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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............................ 8 3.2 BS..................... 8 3.3...................... 9 4 Black Scholes 1 4.1 Brown.............. 1 4.2 Self financing strategy.............................. 14 4.3 Black Sholes BS........... 14 5 BS 16 6 Black Scholes 16 1 1.1

11 1 2 (i) hedge (ii) 1 1 11 1 1 J.Hull short selling, short position, short S T 1 T 1+R F =(1+R)S $T$ $R$ F>(1 + R)S position S

3 +S F (1 + R)S F > (1 + R)S arbitrage F <(1 + R)S F =(1+R)S 1 = S T 1 T 1+R y 1 T 1+R d R y, R d F = 1+R y S interest rate 1+R d parity F> 1+R y 1+R d S S + S 1+R d 1+R d S 1+R d + 1 1+R d 1 1+R d 1 1 +F 1 1+R y 1+R d S F 1+R y 1+R d S >

S = 17.835 (T =1/4) TIBOR Tokyo Inter Bank Offer Rate TIBOR 11 8 36 36.8% 1 1+.8/1 1 4 1+ 3.32834/1 16.967 4 d.897 d discount 17.835.897 = 16.938 S TIBOR 4 1.2

5 $1$ S t T K (S T K) + (K S T ) + C, P C P = S K R 1 T 1+R 1+RC T P T =(S T K) + (K S T ) + = S T K C (S K 1+R )+ C S P,C (S T K) + S T t T S t K t K t (S t ) + 1+R t T 2 2.1 1 2 5 K = 12 8/3 φ ψ 2φ + ψ =8 (1) 5φ + ψ = (2)

φ =8/15 ψ = 8/3 1φ + ψ =8/3 (3) arbitrage 6 (i) 9% (ii) φ ψ (1) 1/3+ (2) 2/3 = (3) (3) 1:2 (iii) 2 3 1 2 5 K = 12 8 1 5/4 φ ψ 2φ + 5 ψ =8 (1 ) 4 5φ + 5 ψ = (2 ) 4 φ = 8 15, ψ = 64 3 1φ + ψ =32 (3 ) (1 ) 4 5 1 2 + (2 ) 4 5 1 2 = (3 ) 32 Step 1. Step 2. Step 3. K 1+R K = 12 ( 1/2+7 1/2)/(5/4) = 28 C P =32 28 = 4, S K 12 = 1 1+R 5/4 =4 $8/3$ Black-Scholes

7 2.2 2, 1, 5 K = 12 4 1 2 25 95 5/4 2 8 5 46 18/5 node 1/2 1 (5/4) (95 1 2 4 +2 1 2 + 1 4 )=18 5 K = 12 2 8 5 7 46 7 156/5 2 156/5 Asian option ( 1 2 S t K) + 3 t=

(S T K) + 1 mint S t>h 11 H = 9 H 8 3 Black Scholes Black Scholes Black Scholes BS PDE martingale BS BS 3.1 BS T n n t = T/n e u e d e nd S S T e nu S e R d <R<u short arbitrage n 1 R R (n) = r t = rt/n e nr(n) = e rt u, d u (n) = µ t+σ t, d (n) = µ t σ t µ t = 1 2 (u(n) +d (n) ) ( ) 2 1 σ t = 2 (u(n) d (n) ) µ σvolatility log S(n) T S Q (n) n N((r 2 σ2 )T,σ 2 T )

9 (K S(n) T P = lim E Q (n)[ )+ ] n e nr(n) = e rt E[ (K S e σ TZ+(r 1 2 σ2 )T ) + ]=Ke rt Φ( d ) S Φ( d + ). BS BS Z d ± = 1 σ T log( S Ke ) ± 1 rt 2 σ T x 1 Φ(x) = e y2 /2 dy 2π C = S Φ(d + ) Ke rt Φ(d ) P, C σ µ 3.2 BS C > (1) σ C =Φ(d + ) > (2) S lim C = S (3) σ lim C =(S Ke rt )) +. (4) σ C σ lim C =(S K) + (5) T HV IV HV historical volatility 2 HV =13.8 13.8% BS σ =.138 IV implid volatility BS C σ C σ imply IV HV HV 1 2 IV IV > 5 Black Scholes BS

1 3.3 V t, t [,T] T K T V T <K (V T K) + min{v T,K} = (K V T ) + + K = + K 2% t 2% V t V = 2 (V T 2 ) + 2 σ C S 4 Black Scholes 4.1 Brown (Ω, F, (F t ) t [,T ],P) 4.1.1. (i) X =(X t ) t [,T ] X :Ω [, 1] (w, t) X(w, t) R F B([,T]) (ii) X =(X t ) t [,T ] adapted ( t) X t F t

4.1.2. Adapted W =(W t ) t [,T ] (F t ) Brown ( B t bond W (i) W = (ii) s t W t W s N(,t s) (iii) s t W t W s F s (iv) W path 1 a.e. w Ω t W (t, w) 11 W path 1 nowhere differentiable finites variation n 1/2 n 1 n Brown Black Scholes B t = e rt S t = S e σwt+µt σ, µ σ µ e W log S t 1 Brown (quadratic variation) 4.1.3. W T = lim n 2 n k=1 (W 2 n kt W 2 n (k 1)T ) 2 = T,in probability 2 1 total vaiation T L 2 dw t = ± dt t 1 t φ t T T φ t (S t S t 1 ) t=1

formal W path Stieltjes 1944 (predictable, previsible) H H u dw u adapted 12 H 2 udu <, a.s., φ t F t 1 t 1 4.1.4. Stieltjes T T W t dw t := lim n f t df t = 1 2 (f 2 t f 2 ) 2 n k=1 W 2 n (k 1)T (W 2 n kt W 2 n (k 1)T ) W 2 n (k 1)T Stieltjes T = lim n W t dw t 2 n k=1 1 2 (W 2 n kt + W 2 n (k 1)T )(W 2 n kt W 2 n (k 1)T ) 2 n k=1 = 1 2 W 2 T 1 2 T. 1 2 (W 2 n kt W 2 n (k 1)T )(W 2 n kt W 2 n (k 1)T ) quadratic vaiation 4.1.5. K X t = X + X t H u dw u + K u du H 2 udu 4.1.6. X t Predictable L t L 2 u H2 u du <, a.s., ( t), L u K u du <, a.s., ( t), L u dx u := L u H u dw u + L u K u du LdX = L 2 u H2 u t LdX = L 2 u d X u t

BS X t t C 1 f : R R C 1 f(x t )=f(x )+ t f (X u )dx u 4.1.7. (i) f : R R C 2 f(w t )=f(w )+ f (W u )dw u + 1 2 4.1.5 X f(x t )=f(x )+ f (X u )dx u + 1 2 f (W u )du, a.s. f (X u )d X u (ii) f : R 2 R C 2 f t f(w t,t)=f(w, ) + x (W f u,u)dw u + t (W u,u)du + 1 2 f 2 x (W u,u)du 2 f(w t )=f(w t ) W t + 1 2 f (W t )( W t ) 2 + o( W t ) 2 ( W t ) 2 = t (dw t ) 2 dt dw t dt, (dt) 2 4.1.4 f(x) =x 2 W t Wt 2 = W 2 +2 W u dw u + t 4.1.4 4.1.4 4.1.4 4.1.8. f(x, t) =S e σx+µt BS f(w t,t) f x = σf, f t = µf, 2 f x = 2 σ2 f, S t = S+σ = S + σ S u dw u + µ S u dw u +(µ + 1 2 σ2 ) S u du + 1 2 σ2 S u du S u du X t = σw t + µt, f(x) =S e x 13

14 (i) short-hand df (W t )=f (W t )dw t + 1 2 f (W t )dt df (X t )=f (X t )dx t + 1 2 f (X t )d X t (ii) 4.1.8 ds t = σs t dw t +(µ + 1 2 σ2 )S t dt S t = S e σwt+µt ds t = σs t dw t +(µ + 1 2 σ2 )S t dt, S = S, (SDE) ds t = σs t dw t + µs t dt, S = S, S t = S e σwt+(µ 1 2 σ2 )t µ BS (iii) Z =(Z t ) t=,1,2, f : Z R f(z t+1 ) f(z t )= 1 2 (f(z t+1) f(z t 1))(Z t+1 Z t )+ 1 2 (f(z t+1) 2f(Z t )+f(x t 1)) Z t+1 ±1 4.2 Self financing strategy. Self financing x t 1 t φ t ψ t t =1, 2, t 1 φ t S t + ψ t B t = φ t+1 S t + ψ t+1 B t, a.s. (1)

t = x = φ 1 S + ψ 1 B (x, φ, ψ) self financing finance V t = φ t S t + ψ t B t (t 1), V = x, (1) t 1 V t := V t V t 1 = φ t S t + ψ t B t. (2) (2) x R, φ =(φ t ) t (,T ] ψ =(ψ t ) t (,T ] (x, φ, ψ) self-financing V t = V + φ u ds u + ψ u db u, t, a.s. V t = φ t S t + ψ t B t, t>, V = x dv t = φ t ds t + ψ t db t 15 4.3 Black Sholes BS Black Scholes BS BS BS S t = S e σwt+µt B t = e rt ds t = σs t dw t +(µ + 1 2 σ2 )S t dt d S t = σ 2 St 2 dt db t = rb t dt t C t C(S t,t) C : (, ) [,T] R Self-financing strategy (x, φ, ψ) V t = C(S t,t) dc(s t,t)= C S ds t + C t dt + 1 2 C 2 S d S 2 t = σs t C S dw t +((µ + 1 2 σ2 )S t C S + C t + 1 2 σ2 S 2 t 2 C S 2 )dt dv t = φ t ds t + ψ t db t = σs t φ t dw t +((µ + 1 2 σ2 )S t φ t + rb t ψ t )dt φ t = C S (S t,t) (*) V t = C t C t (S t,t)+ 1 2 σ2 St 2 2 C S (S C t,t)+rs 2 t S (S t,t) rc(s t,t)=, t, a.s., t > S t deterministic C t (S, t)+1 2 σ2 S 2 2 C C (S, t)+rs (S, t) rc(s, t) =, S2 S BS PDE C(S, T )=(S K) +

16 (i) P t = P (S t,t) PDE P (S, t) =(K S) + BS PDE (ii) (*) 3.2 φ t = C S (S t,t)=φ(d + ) BS (iii) µ PDE PDE PDE $S=e^x$ $S$ $L^2$ 5 BS BS S t = S e σwt+µt W t = W t + λt λ = µ + 1 2 σ2 P σ W Q dq dp = 1 e λwt 2 λ2t Grisanov S t = S e σ W t 1 2 σ2t. (1) t E Q [ S t ]=S E Q [ e σ W t 1 2 σ2t ]=S u t E Q [ S t F u ]=S u S t Q (1) ds t = σs t d W t (2) (S T K) + =E Q [(S T K) + ]+ T H t d W t H (2) (S T K) + =E Q [(S T K) + ]+ T H t σs t ds t E Q [(S T K) + ]

17 (i) H PDE K S min S (ii) Feynman Kac 6 Black Scholes BS log S T implied volatility (volatility smile) (i) Levy jump (Merton 1973) (tick data) (ii)(a) CEV (constant elasticity of variance ) Cox (1975) ds t = σst α dw t + µs t dt α<1 log S T (b) Volatility SDE ds t = σ t S t dw t + µs t dt d log σ t = K(c log σ t )dt + θd W t W W c (c) S t = S e rt e σ(wt+ct) 1 2 σ2t dν(σ) B t = e rt

ν (, ) ν BS P Q Markov S t W t t S t W t W t = W t + ct Q BS S T = f(w T ) S T >K W T >f 1 (K) e σ(wt+ct) 1 2 σ2t >k(σ) k (S T K) + = S e rt (e σ(wt+ct) 1 2 σ2t k(σ)) + dν(σ) BS 18 arbitrage explicit Black Scholes