21 世纪经济管理精品教材 金融学系列 金融经济学原理 [ 美 ] 马成虎著 清华大学出版社 北京
内容简介本书较为系统地介绍了金融经济学基础理论及其在资产定价建模中的应用, 从投资者风险偏好和跨期偏好特征出发, 探讨了不同偏好下的投资者交易行为, 以及由不同投资者构成的 完全竞争市场均衡状态下的证券定价机制 从 Markowitz 静态组合投资理论, 到 Epstein-Zin 递归效用下的动态最优交易策略 ; 从 Sharpe-Lintner 单期均衡资产定价模型 (CAPM) 到影子 -CAPM 动态资产定价模型 ; 从期望效用设定下以社会总消费增长率为定价因子的基于消费的资产定价模型 (C-CAPM) 到非期望效用设定下的跨期多因子资产定价模型 ; 从不依赖于偏好的无套利定价方法到基于偏好的均衡定价理论 ; 从风险偏好到风险测量再到风险管理 ; 从错综复杂的多人经济到基于代表性投资者的单人经济 ; 从股票定价到利率期限结构再到衍生品定价 ; 从股票溢价到无风险利率之谜, 再到期权价格的信息含量贯穿始终的是作者对经典资产定价理论的理解 思考和综合, 其中不乏作者本人在该领域的一些最新研究成果 本书封面贴有清华大学出版社防伪标签, 无标签者不得销售 版权所有, 侵权必究 侵权举报电话 :010-62782989 13701121933 图书在版编目 (CIP) 数据金融经济学原理 /[ 美 ] 马成虎著. 北京 : 清华大学出版社,2016 (21 世纪经济管理精品教材 金融学系列 ) ISBN 978-7-303-44315-5 Ⅰ.1 金 Ⅱ.1 马 Ⅲ.1 金融学 - 高等学校 - 教材 Ⅳ.1F830 中国版本图书馆 CIP 数据核字 (2016) 第 164365 号 责任编辑 : 杜星 封面设计 : 汉风唐韵 责任校对 : 王荣静 责任印制 : 出版发行 : 清华大学出版社 网 址 :http://www.tup.com.cn,http://www.wqbook.com 地 址 : 北京清华大学学研大厦 A 座 邮 编 :100084 社总机 :010-62770175 邮 购 :010-62786544 投稿与读者服务 :010-62776969,c-service@tup.tsinghua.edu.cn 质 量 反 馈 :010-62772015,zhiliang@tup.tsinghua.edu.cn 课 件 下 载 :http://www.tup.com.cn,010-62770175 转 4506 印刷者 : 店装订者 : 店 经 销 : 全国新华书店 开 本 :185mm 260mm 印 张 :16 插 页 :1 字 数 :385 千字 版 次 :2016 年 8 月第 1 版 印 次 :2016 年 8 月第 1 次印刷 印 数 :0~3000 定 价 :39.00 元 产品编号 :063302-01
(Adam Smith, 1776)
ii 20 60 Sharpe-Lintner (Sharpe 1964 Lintner 1965) (CAPM) 70 Black-Scholes (Black and Scholes 1973) 80 90 Ho and Lee (1986) Heath Jarrow and Morton (HJM 1992) 2010 (#71271058) 2015 12
1...1 1.1...1 1.1.1...1 1.1.2...2 1.1.3...2 1.2...8 1.3...10 1.4...13 1.5...14 1.6...15 1.7...17 1.8...19 1.8.1...20 1.8.2...24 1.8.3...25 1.8.4...27 1.9...28 2...29 2.1...29 2.2...33 2.2.1 -...33 2.2.2 1...34 2.2.3...35 2.2.4 - -...37 2.2.5 2...38 2.2.6...39 2.3 CRR...42 2.3.1...42 2.3.2 CRR...42
iv 2.3.3 Laplace...43 2.3.4 Black Scholes...44 2.3.5...46 2.3.6...47 2.3.7...49 2.4 Ho Lee...51 2.4.1...51 2.4.2...52 2.4.3...54 2.4.4 Ho Lee...54 2.4.5...56 2.4.6...57 2.4.7...57 2.4.8...58 2.4.9...58 2.4.10...59 2.5 APT...60 2.6 β...61 2.7...63 3...64 3.1 Stone...64 3.2...65 3.3...66 3.3.1 VaR...66 3.3.2 C-VaR...68 3.4...70 3.4.1 Arrow Pratt...71 3.4.2 Arrow Pratt...71 3.4.3...72 3.5...74 3.6...75 3.6.1 VaR...76 3.6.2...76 3.6.3 C-VaR...78 3.6.4...80 3.7...82 3.8...84
v 4...85 4.1...85 4.1.1...85 4.1.2...86 4.1.3...87 4.2 MPS...89 4.2.1...89 4.2.2...91 4.2.3...91 4.2.4 Black...92 4.2.5...93 4.2.6...95 4.2.7 Tobin...96 4.2.8 HJ...96 4.2.9...98 4.2.10...100 4.3...102 5 MPS CAPM...103 5.1...103 5.2 CAPM...104 5.3 CAPM...105 5.4 CAPM...106 5.4.1 CRR CAPM...107 5.4.2 CAPM...108 5.4.3 CAPM...109 5.5...112 5.6 CAPM...113 5.6.1 CAPM...113 5.6.2 CAPM...114 5.6.3 CAPM...114 5.6.4 APT CAPM...114 5.6.5 CAPM...116 5.7...117 6...118 6.1...118 6.2...121 6.2.1...121 6.2.2...121
vi 6.2.3...125 6.2.4...127 6.2.5...128 6.2.6...132 6.3...134 7 MPS...135 7.1...135 7.2 I-CAPM...136 7.3...138 7.3.1...139 7.3.2...140 7.3.3...141 7.3.4...141 7.3.5 W-CAPM I-CAPM...142 7.3.6 Kreps-Porteus...143 7.4 I-CAPM...144 7.5 I-CAPM...144 7.6...145 8...146 8.1...146 8.2...148 8.3...149 8.3.1...150 8.3.2 Markov...156 8.4 MPS...162 8.4.1...163 8.4.2...165 8.4.3...165 8.5...166 9...167 9.1 MPS -CAPM...167 9.1.1...167 9.1.2...168 9.1.3 -CAPM...170 9.1.4 -CAPM CAPM...171 9.2...172 9.2.1...172 9.2.2...173
vii 9.2.3...175 9.2.4...176 9.2.5 C-CAPM EZ...177 9.3...177 10...178 10.1...178 10.2...179 10.3...181 10.3.1...181 10.3.2...186 10.3.3...189 10.4...191 10.4.1...192 10.4.2...193 10.4.3 Black Scholes...194 10.4.4...195 10.4.5...198 10.4.6...201 10.4.7...203 10.5...203 A...205 A.1...205 A.2...205 A.3...206 A.3.1...206 A.3.2...207 A.3.3...209 A.3.4...209 A.3.5 L p -...210 A.4...210 A.4.1...210 A.4.2 Markov Markov...212 A.4.3 Markov...213 B...214 B.1...214 B.2...215 B.3...216 B.4 Hilbert...217
viii B.5 Riesz...217 B.6...218 B.7...219 B.8 Schwartz...220 B.8.1 S (R) L p (R)...221 B.8.2 Dirac...221 B.8.3...222 B.8.4...222 C...223 C.1 Weierstrass...223 C.2...224 C.3 Kuhn-Tucker...225 C.4...226 D Laplace...228 D.1...228 D.2 Laplace...229 D.3 L{ } L 1 { }...229 D.4 Laplace...230 D.5...231...233
1.1 1.1.1 M1 M2
2 1.1.2 Ω ω Ω T T = {0, 1,,T} T =[0,T] 0 T< F = {F t } t T F t Ω t ( t ) t ω F t (ω) t ω P ( ) Ω T Ω {x t } t T x t t F t - t x t F t - {x t } t T F x P E P [x] = x (ω)dp(ω) t F t E P [x F t ] E [x] E t [x] 1.1.3 ( ) Ω
1 3 T Ω F P j J { δ j (t, ω) } P1 P2 P3 P4 (P1) (P2) (P3)
4 1.1 u (c 0,c 1 )=u (c 0 )+βe [u (c 1 )] (1.1) u > 0 u < 0 0 <β<1 c 1 Ω u (c 0,c 1 ) >u(c 0,c 1) (c 0,c 1 ) (c 0,c 1) β β c 0 c 1 c 0 c 1 u 0 (c 0,c 1 ) u 1 (c 0,c 1 ) ɛ u 0 (c 0,c 1 ) ɛ + o (ɛ) (c 0,c 1 ) MRS u (c 0,c 1 )= u 1 (c 0,c 1 ) u 0 (c 0,c 1 ) u(c0,c 1)= (1.2) (c 0,c 1 ) ɛ MRS u (c 0,c 1 ) ɛ + o (ɛ) (c 0,c 1 ) ρ = dlnmrs u (c 0,c 1 ) dlnc 0 /c 1 u(c0,c 1)= (1.3) ɛ ρɛ + o (ɛ) (c 0,c 1 ) c 1 (c 0, E [c 1 ]) (c 0,c 1 ) 1.1 c 1 E [c 1 ] c 1
1 5 Kreps and Porteus (1978) Epstein and Zin (1989) Chew and Epstein (1989) Kahneman and Tversky 1979 Gul (1991) 1.2 c = {c t } t 0 {V t (c)} V t (c) =A (c t, CE [V t+1 (c)]), t =0, 1, (1.4) A (, ) CE t A (, ) CE[ ] A (c, v) =(c γ 0 + βvγ ) 1/γ, CE [x] =(E [x α ]) 1 α (1.5) β (0, 1), 0 α, γ < 1 KP David Kreps Evan Porteus KP ( ) (1) ρ =1 γ (2) α 1 α KP γ α 1.3 A (c, v) CE[c 1 ] H : R R R H (1) H (x, x) 0 (2) H C 2,1 (R R) H 1 > 0 H 2 < 0 H x CE[x] E [H (x, CE [x])] = 0 (1.6) CE [x] H (x, y) x ( ) x H (x, y) =x δ α α yα α ( [ ] ) 1/α E x α+δ CE [x] = E [x δ ] δ =0 KP dq dp (ω) (ω) =xδ Q E [x δ Q ] CE [x] =(E Q [x α ]) 1/α
6 δ<0 Q x<e [ 1/δ x δ] 1.4 Kahneman Tversky 1979 { x X, x X u (x) = (1.7) λ (x X ), x < X λ 1 X z =1 {x X } Q λp (z =0) Q (z =0)= P (z =1)+λP (z =0) =1 Q (z =1); Q ( z) =P ( z),z {0, 1} λ 1 Q {z =0} λ a = P (z =1)+λP (z =0) [1,λ] E [u (x)] = a (E Q [x] X ) ae Q [x]+(1 a) X, E Q [x] X CE [x] = a ( λ E Q [x]+ 1 a ) (1.8) X, E Q [x] <X λ a [1,λ] CE[x] E Q [x] X Q E Q [x] X [ E P [x]+(λ 1) E P (x X )1 {x<x }], EQ [x] X CE [x] = E P [x]+ 1 λ λ E [ (1.9) P (x X )1 {x X }], EQ [x] <X CE[x] E P [x] Frank Gul 1.5 CE[c 1 ] H H (x, y) = { u (x) u (y), x y λ [u (x) u (y)], x < y (1.10)
1 7 λ 1 u (x) = xα,α 1 α z =1 {x CE[x]} Q λp (z =0) Q (z =0)= P (z =1)+λP (z =0) =1 Q (z =1) Q ( z) =P ( z),z {0, 1} {z =0} Q P λ E [H (x, CE [x])] = 0 CE [x] =(E Q [x α ]) 1 α E Q [x]. (1.11) {z =0} {x <CE [x]} 1.11 Q - 1.11 {x <CE [x]} E Q [x] E [x] CE [x] E [x] K- 1.6 K- K- P Δ Δ Savage 1954 Gilboa and Schmeidler 1989 K- Gilboa-Schmeidler ) CE [x] =u 1 (min Q Δ E Q [u (x)] =min Q Δ CE Q [x] (1.12) CE Q [x] =u 1 (E Q [u (x)]) u Q x Savage Q CE [x] =u 1 (E Q [u (x)]) (1.13)
8 1.3 1.4 1.5 Savage 1.6 1.4 1.5 P λ Λ λ Q Δ={Q (λ) :λ Λ} 1.12 x ) CE [x] =u (min 1 E Q(λ) [u (x)] (1.14) λ Λ Allais [Allais (1953)] Ellsberg [Ellsberg (1961)] Crawford (1986) Epstein (1992) Machina (1992) 1.2 J +1 j =0, 1,,J ( ) 0 δ t = [ ] δt 0,δt 1,,δt J t δ j t (ω) t ω j δ = {δ t } t 1 F p t = [ p 0 t,p1 t, ],pj t t p = {pt } t 0 F t φ t = [ ] φ 0 t,φ 1 t,,φ J t t F φ = {φ t } t 0 j φ j t 0 φ j t Φ
1 9 φ d φ 0 = φ 0 p 0 φ d φ t = φ t 1 (p t + δ t ) φ t p t (1.15) t =1, 2,,n d φ t = φ t 1 (p t p t 1 + δ t ) }{{} [φ t p t φ t 1 p t 1 ] }{{} φ t 1 (p t p t 1 + δ t ) + φ t p t φ t 1 p t 1 d φ t 0 d φ t < 0 φ d φ t =0 φ t t φ t p t = φ 0 p 0 + φ s 1 (p s p s 1 + δ s ) (1.16) s=1 t t D (p, δ) D (p, δ) = { d φ Φ d φ = d } (1.17) D d D d D d D d =(d 0,d 0 ) d 0 M (p, δ) d 0 M φ Φ d φ 0 = d 0 1.7 t =1 {a, b, c} δ 1 = [110, 110, 110] δ 2 =[50, 100, 150] p 1 = p 2 = 100 M δ 1 δ 2
10 e = {e t } t 0 ( ) e 1.1 e c e φ t c t = e t + d φ t (1.18) c φ B (p, δ; e) 1.1 B (p, δ; e) c B (p, δ; e) c e D c φ Φ φ t p }{{} t = t φ s 1 (p s p s 1 + δ s ) + }{{} s=1 t (e s c s ) }{{} s=0 (1.19) 0 t =0, 1,,n =0 s=1 (1.19) t 1.3 c c c c V (c) 0 sup {V (c) :c B (p, δ; e)} (1.20) V (c) =u (c 0,c 0 ) c 0 t =1, 2,,n (c 0,c 0 ) u (c 0,c 0 ) c φ 0 c 0 ɛ j ɛ p j ( ) 0 δj ɛ c 0 ɛ, c 0 + δj ɛ p j 0 p j 0
( ) 0 u c 0 ɛ, c 0 + δj ɛ x y p j 0 u ( c ) 0,c 0 { } ( = u 0 c 0,c ( 0) u 0 c 0,c ) δ j 0 ɛ + o (ɛ) t y t x t y t = t T {0}x ω Ω ( c 0,c 0) p j 0 1 11 x t (ω) y t (ω) ɛ p j 0 = π (c ) δ j (1.21) j π t (c )= u ( t c 0,c 0) ( ) u 0 c 0,c t 0 0 d D d 0 + π (c ) d 0 = 0 (1.22) (1.21) D S1 ψ = {ψ t } t 1 d D d 0 = ψ d 0 (1.23) j =0, 1,,J p j 0 = ψ δj S1 1.1 V (c) =u (c 0,c 0 ) S1 c π (c ) ψ M (1.24) c (1.20) V c B (p, δ; e) 1.24 c B (p, δ; e) c c =(c e) (c e) D S1 c 0 c 0 = ψ ( c 0 c 0) V (c) =u (c0,c 0 ) V (c) V (c ( ) u 0 c 0,c 0) (c0 c ( 0)+u 0 c 0,c ( 0) c 0 c ) 0 ( = u 0 c 0,c 0)[ c0 c 0 + π (c ) ( c 0 c )] 0 ( = u 0 c 0,c 0)[ c0 c 0 + ψ ( c 0 c 0)] =0
12 1.24 c 1.1 S1 S1 ψ ψ max {u (c 0,c 0 ):c 0 e 0 + ψ (c 0 e 0 )=0} (1.25) c (ψ) π (c (ψ)) = ψ c (ψ) e D S1 ψ c = c (ψ ) φ 1.8 t =1 {l, m, h} δ 1 = [110, 110, 110] δ 2 =[50, 100, 150] p 1 = p 2 = 100 e 0 = 1 000 u (c 0,c 1 )=lnc 0 +0.3(lnc 1l +lnc 1m +lnc 1h ) S1 2 ψ l 11 1 ψ m = 12 + ψ h ψ h 11 2 (1.26) 1 0 ψ h 2 11 6 11 1.25 c 1l = 0.3c 0 ψ l,c 1m = 0.3c 0 ψ m,c 1h = 0.3c 0 ψ h (1.27) c 0 = 1000 1.9 [c 1l,c 1m,c 1h ] 0.3c 0 ψ l 0.3c 0 ψ m 0.3c 0 ψ h 110 = x 1 110 + x 2 110 50 100 150 (1.28)
1 13 1 1 1 ψ l 1 1 2 = 0 (1.29) ψ m 1 1 3 1.26 1.29 ψ h = 8+2 ( 7 2 33 11, 6 ) 11 1.27 c 1l = 825 ( 7 1 ),c 1m = 275 ( 5+ 7 ),c 1h = 550 ( 4 7 ) ; 3.8 3.8 1.9 1.28 1 2 x 1 = 11 7 17 0.8 1.9,x 2 = 11 ( ) 11 5 7 4 1.9 ψ h 1.4 S1 1.2 φ t 0 d φ t 0 1 M1 M2 M3 M3 φ d φ t 0 t 1 p φ 0 = φ 0 p 0 t 1 d φ t 0 pφ 0 {r t } t 0 1.2 M3 φ T 1 [ ] p φ 0,pφ T =[ φ 0 p, φ T p T ] 0
14 M3 φ d φ t 0, 1 t T pφ 0 = φ 0p 0 0 φ φ d φ t 0( ) T T T (1 + r t ) (1 + r T 1 ) d φ t S1 1.2 M3 S1 S1 M3 M3 S1 M3 D C + = {c : c t 0, t 0} c = C + = R m + m D ( B.4 A) ϕ R m c C + d D ϕ c ϕ d c = d = c = d D ϕ d 0 d D ϕ d =0 c ϕ c>0 ϕ t ψ t = ϕ t ϕ 0 d 0 = ψ t d t, d D φ p φ 0 = ψ t d φ t t 1 t 1 6 6.1 1.2 ( ) D t=1 1.5 ( E = {Ω, F,δ} ; { ) u i,φ i 1,ei} (1.30) i I Ω δ I i u i i φ i 1 e i δ p φ i c i e i u i F- 1.3 E p { c i} i I { φ i} i I (1) i I φ i = i I φ i 1
1 15 (2) (p, δ) W0 i = p 0 φ i 1 + e i 0 i φ i c i c i max { u i (c) :c B ( p, δ; W i 0,e i)} (1.31) (1) (2) ( c i t e i ( t) = c i 0 e i ) 0 =0 i I i I φ i 1δ t i I 1.3 E F ( W0 i,ei) i I Magill and Quinzii (1996, 10.5) Magill and Quinzii (1996) 1.3 1.1 p ψ S1 p j 0 = ψ δj,j=0, 1,,J (1.32) ( 1.10) 1.6 E = ( {Ω, F,δ} ; { ) u i,φ i 1,ei} i I 1.4 { c i} i I ( c i 0 e i ( 0) 0 c i 0 e i 0) φ i 1 δ i I i I i I { c i} Pareto (P.O.) i i I { d i} ( i I ui d i) u ( i c i) i Pareto { d i} ( i I i ui d i) >u ( i c i) { c i} Pareto i I
16 1.4 { c i} i I ( Pareto i I c i MRS i c i 0,c 0) i ψ i I ( MRS i c i 0,c i 0) = ψ 0 (1.33) ( π i =MRS i c i 0,c 0) i i k (t, ω) πt i (ω) >α>πk t (ω) i k i ɛ k ɛ α t ω ɛ i ( u i c i 0 ɛ, ci 1,,ci t + ɛ ) α,ci t+1, u i ( c i ) 0,ci 0 = u i c 0 ( c i ) ɛ + 1 α ui c t(ω) ( c i ) ɛ + o (ɛ) > 0 k Pareto Pareto ψ 0 D R M F 0 ξ M ψ 0 = ψ+ ɛξ ɛ ɛ ψ 0 0 d M ψ d = ψ 0 d ψ 0 i π i ξ i M π i = ψ+ ξ i 1.4 Pareto c i e i i I Pareto Pareto Pareto 1.5 { c i} Pareto i I { } d i i I (1) d i e i D, i I (2) i u ( i d i) u ( i c i) i 1.5 { c i} i I Pareto ψ 0 ξ i M i MRS i ( c i 0,c i 0) = ψ + ξ i 0 (1.34) { d i} 1.5 i I (1) (2) 2 i
1 17 p { c i} i I S1 ψ 0 1 d i e i D φ i d i 0 = φ i δ + e i 0 0 ψ ( d i 0 e 0) i d i 0 + ψ ( d i 0 ( 0) ) ei u d i >u ( c i) c i d i 0 + ψ ( d i 0 0) ei >p0 φ i 0 + ei 0 = ψ ( φ i δ ) + e i 0 i ( d i 0 e i ( 0) + ψ d i 0 e i 0 φi δ ) i I = i I i I ( d i 0 e i ( 0) + ψ d i 0 e i 0 φ i 1 δ ) > 0 i I { d i} i I ( (1.24) i π i =MRS i c i 0,c 0) i p d 0 M ξ i d 0 = π i d 0 ψ d 0 = p φ 0 pφ 0 =0 φ Φ d 0 i ( MRS i c i 0,c i 0) = ψ + ξ i,ξ i M 1.4 Pareto 1.2 Pareto ψ Pareto 1.5 Pareto 1.7 E = ( {Ω, F,δ} ; { u i,φ i 1,e i} ) i I p u e = i I E r =({Ω, F,δ} ; {u, φ 1,e}) e i φ 1 = i I φ i 1
18 1.6 E p p E r E E r u E E r ( E ) 1.6 { c i} { c i} Pareto i I i I ( Magill and Quinzii 1996) 1.6 (p, δ) { c i} Pareto i I u : C R c C u (c) = max c i C, i { i I 1 u i ( c i) : } c i = c λ i i I (1.35) ( λ i = u ) i c 0 c i > 0 i 0 C ( S1 ψ i π i =MRS i c i 0, c1) i i I ( c i 0, 1) ci (e 0,e 0 + φ 1 δ) MRS u (e 0,e 0 + φ 1 δ) ψ c (e 0,e 0 + φ 1 δ) (1.35) { c i} i I φ 1 (e 0,e 0 + φ 1 δ) (1.35) (1.24) 1.1 (e 0,e 0 + φ 1 δ) { c i} i I (1.35) (e 0,φ δ + e 0 ) Pareto (1.35) (1.35) Pareto Pareto (1.35) 1.7 (p, δ ) { c i} E ψ S1 i I π i = ψ+ ξ i 0 i ξ i M u : L + R
u (c) = max c i C, i { i I 1 u i ( c i) ξ i c i 0 : } c i = c λ i i I 1 19 (1.36) ( λ i = u ) i c 0 c i > 0, i I (1.36) C c C c { c i} c i I (1.36) d D d d i I ξ i d 0 =0 c c + d c + d C u (c + d) i I = i I i I 1 u i ( c i + d ) ξ i ( c i 0 λ + d ) 0 i 1 λ i u i ( c i + d ) ξ i c i 0 1 λ i u i ( c i) ξ i c i 0 = u (c) d 0 u : C R S1 ψ 1.1 c e D (e 0,e 0 + φ 1 δ) π i ξ i = ψ 0 (e 0,e 0 + φ 1 δ) (1.24) 1.1 p (1.20) (e 0,φ 1 δ + e 0 ) φ 1 p ψ Pareto (1.36) ξ =0 (1.35) ψ 1.8 ( )
20 (1) (2) (3) 1.8.1 ( E = Ω,δ, { ) F i,u i,φ i 1,ei} i I i {Ω, F i, P i } ω Ω F i (ω) Ω ω i ( ) F i ω ω F i (ω) =F i (ω ) F i (ω) F i (ω )= i Ω P i F G ω F (ω) G(ω) F G G F ω p (ω) F p (ω) ={ω : p (ω )=p (ω)} ( ) p (ω) =p (F 1 (ω), F 2 (ω),, F I (ω)) i ( 1.1) F i (ω) =F i (ω) F p (ω) A Ω i I A = ω AF i (ω ) A ω A A A ω
1 21 1.1 1.7 ω p (ω) ω F i (ω) F i (ω) i ω F i (ω) =F i (ω) i ω Ω F p (ω) = i I F i (ω) ( ) F i (ω) =F p (ω) = i I F i (ω) p (ω ) p (ω) i F i (ω) F i (ω ) ( 1.2) 1.2
22 1.8 ω p (ω) {F i } i I 1.9 i =1, 2 u (c 0,c 1 ; ω) =c 0 + ω ln c 1 ω {0.5, 1} t =0 1 2 F 1 (ω) ={ω}, F 2 (ω) ={0.5, 1} (t =1) 1 t =0 ω p (ω) ω p (ω) p c 1i i t =0 c 0i = p (1 c 1i ) t =0 c 11 (ω, p) =ω/p c 12 (ω, p) =0.75/p c 11 (ω, p)+c 12 (ω, p) =2 ω p (ω) =3/8+0.5ω p =5/8 2 ω =0.5 p =7/8 ω =1 F 1 (ω) =F 2 (ω) =F p (ω) ={ω} ω c 11 [ω, p (ω)] = c 12 [ω, p (ω)] = ω/p (ω) p (ω) =ω ω {0.5, 1} 1.9
1 23 F (ω) = i I F i (ω) F (ω) F 1.8 {F i } i I ω p (ω) F p (ω) = i I F i (ω) F i (ω) =F i (ω) F p (ω) = i I F i (ω) ω p (ω) {F i } i I 1.8 1.10 1.9 u 1 (c 0,c 1 ; ω) =c 0 + ω ln c 1, u 2 (c 0,c 1 ; ω) =c 0 +(2 ω)lnc 1 ω {0.5, 1} 1 2 c 11 (p, ω) =ω/p (ω), c 12 (p, ω) =(2 ω) /p (ω) p (ω) =1,ω {0.5, 1} 2 p =1 c 12 (p, ω) =1.25 ω +1.25 2 2 c 12 (p, ω) =1.25/p 1 c 11 (p, ω) =ω/p p (ω) =5/8+0.5ω ω {0.5, 1}
24 1.8.2 - REE -REE 1.10 p =1 c 11 (ω) 1 = 1 c 12 (ω) = ω 1 q (ω) ={p (ω), vol (ω), } ω ω F q (ω) ={ω : q (ω )=q (ω)} F i = F i F q i F i F q 1.9 ω F i (ω) F i (ω) ω q (ω) i ω F i (ω) =F i (ω) i ω F q (ω) = i I F i (ω) ω q (ω) 1.10 -REE ω q (ω) {F i } i I ω 1.9 -REE REE -REE 1.11 1.10 -REE p (ω) =1 vol (ω) = ω 1 q (ω) =(p (ω), vol (ω)) -REE ω
1 25 ω c 11 = ω c 12 =2 ω p =1 ω 1 ω vol (ω) (1, ω 1 ) c 11 (p, ω) =ω/p c 12 (p, ω) =1.25/p ( ) 4ω +5 4ω 5, 8 4ω +5 (1, ω 1 ) -REE -REE ( ) 1.11 p =1 -REE 0 0.5 REE 0.25 REE -REE 1.8.3 1.9 1.11 1.9 Fama (1970) ( ) F F
26 S1 1.11 F F ω p (ω) F - ψ j p j (ω) = ψ (ω ) δ j (ω ) (1.37) ω F(ω) F φ ω F (ω) φ (ω )=φ(ω), ω F(ω) ( 1.10) 1.10 F F J ω p (ω) = [ p 1 (ω),p 2 (ω),,p J (ω) ] F- (1.37) F φ p φ (ω) = j φ j (ω) p j (ω) =φ (ω) p (ω) F - d φ (ω )= j φj (ω) δ j (ω ) ω (1.37) ψ (ω ) d φ (ω )=φ(ω) ψ (ω ) δ (ω )=φ(ω) p (ω) ω F(ω) ω F(ω) ω p φ (ω) F F F 1.3 F G F G φ G F G φ F 1.10 F G Fama (1970) Fama
1 27 i I F i 1.10 ( ) i I F i F i F p 1.9 1.8.4
28 ( ) 1.9 Ross (1978) (Dybvig and Ross 1989) Magill and Quinzii (1996) Arrow and Debreu (1954) Debreu (1959) Radner (1972) Arrow Debreu Hart (1975) Duffie and Shafer (1986) Hirsch, Magill and Mas-Colell (1990) Magill and Quinzii (1991) Hart (1975) Duffie and Shafer (1986) Hirsch, Magill and Mas-Colell (1990) Magill and Shafer (1991) Grossman (1977a) Pareto Pareto Gorman (1953) Nigishi (1960) Magill and Quinzii (1996) Ma and Zhang (2013) Pareto Hirshleifer (1973) Grossman and Stiglitz (1976 1980) Grossman (1977b 1981 1989) Kreps (1977) Allen (1986) He and Wang (1995) Blume Easley and O Hara (1994) Hirshleifer and Riley (2002) Brunnermeier (2001)
- - 2.1 ψ φ Φ p φ ψ d φ p (1) d M φ Ψ 0 (d) =ψd (2) 0 d M Ψ 0 (d) 0 Ψ 0 (d) > 0 d 0 (3) d, d M Ψ 0 (d + d) =Ψ 0 (d)+ψ 0 (d ) (4) d M α R Ψ 0 (αd) =αψ 0 (d) (5) d M t 0 t Ψ t (d) F t ψ s d s 1 {Ft(ω)} Ψ t (d, ω)= s t+1 ω F t(ω) (2.1) ψ t (ω )
30 = ψ t+1 [d t+1 +Ψ t+1 (d)] 1 {Ft(ω)} ψ t (ω ) ω F t(ω) Ω x y xy1 {Ft(ω)} ω F t(ω) x (ω ) y (ω ) (2.2) φ φ d φ t ( ) F t (ω) F t (ω) d d s = d [ ] s 1 1{s t} Ft(ω) +[dt +Ψ t (d)] 1 {t}ft(ω) 1 {t}e 1 {s t}e T Ω { 1 s = t, ω E 1 {t}e (s, ω) = 0 { 1 s t, ω E 1 {s t}e (s, ω) = 0 t F t (ω) t +1 t d (2.2) d s = d s ( 1 1{s t+1}ft(ω)) +[dt+1 +Ψ t+1 (d)] 1 {t+1}ft(ω) Ψ 0 (d) =Ψ 0 (d )=Ψ 0 (d ) ψ s d s = ψ s d s + ψ s d s 1 {Ω Ft(ω)} s 1 1 s t + = ω F t(ω) 1 s t s t+1 ψ t (ω )Ψ t (d, ω) ψ s d s + s t+1 ψ s d s 1 {Ω Ft(ω)} + ψ t+1 [d t+1 +Ψ t+1 (d)] 1 {Ft(ω)} (2.1), (6) (Ω, F) Q (t, ω) A F t (ω) Q (A F t (ω)) = ψ t+1 (ω ) ω A ω F t(ω) ψ t+1 (ω ) (2.3)
2 31 Q (Ω, F) t t +1 1 R f t (ω) =1+r t (ω) ω 1 1+r t (ω) = ω F t(ω) ω F t(ω) ψ t+1 (ω ) ψ t (ω ) > 0 (2.4) 2.3 2.4 2.2 d M t 0 Ψ t (d) = 1 E Q [d t+1 +Ψ t+1 (d) F t ] (2.5) 1+r t t Q (7) Rt+1 d = d t+1 +Ψ t+1 (d) t t +1 Ψ t (d) 2.5 Q d E Q [ R d t+1 F t ] =1+rt (2.6) 2.1 ( ) M3 S1 (5) (6) (7) (1) (2) (3) (4) S1 (1) (2) (3) (4) M3 S1 (5) (6) (7) M3 (7) Ψ 0 (d) =E Q d t (2.7) (1 + r 0 )(1+r 1 ) (1 + r t 1 ) t 1 T Ψ T (d) =d T =0 1.2 M3 S1 Q P ψ P S1 ψ 2.1 {a, b, c} δ 1 = [110, 110, 110] δ 2 =[50, 100, 150]
32 p 1 = p 2 = 100 S1 ψ ψ a 2/11 1 ψ b = 12/11 + k 2 0 1 ψ c k (2/11, 6/11) r = 10% q a 1/5 1 q b = 6/5 + q c 2 0 1 q c q c (0.2, 0.6) 2.1 p a = p b = p c =1/3 2.1 δ 3 =[0, 0, 50] k (2/11, 6/11) p 3 =50k p 3 (100/11, 300/11) ( ) p 3 = 200/11 3 3 [x a,x b,x c ] p 3 = 200/11 2.1 k =4/11 [x a,x b,x c ] p x =(2x a +4x b +4x c ) /11 Cox, Ross and Rubinstein (1979) Ho and Lee (1986)
2 33 2.2 ( ) 2.1 1 10% 2 100 δ 3 =[0, 0, 50] p 3 =9 100 10 ( 1) 100 11 9 11 [550, 0, 0] 10 100 11 (100 9) = 1 ( ) 2.2.1 - S 0 d = {d t } T t=1 C 0 (X, T) P 0 (X, T) X T T B 0,T 2.1 T X S 0 + P 0 (X, T) =Ψ 0 (d)+xb 0,T + C 0 (X, T) (2.8) - - {d t } T t=1 X T 2.1 0 T T d T + S T +(X S T ) + (S T X) + X = d T
34 2-1 S 0 d 1 d T 1 d T + S T P 0 (X, T ) 0 0 (X S T ) + C 0 (X, T ) 0 0 (S T X) + XB 0,T 0 0 X Ψ 0 (d) d 1 d T 1 d T Ψ 0 (d) =S 0 + P 0 (X, T) C 0 (X, T) XB 0,T (2.9) - (1) T X (2) X T T X T A S T +(X S T ) + S T T X +(S T X) + B T A B S 0 + P 0 (X, T) =XB 0,T + C 0 (X, T) T d ( S 0 + P 0 (X, T) =Ψ 0 (d)+ψ 0 (S T )+Ψ 0 (X S T ) +) ( =Ψ 0 (d)+ψ 0 S T +(X S T ) +) ( =Ψ 0 (d)+ψ 0 X +(S T X) +) =Ψ 0 (d)+xb 0,T + C 0 (X, T) - 2.3.7 2.2.2 1-2.8 2.9 [T,T ] d (T,T ] Ψ 0 ( d(t,t ]) = P0 (X, T ) C 0 (X, T ) P 0 (X,T)+C 0 (X,T) (2.10) +X B 0,T XB 0,T
2 35 X X (2.10) X = XB 1 0,T B 0,T 2.2 T >T 0 ( Ψ 0 d(t,t ]) = P0 (X, T ) C 0 (X, T ) P 0 (X,T)+C 0 (X,T) (2.11) X 0 T T X 2.9 T B 0,T = P 0 (X,T) P 0 (X, T) C 0 (X,T)+C 0 (X, T) X X (2.12) X X 2.12 X X 2.3 B 0,T = P 0 (X, T) X C 0 (X, T) X (2.13) X>0 2.2.3 2.2 30 5% 33 33 33 33 5% 3 000 30 100 33 100 33 100 3 000 1.05 = 150( ) 31.5 31.5 100 100 31.5 =30 1.05( )
36 S 0 T B 0,T 2.2 2.4 T {d t } T t=1 F 0,T =(S 0 Ψ 0 (d)) B 1 0,T (2.14) F 0,T = S 0 B 1 0,T S T T T S T F 0,T t T 2.1 S T F 0,T Ψ 0 (S T F 0,T )=0 Ψ 0 (S T )=Ψ 0 (F 0,T )=F 0,T B 0,T F 0,T =Ψ 0 (S T ) B 1 0,T (2.15) {d t } T t=1 S 0 =Ψ 0 (d)+ψ 0 (S T ) Ψ 0 (S T )=S 0 Ψ 0 (d) (2.15) (2.14) d t > 0 d t < 0 2.14 Ψ 0 (d) = T d t B 0,t t=1 2.3 T X c i T i T T<T t =0 S 0 = cx B 0,Ti + i XB 0,T Ψ 0 (d) =cx i B 0,Ti 1 {0 Ti T } 2.14 ( F 0,T = XB 1 0,T c ) B 0,Ti 1 {Ti>T } + B 0,T i (2.16) F 0,T = XB 1 0,T B 0,T ETF (d t+1 /S t = κ) (d t+1 /S t+1 = l) 2.4 t 0 d t+1 = κs t S t = κ 1+r S t +Ψ t (S t+1 ) Ψ 0 (S t+1 )= 1+r κ Ψ 0 (S t ) 1+r
2 37 ( ) t 1+r κ Ψ 0 (S t )= S 0 2.15 1+r F 0,T =(1+r κ) T S 0 2.5 2.4 t 0 d t+1 = ls t+1 S t =(1+l)Ψ t (S t+1 ) Ψ 0 (S t )=(1+l)Ψ 0 (S t+1 ) Ψ 0 (S t )=(1+l) t S 0 2.15 2.2.4 - - F 0,T = ( ) T 1+r S 0 1+l S 0 Ψ 0 (d) F 0,T B 0,T - - - 2.2 S T - - P 0 (X, T) C 0 (X, T) =(X F 0,T ) B 0,T (2.17) - - - - S T F 0,T T (X S T ) + (S T X) + X F 0,T T P 0 (X, T) C 0 (X, T) X F 0,T T (X F 0,T ) B 0,T - -
38 F 0,T S T T (S T X) + (X S T ) + F 0,T X T C 0 (X, T) P 0 (X, T) =(F 0,T X) B 0,T - - 2.2.5 2 - - 2.1 F 0,T = X + B 1 0,T [C 0 (X, T) P 0 (X, T)] (2.18) F 0,T X (2.18) {d t } T t=1 (2.18) (2.18) (2.18) X = F 0,T C 0 (F 0,T,T)=P 0 (F 0,T,T) (2.19) 2.19 2.5 C 0 (X, T) =P 0 (X, T) X = F 0,T
2 39 (2.18) X = S 0 F 0,T = S 0 + B 1 0,T [C 0 (S 0,T) P 0 (S 0,T)] (2.20) 2.6 - F 0,T S 0 P 0 (S 0,T) C 0 (S 0,T) 1 F 0,T S 0 - P 0 (S 0,T) C 0 (S 0,T) P 0 (S 0,T) C 0 (S 0,T) 2.12 2.12 (2.18) [ F 0,T = X + 1 C 0 (X,T) P 0 (X ] 1,T) (X X) (2.21) C 0 (X, T) P 0 (X, T) X X X X 2.7 - C 0 (X, T) P 0 (X, T) [ ] 1 dln C0 (X, T) P 0 (X, T) F 0,T = X (2.22) dx X>0 X = F 0,T 2.22 2.22 dln C 0 (X, T) P 0 (X, T) dx F 0,T F 0,T X = F 0,T 2.19 X = F 0,T 2.2.6 2.1 ( )
40 ( ) 2.6 100 20% 120 10% 90 2% 100 102 100 ( ) [0.5 (120 100) + 0.5 0]/1.02 = 9.80( ) C 0 =9.80 9 700 113 170 t = 0 170 9.8 + 9 700 113 100 = 66 ( ) 170 20 9 700 1.02 + 113 120 = 266 170 0 9 700 1.02 + 113 90 = 276 C 0 =9.80 Q Q (2%) q h > 0 q l =1 q h > 0 Q 1.20q h +0.90q l =1.02 q h =0.4 q l =0.6 Q 0.4 (120 100) + 0.6 0 C 0 = =7.84 1.02 Ω={h, l} ( ) Ω={h, l} h l ( ) R T R f =1+r h>1+r>l q h = Q ({h}) =1 q l q h h +(1 q h ) l =1+r q h = 1+r l h l =1 q l (2.23)
2 41 X T Ψ 0 (X T )=(1+r) 1 (q h X h + q l X l ) (2.24) X C 0 =(1+r) 1 [ q h (hs 0 X) + + q l (ls 0 X) +] (2.25) S 0 (2.25) (2.24) 1 d T = κs 0 h + κ>1+r>l+ κ q h (h + κ)+(1 q h )(l + κ) =1+r q h = 1+r l κ h l 2 d T = κs T h> 1+r 1+κ >l (1 + κ)[q h h +(1 q h ) l] =1+r (1 + r) / (1 + κ) l q h = h l (2.25) ( ) (1) r (2) (3) (4) h l μ + σ μ σ σ>0 σ C 0 (5) μ (1) (2) (3) h μ σ (4) (5) μ h
42 2.3 CRR Cox, Ross and Rubinstein (CRR 1979) 2.3.1 CRR Ω Bernoulli {ξ t } t=1 ξ t {0, 1} Ω {ξ t } t=1 ω Ω F t (ω) t ω {ξ t } t=1 ξ t t R f =1+r ( ) {R t } t 1 {ξ t } t=1 R t = l +(h l) ξ t,t=1, 2,,n (2.26) h l h>1+r>l T 0 S 0 S T X (S T ) 2.1 2.1 2.3.2 CRR Q E Q [R t+1 F t ]=1+r E Q [ξ t+1 F t ]= 1+r l h l
2 43 q h (t, ω) =Q [ξ t+1 =1 F t (ω)] = 1 q l (t, ω) (t, ω) q h (t, ω) =E Q [ξ t+1 F t (ω)] = 1+r l (2.27) h l q h q l =1 q h X T 2.8 T X (S T ) T ( ) Ψ 0 (X) =(1+r) T T q m m h qt l X ( h m l T m ) S 0 m m=0 ( ) ( ) T T! T m m!(t m)! 1 0 Ψ 0 (X) =(1+r) T E Q [X (S T )] (2.28) ω Ω ω h l ω m h T m l ω S T (ω) =h m l T m S 0 h l Q T,m T h m ( ) T Q T,m = q m m h qt l, 0 m T m Ψ 0 (X) =(1+r) T T m=0 Q T,m X ( h m l T m S 0 ) 2.28 2.3.3 Laplace C X f X ( ) Laplace m : Z Z ln R t Q m- R t m (s) =q h h s + q l l s,s Z (2.29) z =ln X S 0 z z {, 0, +}
44 2.9 0 X T { C 0 = X (1 + r) T m L 1 T } (s) (z) (2.30) s (s +1) Re (s) =σ< 1 X (S T )=(S T X) + (2.30) CRR (2.28) (2.30) m T (s) = ( q h h s + q l l s) T = { m L 1 T } (s) (z) = s (s +1) = = T m=0 T m=0 T m=0 (m ln h+(t m)lnl)s Q T,m e { } e Q T,m L 1 [m ln h+(t m)lnl]s (z) s (s +1) } + Q T,m {e z+m ln h+(t m)lnl 1 T ( Q T,m h m l T m S 0 X ) + /X m=0 Laplace F (s) = L 1 {F (s)} (x) = ( e x 1 ) + L 1 { F (s)e as} (x) = ( e a x 1 ) + 1, Re (s) < 1 s (s +1) Laplace CRR ( 10 ) Laplace 2.3.4 Black Scholes CRR Black Scholes T>0 [0,T] n T n Rn f =e rt n +o(n 1 ) { } n R (n) k k=1 h n =e μt n +σ T n +o(n 1 ), ln =e μt n σ T n +o(n 1 )
2 45 μ σ σ σ μ n h n >Rn f >l n q (n) h = Rf n l n h n l n (2.31) Q n R (n) k ( ) m n (s) =q (n) h h s n + 1 q (n) h ln s (2.32) Q n m n n ( ) n k=1 R (n) k n CRR { 2.3 CRR (2.30) Laplace C BS 0 = Xe rt L 1 { m T BS (s) s (s +1) C (n) 0 } n=1 } (z) (2.33) m BS (s) =e (r 0.5σ2 )s+0.5σ 2 s 2 Re (s) =σ< 1 z =ln X S 0 Black Scholes C BS 0 = S 0 N (d 1 [z]) Xe rt N (d 2 [z]) (2.34) ( ) r ± 0.5σ 2 T z N (x) d 1,2 [z] = σ T h s n l n s q (n) h h s n l s n q (n) h =e μs T n σs T n +o( 1 n ) =1 ( μs 0.5σ 2 s 2) T n σs =e μs T n +σs T n +o( 1 n) T n + o ( n 1) =1 ( μs 0.5σ 2 s 2) T T n + σs n + o ( n 1) = 1 2 μ r +0.5σ2 T ( 2σ n + o n 1/2) 2.32 m n (s) m n (s) =1 {( r 0.5σ 2) s 0.5σ 2 s 2} T n + o ( n 1) n m n n (s) lim n mn n (s) =e (r 0.5σ2 )Ts+0.5Tσ 2 s 2 = m T BS (s)
46 m T BS (s) N [( r 0.5σ 2) T,σ 2 T ] ( Rn) f n m n n (s) ( ) (2.30) n e rt m T BS (s) lim R f n n n (s) (2.33) 2.33 Laplace (2.34) lim n mn n κ m BS ( ) (2.33) m BS (s) =e (κ r+0.5σ2 )s+0.5σ 2 s 2 (2.35) Black Scholes C BS 0 = S 0 e κt N (d 1 [z]) Xe rt N (d 2 [z]) (2.36) ( ) r κ ± 0.5σ 2 T z d 1,2 [z] = σ z =ln X T S 0 2.3.5 ψ Q CRR (2.30) Laplace m (s) { } m (s) g (z) =L 1 (z),z R s (s +1) mt (s) s (s +1) g g r S 0 T X g G (s) =L{g (z)} (s) m (s) =s (s +1)G (s),s Z 2.10 g m (s) 2.11 X (S T ) Ψ 0 (X) m T (s) Laplace Ψ 0 (X) =(1+r) T R L 1 { m T (s) } (x ln S 0 ) X (e x )dx (2.37)
2 47 Ψ 0 (X) (2.28) m T ( )] Ψ 0 (X)=(1+r) T E Q [X T R t f ( ) =(1+r) T =(1+r) T R R S 0 t=1 X ( e ln S0+x) f (x)dx f (x ln S 0 ) X (e x )dx (2.38) T ln R t Q m (s) R t t=1 T R t m T (s) t=1 f f (x) =L 1 { m T (s) } (x) (2.38) (2.37) m g m g r Ψ 0 ( ) ψ Q m g Rietz Laplace Fourier ( ) 2.3.6 ( ) t CRR ζ t (ω) = ξ s (ω) t ω ζ t {0, 1,,t} t Ψ t ζ t Ψ t {ζ t } Ψ t t ζ t Ψ t =Ψ(t, ζ t ) s=1 Ψ t = 1 1+r E Q [Ψ t+1 F t ] CRR Breeden and Litzenberger (1978) Ma (1992)
48 {Ψ t } t =0, 1,,T Ψ(t, ζ t )=(1+r) 1 [q h Ψ(t +1,ζ t +1) +q l Ψ(t +1,ζ t )] (2.28) T 0 m T Ψ(T,m)=X ( h m l T m S 0 ) 0 t<t 0 m t Ψ(t, m)=(1+r) 1 [q h Ψ(t +1,m+1)+q l Ψ(t +1,m)] Ψ 0 (X) Ψ(0, 0) 0 1. {h, l} t r t t r t r t = r (t, ζ t ) t 0 m {0, 1,,t} h>1+r (t, m) >l t ζ t q h (t, ζ t )= 1+r (t, ζ t) l =1 q h l l (t, ζ t ) (2.39) {Ψ t } {ζ t } Ψ t =Ψ(t, ζ t ) Ψ(t, ζ t )= q h (t, ζ t )Ψ(t +1,ζ t +1) +q l (t, ζ t )Ψ(t +1,ζ t ) 1+r (t, ζ t ) (2.40) {q h (t, ζ t )} X (S T ) T m =0, 1,,T Ψ(T,m)=X ( ) h m l T m S 0 0 t<t m =0, 1,,t Ψ(t, m)= q h (t, m)ψ(t +1,m+1)+q l (t, m)ψ(t +1,m) 1+r (t, m) (2.41) Ψ 0 (X) Ψ(0, 0)
2 49 2. CRR {h, l} {r t } {κ t } {ζ t } r t = r (t, ζ t ) κ t = κ (t, ζ t ) t 0 m {0, 1,,t} h>1+r (t, m) κ (t, m) >l t q h (t, ζ t )= 1+r (t, ζ t) κ (t, ζ t ) l =1 q h l l (t, ζ t ) t m {0, 1,,t} h> 1+r (t, m) 1+κ (t, m) >l q h (t, ζ t )= (1 + r (t, ζ t)) / (1 + κ (t, ζ t )) l =1 q l (t, ζ h l t ) (2.42) {Ψ t } (2.40) 2.3.7 {S t } R f =1+r {X t } T t=0 (t, ω) (t, ω) X t (ω) S X (t, ω) X t (ω) =S t (ω) X T T t ω F t (ω) ω t Ψ t (X; ω) (t, ω) Q 2.4 {Ψ t (X)} T t=0 τ (1) (t, ω) Ψ t (X; ω) =max { } X t (ω), (1 + r) 1 E Q [Ψ t+1 (X) F t (ω)] (2.43)
50 (2) τ (ω) =t Ψ s (X; ω) >X s (ω), s<t Ψ t (X; ω) =X t (ω) τ (ω) =inf{t {0, 1,,T} :Ψ t (X; ω) =X t (ω)} (2.44) τ ( τ F t - ) ω Ω τ (ω) {0, 1,,T} ω τ (ω) =t τ d t [ (ω) =X t (ω) τ (ω) t d t (ω) =0 E Q (1 + r) τ X τ ] d Q [ ] Ψ 0 (X) sup E Q (1 + r) τ X τ 0 τ T ( τ ) [ Ψ 0 (X) sup E Q (1 + r) τ X τ ] 0 τ T τ ] Ψ 0 (X) = max E Q [(1 + r) τ X τ (2.45) 0 τ T (t, ω) [t, F t (ω)] Ψ t (X; ω) T t T τ (ω) >t Ψ t (X; ω) = 1 1+r E Q [Ψ t+1 (X) F t (ω)] τ (ω) =t s<t Ψ s (X; ω) >X s (ω) Ψ t (X; ω) =X t (ω) (1) (2) T X 2.12 T τ = T (t, ω) S t (ω) X (t, ω) ] C t (ω) (1 + r) (T t) E Q [(S T X) + F t (ω) (1 + r) (T t) E Q [S T X F t (ω)] (T t) = S t (ω) X (1 + r) r>0 t<t Ψ t (X; ω) >S t (ω) X t<t S T (ω) >X
2 51 S t (ω) X t >t S t (ω) X {S t } T t=0 {h, l} - r t = r (t, ζ t ) κ t = κ (t, ζ t ) Q (2.42) Ψ t,m t {X t (S t )} T t 0 t m t m t X t,m = X t (h m l t m S 0 ) CRR {Ψ t } T t=0 T m =0, 1,,T Ψ T,m = X T,m 0 t<t m =0, 1,,t { Ψ t,m =max X t,m, q } h (t, m)ψ t+1,m+1 + q l (t, m)ψ t+1,m 1+r (t, m) Ψ 0 (X) Ψ 0,0 τ {Ψ t,m,m (0, 1,,t)} T t=0 ω Ω t [ τ (ω) =t] (1) Ψ (t, ζ t (ω)) = X t (ζ t (ω)) (2) s<t Ψ(s, ζ s (ω)) >X s (S s (ω)) 2.4 Ho Lee 1 B t,t t T ( T + t) T B t,t Ho Lee Ho Lee 2.4.1 y t,t = 1 T ln B t,t y t,t T T y t,t
52 T s t+1,t s t+1,t =lnb t+1,t ln B t,t +1 ln R f t =lnb t+1,t ln B t,t +1 +lnb t,1 (2.46) T s t+1,t T t π t,t = E t [s t+1,t ] s t+1,t = π t,t + ξ t+1,t E t [ξ t+1,t ]=0 y t+1,t y t,t +1 t y t,t +1 y t,1 T Irving Fisher (1930) Fisher Fisher EH-a 1 T (y t,s+1 y t,s ) y t+1,t y t,t +1 T s=1 E t [y t+1,t y t,t +1 ]= y t,t +1 y t,1 (2.47) T EH-b ln R f t t T π t,t =0 ln R f t = E t [ln B t+1,t ln B t,t +1 ] (2.48) 2.4.2 y t+1,t y t,t +1 y t,t +1 y t,1 s t,t
2 53 2.5 t 0 T 1 y t+1,t y t,t +1 = y t,t +1 y t,1 T s t+1,t T (2.49) E t [ξ t+1,t ]=0 y t+1,t y t,t +1 = y t,t +1 y t,1 T π t,t T + ξ t+1,t (2.50) ln B t+1,t ln B t,t +1 ln R f t = Ty t+1,t +(T +1)y t,t +1 y t,1 = T (y t,t +1 y t+1,t )+(y t,t +1 y t,1 ) (2.49) (2.50) (1) y t+1,t y t,t +1 y t,t +1 y t,1 EH-a T (2) EH-a EH-b (3) {s t,t } {y t,t } {y t,t } {s t,t } (4) y t+1,t y t,t +1 (2.50) 1 ( ) (5) σ t [y t+1,t ]= 1 T σ t [s t+1,t ]
54 2.4.3 Fisher Sola and Driffill (1994) Gerlach and Smets (1997) Campbell and Shiller (1991) Backus, Foresi, Mozumdar and Wu (2001) Duffee (2003) y t,t +1 y t,1 y t+1,t y t,t +1 = α 0 + α 1 + ε t+1,t (2.51) T y t+1,t y t,t +1 y t,t +1 y t,1 T EH-a EH-b π t,t 0 (2.51) y t+1,t y t,t +1 = α 0 + α 1 y t,t +1 y t,1 T H 0 : α 0 =0,α 1 =1,α 2 = 1 + α 2 π t,t T + ε t+1,t (2.52) (2.52) π t,t π t,t 2.4.4 Ho Lee CRR {ξ t } t=1 t ξ t 1 0 1 0 Ω {ξ t } t=1 ω Ω F t t ω {ξ t } t=1 ξ t t u : {1, 0} T R + t T B t+1,t = B 1 t,1 B t,t +1u (ξ t+1,t) (2.53) t {B t,t } T =0 F t- B t,0 =1 B t,t t t ζ t ζ t = ξ s 2.2 s=1
2 55 2.2 B t,t = B t,t (ζ t ) t m B t,t (m) t m t T +1 t R t+1,t = B 1 t,t +1 B t+1,t R t+1,t = R f t u (ξ t+1,t) (2.54) R f t = Bt,1 1 t u (ξ,t) F t (1,T)=Bt,1 1 B t,t +1 (t +1 ) T B t+1,t = F t (1,T) u (ξ t+1,t) (2.55) t +1 u (ξ,t) {u (0,T),u(1,T)} 2.13 q (0, 1) γ (0, 1) u u(1,t)= ( q +(1 q ) γ T ) 1, u(0,t)=γ T u(1,t) (2.56) (t, ω) q 1 (t, ω) =Q (ξ t+1 =1 F t (ω)) = 1 q 0 (t, ω) u j (T )=u(j, T ),j =0, 1 (t, ω) T q 1 (t, ω) q 1 (t, ω) u 1 (T )+[1 q 1 (t, ω)] u 0 (T )=1 q 1 (t, ω) = 1 u 0(T ) u 1 (T ) u 0 (T ) =1 q 0 (t, ω) q 1 (t, ω) q {u 1 (T ),u 0 (T )} q = 1 u 0(T ) (0, 1) T =0, 1, (2.57) u 1 (T ) u 0 (T )
56 B 1 t,1 (m) B t,t +1 (m) u 0 (T )=B 1 t,1 (m 1) B t,t +1 (m 1) u 1 (T ) B t+1,t (m) B t,t +1 (m) B t,t +1 (m 1) = B t,1 (m) u 1 (T ) B t,1 (m 1) u 0 (T ) u 1 (T +1) u 0 (T +1) = u 1(1) u 1 (T ) u 0 (1) u 0 (T ) γ = u 0(1) (0, 1) (2.58) u 1 (1) (2.58) u 1 (T )=γ T u 0 (T ) (2.59) (2.57) (2.59) u 1 (T ) u 0 (T ) 2.4.5 2.13 u (0, 1) q γ 2.13 2.14 {B 0,T } T =0 q (0, 1) γ (0, 1) ln B t,t =lnb t,t +(t ζ t ) T ln γ (2.60) { } B t,t t T =0 Δ B 0,T +t u 1 (T + t 1) u 1 (T + t 2) u 1 (T ) B t,t = B 0,t u 1 (t 1) u 1 (t 2) u 1 (0) (2.61) {B 0,T } T =0 u (2.56) t 1,T 0, m =0, 1,,t ζ t = B t,t (m) =B t,t γ (t m)t (2.62) t ξ s t (2.62) s=1 (2.60) B t,t = B t,t γ tt
2 57 {B 0,T } T =0 (2.60) u 1 (T )=[q +(1 q ) γ T ] 1 u 0 (T )=u 1 (T ) γ T Q Q {ξ t =1} = q =1 Q {ξ t =0} T (2.60) t ξ t (q, 1; 1 q, 0) t T (2.60) 2.4.6 E 0 [ln B t,t ]=lnb t,t +(1 q) tt ln γ σ 0 [ln B t,t ]=T ln γ q (1 q) t (2.62) (T =1) R f t R f t (m) =B 0,t B 1 0,t+1 u 1 1 (t) γ m t (2.60) = B 1 t,1 ln R f t =lnrf t +(ζ t t)lnγ, t 1 (2.63) R f t = B 1 t,1 = B 0,t B 1 0,t+1 u 1 1 (t) t R f 0 = B 1 0,1 2.4.7 T t Y t,t = B 1/T t,t [ ( ) ] 1 B t,t = E Q R f t Rf T +t 1 = Y T t,t (2.60) {Y t,t } ln Y t,t = 1 T ln B t,t +(ζ t t)lnγ =lny t,t ln R f t +lnr f t (2.64) Y t,t = B 1/T t,t T ln Y t,t ln Y t,t 1 Ho Lee
58 2.4.8 t Δ 0 T 1 F t (T,Δ) t T +Δ T ( t + T ) T t + T t + T +Δ ft Δ (T,Δ) = Δ F t (T,Δ) Y t,t = f t (0,T) R f t = f t (0, 1) {B t,t } B t,t +Δ = B t,t F t (T,Δ) f t (T,Δ) ln f t (T,Δ) = ln B t,t ln B t,δ+t Δ f t (, ) T =0 B t,δ = f T t (0, Δ) (2.60) ln f t (T,Δ) = ln R f t ln Rf t + ln B t,t ln B t,t +Δ Δ (2.65) 2.4.9 Δ=1 T f t,t Δ = ft (T,1) ln f t,t =lnr f t ln R f t +lnb t,t ln B t,t +1 (2.66) 2.4.1 f t,t ln R f t+t ] EH-c ln f t,t = E t [ln R f t+t Ho Lee EH-c 2.63 2.66 ln R f T ln f 0,T =ln [ q +(1 q ) γ T ] +(ζ T T )lnγ ] E 0 [ln R f T ln f 0,T =ln [ q +(1 q ) γ T ] T (1 q)lnγ 0 2.23 q q ] q q T > 0 E 0 [ln R f T ln f 0,T > 0 q <q ] T > 0 E 0 [ln R f T ln f 0,T > 0 T>T
2 59 2.3 A q q; B: q <q 2.4.10 {X t (m),m (0, 1,,t)} T t=1 (t, m) X t (m) T [ ( ) ] 1 Ψ 0 (X) =max E Q R f 0 τ 0 Rf 1 Rf τ 1 Xτ (2.67) R f 1 =1 2.14 { } Q R f t q,γ (0, 1) q,γ (0, 1) (2.67) Ho Lee q,γ (0, 1) CRR T m {0, 1,,T} Ψ T (m) =X T (m) 0 t<t m {0, 1,,t} R f t (m) = B 0,t γ m t B 0,t+1 u 1 (t) { } Ψ t (m) =max X t (m), q Ψ t+1 (m +1)+(1 q )Ψ t+1 (m) R f t (m) Ψ 0 (X) =Ψ 0 (0) 2.7 980 3 1 000 q =0.5 γ =0.997
60 [B 0,1,B 0,2,B 0,3,B 0,4,B 0,5 ]=[0.982 6, 0.965 0, 0.947 4, 0.929 6, 0.911 9] B 0,t 3 t t =4 m 3 $1 000 B 4,1 (m) T =4 Ψ 4 (m) =X 4 (m) = [980 1 000 B 4,1 (m)] + B 4,1 (m) = B 0,5 u 1 (4)γ 4 m,m=0, 1,, 4 B 0,4 t =0 (2.56) Ψ t (m) = Ψ t+1 (m +1)+Ψ t+1 (m) 2 m =0, 1,,t B 0,t+1 u 1 (t)γ t m B 0,t [u 1 (4),u 1 (3),u 1 (2),u 1 (1)] = [1.006, 1.004 5, 1.003, 1.001 5] [Ψ 4 (0), Ψ 4 (1), Ψ 4 (2), Ψ 4 (3), Ψ 4 (4)] = [4.944, 2.00, 0, 0, 0] 0.75 [Ψ 3 (0), Ψ 3 (1), Ψ 3 (2), Ψ 3 (3)] = [3.39, 0.98, 0, 0] [Ψ 2 (0), Ψ 2 (1), Ψ 2 (2)] = [2.14, 0.48, 0] [Ψ 1 (0), Ψ 1 (1)] = [1.28, 0.24] Ψ 0 (0) = 0.75 2.5 APT APT APT Ross (1976) APT (1) m {f k } m k=1 m r j = μ j + β jk f k + ε j (2.68) k=1 μ j = E [r j ] j ε j j β jk = σ 2 [f k ]Cov(r j,f k ) f k
2 61 {f k } m k=1 {ε j} J j=1 E [ε j f 1,,f m ]=0 (2) dq dp = ξ (f 1,,f m ) (2.69) 2.1 (1) Fama and French (1992) (1) (1) E [ε j ]=0 j m σ 2 [r j ]= βjkσ 2 2 [f k ]+σ 2 [ε j ] (2.70) k=1 σ 2 [ε j ] ε j j σ 2 [f k ] β jk k j (2) Q ε j Q E Q [ε j ]=E [ξ (f 1,,f m ) ε j ]=E [ξ (f 1,,f m ) E [ε j f 1,,f m ]] = 0 (2.71) 2.1 Q r ( 5.21) m r = μ j + β jk E Q [f k ] APT k=1 E [r j ]=r + m β jk λ k (2.72) λ k = E Q [f k ] 2.72 μ j r {β jk } λ k k APT ε j k=1 2.6 β APT β APT
62 (Ω, P) {r j } S1 S2 π>0 j 1 +r π = 2.73 π - 2.6 S2 E [π (1 + r j )] = 1 (2.73) π E [π 2 ] > 0 r j = r + β j [π](r π r)+ε j π (2.74) β j [π] =σ 2 [r π ]Cov(r j,r π ) E [ ( επ] j =Cov rπ,επ) j =0 E [r j ]=r + β j [π](e [r π ] r) (2.75) 2.73 E [r j ] r = Cov (r j,r π ) 1+E [r π ] (2.76) π E [r π ] r = σ2 [r π ] 1+E [r π ] < 0 (2.77) 2.75 j (2.75) (2.74) ε j π Δ = r j r β j [π](r π r) E [ ( επ] j =0 Cov rπ,επ) j =Cov(rπ,r j ) β j [π] σ 2 [r π ]=0 2.74 2.74 APT 2.75 β j [π] E [r π ] r π π 2.74 2.74
2 63 2.7 - - - Cox Ross and Rubinstein (1979) Ho and Lee (1986) Black and Scholes (1973) CRR Heath Jarrow and Morton (HJM 1992) Ho Lee Musiela and Rutkowski (1997) Hull (2000) Sercu and Uppal (1997) APT Ross (1976) APT Fama and French (1992) Fama French Campbell, Lo and MacKinlay (1997) β 2004 2010 Q {r t } Ho Lee (0, 1) q γ Ho and Lee (1986) Campbell, Lo and MacKinlay (1997)
(VaR) (C-VaR) 3.1 Stone Stone (1973) X R ( k, X,X ) = ( ]) E [ X X k 1 k 1 {X X} (3.1) k X X X X X k =2 X = E [X] X = Stone ( ]) E [(X E [X]) 2 1 [ 2 1 {X E[X]} E (E [X] X) +] ( ]) E [ X E [X] k 1 k 1 {X E[X]} Stone
3 65 3.1 X Y 5 000 X X = {0.5, 0; 0.5, 10 000} ; Y f Y (x) = 1 5 000 e x 5 000,x 0 Y X Y X Y X 50% 5 000 Y 5 000 Y Stone 3.2 (Ω, F) A x, y A x + y A a>0 c R,ax+ c A R A R : A R x R (x) x (1) R (ax) =ar (x), a [0, ) (2) x y R (x) R (y) (3) R (x + c) =R (x)+c c R (4) R (x + y) R (x)+ R (y) (5) R [αx +(1 α) y] αr(x)+(1 α) R (y), α [0, 1] c Artzner, Delbaen, Eber & Heath (1999) (1) (4) (5)
66 Artzner 3.1 R (Ω, F) Δ R (x) =supe Q [x] (3.2) Q Δ 3.1 (2012) 3.1 Δ Δ P Δ P Stone 3.2 3.3 Stone ( 3.1) VaR C-VaR Stone J.P. RiskMetrics 3.1 3.3.1 VaR VaR W α (0, 1) VaR(α) α u VaR(α) P {W W 0 u} α (3.3) W 0 X = W 0 W
3 67 VaR VaR 3.2 X Y p F 1 p G { F, α (0,p) VaR X (α) =VaR Y (α) = G, α (p, 1) X + Y 2F, α (0,p 2 ) VaR X+Y (α) = F G, α (p 2, 2p p 2 ) 2G, α ( 2p p 2, 1 ) α ( p, 2p p 2) VaR X+Y (α) =F G> 2G =VaR X (α)+var Y (α) F W (x) = P {W x} F W (α) =inf{x Δ : F W (x) α} VaR VaR (α) =W 0 F W (α) (3.4) α F 1 W (α) VaR (α) =W 0 F 1 W (α) =W 0 E [W ]+c W,α σ [W ] (3.5) { } W E [W ] c W,α P c W,α = α α - σ [W ] E [W ] σ [W ] c W,α c W,α c W,α = c α α- 3.3 W 0 = 10 000, μ =1.1 σ =0.2 α =0.025 c α =1.96 VaR = 10 000 (1 1.1+1.96 0.2) = 2 920 2.5% 2 920
68 VaR 3.1 X Y { 5 000, α (0, 0.5) VaR X (α) = 5 000, α (0.5, 1) VaR Y (α) = 5 000 [1 + ln (1 α)],α (0, 1) α (0, 0.5) ( 1 e 2, 1 ) VaR X (α) > VaR Y (α) α ( 0.5, 1 e 2) VaR X (α) < VaR Y (α) 3.3.2 C-VaR VaR α VaR (α) VaR (α) α (0, 1) α C-VaR C-VaR (α) Δ = E [X X VaR (α)] (3.6) C-VaR (α)=var(α)+e[x VaR (α) X VaR (α)] =VaR(α)+ 1 { E [X VaR (α)] +} (3.7) α + Δ [ α + = FW F W (α)] α x = F W (α) VaR C-VaR α VaR C-VaR 3.1 { 5 000, α (0, 0.5) C-VaR X (α) = 5 000 (1/α 1), α (0.5, 1) C-VaR Y (α) = 5 000 (1 1/α)ln(1 α),α (0, 1) C-VaR X (α) >C-VaR Y (α) α ( 0, 1 e 1) α VaR C-VaR α (0, 0.5) C-VaR X > C-VaR Y VaR X > VaR Y α ( 0.5, 1 e 1) C-VaR X > C-VaR Y VaR X < VaR Y α ( 1 e 1, 1 e 2) C-VaR X < C-VaR Y VaR X < VaR Y α ( 1 e 2, 1 ) C-VaR X < C-VaR Y VaR X > VaR Y
3 69 VaR C-VaR X Y VaR (1) (3) C-VaR C-VaR (4) VaR C-VaR 3.2 VaR C-VaR 3.1 W E [ W ] < α (0, 1) g g (x) Δ = x + E [(X x) +] g VaR(α) C-VaR(α) α +, x R (3.8) C-VaR (α) =ming (x) =g [VaR (α)] (3.9) x R C-VaR(α) =g [VaR (α)] g g (x) =1 P {X x} α + g (x) =0 x =VaR(α) g C-VaR (α) =g (VaR (α)) = min x R g (x) X = W 0 W Y = W 0 W 3.1 X Y α (0, 1) C-VaR X+Y (α) C-VaR X (α)+c-var Y (α)
70 F W +W [ F W +W (α)] [ α = F W F W (α)] [ = F W F W (α)] α (0, 1) α (0, 1) 3.1 x, y E [(X + Y x y) +] C-VaR X+Y (α) x + y + [ F W +W F W +W (α)] E [(X + Y x y) +] x + y + [ α E (X x) +] + E [(Y y) +] x + y + α E [(X x) +] E [(Y y) +] = x + [ F W F W (α)] + y + [ F W F W (α)] x =VaR X (α) y =VaR Y (α) 3.1 C-VaR X+Y (α) C-VaR X (α)+c-var Y (α) 3.4 W W 0 W C E [W ] C W CE [W] W CE[W ] W ζ [W ] W E [W ] ζ [W ] ζ [W ]=E[W] CE [W ] (3.10) CE[W ] < E [W ] u ζ [W ]=E[W] u 1 {E [u (W )]} (3.11) 3.3 u ( )
3 71 3.3 3.4.1 Arrow Pratt x W = x + ε ε 0 σε 2 W ζ [W ]=ζ a [x, σ ε ]+o ( σε 2 ) ζ a [x, σ] = σ2 2 ARA u (x) ARA u (x) = u (x) u (x) ARA u u Arrow Pratt ζ a Arrow Pratt x ε ε ζ a x ζ a [x, σ] x IARA ARA (x) > 0 ζ a [x, σ] x CARA ARA (x) =0 ζ a [x, σ] x DARA ARA (x) < 0 x IARA DARA CARA 3.4.2 Arrow Pratt x W = x (1 + ε) ε 0
72 σ 2 ε x ζ [W ] x = ζ r [x, σ ε ]+o ( σε 2 ) (3.12) ζ r [x, σ] = σ2 2 RRA u (x) RRA u (x) = xu (x) u Arrow-Pratt (x) ζ r Arrow-Pratt x ε ε x ζ r [x, σ] x IRRA RRA u (x) > 0 ζ r [x, σ] x CRRA RRA u (x) =0 ζ r [x, σ] x DRRA RRA u (x) < 0 x ( ) x 3.4.3 i j i j i j 3.1 i( i ) j( j ) W ζ i [W ] ζ j [W ] CE i [W ] CE j [W ] (3.13) i j x i j x i j i j Arrow-Pratt 3.2 u i i u j j x σ 3.2 ζa i [x, σ] ζj a [x, σ] (3.14) (1) x σ ζ i a [x, σ] ζj a [x, σ] (2) x ARA ui (x) ARA uj (x)
(3) u i u j 3 73 (4) W ζ i [W ] ζ j [W ] ( (1) (2) (3) f (z) =u i u 1 j (z) ),z R ζ i a [x, σ] ζ j a [x, σ], x, σ ARA ui (x) ARA uj (x), dln u i (x) u j (x) u j x 0, x dx u i (x) u x j (x) [ f (z) = u i u 1 j (z) ] [ ] u 1 z j (z) f (z) < 0, (3) (4) f z f E {f [u j (W )]} f {E [u j (W )]} E [u i (W )] f {E [u j (W )]} u 1 i {E [u i (W )]} u 1 j {E [u j (W )]} ζ i [W ] ζ j [W ] 3.2 x i j i j i j x i j Arrow-Pratt 3.4 (3) (4) 3.4 u i (x) [0, 2] u j (x) [0, 1] [1, 2] x {0, 1, 2} u i (x) u i u j x [0, 2] u i u j u i u j u i u j u i = f (u j ) u i (0) = f [u j (0)], u i (1) = f [u j (1)], u i (2) = f [u j (2)] u i (0) = u j (0), u i (1) = u j (1), u i (2) = u j (2) f {u i (0),u i (1),u i (2)} 45 f [u i (0),u i (2)] f (x) x [0, 2] u i (x) u j (x) u i u j
74 3.4 3.5 VaR VaR C-VaR C-VaR VaR (α) VaR (α) K = W 0 VaR(α) max {W, K} c [W, K] c [W, K] max {W, K} c [W, K] W CE [max {W, K} c [W, K]] = CE [W ] (3.15) ( ) W W W K W c [W, K] >c[w,k] W W K VaR C-VaR CE[W ] c [W, K] CE [max {W, K} c [W, K]] E [max {W, K}] c [W, K] VaR C-VaR
3 75 c [W, K] E [max {W, K}] CE [W ] = E [max {W, K}] E [W ] }{{} + ζ [W ] }{{} ζ [W ] c [W, α] α 3.5 [ 1 X = 3, 0.2; 1 3, 2; 1 ] 3, 3 [ 2 Y = 3, 0.6; 1 ] 3, 4 E [X] =E [Y ]= 5.2 u (x) = 3 ln x K =2 X Y X Y ζ [X] = 5.2 3 1.2 1 3 > 5.2 3 1.44 1 3 = ζ [Y ] X Y K =2 2ln(2 c [X, 2]) + ln (3 c [X, 2]) = 3 2ln(2 c [Y,2]) + ln (4 c [Y,2]) 3 ln 0.2+ln2+ln3 3 = 2ln0.6+ln4 3 c [X, 2] 1.186 5 <c[y,2] = 1.273 3 K =2 X Y 3.6
76, VaR ( C-VaR) α (0, 1) 3.1 X Y X Y 3.6.1 VaR FSD VaR 3.3 x F X (x) F Y (x) X Y X FSD Y F X (x) =P {X x} X X Y X Y x X Y ( W 0 ) X VaR Y 3.2 X Y E [ X ] < E [ Y ] < X FSD Y VaR X (α) VaR Y (α), α (0, 1) (3.16) X FSD Y α (0, 1), α = F Y [W 0 VaR Y (α)] = F X [W 0 VaR X (α)] F Y [W 0 VaR X (α)] F Y ( ) VaR X (α) VaR Y (α) α (0, 1) VaR X (α) VaR Y (α) α = F X (x) x = F 1 X [F X (x)] = W 0 VaR X [F X (x)] W 0 VaR Y [F X (x)] = F 1 Y [F X (x)] F Y (x) F X (x) x R α (0, 1) VaR(α) X Y FSD 3.6.2