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1、CHAPTER32interestratederivativesInthisChapter...•commonfixed-incomecontractssuchasbondoptions,capsandfloors•howtopriceinterestrateproductsintheconsistentpartialdifferentialequationframework•howtopricecontractsthemarketway•pathdependencyininterestrateproducts,suchastheindexamortizingrat
2、eswap32.1INTRODUCTIONInthefirstpartofthisbookIderivedatheoryforpricingandhedgingmanydifferenttypesofoptionsonequities,currenciesandcommodities.InChapter30Ipresentedthetheoryforzero-couponbonds,boldlysayingthatthemodelmaybeappliedtoothercontracts.Intheequityoptionsworldwehaveseendiffer
3、entdegreesofcomplexity.Thesimplecon-tractshavenopathdependency.Theseincludethevanillacallsandputsandcontractshavingdifferentfinalconditionssuchasbinariesorstraddles.Atthenextstageofcomplexitywefindtheweaklypath-dependentcontractssuchasAmericanoptionsorbarriersforwhich,technicallyspeaki
4、ng,thepathtakenbytheunderlyingisimportant,yetforwhichweonlyneedtosolveintwodimensions(Sandt).Finally,wehaveseenstronglypath-dependentcontractssuchasAsiansorlookbacksforwhichwemustintroduceanewstatevariabletokeeptrackofthekeyfeaturesofthepathoftheunderlying.Manyoftheseideasaremirrored
5、inthetheoryofinterestratederivatives.Inthischapterwedelvedeeperintothesubjectoffixed-incomecontractsbyconsideringinterestratederivativessuchasbondoptions,capsandfloors,swaptions,captionsandfloortions,andmorecomplicatedandpath-dependentcontractssuchastheindexamortizingrateswap.32.2CALLAB
6、LEBONDSAsagentleintroductiontomorecomplexfixed-incomeproducts,considerthecallablebond.Thisisasimplecoupon-bearingbond,butonethattheissuermaycallbackonspecifieddatesforaspecifiedamount.Theamountforwhichitmaybecalledbackmaybetime-dependent.534PartThreefixed-incomemodelingandderivativesThis
7、featurereducesthevalueofthebond;ifratesarelow,sothatthebondvalueishigh,theissuerwillcallthebondback.Thecallablebondsatisfies∂V∂2V∂V+1w2+(u−λw)−rV=0,∂t2∂r2∂rwithV(r,T)=1,andV(r,t−)=V(r,t+)+K,cccacrosscoupondates.IfthebondcanbecalledbackforanamountC(t)thenwehavetheconstraintonthebond’sv
8、alueV(r,t)≤C
