二叉树期权定价方法的一种新推广Ξ

26 3 2006 9 M A TH EM A T ICAL TH EORY AND A PPL ICA T ION S V o l. 26 N o. 3 Sep. 2006Ξ(, , 410083), .A New Expan sion of Binomal Op ition Pr ic ingHou M uzhou Zhou Yaoqiong(Schoo l of M ath. Sci & Comp. T ech. , Central- South U niversity, Changsha, 410083)AbstractA nalyzed the w idesp read used binom al op ition p ricing model currently, expanded a new binom alop ition p ricing model by the random erro r theo ry.Keywordsop ition p ricingrisk- neutral p ricingbinom al op ition p ricing model , . , , . .1 , S , , T f , T S S u, S S d ( : u > 1, d < 1). S u , f u; S d , f d.. , ∃ ., , , S u∃ - f u = S d ∃ - f d∃ = f u - f dS u - S dr , (S u∃ - f u) e - rT(1)Ξ (70471048): 2006 3 28

3 113, , q , 1 - q . n , S n S u j d n- j , j = 0, 1, , n, j , n -, S n = S u j d n- j C nq j j (1 - q) n- j., n , f = nj . , S n [C np j j (1 - p ) n- j m ax (0, S u j d n- j - X ) ] (e - rT ) n (7)j= 0 C n j = (n - j ) ! [n! j ! ], p .(7) , C np j j (1 - p ) n- j , S n = S u j d n- j ,m ax (0, S u j d n- j - X ) . , n , .(7) , j , S u j d n- j , j (j Ε m , m ) , S u j d n- j - X > 0, (7) f = n [C np j j (1 - p ) n- j (S u j d n- j - X ) ] (e - rT ) n , j= mp 3 = p uge rT (8) , 1 - p 3 = (1 - p ) d ge rT . , nnf = S [C np j 3 j (1 - p 3 ) n- j - X (e - rT ) n j= mj= m[C j np j (1 - p ) n- j ], f = SB (n, m , p 3 ) - X (e - rT ) n B (n, m , p ) (9) n , B (n, m , p ) = nj= m[C j np j (1 - p ) n- j ] (10), n, m , p. p , n m , X . B (n, m , p ) = P rob (j Ε m ) , m S u j d n- j > X j., - , . c c = f - S + X (e - rT ) n(9) , n c = - S [1 - B (n, m , p 3 ) ] + X (e - rT ) n [1 - B (n, m , p ) ] (11)u d , p , , u d Ρ , u, d . t, T , n, h = (T - t) gn, , r , R =ex p {rh}.u = exp { (r - Ρ 2 g2) h + Ρh 1g2 } (12)

114 26 d = exp { (r - Ρ 2 g2) h - Ρh 1g2 ) }, (13)n (h 0) , p = (R - d ) g(u - d ) 1g2u = exp {Ρh 1g2 } (14)d = exp {- Ρh 1g2 } (15) h , p p [1 + (r - Ρ 2 g2) gΡ}h 1g2 ]g2 (16)3 S ∃ t p 1 - p S u S d (u > 1, d < 1). , ∃ t S e r∃ t , S e r∃ t = pS u + (1 - p )S d (17)∃S = S Λ∃ t + S Ρ ∃ tΕ (18) Ε.∃ t Q = S + ∃S E (Q 2 ) - E = S 2 Ρ 2 ∃ t (19) (17) (19) S 2 Ρ 2 ∃ t = pS 2 u 2 + (1 - p )S 2 d 2 - S 2 e 2r∃ t (20)a = e r∃ t , (17) (20) a = p u + (1 - p ) d (21)Ρ 2 ∃ t = p u 2 + (1 - p ) d 2 - a 2 (22)p , u, d . . d= 1 u , (21) (22) ∃ t ( ) ∃ t ) , . d =1 u ,(14) (15) . ,: , d = 1 u u d ((24) ) , ; , Ρ, 1 . , r = 0, 1, 2, ∃ t = 0. 1, Ρ =0. 01, (5) p = 2. 4080, 1 - p = - 1. 4080, ..(21) p = (a - d ) g(u - d ) , (22) Ρ 2 ∃ t = p (u 2 - d 2 ) - (a 2 - d 2 )= (a - d ) (u + d ) - (a 2 - d 2 ) p , u, d = (a - d ) (u - a) (23)

3 115p = a - du - d , d = a - Ρ 2 ∃ tu - a , a = er∃ t (24) , u, p d . S , ∃ t S E (S + ∃S ) = S e r∃ṭ, u E (u) = e r∃ ṭ u ∃ t S , u(18) S + ∃S = 1 + Λ∃ t - Ρ ∃ tΕSu Ρ 2 (u) = Ρ 2 ∃ t. Ρ(u) u E (u) L 2 , , u u = E (u) Ρ(u). 0 Φ p Φ 1, (24) u Ε a = E (u). u = E (u) + Ρ(u). (24) p , u, d :p = 1 2 , u = a + Ρ ∃ t, d = a - Ρ ∃ t, a = er∃ t (25):S , p = 1 - p = 1 2 . (25) , , . , u d u d (13) , .[1 ]D ixit A. K, P indyck. R. S. T he op tion app roach to cap itial investm ent [J ]. H arvard Business R eview ,1995, 73: 105- 1151.[2 ]B lack F. Scho lesM. T he p ricing of op tions and co rpo rate liabilities[J ]. J Po l Econ, 1973 (81): 6376591[3 ] 1 1 120 (11): 90931[4 ] 1 1 , 20031[5 ] 1 1 , 20001[6 ] , , 1 ABS [M ]1 : , 19991[7 ]H ull J. Op tion. Futures and O ther derivative Securities[M ]1Second Edition, P rentice H all, 1993.[8 ] 1 1 , 19971