Microsoft PowerPoint - FE

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6 (I) p u d r t Se D = psu + -p Sd ( 1 ) ( 1 ) é ( 1 ) s D = ê + - ë 1 u = d S t ps u p S d S pu p d ù úû 2 6

7 (II) p u d e = u s = e = e rdt -d -d Dt -s Dt -D = é + ( 1 - ) ù êë úû f e r t pf p f u d 7

8 i + 1 j = 0,1,.i 8

9 T T Δt r Δt

10 50, 40% 10%

11 u d s Dt = e = -s Dt = e = rdt e -d p = = u -d 1- p =

12 12

13 N Δt (i, j) j N-j ( ) f = max X -Su d, 0, j Nj = 0,1, 2, ¼.. N ( ) { j i-j -rdt é p f ù} + 1, , f = max X - Su d, e pf êë ij i j i j úû 30 13

14 q ( ) r-q Dt ( ) r-q Dt Dt -s Dt ( 1 ) e = pu + -p d p u d = = e e = e s u -d -d 14

15 r p u d ( ) r-q Dt e -d 1 -d = = u -d u-d s Dt = e = e -s Dt 15

16 j i-j Su d j = 0,1... i ( d) j i-j S 1 - u d j = 0,1... i 16

17 S ( 1 - d ) i u d j i-j 0 17

18 18

19 ,, τ * -r( t-idt) S ( id t) = S( idt) -De * S ( id t) = S( idt) 19

20 iδt, 0 ( t ) * j i - j -r -idt + = 0, S u d De j i S S u d j i * j i-j = 0,

21 p =

22 p = 0.5 u p d p = 0.5 u d = e = e æ 2 ö s r q -- D+ t s Dt ç 2 çè ø æ 2 ö s r q -- D- t s Dt ç 2 çè ø Δt σ 22

23 Δt S u Su S d Sd 23

24 24

25 s 3Dt u = e 1 d = u 2 D t æ s ö 1 p =- r q d 2 12s ç çè 2 ø 6 2 D t æ s ö 1 p = r q u 2 12s ç çè 2 ø 6 2 p = m 3 25

26 Control Variate A B B A B A B A 26

27 f - f = f - f A B A B A A B A f = f + f - f B 27

28 Δt 28

29 BSM é êë (, -D ) = (, ) + ( 1 - ) (, ) r t f S t t pf Su t p f Sd t e -D ù úû 2 f f rs + s S f - rf + o 2 ( D t) = t S 2 S 0 29

32 ds = ( r - q) Sdt +ssdz N Δt ( ) ln ( ) æ 2 s ö ln S t +Dt - S t = r q ç - - D t + se Dt çè 2 ø éæ 2 s ö ù S( t +D t) = S( t) exp r q t se t ç - - D + D 2 êë çè ø úû 32

33 S(t) t S, ε N 33

34 ln S S S t + Δt = T ln S T 34

35 r = 14% σ = 20% Δt = 0.01 S D S = S Se 20 ε 0.52 D S = = ε 1.44 D S = =

36 36

37 ε ε ε 12 = Ri i= Excel NORM.S.INV(RAND()) 37

38 95% 1.96ω 1.96ω μ < f < μ+ M M M μ ω

39 M 10 39

40 40

42 2 f f 1 f rs + s S = rf 2 t S 2 S Grids 42

43 N N M M + 1 (M + 1)(N + 1) (i, j) iδt jδs f(i, j) (i, j) N = T Δt M = S max ΔS 43

44 44

45 f -f f -f f -f,, 和 DS DS 2DS ij, + 1 ij, ij, ij, - 1 ij, + 1 ij, -1 i+ 1, j fi, j f = t Δt æf -f f - f ö f ij, + 1 ij, ij, ij, S S f èç D D ø f + f -2f ij, + 1 ij, -1 ij, = = 2 2 S DS DS 45

46 B-S-M af + bf + cf = f j ij, - 1 j ij, j ij, + 1 i+ 1, j 1 1 ( ) a = r -q jdt - s j Dt j 2 2 b = 1 + s j D t + rdt j 1 1 ( ) 2 2 i = 0,1... N - 1, j = 0,1... M c =- r -q jdt - s j Dt j 46

47 , 47

48 T ( ) f = max X - S, 0 N, j T S = jd S j = 0,1... M T S = 0 f = X i = 0,1... N i,0 f = 0 i = 0,1... N im, 48

49 M 1 af + bf + cf = f j= 1,... M-1 j N-1, j-1 j N-1, j j N- 1, j+ 1 N, j j = 0 时, f = X N -1,0 j = M 时, f = N-1, M,, f 0 49

50 f f - f f f + f -2f = = S 2D S S DS 2 i+ 1, j+ 1 i+ 1, j- 1 i+ 1, j+ 1 i+ 1, j- 1 i+ 1, j, 2 2 PDE f = af + bf + cf * * * i, j i+ 1, j- 1 i+ 1, j i+ 1, j ( - ) D + s D * a = rdt 2 2 * 1 -s j Dt b = 1 + rdt ( r -q) jd t + s j Dt * c = rdt * * * 1 a + b + c = 1 + r D t 2 2 r q j t j t 50

51 a + b + c = + D * * * 1 1 r t 51

52 2 52

53 53

54 54

55 S ln 2 2 f æ s ö f 1 2 f + r - + s = rf 2 t çè 2 ø Z 2 Z Crank-Nicolson f + f = a f + b f + c f + a f + b f + c f * * * ij, i-1, j j i-1, j-1 j i-1, j j i- 1, j+ 1 j ij, - 1 j ij, j ij, + 1 Crank-Nicolson 2 2 ( D, D ) O t S 55

56 56

57 Any Questions?

Microsoft PowerPoint - FE11

Microsoft PowerPoint - FE11 - - 郑振龙陈蓉厦门大学金融系课程网站 http://efinance.org.cn Email: zlzheng@xmu.edu.cn aronge@xmu.edu.cn BSM BSM BSM Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU BSM BSM BSM Copyright 01 Zheng, Zhenlong & Chen, Rong,