基于全最小二乘拟蒙特卡罗方法的可转债定价研究

张卫国,史庆盛,许文坤,510640:基于传统的可转债最小二乘蒙特卡罗模拟定价方法,通过使用随机Faure序列和方差减小技术,有效地降低模型估计结果的误差,使用考虑解释变量和被解释变量误差的全最小二乘回归方法代替普通的最小二乘回归方法,提出可转债的全最小二乘拟蒙特卡罗定价方法,并给出该定价方法的具体算法步骤以2002年10月16日发行的燕京可转换债券为例进行实证分析,从可转债的理论价值计算标准差以及模型的运行时间等几个方面与传统的蒙特卡罗方法进行比较研究结果表明,使用全最小二乘拟蒙特卡罗方法进行计算得到的结果更为合理,且估计误差和计算时间都更少,从而验证了该方法在可转债定价应用上的有效性:可转债定价;全最小二乘;拟蒙特卡罗;Faure序列;对偶变量法:F830.91:A:1672-0334(2011)01-0082-08:2010-04-22:2010-08-22:(70825005);(07JA630048):(1963-),,,,,,:Emai:[e-mail]11843,,,,,,,,,3,,Longstaff[1],Longstaff[1],,(TLSQM)2,Black[2]1977Ingersoll[3]BlackScholes,,,Arak[4],;Mike[5],,;,Gu[6]24120112ISSN1672-0334JournalofManagementScienceVo.l24No.182-89February,2011,,,21[7];[8],,;[9]BlackScholes,,,;[10],,,,;[11],,,,;[12]PLS,Longstaff[1](LSM),;[13][14]LSM,,,,,,,[15][16]HaltonFaureSobol3,3,,,,[17]LSM,,,[18],Faure,,,,(TLSM)3,,,3(),;,,,T,,t=0,I,A,r0,rs,St;Z,Z;Pt,Kt,Pt=0,Kt=;Ht,;N;Vtt;Btt,CttVt=Bt+NCt(1),tTtArBt=!Tt=texp[-(r0+rs)(t-t)]I+exp[-(r0+rs)(T-t)]A(2),,,,3,,3,(1):Ct∀St-Z,,,(2):Ct#Kt-Z,(3):Ct∀Pt-Z,,t,831:Ct=max[Pt-Z,St-Z,min(Kt-Z,Ht-Z)](3),V=B+NC(4)V0=exp[-(r0+rs)](B+NC)(5)44.1,,Longstaff[1](LS),,,,,,(TLS)[18],TLS,(X+d)=Y+e,Xn∃p,dX,Yn∃1,eY,TLS,XY,LSTLS,Y+e=0+1X(LS)Y+e=0+1(X+d)(TLS),0,1,eY,dXLSTLS,X,Y;XY,y=a0+k0x,a0=0.4,k0=1,,LSTLS,12LSTLS,,,,LSTLS12,TLS,LS4.2,,,,1LSFigure1FittingResultsUsingLSMethod2TLSFigure2FittingResultsUsingTLSMethod,,HaltonFaureSobol,Faure,,4.2.1FaureFaure[19]b,b∀2,s,Zb={0,1,%,b-1},[0,1)ssFaure{xn}n=0,1,2,%,nb,nm-1nm-2%n1n0,mn,nm-1&0,ni∋Zb,n=!m-1i=0nibin*=(n0,n1,%,nm-2,nm-1)D1,D2,%,Dsm∃m,D1m,84(JournalofManagementScience)20112D2=00%0100%10%%%%%0100010000D3=D00D01%D0m-10D11%D1m-1%%%%%00%Dm-1m-1Ds=Ds-23(s∀4),DVR=R!V!(R-V)!,0#R#m-1f(t)=mod(Dtn*,b)=(f1,f2,%,fm),mod(Dtn*,b)Dtn*bF(t)n=f1b+f2b2+%+fmbm∋[0,1)s,[0,1)sFaureFn=(F(1)n,F(2)n,%,F(s)n)∋[0,1)s4.2.2,nb,[0,1)sFaureF,,,FaureF,F=[F,-F]FCranleyPatterson,Fsin=Fsin+sisi=1,2,%,2sFsin∋Fsi(0,1)Fsin,FaureF={Fsin}∋(-1,1)sMatlab,[0,1)Faure,5t=0,T,L,0=t0t1%tL=T,Longstaff[1],,1S0,FaureM,SMitMit,Mi=1,2,%,M,t,t=0,1,%,TPitKit2T,BT,BT=I+A,BTCMiTMi,CMiT=max(ST-Z,0),TMiVMiT,VMiT=BT+CMiT3tL-1,HMitL-1=exp(-r0-rs)CMiT(6)0,M,MHMitL-1=!po=1!o(SMiT)o,Mi=1,2,%,M(7),p(7)!1,!2,%,!n,HMitL-1,Mi=1,2,%,MEMitL-1=max(ST-Z,0)HMitL-1,CMitL-1=max(EMitL-1,HMitL-1)0,CMitL-1=HMitL-1tL-1,,EMitL-1∀HMitL-1,BtL-1=I+A(8)EMitL-1HMitL-1,BtL-1=exp[-(r0+rs)]BtL(9)tL-1VMitL-1,VMitL-1=BtL-1+CMitL-143,tk+1Btk+1CMitk+1VMitk+1HMitk=exp(-r0)SMitk+1,M0,MHMitk=!po=1!o(SMitk)o,Mi=1,2,%,MHMitk,Mi=1,2,%,MCMitk=max(EMitk,HMitk),EMitk=max(Stk-Z,0),EMitk∀HMitk,Btk=I+AEMitkHMitk,Btk=exp[-(r0+rs)]Btk+1tkVMitk,VMitk=Btk+CMitk54t1,V0=1M!MMi=1exp[-(r0+rs)](Bt1+NCMit1)(10)6,;,()6.1,dS=rSdt+∀SdW(11)851:,S,r,t(),∀,W,dW0tSt=S0exp[(r-∀22)t+∀dW](12)dW=tx,xSt=S0exp[(r-∀22)t+t∀x](13),SxFaurex,6.2,,,,,,,r0rs,AAA,[20],,350.90%0.98%,r0=2.65%,rs=0.98%,r0=0.0073%,rs=0.0027%,r=r0+rs=0.01%6.3,20021016250,∀=0.0226.4,,(1)20052007,();(2),;(3),;(4);(5),,20021016,3AAA,5,X=10.59,A=100,I0=1.2%,I=0.0033%,S0=8.95121Table1ConvertibleBondParameters()(%)(%)()2002101651.202.6510.592Table2PartofTermsofIssue,20130%,102/(),2070%,2345(/)101102103104注:以上资料来自聚源数据库86(JournalofManagementScience)201127,3,Matlab,(LSM)(LSQM)M=1000,343LSMFigure3StockPricePathUsingLSMMethod4LSQMFigure4StockPricePathUsingLSQMMethod(TLSM)(TLSQM)100,565TLSMFigure5TheoryPriceofConvertibleBondsUsingTLSMMethod6TLSQMFigure6TheoryPriceofConvertibleBondsUsingTLSQMMethod100,105.25104.85,100,5%100,100,787TLSMFigure7StandardDeviationunderTLSMMethod8TLSQMFigure8StandardDeviationunderTLSMMethod871:100,0.510!11,,Faure,,78%,TLSQMTLSM,1,TLSM2183,,TLSQM1256,,TLSQMTLSM(1.71)IntelDualCore,2600M,=1∃,1,TLSQMTLSM8.0198,,,(1)Faure,,;(2),;(3),;(4),5%;,,:[1]LongstaffFA,SchwartzES.ValuingAmericanoptionsbysimulation:Asimpleleastsquaresapproach[J].ReviewofFinancialStudies,2001,14(l):113-147.[2]BlackF,ScholesM.Thepricingofoptionsandcorporateliabilities[J].JournalofPoliticalEconomy,1973,81(3):637-659.[3]IngersollJE,Jr.Acontingentclaimvaluationofconvertiblesecurities[J].JournalofFinancialEconomics,1977,4(3):289-322.[4]ArakM,MartinAL.Convertiblebonds:Howmuchequity,howmuchdebt?[J].FinancialAnalystsJournal,2004,61(2):44-50.[5]MikeS,FarmerJD.Anempiricalbehavioralmodelofliquidityandvolatility[J].JournalofEconomicDynamicsandControl,2008,32(1):200-234.[6]GuGF,ChenW,ZhouWX.EmpiricalregularitiesoforderplacementintheChinesestockmarket[J].PhysicaA:StatisticalMechanicsandItsApplications,2008,387(13):3173-3182.[7],,.[J].,2001,9(4):7-15.YangDakai,DuXinle,LiuQingsheng.ThemechanismofCBpricinganddesigninginChina[J].ChineseJournalofManagementScience,2001,9(4):7-15.(inChinese)[8],.()[J].,2007,4(6):815-819.XuKe,LiXin.Analyzingthepricingofbondwithattachedwarrantempirically[J].ChineseJournalofManagement,2007,4(6):815-819.(inChinese)[9],,.[J].,2009,12(4):135-144.ZhouQiyuan,WuChongfeng,LiuHailong.Analyticvaluationofthecallableconvertiblediscountbonds:Equivalentdecompositionmethod[J].JournalofManagementSciencesinChina,2009,12(4):135-144.(inChinese)[10],.[J].,2009,22(3):89-95.ZhangPu,WuChongfeng.Thestudyofstocksvolatilitypremiumbasedonnonpar