1风险管理与金融衍生品.pptx

1风险管理与金融衍生品.pptx

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liang_,梁进,Fundamentof,FinancialMathematics,-OptionPricing,Chapter1,RiskManagement&FinancialDerivative,Risk,Risk-uncertaintyoftheoutcomebringunexpectedgainscauseunforeseenlossesRisksinFinancialMarketasset（stocks,）,interestrate,foreignexchange,credit,commodity,TwoattitudestowardrisksRiskaversionRiskseeking,FinancialDerivatives,Manyformsoffinancialderivativesinstrumentsexistinthefinancialmarkets.Amongthem,the3mostfundamentalfinancialderivativesinstruments:

ForwardcontractsFutureOptionsIftheunderlyingassetsarestocks,bondsetc.,thenthecorrespondingriskmanagementinstrumentsare:

stockfutures,bondfutures,etc.,RiskManagement,riskmanagement-underlyingassetsMethodhedging-usingfinancialderivativesi.e.holdstwopositionsofequalamountsbutoppositedirections,oneintheunderlyingmarkets,andtheotherinthederivativesmarkets,simultaneously.,Underlyingassetputorcall,Derivativecallorput,=,ForwardContracts,anagreementtobuyorsellataspecifiedfuturetimeacertainamountofanunderlyingassetataspecifiedprice.anagreementtoreplaceariskbyacertaintytradedOTClongposition-thebuyerinacontractshortposition-thesellerinacontractdeliveryprice-thespecifiedpricematurity-specifiedfuturetime,Future,K,K,0,0,Longposition,Shortposition,Futures,sameasaforwardcontracthaveevolvedfromstandardizationofforwardcontractsdifferencesfuturesaregenerallytradedonanexchangeafuturecontractcontainsstandardizedarticlesthedeliverypriceonafuturecontractisgenerallydeterminedonanexchange,anddependsonthemarketdemands,Options,anagreementthattheholdercanbuyfrom（orsellto）theseller（thebuyer）oftheoptionataspecifiedfuturetimeacertainamountofanunderlyingassetataspecifiedprice.Buttheholderisundernoobligationtoexercisethecontract.aright,noobligationtheholderhastopaypremiumforthisrightisacontingentclaimHasamuchhigherlevelofleverage,TwoOptions,Acalloption-acontracttobuyataspecifiedfuturetimeacertainamountofanunderlyingassetataspecifiedpriceAputoption-acontracttosellataspecifiedfuturetimeacertainamountofanunderlyingassetataspecifiedprice.exerciseprice-thespecifiedpriceexpirationdate-thespecifieddateexercise-theactiontoperformthebuyingorsellingoftheassetaccordingtotheoptioncontract,OptionTypes,Europeanoptions-canbeexercisedonlyontheexpirationdate.Americanoptions-canbeexercisedonorpriortotheexpirationdate.OtheroptionsAsiaoptionetc.,TotalGainofanOption,K,K,0,0,Calloption,putoption,p,Totalgain=Gainoftheoptionatexpiration-Premium,OptionPricing,riskyassetspriceisarandomvariablethepriceofanyoptionderivedfromriskyassetisalsorandomthepricealsodependsontimetthereexistsafunctionsuchthatknownHowtofindout,TypesofTraders,Hedger-toinvestonbothsidestoavoidlossSpeculator-totakeactioncharacterizedbywillingtoriskwithonesmoneybyfrequentlybuyingandsellingderivatives（futures,options）fortheprospectofgainingfromthefrequentpricechanges.Arbitrage-basedonobservationsofthesamekindofriskyassets,takingadvantageofthepricedifferencesbetweenmarkets,thearbitrageurtradessimultaneouslyatdifferentmarketstogainrisklessinstantprofits,HedgerExample,In90days,ApaysB1000,000Toavoidrisk,Ahas2plansPurchaseaforwardcontracttobuy1000,000with$1,650,00090dayslaterPurchaseacalloptiontobuy1000,000with$1,600,00090dayslater.Apaysapremiumof$64,000（4%）,SpeculatorExample,StockAis$66.6onApril30,maygrowAspeculatorhas2plansbuys10,000shareswith$666,000onApril30paysapremiumof$39,000USDtopurchaseacalloptiontobuy10,000sharesatthestrikeprice$68.0pershareonAugust22,SpeculatorExamplecont.,SituationI:

Thestock$73.0on8/22.StrategyAReturn=（730-666）/666*100%=9.6%StrategyBReturn=（730-680-39）/39*100%=28.2%SituationII:

Thestock$66.0on8/22.StrategyAReturn=（660-666）/666*100%=-0.9%StrategyBlossallinvestmentReturn=-100%,Chapter2,Arbitrage-FreePrinciple,FinancialMarket,TwoKindsofAssetsRiskfreeassetBondRiskyassetStocksOptions.Portfolioaninvestmentstrategytoholddifferentassets,Investment,Attime0,investSWhent=T,Payoff=Return=Forariskyasset,thereturnisuncertain,i.e.,Sisarandomvariable,APortfolio,arisk-freeassetBnriskyassetsaportfolioiscalledainvestmentstrategyontimet,wealth:

portionofthecor.Asset,ArbitrageOpportunity,Self-financing-during0,TnoaddorwithdrawfundArbitrageOpportunity-Aself-financinginvestment,andProbabilityProb,ArbitrageFreeTheorem,Theorem2.1themarketisarbitrage-freeintime0,T,areany2portfoliossatisfying&,ProofofTheorem,Supposefalse,i.e.,DenoteBisarisk-freebondsatisfyingConstructaportfolioat,ProofofTheoremcont.,rriskfreeinterestrate,att=TThenFromthesupposition,ProofofTheoremcont.,ItfollowsContradiction!

Corollary2.1,Marketisarbitragefreeifportfoliossatisfyingthenforany,ProofofCorollary,ConsiderThenByTheorem,forNamely,ProofofCorollary2.1,InthesamewayThenCorollaryhasbeenproved.,OptionPricing,EuropeanOptionPricingCall-PutParityforEuropeanOptionAmericanOptionPricingEarlyExerciseforAmericanOptionDependenceofOptionPricingontheStrikePrice,Assumptions,Themarketisarbitrage-freeAlltransactionsarefreeofchargeTherisk-freeinterestraterisaconstantTheunderlyingassetpaysnodividends,Notations,-theriskyassetprice,-Europeancalloptionprice,-Europeanputoptionprice,-Americancalloptionprice,-Americanputoptionprice,K-theoptionsstrikeprice,T-theoptionsexpirationdate,r-therisk-freeinterestrate.,Theorem2.2,ForEuropeanoptionpricing,thefollowingvaluationsaretrue:

ProofofTheorem2.2,lowerboundof（upperleavestoex.）considertwoportfoliosatt=0:

ProofofTheorem2.2cont.,Att=T,andByTheorem2.1i.e.,ProofofTheorem2.2cont.cont.,NowconsideraEuropeancalloptioncSinceandByTheorem2.1whentTi.e.Togetherwithlastinequality,2.2proved.,Theorem2.3,ForEuropeanOptionpricing,thereholdscall-putparity,ProofofTheorem2.3,2portfolioswhent=0whent=T,ProofofTheorem2.3cont.,SothatByCorollary2.1i.e.call-putparityholds,Theorem2.4,ForAmericanoptionpricing,ifthemarketisarbitrage-free,then,ProofofTheorem2.4,TakeAmericancalloptionasexample.Supposenottrue,i.e.,s.tAttimet,takecashtobuytheAmericancalloptionandexerciseit,i.e.,tobuythestockSwithcashK,thensellthestockinthestockmarkettoreceiveincash.Thusthetradergainsarisklessprofitinstantly.Butthisisimpossiblesincethemarketisassumedtobearbitrage-free.Therefore,mustbetrue.canbeprovedsimilarly.,AmericanOptionv.s.EuropeanOption,ForanAmericanoptionandaEuropeanoptionwiththesameexpirationdateTandthesamestrikepriceK,sincetheAmericanoptioncanbeearlyexercised,itsgainingopportunitymustbe=thatoftheEuropeanoption.Therefore,Theorem2.5,IfastockSdoesnotpaydividend,theni.e.,theearlyexercisetermisofnouseforAmericancalloptiononanon-dividend-payingstock.,ProofofTheorem2.5,Byaboveinequalities,thereholdsThisindicatesitisunwisetoearlyexercisethisoption,Theorem2.6,IfC,Parenon-dividend-payingAmericancallandputoptionsrespectively,then,ProofofTheorem2.6（rightside）,Itfollowsfromcall-putparty,andTheorem2.5,thustherightsideoftheinequalityinTheorem2.6isproved.,ProofofTheorem2.6（leftside）,ConstructtwoportfoliosattimetIfint,T,theAmericanputoption$P$isnotearlyexercised,then,ProofofTheorem2.6（leftside）cont.1,Namely,whent=T,ProofofTheorem2.6（leftside）cont.2,IftheAmericanputoptionPisearlyexercisedattime,thenByTheorem2.2,2.5,ProofofTheorem2.6（leftside）cont.3,Accordingtothearbitrage-freeprincipleandTheorem2.1,theremustbeThatis,TheTheoremhasbeenproved.,Theorem2.7,LetbethepriceofaEuropeancalloptionwiththestrikepriceK.Forwiththesameexpirationdate,FinancialMeaningofTheorem2.7,For2Europeancalloptionswiththesameexpirationdate,theoptionwithstrikeprice,leavesitsholderprofitroomandisthereforepriced,thedifferencebetweenthetwooptionsshallnotexceedthedifferencebetweenthestrikeprices.,ProofofTheorem2.7（leftside）,LeavetherightsideparttoreaderConstructtwoportfoliosatt:

whent=T:

ProofofTheorem2.7（leftside）cont.1,Case1,ProofofTheorem2.7（leftside）cont.2,Case2So,ProofofTheorem2.7（leftside）cont.3,Case3Thus,whent=TByTheorem2.1&ArbitrageFreePrinciple,for0tT,Theorem2.8,FortwoEuropeanputoptionswiththesameexpirationdate,ifthen,Theorem2.9,Europeancall（put）optionpriceisaconvexfunctionofK,i.e.,ProofofTheorem2.9,Onlyprovethefirstone,thesecondonelefttothereaderConstructtwoportfoliosatt=0Ontheexpirationdatet=T,Discussin4cases,ProofofTheorem2.9cont.1,Case1Case2Therefore,ProofofTheorem2.9cont.2,Case3Case4,ProofofTheorem2.9cont.3,InallCases,whent=TButByArbitrageFreeP.andTheorem2.1,Theorem2.10,Europeancall（put）optionpriceisalinearhomogeneousfunctionoftheunderlyingassetpriceandthestrikepriceK.i.e.for,FinancialMeaningofTheorem2.10,ConsiderbuyingEuropeanoptions,witheachoptiontopurchaseoneshareofastockontheexpirationdateatstrikepriceK;Alsoconsiderbuying1EuropeanoptiontopurchasesharesofthesamestockatstrikepriceKontheexpirationdate;Themoneyspentontheoptionsinthesetwocasesmustbeequal.Proofleavestoexercise,