利率二叉树模型

InterestRateModelsMarkBroadieB8835SecurityPricing:ModelsandComputation3DiscountFunctionComputationΔtq1-q02Δt3ΔtrrurdruurdurddΔt02Δt3Δt1111()uuB3()duB3()ddB3()uB3()dB3()B300123tttindex0123tindexLetBT„x…=priceinstatexof$1paidattimeT.Forsimplicity,useBi„x…ƒBiÑt„x…andBiƒBiÑt„0….TheinitialdiscountfunctionisBi,foriƒ1;2;:::;n:Givenaninterestratelattice,howisthediscountfunctioncomputed?Step1.Use1-steplatticetocomputeB1.Step2.Use2-steplatticetocomputeB2.Step3.Use3-steplatticetocomputeB3.Totalwork:O„n3….2Risk-NeutralPricingrurdrccucdInterestrateAssetpriceΔtΔtqq1-q1-q00ristherisk-freerateofinterestover†0;Ñt‡.Thatis,risaÑt-periodrateand$1investedattime0willbeworth$erÑtattimeÑt.Therisk-neutralprobabilityofanupmoveisq.Therisk-neutralpricingequationis:cƒE†e−rÑtcÑt‡ƒe−rÑt†qcu‚„1−q…cd‡„1…i.e.,thevalueattime0oftherandompayoffcÑtisthediscountedexpectedpayoffundertherisk-neutralmeasure.4ForwardInductionΔt2Δt3ΔtrurdruurdurddΔt02Δt3Δt1Δt02Δtr1000q1-q()-1eqΔ-trueqΔ-trdeuededdedueuueuuueduuedduedddeduLetex„0…ƒexƒpriceattime0of$1paidinstatex(thestatepriceorArrow-Debreuprice).Thepriceattime0ofadiscountbondmaturingat2ÑtisB2ƒeuu‚eud‚edd.Step1.euƒqe−rÑt;edƒ„1−q…e−rÑt=)B1Step2.eudƒ„1−q…e−ruÑteu‚qe−rdÑted;euuƒqe−ruÑteu;eddƒ„1−q…e−rdÑted=)B2Step3.euuuƒqe−ruuÑteuu;edddƒ„1−q…e−rddÑteddeuudƒ„1−q…e−ruuÑteuu‚qe−rudÑteud;etc.Totalwork:O„n2…!5ForwardInduction:ExampleΔt2Δt3ΔtΔt02Δt3Δt109%10%8%11%7%12%6%9%10%8%1/21/2.4570.4570.2067.4177.2109.0926.2835.2892.0983Theinterestratelatticeontheleftgivesthestatepricesontheright.Theinitialdiscountfunctionis:B0ƒ1,B1ƒ0:9139,B2ƒ0:8353,andB3ƒ0:7636.Thecorrespondinginitialyieldcurveis:y1ƒ9%,y2ƒ8:998%,andy3ƒ8:992%.(Ñtƒ1yearinthisexample.)7PricingaCapletinaLatticeΔtq1-q02ΔtΔt02Δt1euededdedueuuLuLdLuuLduLddLSupposethelatticeisconstructedofsimplycompoundedratesandconsideracapletfortheperiod†2Ñt;3Ñt‡,struckatK(withanotionalof$1).Iftherateattime2ÑtisL,apaymentofmax„L−K;0…Ñtismadeat3Ñt.Thevalueat2ÑtisÑt1‚LÑtmax„L−K;0…:Sothevalueofthecapletattime0isXstatesxat2ÑtÑt1‚L„x…Ñtmax„L„x…−K;0…exIfthecontinuouslycompoundedlatticerateisr,converttoLusinge−rÑtƒ11‚LÑtorLƒ1Ñt„erÑt−1…:6PricingEuropeanDerivativeSecuritiesinaLatticeΔtq1-q02Δtrrurdruurdurdd3=ΔTtcuuucduucdducdddΔt02Δt3Δt1euededdedueuueuuueduuedduedddLetc=priceattime0ofcxreceivedinstatexattimeT.Whatisc?Canusebackwardinductionasbefore,e.g.,cuuƒe−ruuÑt†qcuuu‚„1−q…cuud‡;etc.ThisproceduredeterminescinworkwhichisO„n2….Alternatively,ifthestatepriceshavealreadybeencomputed,thencƒXxcxex(thesumisoverthen‚1statesxattimeTƒnÑt).WorkisO„n….8EquivalenceBetweenCapsonRatesandPutsonBondsConsideracapletfortheperiod†mÑt;„m‚1…Ñt‡,struckatK.ThevalueofthecapletatmÑtwhentherateisLisÑt1‚LÑtmax„L−K;0…ƒmax1‚LÑt1‚LÑt−1‚KÑt1‚LÑt;0ƒ„1‚KÑt…max11‚KÑt−11‚LÑt;0i.e.,thecapletisequivalentto„1‚KÑt…putsonaone-perioddiscountbondwithstrike11‚KÑtexpiringatmÑt.Acapletisequivalenttoanoptiontoenterintoaone-periodswap.Sinceacapisasumofcaplets,acapisalsoequivalenttoaportfolioofputsondiscountbonds.9EquivalenceBetweenSwaptionsandOptionsonCouponBonds01234…Discountfactorattime1:()B21()B31()B41Payfixed:KKK1+Kn+1()B+n11Considera1-yearoptiontoenterintoann-yearpayerswap(i.e.,payfixed,receivefloating)withastrikeofK(andanotionalprincipalof$1).Attime1,theswapisworthPV(floating)−PV(fixed).Assumingtheprincipalisexchangedattheendoftheswap,PV(floating)ƒ1.Also,PV„fixed…ƒn‚1Xiƒ2Bi„1…K‚Bn‚1„1…P„1…;i.e.,PV(fixed)ƒvalueofann-yearbondwithacouponofK.Sothepayoffoftheoptionismax„1−P„1…;0…,i.e.,itisequivalenttoaputwithastrikeof1onabondpayingacouponofK.11FlexibleCapsAflexiblecapgivestheholdertherighttoexercisesome(notall)ofthecapletscomprisingthecap.Example:RighttocapatKatmostthreeoutofthenextsevenquarterlyLiborrates.Auto-flexcap:In-the-moneycapletsareexercisedautomaticallyonfixingdatesuntilnoneareleft.Chooser-flexcap:Onfixingdates,theholdercanchoosewhethertoexerciseanin-the-moneycaplet(untilnoneareleft).Auto-flexcapsareEuropean-stylesecurities,butaremorecomplicatedthanplainvanillacapsbecausetheyarepath-dependent.Chooser-flexcapsareAmerican-style(holdercanchooseamongvariousexercisestrategies).10PricingaEuropeanSwaptioninaLatticeq1-q0rrurdruurdurdd01122330123KPuuu+=1KPduu+=1KPddu+=1KPddd+=1PuuPduPddPuPdSuSdSRatelatticeSwappricelatticeCouponbondpricelatticeWhatisthevaluetodayofa1-yearoptiontoenterintoa2-yearpayerswapwithastrikeofK(andanotionalprincipalof$1)?Swappricelattice:Suƒmax„1−Pu;0…,Sdƒmax„1−Pd;0…,andSƒXstatesxat1Sxexƒe−r„qSu‚„1−q…Sd…:Couponbondpricelattice:PuuƒK‚e−ruu„1‚K…,PudƒK‚e−rud„1‚K…;:::;Pdƒe−rd„qPud‚„1−q…Pdd…:12ValuingChooser-FlexCapsinaLattice0123SimplycompoundedratelatticeFlex-cappricelatticeq1-q0123ij()V2,1()V1,1()V1,0()V1,2()V2,2()V3,2()()()()⎜⎜⎝⎜⎛⎟⎟⎠⎟⎞jiVjiVjiVjiV10,,,,nk()()()⎜⎜⎝⎜⎛⎟⎟⎠⎟⎞+++jiVjiVjiV-nkk1,1,1,1()()()⎜⎜⎝⎜⎛⎟⎟⎠⎟⎞++++++jiVjiVjiV-nkk11,11,11,1L1,0L1,1L2,1L1,2L2,2L3,2iNeedtorecordavectorofinformationateachnode:Vk„i;j…ƒvalueofchooser-flexcapwithstrikeKattimeiandratejwithkremainingcaplets(nisthemaximumnumberofcapletsthatcanbeexercised).Vk„i;j…ƒmax„don’texercise;exercise…;don’tƒ11‚Li;jÑtqVk„i‚1;j…‚„1−q…Vk„i‚1;j‚1…
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