Appears in CIFEr '97, March 23-25, New York City HIGH FREQUENCY TIME SERIES ANALYSIS AND PREDICTION USING MARKOV MODELS. Constantine P.

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1 Appears in CIFEr '97, March 23-25, New York City HIGH FREQUENCY TIME SERIES ANALYSIS AND PREDICTION USING MARKOV MODELS Constantine P. Papageorgiou Center for Biological and Computational Learning and Articial Intelligence Laboratory Massachusetts Institute of Technology Cambridge, MA 2139 USA Abstract There has been a surge in interest in the analysis and prediction of high frequency time series in recent years. We consider the problem of predicting the direction of change in tick data of the Swiss franc/u.s. dollar exchange rate. To accomplish this, we show that a Markov model can nd regularities in certain local regions of the data and can be used to predict the direction of the next tick. Predictability seems to decrease in more recent years. With transaction costs, the model is unlikely to be protable. 1 Introduction The analysis and prediction of time series has received much interest in recent years. With respect to nancial time series, understanding the dynamics of these processes is a lucrative challenge; the ability to predict the direction and magnitude of change in these processes could lead to signicant prot. One of the basic premises of our work is that there is indeed some amount of regularity and predictability in nancial time series. The prediction problem that we tackle is formulated as one where we are trying to derive future price patterns in currency exchange trading from the past. Most models have focused on analysis at the level of daily, weekly, or monthly data; the model we present nds patterns at the resolution of ticks. We do not expect to nd a perfect predictor. Rather, we are interested in developing a system that can recognize certain regularities in the data and, when these patterns start emerging in the data, predict what follows at greater than 5% accuracy. Such a system, while possibly losing in the short term, should win in the long term. This approach uses a Markov model, a probabilistic process, to model the increases and decreases in the exchange rate at the tick level. The probabilities in the model are analyzed and used to develop simple prediction rules that are correct more then 5% of the time. The system is trained using a window of data extending N trading days prior to the current trading day. It is from these N days that we are trying to extract information about what might happen in the (N + 1) th day. We show that this simple model uncovers regularities that exist in the tick-by-tick exchange rate data. Using this model, we are able to make predictions in situations where the model has found a pattern; these predictions are correct more than 5% of the time in several dierent periods of data. 2 Foreign Exchange Market The foreign exchange market is a decentralized market where the money exchanged totals approximately one billion dollars each day. Banks and other market makers send their exchange rates to certain companies that disseminate this information to subscribers. Exchange rates actually have two components, the bid price and the ask price. The bid price is the highest price at which the market maker is willing to buy the currency, while the ask price is the lowest price at which the market maker is willing to sell the currency. To trade a currency, a dealer will call a market maker, request the current quote, and decide whether or not to place a trade. 1

2 Period 1 Period 2 Period /12/ /2/1985 4/25/1987 7/3/1989 x 1 5 Figure 1: The bid price time series for the Swiss franc/u.s. dollar exchange rate. The data that we use is the Swiss franc/u.s. dollar data for a single bank from May 2, 1985 to April 12, This series is at the resolution of ticks; a new tick (data point) occurs whenever the bank changes either its bid or ask price. The time between ticks ranges from seconds to minutes. This is an expanded version of the time series used in the 1991 Santa Fe Time Series Competition [7]. 3 Markov Models Markov models are stochastic models that capture variability in a process through time. Markov chains and, in particular, their generalizations, hidden Markov models, have been used heavily in signal processing elds like speech recognition with great success. This section presents a brief introduction to Markov chains; for an excellent introduction to hidden Markov models see Rabiner and Juang [6]. The Markov chain is an integer time process composed of a set of N states and N 2 transition probabilities. We denote the states by Q = fq i g N i=1 and let Q(t) denote the state of the system at time t. Let A(t) = fa ij (t)g N i;j=1 be the matrix of transition probabilities. More formally, a ij (t) = P (Q(t) = jjq(t? 1) = i) (1) is the probability that, given that the process is currently in state i at time t? 1, the process makes a transition to state j at time t. It is important to note that we are not assuming stationarity, so A(t) is time-dependent. The processes that we will be looking at will always have a well dened next state, so: NX a ij (t) = 1 (2) j=1 1 This data was obtained from the University of Colorado at Boulder Time Series Repository ( 2

3 C1 C2 C3 C4 C5 C6 C7 C8 C9 14 number of occurances log difference in bid price x 1 3 Figure 2: Histogram of the log dierences in prices for Period 1; dashed lines denote cluster boundaries. Such a model is characterized by the Markov independence assumption, that the state at time t is dependent only on the state at time t? 1. We can relax this assumption of local dependence to allow the state at time t to depend on the past n states. In this way, our transition probabilities become A(t) = fa in::i1j(t)g N i 1;:::;in;j=1 where, a in::i1j(t) = P (Q(t) = jjq(t? 1) = i1; Q(t? 2) = i2; :::; Q(t? n) = i n ) (3) With the above construction, if we know the current thru (n? 1) th states and the transition probabilities, this n th order Markov model is a complete probability distribution describing the next state. 4 System Architecture We divide the data into three dierent periods { Period 1 (5/2/1985-4/25/1987), Period 2 (4/26/1987-7/3/1989), and Period 3 (7/4/1989-4/12/1991) { shown in Figure (1). This paper analyzes only the bid price time series. Also, as is typically done with nancial time series, the data is preprocessed so that our data points, r t, are the log ratios between consecutive bid prices, p t?1 and p t. r t = ln( p t p t?1 ) (4) We analyze the histogram of the log ratio series for Period 1 and discretize the data into nine states. The nine states are shown in Figure (2) for Period 1. Points at zero signify no change from one tick to the next and constitute a signicant portion of the data set; these are included as a distinct state. To model the dynamics of the system we impose a second order assumption on the tick-by-tick price changes; this means that the state at time t is dependent on the states at time t? 1 and t? 2 and nothing else. Since we are using a second order Markov model, the Markov chain is indexed by the three variables i, j, and k, corresponding to times t? 2, t? 1, and t, respectively. While this model is rather simple, LeBaron[4] shows, for a one year portion of this data set, that there are 3

4 indeed signicant rst-order negative autocorrelations. Our system should be able to model these, and other more complex, patterns. Since we do not assume stationarity, we estimate the transition probabilities for the Markov chain by counting the occurrences of the dierent patterns in a xed window of data. The counts are kept track of in a matrix C. If the current training point is a price dierence from state i to state j to state k, we increment C ijk by one. This process is iterated over each data point in the day and the process is repeated for each day in the training window. We do not model changes between the closing price of one day and the opening price of the next. a ijk = C P ijk 9 x=1 C ijx (5) Hence, given that we know the past two price ratios, the next price ratio is a random variable whose value (actually, the state in which the value falls) is a ijx (t) of the transition matrix. The Markov model then generates trading rules by seeing if the probability values of the probabilities on either side of zero sum to a value greater than some threshold H greater than 5%; in our experiments, we varied this from 5% to 7%. We hope that by modeling the system using the smaller price changes, the predictions will be more reliable; the causes of large price movements have little chance of being modeled by such a simple system. For the purposes of this paper, we use the two states on either side of zero; these four states contain approximately 8% of the data points. If the probability values for a ij6(t) and a ij7(t) sum to some value greater than a threshold H, then we create the following rule to predict the upward direction of change of the next tick: if Q(t? 1) = i and Q(t) = j then Q(t + 1) will be an increase Similarly for decreases, if the sum of the probability values for a ij3 and a ij4 are greater than H, the rule will be: if Q(t? 1) = i and Q(t) = j then Q(t + 1) will be a decrease The state boundaries are dynamically adjusted for each window of training data and maintain the discretization of the price changes into nine states. 5 Experimental Results This process may be non-stationary so, to generate predictions for the current day, the Markov model's probabilities are estimated using a window of data that extends N days in the past. The experiments were run using values of N from 1 to 3. As stated above, the percentage threshold above which the system creates a rule is varied from 5% to 7%. As a benchmark, the Markov-based model is compared with one that uses a simple reversal strategy as in LeBaron[4]. For the next tick, t, this strategy gives a buy signal when r t?1 < and a sell signal when r t?1 >. For a closer comparison of these two strategies, the reversal strategy makes a prediction only when the Markov system makes a prediction. Graphs of a) the number of correct predictions as a percentage of the prediction made using the Markov model, b) the number of correct predictions as a percentage of the prediction made using the reversal strategy, and c) the number of predictions as a percentage of the total number of ticks, each plotted against the training window size and the training threshold size for the current day are plotted for each of the three periods in Figure (3). From the graphs, we see the empirical eects of varying the threshold and training window size. As we would expect, prediction accuracy increases with higher rule thresholds, but the number of predictions that the model makes decreases as the threshold increases. In Period 1, the Markov model seems to be generating robust rules for predicting the next tick; 98.4% (599/9) of the tested congurations are correct over 5% of the time, most are actually correct between and 7% of the time. Just as importantly, the patterns apply to up to 63% of the ticks that are seen. This is important for a high frequency trading system that is based on generating a small prot on each trade { in total, these prots will be substantial. Compared to 4

5 5 Markov model accuracy for Period 1 reversal strategy accuracy for Period 1 number of predictions as a percentage of total ticks for Period number of predictions out of total ticks (%) Markov model accuracy for Period 2 reversal strategy accuracy for Period 2 number of predictions as a percentage of total ticks for Period number of predictions out of total ticks (%) Markov model accuracy for Period 3 reversal strategy accuracy for Period 3 number of predictions as a percentage of total ticks for Period number of predictions out of total ticks (%) Figure 3: Correct predictions as a percentage of predictions made using the Markov model and the reversal strategy, and number of predictions as a percentage of total ticks for, by row, Periods 1, 2, and 3. the reversal strategy, the Markov model is much better at predicting the direction of change during this period; out of the 9 test runs, the system achieves better than 5% accuracy only two times. A possible reason for this could be that the reversal strategy is failing due to the sharp downward trend in this part of the series; the rst-order negative autocorrelations may not hold here. Things are less clear when we look at the Markov model's performance on Period 2. Here, the algorithm makes predictions on fewer ticks (at most 25%), but is still correct at least 5% of the time for 84.2% (513/9) of congurations and, for most congurations, between % and % of the time. Disregarding the higher threshold values where the number of predictions that are made is extremely small, the graph shows that the predictions are more accurate when the smaller sized windows of more recent training data is used to generate the model. This may indicate that the shorter term price patterns are more strongly correlated. The reversal strategy does not do as well as the Markov model; only 46.5% (383/9) of the tests yield performance of better than 5%. In Period 3, the model achieves 5% accuracy in 83.9% (511/9) of the tests with accuracy of between % and % for most of the congurations. The number of ticks that the system makes predictions on drops even more, to a maximum of 18%. The accuracy of the reversal strategy is nearly equivalent to the Markov model over all congurations; the Markov model seems to be uncovering only rst-order negative autocorrelations. To determine statistical signicance, we use the central limit theorem for Bernoulli random variables with = :5 as the level of signicance and measure the signicance of predictions over all tests that yielded accuracy of better than 5%. Signicance is measured separately for buy and 5

6 sell predictions. The tests take into account the actual distribution of increases and decreases for each prediction period; baseline guessing accuracy may dier from 5%. This test is related to the Henriksson-Merton test [3], but this connection has not been fully explored yet. Table (1) shows the results; overall the Markov-based system has a greater number of results where the prediction accuracy is above 5% and a higher percentage of these tests are statistically signicant. percent of tests where predictions are statistically signicant ( = :5) Increases Decreases Period 1 Markov 89.8% (491/547) 87.2% (524/1) reversal. (/2) NA Period 2 Markov 66.7 (316/474) 68.9 (346/52) reversal 33.5 (93/278) 39.3 (17/272) Period 3 Markov 44.2 (242/547) 83.1 (2/37) reversal 44.4 (243/547) 83.4 (256/37) Table 1: Number of statistically signicant tests of increase and decrease predictions for the Markov model and reversal strategy for each of the three periods, measured as a percentage of tests where the prediction accuracy is above 5%. 6 Discussion We present a simple Markov model trained to generate rules to predict the direction of the next tick in Swiss franc/u.s. dollar currency exchange trading. The system nds situations in which it can make predictions at reasonable accuracy and only makes predictions in these situations. For most congurations of the system, the prediction rules it generates are accurate over 5% of the time and often over -% of the time. The rules that the model creates apply to up to 63% of the ticks in Period 1, up to 25% of the ticks in Period 2, and up to 18% of the ticks in Period 3. Prediction accuracy decreases from Period 1 to Period 3. This may indicate that the short-term dynamics of this particular time series are becoming more complex over time, and thereby less predictable using this model. The fundamental aspect of this model that makes it particularly attractive is its simplicity. By discretizing the input space into a set of nine states, all the computations that the system does become trivial. This directly translates into the high speed at which the model is trained and predictions are made; training the model over a window size of 3 days and a threshold of 5% takes less than two seconds on a Sun SPARC 2. The shortcomings of this model are discussed here: A small simulation we performed shows that, when we factor in transaction costs, the model is not protable due to the fact that it trades on very small price dierences. We may want to predict changes in price over a longer time frame of hours or days to allow for larger dierences in prices. The determination of the state space is ad hoc. More experiments should be done to determine the optimal set of states. The model is too simple; assuming that the direction of the next tick can be predicted knowing only the past three tick values is overly restrictive. This system was run over the data from a single bank. This is only a small part of the picture as there are many other market makers and their interactions play a role in determining prices. Only the bid price series was used and this series does not reect prices at which actual trades were executed. 6

7 A direct comparison of this model to other techniques is not straightforward since most other systems predict actual price changes and do not attempt to make \next tick" predictions. We nevertheless present a brief overview of several related systems. Of particular relevance to this paper are those systems that attempt predictions at the level of ticks. Zhang [9] uses a multi-layer perceptron to forecast Swiss franc/u.s. dollar exchange rates based on several inputs from up to 18 minutes prior to the prediction time. These inputs are used to predict the exchange rate 3 minutes in the future. Weigend, Huberman, and Rumelhart [8] predict the Tuesday U.S. dollar/deutsch mark rate using a feedforward neural network over daily data. This system takes as input past measurements for this exchange rate, volatility and trend measures derived exclusively from the exchange rate series, and the current and past exchange rates for other currencies. Mozer [5] uses a recurrent neural network that takes tick data and predicts the exchange rate 1, 15, and minutes in the future. It is also important to note that several researchers have used a similar model to our Markov chain model, but at a much coarser resolution. In particular, Engel and Hamilton [2] and Engel [1] use a Markov switching model to model exchange rates for quarterly data. Engel concludes that the Markov switching model is more eective at predicting the direction of change, rather than the actual value, of exchange rates. This work demonstrates that Markov models may be extremely useful in uncovering and analyzing the hidden dynamics of nancial markets. This paper shows the eectiveness of using a nine state Markov model to create rules that predict the direction of the next tick in Swiss franc/u.s. dollar trading and lays the foundation for future work. References [1] C. Engel. Can the markov switching model forecast exchange rates? Journal of International Economics, 36:151{, [2] C. Engel and J.D. Hamilton. Long swings in the dollar: Are they in the data and do markets know it? The American Economic Review, 8(4):689{713, September 199. [3] R.D. Henriksson and R.C. Merton. On market timing and investment performance. ii. statistical procedures for evaluating forecasting skills. Journal of Business, 54(4):513{533, [4] B. LeBaron. Nonlinear diagnostics and simple trading rules for high-frequency foreign exchange rates. In A.S. Weigend and N.A. Gershenfeld, editors, Time Series Prediction: Forecasting the Future and Understanding the Past, pages 457{74, [5] M.C. Mozer. Neural net architectures for temporal sequence processing. In A.S. Weigend and N.A. Gershenfeld, editors, Time Series Prediction:Forecasting the Future and Understanding the Past, pages 243{64, [6] L.R. Rabiner and B.H. Juang. An introduction to hidden markov models. IEEE Acoustics, Speech, and Signal Processing Magazine, 3(1):4{16, January [7] A.S. Weigend and N.A. Gershenfeld, editors. Time Series Prediction: Forecasting the Future and Understanding the Past. SFI Studies in the Sciences of Complexity. Addison Wesley, [8] A.S. Weigend, B.A. Huberman, and D.E. Rumelhart. Predicting sunspots and exchange rates. In M. Casdagli and S. Eubank, editors, Nonlinear Modeling and Forecasting, pages 395{432, [9] Xiru Zhang. Non-linear predictive models for intra-day foreign exchange trading. Intelligent Systems in Accounting, Finance and Management, 3:293 { 32,