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1、ApproximateOptionPricingforAt-the-MoneyForwardsBobKoppraschThissimplemodel,showntomeinthemid-80’s,canbeusedtopriceputsandcallswhentheyarestruckattheforwardpriceofanyinstrument.[Becauseputsandcallsstruckattheforwardpricehavethesamevalue,themodelisabletovalueeith
2、erputsorcalls.]Afterpricinganat-the-moneyforward,it’seasytoestimatein-orout-ofthemoneypricesaswell,OptionValue=PresentValueof(.4*σ*√t*ForwardPrice)Thevolatilityisexpressedasanannualizednumber,indecimalform,andtimeismeasuredinyears(3months=1/4,etc.)The“model”can
3、beusedtopriceatthemoneyoptionsonfutures,becausefuturesarealreadyattheir“forwardprice.”Asanexample,considertheMarchoptionsontheMarch201010-yearNotefuturescontract,which,asshownintheBloombergscreenshotthatfollows,has71daystillexpiration.OnthemorningofDec.10,2009,
4、thecontractwaspricedat118-10+(118-10.5/32),andthe118.5callshadanimpliedvolatilityof6.39%.Let’sseehowthemodeldoesinmatchingthemarketpricesforthe118.5strikeoptions.First,let'sassumeforaminutethatthecontracthadclosedat118-16,sothatthepriceandstrikearethesame.Theva
5、lueunderthatassumptionwouldbe:PV(.4x.0639x√(71/365)*118.5)=PV(.02556*√.19452*118.5)=PV(1.33586)Withboth2and3monthLIBORatabout24basispointsrightnow,presentvaluingover72dayswillhardlyhaveanyeffect,butforpractice,let’scalculateit:PV(1.33586)=1.33586/(1+.0024*71/36
6、0)=1.3352.Thistranslatesto1-21.45/64.OntheBloombergscreen,notethatthe“Last”priceis1’14,andthebid/offeris1’13/1’16,butwehaven’tyetadjustedforthefactthatthecontractisnotat118.5.Wehavetomakeanadjustmentforthefactthatthefuturesclosedat118.33,not118.50.Thinkofthepro
7、blemthisway:wejustpricedtheat-the-moneyoptionwhenthefutureswerepricedat118.50,andthenthefuturesjustdropped.17points(oralmost11/64)to118.33.Rememberingthatthedeltaofanatthemoney(forward)optionis.5,this.17dropshouldcauseacalltodropby.085points,oralmost5.5/64,andt
8、husourestimatewouldbe1-21.5/64–5.5/64=1-16/64,thecurrentaskingprice.TheMar118.5putswouldhaveclimbed5.5/64fromthe1-21.45/64at-the-moneyfigure,sowe'destimate1-27/64,whichisals
