金融数学讲座2.pdf

Shanghai Normal University Home Page Title Page JJII JI Page 1 of 70 Go Back Full Screen Close Quit 2014 L 7K O Shanghai normal university 7K n M n Shanghai Normal University Home Page Title Page JJII JI Page 2 of 70 Go Back Full Screen Close Quit 1 d l 2 1 2 2 2 3 d 2 4 d 2 5 d Shanghai Normal University Home Page Title Page JJII JI Page 3 of 70 Go Back Full Screen Close Quit 111 ddd lll binary tree method binominal method k dWilliam Sharpe 31978c O d J 1979c Cox Ross Rubinstei 357K L 6 uL d 0 T J l m d dy3 Cox Ross Rubinstein d k uO d 3u I A u n E d K Shanghai Normal University Home Page Title Page JJII JI Page 4 of 70 Go Back Full Screen Close Quit 2 1 2 1 1 ddd q VG one period and two state mod el 3 0 T t 0 1 3 Ft T 1 VG I d 3 5 t T k U G e o L n d b I d S L A d V L 3 t 0 d P S0 O u d L e G o3 5 t T XJ d u G d P Su T d A d P V u T XJ d ueOG d P Sd T A d P V d T 2 1 Shanghai Normal University Home Page Title Page JJII JI Page 5 of 70 Go Back Full Screen Close Quit 2 1 d d y3 E x x x 7L x d 3t 0 d 0 V0 S0 g V0 Shanghai Normal University Home Page Title Page JJII JI Page 6 of 70 Go Back Full Screen Close Quit b x r E d k b 3 duT x oU YE O 3t T T VT ST erT 0 2 1 duI d 3t T k e U G u d u V u T Su T d e d Sd T Sd T Shanghai Normal University Home Page Title Page JJII JI Page 7 of 70 Go Back Full Screen Close Quit F d z 5 d d V u T Su T Sd T Sd T V u T V d T Su T Sd T V S 2 2 Cz d Cz 3 d X 2 1 V0 S0 e rT V u T Su T V0 S0 e rT Sd T Sd T l d V0 S0 e rT V u T Su T 2 3 Shanghai Normal University Home Page Title Page JJII JI Page 8 of 70 Go Back Full Screen Close Quit 2 1 2EEE d 0 E d 2 1 b 7K d x X S x X B XJ 3 S B t T d V d ST BT VT o V EEE replicating Shanghai Normal University Home Page Title Page JJII JI Page 9 of 70 Go Back Full Screen Close Quit 5 b x 3t 0 1 2 1 3t 0 d 0 S0 2 4 o3t T d u Su T erT d e d Sd T erT 2 1 3t T d V d ke Su T erT V u T 2 5 Shanghai Normal University Home Page Title Page JJII JI Page 10 of 70 Go Back Full Screen Close Quit Sd T erT V d T 2 6 2 5 2 6 V u T V d T Su T Sd T V u T Su T e rT 2 4 V0 S0 e rT V u T Su T 2 3 Shanghai Normal University Home Page Title Page JJII JI Page 11 of 70 Go Back Full Screen Close Quit XJ NL L n V0 e rT herTS0 Sd T Su T Sd T V u T Su T erTS0 Su T Sd T V d T i 2 7 2 7 V u T V d T X kA V u T X q erTS0 Sd T Su T Sd T 2 8 KV d T X 1 q Su T erTS0 Su T Sd T 2 9 2 7 V0 e rT h qV u T 1 q V d T i 2 10 p q e 0 xxx 555VVV risk neutral probability Q Shanghai Normal University Home Page Title Page JJII JI Page 12 of 70 Go Back Full Screen Close Quit 2 1 3 xxx 555VVV 3 2 10 L y d 5 d by S n y 2 8 q v 0 q 1 XJq 0 d 2 8 erTS0 1 Kd 2 9 erTS0 Su T 3 3 e 5d Z Su T u 3 S0 vk n d d k0 q 1 n q V 5w 2 2 d 2 1 8 V Q x 5V x 5 Shanghai Normal University Home Page Title Page JJII JI Page 13 of 70 Go Back Full Screen Close Quit L n y 0 q 1 d k Q dV 2 10 V0 e rTEQ VT 2 11 d L 3 x 5V e d y u F d by 2 3 S x B x St Bt 3t x S bbbyyyddd discounted price ddd related price duB x BT erTB0 d 2 1 11 U V0 B0 EQ VT BT 2 12 2 12 L 3 x 5V e byd y u F byd Shanghai Normal University Home Page Title Page JJII JI Page 14 of 70 Go Back Full Screen Close Quit e x 5V Q 3t 0 d V0 d S0 2 11 k S0 e rTEQ ST e rT h qSu T 1 q Sd T i u L O EQ ST BT e rT B0 qSu T 1 q Sd T e rT B0 e rTS0 Sd T Su T Sd T Su T Su T erTS0 Su T Sd T Sd T S0 B0 EQ ST S0 S0 BT B0 S0 S0 S0 BT B0 B0 L 3 x 5V Q e x S 3t T xy d r k 5 7K xxx 555 risk neutral world 3 p k x 2 ky x r l 3 x 5 ed 2 1 10 2 1 11 d x 5d Shanghai Normal University Home Page Title Page JJII JI Page 15 of 70 Go Back Full Screen Close Quit 3 x 5 w y wX x x 5V Q Au e G O u d 5 A d u L d L e u 1 0 d 1 5 ud 1 u 3 e Ft T d U Su T uS0 Sd T dS0 yb 3y V P d V p 3 x 5 d V q 2 2 2 2 e d C Shanghai Normal University Home Page Title Page JJII JI Page 16 of 70 Go Back Full Screen Close Quit 5 b k t 3y L L m t d Cz u b EP ST S0 e t 2 1 13 N 5 b EP ST S0 e t 2 2 t 2 14 d 2 2 a b 2 1 13 2 1 14 puS0 1 p dS0 S0e t 2 15 pu2 1 p d2 pu 1 p d 2 2 t 2 16 u d 2 15 p e t d u d 2 17 2 2 17 2 16 e t u d ud e2 t 2 t 2 18 Shanghai Normal University Home Page Title Page JJII JI Page 17 of 70 Go Back Full Screen Close Quit ex m t9 p 5 ud 1 K 2 18 u e t d e t 3 x 5 XJ V q o 2 2 b K d 2 1 8 x k t n x 5V Q L q er t d u d 2 19 aqu 2 15 2 18 B 3 x 5 d 9 v L O puS0 1 p dS0 S0er t 2 20 er t u d ud e2r t 2 t 2 21 2 15 2 20 9 2 1 18 2 21 w ly x 5 lP Q l A u Cz l C x r E C l x x C x 5 Q P d martingale measure Shanghai Normal University Home Page Title Page JJII JI Page 18 of 70 Go Back Full Screen Close Quit p d0 de 8 F f8x 2 4 m F V P V Q d V 8 A F k P A 0 Q A 0 P 1 Q 1 d O V U U U V o Shanghai Normal University Home Page Title Page JJII JI Page 19 of 70 Go Back Full Screen Close Quit o martingale a18 V61u I 5 5 u L Vg5gu IV 4 P P Levy 1939 c 4 g r n u 1 K I J L Doob n X 1 L k 320 V70 c 7K L 5 d L y3 37K L AO 7K k2 A e l m 2 5 XJ l m L Xn n N v e 1 E Xn 2 E Xn 1 X1 Xn Xn K Xn L d 9 c k e ud K L C S l m Shanghai Normal University Home Page Title Page JJII JI Page 20 of 70 Go Back Full Screen Close Quit c y 3 x 5 Q e byd y u F byd b yd L V B tn EQ V B tn 1 g V0 Vtn o n N p g V0 Vtn L V0 Vtn k Uu 31 O An uud udu duu C O 25 24 44 21 25 T d O cuud 3 10 cudu 3 66 cduu 6 85 n 1n A n O udd dud ddu C O 20 77 20 77 18 06 d T d O cudd cdud 0 cddu 2 71 e A ddd C 15 35 T d cddd 0 Shanghai Normal University Home Page Title Page JJII JI Page 60 of 70 Go Back Full Screen Close Quit 3t 2 2 44 q O 3 d cuu e rt qcuuu 1 q cuud e 0 05 1 2 3 13 02 1 3 3 10 9 25 cud e rt qcudu 1 q cudd e 0 05 1 2 3 3 66 1 3 0 2 33 cdu e rt qcduu 1 q cdud e 0 05 1 2 3 6 85 1 3 0 4 35 cdd e rt qcddu 1 q cddd e 0 05 1 2 3 2 71 1 3 0 1 72 I 5 3 m k cud cdu n 3t 1 O 2 d cu e rt qcuu 1 q cud e 0 05 1 2 3 9 25 1 3 2 33 6 61 cd e rt qcdu 1 q cdd e 0 05 1 2 3 4 35 1 3 1 72 3 31 t 0 k2 v d w d 2 12 c0 e rt qcu 1 q cd e 0 05 1 2 3 6 61 1 3 3 31 5 25 Shanghai Normal University Home Page Title Page JJII JI Page 61 of 70 Go Back Full Screen Close Quit 2 12 k2 v d w d Shanghai Normal University Home Page Title Page JJII JI Page 62 of 70 Go Back Full Screen Close Quit r 6 6uI d 3 k S p k A A k AAA duI d o uI d X d d o u A IO d aq 3 F a 1 vvv ddd w JT K 2 222 vvv ddd w ST JT d A kkk vvv ddd Asian options with fi xed strike price kkk222 vvv ddd Asian op tions with fl oating strike price Shanghai Normal University Home Page Title Page JJII JI Page 63 of 70 Go Back Full Screen Close Quit d d aq C u C 3t tn Jn i k 1 n n X 1 S i Sn in Sn i 3t tn 1 C Jn 1 i ku 1 n 1 n X 1 S i Sn 1 i 1 n n 1J n i k 1 n 1S n 1 i 1 Jn 1 i kd 1 n 1 n X 1 S i Sn 1 i n n 1J n i k 1 n 1S n 1 i uA C 3t tn Jn i k n 1S i 1 n 3t tn 1 C Jn 1 i 1 ku Jn i k n n 1 Sn 1 i 1 1 n 1 Jn 1 i kd Jn i k n n 1 Sn 1 i 1 n 1 Shanghai Normal University Home Page Title Page JJII JI Page 64 of 70 Go Back Full Screen Close Quit 2 6 b cd 25 m T 3 d z e 15 x 5 k v d K 25 wO d K8b S0 25 u 1 15 d 0 85 r 0 05 K 25 T 3 dd x d 2 11 d 2 5 x 5V q 2 3 e OO C d 3 FT 3 1 k uuu C 28 75 33 06 38 02 3 33 28 u T wO d puuu max 25 33 28 0 0 n 31 O An uud udu duu C O 29 97 27 10 24 60 T wO d O puud pudu 0 pduu 0 40 31n A n O udd dud ddu C O 24 65 22 15 20 03 d T wO d O pudd 0 35 pdud 2 85 pddu 4 97 e A ddd C 18 22 T wO d pddd 6 78 Shanghai Normal University Home Page Title Page JJII JI Page 65 of 70 Go Back Full Screen Close Quit 3t 2 2 44 q O 3 d kd O puu e rt qpuuu 1 q puud e 0 05 1 2 3 0 1 3 0 0 pud e rt qpudu 1 q pudd e 0 05 1 2 3 0 1 3 0 35 0 11 pdu e rt qpduu 1 q pdud e 0 05 1 2 3 0 40 1 3 2 85 1 16 pdd e rt qpddu 1 q pddd e 0 05 1 2 3 4 97 1 3 6 78 5 29 I 5 3 m k pud pdu du Jc 3t 2 XJ wO du3 C O 30 91 26 60 22 85 19 66 d 3 d O 0 0 2 15 5 34 kd d K 3t 2 3 wO Sd 1 16 2 15 5 29 5 34 3 d O 0 2 15 5 34 n 3t 1 O 2 d Shanghai Normal University Home Page Title Page JJII JI Page 66 of 70 Go Back Full Screen Close Quit pu e rt qpuu 1 q pud e 0 05 1 2 3 0 1 3 0 11 0 04 pd e rt qpdu 1 q pdd e 0 0