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,,,17,,,194 1 4 ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,, 1 4 1.1........................................ 4 5.1........................................ 5......................................... 8.3.................................... 9 3 11 3.1 ( )................ 11 3.1.1........................... 13 3................................. 18 3..1.................................. 18

3.. CEV (Constant Elasticity of Variance )............. 19 3..3 SV (Stochastics Volatility ).................... 3..4 SV ()................................... 4 3..5................ 7 4 3 4.1.......................... 3 4.1.1.................................. 33 4.1. CEV.................................... 33 5 36 3

1,, T > [, T ] 1.1 (Ω, F, P ) 1.1. ( ) F σ- F t, {F t } t, (Ω, F, P, {F t }), {F t } P -, : F t+ϵ = F t ϵ> 1.. ( ) t [, T ], X t, X = {X t } t [,T ], X = (X 1,, X d ) = {(Xt 1,, Xt d )} t [,T ] d, X, ω Ω,, X t (ω) t [, T ] 1.3. ( ) τ : Ω [, ], {τ t} F t (t ). 1.4. ( ) (Ω, F, P, {F t }) 1. {X t } t [,T ] F t -, t X t F t -. {X t } t [,T ] F t -, X t F t -, t X t, t > s, E[X t F s ] = X s 4

, F t -, F t - 3. {X t } t [,T ] F t -, τ n a.s. τ n, n N, {X t τn } t [,T ] F t - 1.1. ( ) X (Ω, F, P ) p F t -, p F t -, (1) p 1, λ >, λ p P ( sup s [,t] () p > 1, E[ sup X s p ] s [,t] X s λ) E[ X t p ]. ( p p 1 ) p E[ X t p ]., X p, t [, T ] E[ X t p ] < 1.5. ( ) (Ω, F, P ) d w = {w t } t [,T ], (Ω, F, P ) d j = 1,,, d, (1) w j (ω) =, ω Ω () = t < t 1 < < t n n N, {w j t i w j t i 1 } 1 i n,, t i t i 1 (3) {w j } 1 j d, 1 (Ω, F, P, {F t }), j = 1,, d w j F t -, s t, w j t w j s F s, d F t -..1.1. (Doob-Mayer ) X F t -, A a >, {X σ a } σ A lim sup E[ X σ a : X σ a > N] = N σ A, F t - M = {M t } A = F t - A = {A t }, X t = M t + A t (t ) M T := {M = {M t } t [,T ] M F t, M = a.s.} 5

, Jensen M = {Mt } t [,T ], Doob- Meyer (.1),, F t - A = {A t } t [,T ], M A F t - A t M t, A = M M, M, N M T, M, N t := 1 ( M + N t M t N t ), t [, T ], M, N a.s., MN M, N. M, N M N.. (Hilbert ) M, N M T P (M t = N t = 1 t [, T ]) = 1, (M T, E[, T ]) 1 n L = {f = {f t } t [,T ] f t = F i 1 (ti 1,t i ](t), i=1 = t < < t n = T, F i : F ti 1 } f L F t -,.1. n f L, f t = F i 1 (ti 1,t i ](t) ( = t < < t n = T ), I(f) = {I t (f)} t [,T ] i=1 I t (f) = f u dw u := i=1, f w n F i (w ti t w ti 1 t), t [, T ].1. f, g L, I(f) I(f), I(g) t = f u g u du, t [, T ].. ( ) t [, T ], (s, ω) ([, t] Ω, B([, t]) F t ) f s (ω) (R, B(R)), f F t - B(A) A 6

, T L,T := {f = {f t } t [,T ] F t, E[ ft dt] < },f L,T.1. ( ) f L,T, lim f (n) f T = {f (n) } n N L n, f T := E[ T f t dt] f L,T.1, lim n f (n) f T = {f (n) } n N L. f (n), I(f (n) ),.1, I(f (n) ) I(f (m) ), ( 1.1), E[ sup I t (f (n) ) I t (f (m) ) ] 4E[ I T (f (n) ) I T (f (m) ) ] t [,T ] = 4 f (n) f (m) T, {I(f (n) )} n N M T,. M T (a.s. ),, I = {I t (f)} t [,T ], I t (f) = f sdw s, f L,T w, f L,T L.. f, g L,T, I(f) I(f), I(g) t =.3. ( ) f, g L,T, 3 f u g u du, t [, T ] I t (af + bg) = ai t (f) + bi t (g), t L loc := {f = {f t } t [,T ] F t,, f L loc,t, f udu < a.s. t [, T ]} τ n := inf{t f udu n} T 7

inf ϕ T = T f (n) t := f t 1 t τn, m n, τ m τ n, τ n T a.s.(n ), f udw u ( f u dw u )1 t τn := ( f u (n) dw u )1 t τn, n N, f (n) L,T,, m n ( f (n) u dw u )1 t τm = ( f u (m) dw u )1 t τm, w (Ω, F, P ) d F t -, f L loc,t w i (i = 1,, d) I i (f) = f tdwt i.4. d, f, g L,T, I i (f), I j (g) t =, t [, T ], (i j),., E[I i t(f)i j t (g)] = δ ij E[ f u g u du], t [, T ],.3. ( ) M d M, M d F t -, M i, M j t = δ ijt 1 i, j d, t. X X t = X + M t + A t, (, X R, M t := ϕ u dw u, A t := ψ u du, ϕ L loc,t, ψ L loc 1,T ).4. ( ) f C 1, ([, T ] R; R), X., {f(t, X t )} t [,T ], f(t, X t ) = f(, X ) + + f x (u, X u )ϕ u dw u f u (u, X u ) + f x (u, X u )ψ u + 1 f xx(u, X u )ϕ udu 8

.3 σ : [, T ] R Ω R b : [, T ] R Ω R, B([, T ]) B(R) F t -, σ(, x, ), b(, x, ) F t -, x R, dx t (ω) = σ(t, X t (ω), ω)dw t (ω) + b(t, X t (ω), ω)dt, X = x X ω Ω.3. ( ) X = {X t } t [,T ] 1. X, X F F t. {σ(t, X t )} t [,T ] L loc,t, {b(t, X t )} t [,T ] L loc 3. X t = X + σ(s, X s )dw s + 1,T. b(s, X s )ds a.s. t [, T ].5. ( )., T >, 1. σ(t, x) σ(t, y) + b(t, x) b(t, y) L T x y ( t [, T ], x, y R). E[ T ( σ(t, ) + b(t, ) )dt] <. L T >, ( X, X, P (X t = X t, t ) = 1.).1. ( ) a t L loc,t, b t L loc 1,T, dx t = X t (a t dw t + b t dt), X > (1),.5, X t = X e a udw u + (b u 1/a u )du () () (1),, 9

X, Y (1), X (),, d(y t /X t ) = Y t ( dy t dx t dx t dy t X t Y t X t X t Y t, X t = Y t (a.s. ) + dx t dx t Xt ) =, σ, b ω Ω, X Z := {E( θdw) t } t [,T ] (, θ L loc,t ), 1,, E(X) t := e X t 1/ X t, (Ω, F T ) P Q θ Q θ : F T A Q θ (A) := E[E(, θdw) T 1 A ] [, 1] (3).6. ( ) Z, (3) Q θ,, w θ t := w t (Ω, F T, Q θ ) F t - θ u du, t [, T ] 1

3 w (Ω, F, P ), F t = σ(w u ; u [, t]) N,F = F T, N := {A Ω B F B A, P (B) = } S = {S t } t [,T ], S B = {B t } t [,T ] 3.1. ( ) P (Ω, F T ), S,,, Q 3.. ( ) S = {S t } t [,T ], S Q Strict, Strict, S, 3.1 ( ) S, σ L loc,t, µ Lloc 1,T. ds t = S t (σ t dw t + µ t dt), S > ( S t = S exp σ u dw u +, µ u 1 ) σ udu [1] P (σ t t [, T ] ) = 1. [] λ t := σt 1 µ t, λ = {λ t } t [,T ] 11

3.1. ( ) Q, Q(A) := E[E( λdw) T 1 A ], A F T,, w Q t := w t + λ u du, t [, T ] Q (Q- ). [3] Q 3.1 T F, F F T F L 1 (Ω, F T, P ), t x F t : = E Q [F F t ], K t C t (K), P t (K), C t (K) := E Q [(S T K) + F t ] P t (K) := E Q [(K S T ) + F t ] 3.. ( ) S Q, C t (K) P t (K) = S t K 1

., (x K) + (K x) + = (x K), C t (K) P t (K) = E Q [(S T K) + F t ] E Q [(K S T ) + F t ] = E Q [S T F t ] K S Q, E Q [S T F t ] = S t 3.1.1, σ : [, T ] R,, t [, T ], S, ds t = S t σ t dw Q t, S > (4), (4) ( S t = S exp σ u dwu Q 1 ) σudu, F, f : R ++ R, F = f(s T ) S,, V (t, x) := E Q [f(s (t,x) T t )] V (t, S t ) = E Q [F F t ] (5) 13

, S (t,x) t, x, ( u S u (t,x) = x exp σ t+s dws Q 1 u ) σ t+sds, u [, T t], x F t = V (t, S t ) (6), Q, u σ t+sdw Q s,, δ u := u σ t+sds, E Q [S T F t ] = E Q [S (t,s t) T t ( ] δ = x exp T t ( δ ) = x exp T t = S t ) E Q [e T t σ t+s dw Q s ] x=st R 1 πδt t, S Q, (5), (6) ( ) 1 exp δt (y δt t) dy exp t ( δ ) T t x=s t (7) x F t = V (t, S t ) = E Q [f(s (t,x) T t )] x=s t ( ( δ = f x exp T t R )) + y 1 πδt t ( ) y exp δt dy x=st t,, [ ] C t (K) : = V (t, S t ) = = = = x R R ( x exp ( ) ) + y δ T t 1 K ( ( x exp δ T t z δ T t c(t,x) d(t,x) ) K ( ( x exp δ T t z δ T t 1 π exp ( u ) du K = S t Φ( d(t, S t )) KΦ( c(t, S t )). 14 πδ T t ) + 1 π exp ) ) 1 K exp π c(t,x) ( ) y exp δt dy x=st t ( z 1 π exp ( z ) dz x=st ) dz x=st ( z ) dz x=st

[ ] P t (K) = R ( ( )) + ( ) K x exp y δ T t 1 y exp πδt δt dy x=st t t = KΦ(c(t, S t )) S t Φ(d(t, S t ))., c(t, x) := 1 ( log K δ T t x + 1 ) δ T t, d(t, x) := c(t, x) δ T t, C t (K) P t (K) = S t Φ( d(t, S t )) KΦ( c(t, S t )) KΦ(c(t, S t )) + S t Φ(d(t, S t )) = S t (Φ( d(t, S t )) + Φ(d(t, S t ))) K(Φ( c(t, S t )) + Φ(c(t, S t ))) = S t K, x R ξ L loc,t, W (x.ξ) W (x,ξ) t := x + ξ u ds u, t [, T ] ( dw t = ξ t ds t + (W t ξ t S t )dt, W = x, Self-finanncing ) A T := {ξ : [, T ] Ω R F t, σξ L loc,t, lim kq( inf W (x,ξ) t < k) =, x R} k t [,T ] 3.1. x R ξ A T W (x,ξ) Q. x = W = W (,ξ),q, τ n := inf{t > W t > n} T ρ k := inf{t > W t < k} T 15

, τ n T a.s., Fatou {W t τn } t [,T ], s t,, E[W t ρk F s ] lim inf E[W t τ n ρ n k F s ] = lim inf W s τ n n ρ k = W s ρk E[W t ρk : A] E[W s ρk : A], A F s (8), E[W t ρk : A] = E[W t ρk : A {ρ k t}] + E[W t ρk : A {ρ k < t}], k,,,, (8),, = E[W t : A {ρ k t}] kq(a {ρ k < t}) E[W t : A {ρ k t}] E[W t : A], kq(a {ρ k < t}) kq({ρ k < t}) kq( inf u T W u < k) E[W t ρk : A] E[W t : A], (k ) E[W s ρk : A] E[W s : A], (k ) E[W t : A] E[W s : A], A F s, x F : = sup{x R ξ A T F + W x,ξ T } x F : = inf{x R ξ A T F + W x,ξ T } 3.3. ( ) 16

1 x F = x F = x F F, F = W xf,ξf T ξ F A T, (x F, ξ F ) 3 x F. [3] F t - R = {R t } t [,T ] W xf,ξf t = E Q [F F t ] E Q [ sup R t ] < t [,T ] T 3.4. ( ) U := sup E Q [R t ] t [,T ],, U o, U. [3] 17

3.,, (Ω, F, Q), w, E Q [ ] E[ ] 3..1 S = s >. ds t = S t T t dw t, t [, T ), S T = 1 t X t : = (T s) ds + 1 dw s, t [, T ), X = T s f(x) : = se x f(x u ) = s +, = s + f(x s )dx s + 1 f(x s ) T s dw s. f(x s )dx s dx s S t = f(x t ) (9), A t := log(1 t T ) = 1 T sds [4] c(t, ω) >, β t : = c (s, ω)ds, Y t :=, w c(s, ω)dw s Y t = w βt, t [, T ] 18

, c(t) = 1 T s ds,, (9) 1 T s dw s = w At S t = se w A t 1 A t, S t (t T ), [, T ] S S, S = E[S T ] = S = s >, S Strict 3.. CEV (Constant Elasticity of Variance ) S = s > ds t = St dw t, t [, T ].., X t := 1 S t, dx t = 1 St ds t + 1 S 3 t = dw t + 1 X t dt ds t ds t = d w t + 1 X t dt. (, w := w ), w w = x R 3, x = 1 s 3,, d( w t ) = = 3 i=1 3 i=1,.3 dŵ t :=, w i t w t d wi t + 1 3 i=1 w i t w t d wi t + 1 w t dt 3 i=1 (δ ij 1 w t wi t w j t w t )δ ijdt w t i w t dwi t, ŵ d( w t ) = dŵ t + 1 w t dt. 19

, X t = w t, S t = 1 w t,, E[St 1 ] = E[ w t ] = = 1 z (πt) 1 1 tu + x (π) R 3 1 R E[S t ]. (t ) z x e t 3/ dz u e 3/ du, E[S t ]. (t ), E[S T ] < s T >, S, E[S t ] = S = s,, S Strict, 3..3 SV (Stochastics Volatility ) ds t = S t V t dw S t, dv t = ηv t dw V t + µv t dt, S = V = s >. (, η, µ >, w S Q, w Q w S, ρ ( 1, 1), w V := ρw S + 1 ρ w ) SV(Stochastics Volatility) S, dp = S T S dq P, P Q, -, S T /S = exp{ V ssws S 1/ V s ds}, w S,P t := w S t P 3.. V s ds, t [, T ] w P t := ρw S,P t + 1 ρ w t, t [, T ] w P = {w P t } t [,T ] P

. Q, w S w, d(w t S t ) = S t dw t + w t ds t + dw t ds t = S t dw t + w t ds t (7) d(((w t ) t)s t ) = S t d((w t ) t) + ((w t ) t)ds t + d((w t ) t) ds t = S t w t dw t + ((w t ) t)ds t (8) d(wt (w S,P t S t )) = w S,P t S t dwt + wt d(w S,P t S t ) + dwt d(w S,P t S t ) = w S,P t S t dwt + S t (1 + wt S.P V t )dwt S (9), {wt S t }, {((wt ) t)s t }, {wt (w S,P t S t )} Q,, A F s, E P [wt : A] = 1 E Q [wt S T : A] S = 1 E Q [wt S t : A] S = 1 E Q [ws S s : A] ( (7) ) S = E P [ws : A], w P, E P [(w t ) t : A] = 1 S E Q [{(w t ) t}s t : A] = 1 S E Q [{(w s ) s}s s : A] ( (8) ) = E P [(w s ) s : A], w P, E P [w S,P t w t : A] = 1 S E Q [w S,P t w t S t : A] = 1 E Q [ws S,P ws S s : A] ( (9) ) S = E P [w S,P s w s : A], P, w, w S,P t =, w P w S,P, dwt P dwt P = (ρdw S,P t = dt + 1 ρ dw t ) (ρdw S,P t + 1 ρ dw t ), w P P 1

, V, dv t = ηv t dw P t + (µv t + ηρv t )dt. (1) 1 ([5]) 3.5. I := (l, r), l r, σ, b C 1, dx t = σ(x t )dw t + b(x t )dt, X = x X, P x e e(x) := inf{t > X t = l or X t = r} e X c (l, r),, κ(x) := x c e y c b(z) y σ dz (z) c 1 σ (z) e z b(w) c σ dw (w) dzdy s(x) := x c e y c b(z) σ (z) dz dy P x (e < ) = 1, x I κ s (1)κ(r ) <, κ(l+) < ()κ(r ) <, s(l+) = (3)s(r ) =, κ(l+) < κ(x) = x c y s 1 (y) c σ (z) s (z) 1 dzdy, l =, r =, σ(x) = ηx, b(x) = µx + ηρx c (, ), α := η µ > 1, D := e η µ log c+ η ρc >, s (x) = Dx α e βx κ(x) = 1 x y η y α e βy 1 z zα e βz dzdy c c

3.3. 3.5 (). s(+) < x < c, s(x) = c x Dy α e βy dy,, c c y α e βy dy > e βc y α dy x x s(x) = D c c y α e βy dy < De βc y α dy (x +) x x, s(+) =, (), < c < x, κ( ) α y 1 z zα e βz dz = [ 1 β zα e βz ] y c 1 β y c (α )z α 3 e βz dzdy = 1 β yα e βy 1 β cα e βc α β y c e βz z α 3 dz y 1 z zα e βz dz = 1 β yα e βy 1 β cα e βc α y e βz z α 3 dz β c 1 β yα e βy, κ(x) x η y α e βy 1 β eβy y α dy = η 1 β, κ( ) < x c y dy η 1 β c 1 dy <. y 1 < α < y 1 z zα e βz dz = 1 β yα e βy 1 β cα e βc + α β 1 β yα e βy + α y β eβy z α 3 dz = 1 β yα e βy + 1 β eβy (c α y α ) = 1 β eβy c α c y c e βz z α 3 dz 3

, κ(x) x η y α e βy 1 β eβy c α dy = η 1 β cα, κ( ) < x, () c y α dy η 1 β c 1 dy <. yα 3.3, (1) V P Q, V dv t = ηv t dwt V + µv t dt.,, P Q, S S Strict, 3..4 SV () X = {X} t [,T ], f t (X) :=, λ >, η, λe λr log(x t /X t r )dr, (X t r := X t r ) V t (X) := η(f t (X)), ds t = S t V t (S)dw t, S = s >. f t (S), f t (S) = log S t λe λr log S r dr, f t (S), ( ) 3.4. ([6])Lemmma3.1 Z t := log S t, (Z t =, t < ) Y t := f t (S) = λe λr (Z t Z t r )dr 4

, dy t = dz t λy t dt. t [, T ] f(t, x) := e λt x,, e λt Y t = d(e λt Y t ) = e λt dy t + λe λt Y t dt + 1 dy t dy t = e λt dy t + λe λt Y t dt (11) λe λ(t r) dr = Z t λe λl dl λe λl Z l dl, H(t) := λeλl dl, F (t) := λeλl Z l d, g(t, x) := xh(t) F (t),, (11), (1),, 3.4 d(e λt Y t ) = H(t)dZ t + (Z t H (t) F (t))dt = H(t)dZ t (1) H(t)dZ t = e λt dy t + λe λt Y t dt dy t = e λt H(t)dZ t λy t dt = dz t λy t dt dy t = d(log S t ) λy t dt = 1 ds t 1 S t St ds t ds t λy t dt = V t dw t 1 V t dt λy t dt ( = η(y t )dw t λy t + 1 ) η (Y t ) dt (13) 5

, S, dp = S T S dq, Q P -, dw P := dw t V t dt, t [, T ] w P = {w P t } t [,T ] P, dy t = η(y t )dw P t, η (x) := 1 + x 3.5. 3.5 ( λy t 1 ) η (Y t ) dt (14) a, b I, a < x < b, τ := inf{t > X t [a, b]},., P x (X τ = a) = τ s(x t τ ) = s(x) + = s(x) + s(b) s(x) s(b) s(a), P s(x) s(a) x(x τ = b) = s(b) s(a) τ s (X s )dx s + 1 s (X s )σ(x s )dw t τ {s(x t τ )} t, s(x) = E[s(X t τ )] s(x) = E[ lim t s(x t τ )] = E[s(X τ )] = P x (X τ = a)s(a) + P x (X τ = b)s(b), P x (X τ = a) + P x (X τ = b) = 1 s (X s )dx s dx s 3.5 (13) σ (x) = 1 + x, b(x) = {λx + 1 (1 + x )}, D := e λ log(1+c ) c,, c < x, s(x) = x c D(1 + y ) λ e y dy s(x) D x c e y dy = D(e x e c ).(x ) 6

, x < c,, s(x) = c x D(1 + y ) λ e y dy s(x) D c x e y dy = D(e c e x ) De c <.(x ), τ := inf{t > Y t [, ]}, 3.5, Q(Y τ = ) = 1 (15), 3.5 (14) b(x) = {λx 1 (1 + x )}, P (Y τ = ) = 1 (16), (15) (16), P Q S Strict, 3..5., v(x, r) := n ) 3.6. ds t = S t dw t, S = s > x n (n)! h(n) (r), h(r) := e 1/r (, h (n) h n ( ) v xx = v r, (x, t) : = x 1/ e (T t)/8 v(log x, (T t)/) Y t : = (S t, t) (17) 7

,, dy t = x (S t, t)ds t, Y, Y Q Y T =, Y >, Y, 3.6, v xx = n 1 x n (n )! h(n) (r), v r = n, x n (n)! h(n+1) (r) = n 1 x n (n )! h(n) (r) h n, P n (y),, P n, a n k P n y n k, h (n) (r) = P n (1/r)e 1/r h (n+1) (r) = P n(1/r) 1 r e 1/r + P n (1/r) 1 r e 1/r P n+1 (y) = P n(y)( y ) + P n (y)y (18) n 1 P n (y) = y n + a n ky n k (19) k=1 8

, (18), P n+1 (y) = y (P n (y) P n(y)), n 1 n 1 = y (n+1) + a n ky (n+1) k ny n+1 a n l (n l)y n l+1 k=1 n 1 n 1 = y (n+1) + a n ky (n+1) k ny (n+1) 1 a n l (n l)y (n+1) (l+1) k=1 = y (n+1) + {a n 1 y (n+1) 1 ny (n+1) 1 } n 1 +{ k= n 1 a n ky (n+1) k k= = y (n+1) + (a n 1 n)y (n+1) 1 n 1 + l=1 l=1 a n k 1(n k + 1)y (n+1) k } a n n 1(n + 1)y (n+1)+1 {a n k a n k 1(n k + 1)}y (n+1) k a n n 1(n + 1)y (n+1)+1 k=, (), (), (1), (3) a n+1 1 = a n 1 n () a n+1 k = a n k (n k + 1)a n k 1, k n 1 (1) a n+1 n = (n + 1)a n n 1 () a n 1 = n(1 n) (3) a n n 1 = ( 1) n n(n 1) 3a 1 = ( 1) n 1 n! (4) a n a n 1 = (n 3)a n 1 1 = (n 3)(n 1)( n) n a n = a 3 + (k 3)(k 1)( k) = O(n 4 ) l=3, k n a n k = a k+1 k + = O(n k ) n (n 1 k)a l k 1 l=1 9

, (1), (4),, a n n + (n + 1)a n 1 n 3 = an 1 n = ( 1)n (n 1)! a n 1 n 3 + nan n 4 = an n 3 = ( 1)n 3 (n )! : a 4 + 5a 3 1 = a 3 = ( 1) 3! n 3 a n n = ( 1) n (n + 1)! (n + l)! (n l)! + ( 1)n 3 (n + 1)n 5a 3 1 l=1,, k n,,,, a n n = O((n + 1)!) a n k = O((n k 1)!) a n k O(n n ), ( k n/) a n k O((3n/)!), (n/ 1 k n ) P n = O((3n/)!) n! ( n ) n πn e C n = hn (r) (n)! ( ( 3πn 3n e O 4πn( n = O ( ( e e 3/,, 3 ) 3n/ e )n ) ) n ( ) (3n) 3/ n ) 4n

( ) (3n) 3/ Cn O n 4n ( ) 1 = O n, (n ), R = lim n =, 1 n Cn 31

4. 4.1 F = F T, (Ω, F, Q, {F t }) Q- S H = H(S T ) t V t (H) V t (H) = E t [H(S T )] := E[H(S T ) F t ], K, C t (K) := V t ((S T K) + ) P t (K) := V t ((S T K) ) 4.1. S, (5) E t [S T ] < S t (5) C t (K) P t (K) < S t K, (6) lim (P t(k) K + S t ) >. (7) K. S, S E t [S T ] S t (8) S, S Strict, (8) E t [S T ] < S t,(5) (6),(7) C t (K) P t (K) = E t [S T ] K (9) 3

, (5) (6), (9),,(5) (7), P t (K) K + S t = C t (K) E t [S T ] + S t lim C t(k) = K, 3 4.1.1, S, S T =, C t (K) =, P t (K) = K, C t (K) P t (K) < S t K, 4.1. CEV CEV, S, w = x R 3, x = 1 s 3 w S t = 1 w t 4.1. S,. Q(S T dz) = s z 3 1 πt (e 1 T (1/z 1/s) e 1 T (1/z+1/s) ) 33

[3] dr t = dw t + δ 1 R t dt, R = s R = {R t } δ, R t p ν BES (t, s, y),, Γ, p ν BES(t, s, y) = 1 t y ν+1 s ν e 1 t (s +y ) ( sy I ν t ( z ) ν (z/) n I ν (z) : = n!γ(ν + n + 1) ν : = δ/ 1 n ) 1 y δ = 3,, Γ(ν + n + 1) = Γ(n + 3/) ( ) n+1 1 = (n + 1)(n 1) 5 3 π ( z 1/ (z/) I 1/ (z) = ) n n!γ(n + 3/) n = 1 ( z 1/ z 1 π ) z n+1 (n + 1)! n = 1 π z 1/ (e z e z ), p 1/ 1 y BES (t, s, y) = (e 1 t (s y) e 1 (s+y)) t πt s, S T = 1 w T p CEV, w 3, ( p CEV T, 1 ) ( s, z = p 1/ BES T, 1 s, 1 ) 1 z z = s 1 ( z 3 e 1 T (1/z 1/s) e 1 (1/z+1/s)) T πt 34

4.1, C t (K) P t (K) = E t [S T K] = z s 1 ( z 3 e π(t t) = (Φ(θ(S t )) 1) S t K < S t K 1 (T t) (1/z 1/s) e 1, (, Φ( ), θ(s) := 1 s T t ) (T t) (1/z+1/s)) dz s=st K 35

5 T H = H(S T ), T, ; W (x,ξ) T H(S T ) (x, ξ) R A T,,,, t G(S t ), t G(S t ) G S, G, H G S, H(S T ) (x, ξ) R A T, W (x,ξ) t E Q t [W (x,ξ) T ] ( 3.1 W ) E Q t [H(S T )] ( W (x,ξ) T H(S T )) E Q t [G(S T )] G(E Q t [S T ]) ( G Jensen ) = G(S t ) ( S ) (3), S, t [, T ] G(S t ), H G(S t ), S, (3), 5.1. H G t V t (H, G), V t (H, G) : = min{x R ξ A T, W (x,ξ) T H(S T ), W (x,ξ) u G(S u ) u [t, T )}, R = {R t } t [,T ] R t = J(S t, t) := H(S t )1 t=t + G(S t )1 t<t 36

, V t (H, G) R = {R t } t [,T ] t, G(x) 5.1. G β := lim sup x x, H G, V t (H, G) = sup E t [J(S u, u)] u [t,t ] > V t (H, G) = E t [H(S T )] + β(s t E t [S T ]) 5.1, Q S, S, S, V t (H, G) = E t [H(S T )], 5.1 T = 1 (T ) x >, t (, 1),, H x : = inf{u > S u x} 1 H x θ t : = inf{u > t S u x} 1 S(H x θ t ) = {S u Hx θ t } u [,1],, t < 1,,, S t = E t [S Hx θ t ] = E t [S t 1 Hx θ t =t] + E t [x1 t<hx θ t <1] + E t [S 1 1 Hx θ t =1] = S t E t [1 St >x] + xq t ({t < H x θ t < 1}) + E t [S 1 1 Hx θ t =1] S t E t [S 1 1 Hx θ t =1] = S t E t [1 St >x] + xq t ({t < H x θ t < 1}) γ t : = S t E t [S 1 ] = lim (S t E t [S 1 1 Hx θ t =1]) x = lim (S te t [1 St >x] + xq t ({t < H x θ t < 1})) x = lim xq t({t < H x θ t < 1}) x = lim xq t({ sup S u x}) (31) x u [t,1) 37

, G, ϵ >, G(x n ) x n > β ϵ, n N x n.(n ) {x n } n N, H n := H xn θ t, S t < x n n, sup E t [J(S u, u)] E t [J(S Hn, H n )] u [t,1], (31), = E t [H(S 1 )1 Hn =1] + E t [G(S Hn )1 Hn <1] = E t [H(S 1 )1 Hn =1] + G(x n) x n Q t ({H n < 1}) x n E t [H(S 1 )1 Hn =1] + (β ϵ)x n Q t ({ sup S u x n }) u [t,1] E t [H(S 1 )1 Hn =1] + βx n Q t ({ sup S u x n }) u [t,1] sup E t [J(S u, u)] lim E t[h(s 1 )1 Hn =1] + β lim x nq t ({ sup S u x n }) u [t,1] n n u [t,1], t u < 1,, G β = E t [H(S 1 )] + βγ t (3) Y H u := E u [H(S 1 )], Y G u := E u [G(S 1 )], u [t, 1] G β (x) := βx G(x), lim inf x G β (x) x =. E u [βs 1 G(S 1 )] = E u [G β (S 1 )] G(E u [S 1 ]) ( Jensen ) G β (S u ) ( S ) = βs u G(S u ) 38

, βe t [S 1 ] Y G u βs u G(S u ), G(S u ) βγ u + Y G u, u [t, 1), t u < 1, J(S u, u) = G(S u )1 u<1 + H(S u )1 u=1 = G(S u ) ( t u < 1) βγ u + Y G u βγ u + Y H u ( H G) J(S u, u) βγ u + Y H u, u [t, 1) (33) (33), u = 1, (33) u [t, 1], E t [J(S u, u)] E t [βγ u + Y H u ], (3) (34) 5.1. K, α (, 1], = β(e t [S u ] E t [S 1 ]) + E t [E u [H(S 1 )]] βγ t + E t [H(S 1 )] ( S ) sup E t [J(S u, u)] βγ t + E t [H(S 1 )] (34) u [t,1] G α (x) : = α(x K) + H(x) : = (x K) +, H G α t V t (H, G α ), V t (H, G α ) = (1 α)c E t (K) + αc A t (K) (, C E, C A.) 39

., 5.1, H(x) = G(x) = (x k) +, Ct A (K) = Ct E (K) + (S t E t [S T ]), Ct E (K) + α(s t E t [S T ]) = (1 α)ct E (K) + αct A (K) 5.1, H, G α, V t (H, G α ) = E t [(S T K) + ] + α(s t E t [S T ]) = Ct E (K) + α(s t E t [S T ]) = (1 α)ct E (K) + αct A (K) 4

[1] A.M.G.Cox, D.G.Hobson. : Local martingales bubbles and option prices. Finance Stochast 9, 477 49(5) [] : (6) [3] : (7) [4] B. : ( ). (1) [5] S.Watanabe, N.Ikeda : Stochastics Differential Equations and Diffusion Processes. North Holland (1989) [6] D.G.Hobson, L.C.G.Rogers : Complete models with stochastics volatility. Math. Finance 8, 7 48(1998) [7] Heston.S, Loewenstein.M, Willard.G.A. : Options and Bubbles. The review of financial studies, 359 39(7) 41

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