CEV(Constant Elasticity of Variance) FE 1

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1 CEV(Consan Elasiciy of Variance) FE 1

2 1 (Karazas and Shreve[1998] Carr, Jarrow and Myneni[1992] J. Huang, Subrahmanyam and Yu[1996] ) S 1 ds =(r δ)s d + σ(s )dw,s = x(> ) r δ K T sup E[e rτ (K S τ ) + ]= τ T [ [ E e rt (K S T ) +] T + rke [ ] T δe e ru S u 1 {Su Bu}du ] e ru 1 {Su Bu}du B T Taahashi[1995] [1999] () ( [1992] Taahashi[1995] [1999]) (Kuniomo and Taahashi[21]) (Ieda and Waanabe[1989] Yoshida[1992] ) (Kuniomo and Taahashi[23]) 2

3 (Taahashi and Yoshida[21a] [23] Kobayashi, Taahashi and Toioa[21]) (Taahashi and Yoshida[21b]) [23] S CEV(Consan Elasiiciy of Variance) ds =(r δ)s d + σs γ dw, (1/2 γ 1) CEV 5 CEV CEV 2 (Ω, F, {F } T, P) [,T] {F } T (-usual condiions- Karazas and Shreve[1996] Definiion ) W, ( T ) (Ω, F, {F } T, P) 1 P 3

4 S x, ( T ) ds =(r δ)s d + σ(s )dw,s = x(> ) (1) () x,r,δ σ : R R y, z R y, z K 1 > σ(y) 2 K 1 (1 + y 2 ) (2) σ(y) σ(z) h( y z ) h h :[, ) [, ), h() =, ɛ> (,ɛ) h 2 (u)du = 1 [1974] 3.1 Karazas and Shreve[1996] Proposiion ( ) (1) (2) K > < T <x< x p(, x) := sup E[e rτ (K Sτ x ) + ] τ (3) C := {x (, ); p(, x) > (K x) + } (4) sup τ τ a > <l T a l P( inf u l Sx u <a) > (5) c() C =(c(), ) (6) ( ) 6.1 p(, x) x ( K ) (5) ( ) (6) c() c() c() 4

5 2 1 B := c(t ) <T [ ] T sup E[e rτ (K Sτ x ) + ] = E[e rt (K ST x ) + ]+E rk e ru 1 {S x u B u}du τ T [ ] T E δ e ru Su1 x {S x u B u}du (7) ( ) x T ( < T ) 1 2 x T ( <T) p(t, x) = E[e r(t ) (K S x T ) + ]+E[ 1 E[ p(t, x) = sup [e rτ (K Sτ x ) + ] τ T rke ru 1 {S x u B +u}du] δs x ue ru 1 {S x u B +u}du] (8) = E[e r(t ) (K ST x )+ ]+E[ rke ru 1 {S x u c(t u)}du] E[ δsue x ru 1 {S x u c(t u)}du] = E[e r(t ) (K ST x )+ ]+E[ rke ru 1 {S x u B +u}du] E[ δsue x ru 1 {S x u B +u}du]. (8) B B K z = p(t, z) = E[e r(t ) (K ST z ) + ]+E[ rke ru 1 {S z u B +u}du] E[ δsue z ru 1 {S z u B +u}du] (9) z 5

6 3 x> x ds =(r δ)s d + σ (S )dw,s = x (1) S x, T r δ σ : R R (2) 3.1 T K P A P A = sup E[e rτ (K S τ ) + ] (11) τ T P A ( 2 ) P A = P E (T,x) + rke[ e ru 1 {S x u B u}du}] δe[ e ru S x u1 {S x u B u}du] (12) B T K z = P E (T, z)+rke[ e ru 1 {S z u <B u+}du] δe[ e ru Su1 z {S z u <B u+}du] (13) z ( 2 ) P E (, z) :=E[e r (K S z ) + ] (14) S B P A [,T] N := T N,, 2,..., N B (N 1),B (N 2),..., B 6

7 K z = P E (,z) (15) z B (N 1) K z = P E (2,z)+rK e r P(S z <B (N 1) ) δ e r E[S 1 z {S z <B (N 1) }] (16) z B (N 2) B (N 1),B (N 2),..., B (N i+1) i 1 K z = P E (i,z)+rk e r( ) P(S z <B (N i+) ) i 1 =1 δ e r( ) E[S 1 z {S z <B (N i+) }] (17) =1 z B (N i) B (N 1),B (N 2),..., B P A i 1 P A = P E (T,x)+rK e r( ) P(S z <B ) i 1 =1 δ e r( ) E[S z 1 {S z <B }] (18) =1 3.2 P E (, x) P(S <A) E(S 1 {S<A}) A P E (, x) P(S <A) E(S 1 {S<A}) S (1) ɛ S (ɛ) ds (ɛ) =(r δ)s (ɛ) d + ɛσ(s (ɛ) )dw,s (ɛ) = x (19) 7

8 ɛ (, 1] (1) (1) σ (S ) ɛσ(s (ɛ) ) σ (S )=ɛσ(s (ɛ) ) ɛ =. S (ɛ) ɛ =. S () (19) ds (ɛ) =(r δ)s (ɛ) d S (ɛ) = x S () := xe (r δ) (2) S (ɛ) ɛ =. ɛ 2 S (ɛ) = S () + ɛg 1 + ɛ 2 g 2 + o(ɛ 2 ), (21) g 1 := g 2 := σ(s () ) := σ(s(ɛ) e (r δ)( s)) σ(s () s )dw s, e (r δ)( s)) σ(s () s )g 1sdW s, ) S (ɛ) (ɛ) S =S () 1 S () X (ɛ) X (ɛ) X (ɛ) := S(ɛ) ɛ X (ɛ) S (). (22) ɛ = g 1 + ɛg 2 + o(ɛ) g 1 Σ Σ := e2(r δ)( u) σ 2 (S u () )du 1 X (ɛ) ɛ Taahashi[1999] Lemma 2.1(1) E[g 2 g 1 = x] =c x 2 + f, c := 1 Σ 2 f := c Σ, u e (r δ)( u) σ(s u () ) σ(s u () ) X (ɛ) e 2(r δ)( v) σ 2 (S v () )dvdu g (ɛ) X (x) n[x;, Σ ]+ɛ[ d dx {(c x 2 + f )n[x;, Σ ]}] (23) 8

9 ɛ ( Taahashi[1999] ) ( ) 1 x 2 n[x;, Σ ]:= exp, 2πΣ 2Σ X (ɛ) ɛ ((23) 1 ) ɛ ((23) 2 ) X (ɛ) (23) P(S (ɛ) <A) E(S (ɛ) 1 (ɛ) ) {S <A} ( 6.3 ) P(S (ɛ) <A) = P( S(ɛ) S () ɛ = P(X (ɛ) < A S() ) ɛ < A S() ) ɛ N( a ) ɛ(c a 2 + f )n[a ;, Σ ], (24) Σ a := A S(), ɛ y 1 N(y) := exp( x2 2π 2 )dx, E[S (ɛ) 1 (ɛ) {S <A} = E[S(ɛ) 1 (ɛ) {X = E[(S (ɛ) S () = E[(S (ɛ) S () <a } ] = ɛe[ S(ɛ) S () ɛ = ɛe[x (ɛ) )1 (ɛ) + {X <a S() } 1 (ɛ) ] {X <a } )1 (ɛ) {X <a ]+S() } P(X (ɛ) 1 {X (ɛ) <a ]+S() } P(X (ɛ) 1 (ɛ) {X <a ) ɛ( Σn[a ;, Σ ] ɛc a 3 n[a ;, Σ ]) <a ]+S() } P(X (ɛ) <a ) <a ) +S () (N ( a ) ɛ(c a 2 + f )n[a ;, Σ ]), (25) Σ P E (, x) P E (, x) ɛ PE ɛ (, x) PE(, ɛ x) = E[e r (K S (ɛ) ) + ] = E[e r (K S (ɛ) )1 (ɛ) {S <K} ] = e r (KP[S (ɛ) <K] E[S (ɛ) 1 (ɛ) ]) {S <K} e r K(N ( Σ ) ɛ(c 2 + f )n[ ;, Σ ]) 9

10 e r {ɛ( Σ n[ ;, Σ ] ɛc 3 n[ ;, Σ ]) +S () (N ( ) ɛ(c 2 + f )n[ ;, Σ ])}. (26) Σ := K S() ɛ ɛ σ (x) =ɛσ(x) (27) σ(x) Gese and Johnson[1984] (Richardson s approximaion scheme) P A 4 h F (h) F (h) =F () + a 1 h + a 2 h 2 + a 3 h 3 + o(h 3 ) 4 h = T/4 F (ih) ih F (4h) = F () + 4a 1 h +16a 2 h 2 +64a 3 h 3 + o(h 3 ) F (2h) = F () + 2a 1 h +4a 2 h 2 +8a 3 h 3 + o(h 3 ) F (4/3h) = F () a 1h a 2h a 3h 3 + o(h 3 ) F (h) = F () + a 1 h + a 2 h 2 + a 3 h 3 + o(h 3 ) h 3 o(h 3 ) F () F () F () 1 6 F (4h)+4F(2h) 27 2 F (4/3h)+32 F (h) (28) 3 F () P A F (4h) T F (2h) T/2 F (4h/3) T/3 F (h) T/4 4 1

11 4 CEV(Consan Elasiciy Variance) 4.1 CEV CEV CEV ds =(r δ)s d + σs γ dw,s = x (29) γ 1/2 γ 1 r, δ, σ x x > CEV S =(r δ) +( S σ S 1 γ ) W CEV ( 6.4 ) S (ɛ) ds =(r δ)s d + ɛbs γ dw,s = x (3) σ(x) :=bx γ σ (x) =bγx γ 1 α := r δ α Σ Σ = = e 2α( u) σ(s u () ) 2 du e 2α( u) b 2 (xe αu ) 2γ du 11

12 = b 2 x 2γ e 2α e 2(γ 1)αu du = b 2 x 2γ 2(γ 1)α e2α (e 2(γ 1)α 1). (31) c c Σ 2 = u e α( u) b(xe αu ) γ bγ(xe αu ) γ 1 e 2α( v) b 2 (xe αv ) 2γ dvdu u = b 4 x 4γ 1 γe 3α e 2(γ 1)αu e 2(γ 1)αv dvdu = b4 x 4γ 1 γ 2(γ 1)α e3α e 2(γ 1)αu [e 2(γ 1)αu 1]du = b4 x 4γ 1 γ 2(γ 1)α e3α 1 4(γ 1)α (e2(γ 1)α 1) 2 = b 4 x 4γ 1 γ 8(γ 1) 2 α 2 e3α (e 2(γ 1)α 1) 2 Σ 2 = b 4 x 4γ 4(γ 1) 2 α 2 e4α (e 2(γ 1)α 1) 2 c = γ 2xe α (32) 4.2 CEV r =.488 S() = 4 b = 1 γ =.5,.66,.75 T =.833,.3333,.5833,1 K = 35, 4, 452% 3% 4% δ =,.1,.3, ɛ b σ ɛbs() γ = σ S() ɛ δ = γ =.5,.66,.75 T =.833,.3333,.5833,1 K =35, 4, 45 σ =.2,.3,

13 (ame a.e.) Nelson and Ramaswamy [15] (ame lai) (ame lai) (ame a.e.) (error1) 1 γ =.75 K =35 δ =. T =1 σ = % % (eur a.e.) Nelson and Ramaswamy (eur lai) (eur lai) (eur a.e.) (error2) (ame a.e.) (eur a.e.) (premi a.e.) Nelson and Ramaswamy (ame lai) (eur lai) (premi lai) (premi lai) (premi a.e.) (error3) 4 (ame rich.) (ame lai) (ame rich.) (error4) (error1) (error2) (error3) (error4).1 T =.833 K =35 σ = δ =.1,.3, δ = γ =.5,.66,.75 3 (ame a.e.) (ame rich.) (eur a.e.) (average) (rmse) (max) (min) T =.833,.3333,.5833, 1 K =35, 4, 45 σ =.2,.3, δ =.1,.3, δ = γ =.75 T =1 σ =.2 K = 45([A] γ =.75 T =1 σ =.4 K =35 [B] 1 6 S T 6 (ame lai) (ame a.e.) (error1) (error1) 1 (ame rich.) (error4) (error4)

14 (error1) 1 γ T OTM(ou of he money) γ =.75,T =1,K =35 σ = X (ɛ) (γ =.) γ. ɛ = T σ OTM(ou of he money) OTM ( ) (premi lai) (premi a.e.) (error3) (premi lai)(premi a.e.) δ T ITM(in he money) 12 δ = T =1 σ =.2 K =45 ( 1 ) S T 1 6 [B](δ = γ =.75 T =1 σ =.4 K = 35) 6 (error1) [A](δ = γ =.75 T =1 σ =.2 K = 45) [B] δ = γ =.75 T =1 (error1) OTM [B] B 1 2) [A] [B] [A] 14

15 [A] [B] S T ( 3 5 [A] K 4 6 [A] [B] K 5 (Taahashi[22] ) CEV (Taahashi[1999]) (Taahashi and Yoshida[21b]) x y P[S x S y, T ]=1 15

16 Karazas and Shreve [7] Proposiion K 2 > E[ sup S x 2 ] < T (r δ) 2 x 2 + σ(x) 2 K 2 (1 + x 2 ) [2] 3.2.6) e (r δ) S x, T e (r δ) S x = x + e (r δ)u σ(su)dw x u. r δ E[ e 2(r δ)u σ 2 (S x u)du] E[ E[ σ 2 (S x u)du] = K 1 + K 1 E[ K 1 (1 + S x u 2 )du] S x u 2 du] = K 1 + K 1 E[Su x2 ]du K 1 + K 1 E[ sup Ss x 2 ]du s T = K + K 1 E[ sup Ss x 2 ] s T < r δ < e (r δ) S x, T p(, x) (K x) + p(, x) = sup E[e rτ (K Sτ x)+ ] τ E[(K S x)+ ] = (K x) + p(, x) > <x<k K x p(, x) (K x) + > τ 1 := inf{u >; S x u K 2 } 16

17 p(, x) E[e rτ 1 (K S x τ 1 ) + ] E[e rτ 1 K 2 1 {τ 1 <}] = K 2 E[e rτ 1 1 {τ1 <}] >. p(, x) >. 1 p :(,T] (, ) [, ) (33) (,x ) [, ) 2 τ := inf{u [,); p( u, x )=(K S x (u)) + } τ := τ sup E[e rτ (K S x τ )+ ]=E[e rτ (K S x τ )+ ] τ p(, x ) p(,x ) = p(, x ) p(,x ) = E[e rτ (K S x (τ )) + ] p(,x ) E[e rτ (K S x (τ )) + ] E[e rτ (K S x (τ )) + ] = E[e rτ (K S x (τ )) + e rτ (K S x (τ )) + ] E[{e rτ (K S x (τ )) e rτ (K S x (τ ))} + ] E[(e rτ S x (τ ) e rτ S x (τ )) + ] = E[(e r S x ( ) e rτ S x (τ )) + 1 {τ = }] = E[(e r S x ( ) e rτ S x (τ )) + 1 { τ }] E[ sup (e r S x ( ) e ru S x (u)) + 1 { τ }] u E[ sup (e r S x ( ) e ru S x (u)) + ]. u α β α + β + (α β) + 17

18 sup (e r S x ( ) e ru S x (u)) + u 2 sup e ru S x (u) u 2 sup S x (u). u T E[sup u T S x (u) ] < p x <x y x y << τ x := inf{u [,); p( u, x) =(K S(u)) + } x y p(, x) p(, y) p(, x) = sup E[e rτ (K Sτ x ) + ] τ p(, y) = sup E[e rτ (K Sτ y ) + ] τ τ τ S x τ Sy τ e rτ (K S x τ )+ e rτ (K S y τ )+ p(, x) p(, y) p(, x) p(, y) = p(, x) p(, y) = E[e rτx (K S x (τ x )) + ] p(, y) E[e rτx (K S x (τ x )) + ] E[e rτx (K S y (τ x )) + ] = E[e rτx {(K S x (τ x )) + (K S y (τ x )) + }] E[e rτx (S y (τ x ) S x (τ x ))] 18

19 = E[e δτx e (r δ)τx (S y (τ x ) S x (τ x ))] E[e (r δ)τx (S y (τ x ) S x (τ x ))] = y x = x y. u u S x (u) S y (u) e (r δ) S() x >y p x p p(, x) p(,x ) p(, x) p(, x ) + p(, x ) p(,x ) x x + p(, x ) p(,x ) ((, x) (,x ) ) p(, x) (34) x p(, x) (35) x x + p(, x) (36) p(, x) <x y << x y p(, x) p(, y) E[e rτx (K S x (τ x )) + ] E[e rτx (K S y (τ x )) + ] E[e rτx (S y (τ x ) S x (τ x ))] = E[e δτx e (r δ)τx (S y (τ x ) S x (τ x ))] E[e (r δ)τx (S y (τ x ) S x (τ x ))] = y x. p(, x)+x p(, y)+y x C x<y y p(, y) p(, x)+x y > (K x) + + x y (K x)+x y = K y 19

20 p(, y) > p(, y) > (K y) + y C p(, x) C = {x (, ); p(, x) > (K x) + } c() C =(c(), ) Ruowsi[1994] 6.5 A1 X = X + M + V ; T (37) (X ) M M = V V = E[ sup X ] < T V V d v dv d = v d {,T} E[X τ ]=ess sup E[X τ F ], τ T T τ [ ] T E[X τ ]=E[X T ] E 1 {τ u =u}dv u τ (38) 2

21 2 B <K C T =(c(t ), ) p(t, K) >, (K K) + = p(t, K) > (K K) +. K C T B = c(t ) <K e rt (K S T ) + e rt (K S T ) + = (K S ) + + ( rke ru 1 {Su K} + δs u e ru 1 {Su K})du e ru 1 {Su K}σ(S u )dw u + e ru dl K u (S). L K u (S) S K X, M, V X := e r (K S ) + M := V := e ru 1 {Su K}σ(S u )dw u ( rke ru 1 {Su K} + δs u e ru 1 {Su K})du + e ru dl K u (S). σ E[sup T S x 2 ] < Burholder (Karazas and Shreve[1996] Theorem ) E[ sup X ] < T rke r 1 {S K} τ := inf{u [, T ); S u B u } T E[e rτ (K Sτ ) + F ] = ess sup E[e rτ (K S τ ) + F ] for all [,T] τ T 21

22 A1 sup E[e rτ (K S τ ) + ] = E[e rτ (K Sτ ) + ] τ T [ ] T = E[e rt (K S T ) + ] E 1 {τ u =u}dv u [ ] T = E[e rt (K S T ) + ] E 1 {Su Bu}dV u τ [ ] T = E[e rt (K S T ) + ] E 1 {Su Bu}dV u. τ [ ] T E 1 {Su Bu}dV u = E[ ( rke ru 1 {Su K}1 {Su Bu} + δs u e ru 1 {Su K}1 {Su Bu})du + e ru 1 {Su B u}dl K u (S)]. S B B <KL K (S) 1 {Su Bu}1 {Su K} =1 {Su Bu} [ ] T e ru 1 {Su Bu}du sup E[e rτ (K S τ ) + ] = E[e rt (K S T ) + ]+E rk τ T [ E δ e ru S u 1 {Su Bu}du ] 6.3 X (ɛ) P(X (ɛ) <) 1 (ɛ) {X }] E[X (ɛ) < A2 X g (ɛ) X (x) =n[x;, Σ] + ɛ[ d {h(x)n[x;, Σ]}] dx ɛ (, 1] Σ c h(x) :=c x 2 + f f := c Σ ( ) 1 x 2 n[x;, Σ ]:= exp 2πΣ 2Σ P(X (ɛ) <)=N( Σ ) ɛ(c 2 + f )n[;, Σ ], 22

23 E[X (ɛ) 1 (ɛ) ]= Σ {X <} n[;, Σ ] ɛc 3 n[;, Σ ]. N(y) := y 1 2π exp( x2 2 )dx. P(X (ɛ) <) = = g (ɛ) X (x)dx [n[x;, Σ ]+ɛ[ d dx {h(x)n[x;, Σ ]}]dx. n[x;, Σ ]dx = N( ). Σ ɛ[ d dx {h(x)n[x;, Σ ]}]dx = ɛ[h(x)n[x;, Σ ]] x= = ɛ[h()n[;, Σ ]] = ɛ(c 2 + f )n[;, Σ ]. P(X (ɛ) <)=N( Σ ) ɛ(c 2 + f )n[;, Σ ]. E[X1 {X (ɛ) <} ] = = xg (ɛ) X (x)dx { x n[x;, Σ ]+ɛ[ d dx {h(x)n[x;, Σ ]}] } dx. xn[x;, Σ ]dx = ( ) 1 x 2 x exp dx 2πΣ 2Σ = Σ n[x;, Σ ] x= = Σ n[;, Σ ]. x d dx (h(x)n[x;, Σ ])dx = xh(x)n[x;, Σ ] h(x)n[x;, Σ ]dx = = h()n[;, Σ ] = c (c x 2 + f )n[x;, Σ ]dx h(x)n[x;, Σ ]dx h(x)n[x;, Σ ]dx. x 2 n[x;, Σ ]dx + f N( ). Σ 23

24 x 2 n[x;, Σ ]dx = x d dx ( Σ n[x;, Σ ])dx = Σ (xn[x;, Σ ] n[x;, Σ ]dx) = Σ (n[;, Σ ] N( Σ )). E[X (ɛ) 1 (ɛ) ]= Σ {X <} n[;, Σ ] ɛc 3 n[;, Σ ]. 6.4 CEV CEV ( 2) ds = αs d + σ(s ) γ dw,s = x γ 1/2 γ<1 B1γ 1/2 γ 1 x y K x>y x γ y γ K x y γ x γ y γ (x y) γ = 1 uγ (1 u) γ. u := y/x u<1 1 u γ (1 u) γ u= =1 24

25 1 uγ (1 u) γ lim u 1 1 u γ (1 u) γ = u<1 K 1 u γ <K, u<1 (1 u) γ x γ y γ K x y γ x = y x <y S x, < x x S x, <, (P a.s.) [1974] 3.1 B1 Karazas and Shreve[1996] Proposiion x = S =, (Karazas and Shreve[1996] Proposiion ) x> x> P[S S x, < ] =1 P[S x, < ] =1 B2x> a> <l< a l P( inf u l Sx u <a) > 25

26 α> S = x + e α S = x + αs u du + σs γ u dw u σe αu S γ u dw u l M := σe αu S γ udw u P( inf u l Sx u a) =1 M = σ 2 e 2αu S 2γ u du σ 2 e 2αu a 2γ du (P a.s.). s< l σ2 e 2αu a 2γ du s T (s) T (s) := inf{ ; M >s} T (s) l s< l σ2 e 2αu a 2γ du s B s := M T (s), G s := F T (s) B s M T (s) s c P( inf u l M u <c) >. S = e α (x + M ) P( inf u l Sx u <a) > 26

27 P( inf u l Sx u a) =1 α> (Karazas and Shreve[1996] Proposiion ) α P( inf u l Sx u <a) > 6.5 A1 Ruowsi[1994] A1 Z X Z X Jaca [4] η Z Z. Z η = ess sup E[X τ F η ]. η τ T τ = inf{u [, T ] Z u = X u }. Z = Z + N + B N B E[X τ ] = E[E[X τ F ]] = E[ess sup E[X τ F ]] τ T = E[Z ] = E[Z T N T B T ] = E[Z T ] E[B T ] = E[ess sup T τ T E[X τ F T ]] E[B T ] = E[X T ] E[B T ] = E[X T ] E[ 1 {τ u =u}db u ] E[ 1 {τ u >u}db u ]. 27

28 1 {τ u >u}db u = (P a.s.) D := {τ >} = {Z X } Z X D D 1/n L n R n ( =1, 2,..., N n) S n := Ln +1/n τ S n = R n Z S n = ess sup E[X τ F S n S n τ T ] = E(X τ F S n S n ) = E(X R n F S n ) = E(Z R n F S n ). N n D = [S n,rn ). n=1 =1 D n := N n =1 [Sn,Rn ) 1 {τ u >u}db u = = = lim n = lim 1 D (u)db u 1 n=1 D n (u)db u N n n =1 1 Dn (u)db u (B R n B S n ). Z S n = E(Z R n F S n ) E(B R n B S n F S n ) = E(Z R n Z S n F S n ) E(N R n N S n F S n ) =. B B R n B S n B R n B S n = 1 {τ u >u}db u = U := Z X U 28

29 U L (U) L (U) L (U) L (U) =2 1 {} (U u )d(b u V u ) (Proer[1992] ) U L (U) ={τu = u} = {U u =} 1 {τ u =u}db u = 1 {} (U u )db u = 1 2 L (U)+ = 1 2 L (U)+ 1 {} (U u )dv u 1 {τ u =u}dv u. 1 Y = Y +M +C ξ C C i dci = ξ d Y L (Y ) L (Y )= L (Y )=Y L (Y ) = 2 d M u = 2 = 2 1 {} (Y u )dc u 1 {} (Y u )dc i u 1 {} (Y u )ξ u du, T d M u = m u du + d M s u m u W := {u [,T] m u > } W c [,T] W L T (Y ) = 2 1 {} (Y u )ξ u 1 W (u)du +2 1 {} (Y u )ξ u 1 W c(u)du = 2 1 {} (Y u )ξ u 1 W (u)m 1 u (d M u d M s u) +2 = J 1 J 2 + J 3 1 {} (Y u )ξ u 1 W c(u)du 29

30 J 1 := 2 J 2 := 2 J 3 := 2 M = 1 {} (Y u )ξ u 1 W (u)m 1 u d M u 1 {} (Y u )ξ u 1 W (u)m 1 u d M s u 1 {} (Y u )ξ u 1 W c(u)du L z (Y )dz J 1 =2 1 {} (z)ξ u 1 W (u)m 1 u dl z u(y )dz = J 2 = 2 2 = J 1 J 1 = J 2 = 1 {} (Y u )ξ u 1 W (u)m 1 u d M s u 1 {} (Y u )ξ u 1 W (u)m 1 u d( M s u + m u du) M := {u [,T] m u = Y u =} λ λ(m) = J 3 J 3 = λ(m) > g() := 1 M (u)du, T g() g 1 () τ = g 1 () Ỹ := Y τ F := F τ Ỹ F λ{u [,T] Ỹu =} = Ỹ L T (Y )=L g(t ) (Y ) (El Karoui[1978] ) L T (Y )= U V d dv d = v d 1 L (U) = 3

31 E[X τ ] = E[X T ] E[ = E[X T ] E[ 1 {τ u =u}db u ] E[ 1 {τ u >u}db u ] 1 {τ u =u}db u ] = E[X T ]+ 1 2 L (U)+ = E[X T ]+ 1 {τ u =u}dv u 1 {τ u =u}dv u [1] P. Carr, R. Jarrow and R. Myneni, Alernaive characerizaions of American pu opions, Mahemaical Finance, [2] N. Ieda and S. Waanabe, Sochasic Differenial Equaions and Diffusion Processes, Second Ediion, Norh-Holland/Kodansha, Toyo, [3] J. Huang, M. G. Subrahmanyam and G. G. Yu, Pricing and Hedging American Opions: A Recursive Inegraion Mehod and is Implemenaion, Review of Financial Sudies, Spring [4] S. D. Jaca, Local Times, Opimal Sopping and Semimaringales, Annals of Probabiliy, 21, 1993, pp [5] EL Karoui, Sur les Monees des Semi-maringales, Aserisque, 1978, pp [6] R. Gese and H. Johnson, The American Pu Value Analyically, Journal of Finance, 39, [7] I. Karazas and S. E. Shreve, Brownian Moion and Sochasic Calculus, Graduae Texs in Mahemaics 113, Springer-Verlag, 2nd ediion, [8] I. Karazas and S. E. Shreve, Mehods of Mahemaical Finance, Springer, [9] T.Kobayashi, A.Taahashi and N. Toioa, Dynamic Opimaliy of Yield Curve Sraegies, Discussion Paper Series Faculy of Economics Universiy of Toyo, CIRJE-F-141(submied), 21. [1],,, vol 14, 1992, pp

32 [11] N. Kuniomo and A.Taahashi, The Asympoic Expansion Approach o he Valuaion of Ineres Rae Coningen Claims, Mahemaical Finance 11, 21, pp [12] N. Kuniomo and A.Taahashi, On Validiy of he Asympoic Expansion Approach in Coningen Claim Analysis, The Annals of Applied Probabiliy, 13 No.3, Augus, 23(o appear). [13],, - -,, 23( ). [14],,, 2. [15] D. B. Nelson, and K. Ramaswamy, Simple Binomial Processes as Diffusion Approximaions in Financial Models, Review of Financial Sudies, 3, 199. [16] P. Proer, Sochasic Inegraion and Differenial Equaions, Springer, [17] M. Ruowsi, The early exercise premium represenaion of foreign mare American opions, Mahemaical Finance, 4, 1994, pp [18] A.Taahashi, Essays on he Valuaion Problems of Coningen Claims, Ph.D. Disseraion, Universiy of California, Bereley, [19] A.Taahashi, An Asympoic Expansion Approach o Pricing Financial Coningen Claims, Asia-Pacific Financial Mares, 6, 1999, pp [2] A.Taahashi, An Asympoic Expansion Approach o American Opions, Unpublished manuscrip, 22. [21] A.Taahashi and N. Yoshida, An Asympoic Expansion Scheme for Opimal Porfolio for Invesmen, 1215, 21a. [22] A.Taahashi and N. Yoshida, The Asympoic Expansion Approach wih Mone Carlo Simulaions, Preprin, 21b. [23] A.Taahashi and N. Yoshida, An Asympoic Expansion Scheme for Opimal Invexmen Problems, Preprin, 23. [24],,,

33 [25] N. Yoshida, Asympoic Expansions of Maximum Lielihood Esimaor for Small Diffusion via he heory of Malliavin-Waanabe, Probabiliy Theory and Relaed Fields, 92, 1992, pp

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