u τ = 2 u x 2 u(x, 0) = max[e ( 2r σ 2 1)x/2 e ( 2r σ 2 +1)x/2, 0] lim u(x, τ) = x lim u(x, τ) =0 x 1 u(x, τ) V (S, t) V = E 1 2 (1+k) S 1 2 (1 k) e 1

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Download "u τ = 2 u x 2 u(x, 0) = max[e ( 2r σ 2 1)x/2 e ( 2r σ 2 +1)x/2, 0] lim u(x, τ) = x lim u(x, τ) =0 x 1 u(x, τ) V (S, t) V = E 1 2 (1+k) S 1 2 (1 k) e 1"

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1 , Black Scholes Black Scholes Black Scholes Black Scholes Black Scholes ( ) (1 ) Black Scholes Black Scholes Black Scholes C t σ2 S 2 2 C C + rs rc =0, (1) S2 S C(0,t)=0, C(S, t) S as S, C(S, T ) = max(s E, 0) x = log S E u(x, τ) = e( σ2,τ = (T t), 2 2r 2r σ2 1)x/2+( σ 2 +1)2 τ/4 E C(S, t) 1 1

2 u τ = 2 u x 2 u(x, 0) = max[e ( 2r σ 2 1)x/2 e ( 2r σ 2 +1)x/2, 0] lim u(x, τ) = x lim u(x, τ) =0 x 1 u(x, τ) V (S, t) V = E 1 2 (1+k) S 1 2 (1 k) e 1 8 (k+1)2 σ 2 (T t) u ( log (S/E), 1 2 σ2 (T t) ) k = r/ 1 2 σ2 2 Taylor u/ τ u u(x, τ + δτ) u(x, τ) (x, τ) = lim τ δτ 0 δτ δτ 0 δτ u u(x, τ + δτ) u(x, τ) (x, τ) + O(δτ) (2) τ δτ u/ τ (fiite-differece approximatio) u (forward differece) τ O(δτ) δτ 2 O(δτ) u(x, τ + δτ) (x, τ) Taylor u(x, τ + δτ) (x, τ) Taylor (2) 0 λ 1 u(x, τ + δτ) u(x, τ) δτ = u τ (x, τ)+ 2 u (x, τ + λδτ) δτ τ2 (3) 3 u(x, τ + δτ) u(x, ) τ δτ) (x, τ) Taylor (4) (5) O ((δτ) 2 ) 4 (7) O ((δx) 2 2

3 5 (11) 1 ( ) u u(x, τ) u(x, τ δτ) (x, τ) = lim τ δτ 0 δτ u u(x, τ) u(x, τ δτ) (x, τ) + O(δτ) (3) τ δτ u/ τ (backward differece) u u(x, τ + δτ) u(x, τ δτ) (x, τ) = lim τ δτ 0 2 δτ u u(x, τ + δτ) u(x, τ δτ) ( (x, τ) + O (δτ) 2) (4) τ 2 δτ (cetral differeces) 1 3 ( δτ ) 1 ( u/ τ (explicit) (fully implicit) (4) ( ) u τ u(x, τ + δτ/2) u(x, τ δτ/2) δτ ( + O (δτ) 2) (5) Crak Nicolso u x u u(x + δx, τ) u(x δx, τ) ( (x, τ) + O (δx) 2) (6) x 2 δx 1 τ t (4) x S (6) 2 2 u/ x (symmetric cetral-differece) 2 u u(x + δx, τ) 2u(x, τ)+u(x δx, τ) ( (x, τ) x2 (δx) 2 + O (δx) 2) (7) 2 u/ x 2 2 x x 3

4 4

5 2: 3: 3 x- δx (odes) τ δt (x, τ) (mesh poits) 2 ( δx, m δτ) u(x, τ) ( δx, m δτ) 3 u(x, τ) ( δx, m δτ) u m = u( δx, m δτ) (8) 5

6 4: 4 Black Scholes u τ = 2 u x 2 u(x, τ) u (x, τ), u(x, τ) u (x, τ) (x ± ) u(x, 0) = u 0 (x) (9) u (τ), u (τ), u 0 (x) u u/ τ (2) 2 u/ x 2 (7) u m δτ + O(δτ) = um +1 2um + um 1 (δx) 2 + O((δx) 2 ) (10) O(δτ) O ( (δx) 2) = αu m +1 +(1 2α)um + αum 1 (11) α = δτ (δx) 2 (12) ((10) (11) ) m u m um+1 4 u m +1 um, u m 1 4 (11) u m m 1 α (1 2α) 6

7 x δx x <x< x N δx x N + δx N N σ2 T M δτ = 1 2 σ2 T/M N <<N + 0 <m M (11) (9) u m N u m + N u m N = u (N δx, mδτ), 0 <m M, u m N + = u (N + δx, mδτ), 0 <m M (13) (9) u 0 = u 0 (δx), N N + (14) u m Black Scholes Black Scholes x ± u 1 Black Scholes (??) ) δx δτ α δτ α =025 α =05 α =052 (stability) (fiite precisio computer arithmetic) (11) (11) (umerical solutio) (roudig errors) (11) (stable) (11) (ustable) (11) 0 <α 1 ( ) 2 2 C C 7

8 explicit_fd( values,dx,dt,m,nplus,nmius ) a = dt/(dx*dx); for( =Nmius; <=Nplus; ++ ) oldu[] = pay_off( *dx ); for( m=1; m<=m; ++m ) tau = m*dt; ewu[nmius] = u_m_if( Nmius*dx,tau ); ewu[ Nplus] = u_p_if( Nplus*dx,tau ); for( =Nmius+1; <Nplus; ++ ) ewu[] = oldu[] + a*(oldu[-1]-2*oldu[]+oldu[+1]); for( =Nmius; <=Nplus; ++ ) oldu[] = ewu[]; for( =Nmius; <=Nplus; ++ ) values[] = oldu[]; 5: a = α, tau = τ, Nmius = N, Nplus = N + u m oldu[] ewu[] u 0 oldu[] oldu[] values[] 8

9 S α =025 α =050 α = : Black Scholes E = 10, r =005, σ =020 6 α> 1 2 α> 1 2 ( ) 7 ( 1 2α ) 0 < δτ (δx) x x 2 8 δx 0 δτ 0 u m u( δx, m δτ) 9

10 6: 5 0 <α 1 2 x 1 LU SOR 1 O(2N) O(4N) N x Crak Nicolso 6 (implicit fiite-differece method) u/ τ (3) 2 u/ x 2 (7) u m δτ + O(δτ) = um um (δx) 2 + O ( (δx) 2) O(δτ) O((δx) 2 ) α 1 +(1+2α)um+1 α +1 = um (15) (12) α (15) 1, um+1 +1 um 6 x = N δx x = N + δx N N + 10

11 u 0 (14) um+1 N N (4) (15) + m 0 N <<N + (m, ) (15) 1+2α α 0 0 α 1+2α α 0 0 α α 0 0 α 1+2α N +1 0 N + 1 = u m N +1 u m 0 u m N + 1 +α N 0 0 N + = (15) = N + 1 (1 + 2α) N + 1 αum+1 N + 2 = um N αum+1 N + (16) b m N +1 b m 0 b m N + 1 (16) M = b m (17) b m (N + N 1) =( N +1,,um+1 N + 1 ), bm = u m + α(, 0, 0,,0, ) N N + (16) M (N + N 1) α 0 M 8 α α 0 M M 3 M M 1 =M 1 b m (18) u m b m u 0 M (tridiagoal) (super-diagoal) (sub-diagoal) M M 1 2 N M 1 N 2 M 3N 2 M O(N 2 ) M 1 b m O(N 2 ) M (17) O(N) (4N ) 2 LU SOR 11

12 61 LU LU M L U M=LU 1+2α α 0 0 α 1+2α α 0 α 0 = α 0 0 α 1+2α y N +1 z N l N y N zn l N y N + 1 l, y z (19) 2 y N +1 = (1+2α), y = (1+2α) α 2 /y 1, = N +2,,N + 1 (20) z = α, l = α/y, = N +1,,N + 2 (19) (21) y,= N +1,,N y 1 y > 1 M = b m L(U )=b m q m 2 Lq m = b m, U = q m l z (21) α 1 0 q m N +1 b m N +1 y N +1 0 α q m N +2 b m N 0 +2 y N = (22) +2 0 qn m 0 0 α + 2 b m N + 2 qn m b m N + 1 y N

13 y N +1 α y N +2 α yn + 2 α y N + 1 N +1 N +2 N + 2 N + 1 = q m N +1 q m N +2 q m N + 2 q m N + 1 (23) q m = N +1 q m qm 1 qm N +1 q m qm 1 q m q m N +1 = bm N +1, qm = b m + αqm 1 y 1, = N +2,,N + 1 (24) q m (23) u m um+1 N + 1 u m N + 1 = qm N + 1 y N + 1, = qm + αum+1 +1, = N + 2,,N + 1 (25) y (21), (24), (25) LU (21) y 3 b m (24) q m (25) 7 ( Black Scholes ) 10 LU 62 SOR LU 1 (17) (iterative method) ( ) 3 M b m M = b m y 13

14 lu_fid_y( y,a,nmius,nplus ) asq = a*a; y[nmius+1] = 1+2*a; for( =Nmius+2; <Nplus; ++ ) y[] = 1+2*a - asq/y[-1]; if (y[]==0) retur(singular); retur( OK ); lu_solver( u,b,y,a,nmius,nplus ) /* Must call lu_fid_y before usig this */ q[nmius+1] = b[nmius+1]; for( =Nmius+2; <Nplus; ++ ) q[] = b[]+a*q[-1]/y[-1]; u[nplus-1] = q[nplus-1]/y[nplus-1]; for( =Nplus-2; >Nmius; -- ) u[] = (q[]+a*u[+1])/y[]; 7: 3 LU a = α, asq =α 2, Nplus = N +, Nmius = N, b[] =b m, q[] =q m, u[] =, y[] =y Nmius + 1 Nplus 1 Nplus Nmius lu solver lu solver y[] 14

15 4 SOR Successive Over-Relaxatio 5 SOR Gauss Seidel Gauss Seidel Jacobi SOR 2 3 (15) ( (16) (17)) 1 ( ) = b m 1+2α + α(um um+1 +1 ) (26) (15) (16) (17) Jacobi N +1 N + 1 ( u u m ) (26) ( Jacobi,k k,0,k,k+1 = k,k,k+1 (26) 1 ( ) b m + α(,k 1 +,k +1 ), N <<N + (27) 1+2α,k+1,k 2 = ( u m+1,k+1 ),k 2,k+1 α>0 Gauss Seidel Jacobi (27),k+1,k+1 1 Gauss Seidel,k 1 Gauss Seidel Jacobi (27),k+1 = 1 1+2α ( b m + α(um+1,k+1 1 +,k +1 ) ), N <<N + (28) Gauss Seidel Jacobi Jacobi Gauss Seidel (,k,k+1 ) Gauss Seidel Jacobi Gauss Seidel Gauss Seidel Jacobi ( ) Gauss Seidel α 4 LU 4N SOR 4N ( ) 5 ( ) 15

16 SOR_solver( u,b,nmius,nplus,a,omega,eps,loops ) loops = 0; do error = 00; for( =Nmius+1; <Nplus; ++ ) y = ( b[]+a*(u[-1]+u[+1]) )/(1+2*a); y = u[]+omega*(y-u[]); error += (u[]-y)*(u[]-y); u[]=y; ++loops; while ( error > eps ); retur(loops); 8: SOR a = α, Nplus = N +, Nmius = N, b[] =b m, u[] =,k,,k+1, y = y m+1,k+1, omega = ω eps u[ 1], u[ 2],k+1 1,,k+1 2, u[ + 1], u[ + 2],,k +1,,k +2,,k+1,k eps u[ ] SOR Nmius + 1 Nplus 1 Nplus Nmius loops omega 16

17 SOR Gauss Seidel,k+1 =,k +(,k+1,k ),k k (,k+1,k ),k k,k Gauss Seidel SOR y m+1,k+1 1 ( ) = b m + α(,k+1 1 +,k +1 ) 1+2α (29),k+1 =,k + ω(y m+1,k+1,k ) ω>1 (over-correctio parameter, over-relaxatio parameter) (y m+1,k+1 Gauss Seidel,k+1 SOR y m+1,k+1,k,k+1,k ) SOR α>0 0 <ω<2 (15) (0 <ω<1 (over-relaxatio) (uder-relaxatio) 1 <ω<2 ω ω ) ω SOR ω 8 SOR 63 (17), ( (15) (16)) 61 LU SOR LU 9 SOR E =10 σ =04 r =01 Black Scholes x α α =05, α =10, α =50 α> 1 2 (α > 1 2 ) α>0 17

18 implicit_fd1( values,dx,dt,m,nmius,nplus ) a = dt/(dx*dx); for( =Nmius; <=Nplus; ++ ) values[] = pay_off(*dx); lu_fid_y( y,a,nmius,nplus ); for( m=1; m<=m; ++m ) tau = m*dt; for( =Nmius+1; <Nplus; ++ ) b[] = values[]; values[nmius] = u_m_if( Nmius*dx, tau ); values[ Nplus] = u_p_if( Nplus*dx, tau ); b[nmius+1] += a*values[nmius]; b[ Nplus-1] += a*values[ Nplus]; lu_solver( values,b,y,a,nmius,nplus ); 9: LU M a = α lu solver ( ) lu fid y values[] lu solver b[nmius + 1] b[nplus 1] 18

19 implicit_fd2( values,dx,dt,m,nmius,nplus ) a = dt/(dx*dx); eps = 10e-8; omega = 10; domega = 005; oldloops = 10000; for( =Nmius; <=Nplus; ++ ) values[] = pay_off(*dx); for( m=1; m<=m; ++m ) tau = m*dt; for( =Nmius+1; <Nplus; ++ ) b[] = values[]; values[nmius] = u_m_if( Nmius*dx, tau ); values[ Nplus] = u_p_if( Nplus*dx, tau ); SOR_solver( values,b,nmius,nplus,a,omega,eps,loops ); if ( loops > oldloops ) domega *= -10; omega += domega; oldloops = loops; 10: SOR M a = α eps values[] SOR solver b[nmius + 1] b[nplus 1] SOR 19

20 S α =050 α =100 α = : Black Scholes E = 10, r =01,σ =04 3 α = Crak Nicolso Crak Nicolso O((δτ) 2 ) ( O(δτ) ) Crak Nicolso u m δτ + O(δτ) = um +1 2um + um 1 (δx) 2 + O ( (δx) 2) u m δτ + O(δτ) = um um (δx) 2 + O ( (δx) 2) 2 u m δτ + O(δτ) = ( 1 u m +1 2u m + u m 1 2 (δx) 2 (30) + um+1 +1 ) 2um (δx) 2 + O ( (δx) 2) (31) (31) O(δτ) O((δτ) 2 ) Crak Nicolso 1 2 α(um+1 1 2um ) (32) = u m α(um 1 2um + um +1 ) 20

21 α = δτ/(δx) 2, 1 um+1 +1 u m, um +1, um 1 (15) m u m (33) Z m Z m =(1 α)u m α(um 1 + u m +1) (33) (1 + α) 1 2 α(um um+1 +1 )=Zm (34) 2 (15) x = N δx x = N + δx N N + (14) u 0 (4) um+1 N N + 21

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