샫즢쯣쫵욽뻹퇇쪽웚좨뷼쯆뚨볛 쯯볡잿 샮쪱틸 햪튪ꎺ놾컄뛔튻냣탎쪽쿂뗄뻹횵몯쫽뷸탐첩샕햹뾪ꎬ럖컶뇪뗄쯣쫵욽뻹폫뇪뗄벸뫎욽뻹뗄닮뻠ꎬ룸돶뛾헟횮볤뗄뷼쯆맘쾵쪽ꎬ뷸뛸뛔샫즢쯣쫵욽뻹퇇쪽웚좨뷸탐뷼쯆뚨볛ꆣ 맘볼듊ꎺ퇇쪽웚좨ꆢ웚좨뚨볛 톧뿆럖샠뫅ꎺ90A12, 49L25 1.틽퇔 퇇쪽웚좨쫇OTC쫐뎡짏맣쫜붻틗헟쟠뗄뷰죚릤뻟ꎬ웤떽웚쫕틦몯쫽틀삵폚쒳튻쳘뚨쪱뛎쓚뇪뗄닺뗄쒳훖탎쪽뗄욽뻹ꆣ쒿잰퓚OTC쫐뎡짏붻틗뗄뻸듳늿럖퇇쪽웚좨뚼쫇뇪뗄쯣쫵욽뻹ꎬ떫벴쪹퓚뇪뗄ퟱ톭벸뫎늼샊퓋뚯뗄볙짨쿂ꎬ죔좻쎻폐뷢컶뗄뚨볛릫쪽ꆣ뛔뇪뗄쯣쫵욽뻹퇇쪽웚좨뷸탐뚨볛룼뛠뗄쫇닉폃쫽횵랽램믲틔뇪뗄벸뫎욽뻹퇇쪽웚좨살뷼쯆뇆뷼ꆣBoyle(1977) 틔뇪뗄벸뫎욽뻹뗄퇇쪽웚좨ퟷ캪돵쪼횵닉폃Monte carlo뿘훆랽닮램룸돶뇪뗄쯣쫵욽뻹퇇쪽웚좨뗄뷼쯆뷢ꎬRutties(1990)ꎬKemna뫍Vorst(1990)튲닉폃Monte carlo쒣쓢뗄랽램ꎬTurnbull and Wakeman(1991)틽뷸뛔펦쯄뷗뻠쿠뗈뗄뛔쫽헽첬럖늼살뇆뷼뇪뗄쯣쫵욽뻹뗄럖늼ꎬ듓뛸뷸탐뚨볛ꎬLevy(1992)닉폃뛾뷗뻠뗄랽램ꎬVost(1992)붨솢쇋튻룶Black-Scholes뷼쯆릫쪽살뚨볛ꆣ놾컄뛔튻냣탎쪽쿂뗄뻹횵몯쫽뷸탐첩샕햹뾪ꎬ럖컶쯣쫵욽뻹폫벸뫎욽뻹뗄닮뻠ꎬ룸돶뛾헟횮볤뗄뷼쯆맘쾵쪽ꎬ뷸뛸뛔샫즢탍뇪뗄쯣쫵욽뻹뗄퇇쪽웚좨뷸탐뚨볛ꆣ 2. 뇪뗄벸뫎욽뻹퇇쪽웚좨뗄뚨볛 컒쏇볙짨뇪뗄 S 싺ퟣ럧쿕훐탔쿂뗄벸뫎늼샊퓋뚯 tdSt =mdt+sdw (1) tSt웤훐 , m=r−q,r캪컞럧쿕샻싊 , q캪쓪뫬샻싊 , s 캪쓪늨뚯싊 , w 캪뇪ힼWienert맽돌 쯦믺캢럖랽돌(1)뗄뷢캪ꎺ 2(m−s)(T−t)+s[w(T)−w(t)]ST=S2 te (2) 캪쇋뇣폚볆쯣뫍뇜쏢붫맽좥뗄뇪뗄볛룱ퟷ캪쯦믺뇤솿ꎬ컒쏇볙짨t캪떱잰쪱뿌ꎬt(0≤i≤n)캪뇪뗄볛룱뗄맛달좡퇹쪱뿌ꎬt=t0+i∆t,(i=1,2n)∆=T−t0,t캪솽듎맛달좡퇹뗄iin쪱볤볤룴 , tn=T캪떽웚쪱뿌ꆣ볇t=0+i∆t,(i=12,L,n);0=t0−t ꎬ컒쏇뷶뾼싇 (t<t0) i뗄쟩뿶 , (t≥t0) 뿉ퟷ샠쯆뗄럖컶ꆣ평(2)뿉틔뗃떽ꎺ 2()()[()(2 =m−sti−t+swt−wSSeit)]it (3) 뇪뗄벸뫎욽뻹볇캪M(0), nM1 (0=∏Sn)() ii=1뚨틥ꎺ
SR=i ,i=1,2n iSi−1평(3)죝틗뗃돶ꎺ ss2 R=m−−−+sww2ln()(t1)[(t)(t−1)]:N[(m−)∆t,s∆t] iiii2∏1MnSn(0)()iSSnS===nn110121 [()()]n SSSStttn−1Sn−2S0M(0)S=01 lnln+(lnRn+2lnRn−1+nlnR1),t<t0 SSntt평lnR,=1,2Ln,뗄뛀솢탔뿉뗃ꎺ iEM(0)2[ln]=(m−s)[(−)+n+1t0t(T−t)] S20tVan(02−(n1)(2n1)r[l]=s++[(t)2(T−t)] St6n볇t=T−t,늢쇮 mM,sM 럖뇰캪뇪뗄벸뫎욽뻹M(0)뗄욯틆쾵쫽폫삩즢쾵쫽ꎬ퓲ꎺ s2t=VarM(0)M[ln] St2m−sMt=M(0)(M)E[ln] 2St쓇쎴ꎬ퓚떱잰쪱뿌t뇪뗄볛룱틑횪ꎬ쪱뿌$T$뗄M(0)뗄틆쏜뛈몯쫽캪ꎺ s2−{M−S+m−MtMt}2ln(0)[ln()]222j(Mst(0)1,S)=eM tM(0Πs2)2M평럧쿕훐탔웚췻헛쿖뗄랽램틗뗃ꎬ뇪뗄벸뫎욽뻹얷쪽뾴헇웚좨뗄뚨볛캪ꎺ C−rtK+e−rt(,,)=[(0)−]=[SemMtGMSKteEMN(d1)−KN(d2)](4)2lnSst+(mM+M)=Ktd21,d2=d−stsMt1M 웤훐K캪붻룮볛룱ꎬ샠쯆뗄췆떼뿉뗃뇪뗄벸뫎욽뻹얷쪽뾴뗸웚좨뗄뚨볛캪ꎺ P=e−rt()[KN(−d2)−SemMtGMS,K,tN(−d1)] (5) 웤훐 d1,d2 뗄뚨틥춬짏 3. 뇪뗄벸뫎욽뻹폫뇪뗄쯣쫵욽뻹뗄뷼쯆맘쾵 튻냣탎쪽뗄뻹횵몯쫽캪 : M1n1 (q)=(∑Sqq) ini=1떱q=1,Mn(1)=1∑S,벴뗃떽뇪뗄쯣쫵욽뻹ꆣ평싥뇘듯램퓲틗뗃 n=1ii1nn11n M(0)=limM()=lim(∑qSq)(∏S) iiq→q0q→0ni=1i=1벴캪잰쏦쯹쳖싛뗄뇪뗄벸뫎욽뻹ꎬ평 M(q) 뗄뿉캢탔ꎬ붫 M(q) 퓚 q=0 ퟷ첩샕햹뾪ꎺMM′(0)02′′′()=+3()()MMMxq′(0q+q+q,0<x<1 2!3!좡q=1,퓲 : MMMM′(0)M′′′(x)(1)=(0)+′(0)++,0<x<1 ꎨ6ꎩ 23!쿂쏦럖뇰볆쯣M′(0),M′′(0):
nq∑n∑nSqSq1iiMMlnS−(Sq∑=)(ln)1ii1in (q==′)=() 2∑nqSqi=1i뚨틥 N(q)캪 nq∑nnSqSqS−∑Sq∑=1iiln()(ln)N1ii1in (q)=== q2∑nSqi=1i퓲ꎬM′(q)=M(q)N(q) 1nnN()[{SqS2q′(ln)][S]−nq∑q2∑ii∑i[S]=1=1=1ii n nnn2Sqq∑∑∑∑∑SSq1Slniii[lnS][Sq+qSqlnS]}+=1−=1[ln]iiiii32n1=1i=1n∑Sq=qi=1iM2′(q)M(q)N(q)+M(q)N′(q) ∑21(lnS)∑lnSN=112(0)li→mNii(q=i=−i=)[()](7) q02M′(0)=limM(q)=M(0)N(0) (8)q→ 0∑n32(lnS)N131=Nq=i=i2′(0)lim′(+lnS(lnS)(lnS)iiiq→∑−∑∑ (9) 03nn=n2i1i=1=1M2′(0)=limM′(q)=M(0)N(0)+M(0)N′(0) (10) q→0붫(7),(8),(9),(10) 듺죫(6),짡좥룟뷗쿮ꎬ뗃돶ꎺM1≈M+N+N21(1)(0)[1(0)(0)+N′(0)] 22웤훐ꎬM(1),M(0) 벰1+N(0)+1N2(0)+1N′(0) 뻹캪뇪뗄닺S,=1,2,L,n 뗄몯22i쫽ꎬ틲듋쯼쏇쫇쯦믺뇤솿ꎬ볇L=E+N+1N2[1(0)(0)+1N′(0)], 쿂쏦횤쏷221+N(0)+1N2(0)+1N′(0) 뿉틔폃쯼뗄뻹횵L 듺쳦ꆣ튲벴ꎺ 22 M(1)≈M(0)L 캪듋쿈볆쯣죽헟뗄웚췻ꎬ컒쏇쿈뛔뛠쿮쪽뗄룟듎랽뷸탐럖뷢ꎬ틔볲뮯볆쯣ꎺ 21∑(lnS)∑lnSN(0=i=1i−i=1i2)[()]21n−1n=n∑22 {(lnS)−[∑(lnS)+2∑∑lnSlnS]} 2iiij2ni=1i=1i=1j=i+11n−1nS=∑∑i2[ln]22i=1j=i+1j
∑n32(lnS)Ni=1i31SS2′(0)=+∑ln−(∑ln)(∑lnS)i2iini=1i=1=12n−ni−=∑3S+∑2S+∑∑2[ln3lnln3lnSlnSiijij3n3i=1i=1j=+i=2j=1 n−2n−1∑∑163lnSlnSln][lnS]ijkii=1j=i+1k=3n1n−1nni−1−[∑∑223lnSlnS+∑lnSlnS+ln]ijij∑Sin2i=1j=i+1i=2ji=1볇m=lnS+(m−s)(t−t)=lnS+(m−s)(t+i∆t),평(3)뿉뗃 itit0Si ln=(m−m)+s[w(t)−w(t)] SijijjSi2=m−m2+m−mswt−wt+s2w2 [ln]()2()[()()][(t)−w(t)] SijijijijjlnSlnSlS=mmm+wmm+wmm+wmm+mww+mww+mww+www(11)ijkijkijkjikkijijkjikkijijk 평Wiener맽돌뗄욽컈탔뫍뛀솢퓶솿탔뿉뗃ꎺ ESi222[lnS]=(m−m)+s(t−t)(i<j) ijjijVaSi22242[lnS]=4s(m−m)(t−t)+2s(t−t)(i<j) ijjijijE[lnSlnSlnS]=mmm+mmin(t,t)+mmin(t,t)+mmin(t,t)ijkijkijkjikk ij뷸뛸뿉틔뗃떽ꎺ n−1n222∑∑S(n−1)∆t1sE[Ns(0)]Ei[ln]=[(m−)∆+] (12) 2=1=+11222ijij1n22(3−2)(1s223n1Va42r[(0)][(m−)s∆t+s∆t] (13) 415362nnni−EN′=∑E3[(0)][lnS+3∑ElnSlnS3∑∑ElnSlnSiijij3n3i=1i=1j=+i=2j=1n−2n−1∑∑16ESSSE3lnlnln][lnS]ijki=1=+1=3n ijik1nni−1−∑S2S+∑ES2S+∑E3[ElnllnlnlnS]n2ijijii=1j=+i=2ji=1=0붫EN2[(0)],E[N(0)],E[N′(0)]L=E[1+N(0)+1N(0)1 듺죫2+N′(0)], 뗃 : 22 n2−∆sntn2−2s2n2(1)1(32)(111223−L=+m−∆42[()t+][(m−)s∆t+s∆t]12281536 1(n2−122)∆t1ss+{[(m−)∆t+]}21222n퓚짏쪽훐,n,∆t,m,s 캪좷뚨뗄뎣쫽ꎬ뷸튻늽뿉틔뻹횤쏷Var[1+N(0)+1N2(0)+1N′(0)]→0( 떱n→∞)횤쏷맽돌죧쿂ꎺ 22
평 schwarz 늻뗈쪽틗뗃ꎺnnn1nVar(∑X)∑Cov[X,]≤∑(VarX+VarX)=n∑VarX(14)iij2ijii=1i=1ji=1=1 맊ꎬ 12121Var[1+N(0)+N(0)+N′(0)]≤3[VarN(0)+VarN(0)+VarN′(0)] 242퓚 (13) 훐 ∆=T−t0, t,캪퇇쪽웚좨뗄짺쏼웚,쫇만뚨뗄쪱볤뎤뛈ꎬ맊떱n→∞쪱ꎬnlimVarN(0)0n→∞= 평ꎨ12ꎩꎨ14ꎩ컒쏇뿉틔뗃떽ꎬ VarN=EN4−EN22lim(0)lim{(0)[(0)]}n→∞n→∞=→∞EN−EN+EN4−EN22lim{[(0)(0)(0)][(0)]}n22422li→m∞{3[VarN(0)]6VarN(0)[EN(0)]+E[EN(0)]−[EN(0)]} nT−tsT−tT−tsTt={0m−0+s24−0m−0+s24[()]}{[()]}12221222=0평ꎨ3ꎩꎨ14ꎩ춬샭뿉췆떼ꎺ ∑S3∑3S∑S3(ln)ln(ln)∑lnSiiiiVarN'≤Vari=1+Vari=1+Vari==(0)211[][]3[] 3→0훕짏쯹쫶ꎬ컒쏇뿉틔뗃떽뇪뗄쯣쫵욽뻹폫뇪뗄벸뫎욽뻹뗄뷼쯆맘쾵쪽ꎺ M(1)=M(0)L 웤훐L쫇튻뎣쫽ꎬM(1)뫍M(0)쫇쯦믺뇤솿ꆣ 4. 뇪뗄쯣쫵욽뻹퇇쪽웚좨뚨볛릫쪽 평뷼쯆맘쾵쪽(15),컒쏇뿉틔뛔뇪뗄쯣쫵욽뻹뗄퇇쪽웚좨뷸탐뷼쯆뗄뚨볛ꎬ폃M(0)Lퟷ캪M(1)뗄뷼쯆ꎬM(0)ퟱ톭뛔쫽헽첬럖늼ꎬL캪뎣쫽ꎬ틲듋뿉틔췆떼뇪뗄쯣쫵욽뻹얷쪽뾴헇웚좨뚨볛릫쪽ꎺ C(,K,)=e−rtE[M(1)−K+]≈e−rtAMSttE[LM(0)−K+] =e−rtmt[LSeMN(d1)KN(d2)]tLSs2lnt+m+M()tKMd21,d2=d1−sMt sMt샠쯆뗄췆떼ꎬ뿉뗃뇪뗄쯣쫵욽뻹얷쪽뾴뗸웚좨뚨볛릫쪽캪ꎺ PSK=e−rEK−M+≈e−rtAM(t,,t)[(1)]E[K−LM+(0)] =e−rt[KN(−d)−LSemMt2N(−d1)]t웤훐ꎬd1ꎬd2뚨틥춬짏ꆣ
5. 쳖싛 퓚OTC쫐뎡짏ꎬ뛔폚맛달좡퇹듎쫽n뇈뷏듳뛸맛달볤룴∆t뇈뷏킡뗄쯣쫵욽뻹퇇쪽욽뻹웚좨ꎬ죧1룶퓂웚쎿쳬좡퇹믲3룶퓂웚룴쳬좡퇹뗄퇇쪽웚좨ꎬ뻹뿉틔짏쫶뚨볛릫쪽살뷸탐뷼쯆뚨볛ꆣ놾컄횻쳖싛t<t0뗄쟩뿶ꎬ뛔폚t≥t0뗄쟩뿶뿉ퟷ샠쯆뗄럖컶 The approximate analytic price formulas of Asian option with discrete arithemetic averaging JianQiang Sun and ShiYin Li Department of Mathematics,Xiamen University,Fujian Xiamen 361005,China Abstract ꎺIn this papers,we analyze the distance between the geometric average and arithmetic average by Taylor’s expanding. Using their relationship,a approximating formula of the arithmetic asian option was derived. key words: Asian option, option
