Black-Scholes 1 ( )

Transcription

1 Black-Scholes

2 Takaoka[T] Black- Scholes B S α B t = e rt S α t = S e αrt exp σw t + Ct } 2 σ2 t λdσ λ Λ 2 :=,, B, σ 2 λdσ < } W t } Ω, F, P; F t T > t [, T ], α R, r C, S F - S α dst α = αr+cφtst α dt+φtst α σ expσw t + Ct dw t φt 2 := σ2 t}λdσ expσw t + Ct 2 σ2 t}λdσ φtst α dw t α = Takaoka[T] λ Black-Scholes Takaoka[T] λ λ m m = λ M M = Black-Scholes Black-Scholes Takaoka[T] θ α t := C r α S α ds α t = rst α dt + φtst α d W t + φt θ α sds Wt α := W t + θ αsds P α P α

3 P α θ α t α = λ θ α t t α θ α t t λ θ α t Girsanov Girsanov Z α t := exp θ αsdw s t θ 2 αs 2 ds} F t }- P α A := E[ A Z α T ], A F T Wt α := W t + θ αsds Ω, F T, P α X t } F t }- X t F t - E[X t F s ] = X s s t Girsanov Z α t} Fatou Z α t} E[Z α t F s ] Z α s s t Z α t} E[Z α t] α = λ α E[Z α t] Novikov Karatzas and Shreve [KS] Z α t}.. λ Λ 2 m im > Z α t} iim = y = y λ, D = DT y > y σ expσ y + Ct 2 σ2 t}λdσ expσ y + Ct 2 σ2 t}λdσ > D y 2 Z α t} m = λ Λ 2 2 λ ρ ρ 2 2

4 i ii.2. λ Λ 2 m m > t >, µ R, α R lim x x P log Sα t S 2πη exp < x y µ2 2η } dy = η > m 2 t < η < m 2 t m = 2 t >, η >, µ R, α R 3 m = 2 log Sα t S Black-Scholes α Karatzas and Shreve [KS2] X x,π t = xb t + B t πuφudwu α B u A x = π;, X x,π t, t [, T ] a.s} U : [, [, ; V x = sup E[UX x,π T ] 4 π A x 4 π = π π S α X x,π x, π.3. α = λ Λ :=,, B, σλdσ < } π > α < π < λ Λ α < Girsanov Takaoka[T] 3 3

5 4

6 Ω, F, P; F t F t } F t+ := s>t F s = F t F F E P... X = X t } t Ω, F, P; F t martingale i ii X F t }- 5.5 t E[ X t ] < iii E[X t F s ] = X s t s X i, ii iii E[X t F s ] X s t s submartingale iii E[X t F s ] X s t s supermartingale.2. [KS].3.5 Theorem X = X t } t Ω, F, P; F t C := sup t E[X t + ] < X := lim t X t E[ X ] <.2 5

7 .3. [KS].3.6 Problem X = X t } t Ω, F, P; F t X := lim t X t t =.4. Optional Sampling Theorem[KS].3.22 Theorem X = X t } t Ω, F, P; F t X := lim t X t F t } 5.6S T EX T F S X S F S [KS].3.29 Problem X = X t } t Ω, F, P; F t T := inft ; X t = } T < } X T +t =, t <.5.6. X = X t } t Ω, F, P; F t local martingale T n } n= P[lim n T n = ] = n X n t } t := X t Tn } t X = X t } t X. X T n } n= P[lim n T n = ] = n X n t } t := X t Tn } t t s Fatou [ ] E[X t F s ] = E lim inf X t T n n F s lim inf E[X t T n F s ] = lim inf X s T n = X s n n 6

8 Ω, F, P; F t M c 2 := X = X t } t ; X = a.s} X M c 2 Jensen 5.8 X2 - Doob-Meyer 5.6 Xt 2 = M t + A t, t < M = M t } t A = A t } t 5.3, 5.4 X X X M c 2 X quadratic variation X X = X 2 X X, Y M c 2 X + Y 2 X + Y, X Y 2 X Y 4XY [ X + Y X Y ] X, Y.. X, Y M c 2 X, Y X, Y X, Y t := 4 [ X + Y t X Y t ], t < X, Y M c,loc := X = X} t ; X = a.s} X, X, Y.. [KS].5.7 Problem X, Y M c,loc X, Y X Y i X, Y = ii X, Y iii XY X, Y M c,loc.2. [KS].5.7 Problem,.5.2 Exercise X, Y, Z M c,loc α, β R X, X = X, X, Y = Y, X, αx + βy, Z = α X, Z + β Y, Z X, Y X, Y = 7

9 .3. X = X t } t Ω, F, P; F t semimartingale X F t }- X t = X + M t + A t, t <. M = M t } t M M c,loc A = A t } t A = t [, t] X X X t := M t.4. X.2.5. W = W t} t Ω, F, P Brownian motion i ii t < t < < t n W t W t,, W t n W t n s, t, A BR PW s + t W s A = A 2πt exp x2 2t ds iii W = W t 2 W t = t 8

10 .6. [KS] Theorem X = X t } t Ω, F, P; F t F t }- M t := X t X M t = t X. F t } W F W t := σw s; s t} W t F W t- F W t} t [KS] 2.7. Problem augmentation.7. [KS] 2.7 Section W Ω, F, P T > F W t := σw s; s t} N F W T F t := σf W t N }, t [, T ] F t } t T M M c,loc X P M := X = X t } t ; T Xt 2 d M t <, t < } X sdm s Ω, F, P; F t.8. X X t = ξ } t + ξ i ti,t i+ ]t, t < i= X simple t n } n= t = t n ξ n } n= C > sup n ξ n C ξ n F tn - L X X M t := X sdm s X s dm s = ξ i M t ti+ M t ti i= 9

11 .9. M M c 2 L M := X = X} t ; T > [X] T := E T X 2 t d M t 2 < } T > [X Y ] T = X Y L M [ ] [X Y ] := 2 n [X Y ] n n= M M c 2 X L M X sdm s.2. [KS].5.23 Proposition M M c 2 t M t := EX 2 t M := n= 2 n M n M c 2.2. [KS] 3.2 Section B X, Y L, M M c 2 E[X M t F s ] = X M s.2 X M = [X].3 t αx u + βy u dm u = α X u dαm u + βn u = α X M, Y N t = X u dm u + β X u dm u + β Y u dm u.4 X u dn u.5 X u Y u d M, N u [KS] Proposition M M c 2 L L M

12 M M c 2 X L M X sdm s.22 X L M X n } n L.3,.4 [X n X] as n X n M X m M = X n X m M = [X n X m ] as n, m.2 X n M} n= M c 2.2 M c 2 I Mc 2 I X n M as n I M M c 2 X L M X M.23. [KS] 3.2. Proposition M, N M c 2, X, Y L M L N.2,.3,.4,.5, [KS] Proposition M, N M c 2, X L M, Y L N X s Y s dˇξ s X 2 s d M s 2 Ys 2 2 d N s, t < ˇξ s ξ = M, N [, s] M M c,loc X P M M t S n } n= P[lim n S n = ] = n M t Sn } t M c 2 X P M R n } n= } R n = n inf t ; Xs 2 d M s n R n } n= P[lim n R n = ] = T n := R n S n, M n t := M t Tn, X n t := X t Tn >t} M n M c 2, X n L M n X n M n X M X u dm u = X u n dm u n ; t T n

13 . X Y P M Y s dx s := Y s dm s + Y s da s ω Ω.25. [KS] 3.2 Section D M, N M c,loc, X, Y P M P N X M, Y N.4,.5, [KS] Theorem f : R R f C 2 X. fx t = fx + d dx fx sdx s + 2 d 2 dx 2 fx sd X s, t < X f fx.27. [KS] Theorem X t } t Ω, F, P; F t d. ft, X t = f, X d i= d j= t fs, X sds + d i= x i fs, X s dx s 2 x i x j fs, X s d X i, X j s, t <.28. [KS] Problem X, Y. X s dy s = X t Y t X Y Y s dx s X, Y t, t < 2

14 4.29. [KS] Theorem, Problem W Ω, F, P F t } M M = a.s M M c,loc M Y P M M t = Y s dw s.7 Ỹ P M.7 Ỹs Y s ds =.3 W Ω, F, P F t } W T P[ T X2 t dt < ] = X = X t } t [ Z t X := exp X s dw s ] Xs 2 ds 2 Z t X = + Z s XX s dw s.8 ZX = Z t X} t Z X = ZX F T T < P T P T A := E[ A Z T X], A F T,ZX P T ; T < } P T A = P t A, A F t, t T.9 3

15 Ω, F T, P T.3. Girsanov [KS] 3.5. Theorem ZX W = W t } t W t := W t X s ds, t T W Ω, F T, P T ; F } t T.3. Bays [KS] Lemma T < ZX s t T Y F t - Ẽ T Y < Ẽ T [Y F s ] = Z s X E[Y Z tx F s ] P P T Ẽ T P T..9 A F s - A F s [ ] Ẽ T A Z s X E[Y Z tx F s ] = E[ A E[Y Z t X F s ]] = E[ A Y Z t X] A F s - = ẼT [ A Y ].32. [KS] Proposition T ZX M M c,loc M t := M t T X s d M, W s, t T M c,loc M T N M c,loc T Ñ t := N t X s d N, W s, t T M, Ñ = M, N, t T. 4

16 P P T M c,loc T := M = M} t T Ω, F T, P; F t P[M = ] = } M c,loc T := M = M} t T Ω, F T, P T ; F t P T [M = ] = } P. M, N, M, N, ZX X2 s ds ω, t - 2 Xs 2 d M, W s M t Xs 2 ds M Z t X M = = Z s Xd M s + M s dz s X + ZX, M t Z s XdM s + MdZ s X P ZX, M = ZX, M t = t X s Z s Xd M, W s.23 ZX M P s t T Ẽ T [ M t F s ] = Z s X E[Z tx M t F s ] = M s c,loc P P T M M T M t Ñ t M, N t = = M s dñs + Ñ s d M s + M, Ñ M s dn s M s X s d N, W s + P M, Ñ = M, N t t t M, N t Ñ s dm s Ñ s X s d M, W s 5

17 Z t X M t Ñ t M, N t = Z s Xd M s Ñ s M, N s + + ZX, MÑ M, N = + t Z s X M s dn s + Z s XÑsdM s M s Ñ s M, N s dz s X M s Ñ s M, N s dz s X ZX MÑ M, N P Ẽ T [ M t Ñ t M, N t F s ] = M s Ñ s M, N s P P T 2 P T M, Ñ = M, N t, t T t. Ω, F T, P T W.6.32 M = W M = W c,loc W M t.32 W = W t = t, t T t P T.6.8 ZX.8 ZX ZX t E[Z t X] = ZX X ZX 3 3. X t E[Z t X] = ZX 6

18 .34. Novikov [KS] Corollary T > [ T ] E exp Xs 2 ds < 2 E[Z T X] = t T Z t X} [KS] Proposition M = M t } t M c,loc, Z t := E[M t 2 t ], t [ }] E exp 2 M t < E[Z t ] =. 5.9 s T s := inft ; M t > s}, B s := M T s, G s := F T s B s } s G s } b < G s } S b S b := infs ; B s s = b} 5.2, 5.2 [ E exp B Sb ] [ ] 2 S b =, E exp 2 S b = e b Ω, F, P; G s Y = Y s } s, N s = N s } s Y s := exp B s s, N s := Y s Sb N Ω, F, P; G s P[S b < ] = N := lim N s = exp B Sb s 2 S b E[N ] = = E[N ] 5.7 N s = Ω, F, P; G s G s } R [ E exp B Sb }] [ 2 S b = E exp B R Sb }] 2 R S b = 7

19 5.9 t M t G s } b < [ E Sb M t } exp b + ] [ 2 S b + E M t <S b } exp M t ] 2 M t = E[exp M 2 t }] < }] [ E Sb M t } exp b + ] 2 S b [ = e b E exp 2 M t as b [ E M t <S b } exp M t ] 2 M t E[Z t ] = as b [KS] Cororally = t < t < < t n t n } n= [ tn ] E exp Xs 2 ds <, n. 2 t n ZX. X t n := X t [tn,t n]t.34 ZXn E[Z tn X F tn ] = E[Z tn Xn F tn ] = E[Z tn Xn] = E[Z tn X] n E[Z tn X] = lim n t n = t E[Z t X] = 8

20 2 2. Ω, F, P W T terminal time F W t := σw s; s t}, Ft := σf W t N } N F W T S S t := e rt ds t = rs tdt, t [, T ] 2. S S t >, t [, T ] S t := S + bss sds + σss sdw s ds t = bts tdt + σts tdw t, t [, T ] 2.2 T bt} bt dt < a.s σt} T σt 2 dt < a.s bt}, σt} 2.. portfolio processπ, π T T π t + π t dt < a.s 2.3 π tbt r dt < a.s 2.4 T σtπ t 2 dt < a.s 2.5 9

21 π, π gains processg Gt := r[π s+π s]ds+ π s[bs r]ds+ π sσsdw s, t T π i t t [, T ] i t [, T ] i 2.2. π, π G Gt = π t + π t, t T π, π self-financed, self-financing 2.3. excess yield process, over the interest rate processr Rt := bs rdu + σudw u, t T R π, π G Gt = [π s + π s]rdt + π sdrs π, π dgt = Gt S t ds + π tdrt Gt = S t S u π udru, t T , 2.5 F t }- π M π t = S u π udru, t T π tame 2

22 2.6. x R, π, π x, π, π wealth processx Xt := x + Gt, t T 2.8 G π, π Xt = π t + π t, t T π, π x-financed 2.7. x 2.8 x-financed dxt = Xt S t ds t + π tdrt Xt t S t = x + S u π udru, t T π, π tame π, π G GT a.s PGT > > arbitrage viable 2.9. G G = 2.. [KS2].4.2 Theorem i t [, T ] bt r = σtθt a.s θt} 2

23 θt} market price of risk ii t [, T ] bt r = σtθt a.s θt} θt} T θt 2 dt < a.s Z t := exp θsdw s } θs 2 ds, t T E[Z T ] =. ii θt} π, π G 2.6, 2.7 Gt = S t S u π udru = S t W t := W t + S u π uσudw u θsds, t T 2. Rt = σudw u 2.9 Z t P A := E[ A Z T ], A FT 2. W P Gt } Gt.8 S t} P [ E GT ] [ S T G E S ] = E P [ ] GT E S T Z T PGT = PGT > > S t standard i viable 22

24 ii θs} θt 2 dt < a.s T iii 2.9 Z t} 2. P standard martingale measure P E 2.2. P ds t = bts tdt + σts tdw t = rs tdt + σts td W t + = rs tdt + σts tdw t θsds P W , 2.5 F t }- π M π t := S u π uσudw u, t T P π martingalegenerating B B FT - [ S T x := E B S T ] < x-financed i ii B T S T = x + S u π uσudw u 2.2 tame π, π B financeable FT - complete 2.5. i ii [KS2].6.6 Theorem 23

25 [ 2.6. E CT S T ] < FT - CT CT T S T = x + S u π uσudw u 2.3 tame π, π T M π T := S u π uσudw u, t T 2.3 E [M π T ] = 2.ii M π π 2.7. S CT T i ii CT = S T K + ; K S T K CT = K S T + ; K S T K t t x x x π, π X t T XT T S T = x + t S u π uσudw u t x x T CT t [, T ] 24

26 2.8. T CT t V E t, T Ft- Xt = ξ π, π XT S T = Xt S t + T t S u π uσudw u CT S T ξ t Xt t T π, π T CT Xt t Γ -financed t Γt [KS2].3.2 Definition 2.2. [KS2] Proposition T CT t V E t, T [ ] CT V E t, T = S te S T F t. CT π, π CT T S T = x + S u πuσudw u 2.5 x = E [ CT ] S T π, π Xt S t = x + S u πuσudw u X XT = CT [ ] Xt CT S t = E S T F t 2.5 π 2.4 [ ] Xt CT S t E S T F t 2.6 π X

27 3 Black-Scholes Takaoka[T] 3. Black-Scholes B S α Bt = e rt St α = S e 3. αrt expσw t + Ct 2 σ2 t}λdσ W Ω, F, P λ Λ :=,, B, σλdσ < } T > t [, T ], r >, α R, S F - F t } F t := σf W t N }, t [, T ] F W t := σw s ; s t}, N F W T α = Takaoka[T] λ Black-Sholes α R α = Takaoka[T] α = S α [T] Appendix η,, B, σηdσ < exp σw t + Ct } 2 σ2 t ηdσ [ = ηr ++ + σ exp σw u + Cu } ] 2 σ2 u ηdσ dw u + Cu 26

28 t 3.2. σ 2 ηdσ < 3. X t = S e αrt, Y t = expσw t + Ct 2 σ2 t}λdσ dst α = dx t Y t = X t dy t + Y t dx t X t 3. dy t = [ σ expσw t + Ct 2 σ2 t}λdσ]dw t + Ct ds α t [ = X t σ exp σw t + Ct } ] 2 σ2 t λdσ dw t + Ct + αry t X t dt [ = X t Y t σ exp σw t + Ct } ] Y t 2 σ2 t λdσ dw t + Ct + αrx t Y t dt = St α σ expσw t + Ct 2 αr + C σ2 t}λdσ expσw t + Ct dt 2 σ2 t}λdσ +St α σ expσw t + Ct 2 σ2 t}λdσ expσw t + Ct dw t 2 σ2 t}λdσ φt := σ expσw t + Ct 2 σ2 t}λdσ expσw t + Ct 2 σ2 t}λdσ θ α t θ α t = αr + Cσt r φt = r α C φt Z α t := exp θ α sdw s } θ α s 2 ds 2 t T λ Λ m iα = θ t = C λ Λ Z 27

29 iiα m > θ α.34 Z α iiiα m = θ α Z α.36 Z α 3.3. [KS] Corollary σλdσ t = ft, x = R σ expσx+ct 2 σ2 t}λdσ t > R expσx+ct 2 σ2 t}λdσ K T > r α C ft, x K T + x, t T 3.2 Z α t T. T > = t < t < < t nt = T n nt..36 n nt T = T < Y t = X t T.36 T 3.2 r α C σt K T + max W t, t T t T t n < t n K T n θ α s 2 ds t n t n KT 2 + max W t t t T n Dt := exp[ t 4 n t n KT 2 + W t 2 ] Dt} 5.4 [ 2 }] [ ] E exp 2 t n t n K 2 + max W t = E max t T t T Dt2 4E[DT 2 ] t n t n < t T KT 2 n } nt n= E[DT 2 ] <.36 t n } nt n= Zα t T 28 2

30 λ Λ 2 :=,, B, σ 2 λdσ < } ft, x x R λ Λ 2 T > 3.2 K T y = y λ, D = DT y > y ft, y > D y D > 3.3 λ Λ 2 iii λ Λ 2 ρ d > ρ, d] a ρx b b > a > 3.3. t > yσ expσ y + Ct 2 yft, y = σ2 t}λdσ expσ y + Ct 2 σ2 t}λdσ z exp z exp t 2 = z y C2 }ρ z dz y exp z exp t z 2 y C2 }ρ z dz yσ = z y = z exp z exp t z 2 y C2 }ρ z dz + z exp z exp t z y 2 y C2 }ρ z dz y exp z exp t z 2 y C2 }ρ z dz + exp z exp t z y 2 y C2 }ρ z dz y y > z exp z exp t 2 } z z 2 y C ρ dz a exp t2 y C2 z exp zdz = a exp t2 C2 2 e z exp z exp t 2 } z z 2 y C ρ dz y exp z exp t 2 } z 2 y C ρ z y exp z exp t 2 } z 2 y C ρ dz y > yft, y min, ab exp t2 C2 2 e 29 z ρ dz = b y z y dz

31 3.5. λ Λ 2 ρ d > ρ, d] ρax = ρaρxa 3.3. yft, y = = = yσ expσ y + Ct 2 σ2 t}λdσ expσ y + Ct 2 σ2 t}λdσ z exp z exp t z 2 y C2 }ρ z dz y exp z exp t z 2 y C2 }ρ z dz yσ = z y z exp z exp t z 2 y C2 }ρ z dz + z exp z exp t z y 2 y C2 }ρ z dz y exp z exp t z 2 y C2 }ρ z dz + exp z exp t z y 2 y C2 }ρ z dz y y > z exp z exp t 2 } z z 2 y C ρ dz = ρ z exp z exp t 2 } z y y 2 y C ρzdz ρ exp t y 2 C2 zρzdz z exp z exp t 2 } z z 2 y C ρ dz y exp z exp t 2 } z z 2 y C ρ dz = ρ y y ρ y y > yft, y min, exp t 2 C2 exp z exp t 2 } z 2 y C ρ z y dz exp z exp t 2 } z 2 y C ρzdz zρzdz 3.6. λ Λ 2 ρ d > ρ, d] γ x β ρx γ 2 x β β >, γ 2 > γ > 3.3 3

32 . yft, y = = = yσ expσ y + Ct 2 σ2 t}λdσ expσ y + Ct 2 σ2 t}λdσ z exp z exp t z 2 y C2 }ρ z dz y exp z exp t z 2 y C2 }ρ z dz yσ = z y z exp z exp t z 2 y C2 }ρ z dz + z exp z exp t z y 2 y C2 }ρ z dz y exp z exp t z 2 y C2 }ρ z dz + exp z exp t z y 2 y C2 }ρ z dz y y > z exp z exp t 2 } z 2 y C ρ z y dz z exp z exp t 2 } z z 2 y C ρ dz y exp z exp t 2 } z z 2 y C ρ dz y z exp z exp t z 2 γ y exp t β 2 C2 = γ y exp t β 2 C2 β + 2 y C z β+ dz exp z exp t 2 } z 2 y C ρ y > γ yft, y min, γ 2 β + 2 exp t 2 C2 2 } γ z y β dz z y dz exp z exp t 2 } β z z 2 y C γ 2 dz γ 2 y y β 3.7. β < λ γ x β ρx γ 2 x β γ 2 > γ > 3.8. λ Λ 2 ρ lim x ρx = lim x R x ρσdσ xρx lim x ρx = xρx lim x R x ρσdσ R x lim ρσdσ x xρx 3

33 . t > lim y yft, y > yσ expσ y + Ct 2 yft, y = σ2 t}λdσ expσ y + Ct 2 σ2 t}λdσ z exp z exp t 2 = z y C2 }ρ z dz y exp z exp t z 2 y C2 }ρ z dz yσ = z y = z exp z exp t 2 z exp z exp t 2 z exp z exp t z 2 y C2 }ρ z dz + z exp z exp t z y 2 y C2 }ρ z dz y exp z exp t z 2 y C2 }ρ z dz + exp z exp t z y 2 y C2 }ρ z dz y z y C z y C 2 } ρ 2 } ρ z y z y dz exp t } 2 C2 zρ dz exp z exp t 2 } z 2 y C ρ z y z y dz exp z exp t 2 } z 2 y C ρ dz z ρ dz y yft, y exp t 2 C2 } zρ z dz + exp z exp t z y 2 y C2 }ρ z dz y ρ z dz + exp z exp t z y 2 y C2 }ρ z dz y d ρ z y exp t 2 C2 } zρ z y dz ρ z y dz = exp t 2 C2 } x σρσdσ x x ρσdσ z = σ, = x y y lim zρ z dz x y y ρ z dz = lim σρσdσ x x x ρσdσ y = lim x = lim x R x x ρσdσ xρx ρσdσ + xρx xρx + z y dz 2 32

34 3.. λdσ = exp σdσ λdσ =, σdσ Ψx := y2 exp dy, x 2 x > x + x exp x2 Ψx 2 2 x x + 3 exp x x 5 2 t lim y yft, y = 3.5 x2 exp x 2 lim y yft, y = 3.. λ Λ λ Λ 2 λ Z α Z α Z α [K] M = M t } t M c,loc, Z t := E[M t M 2 t ], t [ }] E exp 2 M t < E[Z t ] = E[exp M 2 t }] < [ }] [ ] E exp = E exp exp 2 M t < [ E exp 2 M t 4 M t }] 2 2 M t 4 M t λ Λ 2 λ λ 3.3. m λ Λ 2 m > t >, µ R, α R P log Sα t S lim < x η > m 2 x 2πη exp t } = y µ2 dy < η < m 2 t x 2η 33

35 . t > F x := log e αrt expσx + Ct 2 σ2 t}λdσ F P log Sα x t < x = PW t < F x = exp S 2πt y2 2πt exp F x } R 2 expσ x+ct 2 σ2 t}λdσ R 2t σ expσ x+ct 2 σ2 t}λdσ 2πη exp } x µ2 2η 2t } dy x expσ x + Ct 2 lim σ2 t}λdσ x σ expσ x + Ct 2 σ2 t}λdσ = m 2πt exp F x 2 } 2t 2πη exp x µ2 2η } = exp 2η } η η x µ+ t F x x µ t F x x η x µ + t F x x fx := x µ η F x t η f expσ x + Ct 2 x = + σ2 t}λdσ t σ expσ x + Ct 2 σ2 t}λdσ + η > t m η > m2 t lim x x Plog Sα t S < x 2πη exp y µ2 }dy = 2η + η < t m η < m2 t lim x x Plog Sα t S < x 2πη exp y µ2 2η 34 }dy =

36 3.4. m λ Λ 2 m = 3.3 t >, η >, µ R, α R P log Sα t S < x lim x x 2πη exp } = y µ2 dy. t > F x = log e αrt expσx + Ct 2 σ2 t}λdσ F P log Sα x } t < x = PW t < F x = exp y2 dy S 2πt 2t 2η 2πt exp F x } R 2 expσ x+ct 2 σ2 t}λdσ R 2t σ expσ x+ct 2 σ2 t}λdσ 2πη exp } x µ2 x 3.3 D = DT expσ x + Ct 2 σ2 t}λdσ σ expσ x + Ct 2 σ2 t}λdσ x D 2η 2πt exp F x } R 2 expσ x+ct 2 σ2 t}λdσ R 2t σ expσ x+ct 2 σ2 t}λdσ exp x µ + 2η 2πη exp } x µ2 2η η t F x x µ } η x t F x D x η x µ + t F x x fx := x µ η F x t η f expσ x + Ct 2 x = + σ2 t}λdσ t σ expσ x + Ct 2 σ2 t}λdσ lim x f x = x fx lim x fx = 35

37 3.5. m λ Λ 2 m > t >, µ R P log S t S lim < x η > m 2 x 2πη exp t } = y µ2 dy < η < m 2 t x 2η m = x = x λ, D = DT x > x σ expσ x 2 σ2 t}λdσ expσ x 2 σ2 t}λdσ D x t >, η >, µ R lim x x P log S t S < x 2πη exp } = y µ2 dy P A := E[ A Z T ], A F T 3.6. M λ Λ 2 M < t >, µ R, α R P log Sα t S lim > x η > M 2 x 2πη exp t } = y µ2 dy < η < M 2 t x 2η 3.7. log B t B = rt log Sα t S St α 2η 3.2 B S α λ t dσ := expσw Q t σ2 tλdσ 2 expσw Q t σ2 tλdσ 2 := W t + Ct, P α A = E[ A Z α t], A F t E α P α T K V α,k t, T Φx = x 2π exp y2 dy W Q t 2 36

38 3.8. [T] Proposition 4.2 [ ] S V,K t, T = E T K + ert t F t 3.6 = St Φ ˆx t + σ T t λ t dσ Ke rt t Φ ˆx t T t T t ˆx t = ˆx t W Q t x } St e rt t exp σx σ2 T t λ t dσ = K [MM]Appendices Lemma A. ψ, η Ω, F, P G F σ- ψ G- η E[ hψ, η ] < h : R 2 R E[hψ, η G] = Hψ Hx = E[hx, η] 3.8. kσ := expσˆx σ2 T t 2 S T = S t e rt t exp σw Q T W Q t σ2 T t} λ 2 t dσ } ST K > St e rt t exp σw Q T W Q t σ2 T t λ t dσ > K 2 W Q T W Q t > ˆx t } exp σw Q T W Q t σ2 T t kσ > 2 St e rt t kσλ t dσ = K [ ST K + = St e rt t exp σw Q T W Q t σ2 2 St, W Q t, expσw Q t [ ] S E T K + ert t F t } = St E [exp σw Q T W Q t σ2 T t 2 } = St E [exp σw Q T W Q t σ2 T t 2 Ke rt t P W Q T W Q t > ˆx t F t 37 } + T t kσ] λ t dσ σ2 2 tλdσ F t- S t + ] kσ F t λ t dσ ] T W Q t >ˆx t} F t λ t dσ kσλ t dσ = Ke rt t W Q

39 3.9 } E [exp σw Q T W Q t σ2 T t 2 } = exp σx σ2 T t exp 2πT t ˆx t 2 = Φ ˆx t + σ T t T t W Q T W Q t >ˆxt} x2 ] F t 2T t } dx P W Q T W Q t > ˆx t F t = 2πT t = Φ ˆx t ˆx t T t } exp x2 dx 2T t [ ] S E T K + ert t F t = St Φ ˆx t +σ T t λ t dσ Ke rt t Φ ˆx t T t T t 3.2. [ S V,K t, T = E T K + ert t ] F t C C S St = S,C t [ P,C A := E A exp CW t ] 2 C2 t, A F t P,C W t + C t C C 2 P,C W t + C t A = P,C 2 W t + C 2 t A, A BR } P,C S e rt exp σw t + C t σ2 2 t λdσ A } = P,C 2 S e rt exp σw t + C 2 t σ2 2 t λdσ A λ Black-Scholes, λ 2 Ishimura and Sakaguchi[IS] 38

40 Ω, F, P W T F W t := σw s; s t}, Ft := σf W t N } N F W T S S t := e rt ds t = rs tdt, t [, T ] 4. S S t >, t [, T ] S t := S + bss sds + σss tdw s ds t = bts tdt + σts tdw t, t [, T ] 4.2 T bt} bt dt < a.s, σt} T σt 2 dt < a.s θt := bt r, W σt t := W t + θsds, Z t := exp θsdw s t 2 θs2 ds}, H t := Z t S t P A = E[ A Z T ], A F T x R, π, π x, π, π x-financed X x,π,π t X x,π,π t = xs t + S t S u π uσudw u 39

41 x-financed π, π π π π π x, π X x,π X x,π t = xs t + S t S u πuσudw u 4.. x, π t [, T ] X x,π t π x π is admissible at x 4.2. π x xs t + S t πuσudw S u u S u πuσudw u x, t [, T ] Nt := πuσudw S u u P.8 N P E [NT ] E [N] = [ ] X x,π T E [NT ] = E x Z T = E[X x,π T H T ] x S T E[X x,π T H T ] x budget constraint 4.3. [KS2] Remark τ = T inft [, T ] ; X x,π t = } ω τ < T }. X x.π t, ω =, t [τ ω, T ] H tx x,π t = Z t Xx,π t S t = exp θsdw s 2 } θs 2 ds S u πuσudw u + x + } S u πuσuθudu 4

42 N t := θsdw s, A t := t 2 θs2 ds, Nt 2 := πuσudw S u u, πuσuθudu A 2 t := x + S u dh tx x,π t = H tx x,π tdn t + A t + Z tdn 2 t + A 2 t + 2 H tx x,π d N + Z t td N, N 2 t = H tx x,π t θt}dw t + H tx x,π t 2 } θt2 dt } } +Z t S t πtσt dw t + Z t S t πtσtθt dt + 2 H tx x,π tθt 2 }dt + Z t } S t πtσtθt dt = H tπtσt X x,π tθt}dw t H tx x,π t H X x,π P.8 H X x,π P H X x,π.5 π x 4.4. [KS2] Theorem x ξ F t - E[H T ξ] = x x ξ = X x,π T π. t [, T ] Mt := E[H T ξ F t ] Mt M = x T ψu 2 du < a.s ψt} t T Mt = x + ψudw u Mt M := max t T Mt < a.s κ := max t T < a.s Xt := Mt H t Z t Xt S t = Mt Z t = x + ψudw u exp θsdw s t 2 θs2 ds} Nt := ψudw u, Nt 2 := θsdw s, A 2 t := 2 θs2 ds 4

43 Xt d = S t Z t dn t Mt Z t dn t 2 Mt Z t da2 t Z t d N, N 2 + Mt t 2 Z t d N 2 t = ψt Z t dw t + Mt Z t θtdw t = + M t 2 Z t θt2 dt + Z t ψtθtdt + Mt 2 Z t θt2 dt [ ] ψt + Mtθt} σtdw t S t H tσt πt := ψt + Mtθt} H tσt Xt = xs t + S t S s πsσsdw s 4.3 π 2.4, 2.5 T T πtbt r dt = θt ψt + Mtθt} H t dt bt r = θtσt T σtπt 2 dt = e rt κ ψ 2 θ 2 + θ 2 2 M } < T 2 ψt + Mtθt} H t dt T e 2rT κ 2 ψt + Mtθt 2 dt e 2rT κ 2 [ ψ M ψ 2 θ 2 + M 2 θ 2 2] < Xt = Mt H t 4.4 XT = MT H T = H T E[H T ξ F T ] = ξ a.s , 4.4, 4.5 Utility function p, \} x p x > p ξ U p x = lim p ξ x = p x < 42

44 U x = log x x > x U p p, \} U U p p, [KS2] 3.4 Section i infx R; U p x > } = ii U p + := lim x U p x U p + = iii U p :,, U p I p :,, iv v Ũ p y := supu p x xy}, y > Ũ p y = U p I p y yi p y X y := E[H T I p yh T ], y > X :,, X Y :,, [KS2] Problem Ax := π; X x,π t, t [, T ] a.s} A x := π Ax; E min[, U p X x,π T ] > } V x := sup EU p X x,π T 4.6 π A x 4.6 π A x 4.4 E[U p ξ] E[H T ξ] x F T - ξ y > E[U p ξ]] + yx E[H T ξ] 4.7 F T - ξ E[U p ξ] + yx E[H T ξ] = xy + E[U p ξ yh T ξ] xy + E[ŨpyH T ] U

45 4.8 ξ = I p yh T ξ E[H T ξ] x E[H T I p yh T ] x X y x y = Yx ξ = I p YxH T 4.7 ξ = I p YxH T E[H T ξ] = x F T X x,π T = ξ π Ax X x,π T = I p YxH T π 4.5. [KS2] Theorem X x,π T = I p YxH T π A x π. 4.4 X x,π T = I p YxH T π Ax π π A x U U p X x,π T = U p I p YxH T = ŨpYxH T + YxH T I P YxH T U p Ic cic + YxH T I P YxH T c > U p Ic cic c > E min[, U p X x,π T ] min[, U p Ic cic] c > > π A x U U p I p YxH T YxH T I p YxH T = ŨpYxH T U p X x, π T YxH T X x, π T π A x EU p X x,π T = EU p I p YxH T EU p X x. π T YxE[H T I p YxH T ] E[H T X x, π T ]} π A x = EU p X x, π T Yxx E[H T X x. π T ]} E[H T I p YxH T ] = x EU p X x, π T 44

46 π πt = ψt + Mtθt} H tσt Mt = E[H T I p YxH T F t ] ψt M Mt = x + ψudw u π X x,π X x,π t = H t Mt = H t E[H T I p YxH T F t ] Bt = e rt S α t = S e αrt expσw t + Ct 2 σ2 t}λdσ,w Ω, F, P λ Λ :=,, B, σλdσ < } T > t [, T ], r >, α R F t } F t := σf W t N }, t [, T ] F W t := σw s ; s t}, N F W T p, \}, α = I p y = y p X y = E[H T I p yh T ] = y p E[H T p p ] = y Yx = x X p, Mt = x X E[H T p p Ft ] p X θt = C H T p p = exp p p rt exp p p CW T } p 2 p C2 T = DT mt 45

47 p DT = exp p rt + } p 2 p 2 C2 T mt = exp p p CW T 2 } p C 2 T 2 p DT ω Ω M E[H T p p Ft ] = DT mt X = E[H T p p ] = E[DT mt ] = DT Mt = xmt, X x,π t = xmt H t mt = p C msdw p s Mt = x + xp p Cms dw s ψt = xp p Cmt πt = xp p = X x,π t = H tσt H tσt H t X x,π t πt = xmt H t xcmt p Cmt + xcmt 4. xmt 4. C 4.2 σt p λ Λ 4. S C 4.2 B λ p α = λ t X x,π t fa a >, fa = p exp } p2 log a 2 2πtCa 2tC 2 46

48 E[X x,π t] = x exp r + } p C2 t } V[X x,π t] = x 2 exp 2 r + C2 C 2 t exp p p t 2 } p =, α = I y = y X y = E[H T I yh T ] = y Yx = x, Mt = x ψt = θt = C πt = X x,π t = X x,π t πt = xc H tσt x H t = x exp CW t + x C H t σt r + 2 } C2 t S 4.5 B λ C 4.2. t X x,π t fa a > } log a2 fa = exp 2πtCa 2tC 2 E[X x,π t] = x expr + C 2 t}, V[X x,π t] = x 2 exp2r + C 2 t}expc 2 t 4.2. p Mt ψt 47

49 4.2.3 p, \}, α <, λ α < λ Λ λ σ θt = C := C r α σ 4.2. Mt = x exp := x mt = x + p p CW t 2 xp p C ms dw s 2 p t} C2 p ψt = xp C mt p πt = X x,π t = H t σ H t X x,π t πt = x mt H t x C mt p 4.6 x mt 4.7 C 4.8 σ p 4.6 C > C > r α S α σ σ σ > r α α C σ C < C < r α σ S α σ σ < r α α σ α < λ C 4.7. α λ Λ p, 4.6 Mt Mt = x + ψudw u ψt Ocone and Karatzas[OK] Clark Mt ψt 4.8. [KOL] pp. 27 f : R R exp x2 fx dx < 2T 48

50 t < T E[fW T F t ] = E[fW t ] + V T t, x := pt; x, y := V T x s, W sdw s pt t; x, yfydy } x y2 exp 2πt 2t E[fW T F t ] = E Wt [fw T t ] = pt t; W t, yfydy = V T t, W t V T t, x V T t + 2 V T 2 x = 2 V T t, W t = V T, + V T x t, x = V T x s, W sdw s x y exp 2πT t 3 x y2 2T t } fydy 4.9. Karatzas and Shreve [KS2] Section 3.8 φt ω Ω Hamilton-Jacobi-Bellman φt λ Λ 4.. λ Λ 2 t > φt λ. λ φt φt x R,, B, µ x A µ x A = expσx 2 σ2 t}λdσ expσx, A B, 2 σ2 t}λdσ 49

51 µ x fσ x := fσµ x dσ φt = σ Wt+Ct d σ x = σ σ x 2 x 4.9 dx σ x x a < b W t b Ct σ Wt +Ct σ b W t a Ct σ Wt +Ct σ a W t b Ct W t a Ct P φt = σ Wt +Ct ω Ω σ a = σ b, b > a 4.9 = σ b σ a = b a d dx σ xdx = b a σ σ x 2 x dx σ σ x 2 x x σ σ x 2 x = x R x µ x σ σ x 2 σ σ x = a.e - µ x σ x σ µ x λ 5

52 5 5.. X = X t } t, Y = Y t } t Ω, F, P i ii t P[X t = Y t ] = X Y modification P[ t X t = Y t ] = X Y indistinguishable 5.2. Ω, F, P F F t } i ii t s F s F t F t F t σ- F t } filtration Ω, F, P; F t 5.3. Ω, F, P; F t F t } F t+ := s>t F s = F t F F F t } usual conditions 5.4. Doob s maximal inequality[ks].3.8 Theorem Ω, F, P; F t X = X t } t Ω, F, P ; F t t X t β > α >, p > p p p E sup X t EX p β α t β p 5.5. X = X t } t Ω, F, P; F t 5

53 i ii t, ω X t ω : [, Ω, B[, F R, BR X measurable s, ω X s ω : [, t] Ω, B[, t] F t R, BR t X F t } progressively measurable iii t X t F t - X F t } adapted 5.6. Ω, F, P; F t σ : Ω [, ] stopping time t σ t} F t Ω, F, P; F t σ : Ω [, ] optional time t σ < t} F t 5.7. [KS].2.3 Proposition F t } Ω, F, P; F t σ : Ω [, ] 5.8. Jensen [KS].3.7 ProblemX = X t } t Ω, F, P; F t ϕ : R R t E ϕx t < t s E[ϕX t F s ] ϕe[x t F s ] X ϕx t } 5.9. τ Ω, F, P; F t F τ F τ := A F; t A τ t} F t } 5.. [KS].2.3 Problem τ Ω, F, P; F t F τ σ- τ F τ X = X t } t Ω, F, P X lim sup X t ω Pdω N t X t >N } X uniformly integrable 52

54 5.2. Ω, F, P; F t X = X t } t Pτ < = τ T a > Pτ a = τ T a i ii X τ } τ T X D X τ } τ Ta a > X DL 5.3. A = A t } t Ω, F, P; F t F t } A increasing i ii A = t A t iii t EA t < A := lim t A t EA < A integrable 5.4. Ω, F, P; F t A = A t } t natural M = M t } t t E M s da s = E M s da s,t],t] 5.5. [KS].4.6 Remarks Doob-Meyer Decomposition [KS].4.9 Probrem,.4. Theorem Ω, F, P; F t F t } Ω, F, P; F t X = X t } t DL X X t = M t + A t M = M t } t A = A t } t A 53

55 5.7. [KS].3.25 Problem X = X t } t Ω, F, P; F t X = X t } t t E[X t ] = E[X ] 5.8. [KS].3.24 Problem X = X t } t Ω, F, P; F t τ X τ t } t 5.9. [KS] Theorem Ω, F, P; F t M = M t } t M c,loc lim t M t = s T s := inft ; M t > s}, B s := M T s, G s := F T s B s } s G s } M t = B M t, t t M t G s } 5.2. [KS] Remark 3.5 Section C W = W t } t Ω, F, P; F t T b := inft ; W t = b} b >, α >, t > PT b t = 2 2π b t e x2 2 dx, Ee αt b = e b 2α Z t := exp µw t 2 µ2 t, P µ A := E[ A Z t ], A F t [ P µ [T b < ] = e µb E exp 2 ] µ2 T b 5.2. [KS] Problem µ R, W = W t } t Ω, F, P; F t Ω, F, P; F t τ [ E exp µw τ ] 2 µ2 τ = 54

56 P µ [τ < ] = P µ 5.2 b R, µb < S b := inft ; W t µt = b} P µ [S b < ] = [KS] Theorem W = W t } t Ω, F P x } x R i ii F F x P x F BR/B[, ]- x R P x [W = x] = iii P x W x t s, f : R R E x [fw t+s F s ] = E Ws [fw t ] P x -a.s E x P x 55

57 [D] Richard Durrett, Stochastic Calculus, CRC Press 996 [IS] Naoyuki Ishimura and Toshi-hiko Sakaguchi, Exact solutions of a model for asset prices by K.Takaoka, Asia-Pacific Financial Markets [J] John C Hull, Options, Futures, and Other Derivatives Fifth Edition, Prentice- Hall 22 5 [K] Norihiko Kazamaki, On a problem of Girsanov, Tôhoku Math. Journ [KOL] Ioannis Karatzas, Daniel L. Ocone and Jinlu Li, An extension of Clark s formula, Stochastics and Stochastics Reports, Vol [KS] Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus Second Edition, Springer 99 [KS2] Ioannis Karatzas and Steven E. Shreve, Methods of Mathematical Finance, Springer 998 [KT],,, 23 [M],, 23 [MM] Marek Musiela and Marek Rutkowski, Martingale Methods in Financial Modelling, Springer 997 [N],, 999 [OK] Daniel L. Ocone and Ioannis Karatzas, A generalized Clark representation formula, with application to optimal portfolios, Stochastics and Stochastics Reports, Vol [P] Philip Protter, Stochastic Integration and Differential Equations, Springer

58 [T] Koichiro Takaoka, A complete-market generalization of the Black-Scholes model, Asia-Pacific Financial Markets [T2] Koichiro Takaoka, An equilibrium model of of the short-term stock price behavior [W],,