ꆡ뗚28뻭ꆡ뗚6웚볃톧풺톧놨Vol128ꆡNo162005쓪12퓂ꆡꆡꆡꆡꆡ쪯볒뺭JournalofShijiazhuangUniversityofEconomicsDec12005뿘훆뇤솿벼쫵퓚쏀쪽웚좨뚨볛훐뗄펦폃쇎ퟷ뫨1,2,샮쿾헑2(11커몺샭릤듳톧풴폫뮷뺳릤돌톧풺,뫾놱커몺430070;21붭컷샭릤듳톧펦폃뿆톧톧풺,붭컷룓훝341000)햪ꆡ튪:좫쟲웚좨붻틗듳뛠쫇쏀쪽웚좨,떫쏀쪽웚좨뗄슷뺶틀삵쳘헷떼훂웤뇈얷쪽웚좨뚨볛룼뻟뢴퓓탔ꆣBlack-Scholes뚨볛쒣탍뛔쏀쪽뾴뗸웚좨늻듦퓚뷢컶릫쪽,컞램쟳웤뺫좷뷢ꆣ튻킩뇈뷏돉쫬뗄쫽횵볆쯣랽램뛔쏀쪽웚좨뚨볛폐뻖쿞탔:죧MonteCarlo쒣쓢램쫇잰떼돌탲,뛸쏀쪽웚좨뗄뇟뷧쫇ퟔ평뗄,맊MonteCarlo쒣쓢램늻쓜헽좷뛔웤뚨볛;뛾닦쫷뚨볛쒣탍쯤좻쫊뫏룷훖웚좨맀볛,떫듦퓚좱쿝,죧뺫뛈컊쳢ꆣ뿘훆뇤솿벼쫵쫇튻훖랽닮쯵복랽램,샻폃뿘훆뇤솿벼쫵탞헽뛾닦쫷뚨볛쒣탍뚨볛뗄쏀쪽웚좨볛횵,뿉틔쳡룟쏀쪽웚좨맀횵뗄뺫뛈ꆣ맘볼듊:뿘훆뇤솿벼쫵;쏀쪽웚좨뚨볛;뛾닦쫷뚨볛쒣탍;Black-Scholes훐춼럖샠뫅:F83019ꆡꆡꆡ컄쿗뇪쪶싫:Aꆡꆡ컄헂뇠뫅:100726875(2005)06207842031ꆡ웚좨2ꆡBlack-Scholes뚨볛쒣탍웚좨쫇튻훖퇜짺횤좯,쫇틔풼뚨볛룱퓚풼뚨뗄1973쓪,Black뫍Scholes퓚췪좫쫐뎡뫍막욱쪱웚쓚싲죫믲싴돶쒳훖뇪뗄닺뗄좨샻ꆣ웚좨럖캪볛룱뛔쫽헽첬럖늼뗄볙뚨쳵볾쿂,춨맽붨솢튻룶냼솽샠:뾴헇웚좨,쫇횸퓚맦뚨웚쿞쓚,틔풼뚨볛룱몬웚좨뫍막욱뗄컞럧쿕춷듧,뗃떽믹폚막욱뗄웚좨뫍쫽솿릺싲쒳훖붻틗뇪뗄닺뗄좨샻;뾴뗸웚좨,볛룱뗄캢럖랽돌,쯻쏇돉릦뗘쟳뷢쇋룃캢럖랽돌,쫇횸퓚맦뚨웚쿞쓚,내짌뚨뗄볛룱뫍쫽솿싴돶쒳훖뗃떽얷쪽뾴헇웚좨뫍뾴뗸웚좨뗄뺫좷릫쪽,벴훸쏻붻틗뇪뗄닺뗄좨샻ꆣ내쫇럱뿉쳡잰횴탐뿉럖웚좨뗄Black-Scholes쒣탍,룃쒣탍틑돉캪웚좨붻틗뗄캪:쏀쪽웚좨,쫇횸쓜퓚폐킧웚쓚죎뫎튻쳬뿉쳡잰뇪ힼ릤뻟ꆣ솽룶퓂뫳,Merton뛀솢췪돉쇋샠쯆뗄횴탐뗄웚좨;얷쪽웚좨,쫇횸횻쓜퓚떽웚죕닅쓜횴릤ퟷꆣ본폚웚좨뚨볛샭싛폫랽램퓚뷰죚춶쇬폲뎤탐뗄웚좨ꆣ퓚좫쟲뗄웚좨붻틗훐,뻸듳뛠쫽닉폃쏀듯20뛠쓪뗄맣랺펦폃뫍좡뗃뗄뻞듳돉릦,죰뗤믊쪽웚좨ꆣ볒뿆톧풺붫1997쓪억놴뛻뺭볃톧붱쫚폨솽캻쏀맺퓊탭쳡잰횴탐뗄쏀쪽뾴뗸웚좨뗄볛횵fA싺톧헟Scholes뫍Merton,틔뇭헃쯻쏇뛔헢튻샭싛뗄ퟣ:fAꇝmax(S-X,0),벴폐뛮췢뗄뇟뷧ꆪꆪꆪ붨솢벰펦폃쯹ퟶ돶뗄뾪뒴탔릱쿗ꆣ퓚쎿튻쪱볤짏폐튻풤쿈늻횪뗀뗄막욱볛룱,쯼뇪쏷Black-Scholes쒣탍릲짦벰5룶닎쫽,럖뇰쫇쇋솽룶쟸폲뗄뇟뷧,튻뇟펦룃돖폐웚좨,쇭튻뇟펦뇪뗄닺뗄볛룱S,횴탐볛룱X,컞럧쿕샻싊r,룃횴탐웚좨,맊MonteCarlo쒣쓢램늻쫊틋폚쏀쪽뻠떽웚죕뗄쪱쿞t,뇪뗄닺볛룱뗄늨뚯싊ꛒ,웤웚좨뗄뚨볛;평폚쏀쪽웚좨뗄슷뺶틀삵탔,틔벰뾴헇웚좨뗄볛횵캪:Black-Scholes웚좨뚨볛쒣탍쎻폐뾼싇웚좨뗄쳡잰C=SN(d1)-X-rteN(d2)ꆡ웤훐:횴탐,맊튲늻쫊폃폚쏀쪽뾴뗸웚좨뚨볛ꆣ뛾닦쫷뚨Sꛒ2볛쒣탍쯤좻쫊뫏룷훖웚좨맀볛,떫듦퓚뫜뛠좱쿝ln+r+t,X2죧뺫뛈컊쳢d1=,쯹틔퓚늽쫽N첫킡쪱,믡펰쿬쏀쪽ꛒt웚좨뚨볛뗄뺫뛈,N첫듳쪱,볆쯣솿뫜엓듳ꆲ1ꆳꆣ뿘Sꛒ2훆뇤솿벼쫵쫇튻훖랽닮쯵복랽램,샻폃뿘훆뇤솿벼ln+r-tX2쫵탞헽뛾닦쫷뚨볛쒣탍뚨볛뗄쏀쪽웚좨볛횵,뿉틔d2=ꛒ=d1-ꛒt쳡룟쏀쪽웚좨맀횵뗄뺫뛈,췪짆쏀쪽웚좨뗄뚨볛ꆣtN(x)캪뇪ힼ헽첬럖늼뗄샛볆룅싊럖늼몯쫽ꆣ룹뻝뾴헇뫍뾴뗸웚좨횮볤뗄욽볛맘쾵,뗃떽얷쪽뾴뗸웚좨뗄볛횵캪:쫕룥죕웚:2005-08-04ퟷ헟볲뷩:쇎ퟷ뫨(1970ꆪ),쓐,붭컷벪낲죋,커몺샭릤듳톧퓚뛁늩쪿,붭컷샭릤듳톧뢱뷌쫚,퇐뺿쇬폲캪쪵컯웚좨ꆢ뿳튵뺭볃뫍웳튵뺺헹솦ꆣꆤ784ꆤ' 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved.
P=X-rteN(-d2)-SN(-d1)ퟓ쮵쏷:짨쿬펦뇤솿ꛘ=f(X1ꆢX2ꆢꆭꆭ,Xn),폻뛔웤뻹횵E(ꛘ)=u뷸탐맀횵,캪듋헒튻룶폫3ꆡ뛾닦쫷뚨볛쒣탍ꛘ쿠맘뗄쯦믺뇤솿ꛘ3=f(X31,X32,ꆭꆭ,뛾닦쫷뚨볛쒣탍쫇Cox,Ross,Rubinstein1979X33i),웤훐f퓚쫽톧짏죝틗뒦샭,Xi폫X3i(i=쓪퓚JournalofFinancialEconomics짏랢뇭뗄튻1ꆢ2ꆢꆭꆭn)듦퓚헽쿠맘믲뢺쿠맘뗄쯦믺뇤솿,욪컄헂,뫳살Cox,Rubinstein퓚1985쓪뛔쯼뺫쾸퓲ꛘ3폫ꛘ튲믡듦퓚쿠맘탔,짨쪹폃릲춬쯦믺쫽뛔뮯,쪹쯼맣랺펦폃폚웚좨맀볛ꆣ쯼퓚쎿튻웚붫돶쿖솽훖뿉쓜탔뗄볙짨쿂릹훾뗄쿖뷰쇷믲쒳훖볛룱늨뚯Xi폫X3i뷸탐K듎쒣쓢뗃떽:W1=ꛘKꇆKi,W3i=1뗄쒣탍,쯼듓웚좨폐킧웚뗄쒩캲떹췆믘떽뾪쪼캻훃믱뗃웚좨뗄떱잰볛횵ꆣ틔뛾늽뛾닦쫷춼캪샽1=,죧춼KꇆKꛘi,퓲뛔죎튻쪵쫽a,벴캪u뗄컞욫맀볆,i=11쯹쪾:떱VarꆲE(ꛘ)ꆳ<VarꆲWꆳ쪱,ꛘ3뷐ꛘ뗄뿘훆뇤솿,웤훐a뷐ퟶ폅뮯좨쫽ꆣ톰헒뿘훆뇤솿뗄랽램뿉틔쫇춨맽뛔쒣탍뗄돵늽쫔퇩믲욾뺭퇩ꆣ412ꆡ뿘훆뇤솿벼쫵퓚쏀쪽웚좨뚨볛훐뗄펦폃퓚웚좨뚨볛훐,뿘훆뇤솿벼쫵쫊뫏폚솽훖퇜짺횤좯뗄쟩뿶,웤훐폐튻훖쫇쓜뗃떽뷢컶뗄횤좯,쟒폫쇭튻훖횤좯쿠쯆ꆣ뿉폃뿘훆뇤솿벼쫵살쳡룟뛔쏀쪽웚좨뚨볛뗄뺫뛈,웤뻟쳥닙ퟷ죧쿂:(1)폃춬튻룶뛾닦쫷춼럖뇰볆쯣돶쏀쪽웚좨뗄볛횵fA뫍폫웤뛔펦뗄얷쪽웚좨뗄볛횵fE(볙짨솽헟볆쯣뗄컳닮쿠춬)춼1ꆡ뛾늽뛾닦쫷춼볆쯣쒣탍(2)폃BlackꆪScholes릫쪽볆쯣돶얷쪽웚좨뗄막욱뗄돵쪼볛캪S,웤짏짽뫍쿂붵뗄놶쫽럖뇰볛룱fBSE;쫇uꆢd,P캪짏짽뗄룅싊(캱룅싊)(3)럂헕짏샽톡퓱폃BlackꆪScholes뚨볛쒣탍,1-P캪쿂붵뗄룅싊,뷚뗣짏,쿂뗄쫽뻝럖뇰쫇막욱뗄볛룱뫍쯣돶뗄fBSEퟷ캪뿘훆뇤솿,폅뮯좨쫽짨캪1,뿉뗃웚좨뗄볛횵,쿟짏믲쿂뗄쫽뻝캪막욱뗄볛룱뇤뮯돶쏀쪽웚좨뗄볛횵맀볆횵fBSA캪:fBSA=fA-뗄룅싊ꆣr캪컞럧쿕샻싊,ꚤt캪쎿룶떥늽뛾닦쫷춼(fE-fBSE)뗄쪱볤뎤뛈,퓲폃쿂쇐쪽ퟓ볆쯣웚좨뗄볛횵:fu=-rꚤt5ꆡ쯣샽e[Pfud+(1-P)fdd];fd=-rꚤte[Pfd+(1-P)fd];뾼싇튻룶늻뢶뫬샻막욱뗄쏀쪽뾴뗸웚좨,폐킧rꚤtf=-e[Pfu+(1-P)fd]웚캪1쓪,횴탐볛룱캪18쏀풪,막욱쫐볛캪20쏀뛾닦쫷램췆맣떽뛠늽쪱춬퇹춬짏쫶릫쪽샠쯆뗘풪,컞럧쿕샻싊캪15%,막볛뗄늨뚯싊캪쓪싊볆쯣돶웚좨볛횵ꆣ떫폃뛾닦쫷춼캪쏀쪽웚좨맀볛쪱40%,쿖내캪웚3룶퓂튻룶쪱뛎폃뛾쿮쪽쫷춼캪웤튪퓚쎿룶뷚뗣뒦볬퇩쳡잰횴탐쫇럱ퟮ병,퓚ퟮ뫳뷚맀볛,폃뿘훆뇤솿벼쫵캪웤탞헽ꆣ뗣뗄쏀쪽웚좨폫얷쪽웚좨볛횵쿠춬,퓚뷏퓧뗄튻킩뷢:S=20ꆡX=18ꆡr=0115ꆡꛒ=0140ꆡ뷚뗣,웚좨뗄볛횵쫇좡틔쿂솽헟횮뷏듳헟:ꚤt=0125,퓲:(1)폃릫쪽쟳돶뗄횵;(2)쳡잰횴탐쯹뗃뗄쫕틦ꆣu=ꛒꚤt104ꇁ0125e=0e=11221d=1/u=01819125a=rꚤte=0115ꇁ0e=11038ꆡ1/a=0196344ꆡ뿘훆뇤솿벼쫵벰웤퓚웚좨뚨볛훐뗄펦폃a-dp==015451ꆡꆡ1-p=014549411ꆡ뿘훆뇤솿벼쫵u-d뇤솿횮볤뻟폐쒳훖늻좷뚨탔뗄맘쾵돆캪쿠맘맘샻폃뛾닦쫷릫쪽뿉틔볆쯣돶룷룶뷚뗣뗄웚좨볛쾵ꆣ쓜뿘훆믲헟쯤좻늻쓜뿘훆떫쫇뿉맛닢뗄뇤솿뷐횵럖뇰뇪퓚쿂춼2ꆣퟶ뿘훆뇤솿믲풤놨뇤솿;폫뿘훆뇤솿폐쿠맘맘쾵뗄죧A뗣-0115ꇁ0125,f=eꇁ(4158ꇁ015451+뇤솿뷐ퟶ쿬펦뇤솿ꆣ91002ꇁ014549)=613503쯦믺쒣쓢쫇뛔몬폐뛠룶쯦믺뇤솿(x1ꆢx2퓚A뗣뛔쏀쪽웚좨살붲:18-10199=7101,ꆭꆭ,xn)뗄쾵춳,샻폃쒣탍살짺돉쯹튪뗄뷡맻,맊퓚A뗣톡퓱쳡잰횴탐,쏀쪽웚좨볛횵캪7101믲돆캪쿬펦뇤솿뗄맀횵ꛘ=f(x1ꆢx2ꆢꆭꆭ,퓚BꆢCꆢD뗣뿉폃춬퇹뗄랽램ꆣxn),쳡룟맀횵뺫뛈뗄랽램뫜뛠,뿘훆뇤솿벼쫵뻍AꆢBꆢCꆢD뗣뒦710100,415800,21469쫇웤훐튻훖,웤쮼슷쫇톰헒튻룶폫쿬펦뇤솿쿠맘뗄2,112846캪쏀쪽웚좨볛횵,웤쯼뗣fE폫fA뗄쯦믺뇤솿ퟷ캪뿘훆뇤솿,샻폃솽헟뗄쿠맘탔살믱뗃횵쿠춬ꆣ폃뛾닦쫷춼볆쯣돶뗄쏀쪽웚좨볛횵럖뇰랽닮뗄쯵킡ꆣ솽룶뇤솿x폫y뗄쿠맘쾵쫽rho=ꛑxy캪:=1믲헟rhoꇖ1,뇭쪾솽쯦믺뇤솿쫇헽쿠맘뗄ꆣꇠfE=112478ꆡꆡꆡfA=112846캪쇋룼횱맛뗘샭뷢뿘훆뇤솿램,쿖뻙튻볲떥샽폃BlackꆪScholes릫쪽볆쯣fBSE뗄횵캪:p=ꆤ785ꆤ' 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved.
ꆡꆡd2=d1-ꛒꚤt=018385-014=014385N(-d1)=N(-018385)=0120092;N(-d2)=N(-014385)=0133054폃뿘훆뇤솿벼쫵탞헽쏀쪽뾴뗸웚좨뗄볛횵캪:fBSA=fA-(fE-fBSE)=112846-(112478-117136)=117504쯹틔폃뿘훆뇤솿벼쫵볆쯣돶살뗄횵샫웤헦쪵횵룼볓뷓뷼ꆣ6ꆡ뷡쫸폯MonteCarlo쒣쓢뫍Black-scholes뚨볛쒣탍뻹늻쫊뫏쏀쪽웚좨뗄뚨볛,떱뛾닦쫷춼훐뗄ꚤt럖춼2ꆡ뛾닦쫷릫쪽볆쯣룷룶뷚뗣웚좨볛횵뗃뫜쾸쪱,쓜뗃돶뷏캪뺫좷뗄쏀쪽웚좨볛횵,떫쫇X-rꚤteN(-d2)-SN(-d1)=117136,웤훐:볆쯣솿뫜듳ꆣ폃뿘훆뇤솿벼쫵탞헽뗄뛾닦쫷뚨볛뗄ln(s/x)+(r+ꛒ2/2)ꚤt쏀쪽웚좨뗄볛횵,뿉틔룼볓뷓뷼쏀쪽웚좨뗄헦쪵볛d1=ꛒꚤ=횵,뿉췪짆쏀쪽웚좨뗄뚨볛ꆣ떫쫇튪힢틢뿘훆뇤솿t200142벼쫵뗄쪵훊쫇샻폃뛾닦쫷춼볆쯣얷쪽웚좨뫍쏀쪽웚ln+(0115+)ꇁ1182좨뗄볛룱횮볤뗄닮횵,뛸늻쫇쏀쪽웚좨볛룱놾짭ꆣ018385014ꇁ=1닎뾼컄쿗:ꆲ1ꆳJohnC1Hull1OptionsFutureandotherDerivatives3edꆲ5ꆳR1Merton1TheoryofrationaloptionspricingꆲJꆳ1ꆲMꆳ1NY:PrenticeHall,19971BellJournalofEconomicsandManagementScience,ꆲ2ꆳJ1HullandA1White1TheuseofthecontrolVariant1973,(2):141-1831OptionPricingꆲJꆳ,JournalofFinancialandQuantita2ꆲ6ꆳJ1CoxandS1Ross1Thevaluationofoptionforalterna2tiveAnalysis,1988,237-2511tivestochasticprocessesꆲJꆳ,JournalofFinancialEco2ꆲ3ꆳ튶훐탐,쇖붨훐1쫽샭뷰죚ꆲMꆳ1놱뺩:뿆톧돶냦nomics,1976,(2):145-1661짧,19981ꆲ7ꆳJ1Cox,S1Ross,andM1Rubinstein1Optionpricing:ꆲ4ꆳFisherBlackandMyronScholes1ThepricingofoptionAsimplifiedapproachꆲJꆳ,JournalofFinancialEco2andcorporateliabilitiesꆲJꆳ1JournalofPoliticEco2nomics,1979,229-2631nomic,1973,(81):637-6591AnApplicationoftheControl2variantApproachesintheAmericanOptionsPricingLIAOZuo2hong1,2,LIXiao2zhao2(11WuhanUniversityofTechnology,Wuhan,Hubei430070;21JiangXiUniversityofTechnology,Ganzhou,Jiangxi341000)Abstract:MostofoptionstransactionsareAmerican2typeoptionsintheworld1Duetopath2dependentcharacter,itresultsinmorecomplexAmericanoptionspricingthanEuropeanone1Black2ScholespricingmodelhasnoanalyticalformulaofAmericanputoptions,thusitcannotgetaccuratesolution1Someofrel2ativelymaturenumericalmethodscannotreasonablyestimateAmericanoptions1BecauseMonte2Carlosim2ulationisaforwardinductionprocedureandboundaryofAmericanoptionsisfree,Monte2CarlosimulationisunabletocorrectlyvalueAmericanoptions1Binomialtreecanbeappliedtopricingofallkindsofop2tions,butithasalotofflaws,suchasaccuracy1Sowhenthenumberofstepisverysmall,itaffectstheaccuracyoftheAmericanoptionspricing1Thecontrol2variantapproachesarevariancereductiontechniquesandimprovethevalueofAmericanoptionsvaluedbybinomialtreeandincreaseestimationaccuracy1Keywords:control2varianttechnique;americanoptionspricing;binomialtreevaluingmodel;Black2Schol2es(퓰죎뇠벭ꆡ뛅ꆡ쏴)ꆤ786ꆤ' 1994-2007 China Academic Journal Electronic Publishing House. All rights reserved.
