=325Theresultshouldbeanincreaseinthevalueoftheportfolioof$325.8.16.Consideraone-yearEuropeancalloptiononastockwhenthestockpriceis$30,thestrikepriceis$30,therisk-freerateis5%,andthevolatilityis25%perannum.UsetheDerivaGemsoftwaretocalculatetheprice,delta,gamma,vega,theta,andrhooftheoption.Verifythatdeltaiscorrectbychangingthestockpriceto$30.1andrecomputingtheoptionprice.Verifythatgammaiscorrectbyrecomputingthedeltaforthesituationwherethestockpriceis$30.1.Carryoutsimilarcalculationstoverifythatvega,theta,andrhoarecorrect.Theprice,delta,gamma,vega,theta,andrhooftheoptionare3.7008,0.6274,0.050,0.1135,−0.00596,and0.1512.Whenthestockpriceincreasesto30.1,theoptionpriceincreasesto3.7638.Thechangeintheoptionpriceis3.7638−3.7008=0.0630.Deltapredictsachangeintheoptionpriceof0.6274×0.1=0.0627whichisveryclose.Whenthestockpriceincreasesto30.1,deltaincreasesto0.6324.Thesizeoftheincreaseindeltais0.6324−0.6274=0.005.Gammapredictsanincreaseof0.050×0.1=0.005whichis(tothreedecimalplaces)thesame.Whenthevolatilityincreasesfrom25%to26%,theoptionpriceincreasesby0.1136from3.7008to3.8144.Thisisconsistentwiththevegavalueof0.1135.Whenthetimetomaturityischangedfrom1to1−1/365theoptionpricereducesby0.006from3.7008to3.6948.Thisisconsistentwithathetaof−0.00596.Finally,whentheinterestrateincreasesfrom5%to6%,thevalueoftheoptionincreasesby0.1527from3.7008to3.8535.Thisisconsistentwitharhoof0.1512.8.17.Afinancialinstitutionhasthefollowingportfolioofover-the-counteroptionsonsterling:TypePositionDeltaofOptionGammaofOptionVegaofOptionCall−1,0000.502.21.8Call−5000.800.60.2Put−2,000−0.401.30.7Call−5000.701.81.4Atradedoptionisavailablewithadeltaof0.6,agammaof1.5,andavegaof0.8.(a)Whatpositioninthetradedoptionandinsterlingwouldmaketheportfoliobothgammaneutralanddeltaneutral?(b)Whatpositioninthetradedoptionandinsterlingwouldmaketheportfoliobothveganeutralanddeltaneutral?Thedeltaoftheportfoliois−1,000×0.50−500×0.80−2,000×(−0.40)−500×0.70=−450Thegammaoftheportfoliois−1,000×2.2−500×0.6−2,000×1.3−500×1.8=−6,000Thevegaoftheportfoliois−1,000×1.8−500×0.2−2,000×0.7−500×1.4=−4,000(a)Alongpositionin4,000tradedoptionswillgiveagamma-neutralportfoliosincethelongpositionhasagammaof4,000×1.5=6,000.Thedeltaofthewholeportfolio(includingtradedoptions)isthen:4,000×0.6−450=1,950Hence,inadditiontothe4,000tradedoptions,ashortpositionin£1,950isnecessarysothattheportfolioisbothgammaanddeltaneutral.(b)Alongpositionin5,000tradedoptionswillgiveavega-neutralportfoliosincethelongpositionhasavegaof5,000×0.8=4,000.Thedeltaofthewholeportfolio(includingtradedoptions)isthen5,000×0.6−450=2,550Hence,inadditiontothe5,000tradedoptions,ashortpositionin£2,550isnecessarysothattheportfolioisbothvegaanddeltaneutral.8.18.ConsideragainthesituationinProblem8.17.Supposethatasecondtradedoptionwithadeltaof0.1,agammaof0.5,andavegaof0.6isavailable.Howcouldtheportfoliobemadedelta,gamma,andveganeutral?Letw1bethepositioninthefirsttradedoptionandw2bethepositioninthesecondtradedoption.Werequire:6,000=1.5w10.5w24,000=0.8w10.6w2Thesolutiontotheseequationscaneasilybeseentobew1=3,200,w2=2,400.Thewholeportfoliothenhasadeltaof−4503,200×0.62,400×0.1=1,710Thereforetheportfoliocanbemadedelta,gammaandveganeutralbytakingalongpositionin3,200ofthefirsttradedoption,alongpositionin2,400ofthesecondtradedoptionandashortpositionin£1,710.8.19.(SpreadsheetProvided)ReproduceTable8.2.(InTable8.2,thestockpositionisroundedtothenearest100shares.)Calculatethegammaandthetaofthepositioneachweek.UsingtheDerivaGemApplicationsBuilderstocalculatethechangeinthevalueoftheportfolioeachweek(beforetherebalancingattheendoftheweek)andcheckwhetherequation(8.2)isapproximatelysatisfied.(Note:DerivaGemproducesavalueoftheta“percalendarday.”Thethetainequation8.2is“peryear.”)Considerthefirstweek.Theportfolioconsistsofashortpositionin100,000optionsandalongpositionin52,200shares.Thevalueoftheoptionchangesfrom$240,053atthebeginningoftheweekto$188,760attheendoftheweekforagainof$51,293.Thevalueoftheshareschangefrom52,200×49=$2,557,800to52,200×48.12=$2,511,864foralossof$45,936.Thenetgainis51,293−45,936=$5,357.Thegammaandtheta(peryear)oftheportfolioare−6,554.4and430,533sothatequation(8.2)predictsthegainas430,533×1/520.5×6,554.4×(48.12−49)2=5,742Theresultsforall20weeksareshowninthefollowingtable.WeekActualGain($)PredictedGain($)15,3575,74225,6896,0933−19,742−21,08441,9411,57253,7063,65269,3209,19176,2495,93689,4919,259996187010−23,380−18,992111,6432,497122,6451,3561311,36510,92314−2,876−3,3421512,93612,302167,5668,81517−3,880−2,763186,7646,899194,2955,205204,8044,805