Microsoft PowerPoint - FE11 - PDF Free Download

- - 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, XMU 3

Itô BSM BSM ds = msdt + ssdz æ f f 1 f ö f df = ms s S ç + + dt ssdz + ç è S t S ø S f f 1 f + rs + s S = rf t S S ( ) ( t) -rt- ( ) c = SN d -Xe N d 1 Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 4

BSM BSM BSM Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 5

Brownian Motion 1 Δt Δz Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 6

( ) ( ) ZT - Zt = e Dt ( ) ( ) i= 1 ZT - Zt 0 T t n å i T - t Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 7

1 EMH Quadratic Variation Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 8

x dz dx = adt + bdz () = + + () x t x at bz t 0 adt a x a bdz x x b b b Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 10

D x = ad t + be Dt Δx adt b Dt b Dt T x at b T bt Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 11

Itô Process ( ) 0 (, ) (, ) dx = a x t dt + b x t dz x t = x + ò ads + ò bdz t 0 0 t dz a b x t x a b Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 1

Geometric Brownian Motion ds = msdt + ssdz μ σ Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 13

Itô Lemma x x t G æ G G 1 G ö G dg = a b + + dt + bdz ç è x t x ø x dz Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 14

G G G 1 G D G = D x + D t + Dx x t x G 1 G + DxD t + D t + x t t Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 15

Δt G G D G = D x + Dt x t G G 1 G D G = D x + D t + Dx x t x Δt Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 16

Δx Δt D x = ad t + be Dt G G 1 G G x t b e t D = D + D + D x t x e ~ j 0,1, e = 0, e - é e ù ú = 1 û E E t t ( ) ( ) ( E E ) êe( ) ë ( ) ( e = 1, 因此 e D ) = D ε Δt Δt D = D + D + D x t x G G 1 G G x t b t Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 17

G G 1 G dg = dx + dt + b dt x t x dx = adt + bdz æ G G 1 G ö G dg = a b + + dt + bdz ç è x t x ø x Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 18

x G(x, t) Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 19

11.1 ln S S G = ln S G 1 G 1 G =, =-, = 0 S S S S t G = ln S æ s ö dg = d ln S = m ç - dt + sdz çè ø ds = msdt + ssdz dln S μ dt s dt ds dln S ¹ S σ Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 0

11. F S F = Se ( t) rt ds = msdt + ssdz F rt ( -t) F F e, 0, S t = = =-rf S ( m ) df = - r Fdt + sfdz r Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 1

I ds = msdt + ssdz æ s ö dg = d ln S = m ç - dt + sdz çè ø Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU

II S T t ln ST - ln S - t η h = T éæ s ö s ù h ~ j m, ç - ê ë çè ø T - t úû σ Volatility Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 3

III éæ s ö ù ln S -ln S ~ j m ( T t), s T t T ç - - - ê ë çè ø úû é æ s ö ù ln S ~ j ln S m ( T t), s T t T + t ç - - - ê ë çè ø úû m( T-t) E( ST ) = Se ( T-t) é s ( ) ( ) T-t ù = -1ú ë û m Var S S e e T ê μ Δt Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 4

11.3 I A 50 18% 0% 6 6 S T A 6 Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 5

11.3 II S = 50, μ = 0.18, σ = 0., T t = 0.5 6 S T 0.04 ln S T ~ Φ ln 50 + 0.18 0.5,0. 0.5 ( ) ln S ~ Φ 3.99,0.141 T Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 6

11.3 III 95% 95% 3.71< ln ST < 4.74 40.85 < S < 71.81 T 6 A 40.85 71.81 95% Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 7

11.3 IV A 54.71 60.46 7.78 ( ) E S T 0.18 0.5 50e 54.71 0.18 0.5 ( ) ( 0.04 0.5 ) Var S = 500e e - 1 = 60.46 T = = Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 8

ds S = mdt + se dt Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 9

1900 000 Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 30

μ μ Δt s m - μ Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 31

σ Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 3

/ +1 17.43% 6.17% 5.70% 40.67% 636.6% 44.8% 16.14% 30.34% 8.34% 49.40% 171.11% 64.16% 16.0% 9.16% 6.93% 46.71% 93.44% 63.06% 15.66% 30.40% 7.65% 48.96% 48.96% 6.10% 注 :1. 样本为 1990 年 1 月 19 日 -010 年 4 月 7 日上证指数. 从表中可以看出, 年化收益率的标准差没有经济含义 33

ds = msdt + ssdz G æ G G 1 G ö G dg = ms s S + + dt + ssdz ç è S t S ø S G S dz Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 34

BSM BSM BSM Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 35

μ σ r Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 36

BSM I S ds = msdt + ssdz Δt S ΔS D S = msd t + ssdz Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 37

BSM II f S f S t æ f f 1 f ö f df = ms s S ç + + dt ssdz + ç è S t S ø S Δt f Δf æ f f 1 f ö f D f = ms + + s S t ss z D + D è ç ç S t S ø S Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 38

BSM III Δz f S Π f P=- f + S S Δt ΔΠ DP = -D f + f S DS Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 39

BSM IV Δf ΔS æ f 1 f S t t S s ö DP = - - D ç è ø Π DP = rpdt Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 40

BSM V æ f 1 f f S t r f S t t S s ö æ ö ç + D = - D S èç ø èç ø BSM S f f 1 f + rs + s S = rf t S S Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 41

I BSM f Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 4

II r Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 43

I 10 3 11 9 3 10.5 3 3 11 0.5 3 9 0 Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 45

II 1 Δ 3 11 (11Δ 0.5) 3 9 9Δ 11Δ 0.5 = 9Δ Δ = 0.5 1 0.5 3 11 9.5 Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 46

III 10% 0.1 0.5.5e =.19 10 0.5 - f =.19 f = 0.31元 0.31 Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 47

IV 11 9 Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 48

V 10% P ( ) 0.1 0.5 10 = e 11P+ 9 1 P P= 6.66% 15% ( ) 0.15 0.5 10 = e 11P+ 9 1 P P= 69.11% Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 49

I (T ) ( - ) Eˆ é ê max S X T, 0 ë Ê c ( t) -rt- c = e Eˆ é max ( S X, 0 ) ù ê - ë T úû ù úû Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 50

II é æ s ö ù ln S ~ j ln S r ( T t), s T t T + ç - - - êë çè ø úû Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 51

III ( ) ( t) -rt- ( ) c = SN d -Xe N d 1 ( S X) + ( r + s )( T -t) ln / / d = 1 s T - t ln ( S / X) + ( r -s / )( T -t) d = = d -s T -t 1 s T - t Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 5

BSM I ( t) -rt- c = e Eˆ é max ( S X, 0 ) ù ê - ë T úû é æ s ö ln S ~ j ln S r ( T t), s T t T + ç - - - ê çè ë ø ù úû W = ln S T - m S Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 53

BSM II ç ( ) ( ) ( ln ) æ s ö m = Eˆ ln S = T ln S + ç r - T -t çè ø s = Var S = s T -t T W ( ) ~ N 0,1 W h(w) ( ) hw 1 = e p W - Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 54

BSM Eˆ é ê max ( S ë T - X,0 ) ù úû ò max( S X) f ( S ) ds S f ( S T T T T T) ds Xf T ( ST) ds - òx òx T ln ST sw + m ò ( ln ) ( ln ) ( ln ) ( ln T T ò T T) òln X-m ( ) òln X-m ( ) = - = - = e g S d S - Xg S d S = e h W dw - Xh W dw ln X ln X W s 1 - + m 1 - æ sw + m m - lnx ö = ln X-me e dw - ln X-m Xh( W) d( W) = ln X-me e dw -XN ò ò ò * s p s s p ç s çè ø æ S s ö ln æ r ö ( T t) s + - m X - + çè ø = ln X m e h s ( W) dw XN ò - - - s s T t - ç è ø æ S æ s ö ö æ S æ s ö ö ln r ( T t) ln r + + ( T t) ˆ X - + - X - ç è ø çè ø = E( ST ) N -XN s T t - s T -t ç è ø èç ø s ( W-s) s Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 55

I f N( d ) 1 = S Π Δ ( ) SN d 1 ( ) ( ) -rt-t Xe N d Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 56

II Asset-or-nothing Call Option X Cash-or-nothing Call Option Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 57

III ( ) N d S T X rt ( t) Xe N( d ) S X ( ) T SN d1 N( d 1 ) Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 58

IV N( d ) S T rt ( t) Xe N( d ) ( t) ( ) = ˆ ( ) ( ) rt e SN d1 E ST N d1 ST > X S T SN ( d 1 ) Copyright 01 Zhenlong Zheng 59

c σ T t σ T t = N N S σ T t = N 1 Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 60

c/s Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 61

PCP ( t) -rt- ( ) ( ) p = Xe N -d -SN -d 1 c=p Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 6

C = c ( ) ( t) -rt- ( ) C = SN d -Xe N d 1 Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 63

I S σ Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 64

II I (S I) S q -q( ) S Se T t Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 65

III r Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 66

t n T t n 11.6 Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 67

BSM BSM Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 69

/ Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 70

I 11.7 Generalized Autoregressive Conditional Heteroskedasticity GARCH Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 71

BSM BSM BSM Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 7

BSM BSM BSM BSM Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 73

BSM Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 74

Any Questions Copyright 01 Zheng, Zhenlong & Chen, Rong, XMU 75