CFA三级知识点必备：Derivatives and Currency Management_打印版.docx

CFA直）栗笺亩答BobHong1-272-27CoveredcallstrategyAninvestorcreatescoveredcallpositionbysellingacalloptiononastockthatisownedbytheoptionwriter.YieldenhancementThemostcommonmotivation.BywritinganOTMcalloption.Cashgenerationinanticipationoflimitedupsidemoves.ReducingapositionatafavorablepriceCoveredcallsmightbewritten,whenaninvestorholdsapositioninastockandintendstoreducethatholdinginthenearfuture.(TMcalloption)TargetpricerealizationHybridoftheprevioustwo.Callsarewrittenwithastrikepricejustabovethecurrentmarketprice.(OTMcalloption)CoveredcallstrategyCoveredcall:Inthisstrategy,someonewhoalreadyownssharessellsacalloptiongivingsomeoneelsetherighttobuytheirsharesattheexerciseprice.STSmax0,STX+CConclusion:WhenSTXrwehavemaximumgain5丁Sqmax0,STX+C=(&-$)*X+C=X金+CWhenS干0,wehavemaximumlossST-So-maxO,Sr-X+C=(O-)-O+C=C-SoBreakevenpointST=SLC4-27ProtectiveputstrategyAprotectiveput(alsocalledportfolioinsuranceorahedgedportfolio)isconstructedbyholdingalongpositionintheunderlyingsecurityandbuyingaputoption.Youcanuseaprotectiveputlimitthedownsideriskatthecostoftheputpremium,P0.Youwillseebythediagramthattheinvestorwillstillbeabletobenefitfromincreasesinthestock,sprice,butitwillbelowerbytheamountpaidfortheput,P0.Noticethatthecombinedstrategylooksverymuchlikeacalloption.5-27ProtectiveputstrategyProtectiveput:Someonesimultaneouslyholdsalongpositioninanassetandalongpositioninaputoptiononthatasset.Conclusion:WhenSX,theprofitisunlimited(STS°)+mx0,XSTPWhenSf,wehavemaximumloss(ST-SO)+max0,X-S一P=O-S+X-P=X-S-PooBreakevenpoint:Si=S计PVolatilitySmileWhatisvolatilitysmile?Volatilitysmileisaplotoftheimpliedvolatilityofanoptionasafunctionofitsstrikeprice.Thischapterdescribesthevolatilitysmilesthattradersuseinequityandforeigncurrencymarkets.827VolatilitySmileBasedontheput-callparitysoPmkt+SoqT=Cmkt+e-rT2Conclusions.ThedollarpricingerrorwhentheBlack-ScholesmodelisusedtopriceaEuropeanputoptionshouldbeexactlythesameasthedollarpricingerrorwhenitisusedtopricingaEuropeancalloptionwiththesamestrikepriceandtimetomaturity.TheimpliedvolatilityofaEuropeancalloptionisalwaysthesameastheimpliedvolatilityofaEuropeanputoptionwhenthetwohavethesamestrikepriceandmaturitydate.VolatilitySmileforForeignCurrencyOptionsTheimpliedvolatilityisrelativelylowforat-the-moneyoptions.Itbecomesprogressivelyhigherasanoptionmoveseitherintothemoneyoroutofthemoney.ImpliedvolatilityVolatilityincreasesasoptionsbecomesincreasinglyinthemoneyoroutofthemoney.OutoftheMoneyCallsOutoftheMoneyPutsstrikepriceAtteMoneyOptions10-27ReasonsforSmileinForeignCurrencyOptionsWhyareexchangeratenotIognormallydistributed?TwooftheContidionsforanassetpricetohavealognormaldistributionare:Thevolatilityoftheassetisconstant.Thepriceoftheassetchangessmoothlywithnojumps.Inpractice,neitheroftheseconditionsissatisfiedforanexchangerate.Thevolatilityofanexchangerateisfarfromconstant,andexchangeratesfrequentlyexhibitjumps(sometimesthejumpsareinresponsetotheactionsofcentralbanks).11-27VolatilitySmiles(skew)forEquityOptionsThevolatilityusedtopricealow-strike-priceoption(i.e.,adeepoutofthemoneyputoradeepinthemoneycall)issignificantlyhigherthanthatusedtopriceahigh-strike-priceoption(ie,adeepinthemoneyputoradeepoutofthemoneycall).ImpliedvolatilityOutoftheMoneyCallsOutoftheMoneyPutsstrikepriceAttheMoneyOptionsReasonsfortheSmileinEquityOptions1.everage(equitypriceTvolatility)Asacompany,sequitydeclinesinvalue,thecompanyzsleverageincreases.Thismeansthattheequitybecomesmoreriskyanditsvolatilityincreases.VolatilityFeedbackEffect(volatilityTequityprice)Asvolatilityincreases(decreases)becauseofexternalfactors,investorsrequireahigher(lower)returnandasaresultthestockpricedeclines(increases).Crashophobia(expectedequitypriceTimpliedvolatility)1987stockmarketcrash:higherpremiumsforputpriceswhenthestrikepriceslower.13-27StrategyRelatedtoVolatilitySkewAlongriskreversalcombineslongcallandshortputonthesameunderlyingwithsameexpiration.ForexampleIfatraderbelievesthatputimpliedvolatilityisrelativelytoohigh,comparedtothatforcalls,alongriskreversalcouldbecreatedbybuyingtheOTMcall(underpriced)andsellingtheOTMput(overpriced)forthesameexpiration.However,thiswouldcreatealongexposuretotheunderlying,whichcouldbeproblematic.14-27VolatilitySmileAlternativewaysofcharacterizingthevolatilitysmileThevolatilitysmileisoftencalculatedastherelationshipbetweentheimpliedvolatilityandKS0ratherthanastherelationshipbetweentheimpliedvolatilityandK.ArefinementofthisistocalculatethevolatilitysmileastherelationshipbetweentheimpliedvolatilityandKF0,whereFCistheforwardpirceoftheassetforacontractmaturingatthesametimeastheoptionsthatareconsidered.Anotherapproachtodefiningthevolatilitysmileisastherelationshipbetweentheimpliedvolatilityandthedeltaoftheoption.VolatilitySmileTradersallowtheimpliedvolatilitytodependontimetomaturityaswellasstrikeprice.VolatilitysurfacescombinevolatilitysmileswiththetimetomaturityandK%Impliedvolatilitytendstobeanincreasingfunctionofmaturitywhenshort-datedvolatilitiesarehistoricallylow.Volatilitytendstobeadecreasingfunctionofmaturitywhenshort-datedvolatilitiesarehistoricallyhigh.16-27VolatilityTermStructureandVolatilitySurfaceImpliedvolatilityonthez-axis;maturity(x-axis);andKS0(y-a×is).17-27UsingDerivativestoAlteringAssetAllocationAlteringassetallocationbetweenequityanddebtwithfuturesStep1:CalculatethereallocatingamountStep2:Toreallocateanamountfromequitytobonds:Removeallsystematicriskfromtheposition(beta=O)byshortingequityfutures.Adddurationtotheposition(BPV>O)bygoinglongbondfutures.Step3:Toreallocateanamountfrombondstoequity:Removealldurationfromtheposition(BVP=O)byshortingbondfutures.Addsystematicrisktotheposition(beta>O)bygoinglongequityfutures.equityCashBeta=OBPV=Obonds19-27UsingDerivativestoAlteringAssetAllocationAmendingportfoliobeta&SyntheticstockpositionsNumberofcontractsJalg“PPf'PmultiplierEquity/EquityMid-capequityB；：ToSmalMapEquityequityEquity/DebtCashBeta=ObondsBPV=O20-27VarianceSwapVarianceswapspayoffsarebasedonvarianceratherthanvolatility(standarddeviation).Theseproductsaretermedswapsastheyhavetwocounterparties,onemakingafixedpaymentandtheothermakingavariablepaymentThefixedpaymentistypicallybasedonimpliedvolatility2(impliedvariance)overtheperiodandisknownattheinitiationoftheswap,thisisreferredtoasthevariancestrikeThevariablepaymentisunknownatswapinitiationandisonlyknownatswapmaturity.Itistheactualvarianceoftheunderlyingassetoverthelifeoftheswapandisreferredtoasrealizedvariance.22-27VarianceSwapThefeaturesofvarianceswapnoexchangeofnotionalprincipalandnointerimsettlementperiods.Withavarianceswap,thereisasinglepaymentattheexpirationoftheswapbasedonthedifferencebetweenactualandimpliedvarianceoverthelifeOftheswaptnoumaInemelttesgnolnotionalvariance×(2-Ktnoumatnemelttesgnol=notionalD萌，%(notionalvega=notionalvariance×K223-27VarianceSwapTheMark-to-MarketvalueofvarianceswapThevalueofavarianceswapiszeroatinitiation,butovertime,theswapwilleithergainorlosevalueasrealizedandimpliedvolatilitydiverge.Consideraone-yearswapwherethreemonthshaveelapsedsinceinception,theMtMvalueoftheswapcanbecalculatedasfollow:VarianceSwapTheMark*to-MarketvalueofvarianceswapConsideraone-yearswapwherethreemonthshaveelapsedsinceinception,theMtMvalueoftheswapcanbecalculatedasfollow:Step1:Computeexpectedvarianceatmaturity(thetime-weightedaverageofrealizedvarianceandimpliedvarianceovertheremainderoftheswap'slife).tyituramtoecnairavdetceExp=dK(?TxT'Lr)Step2:Computeexpectedpayoffatswapmaturity:ffyoaPdetceExp=lanoiotnecnairav×(tyituramtoecnairavdetcexpe-Step#flAP½c)payoffatmaturitybacktothevaluationdate.25-27Example1.ukeAmos,anequityfundmanager,haspurchasedaone-yearvarianceswapontheS&P500withveganotionalof$100,000andastrikeof20%.NinemonthshavepassedandtheS&Phasrealizedavolatilityof21%.Thestrikepriceforathree-monthvarianceswapatthistimeisquotedat22%,andthethree-monthinterestrateis2%.Computethecurrentvalueoftheswap.2627ExampleCorrectAnswerStep1:Computetheexpectedvarianceatmaturity:932izx+222×C=330.75+121=451.751212Step2:Computetheexpectedpayoffatmaturity:Variancenotional=Vega犬ional=$1欺00=Expectedpayoffatmaturity=(2-K2)×variancenotional,whereK2=202=400expectedpayoffatmaturity=(451.75-400)×$2,500=$129,375Step3:Discountexpectedpayofffrommaturitytothevaluationdate(3months):Unannualizetheinterestrate=2%×312v=0.5%Currentvalueofswap=$¥/客=$128,731Thisisagaintothepurchaser(long)andalosstotheseller(short).