Black-Scholes [1] Nelson [2] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [2][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-W

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1 Black-Scholes [1] Nelson [] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-Wu Nelson takatoshi-tasaki@nifty.com kabutaro@mocha.fre .ne.jp URL:

2 Black-Scholes Nelson Nelson Fokker-Planck Schrödinger Schrödinger [7][8] t S(t) µ σ W (t) dw (t)dw (s) = δ(t s)dt (1) Wiener µ σ ds(t) = µs(t)dt + σs(t)dw (t) () Black-Scholes µ σ S(t) t µ S(t) t ds(t) = µ(s(t), t)s(t)dt + σs(t)dw (t) (3) ds(t) df(t) f(t + dt) f(t) (4) dx(t) ds(t) S(t) (5) (3) dx(t) = µ(x(t), t)dt + σdw (t) (6) 3 Nelson Nelson [][3][4] Nelson x(t) d x(t) = µ( x(t), t)dt + σd W (t) (7)

3 (6) (6) Nelson Newton Newton Nelson x(t) µ(x(t), t) x(t) t x(t) W (t) Nelson f(t) f(t + t) f(t) Df(t) lim t 0 t f(s) (s t) Wiener W (t) t t 0 f(t) D [7] D x(t) W (t) x(t) (8) Dx(t) = µ(x(t), t) (9) D DDx(t) Newton Nelson f(t) f(t t) D f(t) lim t 0 t f(s) (s t) (6) (10) x(t dt) = x(t) + µ (x(t), t)( dt) + σ(w (t dt) W (t)) (11) µ (x(t), t) µ(x(t), t) x(t) = x(t dt) + µ (x(t), t)dt + σ(w (t) W (t dt)) (1) (4) d f(t) f(t) f(t dt) (13) 3

4 (1) d x(t) = µ (x(t), t)dt + σd W (t) (14) d W (t) W (t) dw (t) x(t) D D x(t) = µ (x(t), t) (15) D D Nelson x(t) α(t) α(t) (DD + D D) x(t) (16) Nelson x(t) Newton m (DD + D D) V (x(t)) x(t) = (t) (17) V (x(t)) m Newton Nelson (9) (15) µ µ µ µ µ µ µ µ Fokker-Planck Markov (17) (9) (15) m (Dµ + D µ) (18) Dµ Dµ µ µ D Dµ µ Dµ (x(t + t), t + t) µ (x(t), t) (x(t), t) lim t 0 t f(s) (s t) x(t + t) = x(t) + x(t) µ (x(t + t), t + t) t Taylor µ (x(t)+ x(t), t+ t) = µ (x(t), t)+ µ µ t+ x(t)+ 1 µ x(t) +o( t) (0) (19) W (t) (1) ( x(t)) σ t Dµ (x(t), t) = µ + µ µ + σ 4 (19) µ (1)

5 D µ D µ(x(t), t) = µ µ + µ σ µ () (17) m { (µ + µ ) + µ µ µ + µ σ (µ µ } ) (3) µ µ (17) m u µ µ v µ + µ { v + v v u u σ } u V (x(t)) = (4) (5) (6) Fokker-Planck u v µ µ Nelson u v 4 Fokker-Planck Nelson u v u v u v Fokker- Planck ( ) 0 t 0 x 0 t 1 x 1 ρ(x 1, t 1 x 0, t 0 ) t 1 ρ(x, t x 0, t 0 )dx = 1 ; (7) Markov ρ(x, t x 0, t 0 ) = ρ(x, t x 1, t 1 )ρ(x 1, t 1 x 0, t 0 )dx 1 (t 0 < t 1 < t ) (8) ; Chapman-Kolmogorov 5

6 Fokker- Planck ρ(x, t) = ρ(x, t x 0, t 0 )ρ(x 0, t 0 )dx 0 (9) D x(t) f(x(t)) t f(x(t + t)) f(x(t + t)) t = f(y)ρ(y, t + t x, t)dy (30) f(x(t)) t = f(y)ρ(y, t x, t)dy (31) f(x(t)) ρ Df(x(t)) = lim t 0 f(y) ρ(y, t + t x, t) ρ(y, t x, t) dy (3) t ρ Markov ρ(x, t x 0, t 0 ) x Chapman-Kolmogorov (8) lim f(y) (ρ(y, t + t x, t) ρ(y, t x, t))ρ(x, t x 0, t 0 ) dxdy t 0 t = lim f(y) ρ(y, t + t x 0, t 0 ) ρ(y, t x 0, t 0 ) dy t 0 t ρ(y, t + t x 0, t 0 ) ρ(y, t x 0, t 0 ) = f(y) lim dy t 0 t = f(y) ρ(y, t x 0, t 0 ) dy = f(x) ρ(x, t x 0, t 0 ) dx (33) Df(x(t))ρ(x, t x 0, t 0 )dx (34) Df(x(t)) (1) t { µ f(x) + σ } f(x) ρ(x, t x 0, t 0 )dx (35) { f(x) µ + σ } ρ(x, t x 0, t 0 )dx (36) 6

7 f(x) ρ(x, t x 0, t 0 ) = { µ + σ } ρ(x, t x 0, t 0 ) (37) Fokker-Planck Kolmogorov ρ ρ(x, t x 0, t 0 ) = { µ σ } ρ(x, t x 0, t 0 ) (38) Kolmogorov (37) (38) µ,µ u,v ( (µ µ )ρ + σ ρ ) = 0 (39) t ρ x 0 0 u, v (4) (5) u = σ 1 ρ ρ = σ (37) (38) ln ρ ρ (40) ρ + (vρ) = 0 (41) (40) (41) ρ u,v (40) t ρ (41) (40) u u = v σ (uv) Fokker-Planck u v (4) 5 Schrödinger u, v (6) (4) Schrödinger Fokker-Planck u + κ v + (uv) = 0 (43) Nelson v u κ 1 (u v ) + 1 V m = 0 (44) 7

8 κ = σ (45) (43) + i (44) = 0 ; i (46) χ u + iv (47) χ i χ + χ κ + 1 χ 1 V m = 0 (48) χ Burgers Cole-Hopf χ = α ln φ (49) α φ { (α ln φ) i + κ (α ln φ) + 1 ln φ) ( (α ) V } = 0 (50) m (ln φ) (ln φ) (ln φ) = φ φ = φ φ = φ φ (φ φ ) { iα φ φ + καφ φ κα(φ φ ) + α (φ φ ) V } m = 0 (51) α = κ { κi φ φ + κ φ φ V } = 0 (5) m t κi φ φ + κ φ φ V + η(t) = 0 (53) m 8

9 η(t) Schrödinger η(t) κi φ ψ + η(t) = κi φ ψ (54) ψ ψ φe i t κ η(τ)dτ (55) ψ (53) κi ψ + κ ψ V m ψ = 0 (56) (45) κ σ iσ ψ + σ4 ψ V m ψ = 0 (57) σ = h m (58) Schrödinger i h ψ = h m ψ + V ψ (59) h π Schrödinger Schrödinger ψ 6 (57) (59) Schrödinger (57) Schrödinger ψ ρ(x, t) Fokker- Planck ψ(x, t) Schrödinger (55) ψ = φ (60) (49) x φ = e 1 α χdx φ φ φ = φ φ = e 1 σ (χ+χ )dx (61) 9

10 (47) (40) (60) χ + χ = u (6) ψ(x, t) = A(t)ρ(x, t) (63) A(t) t ρ(x, t) ψ(x, t) x t V (x) A(t) t ψ(x, t) = ρ(x, t) (64) ψ(x, t) (57) Schrödinger 7 Nelson Schrödinger µ µ Schrödinger Schrödinger 10

11 [9] Schrödinger Korteweg-de Vries (KdV ) ( ) 8 Ornstein-Uhlenbeck [6] 9 (58) σ = h m 11

12 10 Levy Schrödinger Schrödinger Schrödinger ρ(x(t)) 1

13 [1] F. Black and M. Scholes, Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81 (1973), [] E. Nelson, Derivation of the Schrödinger Equation from Newtonian Mechanics, Phys. Rev. 150(1966), [3] E. Nelson, Dynamical Theories of Brownian Motion (Princeton University Press, Princeton, 1967). [4] E. Nelson, Quantum Fluctuations (Princeton University Press, Princeton, 1985). [5] G. Parisi and Y.-S. Wu, Sci. Sin. 4 (1981) 483. [6] G. E. Uhlenbeck and L. S. Ornstein, On the theory of Brownian motion, Phys. Rev. 36 (1930), [7] p.161 [8] p.03 [9] G.L., Jr. [ ] 13

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