The Mathematics Of Financial Derivatives(Solutions)

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1、NotesandSolutionsfor:TheMathematicsofFinancialDerivativesbyPaulWilmott,SamHowison,andJeffDewynneJohnL.Weatherwax∗October8,2006∗wax@alum.mit.edu1Chapter1(OptionsandMarkets)Exercise1(stocksplits)Ifacompanyissuesastocksplit(doublingthenumberofstockshares)Iwoulde

2、xpectthatthevalueofeachshareofstocktodividebytwoandthenumberofstocksharesthatapersonownedtodoublekeepingthenetvalueofallthestockconstant.Iwouldalsoexpecttheoptionsvaluestoalsodividetheirvaluebytwo.Asanexample,ofastocksplitconsiderthevalueofaRolls-Roycecall(t

3、heoptiontobuyonRolls-Roycestockatagivenpriceatafuturedate).AccordingtoFigure1.1fromthebookitsvaluewithaMarchexpirationdateis11p.IfRolls-Royceissuesaone-for-onestocksplitIwouldexpectthatthevalueoftheMarchcalloption(foroneshare)togoto11p/2=5.5p.Foratwo-for-one

4、issue,Iwouldexpectthevalueoftheunderlyingsecuritytodroptoonethirditsoriginalprice,sincetheownerofoneshareendsupowningthreesharesaftertheissue.InthesamewayIwouldexpectthatthevalueofoptionsontheunderlyingwouldbereducedbyafactorofthree.Exercise2(dividends)Iwoul

5、dassumethatthepriceofthestockSbeforethedividendisissuedmustimplicitlyincludethevalueofthedividendinthestocksevaluation.ThusafterthedividendissuethestockpricewoulddecreasebyanamountDtoS−D.Exercise3(thedirectionofuncertainty)Asthevolatilityoftheunderlyingsecur

6、ityincreasesitbecomesmorelikelythattheunderlyingwillobtainitsstrikeprice.Sinceitismorelikelytobeexercisedthereismorevaluetotheoption.Exercise4(zero-sumgame)Whatismeantbythestatement“optionstransactionsareazerosumgame”isthatthereisnolossorgainintheimmediatepu

7、rchaseofanoption.Itcanbesoldimmediatelyafteritspurchaseforthesameprice.Thusoptionshaveanintrinsicvaluethatdoesnotchangeastheyarepurchased.2Chapter2(AssetPriceRandomWalks)Exercise1(stochasticderivatives)Forthisproblem,werequireIto’slemmaforafunctionf(S),whenS

8、isbyastochasticprocessthatsatisfiesdS=µSdt+σSdX,withdXtherandomvariable.Hereweareusingthenotationthatacapitalletterrepresentsarandomvariableandalowercaseletterrepresentsadeterministicvariable.Wit