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3 1 (Black-Scholes) 1.1 (derivative securities) (and/or) (contingent claim) (underlying asset) 1 (contingent claim) 2

4 T S T S T K K (S T K) T max{s T K, 0} 1 1 S T K K S T T max{k S T, 0} 1 2 max{s T K, 0} K 0 K 1 1 S T 0 K 1 2 S T T T T T max{s T K, 0} max{k S T, 0} K S T K S T a S T 3

5 0 K K 2 K 1 S T a 0 K 1 K 2 S T S T S T S T T T S T T t < T T t t S t S T S T 1 7 S T S t S T S T S T S T S t K t 1 7 T Time 4

6 1.2 4 (1) (2) (3) (4) Topix t (0 t < ) S t S t ds t = µ S t dt + σ S t dw t (1-1) S t S T = S t e (µ 1 2 σ2 )(T t)+σ(w T W t ), 0 t < T (1-2) S t+dt = S t e (µ 1 2 σ2 )dt+σ W t (1-3) µ σ W t, (0 t < ) (1) W 0 0 (2) 0 t 1 < t 2 < W t2 W t1 0 t 2 t 1 (3) 0 t 1 t 2 t 3 t 4 < W t4 W t3 W t2 W t1 log(s T /S t ) (µ 1 2 σ2 )(T t) σ 2 (T t) dt 0 dw t W t+dt W t (dw t ) 2 = dt, dw t dt = dt dw t = 0, (dt) 2 = 0 5

7 ds t ds t S t+dt S t ) = S t (e (µ 1 2 σ2 )dt+σ dw t 1 x = 0 e x = 1 + x x2 + [ ( = S t µ 1 ) 2 σ2 dt + σ dw t + 1 {(µ 12 ) } ] 2 2 σ2 dt + σ dw t + = S t (µ dt + σ dw t ) t = 1, 2, log(s t+1 /S t ) t = 1, 2, log(s t+1 /S t ) log(s t+2 /S t+1 ) σ 1900 (Bachelier) S t = S 0 + X 1 + X X t, t = 1, 2, X 1, X 2, S t = S 0 e X 1+X 2 + +X t MIT r 1 S t t C(S t, t) T, K, σ, r 6

8 (1-1) dc C(S t+dt, t + dt) C(S t, t) = C t = C t = C dt + ds t C S t 2 St 2 (ds t ) C 2 t 2 (dt)2 + 2 C S t t (ds t) (dt) C dt + (ds t ) 2 2 C S 2 t ds t + 1 S t 2 ( C t + C µ S t C S t 2 St 2 σ 2 St 2 ) dt + C S t σ S t dw t (ds t ) 2 = (µ S t dt + σ S t dw t ) 2 = µ 2 S 2 t (dt) 2 + 2µ σ S 2 t (dt)(dw t ) + σ 2 S 2 t (dw t ) 2 = σ 2 S 2 t dt µ C 1 ( C C dt + C µ S t C S t 2 St 2 σ C 1 ( ) C σ S t C S t σ 2 S 2 t ) dc C = µ C dt + σ C dw t (1-4) S t K 1 F (S t, t) S t µ F σ F df F = µ F dt + σ F dw t (1-5) 2 C F Π C w C Π F w F Π w C + w F = 1 w C w F Π = w C Π + w F Π = w C Π C C + w F Π F w C Π C F C w F Π F F 7

9 t t + dt dπ Π = Π t+dt Π t Π t = 1 { wc Π dc + w F Π Π C F dc = w C C + w df F F } df (1-6) t t + dt C F w C Π C w F Π t S t dt F 1) (1-4) (1-5) (1-6) dπ Π = (w C µ C + w F µ F )dt + (w C σ C + w F σ F )dw t w C σ C + w F σ F = 0 w C w F w C + w F = 1 σ F wc = σ F σ C wf = σ (1-7) C σ F σ C wc w F Π dπ Π = (w C µ C + w F µ F )dt (1-8) dw t Π dt Π ((wc µ C + wf µ F ) Π dt) r r Π dt Π r Π dt r Π dt (wc µ C + wf µ F r) Π dt (no arbitrage) w C µ C + w F µ F = r (1-9) (1-7) (1-9) σ F µ C σ C µ F = r (σ F σ C ) µ C r σ C = µ F r σ F (1-10) 1) (self-financing portfolio) 8

10 C F (1-10) S t µ C r C λ µ C r σ C = λ (1-11) λ t S t C F (1-1) λ µ r σ σ C = λ (1-12) (1-11) (Sharpe) (σ C ) (r) λ (1-11) µ C σ C (1-4) ( ) 1 C C dt + C S t µ S t σ 2 St 2 r 1 C 2 C S 2 t ( C S t σ S t ) = λ C(S t, t) C t + (µ λ σ)s t C + 1 S t 2 σ2 St 2 2 C St 2 rc = 0 (1-12) µ λ σ = r C t + r S t C + 1 S t 2 σ2 St 2 2 C St 2 rc = 0 (1-13) S t S t C(S t, t) (1-13) ] C(S t, t) = E [g(x t ) e r(t t) X t = S t (1-14) C(S t, T ) = g(s T ) (1-15) (Feynman-Kac) X t dx t = r X t dt + σ X t dv t (1-16) V t W t 9

11 g( ) g( ) g(s T ) = max{s T K, 0} (1-17) (identify) C(S T, t) = E[max{X T K, 0} e r(t t) X t = S t ] (1-16) (1-1) X T = X t e (r 1 2 σ2 )(T t)+σ(v T V t ) X t = S t V T V t 0 (T t) Z 0 1 T t Z [ { C(S t, t) = E max S t e (r 1 2 σ2 )(T t)+σ( }] T t Z) r(t t) K, 0 e C(S t, t) = 1 e 1 2 z2 2π { max S t e (r 1 2 σ2 )(T t)+σ } T t z K, 0 f(z)dz e r(t t) (1-18) { A = z S t exp {(r 12 ) σ2 (T t) + σ } T t z } K > 0 { C(S t, t) = S t e (r 1 2 σ2 )(T t)+σ } T t z f(z)dz K f(z)dz e r(t t) (1-19) A A (1-19) 2 { A = z z > 1 σ T t log K e r(t t) + 1 } 2 σ T t f(z) z = 0 K ht σ T t S t f(z)dz N( ) Φ( ) N( ) K e r(t t) N(h t σ T t) 10

12 h t = 1 σ T t log S t Ke + 1 r(t t) 2 σ T t h t (1-19) 1 e σ T t z f(z) = e σ T t z 1 2π e 1 2 z2 = 1 e 1 2 (z2 2σ T t z) 2π { } 1 = e 1 2 (z σ T t) 2 e 1 2 σ2 (T t) 2π { } σ T t 1 z z = z σ T t 1 S t N(h t ) C(S t, t) = S t N(h t ) K e r(t t) N(h t σ T t) Markowitz K

13 0 0 K K 0 K S T F F F 0 S T 0 F S T 0 S T

14 1.3.2 S t K T S T S t K 1 13 K K 0 K S T 0 S T K S T K K 0 S T 0 K S T K S T (pay later option) t a t T t 13

15 (Thorp) (Kassouf) (convertible bond) 14

16 Π = C b S t, b = c S t (Stochastic Control) (Sequential Analysis) (Merton) (Samuelson) 15

17 2 1 (Vanilla Call, Vanilla Put) S t T T S T F t 1 S T F t 16

18 E[(X T F t ) exp( r(t t)) X t = S t ] F t F t = E[X T X t = S t ] 2 [ ( E X T exp ) ] T r(u)du X t t = S t F t = [ ( E exp )] T r(u)du t [ ( ν ν > t E (X T F t ) exp )] T ν r(u)du X t = S t F t F ν F t S t F t T 1, T 2,..., T n 6 R 1, R 2,..., R n r 1, r 2,..., r n 0 = E[(R 1 F t ) exp(r 1 (T 1 t)...)] + E[(R 2 F t ) exp(r 2 (T 2 t)...)].. + E[(R n F t ) exp(r n (T n t)...)] F t... t (1) 2 (2) 17

19 (3) (4) (5) 1 18

20 30 (6) 3 tree method S S t (S, t) S t (Barrier Option) (S t = a) S 0 S t 19

21 S 0 4 rebate S 0 S 0 2 max {S t K, 0} V (s, t) V (s, t) S t S t 1 V t σ2 S 2 2 V V + rs rv = 0 (2-1) S2 S V (S, T ) = max{s k, 0} (2-2) 1 V (0, t) = 0, S V (S, t) S S t = a V (0, t) = 0 V (a, t) = 0 (2-3) (2-1) V (S, t) = C(S, t) ( ) 2r S σ 2 1 ( ) a 2 C a S, t (2-4) 20

22 S 0 > a τ = inf {t S t a, 0 < t < T } (2-5) τ (stopping time) first exit time S t a S t τ 1 (2-1) V (S t, t) = E [ ] max {X T K, 0} I {τ>t } X t = S t (2-6) I { } { } 1 0 τ σb t + µ t log a = b log S t = x a > 0 } x a g(t : x, a) = { σ 2πt exp (x a µ t)2 3 2σ 2 t > 0 (2-7) t first hitting time (reflection principle) V (S t, t) = E [max {X T K, 0} X t = S t ] E [ max {X T K, 0} I {τ>t } X t = S t ] (2-8) 1 2 K > a E [ ] [ { } ] max {X T K, 0} I {τ>t } X t = S t = E max ae r(t t)+σ(w T W u ) K, 0 τ = u g(u : x, b)du = T t x = log S t b = log a (S t > a ) C(a, T u : K, r, σ) g(u : x, b)du (2-8) 2 = + 1 S t ds 1 = S t (µ D) dt + σ dw t V t + (µ D 0) S V S σ2 S 2 rv = 0 21

23 V (S, T ) = max {S t K, 0} V (0, t) = 0 V (S, t) max {S t K, 0} D S t V S = (1) 1 t [0, t] S u, u [0, t] S(u)du S t 1 t t S(u)du t 0 I t = t 0 S(u)du S I t 3 V (S, I, t) di = S(t)dt ( 1 dv = 2 σ2 S 2 2 V S 2 + µ S V S + V t + S V ) dt + S V I S dw t 1 V t + S V I + rs V S σ2 S 2 2 V S 2 rv = 0 { } V (S T, I T, T ) = min S T 1 T T 0 0 S(u)du, 0 { max S T 1 T T 0 R t = 1 t S(u)du S t 0 S(u)du, 0 } { = S T max 1 R } T T, 0 22

24 R S R t ds t = µ S t dt + σ S t dw t dr t = (1 + R(σ 2 µ))dt σr dw t dr t S t S I t V (S, I, t) V (S T, I T, t) = S t H(R t, t) S t R t V = S H H t + (1 rr) H R σ2 R 2 2 H R 2 = 0 { H(R T, T ) = max 1 R } T T, 0 H(, t) = 0 R 0 R = 0 H t + H R confluent hypergeometric function (2) (α-percentile options) S 1, S 2,, S n S(1) S(2) S(n) [nα] S ([nα]) 2 1 [0, t] m(α, t) S u {u S u m(α, t)} α t m(α, t) m(α T ) 2 α < β m(βcdott ) m(α T ) 2 2 S 1 S 2 S n S ([nα]) 0 t 1 t 2 t n

25 m (α,t) 0 t (risk diversification)

26 75% 25% 2.6 [19] [20] [17] ( ksuzuki/jafee.html) [15] [18] [16] 3 25

27 (Martingale) 2) [0, T ] S t r T max {S T K, 0} [0, T ] r S 1 3) B t = 1 e rt B t 1 t t S B φ t ψ t V t V t = φ t S t + ψ t B t (φ t, ψ t ), t [0, T ] 1 (φ, ψ) V V (φ, ψ) dv t, ds t, db t dv t = φ t ds t + ψ t db t (self-financing ) V t = φ t S t + ψ t B t t t + t V t = φ t S t + ψ t B t V t+ t = φ t+ t S t+ t + ψ t+ t B t+ t 2) Harrison and Pliska [22] [23] Harrison and Kreps [21] 3) 26

28 V t = V t+ t V t = φ t S t + ψ t B t + φ t S t+ t + ψ t B t+ t 3 4 φ t S t+ t + ψ t B t+ t = 0 t φ t t + t S t+ t ψ t B t+ t φ t S t+ t + ψ t B t+ t S t+ t φ t+ t B t+ t ψ t+ t φ t 1 ψ t 1 t [0, T ] V T = S T + B T V t V (φ, ψ) T V T = φ T S T + ψ T B T ( max {S T K, 0}) V V (φ, ψ) Cameron-Martin- Girsanov ) (1) (Ω, F, P ) [0, T ] (path) 1 ω ω Ω 4) [25] [24] 3 27

29 F Ω P Q P (Ω, F, P) (2) F t [0, T ] t t F t F t F (F t F ) u < t F u F t F t u t W u W t F t s > t s F s u < t u F u (3) Ω X F F F t t X X t u (u < t) u X = 0.5 X t u t u 0.5 t t X u X E P = [X F u ] F u X P E P P F u s < u E P [E P [X F u ] F s ] F u F s F E P [E P [X F u ] F s ] = E P [X F s ], s < u X Y a b E P [ax + by F t ] = a E P [X F t ] + b E P [Y F t ] 28

30 (4) t [0, T ] X t, t [0, T ] [0, T ] X M M 0 u t T E P [M t F u ] = M u M t, t [0, T ] (Ω, F, P ) {F t : t [0, T ]} X F F t t [0, T ] M t M t = E P [X F t ] P M t u < t E P [M t F u ] = E P [E[X F t ] F u ] = E P [X F u ] M u (Ω, F) Q P E Q [X F t ] 1 Q (5) (Ω, F, P ) {F t : t [0, T ]} 2 M t N t N t M t N t φ t N t = N 0 + t 0 φ u dm u φ t φ t previsible predictable t φ t M N 1 (Complete market) (6) P X X x P {X < x} (= P {ω X(ω) < x}) X p( ) p(x) = d P {X < x} dx 29

31 X P µ σ 2 p(x) = d P {X < x} dx = 1 e (x µ)2 2σ 2 2πσ P ( X) P Q Q X µ ν σ 2 P Q (Ω, F) {F t : t [0, T ]} P Q Q X q( ) q(x) = d ( Q{X < x} = d ) Q{ω X(ω) < x} dx dx = 1 e 1 (x (µ ν)) 2 2 σ 2 2πσ d q( ) P Q p( ) q( ) p( ) dq dp = dxq{ω X(ω) < x} d dxp {ω X(ω) < x} ω P Q dq dp = q(x) p(x), x (, ) dq x Q P dp (Radon-Nikodym) X P Q X (Ω, F) {W t : t [0, T ]} W t ω Ω dq path {W t : t [0, T ]} dp P Q P W t Q γt t γt t 1 e 1 (x γt) 2 2 t 2π t ( = d ) dx Q{W t < x} P W t t 1 2π t e 1 (x γt) 2 2 t 1 2π t e 1 x 2 2 t = e 1 2γtx+γ 2 t 2 2 t = e γx 1 2 γ2 t t W t P Q {W t : t [0, T ]} P Q W t : t [0, T ] 0 < t 1 < t 2 < < t n x i = W ti (ω) i = 1, 2,, n P, Q 30

32 W ti+1 W ti i = 1, 2,, n q (W t1 (ω) = x 1,, W tn (ω) = x n ) p (W t1 (ω) = x 1,, W tn (ω) = x n ) = q (W n 1 t 1 (ω) = x 1 ) i=1 q ( ) W ti+1 (ω) W ti (ω) = x ti+1 x ti p (W t1 (ω) = x 1 ) n 1 i=1 p ( ) W ti+1 (ω) W ti (ω) = x ti+1 x ti = 1 2π t1 e 1 (x t1 γt 1 ) 2 2 t 1 1 2π t1 e 1 x 2 t1 2 t 1 n 1 = e γx t γ2 t1 i=1 n 1 i=1 e γ x i 1 2 γ2 t i 1 2π ti+1 t e 1 (x ti+1 x ti γ(t i+1 t i )) 2 2 t i+1 t i i n 1 i=1 = e γxt γ2 t1 e γ P n 1 i=1 xi 1 2 γ2 P n 1 i=1 ti e γw T (ω) 1 2 γ2 T (n ) 1 2π ti+1 e 1 (x ti+1 x ti ) 2 2 t i+1 t i t i x i = x ti+1 x ti t i = t i+1 t i t i t i t i+1 x i t i W [0, T ] t 1 < t 2 < < t n t 1 t i n dq ω W t (ω) (ω) dp dq dp (ω) = eγw T 1 2 γ 2 T γ γ t γ dq previsible predictable dp (ω) F T W t Q P P Q W t W t P Q P Q γ W t W t W t = W t + γt, t [0, T ] W t Q { } Q Wt < x = Q {W t + γt < x} = Q {W t < x γt} (P {W t < x}) = 31

33 Q W t γt Q Cameron-Martin-Girsanov 5) (7) C M G R T 0 γ2 t dt ] W t P γ t F t previsible E P [e 1 2 < Q 3 (i) Q P (ii) dq dp = R T e 0 γ tdw t 1 R T 2 0 (iii) Wt = W t + T 0 γ2 t dt γ u du Q Q W t t γ u du t [0, T ] (1) S t r 0 1 r t e rt B t (B t = e rt ) 1 S t B t S t T X X = max {S T K, 0} K f(s T ) f( ) (Ω, F, P ) W t P t [0, T ] W t F t S t ( ds t = S t µ + 1 ) 2 σ2 dt + S t σ dw t S t = S 0 e µt+σ W t S t (2) S t Z t t [0, T ] Z t = B 1 S t B 1 X Z t S t t Z t T 5) C M G 32

34 Z t = Bt 1 S t = B 1 t S 0 e µt+σw t = e rt S 0 e µt+σwt = S 0 e (µ r)t+σwt dz t dz t = Z t ((µ t)dt + σ dw t + 1 ) 2 σ2 dt (( = Z t µ t + 1 ) ) 2 σ2 dt + σ dw t = Z t σ d (W t + µ r + 1 ) 2 σ2 t σ γ = µ r σ2 σ dz t = Z t σ d (W t + γt) dz t (3) dt E P [ dzt Z t ] = S t (µ r + 12 σ2 ) dt [ ] dst E P = (µ + 12 ) S σ2 dt t r dt S t (3) P Q C M G dq dp = eγw T 1 2 γ 2T γ = µ r σ2 σ Q Q W t = W t + γt Z t (2) W t Z t dz t = Z t σ d W t Z t = S 0 e 1 2 σ2 +σ W t [ ] dzt E Q = 0 Z t 33

35 P (µ r + 12 ) σ2 Q r = µ σ2 0 u < t < T E Q [Z t F u ] = E Q [ S 0 e 1 2 σ2 u+σ Wu e 1 2 σ2 (t u)+σ( W t W u) F u ] = Z u E Q [ e 1 2 σ2 (t u)+σ( W t W u) F u ] = Z u 1 Z t Q Q dq C M G dp Q P W t (4) (φ t, ψ t ) 0 t T X = max {S T K, 0} X = f(s t ) [ E t = E Q B 1 T X F ] t E t 3.2 E t, t [0, T ] Q u t E Q [E t F u ] = E u E T = B 1 T X Q Z t E t Z t E t Q 3.2 (previsible, predictable) φ t, t [0, T ] de t = φ t dz t φ t S t B t Z t 1 t (φ t, ψ t ) (φ t, ψ t ) ψ t = E t φ t Z t, t [0, T ] φ t ψ t V V t V (φ t, ψ t ) = (φ t S t + ψ t B t ) E t V t E t = φ t Z t + ψ t 1 dv t = φ t ds t + ψ t db t V t V (φ t, ψ t ) Z t ψ t V t V t = φ t S t + ψ t B t = φ t (B t Z t ) + ψ t B t = φ t B t Z t + (E t φ t Z t )B t = B t E t 34

36 dv t = B t de t + E t db t, (db t de t = 0) = B t φ t dz t + E t db t, (de t = φ t dz t ) = B t φ t dz t + (φ t Z t + ψ t ) db t, (E t = φ t Z t + ψ t 1) = φ t (B t dz t + Z t db t ) + ψ t db t = φ t d(b t Z t ) + ψ t db t, (d(b t Z t ) = B t dz t + Z t db t ) = φ t ds t + ψ t db t, (Z t = B 1 t S t ) V t V T = B T E T = B T (B 1 T X) = X V t V (φ t, ψ t ) T X V t V (φ t, ψ t ) X t [0, T ] replication portfolio V t [0, T ] T X V t V 0 X T r 1 V 0 E t Z t Q t [0, T ] E t = E 0 + t 0 φ u dz u [ E 0 = E Q B 1 T F ] 0 E 0 B 0 = 1 E 0 V 0 = φ 0 S 0 + ψ 0 B 0 = φ 0 S 0 + ψ 0 = φ 0 Z 0 + ψ 0 = E 0 dv t = φ t ds t + ψ t db t V T V 0 = T 0 φ u ds u + 35 T 0 ψ u db u

37 V T (= X) V 0 u [0, T ] (S u, B u ) (φ u, ψ u, ) (φ u, ψ u, ) u [0, T ] (Hedge) X φ t φ t V t = φ t S t + ψ t B t t V t V t [ E t (= B t V t ) = E Q B 1 T X F ] t S t B t S t B t 1 e r(t t) = e rt B t (linear homogeneity) T t = 0 E 0 [ E 0 = E Q B 1 T X F ] 0 [ = E Q e rt ] max {S T K, 0} F 0 { }] = e rt E Q [max S 0 e (r 1 2 σ2 )T +σ W T K, 0 { }] = e rt E Q [max ST K, 0 S t d S t = S t (rdt + σd W t ) W t Q 10 (1-18 ) (T t) S t S 0 φ t 3.4 T (stopping time) path,ω IBM 36

38 [1] P.H.Cootner(ed.) The random characters of stock market prices MIT Press (1964) [2] E.D.Thorp and S.T.Kassouf Beat the Market Random House New York (1967) [3] (1988) [4] P.Wilmott,S.Howison and J.Dewynne The Mathematics of Financial Derivatives Cambridge University Press (1995) [5] F.Black and M.Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, Vol.81, pp (1973) [6] R.C.Merton, Theory of rational option pricing, Bell Journal of Economics and Management Sciences, vol.4, pp (1973) [7],, 8, pp (1995) [8] Peter G.Zhang, Exotic Options: A Guide to Second Generation Options, World Scientific (1997) [9] Israel Nelkin(ed.), The Handbook of Exotic Options: Instruments, Analysis and Applications, IRWIN Professional Publishing (1996) [10] Vineer Bhansali, Pricing and Managing Exotic and Hybrid Options, IRWIN Library of Investment & Finance, McGraw Hill (1998) [11] Eric Briys et al, Options, Futures and Exotic Derivatives: Theory, Application and Practice, John Wiley & Sons (1998) [12] William Shaw, Modelling Finaicial Derivatives with Mathematica,Cambridge University Press (1998) [13] D.R.Cox and H.D.Miller, The Theory of Stochastic Processes, Chapman and Hall (1965) [14] Paul Wilmott et al, Option Pricing: Mathematical Models and Computations, Oxford Financial Press (1993) [15], A Note on look-back option based on order statistics., Hitotsubashi J. Commerce & Management 27, pp (1992) [16], On the price of α-percentile options., Working Paper 24, Faculty of Commerce, Hitotsubashi University (1997) [17], A Note on the Terminal Date Security Prices in a Continuous Time Trading Model with Dividends.,J.of Mathematical Economics 20, pp (1991) [18], Some formula for a new type of path-dependent option., Ann.Appl.Probability, 5, pp (1995) [19], Pricing Options with curved boundaries., Mathematical Finance 2, pp (1992) 37

39 [20], Limit theorems on option replication with transaction costs., Ann.Appl. Probability, 5, pp (1995) [21] J.M.Harrison and D.M.Kreps, Martingales and arbitrage in multi-period securities markets, Journal of Economic Theory, Vol.20, pp (1979) [22] J.M.Harrison and S.R.Pliska, Martingale and stochastic integrals in the theory of continuous trading, Stochastic Processes and its Applications, Vol.11, pp (1981) [23] J.M.Harrison and S.R.Pliska, A Stochastic calculus model of continuous trading: Complete markets, Stochastic Processes and its Applications, Vol.15, pp (1983) [24] M.Baxter and A.Rennie, Financial Calculus; An introduction to derivative pricing, Cambridge University Press (1996) [25] R.J.Elliott and P.E.Kopp, Mathematics of Financial Markets, Springer-Verlag (1999) 38

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untitled 1 25/5/3-6/3 1 1 1.1.................................. 1 1.2.................................. 4 2 5 2.1.............................. 5 2.2.............................. 6 3 Black Scholes 7 3.1 BS............................