モンテカルロ法によるプライシングとリスク量の算出について―正規乱数を用いる場合の適切な実装方法の考察―

Size: px

Start display at page:

Download "モンテカルロ法によるプライシングとリスク量の算出について―正規乱数を用いる場合の適切な実装方法の考察―"

うのすけくまじ
5 years ago
Views:

1 UFJ E-mal: E-mal:

2 CPU Cetral Processg Ut:

3 Excel Falloo Excel McrosoftTM

4 VaR

5 x y x+y x y x x y x y

7 µ Σ α Excel Excel Rad Vsual Basc Rd Rad Rd Rd Exce X ax + c (mod m)a c m m m a,664,525 c,03,904,223 m2 Rd RadMcrosoft Product Support Servces

8 MT Excel Rd Rd(x, y ) (x, x + ),, 2,, 2 Rd Matsumoto ad Nshmura

9 π π π 0, 0, 0,x 0,x (0 x )

10 0, 0,

11 {x, x 2,, x } {{x, x 2, x 3 }, {x 4, x 5, x 6 },, {x 2, x, x }} C CC T C y y y k 2 k k c c c 2 3 c c c c c c x x x k 2 k k, k 3, 6,,, {{y, y 2, y 3 }, {y 4, y 5, y 6 },,{y 2, y, y }} T

12 {{y, y 2, y 3 }, {y 4, y 5, y 6 },,{y 2, y, y }} {0, 0, 0} {0, 0, 0} {{y, y 2, y 3 }, {y 4, y 5, y 6 },,{y 2, y,y }} {0, 0, 0} {{z, z 2, z 3 }, {z 4, z 5, z 6 },,{z 2, z, z }} {0, 0, 0} Excel Rd

13 Excel

14 Excel k {x, x 2,, x k } {x, x 2,, x k/2 } {y, y 2,, y k } {x, x 2,, x k } {x 90, x 95, x 99, x 99.5 }

15 {y, y 2,, y k } {y 90, y 95, y 99, y 99.5 } σ 90 x, σ 95 x, σ 99 x, σ 99.5 x, σ 90 y, σ 95 y, σ y 99, σ 99.5 y k σ y /σ x σ y /σ x σ y /σ x

16 ATM OTM

18 ATM VaR TMATM OTM

20 VaR

21 VaR

27 t quas radom umber Traub ad Werschulz quas Mote Carlo method

28 p(t, X t )X t g(t, X t ) p( t, X ) t E g( t, X t ) g( t, X t ) df ( X, t ) YX t x t F (y) p(t, X t ) g(t,f (y)) dy, 0 g(t, F (y)) dy f (y)dy, 0 0 y, y 2,, y f (y ) A( ) f (y ), p(t, X t )Y A( ) p(t, X t ) A( ) () ( y ) ( y f 2 2 var f var ) { f () y dy p ( t X t) 0 }, var A, f (y) A() var[a()] / ε α Pr [ A( ) p(t, X t ) <ε] α,

29 var[a( )]/ε 2 α f 2( y) dy p 2 (t, X ) 0 t, αε 2 α ε 2 ε α α x α x α ( α) ( x ) 2 α α x α ~ N xα,, f f (x) σ var[ x α ] x α ( p) [ x α σz ( p/ 2), x α + σz ( p/ 2) ], z (p) p α

30 A( ) c c / c lm (/) f (y ) lm (/) f (z ) f (y ) f (z ) {y, y 2,, y } {z, z 2,, z } f (y) {y, y 2,, y } {z, z 2,, z } {y, y 2,, y } lm (/) y f (y) f (y ) f (z ) y z

31 X Y X Y Z X α(y Ε[Y]) g ( t, X t ) {x, x 2,, x } w x {w, w 2,, w } Curra F Moro Joy, Boyke, ad Ta x uφ(x) Φ() x ( 2 exp u2 / π ) ( / 2) du a b c y u 0.5 y Φ ( u ) y ( a0 + a y + a 2 y + a3y )/( + b + b 2 y y + b 6 3 y + b4y ) y >0.42y>0.0 r.0 u y<0.0 Φ ( u ) c c z Φ ( u ) ( c c z ) r uzlog( log(r))

32 g(t, X) g(t,w) g(t, X t ) g(t,x) g(t,w) { x}{x}+{w} x l x ( ) ( ) ( ) [ ] ( ) [ ] ( ) ( ) [ ] ( ) W t g X t g W t g X t g W t g X t g,,, 2cov, var, var 4 2,, var + + +, [ ] ( ) ( ) x x w x x E [ ] ( ) [ ] [ ] ( ) ( ) ( ) + + l l l l l l x x w x E E x E 2 2 ( ) 0 2 l l x x ( ) + l l l x x x 2 x x l l,, ( ) ( ) ( ) t X t p w t g x t g,, lm, lm ( ) ( ) ( ) t X t p w t g x t g, 2,, lm +,

33 {x, x 2,, x } y x / k x k {y, y 2,, y } E[ Y ] x 0 xk x x, k k k y l l E[ ( Y E[ Y] ) ] E[ Y ] E[ ( X E [ X] ) ], l l x l E[(Y ( ) E[Y ( ) ](Y ( j) E[Y ( j) ])] E[Y ( ) Y ( j) ]E[(X ( ) E[X ( ) ])(X ( j) E[X ( j) ])], X ( ) Y ( ) X Y x XY X Y CC T C I

34 {x, x 2,, x } ms m x, s T ( x m )( x m ), s s C C T, y CC x, y {y, y 2,, y } {x, x 2,, x } y CC (x m)+m, y CC x (x m)+, {x, x 2,, x }x {x, x 2,, x } T y {y, y 2,, y } T y Cx+ {x, x 2,, x }

39 Curra, M., Strata gems, Rsk, Vol.7 No.3, 994. Falloo, W., Who ows the deas that derve dervatve?, Rsk, Vol.2 No.2, 999. Johso, N. L., S., Kotz, ad N. Balakarsha, Cotuous Uvarate Dstrbutos Vol. 2d Edto, Joh Wley & Sos, 994. Joro, P., Value at Rsk 2d Edto, McGraw-Hll, 200. Joy, C., P. P. Boyle, ad K. S. Ta, Quas Mote Carlo Methods Numercal Face, Chapter 24 Mote Carlo, Methodologes ad Applcatos for Prcg ad Rsk Maagemet, Rsk, 998. Matsumoto, M., ad T. Nshmura, Mersee Twster, ACM Trascrpt o Modelg ad Computer Smulato, Vol.8 No., 998. Moro, B., The full Mote, Rsk, Vol.8 No.2, 995. Traub, J. F., ad A. G. Werschulz, Complexty ad Iformato, Cambrdge Uversty Press, 998. Wlmott, P., Paul Wlmott o Quattatve Face, Joh Wley & Sos, Mcrosoft Product Support Servces, Excel: Radom Number Geerato, ( com/support/kb/artcles/q86/5/23.asp).

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()