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25120102JOURNALOFSYSTEMSENGINEERINGVol.25No.1Feb.2010doi:10.3969/j.issn.1000-5781.2010.01.0081,2,1(1.,210096;2.,210093):,,.Black2Scholes,,,,.,Black2Scholes.:;;;Black2Scholes;;:F830.9:A:1000-5781(2010)01-0043-07StudyonoptionpricingbyapplyinghybridwaveletnetworksandgeneticalgorithmZHANGHong2yan1,LINHui2,JIANGCai2lou1(1.SchoolofManagement,SoutheastUniversity,Nanjing210096,China;2.BusinessSchool,NanjingUniversity,Nanjing210093,China)Abstract:Theimpliedvolatilityratesofvariedkindsofoptionsaredifferentbecauseofvolatilitysmileeffects.Howtodeterminetheoptimalweightsoftheimpliedvolatilityratesofvariedkindsofoptionsisanimportantissueinoptionpricing.AhybridwaveletneuralnetworkbasedontheBlack2Scholesmodelisproposedinthispaper,andsomehybridforecastingmodelscombiningthehybridwaveletneuralnetworkandgeneticalgorithmarebuilt.Insuchanapproachoptionsareclassifiedaccordingtotheirmoneyness,andtheweightedimpliedvolatilityratesareregardedastheinputoftheneuralnetwork.Ageneticalgorithmisusedtodeterminetheoptimalweightsoftheimpliedvola2tilityratesofdifferentkindsofoptions.CasestudyonHongKongderivativemarketshowsthatthesehybridmodelsarebetterthantheconventionalBlack2Scholesmodelandtheotherneuralnetworkmodels.Keywords:optionpricing;hybridwaveletneuralnetwork;geneticalgorithm;Black2Scholesmod2el;moneyness;impliedvolatilityrate0.Black2Scholes(B2S).B2S,[1].:2007-06-13;:2009-11-08.:(70501013).Akgiray[2]B2S.,.Huchison[3],RBFBP,S()/X(),();C()/X;B2S,Yao[4]Nikkei225,BP,S,X,;C.S,,Schmalensee[5],,S,.,,.Hull[6](moneyness),,,0.9.Latane[7],Chiras[8],Whaley[9],Beckers[10].Black[11],B2S5,B2S.,.(GA).,.,,.,,.1B2S,,,BlackScholes2070(1)B2S[6].C=S(d1)-Xe-r(d2)(1),C;S;();X;t;d1=ln(S/X)+(r+2/2);r;d2=d1-=ln(S/X)+(r-2/2).B2S.[6].2[12]:BP,,,,(,).BP[12-14]Lajbcygier[13],BP,BP[12]:BPB2S,BP,.[15],,,,,BP.4425,B2S.2.1Wf(a,b)=+-f(t)h(a,b,t)dt(2)f(t),h(a,b,t)=1|a|hbasic(t-ba),hbasic(t),1|a|,abh(a,b,t),()f(t),ab,.B2S,1,,B2S,3,,,1,tB2S,Cmarket-CBlack-Scholes.1Fig.1Structureofhybridwaveletneuralnetworkm,g=pk=1wkhmi=1ukixi-bkak(3),wkuki,xi,m,bk,ak,P.wk,uki,bkakE=12Nj=1(gj-tj)2(4),N,gjj,tjj,jCmarket-CBlack-Scholes.(3)Morlet,h(y)=cos(1175y)exp(-y22)(5),Levenberg2Marquardt[16].,,,,.,B2S,,.,S(),X(),()im,i(i),C(),20.2.2B2S,5:S,X,,r,;S,X,,r,:Lajbcygier[13]r,Yao[4]r,17,,r.2.2.1,.(moneyness),,2004-05-042004-06,2004-05-032004-06,0.95S/X1.05,;S/X0.9,.541:,,,.1im,i=prevat,i+(1-)prevdout,i(6)prevat,i=1/n1n1j=1prevat,i,j(7)prevdout,i=1/n2n2j=1prevdout,i,j(8),1im,ii;prevat,ii;prevdout,ii;prevat,i,jji;n1i;prevdout,i,jji;n2i;,01.2.2.2,,,.0.95S/X1.05,;S/X0.95,.S/X1.05,.,,,,,,.2im,i=1prevat,i+2previn,i+(1-1-2)prevout,i(9)prevat,i=1/n1n1j=1prevat,i,j(10)prevout,i=1/n2n2j=1prevout,i,j(11)previn,i=1/n3n3j=1previn,i,j(12)011,021,1+21(13),2im,ii;prevat,ii;previn,ii;prevout,ii;prevat,i,jji;n1i;prevout,i,jji;n2i;previn,i,jji;n3i;1;2.2.3.S,X,,1im,i;S,X,,2im,i.C.2.,,,.2Fig.2Forecastingmodels6425,.1).,,,.,.MAPE(meanabsolutepercentageerror),MAPEMAPE=1NNi=1|y(i)-y^(i)|y(i)100%(14),N;y(i)i;y^(i)i.2):,,,,.,1,2.3),.(normGeomSelect)[17].,(arithXover)[18].,.(multiNonUnifMutation)[18],,.,=10;q=0.08;b=3;=200.2.4[19],S,X,;C.,Levenberg2Marquardt[16],,.3,.1)2004-05-042004-05-2516(HSIcalloption),2004-05-2717.8,2004-05,2004-06,2004-07,2004-09,2004-12,2005-03,2005-06,2005-12..(a),2004-0794002004-05-04,2004-05-032004-079400,.(b)10.2004-05-042004-05-252672.2672,2004-05-27175.2)2,,,,1.1Table1Optimalparameters1S,X,,1im,iC=0.91772S,X,,2im,iC1=0.8543;2=0.02353).4)BPBP.BP3,S,X,;C.tansig,logsig,.741:Levenberg2Marquardt[16].BPLabjcyjier[13],S,X,;C.Black2Scholes,.BPtansig,logsig,.Levenberg2Marquardt[16].S,X,;C.(),(),200.,BP3-11-1,BP3-8-1,3-17-1,3-12-1.B2S2.2(MAPE)Table2Comparisonofvariouskindsofforecastingmodels(MAPE)1S,X,,1im,iC6.3%2S,X,,2im,iC6.84%BPS,X,C10.92%S,X,C9.83%S,X,C7.26%BPS,X,C8.56%B2S15.73%2,B2S,B2S,,.,,,,,.,0.9177,0.9[6],.,,,0.8543,0.0235.,,.4(1),,,,,,,,,,.(2)BPBP,B2S.,B2S,.:[1]PetersEE.FractalMarketAnalysis:ApplyingChaosTheorytoInvestmentandEconomics[M].NewYork:JohnWiley&Sons,1994.[2]AkgiaryV.Conditionalheteroscedasticityintimeseriesofstockreturns:Evidenceandforecasts[J].JournalofBusiness,1989,61(5):55-80.8425[3]HutchisonJM,LoA,PoggioT.Anonparametricapproachtopricingandhedgingderivativeandsecuritiesvialearningnet2works[J].JournalofFinance,1994,49(3):851-889.[4]YaoJT.Optionpriceforecastingusingneuralnetworks[J].Omega,2000,28(4):455-466.[5]SchmalenseeR,RobertR.Commonstockvolatilityexpectationsimpliedbyoptionpremia[J].JournalofFinance,1978,33(2):129-147.[6]HullJC.[M].3.:,1999:221-222.HullJC.Option,Futures,andOtherDerivatives[M].ThirdEdition.Beijing:HuaxiaPress,1999:221-222.(inChi2nese)[7]LataneH,RendlemanRJ.Standarddeviationofstockpriceratioimpliedbyoptionpremia[J].JournalofFinance,1976,31(2):369-382.[8]ChirasDP,ManasterS.Theinformationcontentofoptionpricesandatestofmarketefficiency[J].JournalofFinancialE2conomics,1978,6(1):213-234.[9]WhaleyRE.ValuationofAmericancalloptionsondividend2payingstocks:Empiricaltests[J].JournalofFinancialEco2nomics,1982,10(2):29-58.[10]BeckersS.Standarddeviationsinoptio