金融数学讲座3.pdf

Shanghai Normal University Home Page Title Page JJII JI Page 1 of 33 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 33 Go Back Full Screen Close Quit 1n K a Jc d K 3 1 3 2 3 3 3 4 K 3 5 Shanghai Normal University Home Page Title Page JJII JI Page 3 of 33 Go Back Full Screen Close Quit 111nnn KKK aaa JJJccc ddd KKK 1 21 V 7K u 7K x U Or X7K u 4 T 1 e T Shanghai Normal University Home Page Title Page JJII JI Page 6 of 33 Go Back Full Screen Close Quit 3 f iB3 V Be B T t 4 1 iAw d B X t F T2 b w 3 FT2 v d K K3 FT2w d B X K 5 1 iA V dt d K Q B 0 1 d d Cz B Q 1 B 6 f iBu K1 iAa r A1 a b f iB u K1 iAa r A2 a b d1 iA kf iB 1 iAu1 7 3 Shanghai Normal University Home Page Title Page JJII JI Page 7 of 33 Go Back Full Screen Close Quit 3 E B X t X t 3 t t dt m S Ito d Cz d 1 dB dX B t dt B XdX 1 2 2B X2 dX 2 dX B t dt 1 2 2 j2 t X 2 2B X2dt B X dX d 2 1 B d n k E d dB dX 1 dt 1 B dt r dt r B X dt B t dt 1 2 2 j2 t X 2 2B X2dt B X dX 1 dt 1 B dt r B 4X dt Shanghai Normal University Home Page Title Page JJII JI Page 8 of 33 Go Back Full Screen Close Quit B X dt p B t 1 2 2 j2 t X2 2B X2 rX B X r 1 B 0 e K z b 1 iA d a j t j l 4 B t 1 2 2 j2 X 2 2B X2 rX B X r 1 B 0 B X T X K 1 Shanghai Normal University Home Page Title Page JJII JI Page 9 of 33 Go Back Full Screen Close Quit 4 KKK 1 duf iB U u d1 iA d f iB K x 1 iA d b 1 iA F T2 3d ck T1 T1 lT2 C Kd e m 1 3 0 T1 m S f iB u K3 T1 T2 m Sk i ef iB3 T1 T2 m Su 1 iAa L r A1 a b K3 T1 T2 m S d 1 1 iA d v B1 t 1 2 2 a b j2 X 2 2B1 X2 rX B1 X r a b 1 B1 0 T1 t T2 B1 X T2 X K 2 Shanghai Normal University Home Page Title Page JJII JI Page 10 of 33 Go Back Full Screen Close Quit ii ef iB3 T1 T2 m SE u 1 iAa L r A2 a K3 T1 T2 m S 1 iA d v B2 t 1 2 2 aj2 X 2 2B2 X2 rX B2 X r a 1 B2 0 T1 t T2 B2 X T2 X K 3 Shanghai Normal University Home Page Title Page JJII JI Page 11 of 33 Go Back Full Screen Close Quit e 1 1 T2 t Y lnX W Y B X t K k B t W B X 1 X W Y 2B X2 1 X2 W Y 1 X2 2W Y 2 u 1 C W 2 j2 2 2W Y 2 r 2 j2 2 W Y r 1 W 0 W Y 0 eY K 4 Shanghai Normal University Home Page Title Page JJII JI Page 12 of 33 Go Back Full Screen Close Quit 2 W Y emY n U Y O Y m r 2 j2 2 2 j2 n 1 8 2r 2 j2 2 2 j2 r 1 K 4 C U 2 j2 2 2U Y 2 U Y 0 e mY eY K 5 Black Scholes 5 d 5 1 B X t Xe 1 T2 t N d1 Ke r 1 T2 t N d2 d1 ln X K r 2 j2 2 T2 t 2 j2 T2 t d2 ln X K r 2 j2 2 T2 t 2 j2 T2 t Shanghai Normal University Home Page Title Page JJII JI Page 13 of 33 Go Back Full Screen Close Quit d f iB3 T1 T2 m S 2 B1 X t Xe a b 1 T2 t N d1 Ke r a b 1 T2 t N d2 f iB3 T1 T2 m SE u 3 B2 X t Xe a 1 T2 t N d3 Ke r a 1 T2 t N d4 u 3T1 e 1 iA d O B1 X T1 Xe a b 1 T2 T1 N b d1 Ke r a b 1 T2 T1 N b d2 B2 X T1 Xe a 1 T2 T1 N b d3 Ke r a 1 T2 T1 N b d4 Shanghai Normal University Home Page Title Page JJII JI Page 14 of 33 Go Back Full Screen Close Quit b d1 ln X K r 2 a b j2 2 T2 T1 p 2 a b j2 T2 T1 b d2 ln X K r 2 a b j2 2 T2 T1 p 2 a b j2 T2 T1 b d3 ln X K r 2 aj2 2 T2 T1 p 2 aj2 T2 T1 b d4 ln X K r 2 aj2 2 T2 T1 p 2 aj2 T2 T1 duf iB3 T1 T2 m S 1 iA d K i3 0 T m V P T 1 P T 1 e T KP1 1 e B T2 T1 P2 e B T2 T1 l 3T1 1 iA d B12 X T1 Shanghai Normal University Home Page Title Page JJII JI Page 15 of 33 Go Back Full Screen Close Quit B12 X T1 1 e B T2 T1 h Xe a b 1 T2 T1 N b d1 Ke r a b 1 T2 T1 N b d2 i e B T2 T1 h Xe a 1 T2 T1 N b d3 Ke r a 1 T2 T1 N b d4 i du3 0 T1 m S f iBvk K1 iA v B t 2 aj2 2 X 2 2B X2 rX B X r a 1 B 0 0 t T1 B X T1 B12 X T1 b K 6 Shanghai Normal University Home Page Title Page JJII JI Page 16 of 33 Go Back Full Screen Close Quit T1 t Y lnX B X t W Y emY n U Y m r 2 aj2 2 2 aj2 n 1 8 2r 2 aj2 2 2 aj2 r a 1 b K 1 iA 3T1 v d K 6 z U 2 aj2 2 2U Y 2 U Y 0 e mY B12 Y T1 K XJU Y 0 0 Y Y Kk U Y R 1 2 aj2 2 e Y 2 2 2 aj2 U 0 d R Y 1 2 aj2 2 e Y 2 2 2 aj2 e m B12 Y T1 K I1 I2 I3 I4 I5 Shanghai Normal University Home Page Title Page JJII JI Page 17 of 33 Go Back Full Screen Close Quit e I4 I4 R Y 1 2 aj2 2 e Y 2 2 2 aj2 e m Ke B T2 T1 e r a 1 T2 T1 N b d4 d Ke B T2 T1 e r a 1 T2 T1 2 aj2 2 R Y e Y 2 2 2 aj2 2r 2 aj2 2 2 aj2 1 2 aj2 2 T2 T1 R e w lnK r 2 aj2 2 T2 T1 2 2 2 aj2 T2 T1 dwd Ke B T2 T1 e r a 1 T2 T1 2 2 aj2 T2 T1 R Y R e Y 2 2 2 aj2 w lnK r 2 aj2 2 T2 T1 2 2 2 aj2 T2 T1 2r 2 aj2 2 2 aj2 dwd Shanghai Normal University Home Page Title Page JJII JI Page 18 of 33 Go Back Full Screen Close Quit w z d d dz dw Kk I4 Ke B T2 T1 e r a 1 T2 T1 2 2 aj2 T2 T1 R Y R 0 e Y 2 2 2 aj2 z lnK r 2 aj2 2 T2 T1 2 2 2 aj2 T2 T1 2r 2 aj2 2 2 aj2 dzd Shanghai Normal University Home Page Title Page JJII JI Page 19 of 33 Go Back Full Screen Close Quit H K H Y 2 2 2 aj2 z lnK r 2 aj2 2 T2 T1 2 2 2 aj2 T2 T1 2r 2 aj2 2 2 aj2 T2 T1 2 z2 2 z 2 h Y T2 T1 T2 T1 r 2 aj2 2 r 2 aj2 2 T2 T1 lnK i 2 2 aj2 T2 T1 2 h r 2 aj2 2 T2 T1 lnK i z Y 2 T2 T1 h r 2 aj2 2 T2 T1 lnK i2 2 2 aj2 T2 T1 Shanghai Normal University Home Page Title Page JJII JI Page 20 of 33 Go Back Full Screen Close Quit H f z a0 c0 2 2 a0 c0 b0z d0 b0z d0 2 R L X a0 T 2 T1 b0 T 2 T1 c0 T 2 T1 h Y r 2 aj2 2 i d0 h Y T2 T1 r 2 aj2 2 lnK i R 2 T2 T1 r 2 aj2 2 Y 2 T2 T1 r 2 aj2 2 2 2 Y r 2 aj2 2 2 aj2 z z Y T2 T1 r 2 aj2 2 lnK T 2 T1 2 aj2 Shanghai Normal University Home Page Title Page JJII JI Page 21 of 33 Go Back Full Screen Close Quit K I4 Ke B T2 T1 e r a 1 T2 T1 2 2 aj2 T2 T1 R Y R 0 e a 0 c0 2 2 a0 c0 b0z d0 b0z d0 2 R 2 2 aj2 T2 T1 dzd Ke B T2 T1 e r a 1 T2 T1 2 2 aj2 T2 T1 e r 2 aj2 2 Y 2 aj2 r 2 aj2 2 2 2 2 aj2 R Y R 0 e 2 2 z z 2 2 1 T2 T1 dzd l I4 Ke B T2 T1 e r a 1 T2 T1 e r 2 aj2 2 Y 2 aj2 r 2 aj2 2 2 2 2 aj2 M a4 b4 Shanghai Normal University Home Page Title Page JJII JI Page 22 of 33 Go Back Full Screen Close Quit M a b R a dx R b f X Y X Y dy f X Y X Y 1 2 1 2 e x 2 2 xy y2 2 1 2 a4 Y Y r 2 aj2 2 2 aj2 b4 Y T2 T1 r 2 aj2 2 lnK T 2 T1 2 aj2 n aq I1 I2 I3 I5 5 C l 30 1 iA d B Shanghai Normal University Home Page Title Page JJII JI Page 23 of 33 Go Back Full Screen Close Quit B X 0 Xem nT1Xe a b 1 T2 T1 1 e B T2 T1 Xe r 2 aj2 2 2 aj2 r 2 aj2 2 2 T1 2 2 aj2 M b a1 b b1 Xem nT1Ke r a b 1 T2 T1 1 e B T2 T1 Xe r 2 aj2 2 2 aj2 r 2 aj2 2 2 T1 2 2 aj2 M b a2 b b2 Xem nT1Xe a 1 T2 T1 B T2 T1 Xe r 2 aj2 2 2 aj2 r 2 aj2 2 2 T1 2 2 aj2 M b a3 b b3 Xem nT1Ke a 1 T2 T1 B T2 T1 Xe r 2 aj2 2 2 aj2 r 2 aj2 2 2 T1 2 2 aj2 M b a4 b b4 Xem nT1 b KXe r 2 aj2 2 2 aj2 r 2 aj2 2 2 T1 2 2 aj2 N b d5 Shanghai Normal University Home Page Title Page JJII JI Page 24 of 33 Go Back Full Screen Close Quit b a1 lnX lnX T1 r 2 a b j2 2 T 1 2 a b j2 b b1 lnX lnK T2 r 2 a b j2 2 T 2 2 a b j2 b a2 lnX lnX T1 r 2 a b j2 2 T 1 2 a b j2 b b2 lnX lnK T2 r 2 a b j2 2 T 2 2 a b j2 b a3 lnX lnX T1 r 2 aj2 2 T 1 2 aj2 b b3 lnX lnK T2 r 2 aj2 2 T 2 2 aj2 b a4 lnX lnX T1 r 2 aj2 2 T 1 2 aj2 b b4 lnX lnK T2 r 2 aj2 2 T 2 2 aj2 b d5 lnX lnX T1 r 2 a b j2 2 T 1 2 a b j2 Shanghai Normal University Home Page Title Page JJII JI Page 25 of 33 Go Back Full Screen Close Quit 3 0 T1 m S f iB u K 1 iA Jc 3Jc T1 v d K d i 3 B T1 m S f iB3 B u B 0 T1 1 iA d B X t v B3 t 2 a b j2 2 X 2 2B3 X2 rX B3 X r a b 1 B3 0 B t T1 B3 X T1 X K 7 LBlack Scholes 7 B3 X t Xe a b 1 T1 t N d6 Ke r a b 1 T1 t N d7 K3 B d B3 X B Xe a b 1 T1 B N b d6 Ke r a b 1 T1 B N b d7 Shanghai Normal University Home Page Title Page JJII JI Page 26 of 33 Go Back Full Screen Close Quit ii 3 0 B m S f iB u 1 iA d B X t v B4 t 2 aj2 2 X 2 2B4 X2 rX B4 X r a 1 B4 0 0 t B B4 X B B3 X B 8 n B t Y lnX B X t W Y e e mY e n U Y e m r 2 aj2 2 2 aj2 e n 1 8 2r 2 aj2 2 2 aj2 r a 1 Kk U 2 aj2 2 2U Y 2 U Y 0 e e mYB3 Y B Shanghai Normal University Home Page Title Page JJII JI Page 27 of 33 Go Back Full Screen Close Quit u U Y R 1 2 aj2 2 e Y 2 2 2 aj2 U 0 d KB X t R e e mY e n B t 1 2 aj2 2 B t e Y 2 2 2 aj2 B t U 0 d d 3 0 1 iA d B X 0 Z Z T1 0 Be B T1 B Xe e m e n B 1 p 2 aj2 2 B e lnX 2 2 2 aj2 BU 0 d Bd Shanghai Normal University Home Page Title Page JJII JI Page 28 of 33 Go Back Full Screen Close Quit 5 e 3 e 1 iA d b 1 1 iA d B X r t F T2 2 X 1 iA d 1 iA kf iB f iB 1 iA d A K b 1 iA d X l a Brown dX X dt 1dW1t j t dNt 1 u W1t t 0 IOA Brown Nt t 0 r PoissonL j t d a dNt 2 dt 3 3 x 5 e l k E V asicek drt rt dt 2dW2t 2 u 2 W2t IO A Brown Nt t 0 W2t t 0 p Cov dW1t dW2t dt 1 Shanghai Normal University Home Page Title Page JJII JI Page 29 of 33 Go Back Full Screen Close Quit 4 P x E d P v 6 P t 1 2 2 2 2P r2 r P r rP 0 B X r t 1X t 2P r t d 1 h B t 1 2 2 1X 2 2B X2 1 2 j 2X 2 2B X2 1 2 2 2 2B r2 1 2X 2B X r 2 P t 2 1 2 2 2 2P r2 i dt B X 1 dX B r 2 P r dr d 2 1 B E d 1 1 dt d 2 dt r B 1X 2P dt Shanghai Normal University Home Page Title Page JJII JI Page 30 of 33 Go Back Full Screen Close Quit 1 B X 2 B r P r B t rX B X 1 2 2 1X 2 2B X2 1 2 j 2X 2 2B X2 1 2 2 2 2B r2 1 2X 2B X r r 1 B B r P t 1 2 2 2 2P r2 rP P r 3 x E v P t 1 2 2 2 2P r2 r P r rP 0 K 3 e 1 iA v B t rX B X r B r 1 2 2 2 2B r2 1 2 2 1 j2 X 2 2B X2 1 2X 2B X r r 1 B 0