Black & choles Hull & Whie GARCH GARCH EGARCH EGARCH EGARCH GARCH EGARCH EGARCH GARCH GARCHEGARCH I
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- 缠工 甫
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2 Black & choles Hull & Whie GARCH GARCH EGARCH EGARCH EGARCH GARCH EGARCH EGARCH GARCH GARCHEGARCH I
3 Absrac ubprime morgage leads he financial sysem o be assauled grealy, and cause he global financial sunami. he origin is he derivaives manipulae excessive. In our empirical sudy, Black & choles and Hull & Whie models are used o demonsrae he AIEX opions wih GARCH and EGARCH models. I has hree resulsfirs, before he financial sunami, he pricing bias of call opion is smaller han he period afer he financial sunami. For pu opion, people hink he sock marke will slump, so buying a large amoun of pu opion is a beer sraegy o hedge. I will cause over reflecion on he price of pu opion, so he pricing bias of pu opion is larger in he period before he financial sunami. econd, for call opion, no maer before and afer he financial sunami, he esimaion resul of EGARCH model is relaively good. For pu opion, before he financial sunami, he leverage effec of EGARCH le volailiy esimaion will be more affeced by recen volailiy, so he esimaion resul of GARCH model is relaively good. Afer he financial sunami, he asymmeric effec is more obvious, so he esimaion resul of EGARCH model is relaively good. he las, asymmeric effec can be separaed from adjusmen of he oulier value afer we use he inervenion model in volailiy esimaion. No maer before and afer he financial sunami, he price bias of GARCH model is relaively small. Key wordsopionasymmeric modelinervenion modelgarchegarch II
4 GARCH ARIMAp,d,q III
5 /7/~008/9/ /7/~008/9/ GARCH EGARCH GARCH B& H&W GARCH B& H&W GARCH B& H&W GARCH B& H&W EGARCH B& H&W EGARCH B& H&W EGARCH B& H&W EGARCH B& H&W GARCH B& H&W GARCH B& H&W GARCH B& H&W GARCH B& H&W EGARCH B& H&W EGARCH B& H&W EGARCH B& H&W EGARCH B& H&W I
6 GARCH, EGARCH, ARIMAp,q AICBIC MAPE B& RMPE B& MAPE H&W RMPE H&W MAPE B& RMPE B& MAPE H&W RMPE H&W MAPE B& RMPE B& MAPE H&W RMPE H&W
7 Fama965 ARCH Black976 Chrisie98 & chwer990 GARCH Nelson99 GARCH Exponenial GARCH ModelEGARCH Zakoian994 GARCH hreshold GARCH ModelGARCH
8 008. GARCH EGARCH Black & choles Hull & Whi.. EGARCH GARCH 3.
9 .3.3. Black & choles Hull & Whie GARCH EGARCH Black & choles Hull & Whie 3
10 .3. B& H&W GARCH EGARCH B&+GARCH B&+EGARCH H&W+GARCH H&W+EGARCH 4
11 . Fisher Black & Myron choles 973 Black & choles Black & choles Hull & Whie987 Meron973Amin & Jarrow99 co997 Black & choles Black & choles Corrado & u998 Hull & Whie &P 500 Hull & Whie Black & choles Hull & Whie Black & choles Hull & Whie Black & choles 5
12 . Engle98 Auoregressive Condiional HeeroscedasiciyARCH Bollerslev986 ARCH General ARCH GARCH ARCH ARCH ARCHGARCH GARCH Nelson99 GARCH Exponenial GARCH Model EGARCH Zakoian994 GARCH hreshold GARCH Model GARCH Koumos, Lee & heodossiou994 0 GARCH, GARCH, Cheung & Ng99 EGARCH 6
13 AMEX-NYE Engle & Ng993 news impac curve GARCHEGARCHAGARCHNGARCHGJR OPLX Koumos998 GARCH 9 uleyman Gokcan000 GARCH, EGARCH, GARCH 995 GARCHEGARCHAGARCH NGARCH GARCH GARCH 005 Black & choleshull & Whie Heson GJR-GARCH GJR-GARCH Black & choles 7
14 Black & choles Black & choles Hull & Whie EGARCH EGARCH δ= EGARCH δ=
15 3. Fisher Black & Myron choles 973 Black & choles Black & choles Black & choles 973 Hull & Whie Black & choles Model Black & choles lognormal disribuion Random Walk Process Geomeric Browing Moion Black & choles d d dz 9
16 d d dz d andard Wiener Process f, Ioˆ s Lemma f f f f df d dz Winner Process dz f H f, f H - f, d 3 dh f d - df 4 d d dz f f f f df d dz f f f dh d df d 6 Black & choles r 67 dh r H d 7 0
17 - f - f dh d r f - f, d 8 8 Black & choles, f f rf r f 9 f, X - X, if X 0, if X 0 X Black & choles Fourier Blasck & choles C N d- Xe N d - r - - r - P Xe N- d-- N d C Call opion P Pu opion d d ln - - X r ln X r N - X
18 3.. Hull & Whie Model d d dw 3 d d dz 4 Cov dz, dw d 5 dw dz :andard Wiener Process f,, Ioˆ s Lemma f f f f f f df d d d d d dd!! f f f d d dw d dz f f f 3 d d d f f f f f f 3 d f f dw dz d f dw f dz Wiener Process 67 6
19 f f f f f f 3 E df d 7 df Garman976 f,, i j f f f * ij i j - rf i -i i - r 8 i, j : i i : i i : ij i j r : i j i i * : d i : i N+ CAPM j * i f i -r i 89 i f f 3 f f - rf f * f -r r 9 : d : r *
20 f f 3 f f - rf f f -rs - Hull & Whie 0 0 risk-neural valuaion - r - e f g d C f f,,, e, f, P d - r - - d P P g d h d r - f,, e f g d h d 3 f 0 log ~ - N, r 0 4 Black & choles C Hull & Whie f,, C h d 4 C - N d Xe N d - r - - r - e f g d C 4
21 d ln - r X - d ln r - - X - k - C aylor 4: C C 5 3 f,= C ar kew Boyle & Emanuel 985 μ=0 6 Hull & Whie μ=0ρ=0 4 k -N d dd - -e - k 4 C f, C k -N d dd -3 dd -- d d k k 3 6 e k e 4 8k 6 k k 3 3k 6 4 k -N d dd - -e - k 4 P f, P k -N d dd -3 dd -- d d k k 3 6 e k e 4 8k 6 k k 3 3k 7 5
22 3. GARCH EGARCH 3.. GARCH Engle98 Auoregressive Condiional HeeroscedasiciyARCHp ARCHpBollerslev986 BollerslevARCH ARCH ARCHGeneral ARCHGARCHGARCH Linear GARCH ModelGARCHp,q y ARCH 6
23 3.. Exponenial GARCH Nelson99 Exponenial GARCHEGARCH GARCH EGARCH
24 3.3 Inervenion Oulier Model, IOM 3.3. exogenous evens 0inervenion analysisbox & Jenkins975 8 y f k,, 8 N y f k,, k N 3.3. N y f k,, Auoregressive Inegraed Moving average ModelARIMA9 B By y B N B a 9 a Whie Noise process 0 p B B B... pb q B B B... qb Auoregressivemoving averagepqb B B d30 B d B B B 30 8
25 3.3.3 Box & iao9753 k k j B f,,, Y j j 3 j j j B r s B j B jb... rjb j B jb... sjb rs B Y 3 B B y Y N a B B B Y B 3 j Chang, iao & Chung988 Innovaive Oulier, IOAddiive Oulier, AO 9
26 B IO X Z B B 33 AO X Z 34 X Z 35 m j j j j X v B Z 35 m AO v B j j IO v B B B B j B B B B B B B B... e B X B Z B IO e 36 AO e B 37 Leas squares heoryaoio 3839 IO I e 38 AO A F e n F... F F e n 39 F... n var ˆ I 0
27 var ˆ A 404Chang, iao & Chung988 I A H 0 : 0 IO, ˆ I I 40 H : 0 H0 : 0 AO, ˆ A, A 4 H : IO IO max I, C 4,..., n AO AO max A, C 43,..., n CChang, iao & Chung988C3.0C3.5 C4.0 Fox97AOIO IO I,, I, A, A AO Innovaive Oulier, IO IO Addiive Oulier, AO AO
28 3Level hif, L 4emporary Change, C 3.4. mean absolue percenage error, MAPE n.roo mean squared percenage error, RMPE n
29 Black & chole Hull & Whie EJ R P %
30 4 005/7/008/9/30 005/7/008/9/
31 Bakshi, Cao, & Chen997 A-he-money/K0.97~.03 In-he-money/K.03~.06Ou-of-he-money/K0.94~0.97 Deep-in-he-money/K>.06Deep-ou-of-money /K< call pu GARCH EGARCH GARCH EGARCH EGARCH 4 hp:// 5
32 6 GARCH GARCH EGARCH EGARCH
33 4. ARCH 4.. Jarque & Bera980 J-B -n 6 4 K Jarque-Bera 3 n K 0 3 J-B J-B eries: A ample 806 Observaions 806 Mean Median Maximum Minimum d. Dev kewness Kurosis Jarque-Bera Probabiliy
34 4.. aionary Uni Roo es y y y y p p N 0, Augmen Dickey & Fuller ADF 979 H 0 ADF -8.94% %
35 4..3 ARCH ARCH family Model ARCH Engle 98 Lagrange Muliplier LM LM R x p H : a a a 0 0 H : a i a R j ARCH es R.58Probabiliy P 0.05 ARCH n 9
36 4.3 GARCH GARCH Eviews Hull & Whie = α ξ 5 GARCH ξ 4.3. GARCH GARCH ln c ~0, N - -, 0, 0, 0 - GARCH, Z P-value c ω α β EGARCH
37 EGARCH, Z P-value c ω β ARIMAp,d,q d=0 ARMAp,q ARMA p,q AIC BIC ARIMA3,0, ARIMA,0, AIC BIC BIC ARIMA,0, ARMAp,qAICBIC ARp MAq AIC BIC * * *AICBIC
38 -PLU6.0ARIMA,0, AOL -value 006/6/8 AO /6/0 AO /3/5 AO /7/7 AO /8/ AO /8/6 AO /8/0 AO //8 AO //3 AO //7 AO //7 AO // AO //4 AO /6/ L /7/5 AO /7/7 AO /7/ AO /7/3 AO /9/8 AO /9/6 AO /9/9 AO AO 7 AO L
39
40 MCI 4% % 5%
41 % % % IndyMac
42 % ~ ~ % Fannie MaeFreddie Mac UFJ 5% 3% AIG &P %
43 GARCH EGARCH Black & choles Hull & Whie MAPERMPE Black & choles MAPE RMPE 37
44 EGARCH EGARCH GARCH EGARCH MAPE RMPE GARCH EGARCH GARCH EGARCH EGARCH EGARCH GARCH Hull & Whie Black & choles 6 38
45 MAPE B& MAPEB& GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH MAPEB& GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH EGARCH GARCH EGARCH 39
46 RMPE B& RMPEB& GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH RMPEB& GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH EGARCH GARCH EGARCH 40
47 MAPE H&W MAPEH&W GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH MAPEH&W GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH EGARCH GARCH EGARCH 4
48 RMPE H&W RMPEH&W GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH RMPEH&W GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH EGARCH GARCH EGARCH 4
49 4.5. Black & choles MAPE RMPE MAPE RMPE GARCH EGARCH GARCH Hull & Whie Black & choles Black & choles MAPE RMPE 4 Hull & Whie 43
50 Black & choles MAPE B& MAPEB& GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH MAPEB& GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH 44
51 RMPE B& RMPEB& GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH RMPEB& GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH 45
52 MAPE H&W MAPEH&W GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH MAPEH&W GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH 46
53 RMPE H&W RMPEH&W GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH RMPEH&W GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH 47
54 MAPE B& MAPEB& IOM IOM IOM IOM GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH MAPEB& IOM IOM IOM IOM GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH
55 RMPE B& RMPEB& IOM IOM IOM IOM GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH RMPEB& IOM IOM IOM IOM GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH
56 MAPE H&W MAPEH&W IOM IOM IOM IOM GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH MAPEH&W IOM IOM IOM IOM GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH
57 RMPE H&W RMPEH&W IOM IOM IOM IOM GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH RMPEH&W IOM IOM IOM IOM GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH GARCH EGARCH
58 5. Black & choles Hull & Whie GARCH EGARCH.. EGARCH EGARCH GARCH EGARCH EGARCH 5
59 GARCH GARCH EGARCH GARCH
60 Black & choles Amin, K. and Jarrow, R., Pricing Opions on Risky Asses in a ochasic Ineres Rae Economy, Mahemaical Finance, 7-37, 99.. Black, F. and choles, M., he Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy, , Black, F., udies in ock Price olailiy Changes, Proceedings of he 976 Business Meeing of he Business Economic aisics ecion, American aisical Associaion77-8, Bollerslev,., Generalized auoregressive condiional heeroskedasiciy, Journal of Economerics, , Box, G. E. P., and iao, G. C., Inervenion Analysis Wih Applicaions o Economic and Environmenal Problems, Journal of he American aisical Associaion, , Chang, I., iao, G. C., and Chen, C., Esimaion of ime eries Parameers in he presence of Ouliers, echnomerics, , Charles Corrado, u, An Empirical es of he Hull-Whie Opion Pricing Model, Journal of Fuures Markes, , Cheung, Y. W. and Ng, L. K., ock Price Dynamics and Firm izean Empirical Invesigaion, Journal of Finance, , Chrisie, A. A., he ochasic Behavior of Common ock ariances: alue, Leverage and Ineres Rae Effecs, Journal of Financial Economics, , Engle, R. F., Auoregressive Condiional Heeroscedasiciy wih Esimaes of he ariance of Unied Kingdom Inflaion, Economerica, ,
61 . Engle, R. F. and. M. Ng, Measuring and esing he Impac of News on olailiy, Journal of Finance, , Fama, E., he Behavior of ock Marke Prices, Journal of Business, , Hull, J. C. and Whie, A., he Pricing of Opions on Asses wih ochasic olailiies, Journal of Finance, , Koumos. G., Lee. U. and P. heodossiou, ime-varying Beas and olailiy Persisence in Inernaional ock Markes, Journal of Economics and Business, 460-, Koumos, G., Asymmeries in he Condiional Mean and he Condiional arianceevidence from Nine ock Markes, Journal of Economics and Business, , Meron, R. C., heory of Raional Opion Pricing, he Bell Journal of Economics and Managemen cience, 44-83, Nelson, Daniel B., Condiional Heeroskedasiciy in Asse ReurnsA New Approach, Economerica, , chwer, W. G., ock olailiy and he Crash of 87, he Review of Financial udies, 377-0, co, L. O., Pricing ock Opions in a Jump-diffusion Model wih ochasic olailiy and Ineres Raes Applicaions of Fourier Inversion Mehods, Mahemaical Finance, , uleyman Gokcan, Forescasing olailiy of Emerging ock MarkesLinear ersus Non-linear GARCH Models, Journal of Forecasing, , Zakoian, Jean-Michel, hreshold Heeroskedasic Models, Journal of Economic Dynamics and Conrol, ,
62 56 Black & choles Black & choles Black & choles f r f f rf, X if X if X K f 0,. log r X u x,,, r rx f e y u x e y u x u uu rx u rx x u rx y y e f f y e x x f u u f f y ry y r e x x f u u f f y e r y y e y e r y ry y r e rx u uu rx u rx x u rx rx rx x uu e r y y y e r y 7
63 e rx ry yx yuu ry Black & choles y uu u, x y u, x 0 x X e y u,0 0 u if if X X 0 0. y u, x u W x y uu u, x u W x y u, x u W x uu 3 k k 0 uu u u x W x W x x 3 x k 4 uu u k u Wx x k 5 W x C k C k k C 3 u C k cos xu C k sin ku k x 5 W x C3e y u, x cos u Wsin x C k ku D k ku e 6 k x 6 k 0 k 7 k x C kcos ku D ksin ku e 0 y u, x dk 7 57
64 3. Fourier x 0 y u,0 g u C kcosku D ksin ku dk 0 gu ga Fourier C k, D k C k D k g acos ka da g asin ka da Xe g a 0 u X u 0 u 0 C k, D k 78 k g acos ka dacos ku 0 g asin ka dasin ku e y u, x dk k g acos ka cos ku da 0 g asin ka sin ku da e k g a coska cos ku sin ka sin ku 0 da e 0 g acos k a u da e g a e au x k da x dk dk dk au x g a e x da 8 a u v a u xv x u x v u X e, v x g a G u x v u 0, v x 58
65 59 Black & choles, v y u x g a e da x dv x e a g x v dv e a g v x u v x v u dv e X X e x u v x u v x v u dv X e dv e X e 9 x v z, x u y x u z x x u z rx dz e X dz e e x u y, x u N X x x u N e rx 0 Black & choles, x u N Xe x x u N f rx d N Xe d N r C
66 60 C d g f e r r d g f e x r d g X f e x r x r d g X e d g e ln w Jacobian w ln ln X r w X r dw w g X e dw w g e e ln ln X r w X r dw w g X e dw w g e e r w y Jacobian 3 dy y g e e r X r r ln ln r X r dy y g X e dy e e e y r X r r ln
67 6 r X N X e r ln dy e e e r X y r r ln r K N X e r ln 3 y z Jacobian y z3 4 r X N X e dz e r r X z ln ln r X N X e r X N r ln ln r X N X e r X N r ln ln d N Xe d N r C 4
68 GARCH, GARCH, u i u 4 4 u Winener process dz u E E u u 3 d a L d L 4 d 5 d d d L 5 8 John C. Hull, Opions, Fuures and Oher Derivaives, h 5, assignmen quesions 7.4 6
69 GARCH GARCH Black & choles GARCH Hull & Whie 63
70 GARCH GARCH Black & choles GARCH Hull & Whie 64
71 GARCH GARCH Black & choles GARCH Hull & Whie 65
72 GARCH GARCH Black & choles GARCH Hull & Whie 66
73 EGARCH EGARCH Black & choles EGARCH Hull & Whie 67
74 EGARCH EGARCH Black & choles EGARCH Hull & Whie 68
75 EGARCH EGARCH Black & choles EGARCH Hull & Whie 69
76 EGARCH EGARCH Black & choles EGARCH Hull & Whie 70
77 GARCH GARCH Black & choles GARCH Hull & Whie 7
78 GARCH GARCH Black & choles GARCH Hull & Whie 7
79 GARCH GARCH Black & choles GARCH Hull & Whie 73
80 GARCH GARCH Black & choles GARCH Hull & Whie 74
81 EGARCH EGARCH Black & choles EGARCH Hull & Whie 75
82 EGARCH EGARCH Black & choles EGARCH Hull & Whie 76
83 EGARCH EGARCH Black & choles EGARCH Hull & Whie 77
84 EGARCH EGARCH Black & choles EGARCH Hull & Whie 78
85 79
86 80
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Sock Volailiy Models and he Pricing of Warrans 36005 005 0 Hong & Li 005 Absrac This paper used a lo of popular volailiy models o sudy he dynamic behavior of underlying sock and hen used Hong & Lee (005)
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