Stochastic finance

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Ο στοχαστικός λογισμός και η εφαρμογή του στα χρηματοοικονομικά

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<ul><li> 1. ()</li></ul> <p> 2. Bachelier 1900, , Theorie de la Speculation. O Bachelier Brown , options futures . . 64 PaulSamuelson 1964 Bachelier. 1923 N. Wienner Brown . Brown (s,t) (t-s). . Brown . Brown Bt , t [ 0, 1) B0 =0 2) . Bt 0=t 0t 1...t n1t n=T Bt1 B t0 , Bt2 B t1 ,... 3) 0st Bt B s~ N 0, ts 4) , t B t Brown t-s. . f :[a , b] f [a , b] n f 2 ,b =lim f t i 2 a n 0 i=1 n=supr t i1t ii P n=t 0, t 1,... , t n ={a=t 0 t 1 t 2...t n=b} 3. brown [a , b] 2 ,b=ab L 2 a n2 2 lim E ba =0 tin n 0i=1 n2n 2n n ba =E t i =E 2 t i 2 = E 2 t i 222 1 Eti ti ti t ii=1 i=1i =1i=1nn2 E t 42 t E t 2 t 2= E 3 t i 22 t i t 22t max t i 0 ii i i i=1 i=1 n 0(1) Brown(2) H forth moment .. 0 t it i1 3t 2 i Brown . n f 1 ,b = t a i..i=1 nn t max t t V a , b 1in t 1ini2maxiiii=1i=1max t 0 n 0 i1in n t 2 0 i.. i=1 B .. [a , b] [a , b] Bachelier 1964, A.N. Kolmogorov Bachelier Brown. O Kolmogorov 4. Kolmogorov. (semi-groups) Ito. Kiyosi Ito . Ito Brown Ito . non-anticipating, filtrations - Brown F t= B s , st Brown L 2 . , H 2 [0, T ] . f H =E2 f 2 dt 0 2 2 0 2 [ 0 , ] 0=t 0t 1...t n= . n1 f t= a i 1 t tti i1 i =0 a i F t a i F t ii ai2 . 2 . t 1, t 2 ] [ 0 , ] 0 t2 f = f dB t= t B t1 2t1 [ 0 , ] Tn1 f = f dB t= a i Bt Bt i1 i 0i =0 5. To Ito2 H . T T 2 E [ f t d t ]=E f t2 dt 0 0 2 02 . f 2 f n H 2 0 b 1 f f n2 = f f n2 dt 2 0 n H 2 a 2 Ito f H f n 2 0 b b f t d t=lim f n t d tn a a Ito 2 L 2 f : 2 L2 . b b 2 [I f ]=E [ f t d t ]=E f t2 dt 2 a a I f L = f H 2 2 Ito , 2 . martingale.. f H 2 [ 0 , T ] X t tX t = f s dB s X t martingale0 t X t = f s dB s / F t , 0tT 0 6. I Ito H 2 t3t2t31. f t d t= f t dt f t dt t1t1t2 t2 t2 t22. c 1 f t c 2 g td t=c1 f t dt c 2 f t dt t1 t1 t1t23. E f t dt =0t1 Ito H 2 [ 0 ,T ] . Ito H 2 [ 0 ,T ] localization. loc . P f 2 dt =1 0 localizing sequence, localizing localizing f H 2 [ 0 ,T ] f t 1t H 2 [ 0 ,T ]n localization H 2 [ 0 ,T ] f H loc [ 0 , T ] loc2 n1 n n { } T n=inf t: 0 f 2 t dt=n 0 localizing sequence H 2 [ 0 ,T ] f H 2 [ 0 ,T ] . loc. f H loc [ 0 , T ] localizing { n } 2 T f s 1s dB s n martingale t [ 0 ,T ] mn 0 T T f s 1s dB s= f s 1s dB snm t m 0 0 7. T. f s dB s 0 T TPf s dB s=lim f s1t dB s =1 n [ 0 ,T ] 0n 0 ItoT f s 1t dB s n0 1 L 2 H loc [ 0 ,T ] martingales 2local martingale.Local martingale t filtration F t 0t t local martingale k k 1 k martingale. t = X t X 0k f L2 local martingale loct P X t= f s dB s =10 Ito Bt Brown .. , F , P , Ito X t .tt X t = X 0 a s ds b s dB s (ip1)00 t t a s ds . a H 1 loc 0 8. t b 2 s ds.. bH 2loc 0 Ito dX t =a t dtb t dBt 0= 0. Ito X t (ip1) a H 1 bH X t martingale a=0 .. t02 Formula Ito X t Ito (ip1), t f t , X t , f C 1,2 ttf f 1 2 2 fff t , X t = f 0, X 0 a b 2 dx bdB s0 s x 2 x0 x formula Ito . formulas Ito f X t [ 0 , t ] n 1 f t , X t f 0, X 0 = [ f t i1 , X t i1 f t i , X t i ]i=0 n1 n 1 [f t i1 , X t i1 f t i , X t i 1] [ f t i , X t i 1 f t i , X t i ]i=0i=0 n1 [ f t i1 , X t i1 f t i , X ti 1](ito1) i=0 formula taylor t t i =t i1t i f f t i1 , X t i1 f t i , X t i1=t , X t i1 t it i 9. (ito1) [] n1 tff lim t i , X t i1 t i = s , X s ds n i =0 t0 s n1 [ f t i , X ti1 f t i , X t i ]i=0 taylor X , formula taylor 2 f 1 ff t i , X t i 1 f t i , X t i =t i , X t i i 2 t i , X t i i 2 x 2 x i Ito i =a t t bt Bt Ito Taylor f 1 2 ff t i , X t i 1 f t i , X t i =t i , X t i X i t i , X t i X i2 x 2 x 2f f f t i , X t i 1 f t i , X t i =t i , X t i a t t t , X t i b t B tx x i1 2 f 221 2 f 2 22 ft , X t i a t t , X t i b B t 2 t i , X t i a b t B t2 x2 i 2 x2 ix L 2 n (E ) Martingales X t martingale filtration Bt f H 2 [ 0 , T ] tX t = 0 f s ds [ 0 ,T ]0 martigale 10. Ito a b t, .dX t =a t , X t dtb t , X t dB t tt X t = X 0 a s , X s ds b s , X s dB s0 0 Lipschitz a t , x at , y Kx y b t , xbt , y Kx y a t , x bt , x K 1x2 2 X t , 0t 0 non anticipative E X t 00 . dS t= S t dt S t dB t S t t , drift . historical volatility EWMA (Exponentially Weighting Moving Average) GARCH(Generalized Autoregressive Conditional Heteroskedasticity). formula Ito S t = f t , Bt f, formula Ito f 11. f 1 2 ff dS t ={t , Bt dtt , B t }dtt , Bt dBt t 2 x 2x .2 f1 f f t , x=t , x dtt , x t2 x f f t , x =t , x x x g t f t , x =e f t , x g 1 2 t , x = t 2 g 1 2 t B t 2S t =e Brown (Geometric Brownian Motion). coefficientmatching . coefficient matches Ito t X t=b t{x d s dB s } (cf) 0 formula Ito t dbt dX t ={x d s dB s }dtbt d t dB tdt 0 b 0=1 b t0 t0 (cf) db t/dt dX t =X t dtbtd t dBtX 0=x bt 12. Ornstein-Uhlenbeck coefficient matching . Ornstein-Uhlenbeck Ornstein-Uhlenbeck 1931 . d t =a t X t dt t dBt X 0=x db t/dt = t (cf1) bt b td t= t (cf2) cf1 b 0=1 tb t=e (cf2) d t=e t Ornstein-Uhlenbeck ttX t=et {x e t dB s }=et x e t s dB s0 0 Black Scholes Black Scholes -bill .d t=r t t dt 0=1 (BS 1)dS t= t S t dt t S t dBt (BS 2) S t = t , S t 13. t= 0 t , 1 t . Black Scholes t V t= 0 t t 1 t S t dV t= 0d t 1dS tArbitrage arbitrage V t V 0=0 V T 0 P V T 00Contigent Claim X F T = S T . Contigent Claim . (European Call Option) (European Call Option) x =max xK ,0 T S T (Strike Price) .Look Back Call Option Look Back Call x =max xmin S t ,0 tT 14. Look Back Call option pay off T S T S t=0 t=T (Replicate Portfolio) T contigent claim CallOption V t=max S T K ,0 Girsanov Bt Brown P Q S t =Bt t W C[0,T] QP E [W ]= E [WM T ] M t martingale P 2 t t 2 t =e Formulas Black-Scholes Martingales S t Dt = t martingale Q. (pay off) T 0 X / t=0 (contigent claim) V 0 =E Q [ X / ] pay off contigent claim 0 Q [ X 2 ] . formula t 15. Q X V t = t E [ /F ]B t formula t=T V = , formula , P Q D(t) martingale Q. formula Black Scholes Call Option martingales. Call Option .rTQ V 0 =eE [ max S T K ,0 ] Q Dt =S t / t martingale D(t). Black Scholes D(t) BS 2 BS 1 1 2 t t 12S t S 0e S t 2 r t2 t Dt = = = =S 0e t e rt t formula Ito dD t= r Dt dtDt dBt r dD t=D t dtdBt Girsanov r tBt Bt = Brown Q. H Bt dD t=D t t 16. Q S t Bt 2 TrT t2 S T =S 0e Q 2 TrT trT Q2 V 0 =e E [ max S 0eK ,0] Y = 2 T /2 t 2 T /2 2 T dy 2 rT / 22 /2 2 V 0 =eS 0 e rTY K e Y log K /S 0rT 2 dy dy e Y 2 / 22 /2 2 2 / 22 /2 2 V 0 =S 0e Y e Y erT Klog K / S 0rT 2 log K /S rT0 2 formula Black Scholes . C=S 0 N d 1 Kert N d 2 2 ln S 0 / K r d 1=T d 2=d 1 T formula Black Scholes martingales Black Scholes Formula . Black-Scholes PDE . Formulas Black Scholes 17. Black Scholes BS PDE (Hedged Portfolio) CAPM.1) (Hedged Portfolio) long position shortposition (options) . f t , S t V t =S t1 f / x f t , S t 1 f / x call option o . f t , S t V t = S t 1(HP 1) f / x option formula Ito 2 f1 f f f t , S t=t , S t tt , S t 2 S t2 tt , S t S t (HP 2) t2 x2 x (HP2) (HP1) t t . 2 f t , S t f 1 f S t1 rt=t , S t t , S t 2 S t2 t / f / x f / x t 2 x 2 t Black-Scholes. 18. ff1 2 f t , S t =rf t , S t rS t t , S tt , S t 2 S t 2tx2 x 2 Formula Black Scholes Formula Black Scholes (European Call Option) rt C=S 0 N d 1 KeN d 2 2ln S 0 / K r d 1= d 2=d 1 T T formulas S 0 N d 1 S 0 N d 1 N d 1 N d 1 Call ption . Formulas rt Ke N d 2 (short position) . Formulas 9 S 0=9 strike price =10 3 =0,25 risk free rate r =0.04 volatility EWMA GARCH .2 ln 9/100,040,3 d 1= =0,56 0,3 0,25d 2=0,140,3 0,25=0,71 19. C=9N 0,5610e0,040,25 t N 0,71=0,2249 European Call Option Formula Black Scholes, . (Delta Hedge) , , Formula Black Scholes pean Call 0.2249 . greeks C ==N d 1 =0,29S (short position) 100 (long position) 29 . V =2991000.22=26122=239 9 8 261 298=232 29 hedge . call option formula Black Scholes S=9 S=8 calloption C=0.05 . 29 option 220.05100=17 29 12 . 20. E A C/C++ C/C++ C C main(), C main(). .#include main(){printf("Hello World");return 0;} include stdio.h . printf() stdio.h . Dev-C++ C Dev-C++ IDE (Intergrated Development Enviroment) compilerMinGW. C #include #include using namespace std;int main(int argc, char *argv[]){cout</p>