金融工程导论课件.ppt

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CHAPTERFIVE:OptionsandDynamicNo-Arbitrage—ABriefIntroductionofOptionsAnoptionistherightofchoiceexercisedinfuture.Theholder(buyer,orlonger)oftheoptionhasarightbutnotanobligationtobuyorsellaspecialamountoftheassetwithaspecialqualityatapre-determinedprice.TermsofOptionsCallandputExercisepriceExpirationdateAmericanoptions(CandP)vs.Europeanoptions(candp)1 ThepayoffprofilesofcallandputCallPutLongShort++__XXSTST00LongShort++__XXSTST00In-the-money,out-of-the-money,at-the-money,intrinsicvalueandtimevalue—ABriefIntroductionofOptions(Cont.)2 TheBasicNo-Arbitrage1),2)3),4)If,then,5)3 TheBasicNo-Arbitrage(Cont.)Theunderlyingisanon-dividend-payingstockSuppose,thenArbitrageImmediateCashFlowPositionCashFlowontheexpireddateShortastockLonganEuropeancallLongrisklesssecurityNetcashflowsArbitrageOpportunity!4 TheBasicNo-Arbitrage(Cont.)PropositionIftheperiodtoexpirationisverylong,thevalueofanEuropeancallisalmostequaltoitsunderlying.PropositionAnAmericancallonanon-dividend-payingstockshouldneverbeexercisedpriortotheexpirationdate.5 TherelationshipbetweenAmericanoptionsandEuropeanoptions?andConclusion:6 TheParityofCallandPutTheunderlyingisanon-dividend-payingstockScanbereplicatedbyc,pandrisklesssecuritySupposePositionCashflowatCashflowattimeTtimetBuyashareShortacallLongaputShorttreasuryNetcashflowArbitrage!7 RelationshipbetweenexerciseandforwardpriceNon-dividend-payingstock’sAmericancallandput?8 Non-dividend-payingstock’sAmericancallandput(Cont.)PositionCashflowatCashflowattimewhenputexercisedtimetShortashareLonganAmer.callShortanAmer.putLongtreasuryNetcashflow9 Non-dividend-payingstock’sAmericancallandput(Cont.)Underlyingisdividend-payingstockPresentvalueofdividendsattimetPresentvalueofalongstockforwardpositionPresentvalueofashortstockforwardposition10 Underlyingisdividend-payingstockForEuropeancallandputForAmericancallandputHoldsfornon-dividend-payingstockunderlyingDividendpaidProved!Howtoproveit?Pleaseseethenextpage!11 ProofofPositionCashflowatCashflowattimewhenputexercisedtimetShortashareEffectofdividendsLonganEuro.callShortanAmer.putLongtreasuryNetcashflow12 Proposition!ForanAmericancall,whentherearedividendswithbigamount,thecallmaybeearlyexercisedatatimeimmediatelybeforethestockgoesex-dividend.Question:Iftherearenex-dividenddatesanticipated,what’stheoptimalstrategytoearlyexerciseanAmericancall?Answer:Pleasereadthelastparagraphofpage74ofthetextbook.13 DynamicNo-Arbitraget=0t=1t=2BondABondB14 ReplicationstepbystepUsingBondAandrisklesssecuritywithmarketvaluetoreplicateBondB’svalueintheabovestep15 Replicationstepbystep(Cont.)ReplicatingtheblowbinomialtreebyusingBondAandrisklesssecuritywithmarketvalueReplicatingtheleftbinomialtreebyusingBondAandrisklesssecuritywithmarketvalue16 Self-financingNotes:DynamicreplicationisforwardwhiletheprocedureofpricingisbackwardShortsaleisavailableforself-financing17 OptionPricing—BinomialTrees—One-StepBinomialModelNon-dividend-payingstock’sEuropeancallUsingtheunderlyingstockandrisklesssecuritywithmarketvaluetoreplicatetheEuropeancall?Sensitivityofthereplicatingportfoliotothechangeofthestock.18 Isprobabilityrelevanttooptionpricing?ProbabilitydistributionAnswer:Directly:No!Indirectly:Yes!Noarbitragepricingisnotrelevanttoprobabilitydistribution19 —One-StepBinomialModel(Cont.)NotationNoArbitrageReplicating:Shortsaleofrisklesssecurity20 Risk-Neutrality—Risk-AversionAMiniCase—TossingaCoinHeadTailFairGameFairGameRiskpremiumRiskdiscountInvestmentGamblingInvestors:risk-averseGamblers:risk-preferFromrealeconomybechargedbycasinorisk-neutral21 —Risk-NeutralPricingrisk-neutralprobabilitymeanorexpectationonrisk-neutralprobabilitydiscountedbyrisk-freerateAnalysisbecomesverysimple!andInanimaginaryworldArisk-neutralworld22 —WhatKindofProblemsCanBeResolvedinanImaginaryRisk-NeutralWorld?Proposition:Ifaproblemwithitsresolvingprocedureisfullyirrelevanttopeople’srisk-preference,thenitcanberesolvedinanimaginaryrisk-neutralworldandthesolutionwouldbestillvalidintherealworld.Proposition:No-Arbitrageequilibriuminfinancialmarketsisfullyirrelevanttopeople’srisk-preference.Therefore,risk-neutralpricingisvalidequilibriumpricing.Risk-neutralpricingandno-arbitragepricingmustbeequivalenttoeachother.23 —Risk-NeutralPricing(Multi-StepBinomialModel)t=0t=1t=2TheUnderlyingStockTheCall24 —Risk-NeutralPricing(Cont.)Generalizing:25 —AMiniCaseTheUnderlyingStockTheCallt=0t=1t=2t=0t=1t=2Risk-NeutralPricing:26 —AMiniCase(Cont.)DynamicNo-ArbitragePricing:27 —ImplicationofRisk-NeutralPricingMeanormathematicalexpectationwithprobabilityintherealworldDiscountrateswithriskpremiumRisk-freerateusedasdiscountrateswithoutriskpremiumQuestion:Doesrisk-neutralprobabilityexistandisitunique?Meanormathematicalexpectationwithrisk-neutralprobabilityintheimaginaryworld28 FundamentalTheoremsofFinancialEconomicsTheFirstFinancialEconomicsTheorem:Risk-neutralprobabilitiesexistifandonlyiftherearenorisklessarbitrageopportunities.TheSecondFinancialEconomicsTheorem:Therisk-neutralprobabilitiesareuniqueifandonlyifthemarketiscomplete.TheThirdFinancialEconomicsTheorem:Undercertainconditions,theabilitytorevisetheportfolioofavailablesecuritiesovertimecandynamicallymakeupforthemissingsecuritiesandeffectivelycompletethemarket.29 —ProblemandInverseProblemmanyinvestorsmakeportfoliochangeseachportfolio’schangeislimitedtheaggregationcreatesalargevolumeofbuyingandsellingtorestoreequilibriumimplyingarbitrageopportunityexistseacharbitrageurwantstotakeaslargepositionaspossibleafewarbitrageursbringthepricepressurestorestoreequilibriumInverseProblem:Knowingthemarketpricesofsecurities,determinethemarket’srisk-neutralprobabilities.Problem:Knowingthemarket’srisk-neutralprobabilities,determinethemarketpricesofsecurities.Unfortunately,areactualsecuritiesmarketslikethis?Aretheyincomplete?Soitwouldseemthatwewillnotbeabletosolvetheinverseproblem;thatis,althoughrisk-neutralprobabilitiesmayexist,theyarenotunique.However,in1954,economistKennethArrowsavedthedaybystatingthethirdfundamentaltheoremoffinancialeconomics,thecriticalideabehindmodernsecuritiespricingtheory.30 —EquivalentMartingaleDefinition:Therisk-neutralvaluationapproachissometimesreferredtoasusingequivalentmartingalemeasure,i.e.,therisk-neutralprobabilityisreferredtoanequivalentmartingalemeasure(probabilitydistribution).31 SummaryofChapterFiveNo-ArbitrageTheKeyofFinanceTheory,EspeciallyForDerivativesSuchasOptions.DynamicNo-ArbitragePricingRisk-NeutralPricing.DoesRisk-NeutralProbabilityExistandIsItUnique?TheCoreofFinanceTheory—TheFundamentalTheoremsofFinancialEconomics32