A Partially Observed Model for Micromovement of Asset Prices with Bayes Estimation via Filtering


  • This paper is a revision of the author's doctoral dissertation, Zeng (1999), at Department of Statistics, the University of Wisconsin at Madison, completed under the supervision of Thomas G. Kurtz. I am grateful to Tom for his intuition, inspiration, guidance, and encouragement. I thank Philip Protter for introducing me to the tick-by-tick data, thank Cyrus Ramezani and an associate editor for many detailed comments that improved the exposition, and thank Jie Chen, Chin-Shan Chuang, Robert Elliott, Larry Harris, Michael Kosorok, Yi Lin, Mark Ready, Chris Rogers, Boris Rozovskii, Ruey Tsay, and Kam-Wah Tsui for helpful comments. I thank participants at the Workshop on Stochastic Theory and Control (2001) held at University of Kansas, the Quantitative Risk Management in Finance Conference (2000) held at Carnegie Mellon University, the Symposium on Inference for Stochastic Processes (2000) held at the University of Georgia, and seminar participants at the University of Kansas, University of Missouri at Kansas City, University of Southern California, and University of Wisconsin at Madison for comments. A more detailed version of the paper is available from author upon request.

  • Manuscript received October 2001; final revision received May 2002.

Address correspondence to the author at the Department of Mathematics and Statistics, University of Missouri at Kansas City, Kansas City, MO 64110, USA; e-mail: zeng@mendota.umkc.edu.


A general micromovement model that describes transactional price behavior is proposed. The model ties the sample characteristics of micromovement and macromovement in a consistent manner. An important feature of the model is that it can be transformed to a filtering problem with counting process observations. Consequently, the complete information of price and trading time is captured and then utilized in Bayes estimation via filtering for the parameters. The filtering equations are derived. A theorem on the convergence of conditional expectation of the model is proved. A consistent recursive algorithm is constructed via the Markov chain approximation method to compute the approximate posterior and then the Bayes estimates. A simplified model and its recursive algorithm are presented in detail. Simulations show that the computed Bayes estimates converge to their true values. The algorithm is applied to one month of intraday transaction prices for Microsoft and the Bayes estimates are obtained.