二元树形结构法定价期权的技巧

234Vol123No14200610JOURNALOFSHENZHENUNIVERSITYSCIENCEANDENGINEERINGOct12006:100022618(2006)0420373204:2005206228;:2006203223:(200547):(19652),(),,.E2mail:chenyong828@yahoo1com1,2(1.,518060;2.,610054):.(payoff),,Black2Scholes(ex2trapolation)..:;;;;;:F830159;TB115:A,.,[1]20:dSS=dX+dt(1),dX,,0dt.(risk2neutralworld)r[1]83.BlackScholesBlack2Scholes,[2].,,,.Cox[3]2070,.,.[4].,.1.,S,,.,S,,S,.,t=0t=1,SS,dS,p,1-p,.Cox.1,,:tTM,t=(T-t)/M,,M,M+1SMn=undM-nS,n=0,1,,M.(2)CMn=max(undM-nS-X,0).(3),n=0,1,,M;X..M-1,M-2,,1,0,(4)Cmn=ert[pCm+1n+1+(1-p)Cm+1n].(4),r.,C,M,CM.Coxudp,u=et,d=e-t,p=(ert-d)/(u-d).udp,[1]1842186,[4]:374231Fig11Theprocessofthebinomialtreemethodu=ert(1+e2t-1),d=ert(1-e2t-1),p=1/2.(5)u=e(r-2/2)t+t,d=e(r-2/2)t-t,p=1/2.(6)2,,,.211,:Cn=e-r(T-t)nj=0n!j!(n-j)!pj(1-p)n-1.max[0,ujdn-jS-X].(7)(a;n,p)=nj=aCjnpj(1-p)n-j,(8),alog(X/Sdn)/log(u/d).,1p=(u/r)p,CoxCn=S[a;n,p]-Xe-r(T-t)[a;n,p].(9)an,Cn=0.S,,.Cox.2^Cn,C.g(S),,^Cn=C+O(1/n).(10)3g(S),limn^Cn=C.(11)^Cn=C+O(1/n).(12),[4].,,23,,,.,212B2S,f(S,T-t),f(S,T-t)=C(S,T-t)(f(S,T-t)=C(S,T-t)),fC.,.exotic[4].2(a),2(b)B2S.,B2S,:,B2S,0O(1/n);,,0..213B2SCnO(1/n),..4,:3752B2SFig12ErrorsofthecrudebinomialandB2Sadjustmnt,2C2n-Cn.(13),2C2n-CnC;(14),,.limn+(2C2n-Cn)=2C-C=C,(13)(2C2n-Cn)-C=2(C2n-C)-(Cn-C).(14)3,.S=12,X=10,=40%,r=10%,r;6.udCox.PCMatlab513,CPUCeleron(TM)733MHz,128Mb.12.1n;BTM,Black2Scholes;BSB2S.B2S,,Black2Scholes.n,;BSBTMBSBTM.2(a),2(b)B2S.1Talbe1ErrorsandcuptimeofcomputationbythebinomialtreemethodnBTM/10-6CPU/s32919.10100.010642905.11800.31640.0303.0001281255.63582.31370.1806.000256-55.8444-22.48450.7214.006512224.5485-0.24872.8343.9311024127.04291.767511.5074.060204887.84731.446257.0224.955409640.23782.1832242.9504.261nB2S/10-6CPU/s321520.46070.01064746.03672.03810.0414.1000128377.69611.97520.1804.3902256196.62881.92090.7814.338951296.35892.04063.0643.9232102448.01622.006812.2784.0072204823.66882.028743.9843.5823409611.88561.9914174.7313.9726nExtrapolated/10-6BTMBS3220.991043.785572.433964-28.3873-102.3387-26.28071289.3555134.213640.371625615.5615-3.588612.6356512-3.9110-57.4146-24.63791024-0.3265-389.1054-147.06342048-0.6786-129.4537-34.878940960.1024392.9473116.0703,BTMn,,2;B2S,,n237623,2..1,BS,BTM,,,.,B2S,.,,.,,ud.:[1]PaulWilmott,SamHowison,JeffDewynne.[M].:,1996().[2]BlackF,ScholesMJ.[J].,1973,81:6372659().[3]JohnCCox,StephenARoss,MarkRubinstein.:[J].,1979,7:2292263().[4]SteveHeston,GuofuZhou.[J].,2000,10(1):53257().[5]JohnCHull.[M].4,:PrenticeHall,2000().Abstract:100022618(2006)04203762EAImprovementofusingbinomialtreemethodinpricingoptionsCHENYong1andYEMao21)CollegeofEconomicsShenzhenUniversityShenzhen518060P.R.China2)SchoolofComputerScienceandEngineeringUniversityofElectronicScienceandTechnologyofChina,Chengdu610054P.R.ChinaAbstract:TheBinomialtreemethodisoneofthebasicmethodsofpricingoptions.Sincethepayofffunctionsinfinancialmarketareoftennotsmooth,therateofconvergenceofpricingbythebinomialmodelisreduced.Theimprovementsontheaccuracyofpricingoptionsbybinomialtreemethod,suchasBlack2scholesadjustmentandextrapolation,werepresentedanddiscussedindetail.Byusingthesetechniques,theefficiencyofcomputinginbinomialmethodforpri2cingoptionscanbesignificantlyimproved.Experimentalresultsdemostratetheeffectivessoftheseimprovements.Keywords:financialderivatives;derivativepricing;option;optionpricing;binomialtree;multinomialtreeReferences:[1]PaulWilmott,SamHowison,JeffDewynne.TheMathemat2icsofFinancialDerivatives[M].NY:ThePressSyndicateoftheUniversityofCambridge,1996.[2]BlackF,ScholesMJ.Thepricingofoptionsandcorporateliabilities[J].JournalofPoliticalEconomy,1973,81:6372659.[3]JohnCCox,StephenARoss,MarkRubinstein.Optionpri2cing:asimplifiedapproach[J].JournalofFinancialEco2nomics,1979,7:2292263.[4]SteveHeston,GuofuZhou.Ontherateofconvergenceofdiscrete2timecontingentclaims[J].MathematicalFinance,2000,10(1):53257.[5]JohnCHull.Options,Futures,andOtherDerivatives[M].4thed.NJ:PrenticeHallInc,2000.:;: