衍生工具與風(fēng)險管理 Don M. Chance 14

Chapter14:AdvancedDerivativesandStrategiesWelookateverything.Wedon’tgetscaredbecauseofthecomplexityinvolved.Butweexamineittodeath.Arvind

Sodhani,treasury,IntelBusinessWeek,October21,1994,p.95

D.M.ChanceCh.14:1AnIntroductiontoDerivativesandRiskManagement,6thed.ImportantConceptsinChapter14Theconceptofportfolioinsuranceanditsexecutionusingputs,calls,futuresandt-billsNewandadvancedderivativesandstrategiessuchasequityforwards,warrants,equity-linkeddebt,structurednotes,andmortgagesecuritiesExoticoptionssuchasdigitaloptions,chooseroptions,Asianoptions,lookbackoptions,andbarrieroptionsDerivativesonelectricityandweatherD.M.Chance2AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategiesPortfolioInsuranceWecaninsureaportfoliobyholdingoneputforeachshareofstock.ForaportfolioworthV,weshouldholdN=V/(S0+P)putsandsharesThiswillestablishaminimumofVmin=XV/(S0+P)whereXistheexercisepriceExample:OnSept.26,marketindexis445.75andDec485putis$38.57.ExpirationisDec.19.Risk-freerateis2.99%continuouslycompounded.Volatilityis.155.D.M.Chance3AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)PortfolioInsurance(continued)Wehold100,000unitsoftheindexportfolioforV=$44,575,000.WehaveVmin=(485)(44,575,000)/(445.75+38.57)=44,637,585N=44,575,000/(445.75+38.57)=92,036Thisguaranteesaminimumreturnof1.0014(365/84)-1=.0061peryear,whichmustbebelowtherisk-freerate.D.M.Chance4AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)PortfolioInsurance(continued)OutcomesIndexis510atexpiration92,036sharesworth510=$46,938,36092,036putsworth$0=$0Totalvalue=$46,938,360(>Vmin)Indexis450atexpirationSellstockbyexercisingputssoyouhave92,036(485)=$44,637,460(?

Vmin)D.M.Chance5AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)PortfolioInsurance(continued)SeeFigure14.1,p.503.Ifcallsandt-billsused,NB=Vmin/BT(numberofbills)NC=V/(S0+P)(numberofcalls)SoNB=44,637,585/100=446,376NC=92,036D.M.Chance6AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)PortfolioInsurance(continued)OutcomesIndexis510atexpirationBillsworth$44,637,60092,036callsworth$25=$2,300,900Totalvalue=$46,938,500(>Vmin)Indexis450atexpirationBillsworth$44,637,60092,036callsworth$0Totalvalue=$44,637,600(?

Vmin)SeeFigure14.2,p.505.D.M.Chance7AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)PortfolioInsurance(continued)Dynamichedging:Adynamicallyadjustedcombinationofstockandfuturesorstockandt-billsthatcanreplicatethestock-putorcall-tbill.Thiscanbeeasierbecausethefuturesandt-billmarketsaremoreliquidthantheoptionsmarketsThenumberoffuturesrequiredisSeeAppendix14.Aforderivation.D.M.Chance8AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)PortfolioInsurance(continued)Alternatively,usestockandt-bills(seeAppendix14.Aagainforderivation).SeeTable14.1,p.507

forexampleofdynamichedgeD.M.Chance9AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)EquityForwardsForwardcontractsonstockorstockindicesPreciselylikeallotherforwardcontractswehavecovered.Breakforwardissimilartoanordinarycallbuthasnoup-frontcost.Atexpiration,however,itsvaluecanbenegative,unlikeanordinarycall.SeeTable14.2,p.510.NotethatK=compoundfuturevalueofcallwithexercisepriceFpluscompoundfuturevalueofstock,whichisforwardpriceofstock.D.M.Chance10AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)EquityForwards(continued)ExampleusingAOL:S0=125.9375,T=.0959,rc=.0446,volatility=.83.F=125.9375e.0446(.0959)=126.48OrdinarycallwithX=126.48isworth12.88.K=126.48+12.88e.0446(.0959)=139.41SeeFigure14.3,p.511.Notesimilaritytoforwardcontractandcalloption.Todeterminethevalueofabreakforwardattimetduringitslife,wesimplyvalueitasacallandaloan:D.M.Chance11AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)EquityForwards(continued)Forexample,15dayslater,AOLisat115.75,T–t=20/365=0.0548,andtheotherinputsareunchanged.WeobtainAmoregeneralversionofabreakforwardisapay-lateroption.Inthiscase,thebuyersimplyborrowsthepremiumandhastopayitbackatexpiration.ThisoptionisjustanordinarycallplusaloanofthecallpremiumCe(S0,T,X).Atexpiration,thebuyerdecideswhethertoexercisethecallandineithercasepaysbackD.M.Chance12AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)EquityWarrantsWarrantsissuedbyfirmWarrantstradingonover-the-countermarketsandAmericanStockExchangebasedonvarioussecuritiesandindices.Manyofthesearequantos,whichpayoffbasedontheperformanceofaforeignstockindexbutpaymentismadeinadifferentcurrencythantheoneassociatedwiththecountryoftheforeignstockindex.D.M.Chance13AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)Equity-LinkedDebtAbondthatusuallypaysaminimumreturnplusapercentageofanyincreaseinastockindexExample:One-yearzerocouponbondpaying1%interestand50percentofanygainontheS&P500.Currentlyone-yearzerocouponbondoffers5%compoundedannually.S&P500isat1500withavolatilityof.12andayieldof1.5%.Ifyouinvest$10youreceive$10(1.01)=$10.10forsure.Thepresentvalueofthisis10.10/1.05=9.62(5%isopportunitycost).Thisamountstoalossof$0.38.D.M.Chance14AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedEquityDerivativesandStrategies(continued)Equity-LinkedDebt(continued)Optionpayoffis$10(.5)Max(0,(ST-1500)/1500).Thiscanbewrittenas(5/1500)Max(0,ST-1500),whichis5/1500thofaEuropeancallwithexerciseprice1500.PluggingvaluesintoBlack-Scholesmodelgivescallvalueof$96.81.Multiplyingby5/1500givesavalueof$0.32.Thisislessthantheamountgivenupbyacceptingthelowerrateonthebond($0.38)butmightbeworthwhiletosomeinvestors.D.M.Chance15AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedInterestRateDerivativesStructuredNotesDefinition:anintermediatetermdebtsecurityissuedbyacorporationwithagoodcreditratinginwhichthecouponisalteredbytheuseofaderivative.Examples:FloatingcouponindexedusuallytoLIBORortheCMTrate(e.g.,1.5timestherate).Rangefloater,whichpaysinterestonlyifareferencerate(e.g.,LIBOR)stayswithinagivenrangeoveraperiodoftime.Ifratestayswithinrange,couponwillbehigherthanotherwise.Reverse(inverse)floater,wherecouponmovesoppositetointerestrates,suchas12-LIBORD.M.Chance16AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedInterestRateDerivatives(continued)StructuredNotes(continued)Example:AnissuercouldhedgeitbyaswappayingLIBORandreceivingfixedrateLIBOR<12:-(12-LIBOR)(note)+Fixedrate-LIBOR(swap)=Fixedrate-12LIBOR12:0(note)+Fixedrate-LIBOR(swap)=Fixedrate-LIBOR.IssuercouldbuyacaptopayitLIBORwhileitpaysthestrikerateifitwantedtomakeitrisk-free.Manyinversefloatersareextremelyvolatileduetoleverageintherateadjustmentformula.D.M.Chance17AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedInterestRateDerivatives(continued)Mortgage-BackedSecuritiesSecuritiesconstructedbyofferingclaimsonaportfolioofmortgages,aprocesscalledsecuritization.Mortgage-backedsecuritiesaresubjecttoprepaymentrisk.Mortgagepass-throughsandstripsMortgagepass-through:asecurityinwhichtheholderreceivestheprincipalandinterestpaymentsmadeonaportfolioofmortgages.Mortgagestrip:aclaimoneithertheprincipalorinterestonamortgagepass-through.Calledprincipalonly(PO)orinterestonly(IO).D.M.Chance18AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedInterestRateDerivatives(continued)Mortgage-BackedSecurities(continued)Example:Assumeamortgage-backedsecurityrepresentingasingle$100,000mortgageat9.75%for30years.Assumeannualpaymentsforsimplicity.SeeTable14.3,p.516foramortizationschedule.Annualpaymentwouldbe$100,000/[(1-(1.0975)-30)/.0975]=$10,387.D.M.Chance19AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedInterestRateDerivatives(continued)Mortgage-BackedSecurities(continued)Assumea7percentdiscountrateandthatthemortgageispaidoffinyear12.ValueofIOstrip=9,750(1.07)-1+9,688(1.07)-2+…+8,614(1.07)-12=74,254.ValueofPOstrip=637(1.07)-1+699(1.07)-2+…+(1,773+86,574)(1.07)-12=46,690.Valueofpass-through=$74,254+$46,690=$120,944D.M.Chance20AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedInterestRateDerivatives(continued)Mortgage-BackedSecurities(continued)Letdiscountratedropto6%andassumehomeownerpaysofftwoyearsfromnow.ValueofIO=$9,750(1.06)-1+$9,688(1.06)-2=$17,820,lossof76%ValueofPO=$637(1.06)-1+($699+$98,663)(1.06)-2=$89,033,gainof91%Valueofpass-through=$17,820+$89,034=$106,854,lossof12%D.M.Chance21AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedInterestRateDerivatives(continued)Mortgage-BackedSecurities(continued)Ifthediscountraterisesto8%andthereisnochangeinthepayoffdateofyear12,ValueofIO=$9,750(1.08)-1+$9,688(1.08)-2+...+$8,614(1.08)-12=$70,532,a5%lossValueofPO=$637(1.08)-1+$699(1.08)-2+...+($1,773+$86,574)(1.08)-12=$42,128,a10%lossValueofpass-through=$70,532+$42,128=$112,660,alossofalmost7%.D.M.Chance22AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedInterestRateDerivatives(continued)Mortgage-BackedSecurities(continued)Ifrategoesto8%andprepaymentmovesbacktoyear14,ValueofIO=$9,750(1.08)-1+$9,688(1.08)-2+....+$8,250(1.08)-14=$76,445,againof3%ValueofPO=$637(1.08)-1+$699(1.08)-2+...+($2,136+$82,492)(1.08)-14=$37,276,alossof20%.Valueofpass-through=$76,445+$37,276=$113,721,alossofabout6%.D.M.Chance23AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedInterestRateDerivatives(continued)Mortgage-BackedSecurities(continued)Mortgage-backedsecurityvaluesaretypicallyveryvolatile.CollateralizedMortgageObligations(CMOs)Mortgage-backedsecurityinwhichpaymentsaresplitintopiecescalledtrancheswithdifferentclaimsreflectingdifferentrisks.Sometranchesarepaidfirst,somereceiveonlyinterestandsomereceiveanyresidualafterothertrancheshavebeenrepaid.D.M.Chance24AnIntroductiontoDerivativesandRiskManagement,6thed.AdvancedInterestRateDerivatives(continued)Mortgage-BackedSecurities(continued)CollateralizedMortgageObligations(CMOs)(continued)Thedifferenttranchesreceiveinterest,principalandprepaymentsaccordingtodifferentpriorities.SomeCMOtranchesareextremelyvolatileandothershavelowvolatility.ACMOisgenerallyafairlycomplexsecurity.D.M.Chance25AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptionsDigitalandChooserOptionsDigitaloptions,sometimescalledbinaryoptions,areoftwotypes:Asset-or-nothingoptionspaytheholdertheassetiftheoptionexpiresinthemoneyandnothingotherwise.Cash-or-nothingoptionspaytheholderafixedamountofcash(usually$1)iftheoptionexpiresinthemoneyandnothingotherwise.D.M.Chance26AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)DigitalandChooserOptions(continued)SeeTable14.4,p.520forexampleofaportfoliolongcash-or-nothingsandshortXasset-or-nothings.ThiscombinationisequivalenttoanordinaryEuropeancall.ThevaluesoftheoptionsareD.M.Chance27AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)DigitalandChooserOptions(continued)Example:Asset-or-nothingoptionwrittenonS&P500TotalReturnIndex,at1440.Exercisepriceof1440.Risk-freerateis4.88%,standarddeviationis.11andtimetoexpirationis0.5years.Weobtaind1=.3526,N(.35)=.6368Oaon=1440(.6368)=917For1,440cash-or-nothingoptions,d2=.2748,N(.27)=.6064(1,440)Ocon=1440e-.0488(.5)(.6064)=852.17.D.M.Chance28AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)DigitalandChooserOptions(continued)Avariationofthepreviouslycoveredpay-lateroptionisthecontingent-payoption.Herethepremiumispaidatexpirationbutonlyiftheoptionexpiresin-the-money.Table14.5,p.521showsthatthisoptionisacombinationofastandardoptionandCcpcash-or-nothingcalls.ThevaluemustbezerotodaysoD.M.Chance29AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)DigitalandChooserOptions(continued)SolvingforCcpgivesFortheexamplewehavebeenusingNowmoveforwardtwomonthswhereSt=1440andT-t=4/6=0.333.ThevalueisD.M.Chance30AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)DigitalandChooserOptions(continued)ChooserOptions:Alsocalledas-you-like-itoptions,theyenabletheinvestortodecideataspecifictimeafterpurchasingtheoptionbutbeforeexpirationthattheoptionwillbeacalloraput.Assumethedecisionmustbemadeattimet<TThechooseroptionisidenticaltoanordinarycallexpiringatTwithexercisepriceXplusanordinaryputexpiringattwithexercisepriceX(1+r)-(T-t)Compareandcontrastchooserwithstraddle.D.M.Chance31AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)DigitalandChooserOptions(continued)Example:AOLchooserinwhichchoicemustbemadein20days.Call/putexpiresin35days.S0=125.9375,X=125,=.83,rc=.0446.T=35/365=.0959,t=20/365=.0548soT-t=.0959-.0548=.0411.Exercisepriceonputusedtopricethechooseris125(1.0456)-.0411=124.77.UsingBlack-Scholesmodel,putisworth7.80andcallisworth13.21foratotalof21.01.Straddleisworth13.21(call)+12.09(put)=25.30.D.M.Chance32AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)Path-DependentOptionsPath-dependentoptionsareoptionsinwhichthepayoffisdeterminedbythesequenceofpricesfollowedbytheassetandnotjustbythepriceoftheassetatexpiration.Weshallpricetheseoptionsusingabinomialframework.SeeTable14.6,p.523whichshowsathree-periodproblem.Noteeightpaths,andtheaverage,maximum,andminimumpricesofeachpatharecomputed.Notehowtheprobabilitiesarecalculated.Inpracticethebinomialmodelisdifficulttouseforpath-dependentoptions.MonteCarlosimulation(seeAppendix15.B)isoftenused.D.M.Chance33AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)Path-DependentOptions(continued)Asianoption:anoptioninwhichthefinalpayoffisdeterminedbytheaveragepriceoftheassetduringtheoption’slife.Someareaveragepriceoptionsbecausetheaveragepricesubstitutesfortheassetpriceatexpiration.Othersareaveragestrikeoptionsbecausetheaveragepricesubstitutesfortheexercisepriceatexpiration.Canbecallsorputs.Usefulforhedgingorspeculatingwhentheaverageisacceptableasameasureoftheunderlyingrisk.Alsousefulforcaseswheremarketcanbemanipulated.SeeTable14.7,p.525forexampleofpricingAsianoptions.D.M.Chance34AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)Path-DependentOptions(continued)Lookbackoption:Alsocalledano-regretsoption,itpermitspurchaseoftheassetatitslowestpriceduringtheoption’slifeorsaleoftheassetatitshighestpriceduringtheoption’slife.D.M.Chance35AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)Path-DependentOptions(continued)Lookbackoptions(continued):Fourdifferenttypes. lookbackcall:exercisepricesetatminimumpriceduringoption’slifelookbackput:exercisepricesetatmaximumpriceduringoption’slifefixed-strikelookbackcall:payoffbasedonmaximumpriceduringoption’slife(insteadoffinalprice)comparedtofixedstrikefixed-strikelookbackput:payoffbasedonminimumpriceduringoption’slife(insteadoffinalprice)comparedtofixedstrikeD.M.Chance36AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)Path-DependentOptions(continued)Lookbackoptions(continued):SeeTable14.8,p.526forexampleofpricinglookbackoptions.D.M.Chance37AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)Path-DependentOptions(continued)BarrierOptions:Optionsthateitherterminateearlyiftheassetpricehitsacertainlevel,calledthebarrier,oractivateonlyiftheassetpricehitsthebarrier.Theformerarecalledknock-outoptions(orsimplyout-options)

andthelatterarecalledknock-inoptions

(orsimplyin-options).Ifthebarrierisabovethecurrentprice,itiscalledanup-option.Ifthebarrierisbelowthecurrentprice,itiscalledadown-option.SeeTable14.9,p.528forexampleofpricing.Barrieroptionsarenormallycheaperthanordinaryoptionsbecausetheyprovidepayoffsforfeweroutcomesthanordinaryoptions.D.M.Chance38AnIntroductiontoDerivativesandRiskManagement,6thed.ExoticOptions(continued)Path-DependentOptions(continued)OtherExoticOptions:compoundandinstallmentoptionsmulti-assetoptions,exchangeoptions,min-maxoptions(rainbowoptions),alternativeoptions,outperformanceoptionsshout,cliquetandlock-inoptionscontingentpremium,pay-lateranddeferredstrikeoptionsforward-startandtandemoptionsD.M.Chance39AnIntroductiontoDerivativesandRiskManagement,6thed.SomeImportantNewDerivativesElectricityDerivativesElectricityisanon-storableassetThesederivativesaredifficulttopriceWeatherDerivativesMeasuresofweatheractivityHeatingdegreedaysandcoolingdegreedaysQuantityofrainorsnowFinanciallosscausedbyweatherPricingisdifficultbutnotimpossible;alotofdataareavailableonweatherD.M.Chance40AnIntroductiontoDerivativesandRiskManagement,6thed.SummaryD.M.Chance41AnIntroductiontoDerivativesandRiskManagement,6thed.Appendix14A.:DerivationoftheDynamicHedgeRatioforPortfolioInsuranceStock-FuturesDynamicHedgePortfolioofNsharesandNputsisworthV=N(S+P)SoN=V/(S+P).ChangeinportfoliovalueforasmallchangeinstockpriceisD.M.Chance42AnIntroductiontoDerivativesandRiskManagement,6thed.Appendix14.A:DerivationoftheDynamicHedgeRatioforPortfolioInsurance(continued)Stock-FuturesDynamicHedge(continued)AportfolioofNSsharesandNffuturesisworthtodayV=NSS+NfVfwhereVfisvalueoffutures,whichstartsatzero.ItfollowsthatNS=V/SSetchangeinportfoliovalueforsmallchangeinStoD.M.Chance43AnIntroductiontoDerivativesandRiskManagement,6thed.Appendix14.A:DerivationoftheDynamicHedgeRatioforPortfolioInsurance(continued)Stock-FuturesDynamicHedge(continued)Assumingnodividends,thefuturespriceisSoD.M.Chance44AnIntroductiontoDerivativesandRiskManagement,6thed.Appendix14.A:DerivationoftheDynamicHedgeRatioforPortfolioInsurance(continued)Stock-FuturesDynamicHedge(continued)Aftersubstituting,settingthetwopartialderivativesofVwithrespecttoSequaltoother,recognizingthat1+?P/?Sis?C/?SandN(d1)is?C/?S,weobtainthenumberoffuturescontractsasD.M.Chance45AnIntroductiontoDerivativesandRiskManagement,6thed.Appendix14.A:DerivationoftheDynamicHedgeRatioforPortfolioInsurance(continued)Stock-TbillDynamicHedgeAportfolioofstockandtbillsisworthItssensitivitytoachangeinSisD.M.Chance46AnIntroductiontoDerivativesandRiskManagement,6thed.Appendix14.A:DerivationoftheDynamicHedgeRatioforPortfolioInsurance(continued)Stock-TbillDynamicHedge(continued)Thet-billpriceisnotsensitivetothestockprice.Settingthesensitivityofthestock-tbillportfoliotothatofthestock-futuresportfoliogivesThisisthenumberofsharesofstocktoholdwitht-billstoreplicatethestockandput.D.M.Chance47AnIntroductiontoDerivativesandRiskManagement,6thed.Appendix14.B:MonteCarloSimulationAmethodofusingrandomnumbersdesignedtosimulatetherandomobservationsofpricesofanasset.Thesimulatedseriesofassetpricesatexpirationisthenconvertedtoanequivalentseriesofoptionpricesatexpiration.Thenthecurrentoptionpriceisthediscountedaverageofth