An Extended Hidden Markov Model for Asset Returns

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1 An Extended Hidden Markov Model for Asset Returns Xugang Ye, Johns Hopkins University, USA Huan Wang, Johns Hopkins University, USA ABSTRACT The hidden Markov model (HMM) has been successfully applied to many temporal pattern recognition problems. The authors of this paper consider its application in financial time series data like asset returns. Other than the direct use of the regular HMM, the authors propose an extension to incorporate an input sequence into the state transition mechanism. The authors also explore the inference methods and discuss the generative process for simulation and forecasting. Keywords: Hidden Markov Model, Asset Return, Inference, Forecasting INTRODUCTION The hidden Markov model (HMM), first described by Baum et al. (1966), has become a powerful statistical tool for modeling a wide range of sequential data. Originally applied to speech recognition by Baker (1975) and Bahl and Jelinek (1975), it has also been known for handwritten characters recognition (see Vinciarelli and Luettin, 2000), gesture recognition (see Wu and Huang, 1999), music score following (see Orio and Dechelle, 2001), and biological sequence analysis (see Durbin et al., 1999) etc. The past several years have seen interests of researchers in applying the HMM to financial time series data like stock prices. For examples, Shi and Weigend (1997) proposed a hidden Markov expert model to forecast the change of the S&P Index at half-hour time step, Hassan et al. (2005) made use of the well-established HMM technique to forecast airline stocks, Park et al. (2009) forecasted the change directions of next day s closing price of financial time series using the continuous HMM. The advantages of the HMM include strong statistical foundation, robustness in handling new data, computational convenience, and the efficiency in predicting similar patterns. A thorough explanation of the basics of the HMM can be found in Rabiner (1989), which is a very good tutorial paper. While it can explain a set of data well, the basic HMM is still restricted by the emitted time series data itself. In reality, the stock movements are also affected by information input. Ignoring this issue will incur the oversimplified state transition mechanism and the loss of accuracy in

2 prediction. Based on this point, we aim to extend the regular HMM so that the state transition probabilities are not unconditional but conditional (on the information input). And the state transition probabilities for model extrapolation are the ones integrated over the information input. The rest of the paper is organized as follows: we first present our extended HMM and explore the corresponding inference method. We then present the generative process for simulation and forecasting. Finally, we present some empirical results of analyzing a set of real data. The Model Suppose the value of a market variable at the end of day is for 1,2,, and is the initial value. Then the return at the end of day is r t = ln. Suppose 1,2,, is the index of the type of information input at time, where means there are types of information. The elements of our model are as follows: Sequence of hidden states: 1,2,,, where is the number of hidden states Markov property for : :, :, :, Markov property for : :, :, : Transition mechanism:,, Emission mechanism: ;, exp Initial distribution mechanism:, : Hence the transition mechanism is represented by, for 1,,, 1,,, 1,, and the initial distribution is represented by for 1,,, 1,,. Let,,,,, and,,, then, by the Markov properties for and, the conditional joint likelihood of the history is :, : : ; ;,,. (1) The conditional posterior of : is : :, : ; :, : : ;, (2) : : ; where : : ; is the conditional likelihood of data :. The computation of : : ; requires either the forward or the backward calculation scheme. Define :, : ;. By the initial distribution mechanism, we have, : ;, : ; : ; ;,, (3) and for 2,,, by the Markov properties for and, we have :, : ; :,,, : ;, ;,. (4) By :, : ;, we have : : ;. (5) Define :, : ;, then 1, (6)

3 and for 1,,1, by the Markov properties for and, we have :, : ;, :,, : ;, ;,. (7) (5) and the formula (8) are not explicit, it s difficult to use a general optimization algorithm that is based on gradient ascent. However, it s easy to build an EM-like iterative procedure. Suppose we start from an initial estimate and we have after the -th interation. Obviously, the updating equation for is By :, : ; and the initial distribution mechanism, we have : : ;, :, : ; ;,. (8) The utility of the forward and backward calculation scheme also lies in the computation of the conditional posteriors :, : ;. In fact, by Bayes rule, we have :, : ;, : : ; : : ;, :, : : ; :, :, : ;, : : ; :, : ;, : : ; The INFERENCE. (9) The inference problem is stated as finding that maximizes : : ;. Since the input sequence is given, the job is just to find,,, and, for :, : ;. (10) By maximum likelihood estimation, the updating equations for and are :, ; : :, ; : and respectively. :, ; : :, ; : (11), (12) The analytical formula for updating, is not so trivial as (10), (11), (12). However, sample paths :, : can be generated according to the posteriors (9). And, can be approximated as,,,,,,,,, (13) where, 1 if ;, 0 otherwise. A question on the estimator (13) is how large should be. Note that lim,, for 1,,. Hence a criterion for determining could be 1,,;1,,1. Note that both the formula

4 ,,, (14) where is a preset precision. SIMULATION The purpose of simulation is forecasting. After the process of learning from : and :, a path : could be sampled, where is number of extrapolation days. The segment : is just sampled according to the conditional posteriors (9). Once this segment is generated, the initial state for the draw of the next segment : is determined. But drawing : has to be done without : since the information input beyond time is not available. However, one can use the integrated state transition probabilities,, (15) to generate :, where could be estimated as,. (16) Bahl, L. R. and Jelinek, F. (1975). Decoding for channels with insertions, deletions, and substitutions with applications to speech recognition. IEEE Transaction on Information Theory, Vol. IT-21, pp Baker, J. K. (1975). The dragon system an overview. IEEE Transaction on Acoustic Speech Signal Processing, Vol. ASSP-23(1), pp Baum, L. E. and Petrie, T. (1966). Statistical inference for probabilistic functions of finite state Markov chains. Annals of Mathematical Statistics, Vol. 37, pp Durbin, R., Eddy, S. R., Krogh, A., and Mitchison, G. (1999). Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press. Hassan, M. R., and Nath, B. (2005). Stock market forecasting using hidden Markov model: a new approach. In IEEE International Conference of intelligent Systems Design and Application, pp Orio, N. and Dechelle, F. (2001). Score following using spectral analysis and hidden markov models. In Proceedings of the 2001 International Computer Music Conference. Once a path : is generated, a path : can be drawn by the emission mechanism ;,. After many draws, an empirical distribution of can be obtained for each 1,,. These empirical distributions can then be used to calculate the predicted term structure of values at risk and volatilities. EMPIRICAL RESULTS (to appear ) REFERENCES Park, S., Lee, J., Song, J., and Park T. (2009). Forecasting change directions for financial time series using hidden Markov model. In 4th International Conference on Rough Sets and Knowledge Technology, Vol. 5589, pp Shi, S. and Weigend, A. S. (1997). Taking time seriously: hidden Markov experts applied to financial engineering. In Proceedings of the IEEE/IAFE 1997 Conference on Computational Intelligence for Financial Engineering, pp Vinciarelli, A. and Luettin, J. (2000). Off-line cursive script recognition based on continuous density HMM. Proceedings of the 7th International Workshop on Frontiers in Handwriting Recognition, Amsterdam, pp

5 Wu, Y. and Huang, T. S. (1999). Vision-Based Gesture Recognition: A Review, in Gesture Based Communication. In Human-Computer Interaction, Vol of Springer Lecture Notes in Computer Science, pp (more references to add )

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