An Introduction to Financial Option Valuation Mathematics, Stochastics and Computation.pdf
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An Introduction to Financial Option Valuation Mathematics, Stochastics and Computation.pdf
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ThispageintentionallyleftblankANINTRODUCTIONTOFINANCIALOPTIONVALUATIONMathematics,StochasticsandComputationThisisalivelytextbookprovidingasolidintroductiontofinancialoptionvaluationforundergraduatestudentsarmedwithonlyaworkingknowledgeoffirstyearcalculus.Writtenasaseriesofshortchapters,thisself-containedtreatmentgivesequalweighttoappliedmathematics,stochasticsandcomputationalalgorithms,withnopriorbackgroundinprobability,statisticsornumericalanalysisrequired.DetailedderivationsofboththebasicassetpricemodelandtheBlackScholesequationareprovidedalongwithapresentationofappropriatecomputationaltech-niquesincludingbinomial,finitedifferencesand,inparticular,variancereductiontechniquesfortheMonteCarlomethod.Eachchaptercomescompletewithaccompanyingstand-aloneMATLABcodelistingtoillustrateakeyidea.Theauthorhasmadeheavyuseoffiguresandex-amples,andhasincludedcomputationsbasedonrealstockmarketdata.Solutionstoexercisesaremadeavailableatwww.cambridge.org.DESHIGHAMisaprofessorofmathematicsattheUniversityofStrathclyde.Hehasco-writtentwopreviousbooks,MATLABGuideandLearningLaTeX.In2005hewasawardedtheGermundDahlquistPrizebytheSocietyforIndustrialandAppliedMathematicsforhisresearchcontributionstoabroadrangeofproblemsinnumericalanalysis.ANINTRODUCTIONTOFINANCIALOPTIONVALUATIONMathematics,StochasticsandComputationDESMONDJ.HIGHAMDepartmentofMathematicsUniversityofStrathclydeCAMBRIDGEUNIVERSITYPRESSCambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SoPauloCambridgeUniversityPressTheEdinburghBuilding,CambridgeCB28RU,UKFirstpublishedinprintformatISBN-13978-0-521-83884-9ISBN-13978-0-521-54757-4ISBN-13978-0-511-33704-8CambridgeUniversityPress20042004Informationonthistitle:
www.cambridge.org/9780521838849Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionofrelevantcollectivelicensingagreements,noreproductionofanypartmaytakeplacewithoutthewrittenpermissionofCambridgeUniversityPress.ISBN-100-511-33704-3ISBN-100-521-83884-3ISBN-100-521-54757-1CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurlsforexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate.PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYorkwww.cambridge.orghardbackpaperbackpaperbackeBook(EBL)eBook(EBL)hardbackTomyfamily,Catherine,Theo,SophieandLucasContentsListofillustrationspagexiiiPrefacexvii1Options11.1Whatareoptions?
11.2Whydowestudyoptions?
21.3Howareoptionstraded?
41.4Typicaloptionprices61.5Otherfinancialderivatives71.6Notesandreferences71.7ProgramofChapter1andwalkthrough82Optionvaluationpreliminaries112.1Motivation112.2Interestrates112.3Shortselling122.4Arbitrage132.5Putcallparity132.6Upperandlowerboundsonoptionvalues142.7Notesandreferences162.8ProgramofChapter2andwalkthrough173Randomvariables213.1Motivation213.2Randomvariables,probabilityandmean213.3Independence233.4Variance243.5Normaldistribution253.6CentralLimitTheorem273.7Notesandreferences283.8ProgramofChapter3andwalkthrough29viiviiiContents4Computersimulation334.1Motivation334.2Pseudo-randomnumbers334.3Statisticaltests344.4Notesandreferences404.5ProgramofChapter4andwalkthrough415Assetpricemovement455.1Motivation455.2Efficientmarkethypothesis455.3Assetpricedata465.4Assumptions485.5Notesandreferences495.6ProgramofChapter5andwalkthrough506Assetpricemodel:
PartI536.1Motivation536.2Discreteassetmodel536.3Continuousassetmodel556.4Lognormaldistribution566.5Featuresoftheassetmodel576.6Notesandreferences596.7ProgramofChapter6andwalkthrough607Assetpricemodel:
PartII637.1Computingassetpaths637.2Timescaleinvariance667.3Sum-of-squarereturns687.4Notesandreferences697.5ProgramofChapter7andwalkthrough718BlackScholesPDEandformulas738.1Motivation738.2Sum-of-squareincrementsforassetprice748.3Hedging768.4BlackScholesPDE788.5BlackScholesformulas808.6Notesandreferences828.7ProgramofChapter8andwalkthrough83Contentsix9Moreonhedging879.1Motivation879.2Discretehedging879.3Deltaatexpiry899.4Large-scaletest929.5Long-TermCapitalManagement939.6Notes949.7ProgramofChapter9andwalkthrough9610TheGreeks9910.1Motivation9910.2TheGreeks9910.3InterpretingtheGreeks10110.4BlackScholesPDEsolution10110.5Notesandreferences10210.6ProgramofChapter10andwalkthrough10411MoreontheBlackScholesformulas10511.1Motivation10511.2Whereis?
10511.3Timedependency10611.4Thebigpicture10611.5Changeofvariables10811.6Notesandreferences11111.7ProgramofChapter11andwalkthrough11112Riskneutrality11512.1Motivation11512.2Expectedpayoff11512.3Riskneutrality11612.4Notesandreferences11812.5ProgramofChapter12andwalkthrough12013Solvinganonlinearequation12313.1Motivation12313.2Generalproblem12313.3Bisection12313.4Newton12413.5Furtherpracticalissues127xContents13.6Notesandreferences12713.7ProgramofChapter13andwalkthrough12814Impliedvolatility13114.1Motivation13114.2Impliedvolatility13114.3Optionvalueasafunctionofvolatility13114.4BisectionandNewton13314.5Impliedvolatilitywithrealdata13514.6Notesandreferences13714.7ProgramofChapter14andwalkthrough13715MonteCarlomethod14115.1Motivation14115.2MonteCarlo14115.3MonteCarloforoptionvaluation14415.4MonteCarloforGreeks14515.5Notesandreferences14815.6ProgramofChapter15andwalkthrough14916Binomialmethod15116.1Motivation15116.2Method15116.3Derivingtheparameters15316.4Binomialmethodinpractice15416.5Notesandreferences15616.6ProgramofChapter16andwalkthrough15917Cash-or-nothingoptions16317.1Motivation16317.2Cash-or-nothingoptions16317.3BlackScholesforcash-or-nothingoptions16417.4Deltabehaviour16617.5Riskneutralityforcash-or-nothingoptions16717.6Notesandreferences16817.7ProgramofChapter17andwalkthrough17018Americanoptions17318.1Motivation17318.2Americancallandput173Contentsxi18.3BlackScholesforAmericanoptions17418.4BinomialmethodforanAmericanput17618.5Optimalexerciseboundary17718.6MonteCarloforanAmericanput18018.7Notesandreferences18218.8ProgramofChapter18andwalkthrough18319Exoticoptions18719.1Motivation18719.2Barrieroptions18719.3Lookbackoptions19119.4Asianoptions19219.5Bermudanandshoutoptions19319.6MonteCarloandbinomialforexotics19419.7Notesandreferences19619.8ProgramofChapter19andwalkthrough19920Historicalvolatility20320.1Motivation20320.2MonteCarlo-typeestimates20320.3Accuracyofthesamplevarianceestimate20420.4Maximumlikelihoodestimate20620.5Othervolatilityestimates20720.6Examplewithrealdata20820.7Notesandreferences20920.8ProgramofChapter20andwalkthrough21021MonteCarloPartII:
variancereductionbyantitheticvariates21521.1Motivation21521.2Thebigpicture21521.3Dependence21621.4Antitheticvariates:
uniformexample21721.5Analysisoftheuniformcase21921.6Normalcase22121.7Multivariatecase22221.8Antitheticvariatesinoptionvaluation22221.9Notesandreferences22521.10ProgramofChapter21andwalkthrough225xiiContents22MonteCarloPartIII:
variancereductionbycontrolvariates22922.1Motivation22922.2Controlvariates22922.3Controlvariatesinoptionvaluation23122.4Notesandreferences23222.5ProgramofChapter22andwalkthrough23423Finitedifferencemethods23723.1Motivation23723.2Finitedifferenceoperators23723.3Heatequation23823.4Discretization23923.5FTCSandBTCS24023.6Localaccuracy24623.7VonNeumannstabilityandconvergence24723.8CrankNicolson24923.9Notesandreferences25123.10ProgramofChapter23andwalkthrough25224FinitedifferencemethodsfortheBlackScholesPDE25724.1Motivation25724.2FTCS,BTCSandCrankNicolsonforBlackScholes25724.3Down-and-outcallexample26024.4Binomialmethodasfinitedifferences26124.5Notesandreferences26224.6ProgramofChapter24andwalkthrough265References267Index271Illustrations1.1PayoffdiagramforaEuropeancall.page31.2PayoffdiagramforaEuropeanput.41.3Payoffdiagramforabullspread.51.4MarketvaluesforIBMcallandputoptions.61.5AnotherviewofmarketvaluesforIBMcallandputoptions.71.6ProgramofChapter1:
ch01.m.92.1UpperandlowerboundsforEuropeancalloption.152.2ProgramofChapter2:
ch02.m.182.3Figureproducedbych02.m.193.1DensityfunctionforanN(0,1)randomvariable.253.2DensityfunctionsforvariousN(,2)randomvariables.263.3N(0,1)densityanddistributionfunctionN(x).273.4ProgramofChapter3:
ch03.m.303.5Graphicsproducedbych03.314.1Kerneldensityestimate.364.2Kerneldensityestimatewithincreasingnumberofsamples.374.3Quantilesforanormaldistribution.384.4Quantilequantileplots.394.5KerneldensityestimateillustratingCentralLimitTheorem.394.6QuantilequantileplotillustratingCentralLimitTheorem.404.7ProgramofChapter4:
ch04.m.425.1DailyIBMshareprice.465.2WeeklyIBMshareprice.475.3StatisticaltestsofIBMsharepricedata.475.4ProgramofChapter5:
ch05.m.516.1Twolognormaldensityplots.576.2ProgramofChapter6:
ch06.m.617.1Discreteassetpath.647.2Twodiscreteassetpathswithdifferentvolatility.657.3Twentydiscreteassetpathsandsamplemean.657.4Fiftydiscreteassetpathsandfinaltimehistogram.66xiiixivListofillustrations7.5Thesameassetpathsampledatdifferentscales.677.6Assetpathsandrunningsum-of-squarereturns.697.7ProgramofChapter7:
ch07.m.718.1Assetpathsandrunningsum-of-squareincrements.768.2ProgramofChapter8:
ch08.m.849.1Discretehedgingsimulation:
expiresin-the-money.909.2Discretehedgingsimulation:
expiresout-of-the-money.919.3Discretehedgingsimulation:
expiresalmostat-the-money.929.4Large-scalediscretehedgingexample.939.5ProgramofChapter9:
ch
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