本文采用CGMY和GIG过程对非高斯OU随机波动率模型进行扩展,建立连续叠加Lévy过程驱动的非高斯OU随机波动率模型,并给出模型的散粒噪声(Shot-Noise)表现方式与近似。在此基础上,为了反映的波动率相关性,本文把回顾抽样(Retrospective Sampling)方法扩展到连续叠加的Lévy过程驱动的非高斯OU随机波动模型中,设计了Lévy过程驱动的非高斯OU随机波动模型的贝叶斯参数统计推断方法。最后,采用金融市场实际数据对不同模型和参数估计方法进行验证和比较研究。本文理论和实证研究均表明采用CGMY和GIG过程对非高斯OU随机波动率模型进行扩展之后,模型的绩效得到明显提高,更能反映金融资产收益率波动率变化特征,本文设计的Lévy过程驱动的非高斯OU随机波动模型的贝叶斯参数统计推断方法效率也较高,克服了已有研究的不足。同时,实证研究发现上证指数收益率和波动率跳跃的特征以及波动率序列具有明显的长记忆特性。
The currently most popular models for the volatility of financial time series, non Gaussian Ornstein-Uhlenbeck stochastic processes are extended to more general non Ornstein-Uhlenbeck models driven by the general Lévy process, such as Generalized Inverse Gaussian(GIG)and Tempered Stable distributions(CGMY). In particular, means of making the correlation structure in the volatility process more flexible based on continuous superpositions of the more general non Ornstein-Uhlenbeck models are investigated, which can introduce long-memory into the volatility model. A shot-noise process and approximation for the continuous superpositions process are represented. Inference is carried out in a Bayesian framework, with computation using extended Reversible Jump Markov chain Monte Carlo and dependent thinning to the continuous superposition case. Empirical research demonstrates that the efficient Markov chain Monte Carlo methods appear to be successful in the case of the general GIG and CGMY marginal model, and that those models can be fitted to real share price returns data, and that results indicate that for the series we study, the long-memory aspect of the model is supported.
[1] Barndorff-Nielsen O E, Shephard N. Non-Gaussian OU based models and some of their uses in financial economics [J]. Journal of the Royal Statistical Society.Series B (Statistical Methodology), 2001a, 63(2): 167-241.
[2] Barndorff-Nielsen O E.Shephard N. Modeling by Lévy Processes for Financial Econometrics[M]//Barndorff-Nielse O E, Mikosch T,Resnick S.Lévy Processes-Theory and Applications. Boston: Birkhauser, 2001b: 283-318.
[3] Eraker B, Johannes M, Polson N. The impact of jumps in volatility and returns [J]. Journal of Finance, 2003, 58(3):1269-1300.
[4] Elerian O, Chib S, Shephard N. Likelihood inference for discretely observed non-linear diffusions [J]. Econometrica, 2001, 69(4):959-993.
[5] Jones C. Nonlinear mean reversion in the short-term interest rate [J]. The Review of Financial Studies, 2003, 16(3):793-843.
[6] Roberts G O, Stramer O. On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm [J]. Biometrika, 2001, 88(3):603-621.
[7] Li Haitao, Wells M T, Yu C L.A Bayesian analysis of returns dynamics with Lévy jumps [J]. Review of Financial Studies, 2008, 21(5): 2345-2378.
[8] 朱慧明,黄超,郝立亚,等. 基于状态空间的贝叶斯跳跃厚尾金融随机波动模型研究[J].中国管理科学,2010,18(6):17-25.
[9] Roberts G O, Papaspiliopoulos O, Dellaportas P.Bayesian inference for non-Gaussian Ornstein-Uhlenbeck stochastic volatility processes[J].Journal of the Royal Statistical Society.Series B(Statistical Methodology), 2004, 66(2): 369-393.
[10] Griffin J E, Steel M F J. Inference with non-Gaussian Ornstein-Uhlenbeck processes for stochastic volatility [J].Journal of Econometrics, 2006, 134(2):605-644.
[11] Green P J. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination [J]. Biometrika, 1995, 82(4), 711-732.
[12] Griffin J E, Steel M F J. Bayesian inference with stochastic volatility models using continuous superpositions of non-Gaussian Ornstein-Uhlenbeck processes[J].Computational Statistics and Data Analysis, 2010,54(11):2594-2608.
[13] Gander M P S, Stephens D A. Stochastic volatility modelling with general marginal distributions: Inference, prediction and model selection [J]. Journal of Statistical Planning and Inference, 2007a, 137(10):3068-3081.
[14] Gander M P S, Stephens D A. Simulation and inference for stochastic volatility models driven by Lévy processes [J].Biometrika, 2007b, 94(3): 627-646.
[15] Barndorff-Nielsen O E. Superposition of Ornstein-Uhlenbeck type processes [J]. Theory of Probability and its Applications, 2001, 45(2):175-194.
[16] Cox D R, Isham V. Point processes [M]. London, UK: Chapman & Hall, 1988.
[17] Black F. Studies of stock price volatility changes[C]. Proceedings of the Meetings of the Business & Economics Statistics, 1976:177-181.
[18] Nelson D B. Conditional heteroskedasticity in asset return: a new approach [J].Econometrica, 1991, 59(2): 347-370.
