
-
Previous Article
Combinatorial Hodge theory for equitable kidney paired donation
- FoDS Home
- This Issue
-
Next Article
Approximate Bayesian inference for geostatistical generalised linear models
Particle filters for inference of high-dimensional multivariate stochastic volatility models with cross-leverage effects
Department of Statistics & Applied Probability, National University of Singapore, Singapore, 117546, SG |
Multivariate stochastic volatility models are a popular and well-known class of models in the analysis of financial time series because of their abilities to capture the important stylized facts of financial returns data. We consider the problems of filtering distribution estimation and also marginal likelihood calculation for multivariate stochastic volatility models with cross-leverage effects in the high dimensional case, that is when the number of financial time series that we analyze simultaneously (denoted by $ d $) is large. The standard particle filter has been widely used in the literature to solve these intractable inference problems. It has excellent performance in low to moderate dimensions, but collapses in the high dimensional case. In this article, two new and advanced particle filters proposed in [
References:
[1] |
C. Andrieu, A. Doucet and R. Holenstein,
Particle Markov chain Monte Carlo methods (with discussion), J. R. Statist. Soc. Ser. B, 72 (2010), 269-342.
doi: 10.1111/j.1467-9868.2009.00736.x. |
[2] |
M. Asai, M. McAleer and J. Yu,
Multivariate stochastic volatility: A review, Econ. Rev., 25 (2006), 145-175.
doi: 10.1080/07474930600713564. |
[3] |
L. Bauwens, S. Laurent and J. V. Rombouts,
Multivariate GARCH models: A survey, J. Appl. Econ., 21 (2006), 79-109.
doi: 10.1002/jae.842. |
[4] |
A. Beskos, D. Crisan, A. Jasra, K. Kamatani and Y. Zhou,
A stable particle filter for a class of high-dimensional state-space models, Adv. Appl. Probab., 49 (2017), 24-48.
doi: 10.1017/apr.2016.77. |
[5] |
P. Bickel, B. Li and T. Bengtsson, Sharp failure rates for the bootstrap particle filter in high dimensions, In Pushing the Limits of Contemporary Statistics: Contributions in Honor of J. Ghosh, IMS, 3 (2008), 318–329.
doi: 10.1214/074921708000000228. |
[6] |
S. Chib, F. Nadari and N. Shephard,
Analysis of high-dimensional multivariate stochastic volatility models, J. Econ., 134 (2006), 341-371.
doi: 10.1016/j.jeconom.2005.06.026. |
[7] |
N. Chopin,
Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference, Ann. Statist., 32 (2004), 2385-2411.
doi: 10.1214/009053604000000698. |
[8] |
P. Del Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Springer, New York, 2004.
doi: 10.1007/978-1-4684-9393-1. |
[9] |
P. Del Moral, A. Doucet and A. Jasra,
On adaptive resampling strategies for sequential Monte Carlo methods, Bernoulli, 18 (2012), 252-278.
doi: 10.3150/10-BEJ335. |
[10] |
A. Doucet, On Sequential Simulation-based Methods for Bayesian Filtering, Technical Report, 1998. |
[11] |
A. Doucet and A. Johansen, A tutorial on particle filtering and smoothing: Fifteen years later, In Handbook of Nonlinear Filtering (eds. D. Crisan & B. Rozovsky), Oxford University Press, Oxford, (2011), 656–704. |
[12] |
A. Doucet, M. K. Pitt, G. Deligiannidis and R. Kohn,
Efficient Implementation of Markov chain Monte Carlo when Using an Unbiased Likelihood Estimator, Biometrika, 102 (2015), 295-313.
doi: 10.1093/biomet/asu075. |
[13] |
J. Hull and A. White,
The pricing of options on assets with stochastic volatilities, J. Finan., 42 (1987), 281-300.
doi: 10.1111/j.1540-6261.1987.tb02568.x. |
[14] |
T. Ishihara and Y. Omori,
Efficient Bayesian estimation of a multivariate stochastic volatility with cross leverage and heavy tailed errors, Comp. Statist. Data Anal., 56 (2012), 3674-3689.
doi: 10.1016/j.csda.2010.07.015. |
[15] |
A. Jasra, D. A. Stephens, A. Doucet and T. Tsagaris,
Inference for Lévy driven stochastic volatility models via adaptive sequential Monte Carlo, Scand. J. Statist., 38 (2011), 1-22.
doi: 10.1111/j.1467-9469.2010.00723.x. |
[16] |
N. Kantas, A. Doucet, S. S. Singh, J. M. Maciejowski and N. Chopin,
An overview of sequential Monte Carlo methods for parameter estimation in general state-space sodels, IFAC Proc., 42 (2009), 774-785.
|
[17] |
N. Kantas, A. Doucet, S. S. Singh, J. M. Maciejowski and N. Chopin,
On particle methods for parameter estimation in general state-space models, Statist. Sci., 30 (2015), 328-351.
doi: 10.1214/14-STS511. |
[18] |
S. Kim, N. Shephard and S. Chib,
Stochastic volatility: Likelihood inference and comparison with ARCH models, Rev. Econ. Stud., 65 (1998), 361-393.
doi: 10.1111/1467-937X.00050. |
[19] |
G. Kitagawa,
Monte Carlo filter and smoother for non-Gaussian nonlinear state-space models, J. Comp. Graph. Stat., 5 (1996), 1-25.
doi: 10.2307/1390750. |
[20] |
M. Klaas, N. De Freitas and A. Doucet, Towards practical N2 Monte Carlo: The marginal particle filter, Uncert. A. I., (2005), 308–315. |
[21] |
A. Kong, J. S. Liu and W. H. Wong,
Sequential imputations and Bayesian missing data problems, J. Amer. Statist. Assoc., 89 (1994), 278-288.
doi: 10.1080/01621459.1994.10476469. |
[22] |
C. Naesseth, F. Lindten and T. Schön, Nested sequential Monte Carlo methods, ICML, (2015), 1292–1301. |
[23] |
J. Nakajima,
Bayesian analysis of multivariate stochastic volatility with skew return distribution, Econ. Rev., 36 (2017), 546-562.
doi: 10.1080/07474938.2014.977093. |
[24] |
S. S. Ozturk and J. F. Richard,
Stochastic volatility and leverage: Application to a panel of S & P 500 stocks, Finan. Res. Lett., 12 (2015), 67-76.
|
[25] |
M. K. Pitt, R. Dos Santos Silva, P. Giordani and R. Kohn,
On some properties of Markov chain Monte Carlo simulation methods based upon the particle filter, J. Econom., 171 (2012), 134-151.
doi: 10.1016/j.jeconom.2012.06.004. |
[26] |
M. K. Pitt and N. Shephard,
Filtering via simulation: Auxiliary particle filters, J. Amer. Statist. Assoc., 94 (1999), 590-599.
doi: 10.1080/01621459.1999.10474153. |
[27] |
K. Platanioti, E. McCoy and D. A. Stephens, A Review of Stochastic Volatility Models, Technical Report, 2005. |
[28] |
C. Snyder, T. Bengtsson, P. Bickel and J. Anderson,
Obstacles to high-dimensional particle filtering, Month. Weather Rev., 136 (2008), 4629-4640.
doi: 10.1175/2008MWR2529.1. |
[29] |
C. Vergé, C. Duberry, P. Del Moral and E. Moulines,
On parallel implementation of sequential Monte Carlo methods: The island particle filtering, Stat. Comp., 25 (2015), 243-260.
doi: 10.1007/s11222-013-9429-x. |
show all references
References:
[1] |
C. Andrieu, A. Doucet and R. Holenstein,
Particle Markov chain Monte Carlo methods (with discussion), J. R. Statist. Soc. Ser. B, 72 (2010), 269-342.
doi: 10.1111/j.1467-9868.2009.00736.x. |
[2] |
M. Asai, M. McAleer and J. Yu,
Multivariate stochastic volatility: A review, Econ. Rev., 25 (2006), 145-175.
doi: 10.1080/07474930600713564. |
[3] |
L. Bauwens, S. Laurent and J. V. Rombouts,
Multivariate GARCH models: A survey, J. Appl. Econ., 21 (2006), 79-109.
doi: 10.1002/jae.842. |
[4] |
A. Beskos, D. Crisan, A. Jasra, K. Kamatani and Y. Zhou,
A stable particle filter for a class of high-dimensional state-space models, Adv. Appl. Probab., 49 (2017), 24-48.
doi: 10.1017/apr.2016.77. |
[5] |
P. Bickel, B. Li and T. Bengtsson, Sharp failure rates for the bootstrap particle filter in high dimensions, In Pushing the Limits of Contemporary Statistics: Contributions in Honor of J. Ghosh, IMS, 3 (2008), 318–329.
doi: 10.1214/074921708000000228. |
[6] |
S. Chib, F. Nadari and N. Shephard,
Analysis of high-dimensional multivariate stochastic volatility models, J. Econ., 134 (2006), 341-371.
doi: 10.1016/j.jeconom.2005.06.026. |
[7] |
N. Chopin,
Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference, Ann. Statist., 32 (2004), 2385-2411.
doi: 10.1214/009053604000000698. |
[8] |
P. Del Moral, Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Springer, New York, 2004.
doi: 10.1007/978-1-4684-9393-1. |
[9] |
P. Del Moral, A. Doucet and A. Jasra,
On adaptive resampling strategies for sequential Monte Carlo methods, Bernoulli, 18 (2012), 252-278.
doi: 10.3150/10-BEJ335. |
[10] |
A. Doucet, On Sequential Simulation-based Methods for Bayesian Filtering, Technical Report, 1998. |
[11] |
A. Doucet and A. Johansen, A tutorial on particle filtering and smoothing: Fifteen years later, In Handbook of Nonlinear Filtering (eds. D. Crisan & B. Rozovsky), Oxford University Press, Oxford, (2011), 656–704. |
[12] |
A. Doucet, M. K. Pitt, G. Deligiannidis and R. Kohn,
Efficient Implementation of Markov chain Monte Carlo when Using an Unbiased Likelihood Estimator, Biometrika, 102 (2015), 295-313.
doi: 10.1093/biomet/asu075. |
[13] |
J. Hull and A. White,
The pricing of options on assets with stochastic volatilities, J. Finan., 42 (1987), 281-300.
doi: 10.1111/j.1540-6261.1987.tb02568.x. |
[14] |
T. Ishihara and Y. Omori,
Efficient Bayesian estimation of a multivariate stochastic volatility with cross leverage and heavy tailed errors, Comp. Statist. Data Anal., 56 (2012), 3674-3689.
doi: 10.1016/j.csda.2010.07.015. |
[15] |
A. Jasra, D. A. Stephens, A. Doucet and T. Tsagaris,
Inference for Lévy driven stochastic volatility models via adaptive sequential Monte Carlo, Scand. J. Statist., 38 (2011), 1-22.
doi: 10.1111/j.1467-9469.2010.00723.x. |
[16] |
N. Kantas, A. Doucet, S. S. Singh, J. M. Maciejowski and N. Chopin,
An overview of sequential Monte Carlo methods for parameter estimation in general state-space sodels, IFAC Proc., 42 (2009), 774-785.
|
[17] |
N. Kantas, A. Doucet, S. S. Singh, J. M. Maciejowski and N. Chopin,
On particle methods for parameter estimation in general state-space models, Statist. Sci., 30 (2015), 328-351.
doi: 10.1214/14-STS511. |
[18] |
S. Kim, N. Shephard and S. Chib,
Stochastic volatility: Likelihood inference and comparison with ARCH models, Rev. Econ. Stud., 65 (1998), 361-393.
doi: 10.1111/1467-937X.00050. |
[19] |
G. Kitagawa,
Monte Carlo filter and smoother for non-Gaussian nonlinear state-space models, J. Comp. Graph. Stat., 5 (1996), 1-25.
doi: 10.2307/1390750. |
[20] |
M. Klaas, N. De Freitas and A. Doucet, Towards practical N2 Monte Carlo: The marginal particle filter, Uncert. A. I., (2005), 308–315. |
[21] |
A. Kong, J. S. Liu and W. H. Wong,
Sequential imputations and Bayesian missing data problems, J. Amer. Statist. Assoc., 89 (1994), 278-288.
doi: 10.1080/01621459.1994.10476469. |
[22] |
C. Naesseth, F. Lindten and T. Schön, Nested sequential Monte Carlo methods, ICML, (2015), 1292–1301. |
[23] |
J. Nakajima,
Bayesian analysis of multivariate stochastic volatility with skew return distribution, Econ. Rev., 36 (2017), 546-562.
doi: 10.1080/07474938.2014.977093. |
[24] |
S. S. Ozturk and J. F. Richard,
Stochastic volatility and leverage: Application to a panel of S & P 500 stocks, Finan. Res. Lett., 12 (2015), 67-76.
|
[25] |
M. K. Pitt, R. Dos Santos Silva, P. Giordani and R. Kohn,
On some properties of Markov chain Monte Carlo simulation methods based upon the particle filter, J. Econom., 171 (2012), 134-151.
doi: 10.1016/j.jeconom.2012.06.004. |
[26] |
M. K. Pitt and N. Shephard,
Filtering via simulation: Auxiliary particle filters, J. Amer. Statist. Assoc., 94 (1999), 590-599.
doi: 10.1080/01621459.1999.10474153. |
[27] |
K. Platanioti, E. McCoy and D. A. Stephens, A Review of Stochastic Volatility Models, Technical Report, 2005. |
[28] |
C. Snyder, T. Bengtsson, P. Bickel and J. Anderson,
Obstacles to high-dimensional particle filtering, Month. Weather Rev., 136 (2008), 4629-4640.
doi: 10.1175/2008MWR2529.1. |
[29] |
C. Vergé, C. Duberry, P. Del Moral and E. Moulines,
On parallel implementation of sequential Monte Carlo methods: The island particle filtering, Stat. Comp., 25 (2015), 243-260.
doi: 10.1007/s11222-013-9429-x. |














Standard PF | STPF | Marginal STPF | |
25 | |||
50 | |||
100 | N.A. | ||
200 | N.A. |
Standard PF | STPF | Marginal STPF | |
25 | |||
50 | |||
100 | N.A. | ||
200 | N.A. |
Standard PF | STPF | Marginal STPF | |
25 | |||
50 | |||
100 | N.A. | ||
200 | N.A. |
Standard PF | STPF | Marginal STPF | |
25 | |||
50 | |||
100 | N.A. | ||
200 | N.A. |
Standard PF | STPF | Computation Time | |
100 | |||
200 |
Standard PF | STPF | Computation Time | |
100 | |||
200 |
[1] |
Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004 |
[2] |
Olli-Pekka Tossavainen, Daniel B. Work. Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data. Networks and Heterogeneous Media, 2013, 8 (3) : 803-824. doi: 10.3934/nhm.2013.8.803 |
[3] |
Xin Li, Feng Bao, Kyle Gallivan. A drift homotopy implicit particle filter method for nonlinear filtering problems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 727-746. doi: 10.3934/dcdss.2021097 |
[4] |
Zhiyan Ding, Qin Li. Constrained Ensemble Langevin Monte Carlo. Foundations of Data Science, 2022, 4 (1) : 37-70. doi: 10.3934/fods.2021034 |
[5] |
Pierre Degond, Simone Goettlich, Axel Klar, Mohammed Seaid, Andreas Unterreiter. Derivation of a kinetic model from a stochastic particle system. Kinetic and Related Models, 2008, 1 (4) : 557-572. doi: 10.3934/krm.2008.1.557 |
[6] |
Yuezheng Gong, Jiaquan Gao, Yushun Wang. High order Gauss-Seidel schemes for charged particle dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 573-585. doi: 10.3934/dcdsb.2018034 |
[7] |
Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic and Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291 |
[8] |
Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems and Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81 |
[9] |
Theodore Papamarkou, Alexey Lindo, Eric B. Ford. Geometric adaptive Monte Carlo in random environment. Foundations of Data Science, 2021, 3 (2) : 201-224. doi: 10.3934/fods.2021014 |
[10] |
Annalisa Pascarella, Alberto Sorrentino, Cristina Campi, Michele Piana. Particle filtering, beamforming and multiple signal classification for the analysis of magnetoencephalography time series: a comparison of algorithms. Inverse Problems and Imaging, 2010, 4 (1) : 169-190. doi: 10.3934/ipi.2010.4.169 |
[11] |
Robert J. Elliott, Tak Kuen Siu. Stochastic volatility with regime switching and uncertain noise: Filtering with sub-linear expectations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 59-81. doi: 10.3934/dcdsb.2017003 |
[12] |
Lee DeVille, Nicole Riemer, Matthew West. Convergence of a generalized Weighted Flow Algorithm for stochastic particle coagulation. Journal of Computational Dynamics, 2019, 6 (1) : 69-94. doi: 10.3934/jcd.2019003 |
[13] |
Michele Gianfelice, Marco Isopi. On the location of the 1-particle branch of the spectrum of the disordered stochastic Ising model. Networks and Heterogeneous Media, 2011, 6 (1) : 127-144. doi: 10.3934/nhm.2011.6.127 |
[14] |
Michael B. Giles, Kristian Debrabant, Andreas Rössler. Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3881-3903. doi: 10.3934/dcdsb.2018335 |
[15] |
Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683 |
[16] |
Xia Zhao, Jianping Dou. Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1263-1288. doi: 10.3934/jimo.2018095 |
[17] |
Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control and Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007 |
[18] |
Tony Lyons. Particle paths in equatorial flows. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2399-2414. doi: 10.3934/cpaa.2022041 |
[19] |
Michael C. Fu, Bingqing Li, Rongwen Wu, Tianqi Zhang. Option pricing under a discrete-time Markov switching stochastic volatility with co-jump model. Frontiers of Mathematical Finance, 2022, 1 (1) : 137-160. doi: 10.3934/fmf.2021005 |
[20] |
Laura Martín-Fernández, Gianni Gilioli, Ettore Lanzarone, Joaquín Míguez, Sara Pasquali, Fabrizio Ruggeri, Diego P. Ruiz. A Rao-Blackwellized particle filter for joint parameter estimation and biomass tracking in a stochastic predator-prey system. Mathematical Biosciences & Engineering, 2014, 11 (3) : 573-597. doi: 10.3934/mbe.2014.11.573 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]