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8/11/2019 Investments - Lecture Notes Derivatives 2013-08
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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
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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
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OptionlikeSecurities
Callablebonds
ConvertibleSecurities
Warrants
CollateralizedLoans
ExoticOptions
Asianpayoffdependingonanaverageprice
Barrierunderlyingmustcrossthroughabarrier
Options 84
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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
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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
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*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
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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
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WSJ: 2/19/2009
Futures 92
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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
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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
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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
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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
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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
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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$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
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