FuturesOptions

FNCE30007DerivativeSecurities/Lecture10

Schedule

2

IntroductiontoOptions

PropertiesofStockOptions

TheBinomialModel

TheBlackScholesMertonModel

DividendsandOptionsonOther

InstrumentsTheGreeksFuturesMarkets

HedgingwithFuturesandForwards

ForwardandFuturesPrices FuturesOptions Swaps

Outlineandreadings

3

Outline Futuresoptions PutCallparityforfuturesoptions Boundsforfuturesoptions Valuingfuturesoptions Futuresoptionpricesvs.spotoptionprices

Readings Hull,7th/8th ed.,chapter16

FuturesOption

4

Futuresoption

5

Afuturesoptionistheright,butnottheobligation,toenterintoafuturescontractatacertainfuturespricebyacertaindate. Callfuturesoption:righttoenterintoalongfuturescontract. Putfuturesoption:righttoenterintoashortfuturescontract.

MostfuturesoptionsareAmerican.

Mechanicsofcallfuturesoptions

6

Whenacallfuturesoptionisexercisedtheholderacquires:

Alongpositioninthefutures. Acashamountequaltotheexcessofthefuturespriceat

previoussettlementoverthestrikeprice.

Mechanicsofcallfuturesoptions example

7

ConsiderapositioninaJulycallfuturesoptionongoldwithastrikepriceof$300perounce.Theassetunderlyingonecontractis100ounces.Supposethatthemostrecentsettlementpricewas$325andtheoptionisexercisedwhentheJulygoldfuturespriceis$328.

Theinvestorreceives(325 300)(100)=$2,500. Theinvestorreceivesalongfuturescontract. Iftheinvestorclosesoutthefuturescontractimmediately,

thegainwouldbe(328325)(100)=$300. Totalpayoff=2,500+300=$2,800.

Mechanicsofputfuturesoption

8

Whenaputfuturesoptionisexercisedtheholderacquires:

Ashortpositioninthefutures. Acashamountequaltotheexcessofthestrikepriceoverthe

futurespriceatprevioussettlement.

Mechanicsofputfuturesoption

9

ConsiderapositioninaSeptemberputfuturesoptiononcornwithastrikepriceof$2.00perbushel.Eachcontractison5,000bushels.Supposethatthemostrecentsettlementpricewas$1.89andtheoptionisexercisedwhentheSeptembercornfuturespriceis$1.90.

Theinvestorreceives(2.00 1.89)(5000)=$550 Theinvestorreceivesashortfuturescontract Iftheinvestorclosesoutthefuturescontractimmediately,

thelosswouldbe(1.90 1.89)(5000)=$50 Totalpayoff=550 50=$500

Thepayoffs

10

Ifthefuturespositionisclosedoutimmediately: Payofffromcall=F K (328 300)(100)=$2,800. Payofffromput=K F (2.00 1.90)(5000)=$500.whereFisfuturespriceatthetimeofexercise.

Potentialadvantagesoverspotoptions

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Futurescontractmaybeeasiertotradethanunderlyingasset(liquidityargument).

Exerciseoftheoptiondoesnotleadtodeliveryoftheunderlyingasset(mostfuturescontractsareclosedoutpriortodelivery).

Futuresoptionsmayentaillowertransactionscosts.

PutCallParityforFuturesOptions

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Putcallparityforfuturesoptions

13

Considerthefollowingtwoportfolios: EuropeancallfuturesoptionplusKerT ofcash EuropeanputfuturesoptionpluslongfuturespluscashequaltoF0erT.

TheymustbeworththesameattimeTsothat

c+KerT =p+F0 erT

Putcallparity example

14

SupposethatthepriceofaEuropeancalloptionongoldfuturesfordeliveryinninemonthsis$10.50perouncewhentheexercisepriceis$900.Assumethatthegoldfuturespricefordeliveryinninemonthsiscurrently$875,andtheriskfreerateis%10perannum.FindthepriceofaEuropeanputoption.

PutcallparityforAmericanfuturesoptions

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PutcallparityforAmericanfuturesoptionsisF0 erT K

BoundsforFuturesOptions

16

Boundsforfuturesoptions

17

Europeanoptionsc(F0 K)erT

p(K F0)erT

AmericanoptionsCF0 K

PK F0

ValuingFuturesOptions

18

Valuingfuturesoptions

19

Aonemonthcalloptiononfutureshasastrikepriceof29.

Futuresprice=$30Optionprice=?

Futuresprice=$33Optionprice=$4

Futuresprice=$28Optionprice=$0

Settinguparisklessportfolio

20

ConsiderthePortfolio:long futuresandshort1calloption

Portfolioisrisklesswhen3 4=2 or =0.8

3 4

2

Valuingtheportfolio

21

Therisklessportfoliois:long0.8futures+short1calloption.

Thevalueoftheportfolioin1monthis$1.6. Assumethattheriskfreerateis6%perannum.Then,thevalueoftheportfoliotodayis$1.6e 0.06/12=$1.592.

Valuingtheoption

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Theportfoliothatislong0.8futures+short1optionisworth$1.592.

Thevalueofthefuturesiszero. Thevalueoftheoptionmustthereforebe$1.592.

Generalization

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Generalizationofbinomialtreeexample

24

AderivativelastsfortimeT andisdependentonafutures

F0

F0uu

F0dd

Generalization

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Considertheportfoliothatislong futuresandshort1derivative

Theportfolioisrisklesswhen

0 0u d f

F u F d

F0u F0 u

F0d F0 d

Generalization

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ValueoftheportfolioattimeT is=F0u F0 u

Valueofportfoliotodayis

Hence= [F0u F0 u]erT

Substitutingfor weobtain=[pu +(1 p)d ]erT

where 1 dpu d

Valuingafuturesoption example

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Supposeu=1.1,d=0.9333,r=0.06,T=1month,fu =4andfd =0.Findthevalueofthefuturesoption.

Growthratesforfuturesprices

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Afuturescontractrequiresnoinitialinvestment. Inariskneutralworldtheexpectedreturnshouldbezero.

Theexpectedgrowthrateofthefuturespriceisthereforezero.

Thefuturespricecanthereforebetreatedlikeastockpayingadividendyieldofr.

ValuingEuropeanfuturesoptions

29

WecanusetheBSMformulaforanoptiononastockpayingacontinuousyield.SetS=currentfuturesprice(F0)Setq=domesticriskfreerate(r)

Settingq=rensuresthattheexpectedgrowthoffuturespriceinariskneutralworldiszero.

Blacksmodel

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TheformulasforEuropeanoptionsonfuturesareknownasBlacksmodel.

0 1 2

2 0 1

01

02 1

( ) ( )

( ) ( )

where2ln( / ) /2

2ln( / ) /2

rT

rT

c e F Nd K Nd

p e K N d F N d

F K Td

T

F K Td d T

T

Blacksmodel example1

31

ConsideraEuropeanputfuturesoptiononcrudeoil.Thetimetotheoptionsmaturityisfourmonths,thecurrentfuturespriceis$20,exercisepriceis$20,theriskfreerateis9%perannum,andthevolatilityofthefuturespriceis25%perannum.Findthefuturesoptionsprice.

Blacksmodel example1

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F0 =20,K=20,r=0.09T=4/12 =0.25andln(20/20)=0so,

2

1

2

1

2( 0.09 )(4 /12)

/ 2 0.07216 0.072

0.07216 0.072

( ) ( 0.07) 0.4721( ) (0.07) 0.5279

[(20)(0.5279) (20)(0.4721)] $1.08

T TdTTd

N d NN d Np e

Blacksmodel example2

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Considera6monthEuropeancalloptiononspotgold. 6monthfuturespriceis$620,6monthriskfreerateis5%,strikepriceis$600,andvolatilityofthefuturespriceis20%.

ValueofoptiongivenbyBlacksformulawithF0=$620,K=600,r =0.05,T =0.5,and=0.20is$43.83.

FuturesOptionPricesvs.SpotOptionPrices

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Futuresoptionpricesvs.spotoptionprices

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Iffuturespricesarehigherthanspotprices(normalmarket)then AnAmericancallonfuturesisworthmorethanasimilarAmericancallonspot.

AnAmericanputonfuturesisworthlessthanasimilarAmericanputonspot.

Whenfuturespricesarelowerthanspotprices(invertedmarket)thereverseistrue.