2016 B S option) call option) put option) Chicago Board Option Exchange;CBOE) F.Black M.Scholes Option Pricing Model;OPM) B S 1

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1 206 B S option) call option) put option) Chicago Board Option Exchange;CBOE) F.Black M.Scholes Option Pricing Model;OPM) B S

2 997 Robert Merton A S, max(0, S-) C max(0,s ) C P/L コールオプション購入者の損益 P/L O 株価 (S) 2

3 P/L プットオプション購入者の損益株価 p/l ストラドルホルダーの損益株価 3

4 p/l ストラドルライターの損益株価 P/L 株の買い持ちプロティクティブプット株価プットオプションの買い 4

5 P/L 株の買い持ちカバードコールの損益株価コールオプションの売り Strike Price) ()

6 , S ), r), S,, r () (2) (3) option replicating theory replicate u(.5 d( 0.5 r(. C u C d t t + t { C S B = 0 Cu + us rb = 0 t + C d + ds rb = 0 = C u C d S(u d) B = dc u uc d r(u d) C = r p = r d u d [ ( r d u d )C u + ( u r ] u d )C d C = r [pc u + ( p)c d ] p = u r u d p C u C d 6

7 C : S : : r : σ : N(d) : d price taker) r σ ds = (αs)dt + (σs)dz α σ z E(z) = 0, V ar(z) = E(z 2 ) = t dz = dt f S t f = f(s, t) df = f s ds + f t dt + 2 f ssds 2 + f st dsdt + 2 f ttdt 2 + (2-3) f s = f S, f t = f t, etc. 7

8 df = σsf s dz + ( 2 σ2 S 2 f ss + αsf s + f t )dt + O(dt 3 2 ) O(dt) df = σsf s dz + ( 2 σ2 S 2 f ss + αsf s + f t )dt f df = (α 0 f)dt + (σ 0 f)dz 0 α 0, σ 0 dz = dz 0 α 0 f = 2 σ2 S 2 f ss + αsf s + f t σ 0 f = σsf s W 0, W, W 2 W 0 + W + W 2 = 0 arbitrage portfolio) V dv = W 0 rdt + W (ds/s) + W 2 (df/f) = ( W W 2 )rdt + W (ds/s) + W 2 (df/f) = W (ds/s rdt) + W 2 (df/f rdt) dv = {W (α r) + W 2 (α 0 r) dt + (W σ + W 2 σ 0 )dz W (α r) + W 2 (α 0 r) = 0 W σ + W 2 σ 0 = 0 2 W W σ 0 (α r) = σ(α 0 r) 8

9 2 σ2 S 2 f ss + rsf s + f t rf = 0 f(s, t) 2 3 f(s, T ) = φ(s) 2 3 u t = a 2 u xx (a : t = T t(> 0) f(s, τ) = e rt Y (S, τ) σ2 S 2 Y ss + rsy s Y τ = 0 S, τ ξ(s, τ), η(s, τ) chain rule Y s = ξ s Y ξ + η s Y η Y τ = ξ τ Y ξ + η τ Y η Y ss = ξ ss Y ξ + ξ s ξ s Y ξξ + η ss Y η + η s η s Yηη = ξ ss Y ξ + ξsy 2 ξξ + η ss Y η + ηsyηη Y = Y (S, τ) = Y (ξ(s, τ), η(s, τ)) 2 σ2 S 2 (ξ ss Y ξ + ξ 2 sy ξξ + η SS Y η + η 2 sy ηη ) + rs(ξ S Y ξ + η S Y η ) (ξ τ Y ξ + η τ Y η ) = 0 2 σ2 S 2 ξ 2 SY ξξ + ( 2 σ2 S 2 ξ SS + rsξ S ξ τ )Y ξ + 2 σ2 S 2 η 2 SY ηη + ( 2 σ2 S 2 η ss + rsη s η τ )Y η = 0 (2 7 9

10 2 σ2 S 2 ξ SS + rsξ S ξ τ = σ2 S 2 ξ SS + rsξ S = C(cons tan t) ξ τ = C 9 ξ = C ln S + g(τ) (C = C (r /2σ 2 )) 2 20 g τ =C C = ξ ξ = ln S + (r /2σ 2 )τ η s = 0 η η(τ) η τ = 2 σ2, 2 σ2 Y ξξ η τ Y η = η = 2 σ2 τ 2 24 Y ξξ = Y η τ = 0 f(s, 0) = φ(s) f(s, τ) f(s, τ) = e rτ 2 πη (x ξ)2 φ(x) exp dx η ξ = ln S + (r 2 σ2 )τ η = 2 σ2 τ 0

11 { S (S > ) φ(s) = 0 (S ) { e ξ (ξ > ln ) φ(ξ) = 0 (ξ ln ) 価値満期のコールオプションの価値 O 株価 (S) f(s, τ) = e rτ 2 (e x ) exp πη ln q = x ξ 2η dq = q = ln ξ 2η 2 26 f(s, τ) = e rτ 2 2η πη q = e rτ q dx 2η (x ξ)2 4η (e 2η q+ξ )e (e 2η q+ξ )e q 2 2 dq q 2 2 dq = e rτ (e ξ e q ηq e q2 2 )dq = e rτ+ξ+η q = S q + q 2η 2η e 2 t2 dt e rτ dx 2 26 q e 2 t2 dt e rτ f(s, τ) = SN( q + 2η) e rτ N( q + ) q + ln S 2η = + (r + 2 σ2 )τ σ d τ q e q 2 2 dq e ln S q = + (r 2 σ2 )τ σ = d σ τ τ q 2 2 dq (e rτ+ξ+η = S) f(s, t) = SN(d) e rτ N(d σ τ) 2-27

12 ( S = S t + σs z -28 t S t σs S t σs z σ dz dz = dt Z Z σ (Brownian motion process) Wiener- levy process) S/S ln S/S ln S /S) = µ t + σ Z 2-30 S = Se µ t+σ Z 2 3 µ t + σ Z) e x e x = + x + x2 2 + x3 6 + S /S = + (µ t + σ Z) + (µ t + σ Z) 2 /2 + (2-32) = + σ Z + (µ + σ 2 Z 2 /2) + t) 2, ( 3, 2

13 S /S = + S /S 2-32 S /S = (µ + σ 2 Z 2 /2) + σ Z (µ + σ 2 Z 2 /2) σ z = σ Z -29 us(, ds( C u, C d C = [pc u + ( p)c d ] /r uus uds dus dds C uu C ud C du C dd - C u = [pc uu + ( p)c ud ] /r C d = [pc ud + ( p)c dd ] /r C = [ p 2 C uu + 2p( p)c ud + ( p) 2 C ud ] /r 2 C = (/r) n { n a= [n!/(n a)!a!] pa ( p) n a C u a d n a - C u a d n a = Max [0, u a d n a S ] u a d n a S > 0) m a < m a m a < m C ua d n a C ua d n a = 0, a m = Max [0, u a d n a S ] = u a d n a S C = (/r) n { n a=m [n!/(n a)!a!] pa ( p) n a [u a d n a S ] ( - C = S n a=m p u r )p n! (n a)!a! pa ( p) n a ua d n a r n n r n a=m p =(d/r)( p) C = S n a=m n! (n a)!a! pá ( p ) n a n r n a=m n! (n a)!a! pá ( p ) n a B(m; n, p n a=m n! (n a)!a! pa ( p) n a n! (n a)!a! pa ( p) n a - 3

14 B(m; n, p n n! a=m (n a)!a! pa ( p) n a C = SB(m; n, p r n B(m; n, p) p = (r d)/(u d), p = (u/r)p n B(m; n, p N(d) B(m; n, p) N(d σ τ) S S T τ τ = T t) S D(S ) P/L { S (S > P/L = 0 (S S S E(profit) E profit) = E(S ) = (S )D(S )ds ln S /S S /S y y f(y) = σ y exp (ln y µ)2 2σ 2 (y > 0) 0 (y 0) f a > 0 f(y)dy = N( ln a + µ ) σ a a σ2 µ+ yf(y) = e 2 N( ln a + µ + σ 2 ) σ 4

15 N(x) = x e x2 2 dx ( a 0 0 f(y)dy =, 0 yf(y)dy = exp(µ + σ2 2 ) f exp(µ + σ2 2 ) S /S S S /S D(S ) D(S ) = S f(s S ) = σ τ S exp (ln S /S µτ) 2 2σ 2 τ Ce rτ = (S )D(S )ds = = = = S (S ) S f(s S )ds S (St )f(t)sdt (S /S = t) /S /S /S (St )f(t)dt tf(t)dt /S f(t)dt = S exp(µ + σ2 2 )τn(ln S/ + (µ + σ2 )τ σ ) τ ln S/ + µτ N( σ ) τ rτ = (µ + σ2 σ2 )τ µ = r 2 2 ln S/ + (r + σ2 C = SN( 2 )τ σ τ ln S/ + (r σ2 ) e rτ N( 2 )τ σ τ ) = SN(d ) e rτ N(d 2 ) ln S/ + (r + σ2 d = 2 )τ σ τ d 2 = d! σ τ B S Ce rτ = (S )D(S )ds 5

16 = Se rτ N(d ) N(d 2 ) e rτ S D(S )ds = SN(d ) e rτ D(S )ds = e rτ N(d 2 ) ( (*) f(x)dx = a σ a x exp (ln x µ)2 2σ 2 = σ e t (t µ)2 exp ln a 2σ 2 = σ (t µ)2 exp ln a 2σ 2 dt = σ e k2 2 σdk ( t µ = k) σ (*) a ln a µ σ ln a (µ+σ 2 ) σ dx e t dt(ln x = t) =N( ln a µ ) = N( ln a + µ ) σ σ xf(x)dx = σ (ln x µ)2 exp a 2σ 2 dx = σ (t µ)2 exp ln a 2σ 2 e t dt (ln x = t) = eµ+ σ 2 { 2 { t (µ + σ 2 2 σ exp ln a 2σ 2 dt σ2 µ+ =e 2 σ e k2 σdk ( t (µ + σ2 ) σ σ2 µ+ =e 2 N( ln a + (µ + σ 2 ) ) σ ( e x2 2 dx = α ( α e x2 2 x f(x) dx = N( α)) x y = ϕ(x) g(y) g(y) = f(ψ(y)) dψ(y) dy ψ y = ϕ(x) x = ψ(y)) S /S x = ln S /S N x) = σ (x µ) 2 e 2σ 2 = k) g(y) = D(S ) = f(ψ(s ) dψ(s ) ds (x = ψ(s ) = ln S : /S) = σ exp (ln S /S µ) 2 2σ 2 S S S ) = σ S exp (ln S /S µ) 2 2σ 2 (S = S f(s S ) f 6

17 986 DYNAMIC ASSET ALLOCATION mail: 7

9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x

2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin