HullFund8eCh10ProblemSolutions.docx
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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-freeinterestratesb
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