investments - lecture notes derivatives 2013-08

8/11/2019 Investments - Lecture Notes Derivatives 2013-08

1/23

Ch19 Options Jones

Ch15OptionsMarkets BKM

DerivativeMarketsderivativesmerelyderivetheirvaluefromothersecurities

TheOptionContract

ACALLoptiongivestheholdertheright(butnottheobligation )topurchase...

anASSETforaspecifiedPRICEonorbeforeaspecifiedDATE

ThisassetisreferredtoastheUNDERLYINGasset.

TheoptionholdermaycallawaytheassetfortheEXERCISEPRICEoftheoption.

Thisisthe"X"inouroptionsnotation.

Forthis

right,

the

buyer

of

the

option

pays

aPREMIUM

Sincetheoptionisacontract,theoriginalselleroftheoptioniscalledtheWRITER

ofthecontract.

ThebuyerofthecontractiscalledaHOLDER,andbecausetheoptionisusually

resellable,theholdermaychange.

Whiletheholderoftheoptionhasarightbutnoobligationtoperformonthe

contract,thewriterISobligated (untilhe/shebuysbackthatrightfromaholder).

Werefer

to

the

holder

as

LONG

acall

option.

The

writer

is

SHORT

acall

option.

Theunderlyingassethasasellingprice "S" thatchangesintime

Whentheoptioncontractiswrittenorbought,thepriceisS0

Later(atexpirationorwhenwewanttovalueit)theoptionhasapriceofST

Thevalueoftheoptionisthesellingpriceoftheunderlyingminustheexerciseprice.

ValueCALL=ST

X

Thisisnottheprofit oftheoption.

Thenetprofitonthecallisthevalueoftheoptionminusthepriceoriginallypaid

Profit=(ST X) PricePaid(usuallytheorignalpremium)

Options 78


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HOUSINGExample:

UNDERLYING: Yourneighbor'shouse PREMIUM: 10 K

EXERCISEPRICE: 100 K CURRENTPRICE: 100 K

DATE: 1yearfromtoday OPTIONTYPE: CALL

Wecouldbrieflydescribethebuyer(holder)ofthisoptionas

Longone$100,000,12monthcalloptiononhis/herneighbor'shouse

Doestheholderhavetoperformanythingonthiscontract?

Howaboutthewriter?

Usingnotation,youmightdescribetheoption:

Asset X Time Type Premium S0

House 100$ 12mos Call 10$ 100$

Payoffandpotentialprofit?

Underlying X Time Type Premium ST

House 100$ 12mos Call 10$ 130$

Whatisthepercentageprofittotheholder?

Options 79


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CALLSvs.PUTS

Consideracalloptiona"directionalbet"thattheunderlyingassetwillgoupinvalue

APUTOPTIONisadirectionalbetthattheunderlyingwillDECLINEinvalue

If

the

underlying

declines

in

value,

the

holder

can

"PUT"

the

asset

to

the

writerorthevalueoftheexerciseprice.

Allothertermsapplyasabove,butthedirectionoftheprofitopportunityis

reversedfortheholderofthePUT.

Theholderofaputoptionwantstheassettofallinprice. Thebuyermayalso

wantprotectionfromtheunderlyingdeclininginprice. Thiswouldbeaform

ofINSURANCE,aprotectiveput(mentionedinstrategies,below).

Valuesof

Options

at

Expiration

(or

wheneverwe

chose

to

value

them)

Becausetheprofitdependsonwhetheryouarelongorshorttheoption,it'simportant

toconsiderbothsidesofthedeal

First,wecanrefertothevalueofanoptionasitsPAYOFF

ButthispayoffdependsonwhetheryouaretheHOLDERortheWRITER.

Payofftocallholder= S T X ifS T >X; 0 ifS T X; 0 ifS

T


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VALUEOFAPUT:

Payofftoputholder= $0 ifS T >=X; XS T ifS T =X; (XS T)ifS T X ifS T


5/23


6/23

Optionscontractsarestandardized

Thismeanswhenyoubuyone,youcanselliteasilythroughabrokerandanexchange

Exercisedatesareusuallyonlyonetoseveralmonthsawayfromtoday

LongtermLEAPS(LongtermEquityAnticipationSecurities upto3years)

Americanvs.European

EuropeanmaybeexercisedonlyonexpirationdateAmerican,anytimebefore

Optionsonotherthanstock

Indexoptions,Futuresoptions,Currencies,InterestRateoptions

OptionStrategies notethatJonesdoesn'tcovermanyofthese

ProtectivePut

CoveredCalls

Straddles

Spreads BKMreferstoa"Moneyspread"thatIrefertoas,morespecifically,a

"VerticalCallSpread"or"BullCallSpread"

Howdoyoucreateaspreadofthiskind?

Buyacall

at

astrike

price

at

or

near

the

current

price

of

the

stock

inquestion

IfMSFTistradingat$30.00today,youwouldbuya

callonMSFTstockwitha$30strikeprice.

ThenyouSELL(write)acallonMSFTwitha$35strike

YouthinkMSFTmayrisesoon,butdon'texpectittogomuchbeyond$35

X PREMIUM

35 Sellinganequivalentnumbercallsat $35.00 1.00$ CFCALL

reduces

the

cash

outlay

to

buy

into

the

profitopportunitythatMSFTmayriseabove

$30duringtheoptionperiod

S0 30 CurrentpriceonMSFT $30.00 (2.00)$ CFCALL

Costto"buy"thespreadforMSFT (1.00)$ CFSPREAD

Collars

Options 83


7/23

OptionlikeSecurities

Callablebonds

ConvertibleSecurities

Warrants

CollateralizedLoans

ExoticOptions

Asianpayoffdependingonanaverageprice

Barrierunderlyingmustcrossthroughabarrier

Options 84


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9/23

Ch19 Options Jones

Ch16OptionValuation

OptionvaluationIntrinsicandTimeValues

Timevalueisnotrelatedtotimevalueofmoney

Thinkofitasmorevolatilityvalue

Asintrinsicvaluerises,timevaluedropsbecausethelikelihood

oftheoptionbeingexercisedbecomesacertainty

X Time Type CallPrem0 S0 rf

50.00$ 0.25 Call 5.26$ 50.00$ 50.00% 5.00%

IntrinsicValue $ Moneyness: ATthemoney

TimeValue 5.26$

Determinantsofcalloptionvalues

Ifthisincreases theCallOption

50.00$ StockPrice,S

50.00$ ExercisePrice,X

50.00% Volatility,

0.2500 TimetoExpiration,T

5.00% InterestRate,rf

Dividendpayouts

Increases

Decreases

Increases

Increases

Increases

Decreases

OptionsValuation 86


10/23

WhydoessomethinglikehigherVolatilityincreasethevalueofanoption?

Let'sassumeweareinterestedinvaluingoptionsontwocompaniescurrentlypriced

at$30pershare. X=$30aswell. Onestock'spriceishighlyvolatile. Theotherisnot.

X= $30.00

High

volatility

stock/scenarioExpfuturestockprice $10 $20 $30 $40 $50

Optionpayoff 0.00 0.00 0.00 10.00 20.00

Lowvolatilityscenario

Expfuturestockprice $20 $25 $30 $35 $40

Optionpayoff 0.00 0.00 0.00 5.00 10.00

Forsimplicity,let'sassumeequalprobabilities(p)foreachoutcome

Stockprice 20% 20% 20% 20% 20%

Thenwewouldweighteachoptionpayoffby"p"togetanexpectedpayoff:

HighvolatilityScenario E(payoff)

Weightedpayoff $ $ $ 2.00$ 4.00$ 6.00$

Lowvolatilityscenario E(payoff)

Weightedpayoff $ $ $ 1.00$ 2.00$ 3.00$

Withahigherexpectedpayoff,ithashighervalue.

BinomialOptionPricing

wewillskipthismodel

OptionsValuation 87


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BlackScholesMertonOptionValuation

DevelopedbyBlack,ScholesandMerton earningthemaNobelPrize.

We'veseenwhatdirectlyimpactsthevalueofanoption.

C0 = S0 eT

N(d1) XerT N(d2)

d1 = [ln(S 0 /X)+(r + 2

/2)T]/( T)

d2 = d1 T

C0=currentcalloptionvalue

S0=currentstockprice

N(d)=theprobability(roughly)thattheoptionwillexpireinthemoney

X=exerciseprice

e=naturallogfunction(seebelow,ifyou'renotfamiliarwithit)*

=annualdividendyieldoftheunderlying

r=riskfreerateasadecimal,annualized

T=timeremainingtoexpirationinyears

ln=naturallogfunction

=annualstandarddeviationoftheunderlying,expressedasadecimal

X Time Type Prem0 S0 rf

50.00$ 0.2972 Call 5.47$ 48.89$ 55.56% 0.96%

d1 d2 N(d1) N(d2) CallPrice

0.086746 0.216145 0.5345632 0.4144372 5.47$

IntrinsicValue $ Moneyness: OUTofthemoney

TimeValue 5.47$

NOTE:forpurposesofthisclass,Connectdoesnotprovidegoodpracticeproblemsusing

BlackScholes. Iwillprovidesomefortheexam. Youwillneedtocalculatethis,

however, muchofthecomplexmath(e.g.theprobabilities)can'tbedoneonmany

calculators. Youwillbeprovidedwiththosevalues.

OptionsValuation 88


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PutCallParity

S TX

Payoffofcallheld 0 S TX

MinusPayoffofputwritten (XS T) 0

Total S TX S T X

i. CP=S0XerT

ii. P=CS0+PV(X)+PV(dividends)

iii. PutoptionvaluationviaBlackScholes(skip)

BecauseofPutCallParity,whenwehavethepriceofthecall,wecanvaluethe

putcontract,andviceversa.

Aputoptiononastockwithacurrentpriceof$31hasanexercisepriceof$38.Thepriceofthecorrespondingcalloptionis$2.05.Accordingtoputcallparity,iftheeffective

annualriskfreerateofinterestis8%andtherearethreemonthsuntilexpiration.

Whatshouldbethevalueoftheput?

OptionsValuation 89


13/23

*Whatise ?

e=

"Euler'snumber"(notEuler'sConstant),e ,isamathematicalconstant,likepi,thatisusedin

certainmathematicalcalculations,especiallyinexponentialcalculations. Infact,the

"exponentialfunction"isusuallyjustex. Likepi,e isirrational,havinganonterminating

decimalvalue.

Infinance,itisusedfrequentlyforcompoundinterestcalculations.

Accessthisfunctionorconstantasfollows:

Example: Ifyouneedtocalculatea10%annualpercentage

ratewithcontinuouscompounding

TIBAII+ ".1",2ND,then"ex"then" 1"

HP12c ".1"ENTER,"g"then"ex"then"1"then""

MSExcel =exp(.1) 1

Allthree

of

these

should

produce

avalue

of 0.10517092

TheAPR,annualpercentagerate,maydiffermateriallyfromtheEAR,oreffectiveannual

rate,basedontheperiodofcompounding. Using"e"allowsustocalculatetheeffectiverate

easilyandinsuresweareusingthesameinterestordiscountratesinourcalculations.

Notehowasimple10%APRchangesbasedontheperiodofcompounding,butthe

EARrapidlyconvergesonthecontinuouslycompoundingvalueusinge.

CompoundingtosmallerandsmallerperiodsuntilCONTINUOUScompounding

10%

Comp.

Period NperYear

Rateper

period(N)

Effective

AnnualRate

Year 1 10.0000% 10.000000%

Qtr 4 2.5000% 10.381289%

Month 12 0.8333% 10.471307%

Week 52 0.1923% 10.506479%

Day 365 0.0274% 10.515578%

Hour 8760 0.001142% 10.517029%

Minute 525600 0.0000190% 10.517091%

Second 31536000 0.00000032% 10.517092%

Wecanuse"e"tocalculatetheinterestratewithcontinuousor

instantaneous compounding.*

EAR=er 1 =e^.10 1= 2.71828^.10 1

0.10517092

10.517092%

2.7182818285

OptionsValuation 90


14/23

Ch20 FuturesContracts

Ch17 TheFuturesMarkets&RiskManagement(i.e.Hedging)

Thepricesthatproducersreceivefortheircommoditiesfluctuate.

CORN: Afarmermayproducecornon500acresinIowa

HemightselltoalocalgraneryoncehiscropisharvestedinOctober

(IowacornisnormallyharvestedlateSepttoearlyNovember)

Ifthefarmerwaitsuntilharvesttosellhiscrop,pricevolatilitycouldmeanthe

wholeseasonshiftsfromareasonableprofittoasteeploss.

Hemightwanttosecureabuyerforhiscropashe'splantingit.

AFORWARDcontractcouldbecreatedbetweenthetwoparties,but

thisinvolvesrisksforallparties:

Thegraneryownermaynotwanttocommitattoohighaprice

Apart

from

a

small

market

being

"less

liquid"

(fewer

participants),thecontracthasalimitedresalemarket(ifitistransferableatall).

Thismayfurtherlimithowmuchthegraneryowneriswillingtopay.

Thefarmerisatriskthatthegraneryownermightreneg

Astandardizedfuturescontractaddressestheseconcerns.

Futures 91


15/23

WSJ: 2/19/2009

Futures 92


16/23

FUTURES Terminology,PricingandMarktoMarket

First,afuturescontractisgenerallyanagreementtobuyorsellagoodatafuturedateand

price.

ThepersonbuyingthecontractisLONGthecontractandagreestopurchase

the

asset.

This

is

also

referred

to

as

"taking

delivery."

ThepersonsellingthecontractisSHORTandagreesto"makedelivery."

Eachpartyhasafirmcommitmenttomakegoodontheirsideofthe

contract.

HOWEVER,mostcontracts(asmany98+%onsomefutures)

NEVERexistlongenoughtobe"delivered."

Bothpartiesusuallycloseouttheirlong/shortpositionsby

"offsetting"them.

IftheyboughtafuturescontractandareLONG,theysellacontract,

andthisoffsetsthepreviousobligation. Theresultingcontractdisappears

fromthe"OpenInterest"inthecontract. Itnolongerexists.

Atmaturity,orwhensoldpriortomaturity,theprofit(loss)onthecontractis

ProfittoLong=Spotpriceatmaturity Originalfuturesprice=ST F0

ProfittoShort=Originalfuturesprice Spotpriceatmaturity=F0 ST

Second,thepricingoffutureswillinclude

Theriskfreerate theminimumrateweexpectanassettochangeinvalueovertime.

NOTE:forphysicalcommodities,thiscanalsoincludeafeefor"storage."

F0/(1+rf)T =S0 F0 = S0(1+rf)

T

Dividends becausethepaymentofdividendswillreducethepriceofthestockor

securityifitisonethepaysoutdividends.

F0 = S0(1+rf d)T

Futures 93


17/23

Third,thesizeofthecontract,whichrelatestothetotalmoneyatrisk,willvarybycontract

type.

Commoditieswillbeinhundredsofounces,orthousandsofbushels,andyouneedto

convertthesevaluesfromthe"spot"priceofacommodity(perouceorbushel)

intothetotalamountatriskinthefuturesinvestment.

Financial

futures

will

be

in

a

stated

contract

size,

often

referred

to

as

a

"multiplier"

onanpublishedindex. Thisisn'talwaysthecase,butitquitecommon.

Fourth,asapartofcommittingtoafuturescontract,bothpartiesarerequiredtoputup

"margin"withtheirbrokers.

Thismeansthatanindividualisabletouseextremeleverageintradingfutures

e.g.$5000could"control"$100,000worthofanunderlyingasset.

Thishasmanyadvantages. Italsohasmanyrisks.

Example: "MarktoMarket"onSilvercontract

Assumeafuturescontractis$12.10fordeliveryin5days

F0= $12.10

Thepricesofthefuturescontractareasfollowsoverthenext5days

Day FuturesPrice

0 12.10 purchase

1 12.20

2 12.25

3 12.18

4 12.18

5 12.21 delivery,the"spot"pricemustequalthispricedueto

"convergence" iftheywereunequal,therewouldbearbitrage

Daily(MarkedtoMarket)

Day FuturesPrice Changeinprice proceeds Margin

0 12.10 $21,600

1 12.20 0.83% 0.10 500$ $22,100 2.31%

2 12.25 0.41% 0.05 250$ $22,350 1.13%

3 12.18 0.57% (0.07) (350)$ $22,000 1.57%

4 12.18 0.00%

$

$22,000 0.00%

5 12.21 0.25% 0.03 150$ $22,150 0.68%

0.91% sum 550$ 2.55%

profit

InitialMargin $21,600 550$ 2.5%

Apr2011Initialmargin $11,745 550$ 4.7%

5000ozcontract

http://online.wsj.com/article/SB10001424053111903791504576589221715678498.html

Futures 94


18/23

Example: An Iowa Corn Farmer

Field Name Expected Yield 165 bu./acreAcres 500

Total CostPreharvest machinery Fixed Variable Total All Acres Total per acre $21.40 $21.20 $42.60 $21,300 Total all acres $10,700 $10,600 $21,300 ----

Seed, chemicals, etc. Total ---- $341.92 $341.92 $170,958

Harvest machinery Total per acre $40.63 $69.50 $110.12 $55,060 Total all acres $20,313 $34,748 $55,060 ----

Labor Total $33.06 $0.00 $33.06 $16,530

Land Cash rent equivalent $215.00 ---- $215.00 $107,500

Total fixed, variable and all costs Per acre $310.09 $432.61 $742.70 $371,348

Per bushel $1.88 $2.62 $4.50 ----All acres $155,043 $216,306 $371,348 ----

Return per Acre Over ReturnVariable Costs All Costs All Acres

Gross returns

Expected selling price $5.00 ---- $825.00 $412,500

Government paymentsDirect payment ---- $0.00 $0 Counter Cyclical pymt. ---- $0.00 $0 Expected LDP rate $0.00 ---- $0.00 $0 Total returns ---- $825.00 $412,500Net returns $392.39 $82.30 $41,152

Farmer John, SomewhereIn, Iowa

Cost per Acre

Futures

Corn 95


19/23

Season-average Price Calculator - 2011

Marketing year month

Corn

Soy

beans Corn

Soy

beansSept. 2011 6.37 12.20 Sept. 2011 Sept. 2011Oct. 2011 5.92 11.90 Dec. 2011 6.47 Nov. 2011 12.08

Nov. 2011 Mar. 2012 6.59 Jan. 2012 12.17

Dec. 2011 May 2012 6.65 Mar. 2012 12.27

Jan. 2012 Jul. 2012 6.69 May 2012 12.36

Feb. 2012 Sept. 2012 6.26 Jul. 2012 12.46

Mar. 2012 Aug. 2012 12.44

Apr. 2012 Sept. 2012 12.34

May 2012

Jun. 2012 Soybean

Jul. 2012

Aug. 2012 Estimated season-average pricesCorn 6.22Soybeans 11.80

Monthly farm prices are available from USDA here.

Current futures price by trading contractActual U.S. monthly

farm price

CME Group CornFutures

Futures

Corn 96


20/23

Applications Airlineexample:

SupposeyouaretheCFOofamajorairline. Afteryourlaborcosts,yournextlargestcostof

doingbusinessisfuel. Petroleumisclearlyvolatileinprice. Thiscreatesasignificantriskto

yourfirm'sbottomlineandyourabilitytoproduceconsistentreturnsforyourshareholders.

You

can't

remove

all

the

exposure

your

company

has

to

changes

in

oil

and

gas

prices,

but

youcanmoderateit.

Ifanairlinesellsticketsinadvance(whichtheytendtodo),doesitknowhow

muchgasitwillneedNOWfortheflightsalreadypurchased?

Doesitknowhowmanyflightsitexpectstofly6monthsfromnow? Howabout

12monthsfromnow?

Sinceit'sagoingconcern,itprobablyknowsprettywellhowmuchgasitwill

needforyearsinthefuture.

You,theCFO,believethecurrentpriceof$50oilwillriseintheforeseeablefuture.

Thiswilldirectlyaffectgasandjetfuelprices. Youuseapproximately1billiongallonsof

jetfuelayear(or23.8millionbarrelsat42gallonsperpetroleumbarrel).

WhatcouldyoudoasCFOofthisairline?

Futures

Airline 97


21/23


22/23


23/23

ThisledtothecreationofSYNTHETICCOLLATERLIZEDDEBTOBLIGATIONS.

(MakesureyouunderstandwhatthedifferenceisbetweenaCDS andasyntheticCDO .)

Thesameexampleastheoneabove,butnowtheinvestorisaSPECULATOR.

She

is

not

hedging

the

risk

of

Lehman

bonds.

She

doesn't

own

any

of

them.She'sjustbettingthatLehmanbondswilldefault.

Becausethisisnotahedgedtransaction,therecouldbemanymore

CDOcontractsouttherethanthereareactualbonds.

Thiscreated massivesystematicrisk,andmoreimportantly,becausethese

derivativeproducts

were

not

regulated,

the

amount

of

risk

out

there

was

not

onlynotunderstood,thetotalsizeofallthecontractswasn'tevenknown.

Averygoodbookthatdelvesinto

themortgageCDS/CDOmarketis

"TheBigShort"

MichaelLewisfocusesonthesmall

numberofpeoplewhosawthe

mortgagemeltdowncomingand

wereabletoprofitfromit.

Speculatorin

syntheticCDOs

AIG: Aninsurance

company

Buyerofthesynthetic

CDOisdoingsoasa

wagerthatLehman

hasa"defaultevent"

onitsdebt

Sellerofprotection.

IfLehmandefaults,

AIGpaysasmuchas

theparvalueofthe

bonds"insured."

$

Paymentsmadeasbasis

pointsontheloan

The"defaultevent"triggers$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

investments - lecture notes derivatives 2013-08

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