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1、CFA直）栗笺亩答BobHong1-272-27CoveredcallstrategyAninvestorcreatescoveredcallpositionbysellingacalloptiononastockthatisownedbytheoptionwriter.YieldenhancementThemostcommonmotivation.BywritinganOTMcalloption.Cashgenerationinanticipationoflimitedupsidemoves.ReducingapositionatafavorablepriceCoveredcallsmigh
2、tbewritten,whenaninvestorholdsapositioninastockandintendstoreducethatholdinginthenearfuture.(TMcalloption)TargetpricerealizationHybridoftheprevioustwo.Callsarewrittenwithastrikepricejustabovethecurrentmarketprice.(OTMcalloption)CoveredcallstrategyCoveredcall:Inthisstrategy,someonewhoalreadyownsshare
3、ssellsacalloptiongivingsomeoneelsetherighttobuytheirsharesattheexerciseprice.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(alsocalledportfolioinsuranceo
4、rahedgedportfolio)isconstructedbyholdingalongpositionintheunderlyingsecurityandbuyingaputoption.Youcanuseaprotectiveputlimitthedownsideriskatthecostoftheputpremium,P0.Youwillseebythediagramthattheinvestorwillstillbeabletobenefitfromincreasesinthestock,sprice,butitwillbelowerbytheamountpaidfortheput,
5、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-PooBr
6、eakevenpoint:Si=S计PVolatilitySmileWhatisvolatilitysmile?Volatilitysmileisaplotoftheimpliedvolatilityofanoptionasafunctionofitsstrikeprice.Thischapterdescribesthevolatilitysmilesthattradersuseinequityandforeigncurrencymarkets.827VolatilitySmileBasedontheput-callparitysoPmkt+SoqT=Cmkt+e-rT2Conclusions
7、.ThedollarpricingerrorwhentheBlack-ScholesmodelisusedtopriceaEuropeanputoptionshouldbeexactlythesameasthedollarpricingerrorwhenitisusedtopricingaEuropeancalloptionwiththesamestrikepriceandtimetomaturity.TheimpliedvolatilityofaEuropeancalloptionisalwaysthesameastheimpliedvolatilityofaEuropeanputoptio
8、nwhenthetwohavethesamestrikepriceandmaturitydate.VolatilitySmileforForeignCurrencyOptionsTheimpliedvolatilityisrelativelylowforat-the-moneyoptions.Itbecomesprogressivelyhigherasanoptionmoveseitherintothemoneyoroutofthemoney.ImpliedvolatilityVolatilityincreasesasoptionsbecomesincreasinglyinthemoneyor
9、outofthemoney.OutoftheMoneyCallsOutoftheMoneyPutsstrikepriceAtteMoneyOptions10-27ReasonsforSmileinForeignCurrencyOptionsWhyareexchangeratenotIognormallydistributed?TwooftheContidionsforanassetpricetohavealognormaldistributionare:Thevolatilityoftheassetisconstant.Thepriceoftheassetchangessmoothlywith
10、nojumps.Inpractice,neitheroftheseconditionsissatisfiedforanexchangerate.Thevolatilityofanexchangerateisfarfromconstant,andexchangeratesfrequentlyexhibitjumps(sometimesthejumpsareinresponsetotheactionsofcentralbanks).11-27VolatilitySmiles(skew)forEquityOptionsThevolatilityusedtopricealow-strike-price
11、option(i.e.,adeepoutofthemoneyputoradeepinthemoneycall)issignificantlyhigherthanthatusedtopriceahigh-strike-priceoption(ie,adeepinthemoneyputoradeepoutofthemoneycall).ImpliedvolatilityOutoftheMoneyCallsOutoftheMoneyPutsstrikepriceAttheMoneyOptionsReasonsfortheSmileinEquityOptions1.everage(equitypric
12、eTvolatility)Asacompany,sequitydeclinesinvalue,thecompanyzsleverageincreases.Thismeansthattheequitybecomesmoreriskyanditsvolatilityincreases.VolatilityFeedbackEffect(volatilityTequityprice)Asvolatilityincreases(decreases)becauseofexternalfactors,investorsrequireahigher(lower)returnandasaresultthesto
13、ckpricedeclines(increases).Crashophobia(expectedequitypriceTimpliedvolatility)1987stockmarketcrash:higherpremiumsforputpriceswhenthestrikepriceslower.13-27StrategyRelatedtoVolatilitySkewAlongriskreversalcombineslongcallandshortputonthesameunderlyingwithsameexpiration.ForexampleIfatraderbelievesthatp
14、utimpliedvolatilityisrelativelytoohigh,comparedtothatforcalls,alongriskreversalcouldbecreatedbybuyingtheOTMcall(underpriced)andsellingtheOTMput(overpriced)forthesameexpiration.However,thiswouldcreatealongexposuretotheunderlying,whichcouldbeproblematic.14-27VolatilitySmileAlternativewaysofcharacteriz
15、ingthevolatilitysmileThevolatilitysmileisoftencalculatedastherelationshipbetweentheimpliedvolatilityandKS0ratherthanastherelationshipbetweentheimpliedvolatilityandK.ArefinementofthisistocalculatethevolatilitysmileastherelationshipbetweentheimpliedvolatilityandKF0,whereFCistheforwardpirceoftheassetfo
16、racontractmaturingatthesametimeastheoptionsthatareconsidered.Anotherapproachtodefiningthevolatilitysmileisastherelationshipbetweentheimpliedvolatilityandthedeltaoftheoption.VolatilitySmileTradersallowtheimpliedvolatilitytodependontimetomaturityaswellasstrikeprice.Volatilitysurfacescombinevolatilitys
17、mileswiththetimetomaturityandK%Impliedvolatilitytendstobeanincreasingfunctionofmaturitywhenshort-datedvolatilitiesarehistoricallylow.Volatilitytendstobeadecreasingfunctionofmaturitywhenshort-datedvolatilitiesarehistoricallyhigh.16-27VolatilityTermStructureandVolatilitySurfaceImpliedvolatilityonthez-
18、axis;maturity(x-axis);andKS0(y-ais).17-27UsingDerivativestoAlteringAssetAllocationAlteringassetallocationbetweenequityanddebtwithfuturesStep1:CalculatethereallocatingamountStep2:Toreallocateanamountfromequitytobonds:Removeallsystematicriskfromtheposition(beta=O)byshortingequityfutures.Adddurationtot
19、heposition(BPVO)bygoinglongbondfutures.Step3:Toreallocateanamountfrombondstoequity:Removealldurationfromtheposition(BVP=O)byshortingbondfutures.Addsystematicrisktotheposition(betaO)bygoinglongequityfutures.equityCashBeta=OBPV=Obonds19-27UsingDerivativestoAlteringAssetAllocationAmendingportfoliobeta&
20、SyntheticstockpositionsNumberofcontractsJalg“PPfPmultiplierEquity/EquityMid-capequityB；：ToSmalMapEquityequityEquity/DebtCashBeta=ObondsBPV=O20-27VarianceSwapVarianceswapspayoffsarebasedonvarianceratherthanvolatility(standarddeviation).Theseproductsaretermedswapsastheyhavetwocounterparties,onemakinga
21、fixedpaymentandtheothermakingavariablepaymentThefixedpaymentistypicallybasedonimpliedvolatility2(impliedvariance)overtheperiodandisknownattheinitiationoftheswap,thisisreferredtoasthevariancestrikeThevariablepaymentisunknownatswapinitiationandisonlyknownatswapmaturity.Itistheactualvarianceoftheunderl
22、yingassetoverthelifeoftheswapandisreferredtoasrealizedvariance.22-27VarianceSwapThefeaturesofvarianceswapnoexchangeofnotionalprincipalandnointerimsettlementperiods.Withavarianceswap,thereisasinglepaymentattheexpirationoftheswapbasedonthedifferencebetweenactualandimpliedvarianceoverthelifeOftheswaptn
23、oumaInemelttesgnolnotionalvariance(2-Ktnoumatnemelttesgnol=notionalD萌，%(notionalvega=notionalvarianceK223-27VarianceSwapTheMark-to-MarketvalueofvarianceswapThevalueofavarianceswapiszeroatinitiation,butovertime,theswapwilleithergainorlosevalueasrealizedandimpliedvolatilitydiverge.Consideraone-yearswa
24、pwherethreemonthshaveelapsedsinceinception,theMtMvalueoftheswapcanbecalculatedasfollow:VarianceSwapTheMark*to-MarketvalueofvarianceswapConsideraone-yearswapwherethreemonthshaveelapsedsinceinception,theMtMvalueoftheswapcanbecalculatedasfollow:Step1:Computeexpectedvarianceatmaturity(thetime-weightedav
25、erageofrealizedvarianceandimpliedvarianceovertheremainderoftheswapslife).tyituramtoecnairavdetceExp=dK(?TxTLr)Step2:Computeexpectedpayoffatswapmaturity:ffyoaPdetceExp=lanoiotnecnairav(tyituramtoecnairavdetcexpe-Step#flAPc)payoffatmaturitybacktothevaluationdate.25-27Example1.ukeAmos,anequityfundmanag
26、er,haspurchasedaone-yearvarianceswapontheS&P500withveganotionalof$100,000andastrikeof20%.NinemonthshavepassedandtheS&Phasrealizedavolatilityof21%.Thestrikepriceforathree-monthvarianceswapatthistimeisquotedat22%,andthethree-monthinterestrateis2%.Computethecurrentvalueoftheswap.2627ExampleCorrectAnswe
27、rStep1:Computetheexpectedvarianceatmaturity:932izx+222C=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).
