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Oct 24, 2014

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david-tang

SecondSemester,2011-2012THEUNIVERSITYOFHONGKONGDEPARTMENTOFSTATISTICSANDACTUARIALSCIENCESTAT2807 CORPORATEFINANCEFORACTUARIALSCIENCETutorial10(Finale): OptionPrices Date: April2324,2012WhatisThisFinalTutorialAbout?Inthisnal tutorial, wewill studythefundamentalsof optionpricingtheory. Afterexploring the properties of option prices that the no-arbitrage assumption necessitates, wewill briey introduce the binomial option pricing model, the method of replicating portfolioandrisk-neutral valuation. Theproblemsectionincludesalargenumberof instructiveexercises,whicharecollectedfromawealthofsourcestostrengthenyourunderstanding.1 KeyLearningPointsIntheexamination,candidatesareexpectedto:LP(1) State,proveandmanipulatetheput-callparity.LP(2) Useno-arbitrageargumentstoproveinequalitiesinvolvingoptionprices.LP(3) Detect the existence of arbitrage opportunities, and if they exist, construct a portfolio toreaprisk-freeprot.LP(4) Calculatethevalueof anoptionwithapossiblyunfamiliarpayostructureby(i)themethodofreplicatingportfolio,and(ii)risk-neutralvaluation.#"

!MessagefromAmbroseThisisthelasttutorial. Enjoy!2 ReviewofKeyConcepts2.1 IdentitiesandInequalitiesofOptionPricesTheno-arbitrageassumptioneectivelymakesoptionpricesnon-arbitrary. Asfunctionsof the stock price, strike price, time to maturity, they have to satisfy a number of identitiesandinequalities,including:S&AS: STAT2807 CorporateFinanceforActuarialScience AmbroseLO2012 21. (IMPORTANT!)Put-callparity: CE0+ KerT= S0 + PE0.2. Callpricesarebounded(aboveandbelow): StKer(Tt) CA/Et St.3. Putpricesarebounded: Ker(Tt)St PA/Et Ker(Tt).4. Call prices are decreasingfunctions of the strike price: Ct(K1) Ct(K2) if K1 K2.Interpretation: This is natural as the higher the strike price K, the lower thepayoofacalloption(ST K)+.5. Put prices are increasingfunctions of the strike price: Pt(K1) Pt(K2) if K1 K2.Interpretation: Thepayoofa putoptionis(K ST)+,whichincreaseswithK.6. Optionpricesareincreasinginthetime-to-maturity: IfT1> T2,Ct(T1) Ct(T2) and Pt(T1) Pt(T2).7. Optionpricesareconvex1inthestrikeprice(Problem2(a),Section3.3): Forall [0, 1],Ct[K1 + (1 )K2] Ct(K1) + (1 )Ct(K2)Pt[K1 + (1 )K2] Pt(K1) + (1 )Pt(K2)8. OptionpricesareLipschitz2inthestrikeprice(Assignment3):|C1tC2t| er(Tt)|K1K2|,|P1tP2t| er(Tt)|K1K2|.WhatwillIbeaskedintheFinalExam?IntheFinalExam,youmaybeaskedto:Applyput-callparitytoperformsomecalculations,orProveanyoftheaboveinequalities.Anotherpossibilityistopresentsomeobservedoptionpricesandaskyouwhetherthemarketisarbitrage-free. Almostsurely(otherwise,whatisthepointofthatexamques-tion!?!?) therewillbearbitrageopportunitiesandyoucanfollowthestepsonpage4toconstructanarbitragestrategy.1Recallthatareal-valuedfunctionfdenedonaconvexsetXissaidtobeconvexiff(x1 +(1 )x2) f(x1) + (1 )f(x2)forallx1, x2 Xand [0, 1].2Areal-valuedfunctionfdenedonasetXissaidtobeLipschitz ifthereexistsL > 0suchthat |f(x) f(y)| L|x y|forallx, y X.S&AS: STAT2807 CorporateFinanceforActuarialScience AmbroseLO2012 3Exercise 1.(SOA Exam MFE Sample Q1: Straightforward Put-call Parity Manipu-lations) Consider a European call option and a European put option on a nondividend-payingstock. Youaregiven:Thecurrentpriceofthestockis60.Thecalloptioncurrentlysellsfor0.15morethantheputoption.Boththecalloptionandputoptionwillexpirein4years.Boththecalloptionandputoptionhaveastrikepriceof70.Calculatethecontinuouslycompoundedrisk-freeinterestrate.Solution. Applyingput-callparity,wehaveC0P0= S0KerT0.15 = 60 70e4rr = 0.039 .Exercise2. (SOAExamMFESpring2009)Youaregiven:C(K, T) denotes the current price of a K-strike T-year European call option on anondividend-payingstock.P(K, T) denotes the current price of a K-strike T-year European call option on anondividend-payingstock.Sdenotesthecurrentpriceofthestock.Thecontinuouslycompoundedrisk-freeinterestrateisr.Whichofthefollowingis(are)correct?(I) 0 C(50, T) C(55, T) 5erT(II) 50erT P(45, T) C(50, T) + S 55erT(III) 45erT P(45, T) C(50, T) + S 50erTSolution. (I)istrue. Call priceisadecreasingfunctionof K, soC(50, T) C(55, T).ThesecondinequalityfollowsfromAssignment3(Lipschizity).(II)isincorrect,but(III)istrue. Byput-callparity,P(45, T) C(50, T) + S = [C(45, T) S + 45erT] C(50, T) + S= C(45, T) C(50, T) + 45erT.By(I),0 C(45, T) C(50, T) (50 45)erT,whichisequivalentto45erT C(45, T) C(50, T) + 45erT 50erT.S&AS: STAT2807 CorporateFinanceforActuarialScience AmbroseLO2012 4(IMPORTANT!)HowtoProveTheseIdentities/InequalitiesSystematically?Supposeyouwanttoprove( .Step2. Buylowandsell high.At time0, enter thetransactions withacost of . For example, if =Ct(St, K, T),thenyoushouldbuyacalloption.Atthesametime,shortthetransactionswhichcost . Forexample,if =StK expr(Tt), then you should short sell an asset and lend K expr(Tt).Step3. Verifythatarbitrageprotsexistbyshowingthatthecashowattime0ispositive(non-negative),andthecashowatthematuritydateTisnon-negative.Forclarity,youcanpresentyouranswersinatable:Transaction1 Transaction2 TotalST< K ( 0?)ST K ( 0?)Step4. Arguethatyouhaveconstructedanarbitragestrategy,so(1)musthold.Therearenobetterexercisesthanverifyingasmanyof theidentitiesandinequalitiesaboveonyourown. Checkyourproof withtheoneinthenotes. Totestyourunder-standing,someadditionalinequalitiesareprovidedinProblems1and2inSection3.3.2.2 BinomialOptionPricingModel(OPTIONAL)2.2.1 BasicsUsingbinomial treesisageneral, robust, butcomputationallyintensivemethodtopriceop-tions3. Althoughsimpleexpressionsofoptionpricesmaynotbeavailable, all optionscanbepricedtheoretically.ConstructionofBinomialTrees: Under the binomial option pricing model, the stockpriceinthenexttimeperiodisassumedtomoveeitherupbyafactorofuordownbyafactorofd:S0Su= S0uSd= S0d3Black-Scholespricingformulaewillnotbetreatedinthiscourse.S&AS: STAT2807 CorporateFinanceforActuarialScience AmbroseLO2012 5(Note: Sisnotalwaystheunderlyingriskystock. Itonlyhastobeanassetfromwhichthe value of a derivative can be derived directly. For a compound option(May 2009 ExamQuestion10),Sisactuallythepriceofanotheroption!)Inthiscourse, weshall notpursueissuesconcerningthedeterminationof uandd. Inotherwords,thevaluesofuanddareassumedtobegiven.Objective: Thetime-0priceofaderivativewithpayosfuandfdinthenextperiod.Note that this derivative need not be a standard call or put option. Any payo structures(e.g. May 2010 Exam Question 9 and Problem 1, Section 3.4), regardless of its irregularity,canbetackledbythebinomialtreemethod.Case1. One-periodBinomial TreeMethod1: ReplicatingPortfolio(Goodforone-periodmodel)Thismethodinvolvessolvingapairofsimultaneousequations. Constructaport-folioconsistingof shares of stockandcashof , tomimickthepayoof thederivativeofinterest:_S0u + erT= fuS0d + erT= fd= =? , =? .(Note: You need notremember the expressions of and .)By the law of one price,theoptionpricemustbeequal totheinitial costinconstructingthereplicatingportfolio,whichisS0 + .Method2: Risk-neutral Valuation(Goodfor most cases!)Denetherisk-neutral probabilityq erTdu d.ThentheoptionpricecanberewrittenaserT[qfu + (1 q)fd] = erTEQ[Payo].InterpretationThe price of an option can be evaluated as a discounted expectedpayo,where:1. discounted meansdiscountingbytherisk-freerate,and2. expected means the expectationwhenthestock movesupwitha proba-bilityofq(butnotthetrueprobability!).Case2. Multi-periodBinomial TreeIt is mucheasier toemployrisk-neutral valuation(Method2) whenwehaveseveralperiodsinthebinomialtree. Byworkingbackwardthroughthebinomialtreeandcon-sidering each node as a single-period binomial tree model, the option can be recursivelyvalued.S&AS: STAT2807 CorporateFinanceforActuarialScience AmbroseLO2012 6ImportantSpecial Case: ForEuropeanoptions, earlyexerciseisnotallowed.Therefore,wecansimplydiscounttheexpectedpayoattheendofthebinomialtreebacktotime0.1. Foratwo-periodtree,thepriceofthederivativeiserT[q2fuu + 2q(1 q)fud + (1 q)2fdd].2. Forathree-periodtree,thepricebecomeserT[q3fuuu + 3q2(1 q)fuud + 3q(1 q)2fudd + (1 q)3fddd].Case3. Trinomial Tree (This part is more challenging; suitable for more motivated students)The risk-neutral probabilities for a stock to go up, stay put and go down are more dicultto obtain for a multinomial tree. In this case, the method of replicating portfolio comesto our rescue. The same idea, i.e. constructing a portfolio with available assets such thatthepayoofthederivativecanbereplicated,stillworks,butalargersystemoflinearequations has to be solved. Please try Problem 3 in Section 3.4 for such a nonstandardproblem.2.2.2 AmericanOptionsIdea: ForAmericanoptions, thesamekindof recursiverisk-neutral valuationforEu-ropeanoptions canbeperformed. Theoptionpriceat eachnodeof thetreeis givenbymax {discountedexpectedpayo, payofromearlyexercise} .ImportantSpecial Case: Iftheunderlyingstockpaysnodividends(asisthecaseinourcourse),thenanAmericancallisworththesameasaEuropeancall. Earlyexerciseneednotbeaccountedfor.Exercise3. (SOAExamMFESampleQuestion4: AStandardExercise)Foratwo-periodbinomialmodel,youaregiven:Eachperiodisoneyear.Thecurrentpriceforanondividend-payingstockis20.u = 1.2840.d = 0.8607.Thecontinuouslycompoundedrisk-freeinterestrateis5%.CalculatethepriceofanAmericancalloptiononthestockwithastrikepriceof22.S&AS: STAT2807 CorporateFinanceforActuarialScience AmbroseLO2012 7Solution. Thetwo-periodbinomialtreeisconstructedinFigure1. Therisk-neutralproba-bilityforthestockpricetogoupisq=e0.050.86071.2840 0.8607= 0.450203.AtnodeB:Thevalueofthecalloptionismax___e0.05[q(10.9731) + (1 q)(0.1028)]. .=4.752922, (25.680 22)+. .=3.680___= 4.752922.AtnodeC:Thevalueofthecalloptionbecomesmax___e0.05[q(0.1028) + (1 q)(0)]. .=0.044023, (17.214 22)+. .=0___= 0.044023.Earlyexerciseisnotoptimal.AtnodeA:Finally,thecallpriceismax___e0.05[q(4.752922) + (1 q)(0.044023)]. .=2.0584, (20 22)+. .=0___=2.0584 .Remark 1. Asthestockpaysnodividends, theAmericancall pricemustbeequal totheEuropeancallprice.20A25.68B17.214C32.9731 (10.9731)22.1028 (0.1028)14.8161 (0)Figure1: BinomialtreeforExercise