9 : 3 25 Levy Black Scholes (973)Meron(976)Madan (998)Carr Wu (24)Kim (2822) Levy Levy-Khinchine Levy Φ dl (u) Ε[e iudl ]=e -Ψ (u) dl Ψ dl (u)=-iuυ+ u

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1 3 9 ( 237 ) 系统工程 Vol.3No SysemsEngineering Spe.23 :-498(23) * 3 吴恒煜 2 朱福敏胡根华马晶 3 田海山 (. 63; ; ) : 为研究沪深 3 指数价格随机过程的运动形态捕捉金融资产的非高斯新息波动率集聚和杠杆效应三大特征在时间序列分析的基础上采用非对称 GARCH 模型构建了四种不同跳跃程度的离散时变波动率 Levy 过程并建立了波动率漂移率和跳跃的状态空间模型同时引入粒子滤波方法来研究动态波动率和跳跃类型研究表明 : 沪深 3 指数收益率存在大量的随机跳跃而布朗运动无法刻画这些跳跃且卡尔曼滤波也无法正确追踪跳跃状态而调和稳态过程下粒子滤波对资产的时变活动率水平和跳跃形态上具有最佳的捕获能力 : 沪深 3 指数 ; 粒子滤波 ;N-GARCH 模型 ;Levy 过程 ; 调和稳态过程 :F83 :A Black-Scholes(973) BS VarianceGamma (Madan998)Gamma dorf-nielsen998) (Kim Rachev28) Levy BS Levy ( Meron974;Anderson 2;Carr Wu23~27;Rachev Kim28~2) Levy : CIR (Heson993997) (Duan995;Heson2) Meron (Meron976) NormalInverseGaussian (Barn- Levy 28 GARCH Levy Levy Levy * :23--5; : : (7863;78255;7768); (YJA792;3YJA794); (249877); (22T5726); (JBK324; JBK34) : (97-) : ; (985-) : ; (984-) : ; (965-) : ; (962-) :

2 9 : 3 25 Levy Black Scholes (973)Meron(976)Madan (998)Carr Wu (24)Kim (2822) Levy Levy-Khinchine Levy Φ dl (u) Ε[e iudl ]=e -Ψ (u) dl Ψ dl (u)=-iuυ+ u2 σ 2 (Fama963965; 2 + (-e iudl +iudl ) dl v(dl) R Carr2324;Kim282) (υσv(dl)) Levy Levy Carr(24) ( )Levy (26) (27) (2) h ( h (22) y =ln(s/s-)=μ + 槡 h dl ( -φdl 槡 h ) (2) Levy ( 2) Levy φdl (u)=logε[e iudl ] u=-iu =logε[e udl ] (3) ARMA-GARCH y =μ + 槡 h dl -φdl 槡 h = aiy-i + (GARCH) Levy j= p μ = ai[ μ -i +ε-i -φdl Levy Levy : (υσv(dl)) υ σ ( )v(dl) dl p S =S-e μ d+ 槡 dl -φdl 槡 ) q b jε-j +c+ε q p q ( ) ( = aiy-i [ + b j 槡 h- jdl-j + ) c ] + 槡 h dl j= m n h =α + αi(ε-i -γ 槡 h- i) 2 + (4) βjh-j j= ε ;F Ε[ε Levy F-]= var[ε F-]= h h Engle : N-GARCH ; γ > ;2 ε = 槡 h dl = 槡 h Levy(x;υσv(dL)) 槡 h- i] () (5) βjh-j j= ARMA() GARCH() dl ε dl = Levy(x;υσv(dL)) p μ = ai( μ -i +ε-i -φdl + b jε-j +c+φdl 槡 h j= m n h =α + αi(ε-i -γ 槡 h- i) 2 + 槡 h- i)

3 26 23 q + b jε-j +c+φdl j= h 槡 2.2 Levy N T N q(y x i ; θ) x = {dl; μ ; h 槡 } y =μ + 槡 h dl -φdl 槡 h (7) N ;T Kalman Kalman K = ( Pxy P y ) - Ε[x]= x+k(y- 珚 y) 珔 Σ[x]= Px -KP yk T (8) Levy K Kalman E Σ (BM) Levy VG w (i) = w (i) - p(y x (i) )p(x (i) x -) (i) x -y) (i) (9) ARMA GARCH q(x (i) x (i) - y) 3 珦 w (i) = w(i) w(i) () ARCH q(x (i) x (i) - y)=p(x (i) x (i) - ) ( ) w (i) =p(y x (i) ) (2) m n 槡 h = α + αi(ε-i -γ 槡 h- i) 2 + βjh- 槡 ( /N ( j 6) j= ) Levy ( (6) ) ( UnscenedTransform Levy sigma- ) (MCKF) Levy ; MCKF ( ) p({y} T = θ) =2 N N p(y x i ; - θ)(3) θ ;xy ; 3 3 (JD) VG q(x (i) ARCH p(y x (i) ) ;p(x (i) x (i) ) - (a) (b) ;q(x (i) x (i) - y) (c) (d) Levy GARCH

4 9 : 3 27 Sample Auocorrelaion Sample Parial Auocorrelaions Lag Sample Parial Auocorrelaion Funcion.5 Sample Auocorrelaion Funcion (ACF) Lag (a) Volailiy Levy Noise Drif Sock Marke x Time seps (b). Sock Marke Coefficien wih Reurn Innovaion(Levy) Levy Jumps drif volailiy Condiional variance Time seps.5 x Time seps (c) (d) Coefficien wih volailiy Levy Jumps drif Innovaion 3 ARMA-GARCH ARMA pq a b c Mu Coef Sd GARCH mn Alpha Alpha Bea dela Coef 2.5e Sd 8.6e WhieNoise mean variance skewness kurosis KS(P=.) :p q ARMA m n GARCH ab c ARMA AR() MA() ;Mu ;AlphaAlphaBea dela ARCH() GARCH()

5 Levy Φ W (u) Ε[e iuw ]=e -Ψ (u) W Levy Φ MJ (u) Ε[e iudl ]=e -Ψ (u) dl Levy Carr Wu(24) Kim (2) Levy Φ TS (u) Ε[e iudl ]=e CΓ(-Y)[(M-iu)Y -M Y +(G+iu) Y -G Y ] VG Levy (υσv(dl)) JD (υσv(dl)) VG (υσ) BM (4) = (υσλj 槡 2πσJ (dl -μ ) 2 J e - 2σ 2 J ) (5) = (υσce Gx / x x< +Ce -Mx /x x>) (6) (υσv(dl)) TS = (Ce Gx / x Y+ x< +Ce -Mx /x Y+ x>)(7) =e (iuυ-u 2 2 σ 2 ) (8) u 2 σ 2 =e (iuυ-u 2 2 σ 2 )+λ J (eiu J μ J - 2 -) (9) Φ VG (u) Ε[e iudl ]=e Cln(GM )-Cln(GM+iMu-iGu+u2) (2) (2) VG Levy 2 2 Levy Mean Variance Skewness Kurosis C G M Y BM VG TS JD Mu w Sigma w Lambda J Mu J Sigma J Coef : 3 (BM) (VG) (TS) (JD) CGM Y BM VG TS Mu w Sigma wlambda JMu J Sigma J (JD) Levy Levy 3 Levy (N=4) (22) Kalman ( ) (MCKF) 3 (MCPF)

6 9 : 3 29

3 23 Quaniles of Inpu Sample 5 Normal -5-4 -2 2 4 Sandard Normal Quaniles Variance Gamma Y Quaniles 5-5 Meron Jump 5 TS - -5 5 X Quaniles Tempered Sable 5.5.45.4.35.3.25.

7 3 23 Quaniles of Inpu Sample 5 Normal Sandard Normal Quaniles Variance Gamma Y Quaniles 5-5 Meron Jump 5 TS X Quaniles Tempered Sable PDF of Noise Empirical MJ Normal Y Quaniles 5 Y Quaniles X Quaniles X Quaniles PDF curve Empirical VG Normal PDF of Noise Empirical TS Normal MJ VG TS Levy ( ) 3

8 9 : Jump LinearDrif Volailiy AAE APE MSE RMSE AAE APE MSE RMSE AAE APE MSE RMSE BSKF E E E E BSPF E E E E JDKF E E E E JDPF E E E E VGKF E E E E VGPF E E E E TSKF E E E E TSPF E E E E : (BM) (VG) (TS) (JD) (KF) (PF) BSKF BSPF AAEAPE MSE RMSE 4 3 GARCH raeprocessesimpliciindeuschemarkopions[j]. Levy ReviewofFinancialSudies9969:69~7. [5] Baes D.Maximumlikelihoodesimaionoflaen Levy afineprocesses[j].review offinancialsudies 269:99~965. Levy poraeliabiliies[j].journalofpoliicaleconomy : :637~659. [7] CarrPWu L.Time-changed Levyprocessesand 2 opionpricing[j].journaloffinancialeconomics Levy 247:3~4. [8] ChrisofersenPJacobsK.Theimporanceofhe 3 [6] BlackFScholesM.Thepricingofopionsandcor- 3 lossfuncioninopionvaluaion[j].journaloffi- Levy nancialeconomics2472:29~38. [9] ChrisofersenPJacobsK.Exploringime-varying : [] Ai-SahaliaYMyklandPZhangL.Howofeno sampleaconinuous-imeprocessinhepresenceof markemicrosrucurenoise[j].reviewoffinancial Sudies258:35~46. [2] Andersen T GBenzoniLLund L.Esimaing jump-difusionsforequiyreurns [J].Journalof Finance2257:239~284. [3] Andersen T GBolerslevTDieboldF XEbens H.Thedisribuionofrealizedsockreurnvolaili- y[j].journaloffinancialeconomics26:43 ~76. [4] BaesD.Jumpsandsochasicvolailiy:Exchange jumpinensiies:evidencefrom S&P 5reurns andopions[j].reviewoffinancialsudies28. [] Chrisoferseneal.Opionvaluaion wihcondi- ionalheeroskedasiciy and non-normaliy[j]. ReviewofFinancialSudies223:239~283. [] ErakerB.Dosockpricesandvolailiyjump Rec- oncilingevidencefrom spoandopionprices[j]. JournalofFinance2459:367~43. [2] FamaE.MandelbroandhesablePareianhy- pohesis[j].journalofbusiness96336(4):42 ~429. [3] HesonS.Aclosed-formsoluionforopionswih

9 32 23 sochasicvolailiy wihapplicaionsobondand currencyopions[j].reviewoffinancialsudies 9936:327~343. [4] HesonSNandiS.Aclosed-form GARCH opion pricing model [J].Review offinancialsudies 23:585~626. [5] JacquierEeal.Bayesiananalysisofsochasic volailiymodels[j].journalofbusinessandeco- nomicsaisics9942:~9. [6] JohannesMPolson NSroudJ.Learningabou jumpsandsochasicvolailiy:fileringsochasic diferenialequaionswihjumps[j].reviewoffi- nancialsudies28. [7] Kim YSRachevSTBianchiM LFabozziFJ.A newemperedsabledisribuionandisapplicaion ofinance[c]//bolgrachevstwuerhr. Riskassessmen:Decisionsinbankingandfinance. PhysikaVerlagSpringer28:77~. [8] Kim Yeal.Temperedsableandemperedinfi- niely divisible GARCH models[j].journalof BankingandFinance2349:296~29. [9] Kim Yeal.Timeseriesanalysisforfinancial markemeldowns[j].journalofbanking & Fi- nance235:879~89. [2] LinSMcaleer M.Asympoicheoryforavecor ARMA-GARCH model[j].economerictheory 23(9):28~3. [2] LiuJChenR.SequenialMoneCarlomehodsfor dynamicsysems[j].journalofheamericansa- isicalassociaion99893:32~88. [22] Madan DCarrPChang E.Thevariancegamma processandopionpricing[j].european Finance Review9982:79~5. [23] Meron R.Opionpricing whenunderlyingsock reurnsaredisconinuous[j].journaloffinancial Economics9763:25~44. [24] PanJ.Thejump-riskpremiaimpliciinopions: Evidencefromaninegraedime-seriessudy [J]. JournalofFinancialEconomics2263:3~5. [25] RosinskiJ. Tempering sable processes [J]. SochasicProcessesandheirApplicaions ~77. [26] SabanisS. Sochasicvolailiyandhemeanre- kes2323:33~47. [27]. [J]. 2(4):54~66. [28]. [J]. 2722(4):344~385. [29]. [J] ():43~46. [3]. Levy CF-CGMM [J]. 28. [3].GARCH Levy [J]. 22 (3):327~337. ApplicaionofBoosrapParicleFileringforJumps: Evidencefrom H&S3Index veringprocess [J].TheJournalofFuures Mar- WU Heng-yu 2 ZHU Fu-min HU Gen-hua MAJing 3 TIAN Hai-shan (.SchoolofEconomicInformaionEngineeringSouhwesernUniversiyofFinance & EconomicsChengdu63 China;2.SchoolofBusinessAdminisraionSouhChinaUniversiyofTechnologyGuangzhou564China; 3.SchoolofEconomicManagemenWuxiInsiueofTechnologyWuxi242China) Absrac:ThispapermodelsfourkindsofLevyprocesseswihdiscreeime-varyingvolailiyaccordingodiferende- greesofjumpapplyingasymmericgarch modelbasedonimeseriesanalysisocapurenon-gaussianinnovaions volailiycluseringandleverageefecsofh&s3index.furhermoreibuildshree-dimensionalsaespacemodelof volailiydrifraeandjumpandinroducespariclefilering mehodoanalyzedynamicvolailiyprocessandjumps. TheempiricalresulsshowhahereexissalargenumberofsochasicjumpsinreurnsofH&S3Indexandha neiherbrownianmoionnorkalmanfilercancapurehosejumps.howeverpariclefilerwihemperedsableprocess displaysopimalfiingindescribingime-varyingaciviyandjumpsofasses. Keywords:H&S3Index;ParicleFiler;NH-GARCH Model;LevyNoise;TemperedSableProcess

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untitled 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)