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1、CHAPTER71modelinginflationInthisChapter...•anexaminationofinflationdata•comparisonwithinterestrates•atwo-factormodel•calibration71.1INTRODUCTIONIndex-linkedbondshavecouponsandprincipalthatareincreasedinlinewithameasureofinflation,theConsumerPriceIndex(CPI)intheUSandtheRetailPriceInde
2、x(RPI)intheUK.Thusincomefromthesebondsisnoterodedbyinflation.Inthischapterwewilllookatthevariousinflation-protectedinstrumentsavailable,andalsoexamineinflationdatatotrytoderiveastatisticalmodelforinflationthatcanbeusedforpricingcomplexinflation-linkedproducts.WewillfocusonUKRPIdata.71.
3、2INFLATION-LINKEDPRODUCTS71.2.1BondsInflation-linkedbondshavecouponsandprincipalthatarelinkeddirectlytotheleveloftheinflationindex.1IfwedenotetheleveloftheindexbyIthenthepayoffofthebondwillbejustI,I0thatisinflationscaledwithitsinitiallevelI0.FromnowonI’lldropexplicitmentionofthescali
4、ng,orequivalentlyassumethatI0=1.71.2.2InflationCapsandFloorsThesecontractshavecallorputpayoffsinI,withamaturitydateandastrikeprice.1Theremaybesometimelaginthemeasurementofthedata,suchastheprincipalpayment,dependingontheindexleveleightmonthspreviously.1130PartFiveadvancedtopicsExamp
5、leAbondpaysacouponeveryyearfor30years.Thiscouponstartsat$1butisincreasedbyaspecifiedpercentageeveryyear.ThatpercentageincreaseisIiR=−1Ii−1whereIiistheleveloftheindexattheithsampling(everyyear,say,inthisexample)butwithafloorat3%andacapat6%.71.2.3SwapsInflationswapshaveaperiodicexchang
6、eofafixedrate(representingthemarket’sestimateofaverageinflation)andapaymentdependingontherelativechangeininflationduringthatperiod.TherewillbeaseriesofpaymentsoftheformIi−1−If.Ii−1Therearealsoinflationswaptions,optionstoenterintoaswap.71.2.4BarriersTherearealsobarrier-likecontractswhi
7、chknockinoroutdependingonthelevelofinflation.71.3PRICING,FIRSTTHOUGHTSTheobviouswaytoapproachthepricingofinflation-linkedproductsistotreattheindexIlikeanexchangerateoranequityprice.WewouldwritedownastochasticdifferentialequationmodelforI,dI=···,anotherfortheinterestrater,dr=udt+wdX,
8、and,heypresto!,wehaveapricingmeth
