x3x3c Black–Scholesx48x4x52x53x63x6a
x57x58x7fx59x20 x1cx1dBlack–Scholesx55x71x43x16x4x7dx39x35x56x4fx29 x47x20Black–Scholesx4
x7dx18x48x35x43x16x60x2ex43x5cx23x14x20x27x28x29 x47x20Black–Scholesx4x7dx20Cox–Ross–Robinstein
(x22x70x40x1x52) x73x4dx45xdx29x52x3bx56x20x26x55x7fx18x47x20x48x35x43x16x60x2ex43x5cx20x21x69x7dx5x55x35
x47x7dx20xexfx29
x5ax2ex51x5bx5cx5dx20 x64x3bx56x21x33x32x33x64x44x67x5 x8x2cxex41x4fx7x20x7x76x5cx7ex43x5cx5x28x20
x72x42x38x39x58x53x62x3bx20x55x14x7ax32x33xexfx29x52x53x32x33x60x1bx2bx2x20x1fx70x5 x48x3x1cx3fx5cx20x20
x26xex44x4dx62x67x28x29x2ex4x7x79x6x38x39x58x53x0xbx25 x7dx33xexfx5 x3cx25x27x64x44x4ex23x6x2ex4x7
x5bxdx29
x66x5ex20
? 8.1 Black–Scholesx11x7dx55x2dx55x71x43x16x4x7dx29
? 8.2 Black–Scholesx4x7dx20x4x2dx29
? 8.3 Black–Scholesx4x7dx20Cox–Ross–Robinstein (x22x70x40x1x52)x73x4dx29
? 8.4x3bx56x20x25x7ex2bx68x26x55x7fx18x29
? 8.5x48x35x43x16x60x2ex43x5cx20x21x69x7dx29x7dx20xexfx29
? 8.6x2ax3bx56x20x26x55x7fx18x29x18x37x3ax43x16x52x28x20x5cx28x29
x25x70x51x53x65x63Black–Scholesx3exbx62x4ax69x63x78x17x40x6fx5ex55x49x65x70x51
x2cx19x57x27x28x1fx78xfx2x49x33x18x3ax3ex74x75x2x1x61x70x68x33x6dx3cx61xdx3x50x62xb
x21x1x10x3x69x6ax10x6x9x40x6fx25x22x48x51x36x51x1fx8x15x52x31x32x78x25x18x3ax3ex6cx79
x75x4x0x65x70x51x54x65x63x48x15x52x1fx8x52x53x54Black–Scholesx2ex26x3exbx62x4ax6e
x26x78x1x1fx8x25x70x6ex26x48x3ax3ex1cx7ax73x47x75xdx6dx3cx62x4ax63xfx48x58x2ex47x75xdx72
x73x62x4ax63xfx78
8.1 x3ax3ex79x27x3cx3cx37x34x2Black–Scholesx3exbx6ex26x78x57x4fx54Black–
Scholesx3exbx62x4ax6ex26x2x10x66x2ex26x6ax73x3exbx49x8x54x78x17x6ex26x1
C(S;T) = SN(d1)?Ke?rTN(d2);
x17x2cT x2x19x3ex18x76x48 Sx2x17xbx6ax4ax48 C(S;T)x2x2ex1x17xbx6ax4ax4x19x3ex18
x76x54x65x68x54x2ex26x70x1x3exbx54x4ax4bx57
d1 = 1 pT
?
log SK +
r +
2
2
?
T
?
; d2 = d1 ? pT;
183
K x2x3exbx54x7ax4fx4ax4bx48 rx2x33x69x6ax2cx68x54(x5x18)x20x21x27x48 xcx1x6ax4a
x54x12x13x21 (volatilityx48 (x7fx0x18x76)x20x21x27x54x38x2cx7dx57x18x76x76x2x54x6fx69x75
x77x48x25x2x70x2bx47x75x51x12x54x46x68)x57 N xcx1x6fx2x3bx71x33x6cx65x68x48x62x7ax1
N(d) = 1p2…
Z d
?1
e?y
2
2 dy:
x25x2bx6ex26x17T = 0x18x48x51C(S;0) = (S ?K)+,x17x2c(S ?K)+ x60x27
S ?K x54x3bx32x48x4x17S ?K > 0x18x48x10x9x66S ?K,xfx17S ?K < 0x18x48
x10x1x7dx78x25x67x2x70x1x3exbx49x19x3ex18x54x4ax6x48x10x47x44x66x6ax4ax57x7ax4fx4ax4b
x75x7dx78x58x17T > 0x18x48 C(S;T)x2ex1S x54x65x68x48x17xex44x41x7dx44x77x48xfx6a
x77x2ex30x78x17T x55x24+1x18x48xex44x55x24x66C = Sx45x31xe1)x78
-
6
0 S
C
K
6
T = 0
T = +1
x66 1 x8x43x63x7x70x75x39x36x8x4x45T x79x3bx67
Black–Scholes x6ex26x54x69x66x75x73x49x66x35x6ax4ax54x8x30x27 x19x48x17x62x46
x68x5bx2x18x38x49x42x43x44x23x24x60x19x54x78x79x70x49x67x4bx54x79x7ax56x48 r x2x33x69x6a
x2cx68x54x5x18x20x21x27x48xfx2x49x71x72x6bx12x2cx48x10x18x38x23x24x3bx5fx3ex3cx27x4ex1x78
x45x31x48x31x32S = 18, K = 15, r = 10%, T = 0:25, = 15%,x25x18x4ax4bx24x6b
x38x6bx48x18x76x24x46x6bx48x39x58x44x58x54x78x2bx77x27x5bx5dx54x2x46x27x48x33x34(x24x56x54
x9x49x71x72x44x5bx2x5dxdx9x49)
Ke?rT = 15e?0:1(0:25) = 14:6296;
d1 = log(18=15)+[0:1+(0:15)
2(0:5)]0:25
0:15p0:25 =
0:21013
0:075 = 2:8017;
184
d2 = d1 ?0:15p0:25 = 2:7267:
x5dx25xbx6x4ex1x6ex26x48x68x19
C = 18N(2:8017)?15e?0:1(0:25)N(2:7267):
x3cx3bx6fx2x3bx71x33x6cx65x68x49x772.8017x42.7267x73x54x5dxdx6x48x70x1x3exbx54
x4ax4bx23.3749x48x4
C = 18￠(0:997)?14:6296￠(0:996) = 3:3749:
x21x46x3cx54x6bx12x23x68C = 3:3714.
8.2 Black–Scholes x6ex26x54x8x1ax2ex2x14x7ax54x78x62x63x44x4bx57x51x1fx2dx2e
x72xdx54x6ex26x78x17x2cx15x5x23x38x15x19 1900 x46x55x56x68x4fx59 Bachelier (1870–
1946)x54x4bx4bx63x64x78x25xbx6ex26x2cx51x54x49x40x26x44x19x57x43x7fx38x5dx48xfx2x5bx7b
x49x29x69x44x36x51x4fx2cx75x7ex48x36x51x54x19 Black–Scholes x54x71x3bx78x3ax3ex56x4bx5c
x12x70xbx62x63x44x4bx57x51x1fx54x70xbx6ex26x48x17x2cx21x51x44x49x54x79x7ax5bx57xbx4bx70
x26x48x20x54x0x6fx2ax5cx54x44x49xex4dx4fx70x68x78
1. Bachelier (1900)x2a
C(S;T) = SN
S ?K
pT
?
?KN
S ?K
pT
?
+ pTn
K ?S
pT
?
nx2x3bx71x33x6cx54x60x3bx65x68x78
2. Sprenkle (1961)x2a
C(S;T) = e‰TSN(d1)?(1?A)KN(d2)
x17x2c
d1 = 1 pT
?
log SK +
‰+ 12 2
?
T
?
; d2 = d1 ? pT;
‰x2x6ax73x4ax4bx54x36x72x54x3dx27x48 Ax10x7x69x6ax68x50x6ax3bx78
3. Boness (1964)x2a
C(S;T) = SN(d1)?Ke?‰TN(d2);
x17x2c
d1 = 1 pT
?
log SK +
‰+ 12 2
?
T
?
; d2 = d1 ? pT:
185
4. Samuelson (1965)x2a
C(S;T) = Se?(‰?fi)TN(d1)?Ke?fiTN(d2)
x17x2c
d1 = 1 pT
?
log SK +
‰+ 12 2
?
T
?
; d2 = d1 ? pT:
fix2x3exbx4ax4bx54x36x72x54x3dx27x78
x3ax3ex23x24x3cx19x48x21x51x25xbx6ex26x5bx57x13x27x54Black–Scholesx6ex26x51x2dx2ex7b
xdx54x1ex69x78xfx2x21x51x25xbx6ex26x2cx5bx51x70x2bx6fx2ex2bx46x68x48x10x3ex49x45x66x4f
x1ex31x10x69x6ax6fx10x6ax73x20x21x27x54x7x28x9x78x7bx7ax48 Black–Scholesx6ex26x36x51
x25x26x54x46x68x78x7ax58x48 Black–Scholesx6ex26x2cx51x70x2bxbx7dx36x51x47x48x54x46x68x48
x33x67x2x33x69x6ax2cx68x54x20x21x27 r. x25x2x7bx1xbx7dx5bx36x51x47x48x1ex3dx54xdx18x76
x4ax6xfx78x2fx5ex51x7dx19x57x47x48x19x69x6ax7ex7fx10x6x54x14x25x48xfx26x66x36x51x1fx79x19
x33x69x6ax2cx68x14x25x54x20x21x27x7x8x9x66x33x69x6ax2cx68x54x20x21x27x48x58x20x53x70x16x78
8.3 x56x4bx3ax3ex27x3bx26x76x1f Black–Scholes x6ex26x78x25x70x76x1fx69x55x2
Cox–Ross–Rubinstein [5] x66 1979 x46x20x8x541x78x49x25x70x76x1fx1fx6ax2cx48x3ax3e
x73x3bx19x1ex37x62x4ax40x6fx62x69x48x5ax67x2x70x73x3bx19x33x6dx3cx61xdx48x17x62x5bx2x70
xbx37x1fx54x68x4fx5x12x78x5ex66Black–Scholesx6ax27x54x76x1fx69x55x3ax3ex53x49x24x13
x27x28x78
x3ax3ex79x27x1bx64x65x70x51x54x65x63x78x49x33x18x3ax3ex61x62x6ax73x54x17xbx4ax4bx1
S0,x6ax73x54x50x27x4ax4bx51x78x64x23x25x1dx2a aS0 x4bS0;x3exbx54x7ax4fx4ax4bx1K.
x33x34x3exbx49xdx50x27xfx23x25x51x54x78x64x4ax4bx1Ca = max(aS0 ?K;0)x4Cb =
max(bS0 ?K;0). x49x33x6dx3cx61xdx56x48x51x70x64x23x25x1dx4x6bx48x53x68xdx50x27xfx54
x6ax4ax54x36x72x6x68x28x67x9x66xdx17xbxfx54x6ax4ax78x25x2x7bx1ax4bx2cxcx7ax51x70
x2bx4ex661,x4dx70x2bx16x661,x39x58x66x490x571x75x76x54x68p (x25x27)x53x68
ap+(1?p)b = 1; x4 p = 1?ba?b:
x58xdx17xbxfx54x3exbx4ax4bx7x8x67x2x49x25x64x23x25x1dx4x6bx56x54xdx50x27xfx54x3exb
x4ax4bx54x36x72x6x48x4
C0 = pCa +(1?p)Cb = Ca ?Cb +aCb ?bCaa?b :
11979x20Rendleman–Bartterx30x7ax37[19]x7bx40x7cx7dx37x7ex56x23x7fx0x3fx3ax13x1x2x39x3x4x5[5]
x30x6x37x13x29x5ax7x1dSharpex8x9x7bxax40x7cx7dx37xbxcx23
186
x5dx25x2bx55x56x0x1x1x2x5cx48x3ax3ex61x62x33x69x6a(x22)x3cx272x1R. x33x34x17
xbx54x70x38x5bx19x61x50x27x2ax1(1 + R)x38x78x25x18x33x6dx3cx61xd(x1ex37x62x4ax40x6f
x62x69)x53x1fx5ex2a
pa+(1?p)b = 1+R: (1)
x26x7ex1fx68p = 1+R?ba?b ,x58x6ex26x53x2
C0 = 11+R (pCa +(1?p)Cb) = (1+R)(Ca ?Cb)+aCb ?bCa(1+R)(a?b) :
x1ax49x3ax3ex5dx55x56x13x4ex70x57x1x2x5cx48x61x62x21x47x48x18x76x51 0;1;:::;N
x2bx18x23x48x59x2bx18x23x54x5fx3ex22x4fx3cx27x1cx1 R. x33x34x17xbx54x70x38x5bx19 k x18
x23x13x67x2ax1 (1 + R)k x38x5bx48 k = 1;2;:::;N: x10x18x7fx61x62x59x2bx18x23x54x6a
x4ax2ax5cx1cx7ax73x51x78x64x23x25x48x4x2x18x23x76x54x6ax4ax77x6fx2 a, x6fx2 b, x17x2c
a < 1+R < b. xfx2N x18x23x13x54x6ax4ax2ax5cx70x71x512N x64x78x25x26x54x2ax5cx23
x24x31xe2x2cx33x26x27x60x27x78x25x2x70x2bx49x59x70x18x23x54x59x70x70x71x5bx51x78x2bx33
x53x54xex78x7bx58x25x64x69x55x7ex7fxcx1x1fx17x18x56x5ex78
ft = 0
B
B
B
BN
3
ft = 1
QQ
Qs
3
f
QQ
Qs
*ft = 2
HHHj
*f
HHHj
*f
HHHj
*f
HHHj
ft = 3
f
f
f
f
f
f
f
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
*ft = N ?1
HHHj
*f
HHHj
*f
HHHj
*f
HHHj
ft = N
f
f
f
f
f
f
f
... ... ...
... ... ... 2N x67x4dx6c
... ... ...
x662 xfx19x6ex79x4dx6cx3bx67x662x5xdx3fx37xex52x53x6fx70xfx39x7x42x10x6fx70x45x13x58x11x2x12x13x14x15x16x61x17x41x18x19x37x42x1ax6fx70x45x59x62
1x23
187
x252N x64x70x71x54x6ax4ax57x17xbx54x6ax4ax54x77x6x51xbx2x70x26x54x48x10x3ex33x3fx2
C0N x2baN;
C1N x2baN?1b;
:::::::::; (2)
CkN x2baN?kbk;
:::::::::;
CNN x2bbN:
x7bx7ex48x31x32x5dx77x6x70x26x54x70x71x3cx41x70x26x54x70x71x48x10x54x70x71xex53x31xe 3
(x26x66x10x57x8x66x26x62x69x54x3fx6fx2x7ex7fx70x26x54x48x2ex40xcx10x1xdx8x66x26x70x71
x2ax5cxexf)x2a
*ft = 0
HHHj
*ft = 1
HHHj
*f
HHHj
*ft = 2
HHHj
*f
HHHj
*f
HHHj
*ft = 3 H
HHj
*f
HHHj
*f
HHHj
*f
HHHj
ft = 4
f
f
f
f
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
￠￠￠￠￠￠￠￠￠
*ft = N ?1
HHHj
*f
HHHj
*f
HHHj
*f
HHHj
faNt = N
fC1NaN?1b
fC2NaN?2b2
fCN?2N a2bN?2
fCN?1N abN?1
fbN
... ... ... ...
... ... ... ... N x67x4dx6c
... ... ... ...
x663 xfx73x7cx4dx6cx3bx67x66
(x2ex26)x70x1x3exb(call option)x2x49N x18x23x13x48x24x7ax4fx4ax4bK x70x1
x70x49x6ax73x54xbx3cx78x25x64x1ax36x2cx68x3ax3exaxbx2ex1bx2ax10x17xbx7x8x6x2ex5fx5bx48
xfx2N x18x23x13x48x6ax4aSN x3cx62x13x48x10x54x4ax4bx5ax67x3cx62x61x48x4(SN ?K)
x54x3bx32(SN ?K)+,x10x49SN > Kx18x9x66(SN ?K),x58x49SN < Kx18x1x7dx48
x7bx1x25x18x3exbx1dx51x31x36x7ax2ex4dx12x7ax4fx62x54xbx3cx78x72xdx54x2(x2ex26)x3cx8
188
x3exb(put option),x10x2x49N x18x23x13x24x11x64x35x62x4ax4bK x3cx8x70x49x6ax73x54
xbx3cx78x17N x18x23x13x54x4ax4bx2(K?SN)+. x10x3ex5bx2x49N x18x23x13x512N x64
x23x25x54x50x62xbx21 (x1ax36x2cx68x48x68x4fx44x22N x75x24x6c(dk)k=1;:::;2N)x78x10x3ex54
x62x4ax5fx60x67x2x75x5fx10x17xbx6x2ex5fx78x25x2bx5fx60x3ax3ex19x57x1bx2ax23x3bx57x33x6d
x3cx61xdx9x4ax54x1ex37x62x4ax40x6fx62x69x27x1bx1cx78x25x18x3ax3ex67x23x3cx3bx44x29x54x9
x4ax25x27x51x3bx78x25x70x25x27x51x3bx53x68x6ax4ax20x21x27x2ax5cx54x36x72x6x7cx52x57x33x69
x6ax20x21x27 (x22x4fx3cx27)x70x26x48x39x58x26x10x27x3cx62x1ax36x2cx68x4ax4bx54x36x72x6x67
x2x10x17xbx54(x33x6dx3c)x4ax4bx78x7bx7ex48x1ax36x2cx68x17xbx54x4ax4b ˉdx7x8x9x66
ˉd = 1
(1+R)N E
?[(dk)]
= 1(1+R)N
0
@(1?p)Nd1 +
C1NX
j=1
(1?p)N?1pd1+j +
C2NX
j=1
(1?p)N?2p2d1+C1N+j
+￠￠￠+
CN?1NX
j=1
(1?p)pN?1d1+C1
N+￠￠￠+C
N?2
N +j
+pNd2N
1
A; (3)
x17x2c E? x60x27x10x28x29x2ax6c (dk) x15x44x29x9x4ax25x27x51x3bx39x68x4fx3ex2bx48x10x67
x9x66x1fx14x54x39x4x6x78xbx4bx54x7bx141=(1 + R)N xex2x5dx10x13x1ax1x15x17xbx54
xdx2ex2ax4ax4bxfx27x6bx12x78
x2bx71x44x48x49x44x29x55x56x56x48x2ex26x70x1x3exbx54x17xbx4ax4bC(N)0 x67x7x8x2
C(N)0 = 1(1+R)N
NX
k=0
CkNpN?k(1?p)k?SaN?kbk ?K￠+ ; (4)
x25x18S x2x17xbx54x6ax4a(x3ax3ex37x0x61x56x380)x48x58N x18x23x13x54x6ax4ax2x28x29
x2ax6c (2N x75x24x6c) SN, x10x54 2N x2bx33x6cx26 (2) x26x2cx54x7bx14x57x17xbx54x6ax4a
S x54x16x2x27x3cx62x78x10x26x48x2ex26x3cx8x3exbx54x17xbx4ax4bP(N)0 x7x8x2
P(N)0 = 1(1+R)N
NX
k=0
CkNpN?k(1?p)k?K ?SaN?kbk￠+ : (5)
x26(4), (5), (1)x24x58x8x66x26x62x69x48x7dx4x23x68
C(N)0 ?P(N)0 = 1(1+R)N
NX
k=0
CkNpN?k(1?p)k?SaN?kbk ?K￠ = S? K(1+R)N :
(6)
189
x10xcx1x51x1ax24x52x1bx72x31x76x28x33x1c(call-put parity)x78x25x2bx9x26x5ax23x26x33x6d
x3cx61xdx18x38x1fx8x57x4x17x25x2bx3x4cx26x2ex9xax18x70x62x66x49x6dx3cx29x4dx78x56x4b
x3ax3ex53x7ex1fN !1x27x1fx8Black–Scholesx6ex26x78
x1x7ex3ax3ex61xdx18x76x54x5x3dx3bx2 T, x10x11 N x9x33x78x3cx27 R = rT=N,
x17x2crx2xdx5x18x3cx27xfx48x4
erT = lim
N!1
(1+R)N = lim
N!1
1+ rTN
?N
:
x7fxd
log a1+R = ? TpN; log b1+R = TpN: (7)
x25x26x54x61xdx10a;bx51xcx5ax75x39x78xfx2x25x18x73x2x1x61x28x5cx6bx12x78x2bx71x44x48
x2ex2ex25x26x54xcx5ax61xdx51x2ex13x14x56x4bx54x6bx12x31x32x78x23x24x2cx68x48 2T x2x28x29
x2ax6clog(SN) (x6fx31x70x48x47x6ax4bx54x8x66x33x6cx28x29x2ax6cx48x10x57log(SN)
x7dx70x2bx7fx68)x54x69x7dx17N ! 1x18x54x24x2ax78x17N ! 1x18x48x3ax3ex47x75x8
x66x33x6cx55x24x66x3bx71x33x6cx54xcx5ax2cx21x24x2ax62x69x57x4
x39x3f 8.1 x20(YN)x67x57x21x7bx7cx79x39x3ax3bx3cx7fx7d
YN = XN1 +XN2 +￠￠￠+XNN;
x7ex7fx6dx6ex3dxcN,x39x3ax3bx3cXNi , i = 1;:::;N,x25x1dx50xbx6bx73x5ex78x79x49x6dx6b
x2x24xbx60x0xcx1cf? T=pN; T=pNg,x48x7bx73xaxblimN!1(N?N) = ?x79
x65x66x8x9x1c?N. x5dx5ex42x7f(YN)x1ex49x6dx1fx48x6ex4x1cx1d?,x66x20x51x1d T x79
x68x6cx39x3ax3bx3cx0
x1x2cx68x25x70x62x69x48x73x47x47x48YN x54xcx39x65x68`YN:
`YN(u) = E￡eiuYN? =
NY
j=1
E￡eiuXj?
=
?
E
h
eiuXN1
i·N
=
1+iu?N ?
2
Tu
2
2N +o(1=N)
?N
:
x7bx7ex48 limN!1`YN(u) = exp(iu??( 2Tu2=2));x39x58x0x69x41x7dx78
x1ax49x3ax3ex47
XNj = log
T
j
1+R
?
; YN =
NX
j=1
log
T
j
1+R
?
;
190
x17x2cTj x2x47ax6fbx54x8x66x33x6cx28x29x2ax6cx78x47x48x19x44x4bx54x10a;bx54x61xdx48
XNj x4ex25x62x69x61xdx78x5d(5)x26x2cx62x7ax54x36x72x6x69x1ex1x24E? x60x27x54x68x4f
x3ex2bx48xex10x23x24x45x44x1
P(N)0 =
1+ rTN
??N
E?
2
4
?
K ?S
NY
j=1
Tj
!
+
3
5
= E?
"?
1+ rTN
??N
K ?SeYN
!
+
#
:
x4dx70x69x4bx48x26(7),x3ax3ex7fx51
E?[XNj ] = (1?2p) TpN = 2?e
T=pN ?e? T=pN
e T=pN ?e? T=pN
Tp
N;
x39x58
lim
N!1
NE?[XNj ] = ?
2
T
2 :
x25x26x48x12?(y) = (Ke?rT ?Sey)+,x23x68
flfl
flP(N)0 ?E?[?(YN)]
flfl
fl
=
flfl
flfl
flE
?
"?
1+ rTN
??N
K ?SeYN
!
+
??Ke?rT ?SeYN￠+
#flfl
flfl
fl
6 K
flfl
flfl
fl
1+ rTN
??N
?e?rT
flfl
flfl
fl! 0:
x3cx3bx44x29x0x698.1x48x51x1fx79x19?x2x51x5bx54x7fx0x65x68x48x67x51
lim
N!1
P(N)0 = lim
N!1
E?[?(YN)] = 1p2…
Z +1
?1
Ke?rT ?Se?
2T
2 + Ty
?
+
e?y
2
2 dy:
x13x12
N(d) = 1p2…
Z d
?1
e?x22 dx;
d1 = log
S
K +rT +
2T
2
T ; d2 = d1 ? T;
x15x13x68x19
lim
N!1
P(N)0 = Ke?rTN(?d2)?SN(?d1);
191
x24x58
lim
N!1
C(N)0 = lim
N!1
?
P(N)0 +S ?K
1+ rTN
??N!
= SN(d1)?Ke?rTN(d2):
x4x24x18x23Tx1x19x3ex12x3x17xbx6ax4ax1Sx54x78x64x2ex26x3exbx54x17xbx4ax4bc(S;0)3
x4p(S;0)x33x3fx1
c(S;0) = SN(d1)?Ke?rTN(d2); p(S;0) = Ke?rTN(?d2)?SN(?d1):
x25x67x2Black–Scholesx72x31x36x28x44x21x78
x21x70x71x54Black–Scholesx6ex26x1
c(St;t) = StN(dt1)?Ke?r(T?t)N(dt2);
p(St;t) = Ke?r(T?t)N(?dt2)?StN(?dt1);
x17x2ct 2 [0;T],x58
dt1 = log
St
K +(r +
2
2 )(T ?t)
pT ?t ; d
t
2 = d
t
1 ?
pT ?t:
x10x3ex60x27x49x15x79x18x23x54x3exbx4ax4bx57x6ax4a(x10x2x28x29x2ax6cx54 )x75x76x54x3x4cx78
x17x2cx68x23x3bx7ex7fx70x26x54x5ex7ex78
x25x26x48x3ax3ex67x1fx68x61x52x53x54 Black–Scholes x3exbx62x4ax6ex26x78x79x7ax17
x2cx3bx19x61x70xbx77x2fx29x54x68x4fx26x27x78xfx2x3ax3ex23x24x3cx19x48x17x40x6fx5ex55x57
x3ax3exbx4bx65x63x54xdx17xbx64x50x27xfx54x8x3ex55x56x2cx54x3bx19x54x36x51x6fx60x48x3fx78
x15x3x1dx54x31x32x58x2x57x33x6dx3cx61xdx9x4ax54x1ex37x62x4ax40x6fx62x69x78x44x4bx54x2c
x68x2exfx20x48x61 Black–Scholes x6ex26x54x70x64x2cx68x69x55x48x10x18x48x10x5ax20x48x61
x70x64x6bx12x69x55x78x2bx71x44x48x3cx3bx8x53x33x69x55x48x3cx71x5ax25x49x8x3exbx62x4ax54
x37x1fx6bx12x78
8.4 x24x56x3ax3ex75x5dx44x29x65x63x76x77x19x70x71x54x51x2ax70x71x2ex3ex55x56x74x40x78
x25x2Black–Scholesx3exbx62x4ax69x63x54x8x53x33x69x55x54x70x71x5cx78x10x3ex21x2ex1e
x27x36x3ax3ex4bx20x19x1fx54Harrison–Kreps [9]x54x20x2ex78x2ex3ex57x8x3ex54x48x3fx4c
x49x66x65x63x2cx44x58x54x72x76x75x68x51x21x54x0x78x15x52x68x19x54x31x63x53x2x7ex7fx72xd
x54x78
x57x8x3ex55x56x7bx77x54x6fx60x2ex10x49x66x10x7fx7fx75x39x2ex3cx62x54xdx68x17xfx8x5f
x27x47x48x21x2ex3cx62x54xdx13x17xfx9x9x78x66x2x17x2cx44x58x54x2ex3cx62x1dx53x28x42x18x76
3xbx53x20x55x71x16x17c(S;t)x52x1ax53x12x3bx1x7x20C(S;T ?t). x16x12x5 T = pT.
192
x54x24xbx76x77x4bx27x4bx1x2x78x67x44x3ax3ex49x44x4bx21x3cx19x54x33x26x48x57x1fN x2bx18
x23x24x13x48x23x25x54x70x71x4dx391x64x48 2x64x48x70x18x3fx6fx192N x64x78x25x18x3x1dx54
x25x22x2x21x22xdx2bx3dx33xfx6fxdx7cx6ax59xfx78
x61xdx1ax49x3ax3ex47x48x54x5fx60x51N + 1x2bx18x23x78x39x17xbx54x3cx62x70x71x8
x5fx48x56x70x18x23x23x25x5fx36x40x41x64x70x71x48x39x56x70x18x23x23x25x54x59x70x64x70x71x8
x5fx48x13x56x70x18x23x7fx4dx5fx36x40x41x64x70x71x78x31x7ex9x9x48x70x18x19N x2bx18x23x13x48
x21x51x23x25x5fx36x54x70x71x40x41x70x2bx51x2ax69 (x3ax3ex25x18x73x47x48x51x2a) ?. x1ex2b
x25x26x54x1fx6ax40x41x70x22xdx2bx3dx33xf(x7fxcxdx7cx6ax59xf)x48x17x2cx59x70x2bx51x77x5bx4e
x60x11x18x23x54x11x2bx70x71x57x58x39x59x2bx51x77x8x5fx48x10x7fx5bx25x36x41x8xdx2bx3dx33xf
x54x70x32x33x48x25x32x33x54x52x77x1cx5bx65x1?x10x78x10x18x48x39x2ex10x54x51x77x8x5fx21
x36x41x54xdx2bx3dx33xfx32x33x5ax5bx2x2ex10x54x48x51x6cx25x2ex66x53x78xbx4bx54xe2x2x59
x2bx51x77x73x36x41x78x64x20x70x71x54xdx2bx3dx33xfx54xcx5ax74x40x78
x1ex2bx25x26x54xdx2bx3dx33xfx4ex60x42x70x6ax7cx6ax59x78x44 ?0, ?1, :::, ?N = ?
x1x4ex60x59x18x23x54x74x2bx23x25x54x70x71x51x77x69x25x48x17x2c ?0 x4ex60x17xbx54x70x71
x69x48x7x8x73x51x70x2bx38x46x78x44 ? x54x21x51x14x69x21x40x41x54x69x25x1F, x24x58x70
x71x44?n x54x21x51x14x69x21x40x41x54x69x25x1Fn,x51x5d?n x2cx54x59x70x51x77x57x10x49
?n+k (k 6 N ?n)x2cx23x54x19x54x51x77x7fx1fx3cx2ex70x26x48x33x34x67x51
F0 ‰F1 ‰￠￠￠‰FN?1 ‰FN = F:
x25x70x2bx71x49x25x27x63x2cx51x2bx22x23x53x1excx1 -x22x23(filtration). x17x62x7ax77x25
x18x75x70x71x68x2ex78 xdx2bx3dx33xfx21x36x41x54 -x74x59x2exfx2x70x71x54 -x74x59x54x15x28
x6dx54x74x40x48x5ax1x70x71x54 -x74x59x20x48x61x70x2bx18x76x40x44x78
x3ax3ex2ex40x5d?nx57x10x21x45x6ex54x38x46x2bx68x76x1x70x5fx78x33x34x49x44x29x55x56
x2cx48x70x2bx2ax2bx2cx2dx67x2R? =R?N x2cx54x24x6cx48x4x10x10x59x2bx15x52x70x71x5bx51
x70x2bx68x6x78x10x26x48x10x66x15x1fn 6 N, R?n x2cx54x38x46x5ax23x3cx2ex70x2bx28x29x2a
x6cx48x73x2x10x11x69x1ex1x10x66x59x2bx39 ?n x2cx54x70x71x8x5fx25x49 ? x2cx54x19x54x21
x51x70x71x44x47x10x26x54x6x78x3bx25x27x63x54x7bx11x27x70x48x25x26x54x28x29x2ax6cxcx1x2
Fn-x5dx33x10x48x17x6ex7ax2x25x64x28x29x2ax6cx49x65nx18x23x18x19x45x6ex17x7fx32x7cx6ax78
x16x17x2F0-x23x51x54x28x29x2ax6cx5ax67x2x17xbx54x1dx70x70x71x56x54x3cx62x54x6cx78
x5ex7ex48x3ax3ex58x36x51x0x4ex25x27x78x21x22x25x27x51x3bpx2R?++ (x21x51x33x6cx67
x4bx4ex66x7dx54R? x2cx54x24x6c)x2cx54x38x46x48x6cx10x9xaP!2? p(!) = 1. x59x2bx25
x26x54x25x27x51x3bx5bx11xcx1?x44x54x9x4ax25x27x51x3bx48x7bx1x10x3ex5bx10x23x25x5fx36
x54x70x71x49x4bx43x7dx25x27x78x10x66x28x29x2ax6cz 2R?,
E[z] = p￠z =
X
!2?
p(!)z(!)
xcx1z x54x42x43x72x73x78x10x2x28x29x2ax6cx74x64x23x25x47x54x6x15x17x23x25x1dx4ex16x27
x47x54x36x72x6x78x68x4fx3ex2bx2x70x2bx3cx62x54x68x6x78x6fx31x70x2xdx17xbx54x68x6xfx78
193
x51x18x3ax3ex5ax75x47x48x70x2bx28x29x2ax6cx49x65nx18x23x54x36x72x6x48x26x66x65nx18x23
x54x70x71x49x17xbx3cx27x5ax2x2ex3cx62x54x48x39x58x25x2bx36x72x6x53x2x70x2bFn-x23x51x54
x28x29x2ax6cx78x25x64x36x72x6xcx1x2ax2bx42x43x72x73x78x70x2bFn-x23x51x54x28x29x2ax6cx67
x2x70x2bR?n x2cx54x70x2bx24x6cx48x17x2cx59x2b!n 2 ?n x5bx23x69x1ex1?x2cx54x70x2b
x14x69x48x4x39!nx8x5fx54x49?x2cx23x54x54x70x71x51x77x7fx1fx78x70x2bx28x29x2ax6cx5ax23
x49x25x2bx70x71x51x77x69x44x39x36x72x6x78x66x2x28x29x2ax6cz 2R? x3x66Fn x54x38x3d
x68x4fx3ex2bx62x7ax1
E[zjFn] = (z(!n))!n2?n 2R?n;
z(!n) =
P
!2!n p(!)z(!)P
!2!n p(!)
:
x17x7ax48x68x4fx3ex2bx4x38x3dx68x4fx3ex2bx54x62x7ax5bx47x44x66x2bx79x49x8x54x25x27x51x3bx78
x40x41x2bx28x29x2ax6cx15x6dx28x32x49x70x14x47x48xcx1x2ax2bx24x2ex6fx2ax2bx31x7fx78
x49x3ax3ex47x48x54x5fx60x2cx48x28x29x1fx6ax5x2x57x74x18x23x61x4cx49x70x14x54x48x7bx7ex48
x3ax3ex47x48x54x28x29x1fx6ax5x2x51x56x5cx40x26x2a
x = (xn)n=0;1;:::;N:
x51x6cx3ax3ex24x13x47x48x54x59x2bxn x5bx53x2x2ex75x54x28x29x2ax6cx48 (xn)x2x2ex75x28x29
x1fx6ax78x51x70x64x28x29x1fx6a(Sn),x10x49x65nx18x23x18x54x6x10x65nx18x23x27x70x19x7e
x7fx3cx62x48x4Sn 2R?n x2Fn-x23x51x54x48x33x34x10xcx1x25x43x31x7fx57x24x56x65x63x54
x5fx60x2cx48x2cx68x54x4ax4bx1fx6ax67x2x24x7x1fx6ax48x7bx1x10x49x65nx18x23x54x59x70x70
x71x56x54x6x5bx2x3cx62x54x78x58x51x70x64x28x29x1fx6a(`n),x35x61x17xbx19x48x10x49x65n
x18x23x18x54x6x10x65n?1x18x23x27x70x19x7ex7fx3cx62x48x4`n 2R?n?1 x2Fn?1-x23
x51x54x48x33x34x10xcx1x5dx26x31x7fx78x24x56x65x63x54x5fx60x2cx48x4fx1ex31x59x18x23x4ex4f
x54x2cx68x14x25x66x16x67x2x23x9x1fx6ax48x7bx1x65nx18x23x14x2ex3bx54x2cx68x14x25x66x16
x2x49x65n?1x18x23x18x2ex54x78
x51x61x25xbx68x4fx2cx74x24x13x48x3ax3ex67x23x27x65x63x3ax3ex54x2cx68x66x16x55x56x78
x61x62x55x56x2cx51 K + 1 x64x2cx68x48x17x2cx65 0 x64x2cx68x2x33x69x6ax2cx68x48x4x17x59
x18x23x54x4ax4bx2x70x2bx3cx62x1fx6aS00 = 1, S01 = (1+R), :::, S0N = (1+R)N,x17
x2cRx2x5fx3ex3cx27x78x17x62K x64x2cx68x5bx2x69x6ax2cx68x48x10x3ex59x18x23x54x4ax4bx10
x66x59x18x23x2ex10x54x70x71x5bx2x3cx62x54x48x4x10x3ex5bx2x21x22x24x7x1fx6ax78x44x28x4f
x31x7fx1
Sn = (S0n;S1n;:::;SKn ); n = 0;1;:::;N:
x4fx1ex31x59x18x23x4ex4fx54x2cx68x66x16x2x70x2bx24x6c`n = (`0n;`1n;:::;`Kn ),x17x2cx59
x2bx33x6cx60x27x10x8x64x2cx68x54x66x16x6cx78 ` = (`n)xcx1x4fx1ex31x54x3ex3fx31x7fx48x2e
x1x28x29x1fx6ax10x2x23x9x1fx6ax48x7bx1x10x65nx18x23x14x2ex3bx54xex0x2x49x65n?1
x18x23x2ex8x54x78
194
x4fx1ex31x49x65nx18x23x18x57x2cx6dx6ex54x2cx68x2`n (x10x49x2ex10x54x70x71x56x2
x2ex10x54)x78x26x66x17x18x54x2cx68x4ax4bx1Sn,x46x25x70x2cx68x14x25x54x4ax6x1
Vn(`) = `n ￠Sn =
KX
i=0
`inSin:
x31x32x3ax3ex5bx24x17xbx54xdx2ex2ax4ax4bxfx27x13x1ax48x12 ?Sn = Sn=(1+R)n (x3ax3ex24
x13xcx10x1x61x1bx28x4f),x33x34x25x70x2cx68x14x25x54xdx13x1ax4ax6xfx7x8x2
?Vn(`) = (`n ￠Sn)=(1+R)n = `n ￠ ?Sn:
x47x1fx79x54x2x10x3ex5bx2x24x7x1fx6ax48x4Vn(`); ?Vn(`) 2R?n.
x70x2bxex0`xcx1x2x27x3cx1d(self financing)x54x48x2x5d
`n ￠Sn = `n+1 ￠Sn; n = 0;1;:::;N ?1: (8)
x10x54x6ex7ax2x2ax4fx1ex31x49x65 n x18x23x1x65 n + 1 x18x23x21x2ex8x54xex0x2ex17x2a
x17x21x6dx6ex54x2cx68x14x25x54x5x4ax6x48x4x4fx1ex31x49x7ex1fx6ax2cx47x2ex47x75x54x0x1e
x3dx48x5ax2ex5fx36x77x50x1ex3dx78 (8)x26x74x7ax9x4ax66
`n+1 ￠(Sn+1 ?Sn) = `n+1 ￠Sn+1 ?`n ￠Sn;
x39x58
Vn+1(`)?Vn(`) = `n+1 ￠(Sn+1 ?Sn) x6f ?Vn+1(`) = `n+1 ￠?Sn+1;
x17x2c ?Vn+1(`) = Vn+1(`)?Vn(`), ?Sn+1 = Sn+1 ?Sn. x26x7ex2exfx2cx68x56
x5cx4x60x2a
x59x238.1 x7ex75 ` x51x65x6x26x63x6ex21x7fx5fx60xcx7ax7d
i)
Vn(`) = V0(`)+
nX
j=1
`j ￠?Sj; n = 1;:::;N;
ii)
?Vn(`) = V0(`)+
nX
j=1
`j ￠??Sj; n = 1;:::;N;
x7ex7f??Sj x79x62x59x42x3fx0
195
x49x8x3ex55x56x2cx48x51x36x51x37x34x36x3ex1ex54x25x22x78x2ex3ex55x56x2cx75x21x24x47
x75x25x2bx25x22x2x1x61x53x74x3ex54x4fx1ex2fx30x51x68x3cx54x61x4cx78x26x4x60 8.1 x23x3c
x8x48x25x64x61x4cx2x5dx4fx1ex4ax6x54x17x2ax5bx2x26x66x2cx68x4ax4bx17x2ax21x0x14x54x48
x58x57x4fx1exex0x54x17x2ax33x3x78x8x3ex55x56x2cx2ex47x75x25x2bx25x22x54x6ax7bx49x66x33
x18x73x51x70x3ex4fx1exex0x48x2ex66x49x4fx1exex0x17x2ax54x5fx60x78x10x18x48
x36x3ex1exex0x2x5dx4fx1ex2fx30x57x19x5bxax43x33x3x48x58x2ex2x79x41x42x2cx68x66
x16x6fx50x49x10x70x18x23x2cx2ex13x5bx78x2bx71x44x48x26x66x65 0 x64x2cx68x71x72x44x7bx17x66
x70x2bx23x36x26x66x47x54x22x4fx8x60x48x17x62x2cx68x66x16x21x0x14x54xax2x2ex36x73x5bx23
x26x10x27x14xax78x25x70x2bx71x17x71x19x7ax6ex49x44x29ii)x2cx48x7bx1??S0j x5x1x7dx48x10
x650x64x33x69x6ax2cx68x54xex0(`0n)x17x71x51x2ex8x1ax49ii)x54x60x54x26x2cx78x21x37x1f
x1ex70x48x36x3ex1exex0x54x71x72x75x68x73x51K x75x48x51x6cx23x60x54x1x56x5cx4x60x2a
x59x238.2 x6dx6ex6fx70x19x1axdxexfx10x7ex75 ((`1n;:::;`Kn ))n=0;1;:::;N (x47x28
x28x29)x71x6fx70x29x1cV0, x4fx60x61x3x7ax79x18x19x1axdxex79x7ex75 (`0n)n=0;1;:::;N: (x47
x28x28x29)x6bx4x5x7ex75 ` = (`0;`1;:::;`K)x1dx51x65x6x79x6bx2x73x29x1cV0.
x38x7a `00 x23x7ex1fx56x26x6bx12x68x19x2a
V0 = `00 +`10S10 +￠￠￠+`K0 SK0 :
x1x6bx12x70x71x54`0n,x26x36x3ex1ex38x3dx23x68
?Vn(`) = `0n +`1n ?S1n +￠￠￠+`dn ?SKn
= V0 +
nX
j=1
(`1j??S1j +￠￠￠+`Kj ??SKj );
x39x58`0n x67x23x26x56x26
`0n = V0 +
nX
j=1
(`1j??S1j +￠￠￠+`Kj ??SKj )?`1n ?S1n ?￠￠￠?`Kn ?SKn
x27x1dx70x3cx62x78 2
x56x4bx3ax3ex75x10x25x26x54x42x43x20x8x33x6dx3cx54x25x22x78x25x18x54xdx33x6dx3cxfx74
x75x2x5dxdx50x27x6x5bx54x1ex37x17xbx5ax6x5bxfx78x5ax67x2x70x48x61x4cx3ax3exbx4bx70x1f
x54xdx33x6dx3cx61xdx54x24x2bxex6dxfx48x45x46xbx47x2bxex6dx54x1cx1dx62x4ax55xex17x71x19
x57x0x1x2xdx69x21x17x7axfx54x78x39x58x49x3ax3exbx4bx6bx12x4fx1ex14x25x54x4ax6x18x19x57
x49xdx69x21x17x7axfx1ex49x33x18x5x3bx78x58xdx15x71xex6dxfx54x33x6dx3cx61xdx53x5dx8x25
x70x1cx1dx62x4ax55xexcx2cx2xdx3bxfx54x78x25x64xdx3bxfx1dx48x1ax49x53x24x21x22xdx9x4ax25
x27x9x51x3bxfx54x40x26x27x60x54x78
196
x6cx79x3ax3excxex0`x2x5dx2ax2bx10x48x2x5dx10x2x36x3ex1ex54x48x6cx10x66x15x1f
n = 0;1;:::;N x9xaVn(`) > 0. xcxex0`x2x2dx20x3ex3fx48x2x5dx10x2x23x38x11
xex0x48x6cx9xaV0(`) = 0, VN(`) > 0,x17x2cx13x4bx54>x2x15xbx4bx54x24x6cx79x7a
x27x69x1ex54x48x4VN(`)x49x15x1fx70x71x56x5bx43x50x48xfx5ex5fx49x70x2bx70x71x56x1x3bx78
x31x32x21x65x63x54x42x43x2ex66x49x6dx3cxex0x48x33x34x8x42x43xcx1x5dx2cx63x68x69x78x47
x1fx79x54x2x3ax3ex49x25x18x62x7ax54x6dx3cxex0x75x39x10x2x23x38x11x54x78x17x71x49x25x18
x12x5fx23x38x11x54x75x39x48x10x62x7ax51x33x13x14x78x2bx71x44x48x3ax3ex23x24x2cx68x56x5cx4
x60x2a
x59x238.3 x57x58x41x42x67x47x6x60x79x6bx5dx5ex6fx70x51x65x6x7ex75x24x18x8x61x5
x4dx5fx6bx6cxax60x61x51x65x6x7ex75 (`n),x4x5V0(`) = 0, VN(`) > 0.
x38x7a x2bx71x44x48x31x32x66x49x11x2bx36x3ex1exex0`,x53x68V0(`) = 0, VN(`) >
0,x33x34x5ax51
?VN(`) =
NX
j=1
?
`1j??S1j +￠￠￠+`Kj ??SKj
·
> 0:
x2ex40xd`x2ex2x23x38x11xex0x78x35xex19x26x62x7ax1fx68x5dx23x78x12
n = maxfkj ?Vk(`) 6> 0g:
xexdAn = f! 2 ?j ?Vn(`)(!) < 0g,x51x62x7ax20x54xex0(?n)x1
?k(!) =
‰ 0; k 6 nx6f! =2 A
n;
`k(!); k > nx6c! 2 An:
x33x34x2exfx52x2cx48x25x5ax2x2bx36x3ex1exex0x48x6c
?Vk(?)(!) =
‰ 0; k 6 nx6f! =2 A
n;?
Vk(`)(!)? ?Vn(`)(!); k > nx6f! 2 An:
x26x7ex23x4fx48x25x2x70x2bx23x38x11xex0x48x51x6c
8! 2 An; ?VN(?)(!) > 0:
x7bx7ex48x57x42x43x23x36x66x5dx23x78 2
8.5 x51x61x44x4bx25xbx51x3x51x2ax70x71x2ex3ex55x56x54x40x6fx25x22x24x13x48x3ax3e
x67x23x27x65x63x25x70x70x75x56x54x1ex37x62x4ax40x6fx62x69x78x3ax3ex49xbx4bx19x57x10x51x2a
x70x71x8x3ex55x56x5dx8x1ex37x62x4ax40x6fx62x69x48x10x73x1x48x33x6dx3cx61xdx9x4ax66x66
x49x11x64x9x4ax25x27x51x3bx48x53x68x70x59x70x64x2cx68x54x5bx3cx54x68x4fx3ex2bx6x5bx9x66
197
x66x22x4fx54x5bx3cx78x49x1ax49x54x51x2ax70x71x2ex3ex54x55x56x56x48x0x44x4fx1exex0x54x36
x3ex1ex2ax10x48x21x70x71x54x1ex37x62x4ax40x6fx62x69x53x5dx8x48x25x70x31x63x2exfx75x10x15
x13x18x23x41x7dx48x58x75x10x2cx76x18x23x5ax41x7dx78x1x7ex48x3ax3ex47x75x2d (martingale)
x54x25x22x78
x9x54x25x22x21939x46x26x55x56x68x4fx59Villex6cx79x0x4ex54x78x9x54x79x4bx79x5e
x2x6x2ex2fx48xfx17x55x64x6ax79x58x51x6ex36xdx4bx54x6ex7ax78x51x70x64x70x55x0x1x48 xdx9
(martingale)xfx14x17x66x55x56x54x70x2bxdx30x54x53x79x78x49x3ax3ex44x29x54xex4cx2cx48x9
(Mn)x11x62x7ax1x70x2bx24x7x1fx6ax48xfx2x9x54x25x22x58xcx2cx57x25x27x51x3bx7bx61x4cx48
x7bx1x10x57x68x4fx3ex2bx4x38x3dx68x4fx3ex2bx54x25x22x51x3x78x24x7x1fx6a(Mn)xcx1x9
x2x5dx10x9xax2a
E[Mn+1jFn] = Mn; n = 0;1;:::;N ?1:
x31x32x9x49x17x16 (>), x33x34(Mn)xcx1x9x2d(x32x2d)x78x2exfx3cx8x48 (Mn)x1
x9x54x1dx75x38x3dx5ax23x60x54x1x2a
E[Mn+jjFn] = Mn; n = 0;1;:::;N ?1; j = 0;1;:::;N ?n;
x16x17x2x26x7ex23x1fx8
E[Mn] = E[M0] = M0; n = 0;1;:::;N;
x4x9x49x15x1fx18x23x54x68x4fx3ex2bx6x5bx57x1bx6x70x26x78
x3ax3ex24x13x54x2ex1x1ex37x62x4ax40x6fx62x69x20x40x26x54x74x75x31x63x75x70x2x2ax42
x43x23x36x66x54x1dx75x38x3dx1x66x49x9x4ax25x27x51x3bx48x53x68x13x1ax4ax4bx1fx6ax1x9x78
x7bx7ex48x31x32x3ax3ex49x56x4bx51x3x9x54x70xbx31x63x2cx48x5bx5dx9x69x1ex1x13x1ax4ax4bx48
x33x34x10x3ex54x57x58x79x7ax67x29x33x7bx7cx78x16x17x2x4x60x2cx54x1fx6ax5bx23x69x1ex1x2e
x75x54x78
x59x238.4 x20(Mn)x67x31(x2ax74x63x7)x6b (Hn)x67x47x28x28x29(x7ex75)x6b ?Mn =
Mn ?Mn?1. x5dx5ex57x21x62x59x79x37x6bx28x29 (x2ax74x63x1c) (Xn)x10x67x31x7d
X0 = H0 ￠M0
Xn = H0 ￠M0 +H1 ￠?M1 +￠￠￠+Hn ￠?Mn; n = 1;:::;N:
x38x7a x74x7a(Xn)x2x24x7x1fx6ax78x10x18x48x10x66x15x1fn,x3ax3ex51
E[Xn+1 ?XnjFn] = E[Hn+1 ￠(Mn+1 ?Mn)jFn]
= Hn+1 ￠E[Mn+1 ?MnjFn] = 0;
x25x18Hn+1 x23x20x8x27x2x7bx1x10x2Fn-x23x51x54x78x7bx7ex48
E[Xn+1jFn] = E[XnjFn] = Xn;
198
x4(Xn)x2x9x78 2
x25x70x4x60x79x41x42x2ax73x75x4ax4bx2x9x48x2ex23x4fx1ex31x7ex47x37x34 (x36x3ex1e)
xex0x48x17x57x2cx54x2cx68x14x25x54x6x5ax70x62x2x9x48x4x10x54x68x4fx3ex2bx6x5x2x2e
x2ax54x78x56x70x4x60x70x68x48x25x70x31x63x2fx5ex58x2x4ax4bx1x9x54x1dx75x38x3dx78
x59x238.5 x37x6bx28x29 (x2ax74x63x7) (Mn)x67x31x79x62x4ex5fx60x1dx6dx6ex6fx70
x47x28x28x29 (x7ex75) (Hn),x51
E
" NX
n=1
Hn ￠?Mn
#
= 0: (9)
x38x7a x31x32 (Mn) x2x9x48x33x34 (9) x26x44x29x4x60x23x68x78x7ax75x48x1x2cx68
x26 (9) x23x1fx68 (Mn) x2x9x48x3ax3ex27x4dx48xcx5ax54 (Hn). x2bx71x44x48x1x2cx68
E[Mn+1jFn] = Mn,x5dx28x29x2ax6cx69x1ex1x24x6cx48x73x47x5dx8x48
8!n 2 ?n;
P
!2!n p(!)Mn+1(!)P
!2!n p(!)
= Mn(!n): (10)
x7bx7ex48x3ax3ex10x35x62x54!n 2 ?n,x47
8
<
:
Hk(!) = 0; k 6= n+1;
Hn+1(!) = 1; ! 2 !n;
Hn+1(!) = 0; ! =2 !n:
xex10x25x26x54(Hn), (9)x7dx4x1fx8(10). 2
x1ax49x3ax3ex27x2cx68x56x4bx54x45x75x62x69x2a
x36x3f 8.1 x41x42x47x6x60x79x62x4ex5fx60x1dx60x61x26x63x3dx13x3cx25x6bx4x5x47x73
xdxex79x2ax74x63x7x28x29x1dx31x0
x38x7a x63x4ex76x78x31x32x66x49x9x4ax25x27x51x3bx53x68x21x51x13x1ax4ax4bx1fx6ax5b
x1x9x48x33x34x10x66x15x1fx36x3ex1exex0(`n),x17x7bx7x54x2cx68x14x25x54x13x1ax4ax6x1
?Vn(`) = V0(`)+
nX
j=1
`j ￠??Sj:
x26x4x603.3, (?Vn(`))x10x8x25x27x51x3bx5ax2x9x48x39x58
E
h?
VN(`)
i
= ?V0(`):
199
x31x32 (`n) x2x23x38x11xex0x48x33x34x17 ?V0(`) = 0 x18x48x1fx8 ?VN(`) > 0 x24x58
E[?VN(`)] = 0. x25x4cx17 ?VN(`) = 0x18x78x51x23x25x78x7bx7ex48x42x43x2x23x36x66x54x78
x62x49x76x78x49R? x2cx47x48x69x25
M = fx 2R? j x = ?VN(`) =
NX
j=1
(`1j??S1j +￠￠￠+`Kj ??SKj );
(`n)x1x15x1fx53V0(`) = 0x54x36x3ex1exex0x78 g
x4x69x25
K = fy 2R?jy > 0g:
x33x34x10x3ex2x78x2bx2ex7bx66x54x26x69x48x39x58x26x26x69x33x5bx62x69x66x49? = (?(!))!2? 2
R? x53x68
8x 2 M; 8y 2 K; ?￠y > ?￠x;
x10x18x48x58x51
8x 2 M; 8y 0; ?￠y > ?￠x:
xfx26x66M x71x72x44x2R? x2cx54x14x72x76x48x44x78x26x54x1fx14x5bx73x23x25x1x7dx48x35
xex10x3ex5bx23x25x33x2ax54x4ex48x24x5ex2ex9x26x2ex23x25x41x7dx78x4dx70x69x4bx48x26x66yx54
x59x70x33x6cx5bx23x33x2ax54x4ex48x24x5e?x54x59x70x33x6cx5bxcx2cx1x3bx48x35xex44x29x67
x4bx2ex9x26x2ex23x25x41x7dx78x7bx7ex48 ? 2R?++,x39x58x23x26x10x27x62x7ax25x27x51x3bx1
8! 2 ?; p(!) = ?(!)P
!02? ?(!0)
:
x10x66x25x70x25x27x51x3bx48x26x66x10x66x15x1fx 2 M, x5bx51?￠x = 0, x46x23x1fx68x10
x66x15x1fK x75x23x9x1fx6a(`n) = (`1n;:::;`Kn ),x51
E
" NX
j=1
`j ￠??Sj
#
= 0:
x26x4x608.5, (?Sn) (x58x17x59x70x33x6c)x2x9x78 2
x57xbx4bx54x65x63x70x26x48x23x36x66x42x43x21x1fx8x54x53x21x51x2cx68x13x1ax4ax4bx1
x9x54x9x4ax25x27x51x3bx2ex70x62x2x1dx70x54x78x75x53x25x26x54x9x4ax25x27x51x3bx2x1dx70
x54x48x17x6cx4cx17x44x29x2cx68x2cx54x14x72x76M x54x75x68x1??1. M x4cx4cx2x21x51
x1bx6x1x7dx54x74x64x36x3ex1exex0x21x25x54x19x54x52x6x54x7fx1fx78x31x32x10x54x75x68x1
??1,x33x34x13x61x44x1bx6x2ax5cx21x40x41x54x70x75x72x76x48x10x3ex67x25x0x41x1ex2bR?.
x1bx55x14R? x2cx54x15x1fx38x46x5bx25x11xcx1x1ax36x2cx68x6fx50x62xbx21x48x24x58xbx4b
x3x66x7ex7fx42x43x54x62x7ax48x3ax3ex49x7ex10x26x23x24x62x7ax7ex7fx42x43x54x25x22x78xfx2
200
x1x61x49x57x58x44x21x51x79x7ax48x3ax3ex53x49x25x18x5dx50x62xbx21x2ax10x1x43x50x28x29x2a
x6cx48x51x10x7ex7fx42x43x54x25x22x0x24x21x2ex54x75x39x78
x70x2b (x43x50) x50x62xbx21 h 2 R?+ xcx1x2x5dx32x10x48x2x5dx66x49x70x2bx23x38
x11xex0 `, x53x68 VN(`) = h. x70x2bx42x43xcx1x2x49x4ax68x69x48x2x5dx15x1f (x43
x50) x50x62xbx21x2x23x54x54x78x25x2bx62x7ax49x60x4bx44x3cx27x54x0x61xex0x23x38x11x54
x75x39x48x17x71x10x66x23x36x66x42x43x27x70x48x25x2bx75x39x36x7ax41x7dx78x25x2x7bx1x10x66
x23x36x66x42x43x27x70x48x31x32x10x66x43x50x50x62xbx21 h, x66x49x36x3ex1exex0 `, x53x68
VN(`) = h > 0, x33x34x26x66x25x18x66x49x9x4ax25x27x51x3bx48x53x68 (?Vn(`)) x1x9x48
xcx7ax5ax51 ?Vn(`) = E[?VN(`)jFn] > 0,x4`x2x23x38x11x54x78
x56x4bx3ax3ex49x8x6fx51x54x4dx70x38x40x6fx62x69x58x17x18x38x2cx68x78
x36x3f 8.2 x47x6x60x41x42x7dxbx79x62x4ex5fx60x1dx60x61x3x7ax79x26x63x3dx13x3c
x25x6bx4x5x47x73xdxex79x2ax74x63x7x28x29x1dx31x0
x38x7a x63x4ex76x78x61xdx42x43x2x7ex7fx54x78x33x34x10x66x15x1fx43x50x50x62xbx21
h 2R?+,x66x49x23x38x11xex0`,x53x68h = VN(`),x17x2c`x2x23x38x11xex0x78x31
x32x66x49x78x2bx9x4ax25x27x51x3bp1 x4p2,x53x68(?Vn(`))x1x9x48x33x34x26x25x78x2bx25
x27x51x3bx21x62x7ax54x68x4fx3ex2bE1 x4E2 x53x5bx51
Ei[?VN(`)] = Ei￡h=S0N? = V0(`); i = 1;2:
x26x66hx2x15x79x54x48x25x4cx17p1 = p2 x18x78x51x23x25x78
x62x49x76x78x31x32x23x36x66x42x43x2ex7ex7fx48x33x34x26x40x1V0 +PNn=1 `n ￠??Sn
x54x28x29x2ax6cx7fx1fx0x41x54x72x76R? x54x1cx1dx14x72x76V x2x3bx14x72x76x78x61x62p1
x2x11x9x4ax25x27x51x3bx48x53x68x21x51x2cx68x4ax4bx1fx6ax1x9x78x33x34x26p1 x36x41x54x68
x4fx3ex2bE1 x23x3bx27x10R? x62x7ax10x2x2a
8x;y 2R?; (x;y) = E1[xy]:
x26x66Vx2x3bx14x72x76x48x10x66x25x2bx10x2x48x66x49z 2R?,x53x68zx57Vx3bx66x48x16
x17x2E1(z) = 0,x7bx1x73x73x11x54x28x29x2ax6cx2V x54x38x46x78x62x7ax20x54x25x27
x51x3b
8! 2 ?; p2(!) =
1+ z(!)2max
!2?jz(!)j
?
p1(!):
x33x34x10x74x7ax2x70x2bx57p1 x2ex10x54x9x4ax25x27x51x3bx48x51x6cx26x10x21x62x7ax54x68x4f
x3ex2bE2,x53x51
E2
" NX
n=1
`n ￠??Sn
#
201
= E1
" NX
n=1
`n ￠??Sn
#
+ 12max
!2?jz(!)j
E1
"
z
? NX
n=1
`n ￠??Sn
!#
= 0:
x26x66x44x26x10x15x1fx24x7x1fx6a((`1n;:::;`dn))x5bx41x7dx48x13x26x4x603.5,x4x68(?Sn)
x10x66p2 x5ax2x9x48x4x1dx70x1dx2ex9xax78 2
x5ex7ex48x3ax3ex67x7ex41x61x25x2bx55x56x54x1ex2bx62x4ax69x63xex4cx78x25x67x2x70x48x10
x66x70x2bx7ex7fx42x43x27x70x48x66x49x1dx70x54x9x4ax25x27x51x3bx53x68x21x51x2cx68x13x1ax4a
x4bx1fx6ax2x9x78x58x59x2bx50x62xbx21 hx7fx5bx23x24x3bx11x2bx23x38x11xex0`x27x54x19x48
x53x68VN(`) = h. x7bx1x13x1ax4ax6(?Vn)x10x66x25x70x25x27x51x3bx5ax2x9x48x39x58h
x49x17xbx54x62x4ax67x23x3cx62x1 E[h=(1 + R)N]. x25x26x3ax3ex67x23x10x15x1fx50x62xb
x21x62x4ax78 Cox–Ross–Rubinsteinx55x56x73x2x25x18x65x63x54x55x56x54x15x28x6dx54x74
x40x78
8.6 x49x44x4bx54x65x63x2cx48x26x66x68x4fx44x54xexfx48x3ax3ex47x75x61x62x21x51x50
x27x70x71x54x2bx68x5bx2x51x2ax54x48x39x58x21x44x58x54x21x51x28x29x2ax6cx17x71x5bx2x5bx54
x28x29x2ax6cx48x21x44x58x54x21x51x25x27x70x71x72x76x4 -x74x5bx2x51x2ax69x78x1x65x63x21
x70x71x54x2cx68x42x43x2ex3ex55x56x48x3ax3ex6cx79x47x75x12x35x51x2ax70x71x54x61x62x78x25x18x48
x3ax3ex10x26x51xdx2bx3dx33xfx4 -x74x59x54x25x22x48xfx2x10x3ex54x40x44x19x57x2ex2x70x22
x33x34x33x68x54x33x48x58x75x55x44x1x70x2bx51x60x60x2ax2ax33x35x54xdx13x36xfx78
x3ax3ex39x70x2bx25x27x72x76 (?;F;P)x8x5fx48x25x18 ?x2x70x2bx15x79x54x69x25x48
Fx2x26?x2cx54x14x69x21x4dx41x54 x74x78x7bx7x54 -x74x59x1cx23x60x27x1(Fn)n=0;1;:::;N:,
x25x18x59x2bFn x5bx2?x54x14x69x21x4dx41x54 -x74x48x51x6cx9xax6dx50x1dx2a
0 < m < n < N =) F0 ‰Fm ‰Fn ‰FN = F:
x25x70x62x7ax71x72x44x73x57x23x51x72x76(?;F)x51x3x48x58x57x17x44x62x7ax54x25x27Px33
x3x78x3bx31x3ax3ex49xbx4bx10x66x51x2ax70x71x74x40x65x63x54x33x26x48 -x74x59x54x62x7ax57x25
x27x2x33x3x54x78xfx2x49x70x71x72x76?x2x33x2ax69x18x48x1x61xdx1ex70xbx2excx75x54
x2ax52x48x7ex7fx58x10 -x74x59x75x39?x2cx54x25x27x1x7dx54x14x69x7fx1fx66x66x21x51Fn.
x5ax67x2x70x48x49x21x51x26Fnx21x40x41x54x74x2bx25x27x72x76x2cx48x7dx25x27x2bx3dx2x7bx10
x54x78x7bx7ex48 F0 x5ax67x2ex13x73x51x78x2bx38x46f;;?g (x4ex60xdx1ax62x2ex5fx36x2bx3dxf
x57xdx1ax62x5fx36x2bx3dxf)x48x51x6cx58x45x6ex21x51x25x27x1 1 x54x2bx3dx4x21x51x25x27x1
x7dx54x2bx3dx78x2ex1fx48x3ax3ex57x7fx5dx1ax62x2ex5fx36x2bx3dx57x7dx25x27(xdx3x15x1ax62xf)
x2bx3dx2ex0x48x3fx78
x28x29x2ax6cx54x25x22x17x71x5ax57x25x27x51x3bx33x3x78x21x22x23x51x72x76(?;F)x44
x54x28x29x2ax6c X x2x5dx10x2x62x7ax49 ? x44x54x23x51x65x68x48x4x10x66x15x1fx71x68 c,
f! 2 ?jX(!) < ag‰F,x6fx31x70fX < cgx2x70x2bx2bx3dx78x5dF x43x4ex1x10x54
x14 -x74Fn,x3ax3ex67x23x68x19x7bx7x54Fn-x23x51x28x29x2ax6cx25x22x78x40x41x2bx28x29x2a
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202
x60x2cx48x28x29x1fx6ax5x2x51x56x5cx40x26x2a
x = (xn)n=0;1;:::;N:
x25x18x59x2bxn x10x26x5bx23x24x2x2ex75x28x29x2ax6cx48 (xn)x23x24x2x2ex75x28x29x1fx6ax78
x24x7x1fx6a(Sn)x62x7ax1x10x49x65nx18x23x18x54x6Sn x2Fn-x23x51x54x78x23x9x1fx6a
(`n)x10x26x62x7ax1x48x35x61x17xbx19x48x10x49x65nx18x23x18x54x6`n x2Fn?1-x23x51
x54x78x49x25x64x74x40x56x48x9(Mn)x5ax23x72xdx62x7ax48x4x10x2x24x7x1fx6ax48x51x6cx9
xa
E[Mn+1jFn] = Mn; n = 0;1;:::;N ?1:
xbx4bx65x63x54x31x32x49xaxbx74x40x56x5bx1cx7ax41x7dx48x73x2x21x51x54x9x26x5bx75x69x1e
x1xdx3x15x1ax62(almost sure, a.s.)xfx41x7dx48x4x24x25x271x41x7dx78x2bx71x44x48x3ax3e
x44x4bx54x65x63x2cx47x75x3bx19x70x71x51x2ax54x1ex69x2x49x1ex37x62x4ax40x6fx62x69 8.1 x54
xcx75x1dx2cx68x2cx48x17x62x54x2cx68x5bx2ex47x75x6fx60x54x17x2ax78x62x698.1x54xcx75x1dx2c
x68x2cx74x75x3bx19x26x69x33x5bx62x69x78x25x70x62x69x49x33x2ax75x54x74x75x56x53x11x19x70x62
x54x2ax10x78xfx2 1990 x46x48 Dalang–Morton–Willinger [7] x2cx68x48x25x38x62x69x10
x66x70x71x33x2ax54x25x27x72x76x1cx7ax41x7dx48x17x2cx3bx19x70x38x7bx17x66x26x69x33x5bx62x69
x54x2cx68x7bx17x2fx29x54x25x27x63x64x6fx78x58x17x2cx44x58x54x9x4ax25x27x51x3bxex2x5dx53
x7dx25x27x2bx3dx2ex2ax54x25x27x51x3bx48x6fx31x70x2x10x6ax27x54x25x27x51x3bx52x10x7fx0x54
x25x27x51x3bx78x7ax58x48x31x32x55x56x54x3ex68N x5ax2x33x2ax18x48x33x34x72xdx54x1ex37x62
x4ax40x6fx62x69x67x2ex13x41x7dx78x17x7ax45x23x49[7]x6fPliska [18]x2cx60x19x78
x5ex11x48x3ax3ex1x1fx8Black–Scholesx3exbx62x4ax6ex26x24x58x70x71x54x3exbx62
x4ax6ax69x58x65x63x61x2cx68x42x43x54x2ex3ex55x56x78x17x2cx42xax77x74x75x49x66x70x71x54x1ex37
x62x4ax40x6fx62x69x48x58x10x7ax6ex61xdx54x1cx1dx62x4ax55xex7ex7fx58x15x78x1bx55x14x3ax3ex49
xbx24x51x2cx48x1cx1dx62x4ax55xex4bx53x3ax3ex5bx68x1ex6fx1ex37x62x4ax55x56x3 Markowitz
x2cx68x14x25x7x8x69x63x9x2dx2ex45x75x31x32x48x70x2bx36x7ax54x5fx60x2x2ax1cx1dx62x4ax55
xex24x5ex28x29x13x1ax7bx14x69x63x2x35x10x2cx68x42x43x2ex3ex55x56x5ax25x1fx8x72xdx54x31
x32x7ex25x2bx5fx60x54x1cx6cx7x8x2x1ax62x54x48xfx51x3x54xcxdxdx15x58x2exfx1dx33x78x58
x70xbx1bx57x54x65x63x48x19x57x25x53x3ax3ex10x57x4fx54x69x63x51x21x22x1x54x69x1ex78x2bx71
x44x48x71x72x54x2cx68x42x43x51x43x2x8x3ex54x48x58x2x2ex3ex54x48x2fx5ex2xdx7fx0x3exfx54x78
x8x3ex55x56x53x18x76x7bx46x37x35x5cx48x58x2ex3ex55x56x78x25x23x8x18x76x7bx46x21x14x54x45
x75x38x3dx2ex3bx78
x1ax49x3ax3ex23x24x10x44x29x54 -x74x59x27x62x7axdx50x62xbx21 Hilbertx72x76xfx31
x4dx78x40x6fx54xdx50x62xbx21 Hilbertx72x76xf Mx17x7ax62x7ax1x25x27x72x76(?;F;P)
x44x54x70xbx69x7dx51x2a(x36x69x23x2)x54x28x29x2ax6cx7fx1fx48x10x54x57x58x6ex7ax2N x17
x13x54x50x62xbx21x54x4ax6x28x29x2ax6cx78x10x3ex54x17xbx4ax6x1cx7ax26x11x2bx62x7ax49M
x44x54x62x4ax65x68px27x3cx62x78x31x32x25x2bpx1cx1dx7fx0x48x33x34x66x49m 2 M, x53
x68x10x66x15x1fx 2 M,x51p(x) = E[mx]. x1ax49x54x5fx60x2x3ax3ex58x51x70xbx2cx76
x72x76x75x47x48x78x25xbx2cx76x72x76x2x5dMx2cx54Fn-x23x51x54x7fx1fMn x21x40x41x70
203
xbMx54x14x72x76x78x23x24x2cx68x48x25xbx14x72x76x5bx2x67x54x48x4x10x3ex5bx2Hilbert
x72x76x78x66x2x3ax3ex51
R= M0 ‰M1 ‰￠￠￠‰Mn ‰Mn+1 ‰￠￠￠‰MN = M:
x49x25x2bxex4cx2cx65x63x54xdx50x62xbx21x24x7x1fx6axf (xn)x23x62x7ax1x9xaxn 2
Mn (n = 0;1;:::;N)x54x28x29x1fx6ax48x17x6ex7ax2x50x62xbx21x49x74x2bx18x23x54x6x78
x66x2x3ax3ex2exfx75x47x48x62x7ax49M = MN x44x48x47x6x49M0 = Rx44x54x62x4a
x65x68x48x58x75x47x48x10x660 < s < t 6 N,x62x7ax49Mt x44x47x6x49Ms x44x54x62x4a
x65x68x78x10x54x6ex7ax2x2ax18x23nx54x50x62xbx21x49x18x23mx21x47x54(x2ex3cx62)x6x78
x31x32x25x26x54x62x4ax65x68x5ax2x1cx1dx7fx0x54x48x33x34x10x2x35x5ax51x72xdx54xdx28x29
x13x1ax7bx14xfx66x49x7ex25x2bx5fx60x2ex1x68x4fx5fx60x48x47x75x76x77x57x4fx54Rieszx60x27
x62x69x78x58x25x3bx2 Hansen–Richard [10] x1x61x65x63x38x3dx7cx6ax54x2ex3bx54x74x75
x31x32x78x49x3ax3ex54xex4cx2cx48x25x70x31x32x23x76x29x31x56x2ax200 6 s < t 6 N. x5dx5e
pts:Mt ! Ms x1dx75x76x5fx60x3bx46x79x62x4ex5fx60x1d: x60x61x3x7ax79mts 2 Mt,x4
x5
8xt 2Mt; pts(xt) = E[mtsxtjFs]:
x25x26x70x27x48x10x66x70x2bx71x72x54x3dx3ex5fx60x48x31x32x3ax3ex47x48x50x27x54x78x2b
x18x23x2a tx4t+1,x3ax3ex23x24x65x63x25x26x54x28x29x13x1ax7bx14x69x63x2a
pt(xt+1) = E[mt+1t xt+1jFt] = Et[mt+1t xt+1];
x25x18xt+1 x2t + 1x18x23x54(Ft+1-x23x51)x50x62xbx21x48 Et x60x27x10Ft x54x38x3d
x68x4fx3ex2bx78x25x2bx9x26x57x3ax3ex49x65x63x8x3ex55x56x2cx54x16x6fx2ex10x49x66x10x53x1f
x1ax18x76x7bx46x54x38x3dx7cx6ax2ex3bx48x51x6cx62x4ax65x68x54x47x6x5ax2x2ex3cx62x54x28x29
x2ax6cx78x57x7ex7bx61x4cx54x48x26x7ex8x5fx48x3ax3ex10x26x23x24x68x19x1ex6fx1ex37x62x4ax55
x56x4 Markowitz x72x6x64x69x7dxbx3ax48xfx2x10x3ex5bx53x2x1cx38x3dx7cx6ax54x28x29
x55x56x78x25x26x48x3ax3ex53x21x28x1ex69x1ex57x4fx3dx3ex57x58x4fx71x2cx33x14x78x62x63x44x48x4b
x57x10x1ex6fx1ex37x62x4ax55x56x54x71x2cx7dx52x51x1fx3dx18x3ex54x1ax63x78xfx2x31x32x47x48
x19x38x3dx7cx6ax48x25xbx1ax63x2x51x23x25x68x19x70x2bx25x69x54x1ex4dx54x78
x2bx71x44x48x21x22xdx38x3dx68x4fx3ex2bxfx5ax2x70x2bx28x29x2ax6cx48x10x23x24x69x1ex1
xdx49x2ex10x38x3d(x70x71)x56x48x47x2ex10x6x54xfx54x6cx48xfx10x10xfxfx18x38x2ex12x6bx33
x14x78x7ax58x48x39x68x4fx44x27x70x48x44x29x9x26x9x4ax66x2ax10x66x15x1fFt-x23x51x54x28x29
x2ax6czt,x51E[ztpt] = E[mt+1xt+1zt]:x49x25x15x13x70x2bx9x26x2cx48x68x4fx3ex2bx19x57
x2x33x38x3dx54x48x39x58x67x51x23x25x27x4ex4fx12x6bx6bx12x78x7bx7ex48x1x51x12x38x3dx68x4f
x3ex2bx48x3ax3ex23x24x7ex1fx0x4ex40x41zt x27x4ex4fx78x25x64zt xcx1x38x39x2cx2dx78xaxb
x51x2dx2excxdx38x3dx7cx6ax54x3dx3ex71x2cx33x14x5bx2x7ex1fx25x64x69x55x27x4ex4fx54x78x45
x31x48x3ax3ex4bx57x49xdx4ex0x1xfx2cx20x19x1fx54Fama–Frenchx54x60x7bx14x1ex37x62x4a
x6ex26x48x10x49x42x43x14x25x24x19x48x13x0x1x6ex66x33x55x24x58x6ex66x54x8x4bx6x57x42x6
204
x75x77x2ex1x7bx14x48x67x23x1ex4dx1x10xdx38x3dx1ex6fx1ex37x62x4ax55x56xfx54x12x6bx7dx52x48
x58x2ex2x10x1ex6fx1ex37x62x4ax55x56x54x35x62x78x3x66x25x69x4bx54x4ex70x57x65x63x23x46x3c
Cochrane [4]x78
x58x51x70x2bx5fx60x2ax49x7ex1ex54x33x6dx3cx61xdx56x48xbx4bx19x57x5dx8x48x10x9x4a
x66x10x66x21x51x40x6fx2cx68(x45x46x33x69x6ax2cx68)x54x13x1ax4ax4bx1fx6ax48x66x49x9x4ax25
x27x9x51x3bx78x5dx25x70x31x32x57x28x29x13x1ax7bx14x69x63x7bx77x77x48x3ax3ex23x24x3cx19x48x49
p(x) = E[mx]x2cx54x28x29x13x1ax7bx14mx54xdx54x6xf m(1+R)N x7bx17x66x9x4ax25
x27x9x51x3bx10x6ax25x27x51x3bx54x21x22Radon–Nikodymx1fx68x78x2exfx31x7ex48x26x66
x25x2bx9x4ax25x27x51x3bx51x2ex49x45x66x18x3ex48x46x10x66t (> s)x18x3ex54x50x62xbx21x4a
x6xt x54x13x1ax57x10x49sx18x3ex54x4ax6x62x4apts(xt)x54x13x1ax75x76x7x8x51x56x5cx9
x26x2a
pts(xt)
(1+R)s = E
?
m(1+R)N xt(1+R)t
?
:
x57x44x29Hansen–Richardx21x76x77x54Rieszx60x27x62x69x7bx77x77x48x3ax3ex68x19
pts(xt) = E[mtsxtjFs] = E[m(1+R)N?(t?s)xtjFs]:
x25x67x2x70x48x49x70x62x79x7ax56x48x28x29x13x1ax7bx14x28x18x76x54x2ax5cx7bx17x66x10x70x2b
x28x29x13x1ax7bx14x54xdx54x6xfx2ax5cx78x26x7ex1fx68x54xdx18x2axfx1ex6fx1ex37x62x4ax55x56
x7bx58x5ax67x2ax1x28x6dx54x13x1ax3x4cx78xfx2x25x2x7ex1fx7ex1ex54x33x6dx3cx61xdx41x7d
x58x68x19x54x78x72xdx54x63x2cx2x35x23x18x38x7ex1fx2ex3ex55x56x54x1cx1dx62x4ax55xex27x68
x19xdx15x58x2ex68x1bx78x7ax58x48x39x71x2cx54x76x77x27x3cx48x3ax3ex6fx2dx21x64x2bx51x70x2bx1e
x6fx1ex37x62x4ax55x56x31x1fx21x1x2x1ex28x18x76x2ax5cx54x55x56x78x25x64x55x56x54x70x2bx64
x6fx2x21x22xdx5ax18x1ex6fx1ex37x62x4ax55x56 (Intertemporal Capital Asset Pricing
Model, ICAPM)xfx78x13x31x2Merton [17]x39x7fx0x18x76x3dx3ex4fx54x5ex7ex8x5fx5
x1fx47x48x18x76x54x31x32x48x17x15x52x40x26x31x10x57x4fx54CAPMx28x18x76x2ax5cx13x0x44
x70x66x10x70x71x2ax5cx54x47x48x78x10x23x1x11xbx2ex7bx14x55x56x20x48x49x5ex78
x2fx6a x45x41Black–Scholesx55x71x43x16x4x7d([2])x531973x2xcxfx3x4dx5x7dx7dx7ex7e
x52x11x12x58x59x33x34x11x12x4x2ex65x3bx5dx5x6x29 x3cx47Black–Scholesx4x7dx4ex23x9x47x41x43xfx6ax5c
x4dx20x29x19x1ax418.2x1x7x41x41xdx20x3bx18x48x49xfx34x58x2cx2ax3fx20x3bx18x4x7dx7x5 x48x3x7ax2cx5xb
x3bx4x7dx20x2cx2ax78x2bx47x4ex1ex20x5 x56x47x58x3fx5fx26x33x70x4x2fx26x60x20x44x4cx77x43xex20xax4fx29x25
x27x48x49x64x65x4bx41x45 Briys–Bellalah–Mai–de Varenne [3], x6x7ex53x79x25x2ax1cx5x20x2bx2cx29
Black–Scholesx20x58x1cx4fx2ex7x4ex16x17x2exex3bx18x4x36x20x63x36x29
x25x27Black–Scholesx5cx4fx20x5x6bx34x7x64x31x4x71x17x48x3x4ax10x26x27x30x15x13x4dx69x5bx29x52
x53x32x33x61x20x33x21x4dx2dx5x19xcx20x7x64x47Hull [13]x5 x3cx47x41x25x11x12x58x59x33x1x11x20x6bx36[8],
[9], [12], [14], [15]x60x74x74x25Black–Scholesx5cx4fx20x50xdx29x3ex53x32x33xfx20x5fx5bx5 [12]x34
[14]x60x24x25x38x2x3dx41x5 x56x31x4dx6x3ex3fx20x22x70x40x1x52x29 [8], [9]x34[15]x28x60x4ex4fx50x37x38x41
x2dx29x2bx3f Huang–Litzenberger [12]x20162–166x74x6x7bx1ax20x3cx4dx5cx5x52x3bx14x22x55x7fx18
x7x25x19x52x52x3dx30x22x23x6fx4ax7ax32x20x14x5ex7bx48x70x20x7bx48x78x79xcx30x4ex4dx2cx47 Black–Scholes
x4x7dx29xbx6x53x5cxfx52x36x1ax25x21x20x61x68x29
205
x19x1ax20x4x46x20x5bx2cx3bx5bx6x3dx9x37x58xfx4dx45Lamberton–Lapeyre [16],x47x2fx20x55
x1x6x53x32x33xfx36x5cx37x38x61x62x29x27x44x25x1fx70x36x13x52Black–Scholesx5cx4fx20x32x33xfx20x5dx3b
x2cx1x60x2bx2ax3fx5 x3cx2dx2bx22x7fx55x2dx3fx41x20x25x5ex72x42x5x7ex66x48x3x70x36xex1f Baxter–Rennie
[1]x5x47x2fx52x1ax1cx32xex2bx2cx47x41x25x25x27x20x32x33x5 x3cx47x71x17x24x25x3bx14x5dx17x20x72x42x29x27x53
x76x35x12x14x7fx18x20x55x71x43x16x5cx4fx4ex4fx20x6fx3bx2ex19xcx5x6bx47 Pliska [18]x29 x3cx47x3ex53x47x2f
x36x70x71x76x10x76x35xex7cx13x5x4dx1ex3d Black–Scholesx4x7dx60x24x25x4dx2cx29xbx2bx35x2bx2dx47x3bx14
x5bx55x29
Black–Scholes x4x4dx20x73x4dx47x70x2ax6x3dx61x7fx18x20x60x1bxfx20x5x25x27x4fxdx19x1ax23x6
x3x26x7bx7cx29x4ax22x70x40x1x52x40x2cx55x71x43x16x20x5cx33x72x42x2cx7cx75x53 Cox–Ross–Rubinstein
[5] x3x39 Rendleman–Bartter [19]x29xbx3bx72x42x3cx36x1ax2ax5bx74x62x67 Black–Scholes x4x7d
x20x53x5cx39x35x18x48x35x43x16x60x2ex43x5cx23x14x20x27x28x5x23x44x28x2ex6ex4ex37 Black–Scholes x5cx4f
x2ex3x47x3bxax56x4fx1x52x29 Cox–Rubinstein [6] x2ax47x7bx3bx59x15xbx17x20x45x2fx60x17xax3bx2ex5
x6bx5x28x47x55x71x43x16x5cx4fx7x61x25x20x3bx2ex2bx38x39x58x53x77x4fx25 x3cx2dx25x54x5ax20x32x33x72x42x20x6b
x36x29x52x53x12xex58x53x77x4fx10x24x33x13x20x1fx70x4dx2dx5 x48x35x1fxbx2ex2fx6x3exex49x68x18x29
x48x35x43x16x60x2ex43x5cx52x53x26x55x7fx18x25x47x21x20x69x7dx5 x35x7x20x21x2cx20x47x41x7bx47x7dx20xe
xfx29 x3cx47x19x1ax20x4ex4fx1dx7ex53x25x7ex2bx68xbx69x29x52x53x3bx56xbx69x19x1ax48x3x2fx71x35x32x33xfx20
x50x6cx3ax29xbx62x47x57x2x4dx5fx26x11x12x32x33x3dx6x2bx71x40x32x2cx40xex30x31x20xcx30x29x6x20x26x3bx1
x7x5x19x1ax2bx2cx47x3bx56x20x26x55x7fx18x7x20x48x35x43x16x60x2ex43x5cx29x25x7ex12x55x25 x25x7ex60x2ex72x71
x14x32x48x35x6ex58x47x10x70x47x40x3cx4dx18x2cx6x33x16xex41x7dx22x58x33x16x13x20x5cx7ex29x12x55x70x7e (xax35
x61x7bx3dx61x12x14xbx69)x7cx60x2ex72x71x14x32x70x7ex60x23x3cx50x48x35x43x16x60x2ex43x5cx8x2cx1ax62x29
x15x72x7ex20x70x47x40x3cx4dx1cx59xex37x3ax43x16x52x28x4dx4ex4fx26x55x7fx18x0x47x2dx47x3bxax3dx45x1e
x20x2fx52x29x5x73x6ex58x25x47 Hansen–Richard [10]x20x21x2cx43x5cx5x3cx50x58x53x35x2ax2bx5x5cx4fx48
x7bx0x26x55x7fx18x29 x3cx47xex31x4x37x5bx5xbx1x11x20x5cx4fx30x31x16x17x79x2bx74x77x29
x44x30x23x41x45x23
1. x37x34x2Black–Scholesx3exbx62x4ax6ex26x48x17x2cx54x59x70x2bx46x68x2x37x34x79
x7ax7e
2. x76x3cx3bx6bx12x4ax48x6bx12x40x41x14x2ex26x3exbx54x62x4ax78
3. x26Black–Scholesx6ex26x8x5fx48x23x62x7ax2dx2ex3ax3bx3bx46x68x78x25xbx46x68x7e
x7fx3bx64x3cx79x61x60x27x78x10x66x2ex26x70x1x3exbx27x70x48x25xbx46x68x51
? = @c@S; Γ = @
2c
@S2; Θ =
@c
@T ; V =
@c
@ ; P =
@c
@r; ? =
S
c
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x5dx6fx48 2002x48x68x69x3dx3ex4fx0x63x2ax5bx54x18x76x55x56x48x5x71x56x3ax48x57x58x4e
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