empirical behavior of the spot interest rate

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1、CHAPTER36empiricalbehaviorofthespotinterestrateInthisChapter...•howtoanalyzeshort-terminterestratestodeter-minethebestmodelforthevolatilityandtherealdrift•howtoanalyzetheslopeoftheyieldcurvetogetinformationaboutthemarketpriceofrisk•howtobuildupsensibleone-ortwo-factorspotratemodel

2、s36.1INTRODUCTIONEarlierIdescribedthegeneralframeworkforone-factorinterestratemodelingandwehaveseenseveralpopularmodelsindetail.Theone-factormodelsthatwesawinChapter30forthespotinterestratewereallchosenfortheirniceproperties;formostofthemwewereabletofindsimpleclosed-formsolutionsof

3、thebond-pricingequation.Wediscussedtheprosandconsofthesemodelsinthatchapter.Inthischapterwewillseehowtodeduceamodelforthespotratefromdata.Themethodthatweuseassumesthatthemodelistimehomogeneous,andensuresthatthespotrateiswellbehaved.TheprincipleisthesameasthatinChapter53,wherewefind

4、avolatilitymodelfromvolatilitydata.Here,though,wetaketheideafurther,findingalsothemarketpriceofriskfromyieldcurvedata.Thedownsidetotheresultingmodelisthatwecannotfindclosed-formsolutionsforcontractvalues;therisk-neutraldriftandthevolatilitydon’thaveasufficientlynicestructure.Idescrib

5、etheanalysisandmodelingusingempiricalUSspotinterestratedata,makingcomparisonswiththepopularmodels.Weusethesedatatodeterminewhichmodelsseemtofitrealitybestandtoderiveaspotratemodelthathascertaindesirableproperties:Ithasarealisticvolatilitystructureandameaningfulsteady-stateprobabili

6、tydensityfunction.Bybasingourmodelingonrealdataweinevitablylosetractabilitywhenwecometosolvethebondpricingequation.Ouronlyconcessiontotractabilityistheassumptionoftime-independenceofourrandomwalk.Usingthisassumption,andbyexaminingdataoveralongperiodoftime,we596PartThreefixed-income

7、modelingandderivativeshopetomodel,inasense,the‘average’behaviorofthespotrate.Suchamodelcould,forexample,beusedtopricelong-datedinstruments.Iconcludewithsomesuggestionsforhowtobuildupatwo-factormodelforinterestrates.36.2POPULARONE-FACTORSPOT-RATEMODELSBecauseofthecomplexityoffixed-i

8、ncomeinstrumentsthemostpopularoft