ÍÚÈÔ (È È È Ï Ê Ø2017 ÏÑ) Ñ(1) È 2017 10 4, 11 1 / 49
ÏÑÒ ØËÌÙ Õ Õ Ø 2 / 49
ÏÑÒ ØËÌÙ ÏÑÒ ØËÌÙ Õ Õ Ø (I) Í(Ù ) ØÙ Ù ÚÓ (II) Black-Scholes-Merton Ð Ú ÌÑØÎÔ Ð Ô Black-Scholes (III) ØÕÊÔ ÕÚÔÍ(ÓÙÊØ Ú ) È ÎÑØÎÔ Ð ÍÒËÒ ÌÒ ÚÔÍ (IV) ÚÚ ÏË Ð, or/and, ÈÔÈË Ð 3 / 49
Õ Õ Ø ÏÑÒ ØËÌÙ Õ Õ Ø (I): ÍØÙ Ù Ê Í ÚÓ Ê É ÏÑÏ Í ÚÓÑØ Ù2 Ù Ì È (ÊÏ Ô) ÍÚ ÉÊÖ Ó (ÓÙ ) ÍÑ( Õ ) Î Ó( ÕÕ ) Ø ÌChung and Williams (Birkhäuser), Durret (CRC Press), Ikeda and Watanabe (North-Holland), Karatzas and Shreve (Springer), Øksendal (Springer), Revuz and Yor (Springer), Rogers and Williams (Cambridge), Steele (Springer), etc. (II)-(III): Ë ÙÒÈÎLamberton and Lapeyre(Champan & Hall), Shreve(Springer). Õ ÏÑÚÔ ÉÍ 4 / 49
Gaussian r.v. Ì ÚÊÙÓÕÏ Ô ÔÊÍÈ ØÚ Ô 0. Ú ( ) ÍÈÔ 5 / 49
Ê ÓØÚ Ù ÍÈÔ Gaussian r.v. Ì ÚÊÙÓÕÏ Ô ÔÊÍÈ ØÚ Ô Z N(m,σ 2 ) :ÎÈm, Ûσ 2 (> 0) Ú Í ÍÈÔ. P(a < Z b) = P(a < Z < b) = 1 2πσ 2 e (x m)2 2σ 2 : Ú Ï Ô. b a 1 2πσ 2 e (x m)2 2σ 2 dx. 6 / 49
Ì Gaussian r.v. Ì ÚÊÙÓÕÏ Ô ÔÊÍÈ ØÚ Ô Ef(Z) = 1 f(x) 2πσ 2 e (x m)2 2σ 2 dx. f(x) = x,(x m) 2,1 (a,b] (x) ÕÕÏ m = σ 2 = P(a < Z b) = 1 x 2πσ 2 e (x m)2 2σ 2 dx, (x m) 2 1 2πσ 2 e (x m)2 2σ 2 dx, 1 1 (a,b] (x) 2πσ 2 e (x m)2 2σ 2 dx. 7 / 49
ÚÊÙÓÕÏ Ô Gaussian r.v. Ì ÚÊÙÓÕÏ Ô ÔÊÍÈ ØÚ Ô Λ(λ) := Ee λz = = e λx 1 2πσ 2 e (x m)2 2σ 2 dx 1 2πσ 2 e (x m λσ2 ) 2 2σ 2 +λm+ λ2 σ 2 2 dx =exp (λm+ λ2 σ 2 ). 2 EZ k = Λ (k) (0): ØÒmÕσ 2 ÔËÚÏÍ 8 / 49
ÔÊÍÈ Gaussian r.v. Ì ÚÊÙÓÕÏ Ô ÔÊÍÈ ØÚ Ô Y N(0,1) Z = m+σy N(m,σ 2 ). φ(y) := 1 2π e y2 2 È Ò Φ(d) := d Ef(Z) = Ef(m+σY) = φ(y)dy È Ò f(m+σy)φ(y)dy. ( a m P(a < Z < b) =P < Y < b m ) = σ σ ( ) ( ) b m a m =Φ Φ. σ σ b m σ a m σ φ(y)dy 9 / 49
ØÚ Ô ÙÙ Gaussian r.v. Ì ÚÊÙÓÕÏ Ô ÔÊÍÈ ØÚ Ô Φ(x) := 10 7 Ô ÙÙÓÌ x 1 2π e z2 /2 dz 10 / 49
ØÚ Ô ÙÙ Gaussian r.v. Ì ÚÊÙÓÕÏ Ô ÔÊÍÈ ØÚ Ô Φ(x) := x 1 2π e z2 /2 dz 10 7 Ô ÙÙÓÌ x > 0 Õ, ÎÐ Φ(x) 1 1 2π e x2 /2 ( b 1 t+b 2 t 2 +b 3 t 3 +b 4 t 4 +b 5 t 5), t := 1 1+px, p := 0.231641900, b 1 := 0.319381530, b 2 := 0.356563782, b 3 := 1.781477937, b 4 := 1.821255978, b 5 := 1.330274429. Øx < 0 Õ Ô ÓΦ(x) = 1 Φ( x) ÕÊ Ù 10 / 49
ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) 11 / 49
ÌÑ ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) ØÊ ÉÎ ÍÙÚw := (w t ) t [0,T] ( Ø) ÏËÊÓÎÓÕ Ê(Ð É T > 0 time horizon ÕÒ ÏÍ) (1) t w t. (2) w 0 = 0. (3) ÕÏ Ø Ñ 0 = t 0 < t 1 < < t n T Ù Ò w ti w ti 1, i {1,...,n} ÕÏ (4) Ú Í: w t w s N(0,t s). 12 / 49
BM ÙÙ ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) BM: (w t ) t [0,T] ÙÙÙÚ (W (N) t ) t [0,T] ØÊÔË Ì (1) Ë Ñ ÉÏ ØÌ 0 = t 0 < t 1 < < t n < t n+1 < < t N = T. (2) N ÕÏÚ ÍÈÔ(Y i ) i {1,...,N}, Y i N(0,1) Õ Ò Ì W (N) t i W (N) t i 1 := t i t i 1 Y i ÌÑ Í (3) t i 1 Õt i Ï ÉÑÙ Ï ÍÌ W (N) t = W (N) t i 1 + t t ( i 1 W (N) t t i t i i 1 W (N) t i 1 ), t [t i 1,t i ]. 13 / 49
ÚÓÑÍÈ (Sample Paths) Í ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) w 1 0.5 0-0.5-1 -1.5 Sample-Paths of Simple Random Walk(10000-Steps) line 1 line 2 line 3-2 0 0.2 0.4 0.6 0.8 1 t 14 / 49
ÕÍ Î ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) BM Markov Ô Í Ø Ñ Î 0 s t T Ù Ò E[f(w t ) (w u ) u [0,s] ] =E[f( É ) ËÖ + Ù ] =E[f( É ) ËÖ ] =E[f(w t ) w s ]. E[f(w t ) w s ] =E[f (w s +(w t w s )) w s ] =E[f (x+(w t w s ))] { x=ws 1 = f(x+y) e y2 2π(t s) R 2(t s)dy} x=ws. 15 / 49
Martingale ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) BM Ó (w s ;s [0,t]) (t 0) Ù ÒÕÍÑÓÕÊÍÔ ÍÌ i.e., E s [w t ] = w s, 0 s t T, (E s [ ] := E[( ) (w u ;u [0,s])]). 16 / 49
Martingale ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) BM Ó (w s ;s [0,t]) (t 0) Ù ÒÕÍÑÓÕÊÍÔ ÍÌ i.e., E s [w t ] = w s, 0 s t T, Ð: (E s [ ] := E[( ) (w u ;u [0,s])]). E s [w t ] =E s [w s +(w t w s )] =w s +E[w t w s ] = w s. 16 / 49
Martingale (2)Ì Ì ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) Q t := w 2 t t (t 0) Ó (w s ;s [0,t]) Ù ÒÕÍÑÓ ÕÊÍÔ ÍÌ i.e., E s [Q t ] = Q s, 0 s t T. 17 / 49
Martingale (2)Ì Ì ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) Q t := w 2 t t (t 0) Ó (w s ;s [0,t]) Ù ÒÕÍÑÓ ÕÊÍÔ ÍÌ i.e., Ð: E s [Q t ] = Q s, 0 s t T. E s [Q t ] =E s [Q s +(Q t Q s )] =Q s +E s [ (wt w s ) 2 +2w s (w t w s ) (t s) ] =Q s +E s [ (wt w s ) 2] +2w s E s [w t w s ] (t s) =Q s. 17 / 49
Ó Ë ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) t w t ( Í1Ô ÌÍÌ) Ì ËÔ Í È ÈÓ Ò Í Ï ÎÑÈÌ É Ò d dt f(w t) = f (w t ) dw t dt, f(w t ) f(w 0 ) = = t 0 t 0 f (w s ) dw s ds ds f (w s )dw s. 18 / 49
Exponential Brownian Motion (1) ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) µ R, σ > 0: ÈËÙÊÎ { ) Z t := exp σw t + (µ σ2 2 } t, t 0. 19 / 49
Exponential Brownian Motion (1) ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) µ R, σ > 0: ÈËÙÊÎ { ) Z t := exp σw t + (µ σ2 2 ÎÌ. Z := (Z t ) t [0,T] } t, t 0. ÔÕÍÑÓÕÊÍ µ 0, ÕÍÑÓÕÊÍ µ = 0, ÉÕÍÑÓÕÊÍ µ 0. 19 / 49
Exponential Brownian Motion (2) ÌÑ BM ÙÙ ÚÓÑÍ Í ÕÍ Î Martingale Martingale (2) Ó Ë Ô BM(1) Ô BM(2) Ð: E s [Z t ] =E s [ Z s Z t =Z s e =Z s e Z s (µ σ2 2 (µ σ2 2 =Z s e µ(t s) ] Z s, if µ 0, =Z s, if µ = 0, Z s, if µ 0. )(t s) Es [e σ(w t w s ) ] ) (t s) e σ 2 2 (t s) 20 / 49
BS ÚÔÍ ÎÈ ÑÍ ÎÈ ÑÍ (2) ÒËÒ ÌÒ ÒËÒ ÌÒ (2) 2. Black-ScholesÚÔÍ 21 / 49
Black-Scholes ÚÔÍ BS ÚÔÍ ÎÈ ÑÍ ÎÈ ÑÍ (2) ÒËÒ ÌÒ ÒËÒ ÌÒ (2) Ì ÚÙÚ(bank-account process): S 0 := (St) 0 t [0,T], ÐÔ ÑØÙÚ(risky asset price process): S := (S t ) t [0,T], ÔÓ(stock) Ó ÛÖ(currency), ÚÔ Ò (commodity)ì Ð Ð Ø S 0 t =e rt, S t =S 0 exp { σw t + ) (µ σ2 2 } t. r 0: ÚÚØ ÍÙ σ > 0: ÒËÒ ÌÒ Ø ÍÙ µ R: Ì ÑÍØ ÍÙ S 0 > 0: Ë ÑØ w := (w t ) t [0,T] : BM. 22 / 49
ÎÈ ÑÍ(mean-return-rate): µ BS ÚÔÍ ÎÈ ÑÍ ÎÈ ÑÍ (2) ÒËÒ ÌÒ ÒËÒ ÌÒ (2) E t [ St+h S t S t when h 1. ] =E t exp =e µh 1 µh, { σ(w t+h w t )+ ) (µ σ2 2 } h 1 23 / 49
ÎÈ ÑÍ(mean-return-rate)(2) BS ÚÔÍ ÎÈ ÑÍ ÎÈ ÑÍ (2) ÒËÒ ÌÒ ÒËÒ ÌÒ (2) 2 1.8 1.6 1.4 1.2 b=-0.5 b=0 b=0.5 1 0.8 0.6 0.4 0 0.2 0.4 0.6 0.8 1 24 / 49
ÒËÒ ÌÒ (volatility): σ BS ÚÔÍ ÎÈ ÑÍ ÎÈ ÑÍ (2) ÒËÒ ÌÒ ÒËÒ ÌÒ (2) when h 1. V t [ St+h S t S t ] [ {St+h } ] S 2 t =E t (e µh 1) S t =E t [ { e σ(w t+h w t ) σ2 2 h 1 = ( ) e σ2h 1 e 2µh (σ 2 h)(1+2µh) σ 2 h, } 2 e 2µh ] 25 / 49
ÒËÒ ÌÒ (volatility)(2) BS ÚÔÍ ÎÈ ÑÍ ÎÈ ÑÍ (2) ÒËÒ ÌÒ ÒËÒ ÌÒ (2) 2 1.8 1.6 1.4 1.2 a=0.1 a=0.5 a=1.0 1 0.8 0.6 0.4 0 0.2 0.4 0.6 0.8 1 26 / 49
0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ 27 / 49
0th Step (1) 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ w := (w t ) 0 t T : BM. F t := F((w s ) 0 s t ): É tõô BM (w s ) 0 s t ÔÌÕ Ò Í ÍÈÔ. F t := BM É tõô Ó F t F t - Õ ÊÐ Í. 28 / 49
0th Step (1) 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ w := (w t ) 0 t T : BM. F t := F((w s ) 0 s t ): É tõô BM (w s ) 0 s t ÔÌÕ Ò Í ÍÈÔ. F t := BM É tõô Ó F t F t - Õ ÊÐ Í. ÎÌ. 0 t 1 < t 2 T Õ Í ÕÍÑÓÕÊÍ M t :=F t1 (w t t2 w t t1 ) (0 t T) 0 if t t 1, = F t1 (w t w t1 ) if t 1 t t 2, F t1 (w t2 w t1 ) if t 2 t T, 28 / 49
0th Step (2): Ù 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ (w t ) t [0,T] : BM, ÑØÙÚ, É t 1 ÙF t1 (È) É, É t 2 ( t 1 ) Ù È, É t( [0,T]) Ô ÑØÚÚ0 ÍÌÙÌ M t :=F t1 (w t t2 w t t1 ) (0 t T) 0 if t t 1, = F t1 (w t w t1 ) if t 1 t t 2, F t1 (w t2 w t1 ) if t 2 t T, ÑÙÚ(M t ) t [0,T] ÕÍÑÓÕÊÍ. 29 / 49
0th Step (3) 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ Ð: (A) 0 s t t 1 Õ M t :=F t1 (w t t2 w t t1 ) 0 if 0 t t 1 = F t1 (w t w t1 ) if t 1 t t 2 F t1 (w t2 w t1 ) if t 2 t T. E s M t = E s 0 = 0 = M s. 30 / 49
0th Step (4) 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ (B-i) 0 s t 1 t t 2 Õ (B-ii) 0 t 1 s t t 2 Õ E s M t =E s F t1 (w t w t1 ) =E s [E t1 [F t1 (w t w t1 )]] =E s [F t1 E t1 [w t w t1 ]] =0 = M s. E s M t =E s F t1 (w t w t1 ) =F t1 {E s [w t ] w t1 } =F t1 (w s w t1 ) = M s. 31 / 49
0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ 0th Step (5) (C-i) 0 s t 1 t 2 t T Õ E s M t =E s F t1 (w t2 w t1 ) (C-ii) 0 t 1 s t 2 t T Õ =E s [E t1 [F t1 (w t2 w t1 )]] = 0 = M s. E s M t =E s F t1 (w t2 w t1 ) (C-iii) 0 t 1 t 2 s t T Õ =F t1 {E s [w t2 ] w t1 } = F t1 (w s w t1 ) = M s. E s M t =E s F t1 (w t2 w t1 ) = F t1 (w t2 w t1 ) = M s. 32 / 49
0th Step (6) 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ w: BM. 0 = t 0 < t 1 < t 2 <... < t n = T F ti : F ti - ÍÈÔ, EF 2 t i <. 33 / 49
0th Step (6) 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ w: BM. 0 = t 0 < t 1 < t 2 <... < t n = T F ti : F ti - ÍÈÔ, EFt 2 i <. ÎÌ M t := n F ti 1 (w t ti w t ti 1 ) i=1 (2 É È ) ÕÍÑÓÕÊÍ, i.e., E s M t = M s 0 s t T. 33 / 49
0th Step (7): Ù 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ (w t ) t [0,T] : BM, ÑØÙÚ, É t i 1 ÙF ti 1 (È) É(i = 1,...,n), É t i (> t i 1 ) Ù È(i = 1,...,n), É t( [0,T]) Ô ÖÈ ÑØÚÚ0 ÍÌÙÌ M t := n F ti 1 (w t ti w t ti 1 ) i=1 ÑÙÚ(M t ) t [0,T] ÕÍÑÓÕÊÍ. 34 / 49
1st Step: Ô ÙÚÙ Í ÍÈ 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ f := (f t ) t [0,T] : Ô ÙÚØ ØÙÚÙ n f t := i=1 F ti 1 1 [ti 1,t i )(t), F ti 1 : F ti 1 -. ÍÈ ÌI : L 0 f I(f) := (I(f) t ) t [0,T] M, I(f) t = t 0 n f t dw t := f ti 1 (w t ti w t ti 1 ) = i=1 k f ti 1 (w ti w ti 1 )+f tk (w t w tk ), i=1 if t k t t k+1. (L 0 : Ô ÙÚ Ø M: 2 É È ÕÍÑÓÕÊÍ Ø ) 35 / 49
ÑØ ÚØ ÐÎÈÚÔÍ 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ w := (w t ) t [0,T] : ÔÑÚÔÍ ÕÍÑÓÕÊÍØ ÚÎØÉÔ ÚÔÍÙ f t := n F ti 1 1 [ti 1,t i )(t) (t [0,T]): Ô Ó i=1 É t [t i 1,t i ) Ô F ti 1 ÈÔÓ Ó. É t i 1 ÙF ti 1 È ÔÓ ÔÑw ti 1 Ô É É t i ÙÔÑ w ti ÕØÍÕF ti 1 ( wti w ti 1 ) Ñ É t 0 = 0 ÑËt [t k,t k+1 ] ÕÔ ÎÈÔ ÕÏÍÖÈ Ñ: = k ( ) F tj 1 wtj w tj 1 +Ftk (w t w tk ) j=1 k ( ) f tj 1 wtj w tj 1 +ftk (w t w tk )= j=1 t 0 f u dw u. 36 / 49
0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ ÎÌ. recalling I(f) 2 M :=E [ I(f) 2 ] T [ T ] =E ft 2 dt =: f 2 L 2 (dt P), 0 E { f ti 1 (w t ti w t ti 1 ) } 2 = Ef 2 ti 1 i ( i := t i t i 1 ). E [ f ti 1 (w t ti w t ti 1 )f tj 1 (w t tj w t tj 1 ) ] = 0, if i j. 37 / 49
0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ ÎÌ. recalling I(f) 2 M :=E [ I(f) 2 ] T [ T ] =E ft 2 dt =: f 2 L 2 (dt P), 0 E { f ti 1 (w t ti w t ti 1 ) } 2 = Ef 2 ti 1 i ( i := t i t i 1 ). E [ f ti 1 (w t ti w t ti 1 )f tj 1 (w t tj w t tj 1 ) ] = 0, if i j. (f (n) ) n N : Cauchy in L 0 ( L 2 (dt dp)) (I(f (n) )) n N : Cauchy in M ( L 2 (P,F T )). 37 / 49
2nd Step 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ w: BM. (f t ) t [0,T] : adapted process, i.e., f t : F t - for t [0,T], s.t. [ ] T E <. 0 f2 t dt f Ô ÙÚÔÙÙÔ ÍÌ 0 = t 0 < t 1 < t 2 <... < t n = T. n := max 1 i n i 0 as n. f (n) t := n i=1 f t i 1 1 [ti 1,t i ). f (n) f as n. I(f) := (I(f) t ) t [0,T] M I(f) := lim n I(f(n) ) ÔÌÑ Í 38 / 49
ÍÈ Ñ 0th Step (1) 0th Step (2): Ù 0th Step (3) 0th Step (4) 0th Step (5) 0th Step (6) 0th Step (7): Ù 1st Step SF Investor 2nd Step ÍÈ Ñ ÎÌ. I(f) t := t 0 f s dw s ØÊ ÉÎ (1) (I(f) t ) 0 t T Ø2 É È ÙÕÍÑÓÕÊÍ. Ù EI(f) t = E (2) Ï. i.e., [ ( T E 0 t 0 f s dw s = 0 = I(f) 0. ) 2 ] [ T f s dw s = E 0 ] fsds 2. 39 / 49
È Ô ÚÓ Ù ÚÓ (1) (2) (3) (4) Ù ÙÚ Ù ÚÓ (2) Í 40 / 49
È Ô ÚÓ È Ô ÚÓ Ù ÚÓ (1) (2) (3) (4) Ù ÙÚ Ù ÚÓ (2) Í f C 1,1 ([0,T] R). w t : t Ù Ò ËØ Ô È Ô ÚÓÌ Í È ÑÔ Ò d dt f(t,w t) = t f(t,w t )+ x f(t,w t ) dw t dt f(t,w t ) = f(0,w 0 )+ t 0 x f(s,w s )dw s + t 0 s f(s,w s )ds. 41 / 49
Ù ÚÓ È Ô ÚÓ Ù ÚÓ (1) (2) (3) (4) Ù ÙÚ Ù ÚÓ (2) Í Ì. f C 1,2 ([0,T] R). (w t ) t [0,T] : BM (t Ù ÒÌÍÌ Ì Ë). ÑÔ t f(t,w t ) =f(0,w 0 )+ x f(s,w s )dw s 0 t { + s f(s,w s )+ 1 } 2 xxf(s,w s ) ds. df(t,w t ) = x f(t,w t )dw t + 0 { t f(t,w t )+ 1 } 2 xxf(t,w t ) dt. 42 / 49
(1) È Ô ÚÓ Ù ÚÓ (1) (2) (3) (4) Ù ÙÚ Ù ÚÓ (2) Í f(t,x) := x 2 t. f(t,w t ) = wt 2 tìõíñóõêíô Í Õ Ù ÑÙÎ Ù ÚÓ È ÍÕ f(t,w t ) = f(0,w 0 )+2 t 0 w s dw s. 43 / 49
(2): ÔBM È Ô ÚÓ Ù ÚÓ (1) (2) (3) (4) Ù ÙÚ Ù ÚÓ (2) Í { ( f(t,x) := exp σx+ X t := f(t,w t ) = exp µ σ2 2 { σw t + ) ( } t. µ σ2 2 Recall x f = σf, xx f = σ 2 f, t f = Ù ÚÓ ÈÌ f(t,w t ) = 1+ ÑÔ Ô t 0 ) } t f(s,w s )σdw s + ( µ σ2 2 : Exponential BM. ) f. t dx t = X t (σdw t +µdt). 0 f(s,w s )µds. 44 / 49
(3): Black-Scholes model È Ô ÚÓ Ù ÚÓ (1) (2) (3) (4) Ù ÙÚ Ù ÚÓ (2) Í BS-model: S t := S 0 exp { σw t + S t = S 0 + t 0 S sσdw s + t 0 S sµds, ( ) } µ σ2 2 t. 45 / 49
(3): Black-Scholes model È Ô ÚÓ Ù ÚÓ (1) (2) (3) (4) Ù ÙÚ Ù ÚÓ (2) Í BS-model: S t := S 0 exp { σw t + S t = S 0 + t 0 S sσdw s + t 0 S sµds, ( ) } µ σ2 2 t. ds t = S t (σdw t +µdt) :(S Ù Ò ÚÓÕ ÒÑÒ) Í ÚÓ. 45 / 49
(4) È Ô ÚÓ Ù ÚÓ (1) (2) (3) (4) Ù ÙÚ Ù ÚÓ (2) Í f(t,x): t f + 1 2 xxf = 0 ÉÎ Õ Í Ø Ñ df(t,w t ) = x f(t,w t )dw t + = x f(t,w t )dw t. f(t,w t ) = f(0,0)+ { t f + 1 } 2 xxf (t,w t )dt t 0 x f(u,w u )dw u. (E T 0 xf(t,w t ) 2 dt < ØË) (f(t,w t )) t [0,T] (2 É È ) ÕÍ ÑÓÕÊÍ 46 / 49
Ù ÙÚ È Ô ÚÓ Ù ÚÓ (1) (2) (3) (4) Ù ÙÚ Ù ÚÓ (2) Í X t := X 0 +M t +A t, X 0 R, M t := A t := t 0 t (φ,ψ: ÈÙÚ). Í ÑÔ 0 φ s dw s, ψ s ds, dx t = dm t +da t = φ t dw t +ψ t dt. 47 / 49
Ù ÙÚÙ Ò Ù ÚÓ È Ô ÚÓ Ù ÚÓ (1) (2) (3) (4) Ù ÙÚ Ù ÚÓ (2) Í Ì. f C 1,2 ([0,T],R) Ù Ò ÎÐ f(t,x t ) =f(0,x 0 )+ t 0 t 0 + t 0 t 0 x f(s,x s )dx s s f(s,x s )ds+ 1 2 x f(s,x s )dx s = xx f(s,x s )d X,X s = t 0 t 0 t 0 xx f(s,x s )d X,X s. x f(s,x s )(φ s dw s +ψ s ds), xx f(s,x s )φ 2 sds. 48 / 49
Í È Ô ÚÓ Ù ÚÓ (1) (2) (3) (4) Ù ÙÚ Ù ÚÓ (2) Í X: Ù ÙÚ (dx = dm +da, dm = φdw, da = ψdt). Ù Ò (i) dxdx := d X,X = dmdm = d M,M = φ 2 dt. (ii) dxdt = dtdt = 0. df(t,x t ) Taylor Ñ 2 ÕÔ Ø ÑÈ ÎÚ Ù ÚÓÌ df(t,x t ) = x f(t,x t )dx t + t f(t,x t )dt + 1 2 xxf(t,x t )dx t dx t. 49 / 49