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1、NotesandSolutionsfor:TheMathematicsofFinancialDerivativesbyPaulWilmott,SamHowison,andJe?DewynneJohnL.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、isbyastochasticprocessthatsatis?esdS=μSdt+σSdX,withdXtherandomvariable.Hereweareusingthenotationthatacapitalletterrepresentsarandomvariableandalowercaseletterrepresentsadeterministicvariable.Wit
