Skip to main content
SpringerLink
Log in
Book cover

State-Space Models pp 321–344Cite as

Parameter Estimation via Particle MCMC for Ultra-High Frequency Models

  • Chapter
  • First Online:

  • 3462 Accesses

Part of the book series: Statistics and Econometrics for Finance ((SEFF,volume 1))

Abstract

In this chapter, particle Markov Chain Monte Carlo (PMCMC) method is applied to estimate ultra-high frequency data models. The class of models is proposed by Zeng (2003), who considers the explicit structure of market microstructure noise. Although the model is able to capture stylized facts of tick data, the nonlinear state-space model structure makes parameter estimation a challenge. We use PMCMC to estimate a couple models when the underlying intrinsic value processes follow a geometric Brownian motion or a jump-diffusion process, under 1/8 and 1/100 tick size rules. Moreover, some numeric methods that are able to enhance the algorithm efficiency are discussed. Numerical studies through simulation and real data show that PMCMC method is able to yield reasonable estimates for model parameters.

1 There is a difference between high frequency data in the literature related to realized variance, which are equally spaced in time and ultra-high frequency data which are irregularly spaced. In this chapter, we use high frequency data to denotes ultra-high frequency data to keep in line with other literature. But readers should notice the difference.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    The number of dimensions is equal to the number of data points in high frequency data set.

References

  1. Aït-Sahalia Y., Mykland P. A. and Zhang L. How often to sample a continuous-time process in the present of market microstructure noise. Review of Financial Studies, 18: 351-416. 2005.

    Article  Google Scholar 

  2. An S. and Schorfheide F. Bayesian analysis of DSGE models. Econometric Reviews, 26: 113-172. 2007.

    Article  MathSciNet  MATH  Google Scholar 

  3. Andrieu C., Doucet A. and Holenstein R. Particle Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, Series B, 72: 1-33. 2010.

    Article  MathSciNet  Google Scholar 

  4. Asparouhova E. N., Bessembinder H. and Kalchevab I. Liquidity biases in asset pricing tests. Journal of Financial Economics, 96: 215-237. 2010.

    Article  Google Scholar 

  5. Bandi F. M. and Russell J. R. Separating microstructure noise from volatility. Journal of Financial Economics, 79: 655-692. 2006.

    Article  Google Scholar 

  6. Black F. Noise. Journal of Finance, 41: 529-543. 1986.

    Google Scholar 

  7. Bolstad W. M. Understanding computational Bayesian Statistics. New York: Wiley. 2010.

    MATH  Google Scholar 

  8. Carvalho C. M. and Lopes H. F. Simulation-based sequential analysis of Markov switching stochastic volatility models. Computational Statistics & Data Analysis, 51: 4526-4542. 2007.

    Article  MathSciNet  MATH  Google Scholar 

  9. Chorin A. J., Morzfeld M. and Tu X. M. A survey of implicit particle filters for data assimilation. this volume, 63–88. 2013.

    Google Scholar 

  10. Christoffersen P., Jacobs K. and Mimouni K. Volatility dynamics for the S&P500: evidence from realized volatility, daily returns, and option prices. Review of Financial Studies, 23: 3141-3189. 2010.

    Article  Google Scholar 

  11. Cohen K., Hawawini G., Maier S., Schwartz R. and Whitcomb D. Implications of microstructure theory for empirical research on stock price behavior. Journal of Finance, 35: 249-257. 1980.

    Article  Google Scholar 

  12. Creal D. A survey of sequential Monte Carlo methods for economics and finance. Working Paper. 2009.

    Google Scholar 

  13. Crisan D. and Doucet A. A Survey of convergence results on particle filtering methods for practitioners. IEEE Transactions on Signal Processing, 50: 736-746. 2002.

    Article  MathSciNet  Google Scholar 

  14. Del Moral P., Doucet A. and Jasra A. On adaptive resampling procedures for sequential Monte Carlo methods. Bernoulli, 18: 252-278. 2012.

    Article  MathSciNet  MATH  Google Scholar 

  15. Duan, J.C. and Fulop, A. How Frequently Does the Stock Price Jump? - An Analysis of High-Frequency Data with Microstructure Noises. Working Paper. (2007)

    Google Scholar 

  16. Duan J. C. and Fulop A. Estimating the structural credit risk model when equity prices are contaminated by trading noises. Journal of Econometrics, 150: 288-296. 2009.

    Article  MathSciNet  Google Scholar 

  17. Douc R. and Moulines E. Limit theorems for weighted samples with applications to sequential Monte Carlo methods. The Annals of Statistics, 36: 2344-2376. 2008.

    Article  MathSciNet  MATH  Google Scholar 

  18. Doucet A. and Johansen A. M. A tutorial on particle filtering and smoothing: Fifteen years later. In Crisan D. and Rozovsky B., editors, Handbook of Nonlinear Filtering. Oxford University Press, 2011.

    Google Scholar 

  19. Fernández-Villaverde J. and Rubio-RamÍrez J. F. Estimating macroeconomic models: A likelihood approach. Review of Economic Studies, 74: 1059-1087. 2007.

    Article  MathSciNet  MATH  Google Scholar 

  20. Gordon N., Salmond D. and Smith A. F. M. Novel approach to nonlinear/nongaussian Bayesian state estimation. IEEE: Radar and Signal Processing, 140: 107-113. 1993.

    Article  Google Scholar 

  21. Harris L. Stock price clustering and discreteness. Review of Financial Studies, 4: 389-415. 1991.

    Article  Google Scholar 

  22. Hasbrouck J. Trades, quotes, inventories and information. Journal of Financial Economics, 42: 229-252. 1988.

    Article  Google Scholar 

  23. Hasbrouck J. Security bid / ask dynamics with discreteness and clustering: Simple strategies for modeling and estimation. Journal of Financial Markets, 2: 1-28. 1999.

    Article  Google Scholar 

  24. Johannes M., Polson N. and Stroud J. Optimal filtering of jump-diffusions: Extracting latent states from asset prices. Review of Financial Studies, 22: 2759-2799. 2009.

    Article  Google Scholar 

  25. Liu J. and West M. Combined parameter and state estimation in simulation-based filtering. In Doucet A., De Freitas J. F. G. and Gordon N. J., editors, Sequential Monte Carlo Methods in Practice. Springer-Verlag, 2001.

    Google Scholar 

  26. Lopes H. F. and Tsay R. S. Particle filters and Bayesian inference in financial econometrics. Journal of Forecasting, 30: 169-209. 2011.

    Article  MathSciNet  Google Scholar 

  27. Malik S. and Pitt M. K. Particle filters for continuous likelihood evaluation and maximisation. Journal of Econometrics, 165: 190-209. 2011.

    Article  MathSciNet  Google Scholar 

  28. Merton R. C. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29: 449-470. 1974.

    Google Scholar 

  29. Merton R. C. Option Pricing when Underlying Stock Returns are Discontinuous. Journal of Financial Economics, 3: 125-144. 1976.

    Article  MATH  Google Scholar 

  30. Metropolis N., Rosenbluth A. W., Rosenbluth M. N., Teller A. H., Teller E. Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21: 1087-1092. 1953.

    Article  Google Scholar 

  31. Pitt M. K. Smooth particle filters for Likelihood Evaluation and Maximization. Working Paper. 2002.

    Google Scholar 

  32. Pitt M. K. and Shephard N. Filtering via simulation: Auxiliary particle filters. Journal of the American Statistical Association, 94: 590-599. 1999.

    Article  MathSciNet  MATH  Google Scholar 

  33. Pitt M. K., Silva R. S., Giordani P. and Kohn R. Auxiliary particle filtering within adaptive Metropolis-Hastings sampling. Working Paper. 2010.

    Google Scholar 

  34. Rios M. P. and Lopes H. F. The extended Liu and West filter: Parameter learning in Markov switching stochastic volatility models, Chapter 2, in this volume.

  35. Spalding R., Tsui Kam-Wah and Zeng Y. A Micromovement Model with Bayes Estimation via Filtering: Application to Measuring Trading Noises and Trading Cost. Nonlinear Analysis: Theory, Methods and Applications, 64: 295-309. 2006.

    Google Scholar 

  36. Storvik G. Particle filters in state space models with the presence of unknown static parameters. IEEE: Transactions of Signal Processing, 50: 281-289. 2002.

    Article  MathSciNet  Google Scholar 

  37. Xiong J. and Zeng Y. A branching particle approximation to the filtering problem with counting process observations. Statistical Inference for Stochastic Processes, 14: 111-140. 2011.

    Article  MathSciNet  MATH  Google Scholar 

  38. Zeng Y. A partially observed model for micromovement of asset prices with Bayes estimation via filtering. Mathematical Finance, 13: 411-444. 2003.

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhu C. Parameters estimation for jump-diffusion process based on low and high frequency data. Master Thesis. The Hong Kong Polytechnic University. 2011.

    Google Scholar 

Download references

Acknowledgments

We thank Professor Yong Zeng and one anonymous referee for numerous suggestions and insights; thank Louis Schenck and Taisuke Nakata for the R-codes to produce Fig. 15.1, and Eric Goldlust for providing the formulae for Table 15.1. The second author acknowledges the support from Hong Kong RGC Earmaked grant 500909.

Author information

Authors and Affiliations

  1. Department of Finance, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

    Cai Zhu

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Jian Hui Huang

Authors
  1. Cai Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Hui Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cai Zhu .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Missouri at Kansas City, Kansas City, Missouri, USA

    Yong Zeng

  2. Department of Economics, The University of Kansas, Lawrence, Kansas, USA

    Shu Wu

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Zhu, C., Huang, J.H. (2013). Parameter Estimation via Particle MCMC for Ultra-High Frequency Models. In: Zeng, Y., Wu, S. (eds) State-Space Models. Statistics and Econometrics for Finance, vol 1. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7789-1_15

Download citation

Publish with us

Policies and ethics

Search

Navigation