mf.dvi

21 9 29 1 2 3....................................... 3 :......................... 3....................................... 4................................ 4..................................... 5................................ 5 3.................................... 8..................................... 8 2 =CRR............................ 1 CRR........................................ 12.......................................... 13 2 14......................................... 14.......................................... 16......................................... 18..................................... 21 (EMM)........................... 23......................................... 25......................................... 25................................ 26................................ 26 29 9 14 ( ) 17 ( ) e-mail: ichiro@math.kyoto-u.ac.jp, URL: http://www.math.kyoto-u.ac.jp/~ichiro/ 1

3 Black-Scholes 27....................................... 27 Y k........................................ 28 Black-Scholes.................................. 3 4 32......................................... 32................................ 33...................................... 34 CRR.................................. 36 5 38................................. 38...................................... 39 Doob........................................ 39......................................... 4........................... 41 6 42...................................... 42......................................... 43......................................... 45....................................... 45.................................... 46.................................... 47 7 Black-Scholes 47.......................................... 47 Black-Scholes.................................. 48...................................... 48.................................. 49......................................... 5 2

1. 2 2 t =,1,..., T S t S t.5 : T K : T K 1 S T S T K S T K S T >K K K S T S T K (S T K) + : H =(S T K) + : H =(K S T ) + π(h) (derivative secureity) (contingent claim) : 1 {S t }, t =,1 S =1 { 2, p S 1 = 7.5, 1 p H =(S 1 K) + K =15 E[(S 1 K) + ] = (2 15) p + (1 p) =5p p =.5 2.5 1 3

viable (no arbitrage) viable (arbitrage) (no arbitrage) no free lunch (S T K) + (K S 1 ) + (1 + ρ) t t C t, P t ( ) C t P t = S t (1 + ρ) (T t) K. (1.1) t 1. 2. 1 (1 + ρ) (T t) K T 1. C T P T =(S T K) + (K S T ) + 2. S T (1 + ρ) (T t) (1 + ρ) (T t) K S T K (1.1) t 4

S t B t B t (riskless security) (risky security) B t B t B t B t =(1+ρ) t ρ B t η t S t θ t (η t,θ t ) V t = η t B t + θ t S t η t B t + θ t S t = η t+1 B t + θ t+1 S t (self financing) V T H =(S T K) + (duplication) V T = H (η t,θ t ) V B t =1 T =1 (η 1,θ 1 ) (η, θ) B t =1. V = η + θs V 1 = η + θs 1 H = V 1 (η, θ) S 1 S 1 (ω +) =2, S 1 (ω ) =7.5, H(ω +) =5, H(ω ) = H(ω) =η + θs 1 (ω) 5

{ 5=η +2θ, =η +7.5θ S S 1 H 1 2 5 5 = η +2θ 7.5 = η +7.5θ η = 3, θ =.4 V = η + θs V = 3+.4 1 = 1 (writer) t = 1 1 3 3.4.4 1 4 t =1 S 1 =2 15 15.6.6 2 12 3 3 1 S 1 =7.5.4.4 7.5 3 3 3 π(h) > 1 1 π(h) 1 π(h) > 1 (buyer) 6

t =.4 =.4 ().4 1 4 3 3 1 1 t =1 S 1 =2 15 15 15 1 15 1.4 =.6.6 2 12 12 12 S 1 =7.5 3 3.4.4 7.5 3 π(h) < 1 1 π(h) π(h) < 1 { 2, p S 1 = 5, 1 p { 5=η +2θ, =η +5θ V = η + θs η = 5 3, θ = 1 3 V = 5 3 + 1 3 1 = 5 3. 7

3 S =1 2, p 1 S 1 = 1, p 2 7.5, p 3 K =15 S 1 K + 2 5=η +2θ, =η +1θ, =η +7.5θ. B t =(1+ρ) t β =(1+ρ) 1 1 S, S 1 S 1 S 1 (ω +), S 1 (ω ) P (ω +) =p, P (ω ) =1 p p H H V 1 = H H = β 1 η + θs 1. V = η + θs V 1 = β 1 η + θs 1. H(ω +) =β 1 η + θs 1 (ω +) (1) H(ω ) =β 1 η + θs 1 (ω ) (2) θ = H(ω +) H(ω ) S 1 (ω +) S 1 (ω ) 8

(1) S 1 (ω ) (2) S 1 (ω +) V S 1 (ω )H(ω +) S 1 (ω +)H(ω ) =β 1 η(s 1 (ω ) S 1 (ω +)) η = β(s 1(ω +)H(ω ) S 1 (ω )H(ω +)) S 1 (ω +) S 1 (ω ) V = η + θs = β(s 1(ω +)H(ω ) S 1 (ω )H(ω +)) S 1 (ω +) S 1 (ω ) = S βs 1 (ω ) = β + H(ω +) H(ω ) S 1 (ω +) S 1 (ω ) S S 1 (ω +) S 1 (ω ) H(ω +)+ βs 1(ω +) S H(ω ) S 1 (ω +) S 1 (ω ) { β 1 S S 1 (ω ) S 1 (ω +) S 1 (ω ) H(ω +)+ S 1(ω +) β 1 S H(ω ) S 1 (ω +) S 1 (ω ) = β(qh(ω +)+(1 q)h(ω )). } q = β 1 S S 1 (ω ) S 1 (ω +) S 1 (ω ) ( H ) Q (1.2) Q({ω +}) =q, Q({ω }) =1 q π(h) =V = E Q [βh]. E Q [βs 1 ] E Q [βs 1 ]=βs 1 (ω +)q + βs 1 (ω )(1 q) = βs 1 (ω +) β 1 S S 1 (ω ) S 1 (ω +) S 1 (ω ) + βs 1(ω ) S 1(ω +) β 1 S S 1 (ω +) S 1 (ω ) = S 1(ω +)S βs 1 (ω +)S 1 (ω )+ βs 1 (ω )S 1 (ω +) S 1 (ω )S S 1 (ω +) S 1 (ω ) = (S 1(ω +) S 1 (ω ))S S 1 (ω +) S 1 (ω ) = S S, βs 1 ( ) 9

2 =CRR S, S 1, S 2,..., S T { (1 + b)st 1 p S t = (1 + a)s t 1 1 p (1 + b) 3 S (1 + b) 2 S (1 + b)s (1 + a)(1 + b) 2 S p S (1 + a)(1 + b)s 1 p (1 + a)s (1 + a) 2 (1 + b)s (1 + a) 2 S (1 + a) 3 S 1: CRR (1.2) β =(1+ρ) 1, q = β 1 S (1 + a)s = ρ a (1 + b)s (1 + a)s b a 1

2 V = E Q [βh]=β(qh b +(1 q)h a ) H bb V b q V H ba 1 q H ab V a H aa 2: 2 CRR H T 2 V T 2 T 1 V a, V b V b = β(qh bb +(1 q)h ba ) (1.3) 11

V a = β(qh ab +(1 q)h aa ). (1.4) V b, V a T 2 V = β(qv b +(1 q)v a ) (1.5) (1.5) (1.3), (1.4) V = β 2 (q 2 H bb + q(1 q)h ba +(1 q)qh ab +(1 q) 2 H aa ) (1.6) S T 2 = S H =(S T K) + CRR V = β 2 {q 2 ((1 + b) 2 S K) + +2q(1 q)((1 + a)(1 + b)s K) + +(1 q) 2 ((1 + a) 2 S K) + } (1.7) T ( ) T V = β T q t (1 q) T t ((1 + b) t (1 + a) T t S K) t + t= T ( ) T = S (1 + ρ) T q t (1 q) T t (1 + b) t (1 + a) T t K(1 + ρ) T t = S T t=a t=a ( ) ( ) t ( T 1+b 1+a q t (1 q) T t t 1+ρ 1+ρ ) T t K(1 + ρ) T T A = min{k; S (1 + b) k (1 + a) T k >K}. q 1+b +(1 q)1+a 1+ρ q = ρ a b a, q = q 1+b 1+ρ 1+ρ = q b a 1+ρ + 1+a 1+ρ = ρ a b a b a 1+ρ + 1+a 1+ρ = ρ a b a 12 t=a ρ a 1+ρ + 1+a 1+ρ =1 T ( ) T q t (1 q) T t t t=a ( T t ) q t (1 q) T t

V = S T t=a q (, 1), 1 q =(1 q) 1+a 1+ρ ( ) T T (q ) t (1 q ) T t K(1 + ρ) T t t=a ( ) T q t (1 q) T t t = S Ψ(A; T,q ) K(1 + ρ) T Ψ(A; T,q) (1.8) Ψ(m; n, p) = n j=m ( ) n p j (1 p) n j. j (1.8) Cox-Ross-Rubinstein (CRR) t T t ( ) T t V t = β T t q s (1 q) T t s ((1 + b) s (1 + a) T t s S t K) s + s= = S t Ψ(A t ; T t, q ) K(1 + ρ) (T t) Ψ(A t ; T t, q) (1.9) A t = min{k; S t (1 + b) k (1 + a) T t k >K}. [t 1,t] (η t,θ t ) V t = η t (1 + ρ) t + θ t S t V t S t S t S t 1 [t 1,t] 2 S t =(1+b)S t 1 S t =(1+a)S t 1 Vt b, V t a V b t = η t (1 + ρ) t + θ t (1 + b)s t 1, V a t = η t (1 + ρ) t + θ t (1 + a)s t 1 θ t = V b t Vt a, η t = (b a)s t 1 (1 + b)v a t (1 + a)v b t (1 + ρ) t (b a) (1.1) 13

Vt b, Vt a V b t V a t T t ( ) T t = β T t q s (1 q) T t s ((1 + b) s (1 + a) T t s S t 1 (1 + b) K) s + s= T t ( ) T t = β T t q s (1 q) T t s ((1 + b) s (1 + a) T t s S t 1 (1 + a) K) s + s= θ t, η t S t 1 S T K S t K ( ) (hedge) 2. (Ω, F,P) T = {, 1,...,T} (S t ) t σ-field F t d- S 1,S 2,...,S d S S =(S,S 1,S 2,...,S d ) S S 1,S 2,...,S d (riskless security) (risky security) β t = 1 (discount factor) S St t =(1+ρ)t β t =(1+ρ) t S deterministic β t t 1 β t t S, S 1,...,S d θ t =(θ t,θ1 t,...,θd t ) (trading strategy) (value process) V (θ) =θ 1 S (2.1) d V t (θ) =θ t S t = θt i Si t, t 1 (2.2) 14 i=

t 1 θ t (t 1,t] t θ t S t θ t+1 t θ t+1 θ t+1 t θ t F t 1 σ-field (predictable) t 1 θ t S t 1 S t t θ t S t θ t S t θ t S t 1 ( ) (X t ) ΔX t = X t X t 1 θ t ΔS t (2.3) (gain process) G : G (θ) =, (2.4) G t (θ) =θ 1 ΔS 1 + θ 2 ΔS 2 + + θ t ΔS t. (2.5) 2.1. θ (self-financing) (): θ t S t = θ t+1 S t, 1 t T 1. (2.6) t θ t S t θ t+1 S t self-financing θ t+1 S t θ t S t =Δθ t+1 S t Δθ t S t 1 =, 2 t T. (2.7) self-financing 2.2. : Δθ t S t 1 =, 2 t T, (2.8) ΔV t (θ) =θ t ΔS t, t =1,...,T. (2.9) V t (θ) =V (θ)+g t (θ), t =, 1,...,T. (2.1) (2.8) (2.9) (2.8) t =2,...,T θ t S t 1 = θ t 1 S t 1 ΔV t (θ) =θ t S t θ t 1 S t 1 = θ t S t θ t S t 1 = θ t ΔS t 15

t =1 ΔV 1 (θ) =V 1 (θ) V (θ) =θ 1 S 1 θ 1 S = θ 1 ΔS 1 (2.9) (2.9) (2.1) (2.9) V t (θ) =V (θ)+ = V (θ)+ = V (θ)+ t ΔV s (θ) s=1 t θ s ΔS s s=1 t ΔG s (θ) s=1 = V (θ)+g t (θ) G (θ) = V (θ)+g t (θ). 3 ΔG s (θ) =θ s ΔS s (2.1) (2.1) (2.8) (2.1) t 1 t 2 t 2 (2.8) ΔV t (θ) =ΔG t (θ) =θ t ΔS t. ΔV t (θ) =θ t S t θ t 1 S t 1. θ t S t θ t 1 S t 1 = θ t ΔS t = θ t S t θ t S t 1. (θ t θ t 1 ) S t 1 = (Z t ) (numéraire) S t Z t S t self-financing Δθ t S t 1 = Δθ t (Z t 1 S t 1 )= (Z t ) (Z t ) 16

Z t =(S t ) 1 = β t (S 1 t ) 1 S Z t = β t S t := β t S t (S t ) (discounted security price process) (S t ) V, G : V (θ) =θ 1 S (2.11) V t (θ) =θ t S t, t 1 (2.12) G (θ) =, (2.13) G t (θ) =θ 1 ΔS 1 + θ 2 ΔS 2 + + θ t ΔS t. (2.14) 2.3 S Δθ t S t 1 =, 2 t T, (2.15) ΔV t (θ) =θ t ΔS t, t =1,...,T, (2.16) V t (θ) =V (θ)+g t (θ), t =, 1,...,T. (2.17) (2.15) self-financing self-financing V V t (θ) =θ t S t = θ t β t S t = β t θ t S t = β t V t (θ) G t (θ) G t (θ) θ self-financing (2.17) G t (θ) =V t (θ) V (θ) = β t V t (θ) β V (θ) = β t (V (θ)+g t (θ)) β V (θ) = β t G t (θ)+(β t β )V (θ) V (θ) = G t (θ) =β t G t (θ) S =1 V (θ) =V (θ) G(θ) θ t θ t V (θ) θ i, i =1,...,d 2.3. V predictable process θ 1,...,θ d predictable process θ θ =(θ,θ 1,...,θ d ) 17

self-financing θ : t 1 θt = V + (θuδs 1 1 u + + θuδs d d u) (θt 1 S 1 t 1 + + θt d S d t 1). (2.18) u=1 θ =(θ,θ 1,...,θ d ) self-financing V t (θ) =θt + θt 1 S 1 t + + θt d S d t = V + G t t = V + (θu 1 ΔS1 u + + θd u ΔSd u ). u=1 θ t = V + t (θu 1 ΔS1 u + + θd u ΔSd u ) θ1 t S1 t + + θd t Sd t u=1 t 1 = V + (θu 1 ΔS1 u + + θd u ΔSd u )+(θ1 t ΔS1 t + + θd t ΔSd t ) θ1 t S1 t + + θd t Sd t u=1 t 1 = V + (θu 1 ΔS1 u + + θd u ΔSd u ) θ1 t S1 t 1 + + θd t Sd t 1 u=1 (2.18) (2.18) θ t predictable V t(θ) =V +G t (θ) self-financing self-financing strategy Θ self-financing strategy θ (admissible) V t (θ), t T self-financing strategy Θ a 2.4. (admissible strategy) (arbitrage opportunity) V (θ) =, V t (θ) t T, E[V T (θ)] >. E[V T (θ)] > P (V T (θ) > ) > 2.5. (viable θ Θ a V (θ) = V T (θ) = 18

2.4 V t (θ) t T V (θ) =, V T (θ), E[V T (θ)] > 2.6. θ t V t (θ) t<t A F t P (A) > θ t S t < on A, θ u S u for u>t φ A c φ u = A φ u (ω) =, u t φ u(ω) =θ u(ω) θ t S t S t (ω), φi u(ω) =θ u(ω), i =1,...,d, u>t φ predictable self-financing A c V u (φ) = A Δφ t+1 S t = (Δφ u Δθ u u = t +1) A Δφ t+1 = φ t+1 = θt+1 θ t S t, St Δφ i t+1 = θt+1, i i =1,...,d. Δφ t+1 S t =Δφ t+1s t + d i=1 Δφ i t+1s i t =(θt+1 θ t S t d )S St t + i=1 d = θt+1s t θ t S t + = θ t+1 S t θ t S t = θ self-financing i=1 θ i t+1s i t θ i t+1s i t 19

V u (φ) P (V T (φ) > ) > A c V u (φ) = A u t V u (φ) = u >t V u (φ) =φ u S u = θ u S u (θ t S t )S u S t = θ u S u (θ t S t ) S u. St + d i=1 θ i u Si u θ u S u, u>t (θ t S t ) <, S V u (φ) A V T (φ) > θ t S t < H 1 H F T H θ Θ a V T (θ) =H (2.19) (duplicated) viable V t (θ) θ, θ Θ a V T (θ) =V T (θ )=H V t (θ) =V t (θ ) t 2.7. viable market H θ, φ V T (θ) =V T (φ) =H V (θ) V (φ) t<t V u (θ) =V u (φ), V t (θ) V t (φ) u < t t A = {V t (θ) >V t (φ)} P (A) > X = V t (θ) V t (φ) F t ψ A A c ψ u = θ u φ u, u t ψu = β t X, ψu i =,i=1,...,d, u>t. ψ u = θ u φ u, u T. ψ predictable. self-financing u <t θ, φ self-financing u >t A c self-financing u >t A ψ u+1 = ψ u 2

u = t ψ t S t = V t (θ) V t (φ). ψ t+1 S t =1 A c(θ t+1 φ t+1 ) S t +1 A β t XSt =1 A c(θ t+1 φ t+1 ) S t +1 A (St ) 1 (V t (θ) V t (φ))st =1 A c(θ t+1 φ t+1 ) S t +1 A (V t (θ) V t (φ)) = V t (θ) V t (φ) =ψ t S t. ψ self-financing V (ψ) = V T (ψ) =1 A β t XS T A viable σ-field G X L 1 A G E[X1 A ]=E[Y 1 A ] G Y X G E[X G ] G disjoint A 1,...,A K j A j =Ω G A 1,...,A K E[X G ]= j 1 P (A j ) E[X1 A j ]1 Aj 1. E[αX + βy G ]=αe[x G ]+βe[y G ] 2. G 1 G 2 E[E[X G 2 ] G 1 ]=E[X G 1 ] 3. E[E[X G ]] = E[X] 4. X G - E[XY G ]=XE[Y G ]. 5. X G E[X G ]=E[X]. σ-fields {F t } t T (M t ) (F t )- adapted E[M t+1 F t ]=M t, t =1, 2,...,T 1 (2.2) 21

E[ΔM t+1 F t ]=, t =1, 2,...,T 1 E[ΔM t+1 ]= E[M t+1 ]=E[M t ] (2.2) E[M t+1 F t ] M t, t =1, 2,...,T 1 E[M t+1 F t ] M t, t =1, 2,...,T 1 2.8. M =(M t ) predictable process φ =(φ t ) X t = φ 1 ΔM 1 + φ 2 ΔM 2 + + φ t ΔM t (2.21) X M φ (martingale transform) X = X φ M φ =(φ t ) predictable φ t+1 ΔM t+1 E[ΔX t+1 F t ]=E[φ t+1 ΔM t+1 F t ]=φ t+1 E[ΔM t+1 F t ]= X 2.9. (F t )-adapted (M t ) predictable process φ E[(φ M) t ]=E[ t φ u ΔM u ] = (2.22) M X = φ M X = E[(φ M) t ]=E[X ]= (2.22) predictable process φ A F t φ t+1 =1 A u φ u = u=1 =E[(φ M) T ]=E[1 A ΔM t+1 ]. A E[ΔM t+1 F t ]= M 22

(EMM) discounted (S) (S) Q E Q [ΔS i t F t 1] =, i =1,...,d V t (θ) =V (θ)+g t (θ) =θ 1 S + t θ u ΔS u = θ1 S + ( θ1 i Si + i=1 u=1 t ) θu i ΔSi u (V t (θ)) θ V (θ) = V T (θ) (V t (θ)) Q E[V T (θ)] = E[V (θ)] = V T (θ) = Q-a.e. Q P (i.e., P (A) = Q(A) =) V T (θ) = P -a.e. 2.1. P Q (S) viable 1 2.11. (S) P (equivalent martingale measure = EMM) viable V t (θ) Q 2.12. Q H θ H = V T (θ) β T H Q- V t (θ) u=1 V t (θ) =β 1 t E Q [β T H F t ] (2.23) 23

H = V T (θ) V = V (θ) ( ) V t t = T V T = β T H V t n N A n = { θ t n, V t 1 n} A n F t 1 θ self-financing V t = V t 1 + θ t ΔS t. 1 An V t 1 An = V t 1 1 An + θ t ΔS t 1 An. (2.24) V t 1 1 An θ t ΔS t 1 An. V t 1 1 An = E[V t 1 1 An F t 1 ] E[θ t ΔS t 1 An F t 1 ]= 1 An θ t E[ΔS t F t 1 ]=. n V t 1 Q-a.e. (2.24) E[V t 1 An ]=E[V t 1 1 An ]+E[θ t ΔS t 1 An ]=E[V t 1 1 An ]. n E[V t ]=E[V t 1 ] V t V t (V t ) A F t 1 (2.24) 1 A E[V t 1 An 1 A ]=E[V t 1 1 An 1 A ]+E[θ t ΔS t 1 An 1 A ]=E[V t 1 1 An 1 A ]. n E[V t 1 A ]=E[V t 1 1 A ]. V t (θ) =E Q [β T H F t ] V t (θ) =β 1 t E Q [β T H F t ] 24

( T ) H ( ) π(h) π(h) =V (θ) =E Q [β T H F ]=E Q [β T H] (2.25) H V (θ) 2.13. H x V T (θ) H θ (x, H)- (superhedging) H V T (θ) =H θ (minimal hedge) (seller s price) π s = inf{z ; θ Θ s.t. V T (θ) =z + G T (θ) H} (buyer s price) π b = sup{y ; θ Θ s.t. y + G T (θ) H} 2.14. viable : π b E Q [β T H] π s. (2.26) V T (θ) =z + G T (θ) H z = V (θ) V T = V + G T S E Q [G T ]= inf π b E Q [β T H] z = V (θ) =E Q [V T ]=E Q [β T V T ] E Q [β T H] π s E Q [β T H] viable H π s = π b = E Q [β T H] 25

β t C t = E Q [β T (S T K) + F t ] β t P t = E Q [β T (K S T ) + F t ] C t P t = βt 1 E Q [β T (S T K) + F t ] βt 1 E Q [β T (K S T ) + F t ] = βt 1 E Q [β T (S T K) F t ] = βt 1 E Q [β T S T F t ] βt 1 E[β T K F t ] = βt 1 β t S t (1 + ρ) t (1 + ρ) T K = S t (1 + ρ) (T t) K (1.1) Q V t V t =(1+ρ) (T t) E Q [((S T K) + F t ] (2.27) E Q [ F t ] Q R t = S t S t 1 Q R 1, R 2,..., R T Q(R t =1+b) =q, Q(R t =1+a) =1 q q = ρ a b a E Q [R t ] = (1 + b) ρ a b a +(1+a)b ρ b a ρ a + bρ ba + b ρ + ab aρ = (b a)(1 + ρ) = b a =1+ρ. b a 26

E Q [βs t+1 F t ]=βe Q [S t R t+1 F t ] = βs t E Q [R t+1 F t ] = βs t E Q [R t+1 ] ( R t+1 F t ) = βs t (1 + ρ) =S t. {β t S t } : Q E Q [β t+1 S t+1 F t ]=β t S t. (1 + ρ) (T t) E Q [(S T K) + F t ] =(1+ρ) (T t) E Q [(S t R t+1...r T K) + F t ] T t ( ) T t =(1+ρ) (T t) q s (1 q) T t s (S t (1 + b) s (1 + a) T t s K) s + = V t. s= (1.9) 3. Black-Scholes 2 Black-Scholes [,T] N h N = T N {,h N, 2h N,...,Nh N } N 2 a, b, ρ N a N a h N h r, σ> a, b, ρ N ρ = rh ( ) 1+b log = σ T h = σ 1+ρ N, ( ) 1+a log = σ T h = σ 1+ρ N. ρ ( lim (1 + N ρ)n = lim 1+ rt ) N = e rt N N 27

u, d ( N ) ( u =1+b = 1+ rt ) e σ T N N ( d =1+a = 1+ rt ) e σ T N. N kh S k R k = S k S k 1 ( N ) Q(R k =1+b) =q = ρ a b a, Q(R k =1+a) =1 q = b ρ b a. {Y k } k=1,...,n ( ) Rk Y k = log. 1+ρ Z N = N Y k = k=1 N log R k N log(1 + ρ) k=1 T = Nh S N = S N k=1 { N R k = S (1 + ρ) N exp k=1 Y k } = S (1 + ρ) N e Z N C =(S N K) + V (C) =β N E Q [(S N K) + ] = β N E Q [(S (1 + ρ) N e Z N K) + ] = E Q [(S e Z N (1 + ρ) N K) + ] (3.1) N Y k Y k μ, v ( )] E Q [Y k ]=E [log Q Rk 1+ρ ( ) ( ) 1+b 1+a = log q + log (1 q) 1+ρ 1+ρ 28

= σ hq σ h(1 q) =(2q 1)σ h. 2 ( )] E Q [Yk [log 2 Rk ]=EQ 1+ρ { ( )} 2 { ( )} 2 1+b 1+a = log q + log (1 q) 1+ρ 1+ρ = σ 2 hq + σ 2 h(1 q) =σ 2 h v = E Q [Y 2 k ] EQ [Y k ] 2 = σ 2 h (2q 1) 2 σ 2 h. q 1 q = b ρ b a 1+b (1 + ρ) = 1+b (1 + a) = (1 + ρ)e σ h (1 + ρ) (1 + ρ)e σ h (1 + ρ)e σ h h 1 = eσ e σ h e σ h 2q 1=1 2(1 q) e σ h 1 =1 2 e σ h e σ h h e σ h 2e σ h +2 = eσ e σ h e σ h = 2 h eσ e σ h e σ h e σ h = 1 cosh σ h sinh σ h 1 2 σ h. Nμ = N(2q 1)σ ( h N 1 ) 2 σ h σ h = 1 2 σ2 Nh = 1 2 σ2 T. Nv = Nσ 2 h N(2q 1) 2 σ 2 h = σ 2 T (2q 1) 2 σ 2 T σ 2 T. 29

3.1. N N {Yk N} k=1,...,n μ N, σn 2 Nμ N μ, NσN 2 Σ2 Z N = N k=1 Y k N μ, Σ 2 Z N 1 2 σ2 T, σ 2 T Z (3.1) N Black-Scholes V (C) =E Q [(S e Z e rt K) + ] (3.2) X = 1 ( σ Z + 1 ) T 2 σ2 T X N(, 1) V (C) V (C) = Z = σ TX 1 2 σ2 T (S e 1 2 σ2 T +σ Tx e rt K) + 1 2π e 1 2 x2. x ( ) K log =(r 1 S 2 σ2 )T + σ Tx x = log( K S ) (r 1 2 σ2 )T σ T. γ V (C) = γ = S (S e 1 2 σ2 T +σ Tx e rt 1 K) e 1 2 x2 dx 2π γ = S γ = S γ σ T e 1 2 σ2 T +σ Tx 1 2π e 1 2 x2 dx e rt K e 1 2 (x σ T ) 2 1 2π dx e rt K(1 Φ(γ)) e 1 2 x2 1 2π dx e rt K(1 Φ(γ)) 3 γ 1 2π e 1 2 x2 dx

= S (1 Φ(γ σ T )) e rt K(1 Φ(γ)). Φ N(, 1) Φ(x) = x d = γ, d + = d + σ T 1 2π e 1 2 y2 dy. 1 Φ(γ) =Φ( γ) =Φ(d ) 1 Φ(γ σ T )=Φ(d + ) d ± = log( K S ) (r ± 1 2 σ2 )T σ T V (C) =S Φ(d + ) e rt KΦ(d ). Black-Scholes t V t (C) =S t Φ(d + t ) e r(t t) KΦ(d t ). (3.3) d ± t = log( K S t ) (r ± 1 2 σ2 )(T t) σ T t (3.3) c c S t, t, K, T, r, σ c 1. c S t t V t (C) = c(s t,t) c(x, t) Black-Scholes : c t + 1 2 σ2 x 2 2 c c + rx rc =. (3.4) x2 x 2. lim c(s t,t)=(s t K) + T t c σ > lim σ c(s t,t)=s t lim c(s t,t)=(s t Ke r(t t) ) + σ c =Φ(d +) x 31

4. viable viable Ω Banach 4.1. L R n K L K L φ: R n R L φ(x) = K φ(x) > claim 1 C R n : C ψ ψ c>onc. B = B(,r)={x; x <r} B C z B C. C x C, λ [, 1] λ y = λx +(1 λ)z C z 2 λx +(1 λ)z 2 = λ 2 x 2 +2λ(1 λ)x z +(1 λ) 2 z 2. λ 2 x 2 +2λ(1 λ)x x +(λ 2 2λ) z 2 λ x 2 + 2(1 λ)x z +(λ 2) z 2 x z z 2. ψ(x) =x z C ψ z // claim 2 C = K L = {k l; k K, l L} C C. x n = k n l n x k n k nj k K l nj = k nj x nj k x L k x L x = k l K L. // claim 2 C claim 1 ψ c on K L x = k λl ψ(k) λψ(l) >c λ or λ ψ(l) = ψ(k) >c. 32

(Ω, F,P) S S 1,...,S d Ω n Ω X Ω R n C C = {X :Ω R; X ω Ω X(ω) > }. θ Θ a V t (θ) =G t (θ) C if V (θ) = (self-financing strategy) θ =(θ,θ 1,...,θ d ) ˆθ =(θ 1,...,θ d ) ˆθ t =1,2,...,T G t (ˆθ) = t θ u ΔS(u) = u=1 t d θu i ΔSi u. u=1 i=1 G T (ˆθ) C β t =(S t ) 1 V T (θ) =β 1 T V T (θ) =β 1 {V (θ)+g T (θ)} = β 1 G T (ˆθ) T θ viability G T (ˆθ) C 4.2. viable predictable R d - ˆθ G T (ˆθ) C 4.3. viable viable C ω F (ω) ω F (ω ) > Ω viable 4.2 G T (ˆθ) C L = {G T (ˆθ); ˆθ =(θ 1,...,θ d ),θ i (i =1,...,d) predictable} Ω Ω =ω 1,...,ω n p i = P ({ω i }) > T 33

L C K = {X C; E P [X] =1} Ω L K f f : f(x) =x q = n x i q i. ξ i =(,...,, 1 p i,,...,) E P [ξ] =1 ξ i K f(ξ) = q i p i > q i > g = f, α = n α i=1 q i p i = q i α P P P g(x) = 1 f(x) =onl α EP [G T (ˆθ)] = ˆθ self-financing strategy θ V (θ) = E P [G T (θ)] = [ T ] E P θu i ΔSi u =. u=1 i=1 2.9 S i P 4.4. 4.5. EMM Q viable (Q, (F t ))- M predictable process γ M t = M + t γ u ΔS u = M + u=1 t u=1 d γu i ΔSi u. (4.1) i=1 M (Q, (F t ))- M H = M T S T θ Θ a V T (θ) =H V T (θ) =M T V Q- V t (θ) =E Q [V T (θ) F t ]=E Q [M T F t ]=M t. M t = V t (θ) =V (θ)+ t θ u ΔS u = M + u=1 34 t θ u ΔS u u=1

(4.1) H M M t = E Q [β T H F t ] M t (4.1) θ i t = γ i t, i =1,...,d θt = M t γ t S t θ self-financing Δθ t S t 1 =. d Δθ t S t 1 = St 1(ΔM t Δ[ γts i i t]) + = = = d i=1 i=1 S i t 1Δγ i t d (St 1[γ tδs i i t (γts i i t γt 1S i i t 1)] + St 1Δγ i t) i i=1 d (St 1[ γts i i t γts i i t 1 ( γts i i t γt 1S i i t 1)] + St 1Δγ i t) i i=1 d St 1(Δγ i t i Δγt)=. i i=1 V t (θ) =θ t S t = θ t + γ t S t = M t V T (θ) =M T = β T H V T (θ) =H EMM 4.6. viable EMM Q Q H θ Θ a β T H = V T (θ) =V (θ)+ S Q Q T θ u ΔS u. u=1 E Q [β T H]=V (θ) =E Q [β T H]. 35

E Q [H] =E Q [H] Q = Q viable X T L = {c + ˆθ u ΔS u : ˆθ =(θ 1,...,θ d ) is predictable}. u=1 Ω L (Ω,P) L (Ω,P) R n β T X L β T X L β T X = V T (ˆθ) θ =(θ, ˆθ) selffinancing θ ((2.18)) β T X = V T (θ), X = V T (θ) X L L (Ω,P) L Z E Q [YZ]=, Y L. 1 L E[Z] = R =1+ Z 2 Z Q = RQ R 1 2. Q Q Y = c + T u=1 ˆθ u ΔS u L E Q [Y ]=E Q [RY ]=E Q [Y ]+ c = E Q [Y ]= E Q [ 1 2 Z E Q [YZ]=E Q [Y ]=c. T ˆθ u ΔS u ]=. u=1 Q CRR CRR St =(1+ρ) t S t = R t S t 1 R t i.i.d. { 1+b, q = ρ a b a R t = 1+a, 1 q = b ρ b a 1 <a<ρ<b viable EMM u = 1 + b, d = 1 + a, E Q [R t ] = w F t = σ{r u ; u t} t m t = (R u w) (m t ) u=1 36

4.7. M = Q- M t = t θ u Δm u (4.2) u=1 θ =(θ t ) predictable M t F t M t = f t (R 1,...,R t ) (4.2) ΔM t = θ t Δm t ft u ft d = f t (R 1,...,R t 1,u) = f t (R 1,...,R t 1,d) ft u f t 1 = θ t (u w), ft d f t 1 = θ t (d w) θ t = f t u f t 1 u w = f d t f t 1 d w E Q [ΔM t F t 1 ]= qf u t +(1 q)f v t = f t 1 = qf t 1 +(1 q)f t 1. f u t f t 1 q 1 = f d t f t 1 q w = E Q [R 1 ]=uq + v(1 q) q = w d u d, u q 1=w u d 37

4.5 M t = M + t u=1 γ u ΔS u Δm u w =1+ρ E Q [R t ] = (1 + b)q +(1+a)(1 q) =(1+b) ρ a b a +(1+a)b ρ b a ρ(1 + b 1 a) a ba + b + ab = b a (1 + ρ)(b a) = b a =1+ρ. ΔS t =(1+ρ) t S t (1 + ρ) t+1 S t 1 =(1+ρ) t (S t 1 R t (1 + ρ)s t 1 ) =(1+ρ) t S t 1 (R t (1 + ρ)) =(1+ρ) t S t 1 (R t w) =(1+ρ) t S t 1 Δm t 5. t =,1,..., T S, S 1,...,S d β t =(St ) 1 S =(S,S 1,...,S d ) t σ-field F t t T f t (S) viable Q Q τ t {τ t} F t (5.1) Z +- Z +- T T- τ 38

f τ (S) E[β τ f τ (S)] τ T- x = sup E[β τ f τ (S)] (5.2) τ hedge θ t V t (θ) f t (S) (5.3) V (θ) =E[V τ (θ)] = E[β τ V τ (θ)] E[β τ f τ (S)] V (θ) x θ (5.3) V (θ) =x (5.2) τ σ-fileld F τ F τ = {A F ; A {τ t} F t t} (5.4) τ Doob 5.1. (X t ) σ, τ σ τ E[X τ F σ ] X σ (5.5) X (5.5) 5.2. X τ τ X τ X τ t = X τ t 5.3. X ( ) τ X τ ( ) Doob 5.4. X X X t = X + M t A t. (5.6) (M t ) M = (A t ) A = predictable 39

A t = A t 1 + X t 1 E[X t F t 1 ], M t = M t 1 + X t E[X t F t 1 ]. 5.5. X Z (Snell envelope) Z T = X T Z t 1 = max{x t 1,E[Z t F t 1 ]} t =1, 2,...,T. 5.6. (Z t ) (X t ) 1. Z X 2. τ = min{t ; Z t = X t } Z τ Z t X t Z t 1 E[Z t F t ] Z Y =(Y t ) Y t X t Y T X T = Z T Y t Z t Y t 1 E[Y t F t 1 ] E[Z t F t 1 ]. ( Y ) ( ) Y t 1 X t 1 Y t 1 max{x t 1,E[Z t F t 1 ]} = Z t 1. Z τ φ t =1 {τ t} φ predictable Z τ t = Z + t φ u ΔZ u. u=1 Z τ t Z τ t 1 = φ t (Z t Z t 1 )=1 {τ t}(z t Z t 1 ). τ (ω) t Z t 1 (ω) >X t 1 (ω) Z t 1 (ω) =E[Z t F t 1 ](ω) E[Z τ t Z τ t 1 F t 1] =E[1 {τ t}(z t E[Z t F t 1 ]) F t 1 ] =1 {τ t}e[(z t E[Z t F t 1 ]) F t 1 ]=. Z τ 4

5.7. τ X 5.8. (Z t ) (X t ) τ X E[X τ ] = sup E[X σ ] (5.7) σ τ = min{t ; Z t = X t } (5.8) 5.6 Z τ Z = E[Z τ ]=E[Zτ T ]=E[Z τ ]=E[X τ ]. τ Z τ τ f t discount process f Z : Z = E[Z τ ] E[Zτ T ]=E[Z τ] E[X τ ]. f t = β t f t Z T = f T, Z t 1 = max{f t 1,E[Z t F t 1 ]} Z t t 1 Z t t 1 f t 1 Z t Z t Z τ = min{t ; Z t = f t } 5.6 Z = sup E[f τ ]=E[f τ ] τ (Z t ) Z t = Z + M t A t Z + M T = Z T + A T θ V t (θ) =Z + M t 41

Z τ t τ Z t = Z + M t Z τ = Z + M τ = V τ (θ) V (θ) =E[V τ (θ)] = E[Z τ ]=E[f τ ] = sup E[f τ ]. τ V (θ) =Z = sup E[f τ ]. τ 3 sup E[f τ ] τ θ hedge 6. (Ω, F,P) R + =[, ) t [, ) X t F - t ω 2 (t, ω) X t (ω) σ-fields (F t ) t X t F t (F t )- 6.1. (W t ) (Wiener ) 1. W = 2. n t <t 1 <t n n W t1 W t,w t2 W t1,...,w tn W tn 1 3. s<t W t W s N(,t s) 4. 1 t W t Brown Wiener Wiener σ-filed (F t ) (W t ) (F t )- s <t F s W t W s F t W t σ-field F t = σ{w u ; u t}. 42

Wiener (W t ) H sdw s (W t ) Stieltjes 6.2. (H t ) : H t = i=1 φ i 1 (ti 1,t i ](t). (6.1) =t <t 1 <t n φ i F ti 1 - H H s dw s H s dw s = φ i (W t ti W t ti 1 ) (6.2) i=1 6.3. H 1. H s dw s [( ) 2 ] [ ] 2. E H s dw s = E Hs 2 ds [ ( ) 2 ] [ T ] 3. E H s dw s 4E Hs 2 ds. sup t T 1. s <t, t j 1 <s t j [ ] [ ] E H u dw u F s = E φ i (W t ti W t ti 1 ) F s i=1 j 1 = φ i (W t ti W t ti 1 )+E[φ j (W t tj W t tj 1 ) F s ] i=1 + i=j+1 E[E[φ i (W t ti W t ti 1 ) F ti 1 ] F s ] j 1 = φ i (W s ti W s ti 1 )+φ j (W s tj W s tj 1 ) = i=1 + i=j+1 s E[φ i E[(W t ti W t ti 1 ) F ti 1 ] F s ] H u dw u. 43

2. t n 1 <t t n X i =(W t ti W t ti 1 ) X 1,...,X n, t t i t t i 1 [( ) 2 ] E H u dw u = = n i=1 n n E[φ i φ j X i X j ] j=1 E[φ 2 i Xi 2 ]+2 i=1 i<j n = E[φ 2 i ]E[X2 i ]+2 i=1 i<j n = E[φ 2 i ](t t i t t i 1 ) i=1 [ ] = E Hu 2 du 3. Doob E[E[φ i φ j X i X j F tj 1 ]] E[φ i φ j X i E[X j F tj 1 ]] 6.3 H [ M ] H = {H =(H t ); (F t )- M E Ht 2 dt < } (6.3) predictable (F t )- predictable predictable H H H s dw s 2 2 M t H H M t = M + H s dw s τ T τ H s dw s = T 1 {s τ} H s dw s (6.4) H H = {H =(H t ); (F t )- M T Ht 2 dt < P -a.e.} (6.5) 44

τ n τ n E[ τ n H 2 s ds] < t<τ n H s dw s = 1 {s τn}h s dw s H s dw s H τ n τ n 6.4. (X t ) X t = X + K s ds + H s dw s. (6.6) X F - K =(K t ), H =(H t ) F t - M M K s ds < P -a.s. M M H s 2 ds < P -a.s. 6.5. X X t = X + K s ds + H s dw s. f(t, x) x 2 t f(t, X t ) f(t, X t )=f(,x )+ f t (s, X s )ds + f x (s, X s )dx s + 1 2 45 f xx (s, X s )d X, X s. (6.7)

f x (s, X s )dx s = X, X t = f x (s, X s )K s ds + H 2 s ds f x (s, X s )H s dw s M t = H s dw s M 2 t = 2M s H s dw s + M,M t M 2 t M,M t M,M t 2 (quadratic variation) Wiener W, W t = t W t Wiener Lévy S t = x exp{(μ σ 2 /2)t + σw t } (6.8) Black-Scholes f(t, x) =x exp{(μ σ 2 /2)t+σx} S t = f(t, W t ) f t =(μ σ 2 /2)f, f x = σf, f xx = σ 2 f f(t, W t )=f(,x )+ = x + (μ σ 2 /2)f(s, W s )ds + σs s dw s + S t = S + μs s ds σs s dw s + σf(s, W s ) dw s + 1 2 ds t = σs t dw t + μs t dt 46 μs s ds σ 2 f(s, W s )ds

(Girsanov) 6.6. (θ t ) T θ2 sds < P -a.s. (F t )- { L t = exp θ s dw s 1 } θs 2 2 ds (6.9) Q = L T P Q P Q B t = W t + θ s ds Wiener (L t ) { 1 T } exp θs 2 2 ds L 1 (P ) (Novikov ) 7. Black-Scholes Black-Scholes [,T] T e rt ( ) S t S 6 ds t = σs t dw t + μs t dt. (7.1) S t = S exp{(μ σ 2 /2)t + σw t } (7.2) S S t = e rt S t ds t = re rt S t dt + e rt ds t = rs t dt + e rt (σs t dw t + μs t dt) =S t ((μ r)dt + σdw t ) 47

B t = μ r t + W σ t ds t = σs t db t (7.3) θ t =(μ r)/σ Girsanov P = L T P (B t ) S S S t = S exp{σb t σ 2 t/2}. (7.4) P Black-Scholes H =(S T K) + π(h) e rt (S T K) + =(S T e rt K) + π(h) =E P [(S T e rt K) + ]=E P [(S exp{σb T σ 2 T/2} e rt K) + ] = E P [(S e Z e rt K) + ]. Z = σb T σ 2 T/2 P Z σ 2 T/2, σ 2 T (7.2) φ =(η, θ) V t (φ) =η t S t + θ ts t (7.5) self-financing strategy ΔV t (φ) =φ t ΔS t T dv t (φ) =η t ds t + θ t ds t (7.6) η t dt <, T θ t 2 dt < P -a.s. (7.7) (F t )- φ (7.6), (7.7) self-financing strategy S t = e rt S t 7.1. φ (7.7) V t (φ) (7.5) V t (φ) =e rt V t (φ) φ self-financing V t (φ) =V (φ)+ t [,T] 48 θ t ds t (7.8)

V (φ) dv t (φ) = rv t (φ) dt + e rt dv t (φ) dv t (φ) = re rt ( η t e rt + θ t S t ) dt + e rt η t d(e rt )+e rt θ t ds t = θ t (( re rt S t ) dt + e rt ds t ) = θ t ds t (7.8) 7.2. self-financing strategy φ (admissible) V t (φ) =η t + θ t S t t P 2 H ( ) T 7.3. H P 2 H t V t = E P [e r(t t) H F t ] (7.9) H φ V t (φ) =η t S t + θ t S t V T (φ) =H V t (φ) =e rt V t (φ) V t (φ) =η t + θ t S t. self-financing 7.1 V t (φ) =V (φ)+ θ s ds s = V (φ)+ θ s σs s db s. V t (φ) P 2 V t (φ) 2 V t (φ) =E P [V T F t ]. 49

V t (φ) =E P [e r(t t) H F t ]. H M t = E P [e rt H F t ] 2 E P [ T K2 s ds] < (F t )- M t M t = M + K s db s θ t = K t /σs t, η t = M t θ t S t φ =(η, θ) V t (φ) =η t + θ t S t = M t = M + = M + = M + σθ s S s db s θ s ds s. ( (7.3)) K s db s 7.1 φ =(η, θ) self-financing V t (φ) =e rt M t = E P [e r(t t) H F t ]. V t (φ) 2 V T (φ) =H H f(s T ) f :(, ) [, c >,k 1 >,k 2 > f(x) c(1 + x) k 1 x k 2. 7.4. H = f(s T ) φ =(η, θ) : θ t = e r(t t) F x (T t, S t ) (7.1) η t = e rt (F (T t, S t ) F x (T t, S t )S t ). (7.11) F (T t, x) = 1 2π R f(x exp{σy T t +(r σ 2 /2)(T t)})e y2 /2 dy. 5

φ H S t = S exp{(r σ 2 /2)t + B t } (S t ) : t>s S t = S s exp{(r σ 2 /2)(t s)+σ(b t B s )} V t (φ) =E P [e r(t t) f(s T ) F t ] = E P [e r(t t) f(s t exp{(r σ 2 /2)(T t)+σ(b T B t )) F t ] = e r(t t) F (T t, S t ). (7.12) F (T t, x) =E P [f(x exp{(r σ 2 /2)(T t)+σ(b T B t ))] = 1 f(x exp{σy T t +(r σ 2 /2)(T t)})e y2 /2 dy 2π R = 1 f(y)g(t t, y/x, r σ 2 /2,σ) dy. x R g(t, z, α, β) = } 1 { βz 2π exp (log z αt)2. 2βt f F (T t, x) (t, x) G(t, x) =F (t, e rt x) V t (φ) =V t (φ)e rt = e rt G(t, e rt S t )=e rt G(t, S t ). d(v t (φ)) = e rt d(g(t, S t )) = e rt G x (t, S t ) ds t + e rt G t (t, S t ) dt + 1 2 e rt G xx (t, S t ) d S,S t = e rt G x (t, S t )σs t db t + e rt G t (t, S t ) dt + 1 2 e rt G xx (t, S t )σ 2 S 2 t dt. E P [e rt f(s T ) F t ]=V t (φ) = E P [e rt f(s T )] + e rt G x (t, S t )σs t db t 51

+ e rt {G t (t, S t )+ 1 2 G xx(t, S t )σ 2 S 2 t }dt. G t + 1 2 G xxσ 2 x 2 V t (φ) =E P [e rt f(s T )] + e rt G x (t, S t ) ds t = E P [e rt f(s T )] + e rt e rt F x (t, S t ) ds t. G x (t, x) =e rt F x (t, e rt x) (7.8) θ t = e r(t t) F x (T t, S t ) η V t (φ) =η t e rt + θ t S t η t = e rt V t (φ) e rt θ t S t = e rt e r(t t) F (T t, S t ) e rt e r(t t) F x (T t, S t )S t = e rt (F (T t, S t ) F x (T t, S t )S t ). [1] R. J. Elliott and P. E. Kopp, Mathematics of financial markets, Springer-Verlag, New York, 1999. [2] I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, Second edition, Springer-Verlag, New York, 1991. [3] I. Karatzas and S. E. Shreve, Methods of mathematical finance, Applications of Mathematics, 39, Springer-Verlag, New York, 1998. [4] D. Lamberton and B. Lapeyre, Introduction to stochastic calculus applied to finance, Second edition, Chapman & Hall/CRC, Boca Raton, FL, 28. [5] D. Revuz and M. Yor, Continuous martingales and Brownian motion, Third edition, Berlin- Heidelberg-New York, Springer-Verlag, 1999. [6] R. J. Williams, Introduction to the mathematics of finance, Graduate Studies in Mathematics, 72, American Mathematical Society, Providence, RI, 26. 52

1. 1 1 1.1 1 1 16.6 8.4 12 2. CRR, 1, 2,...,T S t = (1 + ρ) t, ρ> 1 (S t ) ( 1 <a<b) (1 + b)s t p S t (1 + a)s t 1 p (1) (S t /S t ) p (2) viable ρ (a, b) (3) ρ a (4) t Z t Z t = A(t, S t ) t A(t, x) 3. (M t ) t=,1,...,t ( ) τ M τ ( )