HullFund8eCh10ProblemSolutions.docx

HullFund8eCh10ProblemSolutions.docx

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CHAPTER10

PropertiesofStockOptions

PracticeQuestions

Problem10.8.

Explainwhytheargumentsleadingtoput–callparityforEuropeanoptionscannotbeusedtogiveasimilarresultforAmericanoptions.

Whenearlyexerciseisnotpossible,wecanarguethattwoportfoliosthatareworththesameattimeTmustbeworththesameatearliertimes.Whenearlyexerciseispossible,the

argumentfallsdown.SupposethatP+S>C+Ke-rT.Thissituationdoesnotleadtoan

arbitrageopportunity.Ifwebuythecall,shorttheput,andshortthestock,wecannotbesure

oftheresultbecausewedonotknowwhentheputwillbeexercised.

Problem10.9.

Whatisalowerboundforthepriceofasix-monthcalloptiononanon-dividend-paying

stockwhenthestockpriceis$80,thestrikepriceis$75,andtherisk-freeinterestrateis10%perannum?

Thelowerboundis

Problem10.10



80-75e-0.1´0.5=$8.66

Whatisalowerboundforthepriceofatwo-monthEuropeanputoptionona

non-dividend-payingstockwhenthestockpriceis$58,thestrikepriceis$65,andtherisk-freeinterestrateis5%perannum?

Thelowerboundis

Problem10.11.



65e-0.05´2/12-58=$6.46

Afour-monthEuropeancalloptiononadividend-payingstockiscurrentlysellingfor$5.The

stockpriceis$64,thestrikepriceis$60,andadividendof$0.80isexpectedinonemonth.Therisk-freeinterestrateis12%perannumforallmaturities.Whatopportunitiesarethereforanarbitrageur?

Thepresentvalueofthestrikepriceis

60e-012´4/12=$57.65.Thepresentvalueofthe

dividendis

0.80e-0.12´1/12=0.79.Because

5<64-57.65-0.79

theconditioninequation（10.8）isviolated.Anarbitrageurshouldbuytheoptionandshort

thestock.Thisgenerates64–5=$59.Thearbitrageurinvests$0.79ofthisat12%foronemonthtopaythedividendof$0.80inonemonth.Theremaining$58.21isinvestedforfourmonthsat12%.Regardlessofwhathappensaprofitwillmaterialize.

Ifthestockpricedeclinesbelow$60infourmonths,thearbitrageurlosesthe$5spentontheoptionbutgainsontheshortposition.Thearbitrageurshortswhenthestockpriceis$64,hastopaydividendswithapresentvalueof$0.79,andclosesouttheshortpositionwhenthe

stockpriceis$60orless.Because$57.65isthepresentvalueof$60,theshortpositiongeneratesatleast64–57.65–0.79=$5.56inpresentvalueterms.Thepresentvalueofthearbitrageur’sgainisthereforeatleast5.56–5.00=$0.56.

Ifthestockpriceisabove$60attheexpirationoftheoption,theoptionisexercised.Thearbitrageurbuysthestockfor$60infourmonthsandclosesouttheshortposition.Thepresentvalueofthe$60paidforthestockis$57.65andasbeforethedividendhasapresentvalueof$0.79.Thegainfromtheshortpositionandtheexerciseoftheoptionisthereforeexactly 64–57.65−0.79=$5.56.Thearbitrageur’sgaininpresentvaluetermsis5.56–5.00=$0.56.

Problem10.12.

Aone-monthEuropeanputoptiononanon-dividend-payingstockiscurrentlysellingfor

$2.50.Thestockpriceis$47,thestrikepriceis$50,andtherisk-freeinterestrateis6%perannum.Whatopportunitiesarethereforanarbitrageur?

Inthiscasethepresentvalueofthestrikepriceis

50e-0.06´1/12=49.75.Because

2.5<49.75-47.00

theconditioninequation（10.5）isviolated.Anarbitrageurshouldborrow$49.50at6%foronemonth,buythestock,andbuytheputoption.Thisgeneratesaprofitinallcircumstances.Ifthestockpriceisabove$50inonemonth,theoptionexpiresworthless,butthestockcanbesoldforatleast$50.Asumof$50receivedinonemonthhasapresentvalueof$49.75today.Thestrategythereforegeneratesprofitwithapresentvalueofatleast$0.25.

Ifthestockpriceisbelow$50inonemonththeputoptionisexercisedandthestockownedissoldforexactly$50（or$49.75inpresentvalueterms）.Thetradingstrategythereforegeneratesaprofitofexactly$0.25inpresentvalueterms.

Problem10.13.

GiveanintuitiveexplanationofwhytheearlyexerciseofanAmericanputbecomesmoreattractiveastherisk-freerateincreasesandvolatilitydecreases.

TheearlyexerciseofanAmericanputisattractivewhentheinterestearnedonthestrikepriceisgreaterthantheinsuranceelementlost.Wheninterestratesincrease,thevalueoftheinterestearnedonthestrikepriceincreasesmakingearlyexercisemoreattractive.Whenvolatilitydecreases,theinsuranceelementislessvaluable.Again,thismakesearlyexercisemoreattractive.

Problem10.14.

ThepriceofaEuropeancallthatexpiresinsixmonthsandhasastrikepriceof$30is$2.Theunderlyingstockpriceis$29,andadividendof$0.50isexpectedintwomonthsandagaininfivemonths.Thetermstructureisflat,withallrisk-freeinterestratesbeing10%.WhatisthepriceofaEuropeanputoptionthatexpiresinsixmonthsandhasastrikepriceof

$30?

Usingthenotationinthechapter,put-callparity,equation（10.10）,gives

c+Ke-rT+D=p+S0

or

Inthiscase

p=c+Ke-rT+D-S0

p=2+30e-0.1´6/12+（0.5e-0.1´2/12+0.5e-0.1´5/12）-29=2.51

Inotherwordstheputpriceis$2.51.

Problem10.15.

ExplaincarefullythearbitrageopportunitiesinProblem10.14iftheEuropeanputpriceis

$3.

Iftheputpriceis$3.00,itistoohighrelativetothecallprice.Anarbitrageurshouldbuythecall,shorttheputandshortthestock.Thisgenerates−2+3+29=$30incashwhichisinvestedat10%.Regardlessofwhathappensaprofitwithapresentvalueof3.00–2.51=

$0.49islockedin.

Ifthestockpriceisabove$30insixmonths,thecalloptionisexercised,andtheputoptionexpiresworthless.Thecalloptionenablesthestocktobeboughtfor$30,or

30e-010´6/12=$28.54

inpresentvalueterms.Thedividendsontheshortpositioncost

0.5e-01´2/12+0.5e-01´5/12=$0.97

inpresentvaluetermssothatthereisaprofitwithapresent

valueof 30–28.54−0.97=$0.49.

Ifthestockpriceisbelow$30insixmonths,theputoptionisexercisedandthecalloptionexpiresworthless.Theshortputoptionleadstothestockbeingboughtfor$30,or

30e-0.10´6/12=$28.54

inpresentvalueterms.Thedividendsontheshortpositioncost

0.5e-01´2/12+0.5e-01´5/12=$0.97

inpresentvaluetermssothatthereisaprofitwitha

presentvalueof30–28.54−0.97=$0.49.

Problem10.16.

ThepriceofanAmericancallonanon-dividend-payingstockis$4.Thestockpriceis$31,thestrikepriceis$30,andtheexpirationdateisinthreemonths.Therisk-freeinterestrateis8%.DeriveupperandlowerboundsforthepriceofanAmericanputonthesamestockwiththesamestrikepriceandexpirationdate.

Fromequation（10.7）

Inthiscaseor

or



S0-K£C-P£S0-Ke-rT

31-30£4-P£31-30e-0.08´0.25

1.00£4.00-P£1.59

2.41£P£3.00

UpperandlowerboundsforthepriceofanAmericanputaretherefore$2.41and$3.00.

Problem10.17.

ExplaincarefullythearbitrageopportunitiesinProblem10.16iftheAmericanputpriceisgreaterthanthecalculatedupperbound.

IftheAmericanputpriceisgreaterthan$3.00anarbitrageurcanselltheAmericanput,shortthestock,andbuytheAmericancall.Thisrealizesatleast3+31–4=$30whichcanbeinvestedattherisk-freeinterestrate.Atsomestageduringthe3-monthperiodeithertheAmericanputortheAmericancallwillbeexercised.Thearbitrageurthenpays$30,receivesthestockandclosesouttheshortposition.Thecashflowstothearbitrageurare+$30attimezeroand−$30atsomefuturetime.Thesecashflowshaveapositivepresentvalue.

Problem10.18.

Provetheresultinequation（10.7）.（Hint:

Forthefirstpartoftherelationshipconsider（a）aportfolioconsistingofaEuropeancallplusanamountofcashequaltoKand（b）aportfolioconsistingofanAmericanputoptionplusoneshare.）

AsinthetextweusecandptodenotetheEuropeancallandputoptionprice,andCandPtodenotetheAmericancallandputoptionprices.BecauseP³p,itfollowsfromput–callparitythat

andsincec=C,or

P³c+Ke-rT-S0P³C+Ke-rT-S0C-P£S0-Ke-rT

ForafurtherrelationshipbetweenCandP,consider

PortfolioI:

OneEuropeancalloptionplusanamountofcashequaltoK.PortfolioJ:

OneAmericanputoptionplusoneshare.

Bothoptionshavethesameexercisepriceandexpirationdate.AssumethatthecashinportfolioIisinvestedattherisk-freeinterestrate.IftheputoptionisnotexercisedearlyportfolioJisworth

max（ST,K）

attimeT.PortfolioIisworth

max（S-K,0）+KerT=max（S,K）-K+KerT

T T

atthistime.PortfolioIisthereforeworthmorethanportfolioJ.SupposenextthattheputoptioninportfolioJisexercisedearly,say,attimet.ThismeansthatportfolioJisworthK

attimet.However,evenifthecalloptionwereworthless,portfolioIwouldbeworthattimet.ItfollowsthatportfolioIisworthatleastasmuchasportfolioJinallcircumstances.Hence

Kert

Sincec=C,or

c+K³P+S0

C+K³P+S0C-P³S0-K

CombiningthiswiththeotherinequalityderivedaboveforC-P,weobtain

0 0

S-K£C-P£S-Ke-rT

Problem10.19.

Provetheresultinequation（10.11）.（Hint:

Forthefirstpartoftherelationshipconsider（a）

aportfolioconsistingofaEuropeancallplusanamountofcashequaltoD+KportfolioconsistingofanAmericanputoptionplusoneshare.）

and（b）a

AsinthetextweusecandptodenotetheEuropeancallandputoptionprice,andCandPtodenotetheAmericancallandputoptionprices.Thepresentvalueofthedividends