Derivativesand

Currency

Management

CFAй???畝?

???BobHong

1-27

1.Coveredcall

&protective

put

2-27

Coveredcallstrategy

?Aninvestorcreatescoveredcallpositionbysellingacalloptiononastock

thatisownedbytheoptionwriter.

zYieldenhancement

9Themostcommonmotivation.BywritinganOTMcalloption.Cash

generationinanticipationoflimitedupsidemoves.

zReducingapositionatafavorableprice

9Coveredcallsmightbewritten,whenaninvestorholdsapositionin

astockandintendstoreducethatholdinginthenearfuture.(ITM

calloption)

zTargetpricerealization

9Hybridoftheprevioustwo.Callsarewrittenwithastrikepricejust

abovethecurrentmarketprice.(OTMcalloption)

3-27

Coveredcallstrategy

?Coveredcall:Inthisstrategy,someonewhoalreadyownssharessellsacall

optiongivingsomeoneelsetherighttobuytheirsharesattheexercise

price.

??????????????

?

?

?

?Conclusion:

zWhenS>X,wehavemaximumgain

T

????????????????????????????????

?

?

?

?

?

?

?

zWhenS=0,wehavemaximumloss

T

????????????????????????????

?

?

?

?

?

zBreakevenpoint?????

?

?

4-27

Protectiveputstrategy

?Aprotectiveput(alsocalledportfolioinsuranceorahedgedportfolio)is

constructedbyholdingalongpositionintheunderlyingsecurityand

buyingaputoption.

zYoucanuseaprotectiveputlimitthedownsideriskatthecostof

theputpremium,P.

0

zYouwillseebythediagramthattheinvestorwillstillbeabletobenefit

fromincreasesinthestock’sprice,butitwillbelowerbytheamount

paidfortheput,P.

0

zNoticethatthecombinedstrategylooksverymuchlikeacalloption.

5-27

Protectiveputstrategy

?Protectiveput:Someonesimultaneouslyholdsalongpositioninanasset

andalongpositioninaputoptiononthatasset.

?Conclusion:

zWhenS>X,theprofitisunlimited

T

????????????????

?

?

?

zWhenS=0,wehavemaximumloss

T

??????????????

????????????????

??

?

?

?

zBreakevenpoint:?????

?

?

6-27

2.Volatility

Smile

7-27

VolatilitySmile

?Whatisvolatilitysmile?

zVolatilitysmileisaplotoftheimpliedvolatilityofanoptionasa

functionofitsstrikeprice.

9Thischapterdescribesthevolatilitysmilesthattradersuseinequity

andforeigncurrencymarkets.

8-27

VolatilitySmile

?Basedontheput-callparity?

?????????????

?????????

?

???????????????

?????????

?

z(1)-(2):wecanget:?????????????????

?Conclusions?

zThedollarpricingerrorwhentheBlack-Scholesmodelisusedtopricea

Europeanputoptionshouldbeexactlythesameasthedollarpricing

errorwhenitisusedtopricingaEuropeancalloptionwiththesame

strikepriceandtimetomaturity.

zTheimpliedvolatilityofaEuropeancalloptionisalwaysthesameas

theimpliedvolatilityofaEuropeanputoptionwhenthetwohavethe

samestrikepriceandmaturitydate.

9-27

VolatilitySmileforForeignCurrencyOptions

?Theimpliedvolatilityisrelativelylowforat-the-moneyoptions.Itbecomes

progressivelyhigherasanoptionmoveseitherintothemoneyoroutof

themoney.

Implied

volatility

Volatilityincreasesasoptions

becomesincreasinglyinthe

moneyoroutofthemoney.

OutoftheMoneyCalls

OutoftheMoneyPuts

strikeprice

AttheMoneyOptions

10-27

ReasonsforSmileinForeignCurrencyOptions

?Whyareexchangeratenotlognormallydistributed?Twoofthecontidions

foranassetpricetohavealognormaldistributionare:

zThevolatilityoftheassetisconstant.

zThepriceoftheassetchangessmoothlywithnojumps.

?Inpractice,neitheroftheseconditionsissatisfiedforanexchangerate.

Thevolatilityofanexchangerateisfarfromconstant,andexchangerates

frequentlyexhibitjumps(sometimesthejumpsareinresponsetothe

actionsofcentralbanks).

11-27

VolatilitySmiles(skew)forEquityOptions

?Thevolatilityusedtopricealow-strike-priceoption(i.e.,adeepoutofthe

moneyputoradeepinthemoneycall)issignificantlyhigherthanthatused

topriceahigh-strike-priceoption(i.e.,adeepinthemoneyputoradeep

outofthemoneycall).

Implied

volatility

OutoftheMoneyCalls

OutoftheMoneyPuts

strikeprice

AttheMoneyOptions

12-27

ReasonsfortheSmileinEquityOptions

?Leverage(equityprice→volatility)

zAsacompany’sequitydeclinesinvalue,thecompany’sleverage

increases.Thismeansthattheequitybecomesmoreriskyandits

volatilityincreases.

?VolatilityFeedbackEffect(volatility→equityprice)

zAsvolatilityincreases(decreases)becauseofexternalfactors,investors

requireahigher(lower)returnandasaresultthestockpricedeclines

(increases).

?Crashophobia(expectedequityprice→impliedvolatility)

z1987stockmarketcrash:higherpremiumsforputpriceswhenthestrike

priceslower.

13-27

StrategyRelatedtoVolatilitySkew

?Alongriskreversalcombineslongcallandshortputonthesame

underlyingwithsameexpiration.

?Forexample

zIfatraderbelievesthatputimpliedvolatilityisrelativelytoohigh,

comparedtothatforcalls,alongriskreversalcouldbecreatedby

buyingtheOTMcall(underpriced)andsellingtheOTMput(overpriced)

forthesameexpiration.

zHowever,thiswouldcreatealongexposuretotheunderlying,which

couldbeproblematic.

14-27

VolatilitySmile

?Alternativewaysofcharacterizingthevolatilitysmile

zThevolatilitysmileisoftencalculatedastherelationshipbetweenthe

impliedvolatilityandK/Sratherthanastherelationshipbetweenthe

0

impliedvolatilityandK.

9Arefinementofthisistocalculatethevolatilitysmileasthe

relationshipbetweentheimpliedvolatilityandK/F,whereFis

0

0

theforwardpirceoftheassetforacontractmaturingatthesame

timeastheoptionsthatareconsidered.

zAnotherapproachtodefiningthevolatilitysmileisastherelationship

betweentheimpliedvolatilityandthedeltaoftheoption.

15-27

VolatilitySmile

?Tradersallowtheimpliedvolatilitytodependontimetomaturityaswellas

strikeprice.

?Volatilitysurfacescombinevolatilitysmileswiththetimetomaturityand

K/S.

0

zImpliedvolatilitytendstobeanincreasingfunctionofmaturitywhen

short-datedvolatilitiesarehistoricallylow.

zVolatilitytendstobeadecreasingfunctionofmaturitywhenshort-

datedvolatilitiesarehistoricallyhigh.

16-27

VolatilityTermStructureandVolatilitySurface

Z

X

Y

Impliedvolatilityonthez-axis;maturity(x-axis);andK/S(y-axis).

0

17-27

3.AlteringAsset

Allocation

18-27

UsingDerivativestoAlteringAssetAllocation

?Alteringassetallocationbetweenequityanddebtwithfutures

zStep1:Calculatethereallocatingamount

zStep2:Toreallocateanamountfromequitytobonds:

9Removeallsystematicriskfromtheposition(beta=0)byshorting

equityfutures.

9Adddurationtotheposition(BPV>0)bygoinglongbondfutures.

zStep3:Toreallocateanamountfrombondstoequity:

9Removealldurationfromtheposition(BVP=0)byshortingbond

futures.

9Addsystematicrisktotheposition(beta>0)bygoinglongequity

futures.

Cash

Beta=0

BPV=0

equity

bonds

19-27

UsingDerivativestoAlteringAssetAllocation

Numberofcontracts

Amendingportfoliobeta&

Syntheticstockpositions

§E

¨

?

E·§

?¨

?¨

·

?

VP

target

E

p

¨

?

1

Pmultiplier

f

1?

f

Equity/Equity

Cash

Beta=0

Mid-capequity

Small-capequity

Equity/Debt

Cash

Beta=0

BPV=0

equity

bonds

20-27

4.Variance

Swap

21-27

VarianceSwap

?Varianceswapspayoffsarebasedonvarianceratherthanvolatility

(standarddeviation).

?Theseproductsaretermedswapsastheyhavetwocounterparties,one

makingafixedpaymentandtheothermakingavariablepayment.

zThefixedpaymentistypicallybasedonimpliedvolatility2(implied

variance)overtheperiodandisknownattheinitiationoftheswap,this

isreferredtoasthevariancestrike.

zThevariablepaymentisunknownatswapinitiationandisonlyknownat

swapmaturity.Itistheactualvarianceoftheunderlyingassetoverthe

lifeoftheswapandisreferredtoasrealizedvariance.

22-27

VarianceSwap

?Thefeaturesofvarianceswap

znoexchangeofnotionalprincipalandnointerimsettlementperiods.

zWithavarianceswap,thereisasinglepaymentattheexpirationofthe

swapbasedonthedifferencebetweenactualandimpliedvarianceover

thelifeoftheswap

9????????????????????=?????????????????(?????)

?????

9????????????????????=?????????????(

?????????????=???????????????????

)

??

23-27

VarianceSwap

?TheMark-to-Marketvalueofvarianceswap

zThevalueofavarianceswapiszeroatinitiation,butovertime,theswap

willeithergainorlosevalueasrealizedandimpliedvolatilitydiverge.

zConsideraone-yearswapwherethreemonthshaveelapsedsince

inception,theMtMvalueoftheswapcanbecalculatedasfollow:

24-27

VarianceSwap

?TheMark-to-Marketvalueofvarianceswap

zConsideraone-yearswapwherethreemonthshaveelapsedsince

inception,theMtMvalueoftheswapcanbecalculatedasfollow:

9Step1:Computeexpectedvarianceatmaturity(thetime-weighted

averageofrealizedvarianceandimpliedvarianceoverthe

remainderoftheswap’slife).

?

?

?

????

?????

??????????????????????????????

????

?

?

9Step2:Computeexpectedpayoffatswapmaturity:

???????????????=????????????????

?(??????????????????????????????????????????)

9Step3:Discountexpectedpayoffatmaturitybackto