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1、CHAPTER52uncertainparametersInthisChapter...•whytheparametersintheBlack–Scholesmodelarenotreliable•thedifferencebetween‘random’and‘uncertain’•howtopricecontractswhenvolatility,interestrateanddividendareuncertain52.1INTRODUCTIONTheBlack–Scholesequation∂V∂2V∂V+1σ2S2+(r
2、−D)S−rV=0∂t2∂S2∂Sisaparabolicpartialdifferentialequationintwovariables,Sandt,withthreeparameters,σ,randD,nottomentionotherparameterssuchasstrikeprice,barrierlevelsetc.specifictothecontract.Outofthesevariablesandparameters,whichonesareeasilymeasurable?•Assetprice:Theas
3、setpriceisquotedandthereforeeasytomeasureintheory.Inpractice,twopricesarequoted,thebidandtheaskprices;andeventhesepriceswilldifferbetweenmarketmakers.ThisissueoftransactioncostsandtheireffectonoptionpriceswasdiscussedinChapter48.•Timetoexpiry:Today’sdateandtheexpiryd
4、atearetheeasiestquantitiestomeasure.(Thereissomequestionabouthowtotreatweekends,butthisismoreaquestionofmodelingassetpricemovementsthanofparameterestimation.)•Volatility:Therearetwotraditionalwaysofmeasuringvolatility:Impliedandhistorical.Whicheverwayisused,theresult
5、cannotbethefuturevalueofvolatility;eitheritisthemarket’sestimateofthefutureoranestimateofvaluesinthepast.Thecorrectvalueofvolatilitytobeusedinanoptioncalculationcannotbeknownuntiltheoptionhasexpired.Atimeseriesplotofhistoricalvolatility,say,mightlooksomethinglikeFigu
6、re52.1,andit870PartFiveadvancedtopics0.8VolatilityofShangri-laAsia0.70.60.50.40.30.20.1015-Jun-9420-Jul-9523-Aug-9627-Sep-97Figure52.1Atypicaltimeseriesforhistoricalvolatility;animpliedvolatilitytimeserieswouldlooksimilar.iscertainlynotconstantasassumedinthesimpleBla
7、ck–Scholesformulae.Wecanseeinthisfigurethatvolatilityforthisstocktypicallyrangesbetween20and60%.TheexceptiontothiswasduringtheOctober/November1997crash,forwhichjump/crashmodelsareperhapsmorerelevant.•Risk-freeinterestrate:Supposewearevaluinganoptionwithalifespanofsixm
8、onths.Wecaneasilyfindtheyieldtomaturityofasix-monthbondandthiscouldbeourvaluefortherisk-freerate.However,becausethehedgedportfolioea
