Lévy框架下SVJ模型定价效率的比较研究
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摘要
1973年提出的BSM期权定价公式受到大量的质疑,主要表现在以下几个方面:(1)资产价格的对数收益率存在尖峰厚尾和负偏现象,并且会产生一些跳跃行为;(2)波动率不是常数,存在一定的随机性和波动率聚类特征,并与收益率之间存在负相关关系;(3)波动率偏斜不因为期权到期日的延长变得平整。理论界提出了很多改进办法,主要有两个方向:一是对波动率的改进,包含确定性的波动率模型和随机波动率模型;另外一个是加入跳跃,包括复合泊松过程跳跃和无限活性的跳跃。这些模型基本上都可以归结到L(?)vy框架下。
本文利用香港恒生指数期权市场的数据,首先研究了隐含波动率曲面的特征,然后选择了L(?)vy框架下四个代表性的既包含随机波动率,又包含跳跃的SVJ模型做定价效率实证检验。包括:(1)跳跃为从属过程,随机波动率服从伽玛O-U过程的BNS模型;(2)跳跃为复合泊松过程,随机波动率为CIR模型的SVMJ模型;(3)跳跃服从方差伽玛过程,随机波动率为CIR模型的SVVG模型;(4)跳跃服从有限矩对数稳定过程,随机波动率为CIR模型的SVLG模型。后三者的随机波动率采用CIR时钟变换的(time-changed)技术来实现。前两者为有限活性跳跃,后二者为无限活性跳跃。
本文根据模型的特征函数,利用看跌期权的快速傅立叶变换算法来校准结果,结论表明:(1)一年期样本内恒生指数期权隐含波动率期限结构倒挂,并且隐含波动率偏斜随着时间增加变得更加陡峭;(2)无限活性的L(?)vy跳跃模型,优于有限活性的L(?)vy跳跃模型;(3)不受中心极限定理约束的SVLG模型能更好地拟合波动率偏斜期限特征,有着最高的期权定价效率。在参数较少情况下,使用O-U过程驱动随机波动率的办法并不比CIR模型驱动的好。
BSM option pricing model, which was aroused in 1973, has greatly been challenged in the following aspects: Firstly, the log return of assets price is non-normal distribution with negative skewness, fat tails and excess kurtosis; Secondly, the volatility is not a constant, but appears to be changing stochastically and has volatility clusters; Thirdly, the implied volatility smirk does not flatten out as maturity increases within several years. Because of problems above, the hypothesis of BSM model have been released in theory field, which are improved in two main directions. One is the improvement of volatility, including deterministic volatility models and stochastic volatility models. The other one is to add jumps, containing the finite activity compound Poisson process, infinite activity Lévy jump. These models can be classified as Lévy model.
By the data of Hangseng index option market of Hongkong, the thesis has done research on the characteristics of the implied volatility surface and then selects four representative models to test their empirical performance. The models contains: (1) the BNS models whose stochastic volatility abided by Gamma O-U process, during which jumps structure is drove by subordinator; (2)the SVMJ model whose jumps structure is droved by compound poisson process;(3) the SVVG model whose jumps structure is droved by variance gamma process;(4) the SVLS model whose jumps structure is droved by the finite moment log stable process. The latter three models utilize the CIR time-changed technology to realize stochastic volatility. The former two models have finite activity jumps, while the latter two have infinite activity jumps.
The thesis makes use of the Fast Fourier Transform to calibrate parameters, and its result is as follows: the Hangseng index option market's implied volatility term structure was negative to maturity, and the implied volatility smirk became steeper as maturity increases up to the observable horizon of one years. The infinite activity Lévy model is better to the finite activity Lévy model. Without the limitation of the central limit theorem, the SVLG model has the best performance among the four models owing to its better reflecting of the characteristics of the volatility smirk. Under the situation of less parameters, the stochastic volatility drove by the O-U process is not better than the CIR process.
本文利用香港恒生指数期权市场的数据,首先研究了隐含波动率曲面的特征,然后选择了L(?)vy框架下四个代表性的既包含随机波动率,又包含跳跃的SVJ模型做定价效率实证检验。包括:(1)跳跃为从属过程,随机波动率服从伽玛O-U过程的BNS模型;(2)跳跃为复合泊松过程,随机波动率为CIR模型的SVMJ模型;(3)跳跃服从方差伽玛过程,随机波动率为CIR模型的SVVG模型;(4)跳跃服从有限矩对数稳定过程,随机波动率为CIR模型的SVLG模型。后三者的随机波动率采用CIR时钟变换的(time-changed)技术来实现。前两者为有限活性跳跃,后二者为无限活性跳跃。
本文根据模型的特征函数,利用看跌期权的快速傅立叶变换算法来校准结果,结论表明:(1)一年期样本内恒生指数期权隐含波动率期限结构倒挂,并且隐含波动率偏斜随着时间增加变得更加陡峭;(2)无限活性的L(?)vy跳跃模型,优于有限活性的L(?)vy跳跃模型;(3)不受中心极限定理约束的SVLG模型能更好地拟合波动率偏斜期限特征,有着最高的期权定价效率。在参数较少情况下,使用O-U过程驱动随机波动率的办法并不比CIR模型驱动的好。
BSM option pricing model, which was aroused in 1973, has greatly been challenged in the following aspects: Firstly, the log return of assets price is non-normal distribution with negative skewness, fat tails and excess kurtosis; Secondly, the volatility is not a constant, but appears to be changing stochastically and has volatility clusters; Thirdly, the implied volatility smirk does not flatten out as maturity increases within several years. Because of problems above, the hypothesis of BSM model have been released in theory field, which are improved in two main directions. One is the improvement of volatility, including deterministic volatility models and stochastic volatility models. The other one is to add jumps, containing the finite activity compound Poisson process, infinite activity Lévy jump. These models can be classified as Lévy model.
By the data of Hangseng index option market of Hongkong, the thesis has done research on the characteristics of the implied volatility surface and then selects four representative models to test their empirical performance. The models contains: (1) the BNS models whose stochastic volatility abided by Gamma O-U process, during which jumps structure is drove by subordinator; (2)the SVMJ model whose jumps structure is droved by compound poisson process;(3) the SVVG model whose jumps structure is droved by variance gamma process;(4) the SVLS model whose jumps structure is droved by the finite moment log stable process. The latter three models utilize the CIR time-changed technology to realize stochastic volatility. The former two models have finite activity jumps, while the latter two have infinite activity jumps.
The thesis makes use of the Fast Fourier Transform to calibrate parameters, and its result is as follows: the Hangseng index option market's implied volatility term structure was negative to maturity, and the implied volatility smirk became steeper as maturity increases up to the observable horizon of one years. The infinite activity Lévy model is better to the finite activity Lévy model. Without the limitation of the central limit theorem, the SVLG model has the best performance among the four models owing to its better reflecting of the characteristics of the volatility smirk. Under the situation of less parameters, the stochastic volatility drove by the O-U process is not better than the CIR process.
引文
[1]Anderson,T.,L.Benzoni,and J.Lund.An empirical investigation of continuous-time equity return models [J].Journal of Finance,2002,(57):1239-1284.
[2]Backus,D.K.,F.Silverio,and L.Wu.Accounting for biases in Black-Scholes [R].Working paper.New York University,1997.
[3]Bakshi,G.,C.Cao,and Z.Chen.Empirical performance of alternative option pricing models" [J].Journal of Finance,1997,(52):2003-2049.
[4]Barndorff-Nielsen,O.E..Normal inverse Gaussian distributions and the modeling of stock returns [R].Department of Theoretical Statistics,Aarhus University,1995,Research Report,300.
[5]Barndorff-Nielsen,O.E..Processes of normal inverse Gaussian type [J].Finance and Stochastics,1998,41-68.
[6]Barndorff-Nielsen,O.E.,and N.Shephard.Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics [J].Journal of the Royal Statistical Society,2001,B(63):167-241.
[7]Barndorff-Nielsen,O.E.,E.Nicolato,and N.Shephard.Some recent developments in stochastic volatility modelling [J].Quantitative Finance,2002,11-23.
[8]Barndorff-Nielsen,O.E.,and N.Shephard.Integrated OU Processes and non-Gaussian OU-based Stochastic Volatility Models [J].Scandinavian Journal of Statistics,2003.
[9]Barndorff-Nielsen,O.E..Normal inverse Gaussian distributions and stochastic volatility models [J].Scandinavian Journal of Statistics,1997,(24):1-13.
[10]Bates,D.Jumps and stochastic volatility:Exchange rate processes implicit in Deutsche mark options [J].Review of Financial Studies,1996,(9):69-107.
[11]Bates,D.Post-'87 crash fears in S&P 500 futures options [J].Journal of Econometrics 2000,(94):181-238.
[12]Black,F.,and M.Scholes.The pricing of options and corporate liabilities [J].Journal of Political Economy,1973,(81):631-659.
[13]Carr,P.,and D.Madan,Option valuation using the fast Fourier transform [J].Journal of Computational Finance,1998 ,(2):61-73.
[14]Carr,P.,H.Geman,D.Madan,and M.Yor.The fine structure of asset returns:investigation [J].Journal of Business ,2002,(75):305-332.
[15]Carr,P.,H.Geman,D.Madan,and M.Yor.Stochastic volatility for Levy processes [J].Mathematical Finance,2003,(13):345-382.
[16]Carr,P.,H.Geman,D.Madan,and M.Yor.From Local Volatility to Local Levy Models [J].Quantitative Finance,2004,(5).
[17]Carr,P.,and L.Wu.The finite moment log stable process and option pricing [J].Journal of Finance ,2003,(58):753-777.
[18]Cont,Rama,J.Fonseca,and V.Durrleman.Stochastic models of implied volatility surfaces [J].Economic Notes .2002,(31):361-377.
[19]Chernov,M.,and E.Ghysels.A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation [J].Journal of Financial Economics ,2000,(56):407-458.
[20]Clark,P.A subordinated stochastic process model with finite variance for speculative prices [J].Econometrica,1973,(41):135-156.
[21]Cont,R.,and P.Tankov.Financial Modeling With Jump Processes [M].CRC Press,2004.
[22]Derman,E.,and I.Kani.Riding on a smile [J].Risk,1994.(7):32-39.
[23]Derman,E.,and I.Kani.Stochastic implied trees:Arbitrage pricing with stochastic term and strike structure of volatility [J].International Journal of Theoretical and Applied Finance,1998,(1):61-110.
[24]Dupire.Pricing with a smile [J].Risk,1994.( 7):18-20.
[25]Eberlein,E.,U.Keller,and K.Prause.New insights into smile,mispricing and Value at Risk:The hyperbolic model [J].Journal of Business,1998,(71):371-405.
[26]Eraker,B.Do stock prices and volatility jump? Reconciling evidence from spot and option prices manuscript [R].Duke University,2001.
[27]Eraker,B.,M.Johannes,and N.Poison.The impact of jumps in equity index volatility and returns [J].Journal of Finance ,2003,(58):1269-1300.
[28]Heston S.A closed-form solution for options with stochastic volatility,with application to bond and currency options [J].Review of Financial Studies,1993,(6):327-343.
[29]Hull J.Options,Futures And Other Derivatives [M].6th,Prentice Hall,2006.
[30]Jim G.The Volatility Surface[M].John Wiley&Sons,Inc.2006.
[31]Konikov,M.,and D.Madan.Pricing options of all strikes and maturities using a generalization of the VG model[R].Working paper,University of Maryland.2000.
[32]Kou,S.A Jump-Diffusion Model for Option Pricing[J].Management Science,2002,48(8):1086-1101.
[33]Madan,D.,and E.Seneta.Chebyshev polynomial approximations and characteristic function estimation[J].Journal of the Royal Statistical Society,1987,B49(2):163-169.
[34]Madan,D.,and F.Milne.Option pricing with VG martingale components[J].Mathematical Finance,1991,(1):39-56.
[35]Madan,D.,P.Carr,and E.Chang.The variance gamma process and option pricing[J].European Finance Review,1998,(2):79-105.
[36]Merton,R.Option pricing when underlying stock returns are discontinuous[J].Journal of Financial Economics,1976,(3):125-144.
[37]Pan,J.The jump-risk premia implicit in options:Evidence from an integrated time-series study[J].Journal of Financial Economics,2002,(63):3-50.
[38]Rebonato,R.Volatility and Correlation[M].2nd section,John Wiley & Sons,2004,(6):427-440.
[39]Sato,K.L(?)vy Processes and Infinitely Divisible Distributions.Cambridge Studies in Advanced Mathematics[M].Cambridge University Press.1999,(68).
[40]Sch(o丨¨)nbucher,A.Philipp,market model of stochastic implied volatility[J].Philosophic Transactions of the Royal Society,1999,Series A(357):2071-2092.
[41]Schoutens,W.L(?)vy Processes in Finance:Pricing Financial Derivatives[M].John Wiley &Sons,Ltd.,2003.
[42]Vuong,H.Quang.Likelihood ratio tests for model selection and non-nested hypotheses[J].Econometrica,1989,(57):307-333.
[43]冯彬.基于香港市场的期权定价模型定价效率实证检验[D].厦门大学硕士论文,2008.
[44]杜澄楷.双指数跳跃扩散模型在中国的实证研究[D].厦门大学硕士论文,2008.
[45]刘次华.随机过程及其应用[M].北京:高等教育出版社,2004.
[46]张渭滨.数学物理方程[M].北京:清华大学出版社,2007.
[2]Backus,D.K.,F.Silverio,and L.Wu.Accounting for biases in Black-Scholes [R].Working paper.New York University,1997.
[3]Bakshi,G.,C.Cao,and Z.Chen.Empirical performance of alternative option pricing models" [J].Journal of Finance,1997,(52):2003-2049.
[4]Barndorff-Nielsen,O.E..Normal inverse Gaussian distributions and the modeling of stock returns [R].Department of Theoretical Statistics,Aarhus University,1995,Research Report,300.
[5]Barndorff-Nielsen,O.E..Processes of normal inverse Gaussian type [J].Finance and Stochastics,1998,41-68.
[6]Barndorff-Nielsen,O.E.,and N.Shephard.Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics [J].Journal of the Royal Statistical Society,2001,B(63):167-241.
[7]Barndorff-Nielsen,O.E.,E.Nicolato,and N.Shephard.Some recent developments in stochastic volatility modelling [J].Quantitative Finance,2002,11-23.
[8]Barndorff-Nielsen,O.E.,and N.Shephard.Integrated OU Processes and non-Gaussian OU-based Stochastic Volatility Models [J].Scandinavian Journal of Statistics,2003.
[9]Barndorff-Nielsen,O.E..Normal inverse Gaussian distributions and stochastic volatility models [J].Scandinavian Journal of Statistics,1997,(24):1-13.
[10]Bates,D.Jumps and stochastic volatility:Exchange rate processes implicit in Deutsche mark options [J].Review of Financial Studies,1996,(9):69-107.
[11]Bates,D.Post-'87 crash fears in S&P 500 futures options [J].Journal of Econometrics 2000,(94):181-238.
[12]Black,F.,and M.Scholes.The pricing of options and corporate liabilities [J].Journal of Political Economy,1973,(81):631-659.
[13]Carr,P.,and D.Madan,Option valuation using the fast Fourier transform [J].Journal of Computational Finance,1998 ,(2):61-73.
[14]Carr,P.,H.Geman,D.Madan,and M.Yor.The fine structure of asset returns:investigation [J].Journal of Business ,2002,(75):305-332.
[15]Carr,P.,H.Geman,D.Madan,and M.Yor.Stochastic volatility for Levy processes [J].Mathematical Finance,2003,(13):345-382.
[16]Carr,P.,H.Geman,D.Madan,and M.Yor.From Local Volatility to Local Levy Models [J].Quantitative Finance,2004,(5).
[17]Carr,P.,and L.Wu.The finite moment log stable process and option pricing [J].Journal of Finance ,2003,(58):753-777.
[18]Cont,Rama,J.Fonseca,and V.Durrleman.Stochastic models of implied volatility surfaces [J].Economic Notes .2002,(31):361-377.
[19]Chernov,M.,and E.Ghysels.A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation [J].Journal of Financial Economics ,2000,(56):407-458.
[20]Clark,P.A subordinated stochastic process model with finite variance for speculative prices [J].Econometrica,1973,(41):135-156.
[21]Cont,R.,and P.Tankov.Financial Modeling With Jump Processes [M].CRC Press,2004.
[22]Derman,E.,and I.Kani.Riding on a smile [J].Risk,1994.(7):32-39.
[23]Derman,E.,and I.Kani.Stochastic implied trees:Arbitrage pricing with stochastic term and strike structure of volatility [J].International Journal of Theoretical and Applied Finance,1998,(1):61-110.
[24]Dupire.Pricing with a smile [J].Risk,1994.( 7):18-20.
[25]Eberlein,E.,U.Keller,and K.Prause.New insights into smile,mispricing and Value at Risk:The hyperbolic model [J].Journal of Business,1998,(71):371-405.
[26]Eraker,B.Do stock prices and volatility jump? Reconciling evidence from spot and option prices manuscript [R].Duke University,2001.
[27]Eraker,B.,M.Johannes,and N.Poison.The impact of jumps in equity index volatility and returns [J].Journal of Finance ,2003,(58):1269-1300.
[28]Heston S.A closed-form solution for options with stochastic volatility,with application to bond and currency options [J].Review of Financial Studies,1993,(6):327-343.
[29]Hull J.Options,Futures And Other Derivatives [M].6th,Prentice Hall,2006.
[30]Jim G.The Volatility Surface[M].John Wiley&Sons,Inc.2006.
[31]Konikov,M.,and D.Madan.Pricing options of all strikes and maturities using a generalization of the VG model[R].Working paper,University of Maryland.2000.
[32]Kou,S.A Jump-Diffusion Model for Option Pricing[J].Management Science,2002,48(8):1086-1101.
[33]Madan,D.,and E.Seneta.Chebyshev polynomial approximations and characteristic function estimation[J].Journal of the Royal Statistical Society,1987,B49(2):163-169.
[34]Madan,D.,and F.Milne.Option pricing with VG martingale components[J].Mathematical Finance,1991,(1):39-56.
[35]Madan,D.,P.Carr,and E.Chang.The variance gamma process and option pricing[J].European Finance Review,1998,(2):79-105.
[36]Merton,R.Option pricing when underlying stock returns are discontinuous[J].Journal of Financial Economics,1976,(3):125-144.
[37]Pan,J.The jump-risk premia implicit in options:Evidence from an integrated time-series study[J].Journal of Financial Economics,2002,(63):3-50.
[38]Rebonato,R.Volatility and Correlation[M].2nd section,John Wiley & Sons,2004,(6):427-440.
[39]Sato,K.L(?)vy Processes and Infinitely Divisible Distributions.Cambridge Studies in Advanced Mathematics[M].Cambridge University Press.1999,(68).
[40]Sch(o丨¨)nbucher,A.Philipp,market model of stochastic implied volatility[J].Philosophic Transactions of the Royal Society,1999,Series A(357):2071-2092.
[41]Schoutens,W.L(?)vy Processes in Finance:Pricing Financial Derivatives[M].John Wiley &Sons,Ltd.,2003.
[42]Vuong,H.Quang.Likelihood ratio tests for model selection and non-nested hypotheses[J].Econometrica,1989,(57):307-333.
[43]冯彬.基于香港市场的期权定价模型定价效率实证检验[D].厦门大学硕士论文,2008.
[44]杜澄楷.双指数跳跃扩散模型在中国的实证研究[D].厦门大学硕士论文,2008.
[45]刘次华.随机过程及其应用[M].北京:高等教育出版社,2004.
[46]张渭滨.数学物理方程[M].北京:清华大学出版社,2007.